相关与回归分析 100 个数据分析案例!
相关与回归分析 100 个数据分析案例!
相关与回归分析都是研究变量间关系的统计工具，但侧重点不同：相关分析看“关系有多紧”，回归分析看“怎么用一个变量预测另一个”。你可以先通过相关分析了解关联强度，再用回归分析建立预测模型。
这俩到底有啥不一样
变量角色不同：相关分析把两个变量当“平等伙伴”，不区分谁影响谁；回归分析必须明确“自变量”（解释变量）和“因变量”（被解释变量），方向是单向的。
目标不同：相关分析只告诉你“这两个变量有没有关系、关系多强”；回归分析能算出具体方程，比如“学习时间每增加1小时，成绩预计提高5分”，还能用来预测未来值。
输出形式不同：相关分析输出一个相关系数r（范围在-1到1之间）；回归分析输出一个方程，比如y=ax+b。
怎么判断两个变量关系紧不紧
看相关系数r的值：r在0到1之间是正相关（一个变量增加，另一个也增加），r在-1到0之间是负相关（一个变量增加，另一个减少），r绝对值越接近1关系越强，越接近0关系越弱。
画散点图辅助判断：把数据点在直角坐标系上描出来，点集呈斜向上趋势说明正相关，斜向下趋势说明负相关，点分布杂乱说明不相关。
注意r²和决定系数的关系：相关系数r的平方等于回归模型中的决定系数R²，表示自变量能解释因变量变异的比例。
实际用的时候怎么选
先用相关分析筛选变量：如果想研究多个变量，先做相关分析找出哪些变量之间关系密切，再把相关的变量纳入回归模型。
预测场景用回归分析：如果你需要根据已知数据预测未知值，比如根据广告投入预测销售额、根据客户信用评分预测违约概率，就用回归分析建模。
两者常配合使用：如果相关分析显示变量间关系非常密切，说明求得的回归方程式更可信，实际分析中一般先做相关分析再做回归建模。
说明：本报告含 Pearson 相关(40)、简单线性回归(35)、多元线性回归(10)、逻辑回归(15) 共 100 例。
每个案例原始样本量 n∈[300,1000] 且全部数据完整列出；所有统计量（相关系数 r、回归系数、优势比 OR、t/F/z 的 p 值）均由 Python 从数学原理精确计算。
重要提醒：大样本会显著提升检验功效，使 p 值极小——统计显著 ≠ 实际意义重大，解读时请结合效应量（r、R²、OR、伪 R²）。
第1类：Pearson相关（案例 1–40）
案例 1. 广告投入(万元)与产品销量(千件)
Pearson相关广告费越高销量越高
① 原始数据：
样本 n=928，逐条 (x, y)：[(39.72, 225.7), (60.51, 231.91), (25.37, 192.59), (38.36, 209.56), (57.44, 224.23), (44.02, 154.42), (47.0, 164.09), (39.99, 182.63), (50.81, 210.61), (58.11, 242.32), (29.43, 136.13), (32.8, 126.83), (49.84, 195.92), (63.4, 203.33), (44.09, 193.79), (53.16, 214.57), (72.09, 268.72), (31.76, 201.44), (70.85, 235.04), (52.44, 220.71), (67.04, 224.51), (30.7, 224.99), (64.9, 279.04), (66.05, 231.91), (29.19, 227.04), (31.37, 175.71), (57.41, 176.23), (63.5, 179.25), (57.46, 220.31), (37.49, 192.39), (53.44, 192.02), (77.32, 246.65), (43.19, 169.36), (54.64, 164.82), (58.3, 197.07), (52.43, 185.84), (42.19, 189.91), (47.18, 263.6), (58.72, 216.17), (49.13, 196.13), (39.74, 204.53), (50.54, 209.53), (34.62, 117.52), (54.36, 234.38), (49.16, 153.15), (46.12, 168.91), (75.26, 230.86), (57.29, 175.98), (66.97, 292.68), (35.29, 154.12), (38.83, 134.74), (55.12, 224.75), (45.14, 203.23), (47.51, 179.94), (34.81, 144.52), (49.83, 149.14), (40.56, 179.31), (66.95, 207.87), (19.79, 143.0), (57.94, 179.76), (42.07, 197.48), (54.29, 210.99), (72.34, 247.98), (65.71, 261.64), (48.3, 215.44), (63.9, 218.96), (67.45, 257.13), (29.89, 104.62), (53.98, 180.24), (42.02, 176.06), (41.14, 210.76), (52.3, 258.26), (53.32, 178.82), (56.05, 184.12), (54.61, 228.55), (67.26, 262.76), (48.1, 206.02), (65.51, 241.73), (57.7, 199.35), (60.75, 173.38), (60.94, 231.93), (71.26, 214.01), (48.05, 192.06), (50.56, 246.46), (56.42, 227.1), (65.94, 245.64), (58.63, 265.72), (70.17, 259.6), (60.42, 210.82), (30.33, 195.72), (54.92, 242.05), (57.95, 238.6), (45.63, 191.59), (62.25, 203.39), (55.87, 217.26), (49.51, 209.43), (47.2, 175.97), (32.41, 165.17), (49.79, 240.25), (35.03, 205.46), (29.49, 148.39), (74.37, 232.87), (60.36, 171.37), (52.62, 174.89), (17.03, 228.84), (43.6, 216.51), (48.25, 207.3), (38.35, 170.85), (65.29, 282.63), (45.37, 223.42), (86.88, 262.69), (69.69, 213.16), (42.05, 208.07), (55.09, 181.13), (54.9, 188.84), (32.31, 110.42), (46.99, 195.96), (46.33, 211.19), (45.04, 165.44), (38.18, 193.34), (54.9, 176.74), (58.71, 218.09), (59.49, 254.9), (51.74, 230.21), (65.72, 203.44), (57.07, 184.98), (43.2, 188.42), (61.67, 247.33), (49.49, 192.44), (59.8, 253.15), (50.42, 226.38), (32.25, 138.84), (55.71, 214.69), (39.84, 185.87), (58.31, 235.27), (28.5, 164.29), (91.59, 262.7), (55.61, 186.53), (58.65, 230.01), (58.09, 264.63), (54.16, 171.19), (60.45, 227.99), (40.61, 162.74), (34.69, 205.83), (62.69, 223.86), (31.61, 157.26), (43.66, 185.58), (58.62, 227.38), (82.42, 277.07), (53.62, 199.14), (52.55, 189.6), (62.24, 226.6), (79.59, 275.02), (32.67, 215.16), (19.34, 216.1), (68.66, 156.39), (37.89, 156.1), (65.75, 221.06), (54.68, 216.29), (38.58, 161.6), (44.55, 190.14), (7.31, 91.75), (49.21, 239.03), (21.75, 101.13), (20.72, 173.15), (47.0, 240.47), (44.61, 215.84), (72.5, 214.43), (20.78, 181.33), (43.2, 252.37), (81.18, 263.52), (28.88, 158.62), (57.0, 201.89), (58.62, 260.51), (68.79, 201.65), (45.15, 205.79), (38.77, 153.82), (24.97, 199.76), (65.66, 200.18), (30.64, 152.9), (62.69, 204.92), (70.02, 219.6), (69.52, 252.88), (38.08, 186.6), (53.3, 205.58), (86.13, 306.06), (22.7, 161.72), (28.57, 123.48), (50.86, 201.72), (42.78, 146.88), (43.3, 187.09), (61.75, 226.16), (65.17, 265.22), (44.8, 204.52), (64.2, 230.24), (78.1, 298.42), (67.17, 262.91), (32.31, 174.68), (84.49, 269.62), (50.79, 200.74), (38.25, 159.53), (60.18, 211.71), (35.14, 208.99), (50.51, 202.07), (70.84, 222.04), (36.19, 118.04), (33.05, 190.84), (56.17, 242.75), (90.97, 306.81), (70.65, 272.8), (38.38, 206.07), (54.66, 224.48), (45.81, 154.86), (50.04, 178.58), (38.35, 173.27), (56.18, 241.82), (47.7, 222.45), (81.4, 197.78), (69.92, 271.01), (73.51, 214.98), (21.43, 183.7), (42.87, 172.53), (45.01, 203.02), (54.46, 275.57), (49.3, 206.15), (57.28, 200.54), (57.48, 174.0), (21.61, 157.8), (38.03, 205.18), (72.14, 218.73), (29.24, 161.63), (28.68, 251.51), (1.63, 109.99), (61.33, 206.87), (39.23, 226.86), (6.74, 106.71), (28.98, 171.86), (47.81, 259.2), (64.49, 227.31), (45.06, 233.43), (50.3, 193.74), (52.54, 233.38), (78.29, 206.27), (32.99, 184.35), (40.3, 168.04), (65.74, 241.56), (57.2, 203.84), (61.5, 184.27), (38.2, 189.37), (69.86, 263.91), (42.4, 142.37), (43.49, 199.59), (30.89, 149.63), (32.79, 148.93), (77.82, 205.37), (19.02, 110.94), (72.36, 284.78), (57.74, 203.28), (62.92, 240.94), (35.9, 197.7), (68.65, 229.66), (68.59, 239.36), (66.27, 218.55), (67.07, 273.76), (55.2, 206.48), (60.67, 259.27), (41.79, 199.45), (35.24, 132.86), (51.14, 216.87), (99.59, 299.17), (30.21, 151.07), (45.33, 198.16), (59.52, 218.52), (28.15, 178.83), (72.92, 238.59), (58.73, 233.31), (47.18, 222.81), (44.14, 182.16), (51.11, 139.23), (24.03, 118.19), (40.18, 215.64), (40.87, 169.24), (42.98, 215.39), (51.95, 246.49), (44.49, 126.17), (23.16, 139.23), (57.72, 178.27), (29.28, 136.76), (16.32, 153.02), (77.11, 229.13), (67.02, 244.88), (82.99, 224.06), (65.74, 245.96), (72.27, 278.3), (57.19, 220.88), (31.32, 176.38), (47.44, 200.65), (33.39, 227.19), (33.79, 175.44), (62.6, 179.84), (17.13, 111.79), (62.62, 253.81), (44.07, 238.51), (48.74, 213.82), (59.26, 227.88), (60.05, 211.25), (48.93, 270.0), (64.39, 203.49), (30.63, 197.67), (78.38, 266.6), (55.97, 168.9), (49.32, 240.99), (40.56, 157.74), (36.25, 149.06), (36.77, 175.44), (48.63, 215.39), (64.25, 263.75), (55.5, 256.77), (40.19, 201.75), (46.8, 217.08), (50.13, 206.42), (67.84, 222.02), (60.24, 232.58), (53.26, 234.87), (49.66, 216.2), (46.62, 156.74), (73.14, 253.4), (46.33, 177.98), (19.47, 150.76), (46.77, 233.57), (28.83, 143.82), (34.9, 159.68), (40.41, 178.01), (54.77, 225.87), (43.48, 164.9), (50.25, 195.48), (54.15, 222.55), (60.01, 208.39), (38.92, 166.79), (36.56, 172.67), (60.29, 250.61), (15.37, 121.22), (53.14, 233.75), (20.05, 132.94), (82.32, 295.58), (24.45, 178.9), (30.23, 149.41), (41.13, 169.18), (77.85, 230.46), (66.38, 259.02), (65.45, 193.07), (51.22, 229.03), (44.76, 224.06), (56.06, 196.51), (67.61, 266.35), (60.69, 182.82), (62.69, 196.75), (24.65, 171.02), (48.99, 189.93), (62.73, 229.01), (42.61, 178.06), (28.46, 177.15), (47.49, 192.8), (55.27, 197.32), (79.61, 239.38), (69.82, 205.66), (54.41, 202.36), (57.91, 279.16), (45.28, 177.92), (48.06, 249.0), (41.75, 198.24), (53.1, 194.61), (41.41, 136.42), (43.86, 161.37), (58.5, 236.2), (58.91, 231.69), (68.34, 187.33), (57.17, 224.16), (43.98, 193.81), (61.68, 184.07), (76.29, 210.82), (50.36, 186.08), (25.66, 174.21), (73.77, 272.51), (44.36, 183.05), (35.15, 168.12), (62.82, 256.83), (79.95, 239.69), (56.4, 217.96), (74.01, 230.05), (50.97, 243.18), (36.67, 171.93), (54.09, 189.86), (42.84, 160.76), (41.96, 204.12), (81.21, 316.81), (50.83, 206.55), (17.71, 156.25), (58.34, 235.44), (68.31, 224.62), (45.45, 228.79), (70.34, 263.6), (54.53, 191.67), (23.0, 137.65), (32.8, 167.61), (40.22, 193.53), (79.38, 218.35), (51.54, 149.43), (42.06, 158.5), (50.94, 198.66), (57.22, 244.28), (41.43, 139.85), (67.86, 233.56), (46.84, 217.3), (44.05, 176.63), (28.56, 203.45), (53.54, 233.88), (30.03, 178.85), (54.47, 184.77), (41.38, 184.01), (45.73, 224.07), (53.47, 227.78), (56.42, 257.22), (57.55, 209.38), (57.52, 168.93), (70.92, 190.45), (48.12, 196.09), (72.33, 286.23), (59.93, 226.18), (42.29, 172.85), (72.94, 274.9), (66.66, 205.27), (60.92, 237.21), (41.68, 143.41), (40.88, 194.6), (56.38, 224.31), (22.97, 148.83), (37.96, 135.84), (53.55, 201.28), (23.96, 128.77), (56.58, 206.5), (43.44, 165.37), (41.99, 126.22), (34.94, 162.0), (82.48, 289.68), (65.26, 283.75), (47.52, 248.92), (20.87, 169.89), (46.7, 214.42), (60.59, 240.35), (69.09, 184.14), (63.12, 260.32), (42.49, 149.37), (102.38, 312.74), (30.44, 154.21), (42.09, 218.9), (40.16, 197.23), (9.07, 128.09), (56.88, 219.72), (32.71, 155.67), (44.25, 192.99), (34.17, 174.34), (59.99, 243.43), (62.22, 266.34), (44.23, 176.24), (38.89, 153.78), (65.7, 236.58), (60.14, 211.79), (49.96, 174.52), (26.91, 171.8), (63.2, 222.5), (63.19, 217.05), (43.61, 196.17), (43.19, 212.34), (76.07, 277.97), (67.72, 164.19), (80.85, 284.62), (56.67, 210.01), (56.16, 172.04), (49.06, 155.75), (67.37, 259.09), (30.55, 167.68), (70.39, 232.72), (40.11, 176.1), (59.29, 216.59), (25.44, 160.89), (38.79, 195.31), (39.42, 192.34), (36.52, 209.21), (55.46, 215.71), (48.87, 133.83), (63.53, 207.12), (23.78, 158.69), (56.05, 232.74), (49.86, 178.73), (44.3, 163.35), (55.01, 207.18), (65.56, 216.11), (29.06, 157.09), (39.76, 173.87), (39.07, 164.74), (52.91, 236.88), (59.35, 230.85), (49.55, 227.9), (74.39, 220.61), (48.32, 179.58), (60.51, 220.2), (48.97, 235.18), (43.29, 156.04), (49.3, 184.12), (57.61, 208.51), (33.42, 151.32), (45.92, 144.48), (32.44, 201.31), (61.24, 194.21), (73.86, 284.64), (40.37, 159.57), (56.67, 180.04), (62.23, 267.89), (38.46, 169.47), (33.72, 205.3), (56.46, 191.29), (59.91, 189.91), (26.82, 174.73), (14.49, 158.56), (50.12, 200.26), (34.55, 151.27), (57.14, 203.91), (30.09, 165.06), (32.1, 142.99), (52.49, 182.01), (52.09, 201.08), (63.67, 241.76), (24.1, 155.82), (57.94, 213.33), (34.38, 152.25), (63.75, 203.91), (64.42, 197.6), (26.34, 166.88), (78.73, 292.36), (48.19, 205.03), (48.85, 213.84), (70.85, 242.85), (50.0, 224.88), (31.84, 175.45), (50.26, 196.4), (34.42, 160.37), (42.23, 215.59), (59.96, 195.24), (22.84, 190.85), (58.61, 180.73), (67.28, 262.85), (56.14, 222.47), (28.87, 127.29), (64.35, 245.83), (54.07, 204.76), (52.28, 247.66), (38.99, 189.8), (55.94, 177.11), (43.96, 210.51), (42.84, 216.95), (72.66, 266.02), (53.06, 188.89), (44.24, 241.93), (37.05, 181.46), (83.26, 270.82), (49.83, 195.42), (63.6, 245.6), (58.27, 245.13), (26.82, 99.51), (56.86, 187.84), (63.6, 202.61), (50.17, 181.04), (44.61, 183.34), (48.19, 147.3), (22.97, 122.79), (64.12, 254.38), (74.93, 225.13), (30.7, 139.4), (53.56, 142.94), (42.25, 181.54), (71.95, 245.35), (57.62, 255.5), (68.14, 222.17), (43.94, 175.37), (26.39, 145.67), (48.07, 256.02), (43.96, 159.28), (56.19, 213.15), (22.11, 204.4), (51.69, 151.31), (60.52, 203.78), (46.58, 186.88), (22.77, 105.9), (52.42, 166.87), (23.71, 152.1), (41.53, 199.63), (67.2, 249.98), (46.49, 229.43), (51.22, 177.85), (56.82, 225.1), (33.7, 152.94), (74.45, 227.81), (28.6, 107.33), (33.4, 199.58), (54.43, 205.47), (33.61, 196.73), (39.09, 143.27), (33.46, 183.2), (59.74, 240.73), (52.57, 182.42), (49.72, 159.87), (73.84, 214.0), (65.54, 292.02), (57.16, 155.64), (49.67, 189.17), (39.26, 174.8), (64.44, 205.17), (64.58, 216.25), (44.93, 155.29), (25.43, 96.64), (30.69, 140.2), (41.45, 196.1), (48.65, 216.65), (28.26, 226.98), (65.57, 234.07), (74.85, 298.72), (50.95, 201.08), (73.59, 269.72), (60.96, 172.69), (66.79, 266.04), (32.09, 179.96), (43.43, 177.94), (46.58, 224.06), (69.1, 251.44), (59.84, 202.7), (40.92, 213.27), (60.33, 205.27), (51.06, 200.1), (39.35, 193.71), (52.46, 223.92), (42.23, 155.49), (51.84, 260.75), (54.65, 242.38), (65.82, 217.87), (87.3, 247.4), (46.53, 153.49), (65.12, 235.69), (56.29, 179.31), (39.18, 167.11), (48.78, 209.77), (45.63, 186.61), (61.89, 235.19), (25.35, 178.69), (52.74, 200.73), (40.39, 200.75), (40.75, 201.48), (57.02, 166.39), (58.4, 175.65), (48.5, 229.97), (45.61, 207.36), (77.76, 276.03), (60.79, 210.66), (56.0, 221.78), (58.03, 184.09), (63.95, 221.0), (54.12, 208.29), (67.59, 187.42), (66.85, 202.89), (63.48, 213.37), (60.83, 223.64), (56.01, 181.12), (49.77, 182.14), (57.7, 198.9), (65.2, 262.18), (49.8, 171.23), (54.4, 182.53), (51.98, 242.07), (77.51, 279.36), (26.11, 149.39), (72.18, 234.31), (17.82, 170.29), (44.26, 223.97), (55.67, 191.51), (30.01, 135.03), (47.66, 213.02), (47.8, 158.95), (56.31, 203.29), (53.37, 168.29), (61.34, 275.46), (32.42, 178.26), (27.43, 179.02), (51.91, 214.41), (36.41, 176.6), (71.4, 274.36), (48.35, 193.29), (51.27, 199.36), (61.31, 212.83), (68.06, 212.12), (42.1, 211.79), (56.13, 236.45), (72.47, 247.03), (30.47, 148.04), (54.74, 189.86), (45.12, 166.25), (31.24, 158.87), (76.2, 261.64), (39.88, 112.95), (39.55, 211.8), (42.55, 153.14), (34.41, 213.93), (48.34, 172.67), (50.04, 149.98), (62.31, 165.24), (65.8, 249.5), (52.46, 204.07), (54.75, 268.76), (22.88, 148.81), (53.97, 196.89), (66.84, 247.26), (29.94, 166.54), (47.45, 197.82), (48.82, 204.26), (59.33, 230.88), (58.8, 201.73), (62.12, 225.63), (33.9, 130.62), (38.66, 135.08), (77.44, 212.54), (47.68, 177.26), (40.11, 154.6), (35.07, 210.87), (47.95, 231.57), (41.11, 167.45), (50.48, 209.94), (58.59, 282.77), (38.52, 177.41), (71.42, 257.89), (60.37, 174.38), (39.29, 210.36), (48.89, 173.32), (41.19, 208.36), (55.76, 190.47), (38.89, 180.53), (34.23, 127.93), (69.5, 195.33), (59.69, 228.03), (44.36, 194.02), (55.57, 217.83), (68.98, 213.76), (36.04, 218.27), (58.38, 193.37), (68.49, 240.06), (38.85, 150.57), (78.9, 294.68), (34.34, 217.87), (27.67, 199.51), (52.75, 255.19), (32.37, 225.2), (71.68, 250.18), (71.51, 333.42), (59.32, 180.76), (28.23, 131.77), (55.57, 196.49), (39.01, 188.43), (59.33, 205.78), (36.9, 168.0), (69.52, 291.33), (38.79, 174.28), (39.97, 196.49), (57.97, 155.67), (52.96, 214.58), (69.67, 250.82), (67.02, 180.19), (29.97, 135.05), (12.49, 147.27), (24.04, 136.13), (48.66, 207.48), (57.75, 171.53), (51.59, 225.8), (53.43, 207.39), (40.54, 201.16), (43.9, 235.27), (47.29, 188.84), (49.13, 213.52), (35.51, 141.18), (44.57, 203.59), (56.95, 194.88), (16.55, 200.83), (45.59, 161.86), (34.49, 169.48), (48.74, 206.72), (66.07, 212.3), (20.92, 129.84), (48.12, 219.83), (53.82, 204.57), (64.55, 195.62), (51.66, 135.25), (48.62, 192.86), (39.14, 185.73), (43.37, 163.1), (39.25, 220.72), (50.37, 184.78), (36.93, 198.2), (40.9, 221.43), (52.95, 268.34), (58.32, 187.98), (56.6, 258.1), (37.37, 196.07), (40.04, 152.27), (54.38, 200.39), (24.85, 159.61), (41.16, 201.75), (80.15, 274.71), (41.1, 182.45), (47.7, 158.23), (45.12, 110.78), (63.63, 224.73), (74.21, 231.28), (80.46, 304.0), (44.81, 144.82), (63.48, 219.4), (60.3, 259.76), (68.64, 231.76), (61.46, 249.57), (48.06, 157.54), (41.12, 176.14), (60.89, 226.11), (39.71, 165.19), (61.39, 246.97), (59.73, 165.05), (23.73, 152.02), (67.45, 237.0), (62.01, 220.63), (21.22, 169.67), (55.19, 160.77), (35.97, 173.94), (41.65, 197.75), (44.4, 213.98), (34.14, 182.24), (68.97, 223.39), (64.34, 220.75), (40.16, 203.84), (48.66, 242.29), (56.51, 149.41), (56.63, 236.34), (55.2, 220.97), (68.82, 231.43), (61.19, 185.1), (31.51, 169.36), (62.06, 227.99), (40.68, 230.65), (79.97, 229.46), (55.06, 197.93), (69.76, 191.57), (66.1, 227.4), (47.25, 177.67), (74.05, 240.04), (53.31, 248.23), (76.04, 255.08), (82.88, 284.43), (14.95, 170.04), (67.36, 214.8), (27.96, 102.27), (44.24, 224.09), (37.85, 204.95), (14.14, 133.22), (39.5, 171.33), (58.99, 239.23), (39.72, 212.24), (55.2, 169.88), (29.47, 182.52), (55.85, 305.67), (50.31, 191.45), (34.8, 224.93), (47.71, 223.99), (63.75, 203.63), (37.85, 166.87), (65.2, 232.82), (24.44, 181.72), (30.05, 148.53), (49.84, 208.62), (53.05, 183.28), (43.89, 161.27), (51.08, 212.27), (49.3, 236.03), (63.72, 246.39), (43.41, 202.75), (65.78, 207.33), (55.72, 225.68), (32.72, 180.13), (50.03, 206.65), (63.3, 232.1), (80.9, 196.76), (51.93, 239.75), (18.11, 163.18), (37.02, 166.01), (49.11, 188.67), (58.47, 186.3), (43.73, 191.97), (39.61, 189.13), (48.36, 221.3), (40.33, 180.56), (56.21, 266.03), (57.49, 218.78), (48.26, 181.47), (40.33, 141.44), (61.18, 211.86), (36.97, 187.1), (69.48, 236.69), (14.16, 119.15), (41.22, 179.19), (61.12, 231.76), (42.38, 157.76), (36.91, 183.55), (44.67, 187.42), (39.88, 197.45), (45.56, 184.7), (53.92, 212.61), (52.56, 227.87), (20.13, 127.97), (40.36, 191.19), (75.99, 213.69), (51.59, 251.91), (61.18, 228.39), (44.89, 212.7), (49.73, 215.73), (50.84, 167.11)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=928；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6927；
t = r·√((n−2)/(1−r²)) = 0.6927·√((928−2)/(1−0.4798)) = 29.2268；df=926；p = -2.95e+59。
95% CI（Fisher z）：tanh(z±30.430·1/√(n−3)) = [-0.1464, 0.9521]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=928；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6927；
t = r·√((n−2)/(1−r²)) = 0.6927·√((928−2)/(1−0.4798)) = 29.2268；df=926；p = -2.95e+59。
95% CI（Fisher z）：tanh(z±30.430·1/√(n−3)) = [-0.1464, 0.9521]。
③ 计算结果：
r = 0.6927，t = 29.2268，p = -2.95e+59，95%CI = [-0.1464, 0.9521]
④ 结论：在 α=0.05 水平下，p=-2.95e+59 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.6927，方向正），广告费越高销量越高。
案例 2. 学习时长(小时)与考试成绩(分)
Pearson相关学习时间对成绩正向影响
① 原始数据：
样本 n=772，逐条 (x, y)：[(9.72, 55.16), (16.25, 85.55), (6.34, 66.87), (10.83, 60.76), (12.15, 84.97), (9.99, 71.63), (7.75, 64.72), (6.89, 55.15), (13.29, 75.71), (6.33, 68.13), (7.55, 61.84), (13.38, 91.74), (7.45, 84.01), (11.8, 82.8), (12.48, 64.37), (11.81, 75.81), (9.81, 73.93), (12.09, 77.02), (15.16, 83.31), (9.5, 67.61), (7.42, 68.82), (9.72, 54.24), (15.32, 91.18), (12.06, 76.62), (13.74, 81.75), (8.89, 61.12), (9.42, 67.92), (14.1, 79.47), (10.99, 88.78), (7.26, 77.25), (8.05, 65.37), (12.82, 76.48), (10.26, 66.61), (8.94, 76.63), (9.18, 74.15), (6.88, 65.17), (11.64, 72.13), (11.45, 62.28), (11.02, 77.94), (6.45, 66.91), (6.92, 61.12), (5.38, 58.53), (10.91, 68.59), (10.91, 68.41), (11.01, 79.25), (10.81, 82.63), (5.24, 63.54), (9.9, 78.99), (8.41, 66.16), (5.44, 67.68), (10.54, 73.81), (11.59, 67.26), (8.4, 66.08), (8.91, 65.92), (10.37, 79.56), (10.25, 74.43), (8.27, 69.29), (16.45, 95.45), (6.21, 46.54), (6.49, 50.78), (8.36, 51.1), (14.03, 73.46), (5.3, 53.24), (8.04, 68.61), (7.8, 56.47), (12.27, 66.82), (14.33, 88.4), (4.87, 53.54), (8.18, 58.36), (12.97, 69.86), (11.38, 90.28), (9.64, 75.94), (6.39, 86.91), (10.09, 68.12), (16.02, 89.94), (5.89, 67.95), (11.64, 78.56), (9.79, 83.45), (5.88, 69.46), (10.92, 80.42), (10.26, 74.54), (6.89, 47.35), (10.38, 75.11), (16.25, 86.38), (8.8, 64.27), (15.96, 82.3), (12.7, 88.8), (11.65, 77.74), (10.06, 54.94), (11.13, 78.53), (3.47, 66.21), (12.79, 85.11), (9.48, 77.09), (13.01, 94.51), (13.87, 81.05), (9.54, 72.9), (3.55, 44.04), (16.19, 90.51), (7.04, 73.48), (11.74, 83.5), (6.8, 67.21), (9.56, 86.49), (11.38, 74.22), (6.31, 68.53), (16.02, 77.21), (12.43, 69.06), (12.12, 83.05), (6.27, 56.29), (8.7, 57.27), (9.58, 60.05), (9.91, 63.26), (5.57, 71.17), (11.2, 75.13), (4.91, 57.78), (10.83, 65.97), (11.9, 69.24), (8.4, 64.94), (4.9, 77.75), (8.59, 73.81), (11.44, 75.15), (11.99, 71.98), (12.99, 64.65), (8.18, 67.05), (6.36, 54.08), (8.12, 61.53), (13.98, 88.46), (7.52, 62.97), (9.96, 80.14), (10.34, 64.79), (9.98, 76.19), (13.48, 75.54), (9.74, 63.86), (14.09, 89.72), (8.73, 81.67), (13.04, 76.13), (10.67, 90.29), (9.41, 72.07), (3.35, 48.1), (10.89, 70.47), (7.54, 72.51), (10.75, 59.72), (13.98, 73.73), (3.9, 44.35), (12.57, 71.95), (14.98, 95.67), (9.8, 69.6), (7.87, 66.9), (18.09, 86.72), (8.25, 71.82), (7.02, 51.97), (11.54, 70.59), (6.69, 52.92), (11.99, 74.31), (7.02, 61.68), (9.26, 72.07), (12.05, 81.42), (15.11, 94.46), (6.43, 56.82), (8.77, 63.46), (11.3, 71.51), (7.77, 74.3), (8.14, 82.44), (12.1, 81.21), (14.07, 86.1), (9.87, 74.33), (16.51, 102.88), (12.16, 65.97), (13.63, 71.99), (12.35, 74.93), (7.54, 62.48), (9.08, 65.48), (10.96, 74.53), (9.61, 63.62), (12.07, 65.0), (8.66, 56.03), (7.33, 50.31), (9.87, 71.05), (10.22, 65.52), (12.12, 85.23), (14.12, 94.53), (7.34, 74.78), (12.4, 86.52), (12.9, 84.74), (11.17, 75.39), (8.14, 77.12), (12.41, 69.71), (11.51, 60.99), (14.84, 81.1), (10.29, 71.07), (11.55, 87.28), (11.11, 69.05), (12.23, 74.64), (11.42, 72.77), (17.27, 83.45), (8.83, 73.39), (12.43, 70.15), (14.58, 86.66), (9.21, 74.77), (5.0, 63.26), (10.81, 79.89), (12.45, 81.35), (12.83, 72.18), (16.5, 88.37), (8.46, 74.07), (12.35, 68.34), (11.12, 77.08), (14.41, 66.74), (13.84, 89.8), (8.48, 53.2), (9.09, 73.93), (7.59, 65.66), (10.09, 74.0), (10.81, 78.3), (8.58, 79.38), (7.29, 66.83), (18.37, 86.8), (11.77, 70.36), (11.1, 79.35), (8.09, 67.28), (7.53, 59.13), (10.78, 67.48), (14.86, 81.91), (8.75, 74.38), (8.88, 63.2), (11.16, 70.79), (10.86, 76.32), (8.05, 49.29), (10.38, 76.71), (12.87, 96.09), (6.9, 57.76), (9.15, 45.0), (13.35, 76.99), (13.34, 78.78), (6.67, 80.4), (10.15, 63.14), (7.15, 67.5), (11.39, 85.62), (6.77, 57.19), (9.4, 70.54), (16.26, 83.67), (5.91, 58.32), (11.7, 75.33), (11.68, 80.68), (11.88, 69.1), (14.84, 89.91), (9.86, 72.99), (9.34, 68.97), (8.96, 70.06), (11.24, 52.93), (7.55, 71.91), (11.9, 76.31), (11.48, 76.71), (12.3, 80.0), (7.47, 79.04), (11.14, 71.69), (5.35, 64.27), (8.67, 63.39), (6.63, 71.54), (10.51, 73.56), (8.03, 54.84), (15.87, 91.02), (3.48, 45.88), (7.67, 58.96), (9.1, 61.45), (8.1, 76.97), (7.57, 55.46), (13.93, 85.16), (9.34, 63.2), (8.15, 55.63), (13.11, 85.33), (10.83, 85.23), (7.94, 50.82), (6.85, 66.59), (13.45, 78.35), (14.32, 88.71), (9.83, 76.78), (9.34, 71.55), (11.32, 82.02), (9.67, 74.14), (8.28, 70.46), (7.83, 75.23), (11.65, 74.92), (13.5, 89.04), (11.85, 75.58), (8.23, 57.78), (11.22, 82.42), (16.8, 104.23), (7.39, 67.6), (7.09, 73.76), (12.75, 94.18), (13.62, 77.16), (13.03, 65.58), (12.65, 78.61), (14.37, 91.46), (14.1, 84.42), (15.41, 89.83), (13.84, 84.77), (9.12, 71.51), (4.26, 63.83), (10.24, 71.6), (12.35, 82.68), (9.86, 51.79), (3.58, 54.9), (10.14, 58.65), (10.35, 83.12), (9.23, 72.98), (15.91, 88.06), (10.86, 74.3), (12.18, 75.97), (11.25, 79.76), (9.4, 83.86), (13.57, 77.37), (9.01, 64.89), (10.79, 79.35), (4.94, 70.19), (10.47, 64.81), (9.32, 57.37), (9.4, 81.1), (13.52, 87.05), (5.01, 69.4), (10.55, 78.85), (11.47, 74.66), (9.57, 63.16), (14.19, 89.02), (6.8, 75.55), (9.25, 70.69), (11.71, 88.7), (9.5, 74.23), (6.8, 67.44), (9.53, 69.91), (7.52, 72.12), (17.83, 96.73), (12.5, 83.07), (12.71, 74.78), (13.23, 86.01), (6.84, 81.73), (10.94, 85.66), (12.97, 66.9), (17.33, 93.59), (12.86, 77.06), (10.52, 64.48), (12.23, 71.54), (7.69, 62.07), (11.32, 93.67), (11.57, 72.96), (13.19, 72.54), (11.42, 76.21), (10.62, 79.75), (12.43, 91.6), (9.2, 67.55), (10.55, 68.12), (10.74, 65.35), (9.24, 60.56), (10.27, 66.83), (6.38, 74.45), (8.35, 70.37), (12.39, 86.13), (4.98, 72.32), (9.14, 67.04), (11.76, 87.59), (8.15, 77.96), (8.36, 70.64), (10.11, 58.11), (10.06, 87.06), (8.89, 58.91), (12.47, 99.15), (5.36, 61.09), (6.13, 67.88), (9.81, 61.23), (16.74, 86.34), (9.54, 64.36), (14.66, 90.97), (9.54, 65.3), (9.25, 76.68), (9.48, 82.61), (10.47, 70.16), (8.79, 70.83), (7.29, 63.47), (7.5, 64.1), (10.08, 75.32), (6.66, 58.55), (11.26, 85.13), (10.57, 76.91), (11.74, 68.39), (16.33, 81.91), (6.77, 63.66), (11.48, 82.14), (6.75, 73.73), (7.71, 74.07), (5.71, 53.26), (15.46, 91.89), (9.58, 64.13), (13.78, 73.03), (7.82, 71.78), (10.17, 61.85), (7.2, 59.8), (9.51, 75.93), (13.57, 92.94), (8.91, 79.13), (14.49, 89.27), (11.95, 71.17), (16.35, 88.59), (8.78, 74.81), (8.08, 70.49), (10.78, 76.88), (10.34, 75.6), (12.49, 91.95), (12.46, 81.01), (17.29, 105.74), (12.0, 79.05), (12.5, 95.6), (12.61, 101.19), (12.72, 64.54), (8.56, 60.58), (5.5, 61.92), (14.72, 95.91), (10.31, 81.86), (6.83, 67.75), (11.84, 89.95), (8.14, 65.88), (10.55, 79.91), (10.47, 89.23), (12.8, 72.79), (5.74, 64.54), (10.07, 71.52), (12.07, 100.25), (10.2, 78.23), (6.3, 63.69), (9.85, 68.72), (4.52, 66.52), (12.89, 84.78), (15.32, 80.19), (11.83, 78.37), (9.4, 69.47), (8.69, 86.52), (12.52, 81.76), (10.15, 80.06), (12.82, 81.97), (13.27, 86.53), (10.38, 64.73), (7.55, 79.99), (6.95, 61.85), (9.71, 79.82), (12.02, 77.16), (10.31, 73.67), (12.44, 81.2), (9.43, 64.94), (7.34, 65.17), (15.83, 84.14), (9.11, 74.98), (7.24, 68.59), (8.25, 47.35), (10.59, 66.88), (8.3, 81.42), (13.97, 77.93), (10.97, 72.54), (11.88, 88.79), (12.0, 76.01), (8.9, 58.45), (7.07, 65.71), (5.24, 62.1), (11.47, 57.95), (6.76, 60.0), (7.6, 63.39), (2.98, 39.37), (14.15, 81.89), (8.43, 54.51), (11.08, 73.76), (2.56, 53.97), (12.09, 77.32), (9.61, 75.96), (11.82, 69.08), (15.02, 99.88), (11.82, 79.17), (8.83, 75.52), (16.74, 79.41), (7.45, 77.44), (14.49, 83.14), (9.63, 69.27), (13.48, 79.46), (13.0, 69.03), (11.42, 80.88), (15.22, 99.95), (6.31, 61.49), (9.69, 84.11), (8.76, 64.89), (11.14, 67.03), (11.19, 76.55), (12.14, 79.09), (9.9, 67.69), (17.4, 90.79), (9.74, 57.26), (3.76, 62.44), (8.26, 60.46), (6.35, 72.05), (11.17, 69.32), (7.08, 72.86), (10.32, 74.49), (11.83, 86.37), (10.57, 75.22), (16.87, 77.58), (6.34, 58.79), (10.92, 69.37), (6.83, 56.26), (10.34, 63.65), (11.14, 80.39), (3.85, 48.85), (10.21, 58.42), (15.29, 79.52), (4.0, 52.81), (11.37, 76.86), (7.73, 54.67), (10.12, 72.56), (11.78, 79.15), (9.27, 54.64), (10.95, 66.06), (11.35, 69.91), (8.39, 67.94), (13.75, 81.81), (8.78, 75.91), (11.21, 79.42), (11.33, 96.08), (9.99, 71.39), (6.59, 53.03), (8.62, 58.77), (8.9, 59.29), (8.25, 59.14), (8.11, 78.58), (9.2, 69.94), (9.2, 71.47), (9.48, 89.32), (9.54, 69.28), (8.92, 55.23), (8.7, 60.62), (7.11, 61.36), (7.36, 65.35), (8.43, 73.17), (11.54, 81.59), (10.76, 79.69), (10.78, 78.93), (12.22, 86.5), (11.6, 89.88), (12.24, 76.65), (5.14, 56.39), (7.49, 60.03), (5.24, 51.49), (9.15, 65.15), (15.71, 84.2), (8.41, 64.83), (7.59, 75.97), (11.48, 70.07), (10.06, 79.72), (11.65, 79.8), (10.31, 75.29), (11.83, 64.27), (10.64, 76.42), (9.57, 71.42), (9.08, 75.82), (8.09, 57.5), (6.97, 75.55), (6.46, 46.68), (6.38, 44.61), (12.57, 95.9), (10.34, 71.73), (15.96, 87.29), (13.38, 88.86), (17.46, 87.89), (10.78, 70.88), (10.84, 82.06), (12.8, 63.44), (5.26, 63.79), (7.15, 63.04), (3.51, 63.3), (9.66, 44.76), (11.62, 85.64), (4.32, 51.68), (10.53, 72.11), (7.18, 55.43), (7.11, 53.65), (7.13, 61.61), (13.31, 84.22), (12.32, 71.4), (12.62, 86.03), (15.25, 85.9), (14.63, 90.59), (5.78, 61.75), (11.31, 92.18), (6.72, 69.47), (6.18, 57.12), (7.59, 78.95), (12.72, 92.07), (6.51, 67.16), (13.48, 70.47), (11.67, 77.98), (6.43, 58.76), (9.93, 65.85), (10.63, 77.6), (6.6, 56.2), (10.89, 78.8), (12.61, 66.62), (6.9, 75.6), (8.76, 65.23), (9.48, 50.05), (8.97, 69.21), (7.88, 75.9), (6.9, 57.54), (11.51, 89.63), (8.93, 57.62), (11.44, 70.13), (17.23, 81.77), (11.79, 88.63), (7.77, 62.62), (24.79, 123.43), (4.1, 46.81), (12.52, 76.6), (8.74, 64.71), (10.21, 57.76), (6.27, 56.58), (5.55, 68.06), (13.61, 90.92), (16.38, 82.83), (7.2, 61.56), (11.63, 66.36), (12.07, 86.36), (14.29, 79.81), (9.86, 58.56), (7.89, 66.97), (11.19, 66.69), (9.17, 76.35), (11.83, 81.69), (12.54, 74.15), (11.91, 71.74), (14.27, 82.66), (9.09, 70.55), (11.38, 76.28), (6.33, 59.27), (10.12, 81.29), (9.98, 66.93), (8.65, 65.6), (7.25, 77.99), (5.48, 67.2), (8.2, 62.79), (9.04, 78.91), (6.49, 64.39), (9.7, 72.16), (10.27, 87.3), (5.81, 69.9), (4.19, 58.96), (8.39, 70.93), (8.37, 63.61), (14.13, 90.44), (6.09, 71.2), (12.88, 94.17), (11.84, 77.03), (7.41, 56.6), (14.43, 81.13), (5.52, 63.57), (10.6, 90.1), (13.62, 86.09), (10.7, 75.47), (15.11, 79.98), (13.78, 84.0), (10.54, 68.27), (14.49, 84.33), (10.5, 83.01), (11.62, 77.3), (15.61, 86.15), (13.54, 66.41), (9.71, 65.47), (8.31, 78.64), (12.23, 80.81), (12.09, 78.58), (5.55, 67.34), (6.95, 64.27), (0.61, 51.05), (10.34, 50.74), (12.03, 78.33), (9.88, 68.88), (9.24, 60.95), (11.7, 88.39), (10.76, 52.48), (6.32, 57.06), (7.13, 63.22), (8.47, 67.96), (5.22, 59.5), (10.32, 66.91), (8.57, 56.28), (6.5, 65.86), (13.89, 89.09), (9.11, 63.84), (15.04, 74.52), (11.02, 68.5), (14.6, 76.84), (13.93, 68.12), (14.43, 91.24), (13.66, 75.42), (10.24, 63.12), (8.86, 66.29), (12.35, 73.98), (10.61, 77.9), (7.73, 63.05), (14.32, 105.49), (11.95, 75.97), (15.06, 86.23), (10.47, 74.33), (7.18, 81.04), (13.98, 80.2), (11.81, 83.34), (7.52, 67.18), (9.63, 75.44), (6.8, 60.14), (8.36, 53.29), (3.56, 62.05), (14.96, 86.45), (7.04, 76.14), (6.38, 47.07), (9.69, 65.52), (11.9, 75.48), (13.25, 92.56), (8.27, 82.99), (8.83, 62.74), (8.02, 65.2), (11.59, 72.92), (10.2, 58.06), (10.77, 79.54), (13.89, 97.14), (15.12, 80.36), (13.34, 71.01), (8.69, 71.3), (10.73, 58.59), (13.48, 73.91), (16.73, 87.47), (5.09, 66.93), (11.83, 76.09), (11.78, 84.22), (7.65, 54.92), (10.18, 69.64), (7.54, 55.99), (8.8, 83.91), (6.74, 60.98), (4.22, 58.2), (9.35, 59.29), (9.01, 63.23), (10.35, 82.75), (7.51, 68.38), (7.47, 53.72), (10.16, 66.94), (7.36, 59.35), (14.44, 70.75), (11.34, 66.55), (6.79, 64.66), (15.29, 81.3), (9.96, 79.94), (11.47, 72.71), (13.47, 79.27), (7.62, 66.89), (5.69, 65.73), (8.82, 44.92), (7.66, 63.93), (11.8, 79.2), (9.25, 84.11), (12.3, 84.97), (9.12, 69.67), (5.98, 72.58), (5.8, 58.88), (11.02, 63.68), (12.05, 74.36), (13.19, 93.6), (7.58, 84.96), (9.37, 80.41), (10.73, 77.78), (4.08, 38.39)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=772；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6898；
t = r·√((n−2)/(1−r²)) = 0.6898·√((772−2)/(1−0.4758)) = 26.4363；df=770；p = -5.33e+46。
95% CI（Fisher z）：tanh(z±20.956·1/√(n−3)) = [0.0916, 0.9222]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=772；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6898；
t = r·√((n−2)/(1−r²)) = 0.6898·√((772−2)/(1−0.4758)) = 26.4363；df=770；p = -5.33e+46。
95% CI（Fisher z）：tanh(z±20.956·1/√(n−3)) = [0.0916, 0.9222]。
③ 计算结果：
r = 0.6898，t = 26.4363，p = -5.33e+46，95%CI = [0.0916, 0.9222]
④ 结论：在 α=0.05 水平下，p=-5.33e+46 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.6898，方向正），学习时间对成绩正向影响。
案例 3. 气温(℃)与冰淇淋日销量(个)
Pearson相关夏季温度越高冰淇淋卖得越多
① 原始数据：
样本 n=663，逐条 (x, y)：[(27.15, 300.13), (38.13, 467.76), (19.0, 98.74), (29.73, 327.4), (28.11, 196.2), (18.68, 265.78), (24.77, 357.14), (26.45, 318.92), (31.38, 288.92), (28.55, 227.86), (32.05, 269.02), (31.11, 295.41), (19.07, 171.53), (33.47, 309.96), (32.24, 295.14), (30.96, 307.29), (43.58, 423.03), (38.32, 350.14), (33.98, 362.84), (25.22, 275.46), (28.31, 322.04), (24.18, 244.33), (24.0, 197.16), (35.29, 240.99), (29.52, 338.56), (33.8, 378.02), (35.19, 293.71), (19.3, 166.34), (24.13, 216.91), (27.43, 319.86), (21.69, 160.73), (28.08, 304.23), (24.87, 253.19), (30.84, 401.42), (28.14, 328.43), (27.55, 292.79), (25.41, 297.05), (26.0, 313.32), (36.16, 330.9), (22.83, 249.15), (21.69, 281.08), (23.63, 358.78), (38.21, 381.3), (32.36, 451.87), (26.22, 287.5), (26.5, 314.96), (40.02, 446.83), (28.07, 268.32), (33.88, 291.54), (28.18, 292.49), (33.98, 349.65), (25.78, 217.56), (27.7, 269.94), (34.55, 379.81), (28.8, 388.6), (34.18, 220.97), (22.29, 217.15), (28.2, 355.87), (31.52, 273.36), (30.31, 299.63), (15.12, 228.7), (30.49, 336.25), (30.14, 270.32), (21.64, 307.82), (29.54, 287.21), (25.12, 341.3), (36.13, 309.78), (35.4, 449.64), (31.14, 328.15), (28.07, 240.19), (20.69, 169.08), (23.08, 265.98), (18.95, 211.76), (27.3, 257.64), (18.66, 201.92), (24.39, 397.91), (22.54, 237.85), (28.12, 390.0), (30.27, 466.22), (27.12, 279.7), (31.16, 261.51), (35.29, 412.27), (27.12, 249.48), (26.32, 241.01), (28.09, 316.11), (35.28, 424.0), (25.71, 234.57), (32.81, 316.86), (25.5, 292.99), (30.03, 238.99), (29.34, 230.25), (42.37, 519.48), (37.07, 379.45), (22.32, 296.56), (29.49, 261.42), (36.51, 393.84), (28.04, 308.34), (30.93, 294.5), (28.95, 323.09), (34.48, 379.18), (30.8, 358.93), (21.02, 208.37), (30.46, 302.51), (31.3, 347.08), (32.74, 289.34), (20.85, 256.77), (16.27, 230.7), (29.76, 348.82), (26.48, 321.11), (33.21, 351.46), (38.13, 402.87), (31.66, 310.09), (29.75, 239.1), (34.61, 322.8), (34.15, 414.59), (34.66, 312.06), (29.99, 238.22), (32.67, 363.66), (25.23, 226.49), (20.94, 230.71), (28.87, 252.61), (24.94, 230.99), (25.24, 281.07), (20.89, 330.63), (17.7, 178.15), (22.41, 247.15), (25.54, 249.88), (29.9, 361.31), (24.23, 297.03), (22.28, 269.15), (24.66, 202.26), (37.86, 362.89), (29.48, 289.87), (31.07, 276.65), (32.55, 398.02), (22.93, 273.38), (31.75, 365.79), (33.71, 300.54), (22.42, 297.36), (19.07, 200.42), (28.61, 295.89), (24.65, 280.46), (29.52, 291.1), (37.69, 353.13), (31.39, 299.3), (21.66, 221.21), (28.56, 295.31), (23.19, 240.71), (35.81, 357.47), (26.93, 203.87), (35.65, 354.66), (35.56, 359.78), (32.77, 377.94), (31.49, 350.63), (23.78, 250.89), (25.38, 243.52), (24.18, 250.03), (21.52, 226.47), (30.9, 222.18), (20.07, 268.48), (21.87, 174.0), (19.94, 196.82), (30.3, 418.75), (31.7, 299.7), (21.95, 285.83), (26.42, 279.4), (24.84, 358.98), (24.17, 140.89), (27.37, 321.05), (19.8, 309.87), (26.81, 287.7), (25.93, 309.85), (23.46, 359.54), (36.83, 220.21), (15.77, 227.31), (21.55, 242.2), (37.06, 394.92), (32.9, 317.04), (26.37, 351.63), (26.72, 246.54), (35.18, 295.11), (18.12, 170.4), (28.22, 311.95), (31.19, 291.42), (25.14, 227.32), (41.61, 390.15), (32.11, 257.42), (22.39, 221.01), (15.8, 241.71), (16.72, 199.08), (23.18, 308.2), (37.98, 458.41), (29.99, 293.94), (30.94, 309.13), (26.9, 309.74), (30.05, 268.03), (22.37, 280.92), (27.91, 249.09), (26.88, 276.14), (42.98, 358.07), (26.14, 246.81), (32.58, 413.76), (28.24, 291.46), (11.71, 193.28), (20.35, 164.79), (22.63, 187.29), (31.08, 435.86), (23.25, 300.23), (35.62, 393.52), (33.41, 321.52), (23.32, 264.4), (35.05, 395.15), (28.25, 329.62), (29.78, 189.58), (21.26, 164.58), (31.08, 285.31), (30.3, 334.74), (33.06, 420.02), (34.87, 418.26), (23.02, 218.16), (26.9, 323.05), (20.52, 197.9), (33.68, 370.44), (28.5, 338.14), (31.11, 269.16), (35.45, 399.35), (33.11, 298.44), (19.24, 290.78), (25.12, 329.11), (21.21, 241.97), (22.56, 274.79), (24.57, 332.51), (23.92, 242.99), (27.68, 296.43), (19.22, 279.0), (29.69, 366.6), (37.18, 393.28), (39.27, 268.06), (20.21, 195.48), (36.84, 454.54), (27.38, 312.02), (30.49, 223.01), (33.14, 423.57), (25.37, 182.09), (28.85, 292.65), (16.36, 200.9), (33.94, 368.41), (25.17, 226.02), (28.57, 344.79), (28.64, 307.67), (22.23, 279.83), (31.7, 384.21), (22.08, 227.33), (19.25, 290.51), (24.33, 286.54), (22.63, 309.97), (23.77, 305.09), (35.74, 362.81), (36.26, 387.48), (30.23, 322.34), (40.51, 382.96), (25.46, 279.95), (20.24, 168.73), (33.54, 421.58), (20.15, 241.65), (39.46, 373.53), (24.43, 295.71), (27.4, 214.57), (30.29, 325.89), (27.39, 277.07), (38.77, 370.9), (28.13, 313.18), (26.03, 292.13), (26.85, 218.1), (25.63, 217.18), (24.04, 285.97), (31.73, 352.42), (31.69, 357.25), (28.01, 321.9), (31.46, 320.01), (25.47, 254.75), (33.77, 339.81), (30.08, 332.69), (21.44, 199.3), (29.02, 241.99), (34.7, 431.01), (29.97, 384.31), (27.16, 251.66), (35.23, 318.74), (31.68, 247.8), (27.24, 283.31), (32.3, 381.32), (31.18, 361.11), (33.82, 382.07), (29.81, 342.0), (17.05, 200.05), (32.57, 312.78), (25.83, 286.68), (32.38, 345.65), (29.1, 230.52), (28.01, 353.93), (26.76, 260.08), (28.85, 377.6), (19.63, 315.69), (36.42, 371.33), (28.78, 256.19), (26.1, 221.22), (35.11, 369.01), (27.21, 349.27), (27.91, 383.09), (33.25, 192.52), (19.42, 208.42), (23.07, 291.42), (26.23, 423.78), (28.92, 320.78), (18.41, 202.32), (26.63, 305.92), (27.7, 275.83), (27.9, 297.12), (34.29, 376.87), (33.29, 342.23), (23.96, 237.45), (27.18, 382.7), (21.99, 292.12), (25.23, 254.57), (31.37, 277.25), (26.25, 288.56), (24.95, 312.36), (22.99, 210.64), (26.51, 383.72), (34.36, 358.1), (23.1, 189.2), (20.31, 198.53), (34.13, 357.31), (26.42, 301.58), (31.0, 292.32), (22.93, 242.18), (26.4, 344.72), (26.1, 332.95), (20.07, 195.59), (14.21, 99.48), (26.73, 260.21), (26.76, 307.6), (25.83, 347.84), (26.11, 238.2), (27.44, 328.14), (39.4, 474.77), (21.97, 237.66), (24.65, 220.13), (34.14, 423.31), (22.46, 306.8), (39.28, 389.41), (28.33, 301.87), (30.7, 328.91), (29.44, 365.42), (32.19, 284.71), (21.92, 325.1), (30.19, 225.15), (36.48, 388.17), (34.77, 377.2), (28.37, 270.4), (34.87, 410.36), (29.13, 305.37), (26.67, 287.07), (38.85, 433.07), (34.78, 372.95), (28.94, 347.24), (26.09, 305.92), (23.53, 251.72), (29.4, 246.41), (21.41, 224.41), (23.71, 271.3), (19.61, 173.85), (39.05, 330.51), (32.5, 364.77), (18.41, 201.29), (26.4, 318.37), (24.66, 280.82), (28.76, 313.84), (29.08, 250.06), (32.04, 320.82), (36.34, 387.85), (41.99, 464.11), (35.9, 338.09), (25.47, 208.7), (22.78, 224.37), (23.42, 268.82), (25.47, 295.55), (20.77, 206.36), (31.91, 348.38), (30.81, 283.39), (25.32, 229.42), (25.69, 287.7), (31.58, 359.57), (26.14, 189.99), (26.9, 266.72), (17.77, 289.96), (28.95, 296.31), (31.36, 338.04), (28.33, 211.51), (27.22, 240.49), (37.65, 331.07), (22.89, 290.89), (27.28, 290.1), (29.61, 364.37), (39.76, 392.97), (20.5, 269.71), (20.94, 330.22), (29.84, 288.56), (24.66, 234.56), (27.32, 398.01), (26.3, 255.27), (27.99, 272.7), (31.28, 377.6), (22.95, 273.38), (28.04, 234.67), (27.82, 325.36), (26.64, 289.16), (19.71, 183.35), (24.37, 326.59), (26.25, 180.44), (28.91, 338.85), (32.21, 360.9), (28.46, 281.3), (32.74, 430.62), (30.86, 313.53), (18.12, 225.15), (30.65, 299.02), (29.46, 319.53), (35.82, 343.73), (19.04, 202.88), (29.0, 331.69), (20.73, 202.89), (37.18, 419.14), (27.36, 311.49), (25.88, 310.83), (39.59, 298.96), (32.1, 314.85), (26.28, 222.3), (28.82, 266.8), (34.24, 349.99), (25.36, 306.27), (26.61, 247.26), (37.74, 372.59), (23.2, 269.11), (24.65, 248.81), (30.99, 307.34), (20.69, 221.54), (27.87, 353.44), (25.04, 300.31), (29.85, 316.36), (21.69, 215.61), (18.82, 243.45), (30.14, 309.75), (31.62, 322.18), (34.61, 459.64), (22.17, 144.18), (23.88, 291.26), (30.1, 285.9), (33.09, 412.77), (38.34, 329.58), (39.77, 377.82), (31.84, 253.09), (21.06, 209.2), (20.76, 193.33), (20.78, 223.57), (33.03, 322.76), (23.6, 267.71), (31.29, 340.71), (29.07, 291.1), (31.96, 377.0), (27.83, 267.37), (32.68, 353.07), (24.68, 243.27), (33.39, 416.91), (28.89, 284.67), (29.35, 270.87), (34.96, 415.05), (31.72, 363.12), (31.83, 421.29), (30.5, 340.37), (30.78, 236.25), (35.44, 362.86), (23.82, 323.3), (17.83, 224.04), (27.27, 269.08), (27.79, 347.81), (35.8, 391.68), (32.77, 317.11), (33.48, 308.27), (26.94, 322.09), (34.0, 372.21), (39.53, 423.8), (24.45, 240.94), (37.64, 410.34), (26.82, 373.55), (27.49, 308.59), (32.54, 341.75), (36.76, 448.66), (38.69, 371.17), (36.49, 337.92), (24.6, 268.77), (25.59, 263.24), (26.44, 250.74), (31.2, 317.88), (40.63, 420.89), (17.58, 214.45), (23.05, 284.91), (23.8, 241.39), (29.29, 276.29), (26.68, 251.09), (41.88, 411.76), (29.08, 340.75), (20.62, 214.15), (12.06, 157.28), (27.69, 328.01), (28.86, 353.68), (32.2, 329.8), (23.27, 203.11), (34.13, 323.02), (34.6, 392.98), (24.26, 249.24), (31.04, 254.62), (31.28, 285.16), (19.44, 219.52), (41.16, 376.35), (18.14, 263.87), (22.94, 276.03), (30.18, 254.14), (24.49, 323.5), (30.53, 306.92), (39.94, 429.8), (20.43, 237.1), (24.04, 182.02), (26.97, 302.21), (29.92, 391.99), (34.62, 362.48), (29.71, 283.28), (35.93, 321.8), (20.29, 240.1), (27.41, 302.05), (27.2, 277.24), (29.25, 315.15), (31.23, 286.82), (33.32, 263.7), (29.1, 389.71), (19.27, 299.92), (36.71, 345.67), (7.25, 215.06), (33.71, 367.19), (25.29, 351.23), (29.35, 423.13), (26.37, 287.4), (27.61, 350.73), (26.02, 229.75), (19.42, 172.78), (22.19, 266.36), (35.77, 334.81), (32.42, 292.12), (35.8, 285.52), (21.4, 230.71), (36.96, 408.91), (24.11, 241.1), (17.34, 159.04), (19.62, 296.51), (27.38, 369.81), (21.88, 268.85), (29.4, 328.08), (22.96, 291.46), (34.41, 367.0), (33.5, 344.88), (32.37, 324.22), (28.31, 358.79), (36.63, 408.76), (25.88, 312.77), (30.67, 279.84), (27.81, 349.71), (20.64, 220.74), (29.49, 321.73), (25.16, 271.68), (22.46, 307.04), (24.19, 206.28), (31.29, 301.85), (26.33, 303.33), (35.19, 354.87), (28.35, 271.25), (36.34, 366.85), (28.26, 243.97), (31.68, 293.64), (39.55, 387.22), (23.89, 352.45), (26.09, 275.58), (24.76, 302.56), (28.27, 380.11), (32.97, 302.87), (36.67, 334.02), (21.52, 216.89), (28.48, 275.45), (25.96, 277.95), (17.89, 286.72), (20.21, 232.73), (30.92, 400.46), (27.22, 333.11), (20.2, 226.86), (34.97, 322.91), (35.1, 334.93), (25.5, 220.31), (35.36, 299.14), (16.87, 199.08), (39.96, 430.69), (25.63, 299.19), (23.84, 258.38), (33.28, 382.72), (35.43, 333.89), (29.54, 240.9), (24.56, 162.45), (34.86, 392.89), (30.03, 290.0), (23.04, 312.56), (26.04, 278.6), (30.0, 343.82), (10.32, 114.19), (26.65, 328.97), (29.19, 230.17), (20.14, 205.1), (34.05, 370.86), (23.79, 228.51), (32.15, 371.66), (17.44, 205.79), (28.11, 316.87), (33.91, 421.77), (27.0, 391.61), (28.7, 269.96), (24.68, 219.76), (15.95, 237.84), (28.15, 258.05), (14.99, 242.17), (37.85, 314.69), (35.98, 463.64), (28.6, 260.21), (35.75, 342.51), (27.97, 332.36), (26.67, 202.82), (27.62, 286.44), (28.75, 332.65), (32.34, 354.06), (20.37, 272.89), (26.68, 291.2), (29.85, 323.54), (32.02, 293.05), (35.93, 340.72), (29.66, 252.73), (30.02, 333.76), (19.08, 231.23), (31.26, 334.82), (28.95, 338.46), (24.84, 236.98), (24.04, 265.7), (34.24, 390.65), (19.85, 157.95), (21.54, 231.12), (37.44, 420.58), (17.33, 179.21)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=663；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.7129；
t = r·√((n−2)/(1−r²)) = 0.7129·√((663−2)/(1−0.5082)) = 26.1340；df=661；p = 6.03e-104。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [0.6732, 0.7485]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=663；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.7129；
t = r·√((n−2)/(1−r²)) = 0.7129·√((663−2)/(1−0.5082)) = 26.1340；df=661；p = 6.03e-104。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [0.6732, 0.7485]。
③ 计算结果：
r = 0.7129，t = 26.1340，p = 6.03e-104，95%CI = [0.6732, 0.7485]
④ 结论：在 α=0.05 水平下，p=6.03e-104 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.7129，方向正），夏季温度越高冰淇淋卖得越多。
案例 4. 社交媒体粉丝数(万)与帖子互动量(次)
Pearson相关粉丝多互动高
① 原始数据：
样本 n=488，逐条 (x, y)：[(23.06, 3407.01), (16.74, 4381.42), (7.63, 3759.06), (9.63, 2747.17), (20.06, 5700.22), (15.78, 4895.74), (26.89, 6939.22), (18.48, 4690.94), (20.77, 4662.6), (17.11, 3799.39), (22.15, 6397.28), (21.89, 7079.72), (20.19, 4750.77), (19.01, 4763.25), (10.1, 5056.87), (35.54, 7522.64), (19.53, 5981.69), (12.1, 3519.17), (27.84, 6394.0), (31.0, 7109.07), (20.78, 4803.33), (21.87, 5894.71), (18.38, 6472.37), (28.63, 5026.04), (34.84, 7188.54), (17.66, 5637.03), (23.98, 5283.78), (26.22, 6990.0), (23.5, 5846.23), (20.56, 4801.25), (27.0, 5810.47), (21.52, 5146.32), (24.34, 4311.64), (17.29, 3906.89), (19.6, 3929.29), (13.46, 5029.5), (28.42, 7806.46), (33.96, 9230.27), (26.41, 3610.07), (18.76, 6687.33), (31.89, 4527.05), (31.79, 7265.42), (37.73, 5287.64), (13.25, 4761.67), (23.8, 6552.71), (20.03, 5226.92), (31.15, 7158.51), (21.45, 6691.55), (31.49, 6258.11), (24.42, 4376.86), (26.87, 5652.53), (29.68, 6919.1), (39.88, 6819.86), (28.06, 5176.7), (18.8, 5733.15), (16.34, 4417.17), (3.58, 3151.63), (22.26, 4328.35), (9.73, 4238.84), (44.96, 9818.63), (16.93, 2548.59), (14.28, 2818.38), (5.8, 3471.33), (12.51, 2292.44), (17.91, 5197.15), (13.27, 6026.65), (15.24, 4246.37), (18.81, 5810.94), (24.53, 5377.8), (4.77, 4470.76), (18.92, 4544.05), (3.4, 1885.33), (4.05, 4346.56), (18.65, 2842.31), (9.23, 3625.86), (23.41, 5310.84), (16.36, 3621.64), (15.47, 3941.17), (19.64, 5864.44), (30.24, 7709.15), (18.63, 6008.69), (-2.03, 4706.65), (20.77, 3630.42), (9.67, 2909.63), (28.87, 3997.65), (16.64, 5722.18), (18.25, 4511.9), (27.11, 6187.14), (11.22, 4066.84), (19.15, 5626.0), (21.05, 5520.47), (17.78, 5058.26), (10.27, 4791.9), (31.23, 6606.15), (26.53, 5189.52), (6.67, 3177.49), (14.64, 4982.12), (16.06, 3516.23), (27.69, 6185.36), (36.82, 5481.59), (29.32, 6387.67), (35.2, 8374.66), (16.32, 2515.2), (21.53, 4947.05), (11.61, 4520.74), (14.88, 5094.29), (28.6, 5788.57), (37.12, 6168.98), (20.43, 5910.4), (13.19, 4006.39), (19.62, 5518.22), (15.88, 4570.46), (8.56, 2866.09), (22.6, 4880.1), (36.67, 6375.83), (33.78, 7600.1), (16.67, 3730.3), (7.15, 2332.66), (10.96, 4770.42), (20.83, 3798.34), (18.74, 4580.12), (26.82, 4647.63), (17.96, 3282.5), (42.27, 8100.92), (23.56, 5129.05), (7.22, 4189.82), (17.57, 4634.98), (25.58, 6151.56), (23.4, 4638.21), (13.0, 4691.55), (16.61, 3494.88), (27.08, 5356.92), (11.05, 3886.85), (25.02, 5365.75), (7.81, 2312.92), (20.83, 3529.98), (27.97, 5710.8), (9.46, 3418.54), (20.93, 3931.42), (16.74, 3956.02), (17.54, 2350.87), (21.71, 5765.26), (24.05, 3009.22), (-4.29, 2604.02), (8.34, 2313.86), (24.97, 6959.27), (18.19, 4554.78), (19.47, 4196.73), (23.47, 5282.35), (18.87, 6727.98), (23.08, 5522.75), (18.95, 3023.47), (22.72, 4881.51), (23.36, 4146.82), (20.15, 5464.42), (17.97, 4086.21), (25.76, 4903.97), (6.02, 3513.56), (35.97, 8026.43), (6.34, 4510.74), (18.19, 4092.98), (20.17, 6805.42), (15.37, 4629.0), (28.17, 7284.93), (34.63, 4877.89), (9.02, 3726.65), (9.64, 4467.04), (34.39, 6428.85), (23.65, 6405.02), (24.86, 5901.22), (17.08, 3931.91), (16.57, 3830.29), (19.65, 3597.89), (15.27, 3085.31), (22.18, 4141.88), (20.07, 1236.0), (23.68, 5414.42), (22.42, 4991.64), (7.2, 2711.05), (36.03, 6360.89), (21.49, 4805.29), (27.14, 3753.12), (17.83, 5441.69), (25.24, 4729.55), (33.72, 6747.76), (15.91, 2716.57), (21.62, 6268.5), (5.23, 3935.7), (21.81, 7176.73), (23.38, 5240.57), (5.23, 4911.94), (26.8, 6153.21), (20.59, 6078.85), (34.73, 7665.82), (23.25, 4554.94), (26.92, 6898.08), (24.65, 5708.47), (19.91, 5677.65), (32.14, 7685.12), (24.48, 8306.3), (14.33, 4220.2), (20.58, 5747.91), (9.9, 2994.2), (15.49, 3497.9), (26.03, 4279.72), (20.11, 3681.02), (35.9, 6836.53), (25.01, 7224.99), (21.76, 5817.6), (23.54, 5860.11), (36.4, 6701.22), (19.62, 4226.35), (19.48, 5682.84), (22.7, 7199.4), (21.17, 6604.34), (29.02, 6245.99), (22.22, 6281.31), (13.32, 3157.86), (21.44, 6691.02), (11.06, 4269.36), (32.56, 5193.22), (14.63, 4682.36), (13.97, 3914.13), (19.21, 5078.48), (19.72, 5823.72), (17.49, 3762.38), (35.0, 6301.86), (15.48, 4520.74), (29.33, 5299.06), (22.93, 4502.21), (21.42, 5513.81), (19.74, 4224.54), (15.83, 4362.77), (19.82, 3916.95), (13.6, 2295.82), (16.17, 4748.55), (29.49, 6922.86), (8.15, 4607.42), (23.86, 5762.58), (25.92, 5204.81), (30.82, 6630.0), (10.24, 3928.25), (4.44, 2855.37), (-2.92, 749.4), (16.64, 2404.45), (18.44, 4404.39), (10.47, 3794.04), (36.14, 5534.4), (6.47, 4393.17), (25.81, 5726.23), (-1.88, 2362.08), (16.35, 4340.73), (33.68, 6010.07), (25.95, 4832.14), (22.54, 6060.84), (19.15, 4365.81), (16.49, 4462.49), (18.1, 6632.52), (34.02, 6455.46), (36.87, 8686.46), (31.23, 6036.06), (22.76, 4609.98), (10.33, 1892.37), (8.56, 3364.04), (16.15, 4465.42), (22.4, 3838.34), (28.27, 3189.93), (22.91, 4986.47), (15.64, 4306.41), (24.25, 4394.35), (28.43, 5746.94), (14.38, 4138.16), (21.71, 4194.42), (23.42, 7478.64), (25.68, 5618.86), (22.53, 4972.27), (8.99, 3749.44), (29.07, 3542.44), (5.43, 3144.91), (6.21, 4049.76), (11.78, 3907.74), (17.37, 4218.0), (27.41, 5643.38), (22.22, 4652.61), (7.66, 2053.14), (34.13, 5944.12), (13.56, 3326.33), (3.49, 3962.56), (9.17, 6705.19), (21.96, 4381.18), (10.32, 3535.57), (24.37, 7584.69), (21.32, 4937.17), (26.79, 6208.96), (27.79, 5379.68), (26.77, 5179.06), (17.47, 6310.31), (28.83, 6581.97), (23.54, 4539.72), (13.54, 3555.72), (26.48, 3243.48), (43.22, 7995.09), (19.86, 6055.05), (21.55, 5688.03), (24.5, 4866.37), (30.48, 6824.37), (23.32, 5780.07), (16.66, 4670.89), (27.44, 5810.92), (17.39, 3811.83), (30.3, 8462.91), (33.18, 5136.85), (16.79, 5338.53), (9.09, 2967.9), (21.46, 5442.79), (22.62, 4306.0), (1.19, 707.2), (17.21, 2149.12), (14.99, 4418.61), (26.14, 5515.93), (19.44, 5315.42), (2.97, 3400.1), (8.8, 5969.28), (25.26, 5308.01), (25.5, 6575.36), (29.15, 7803.38), (9.63, 3926.24), (0.75, 1839.21), (8.87, 3980.63), (11.18, 3184.12), (5.43, 3770.58), (26.14, 3399.13), (21.06, 4496.85), (13.06, 5087.76), (23.34, 4466.44), (12.2, 5540.81), (12.21, 5275.86), (16.15, 4936.7), (11.13, 5316.51), (16.15, 4259.24), (30.8, 5934.97), (16.82, 4445.38), (18.95, 3453.41), (12.04, 3240.27), (36.09, 5940.01), (23.67, 5685.2), (24.92, 6492.72), (10.55, 3738.79), (24.63, 5842.1), (20.44, 3913.64), (25.94, 5348.26), (31.53, 6249.41), (14.2, 5486.87), (27.15, 6319.53), (23.37, 3239.73), (19.93, 5034.73), (24.52, 5777.11), (33.7, 7661.32), (17.52, 5781.98), (8.59, 3335.67), (12.7, 5460.84), (13.84, 3542.67), (20.2, 4793.48), (30.45, 6373.13), (18.95, 3236.85), (25.74, 7137.51), (14.03, 3241.16), (25.7, 5255.78), (23.95, 5231.91), (31.24, 5158.01), (25.69, 4425.24), (8.99, 4746.9), (27.7, 4903.08), (10.22, 5288.84), (18.34, 4652.25), (10.37, 3980.78), (-1.74, 1492.23), (6.49, 3574.63), (27.65, 6908.31), (24.86, 4773.42), (28.39, 5856.58), (14.79, 3882.99), (23.8, 5433.56), (24.48, 6046.57), (21.69, 5893.48), (13.74, 6949.77), (20.63, 7937.02), (10.06, 5558.3), (22.68, 4312.56), (14.8, 2976.84), (10.71, 1650.63), (26.46, 4045.22), (26.08, 8407.3), (17.96, 4707.14), (34.86, 5270.25), (15.21, 4913.9), (26.18, 6365.09), (21.16, 4379.14), (26.45, 5730.04), (18.58, 4794.94), (12.9, 6436.47), (19.02, 7296.1), (30.58, 6825.6), (25.42, 7995.26), (29.96, 5990.64), (14.12, 5309.06), (32.65, 6834.01), (20.16, 6090.62), (16.81, 3939.24), (17.28, 3870.06), (28.05, 6377.98), (25.24, 6592.53), (21.36, 4333.07), (27.61, 6767.13), (7.18, 3421.26), (5.58, 3339.04), (22.19, 3659.13), (2.13, 4537.36), (10.52, 3053.17), (22.18, 6602.63), (20.5, 5789.57), (14.78, 3535.27), (25.78, 4117.34), (29.37, 5133.37), (18.55, 4125.41), (19.29, 3456.88), (-0.66, -1039.49), (8.2, 4613.1), (32.57, 6371.13), (18.28, 6000.3), (26.51, 7335.55), (18.91, 3444.88), (21.76, 5027.89), (38.33, 10256.74), (21.5, 4824.42), (23.73, 4579.4), (22.55, 6328.38), (12.71, 4744.97), (8.22, 3928.56), (1.48, 2188.62), (27.14, 3794.51), (15.24, 5049.3), (11.84, 4552.94), (26.7, 6989.25), (25.67, 6654.03), (16.3, 4327.17), (20.83, 4805.67), (24.41, 4921.67), (14.2, 6073.53), (25.27, 8583.8), (23.44, 5613.86), (11.31, 2270.56), (21.51, 6393.76), (10.56, 3507.48), (17.1, 3641.92), (18.21, 4185.63), (14.32, 4707.49), (33.61, 5789.83), (17.25, 3708.61), (5.84, 4628.2), (7.73, 4054.76), (36.25, 6333.46), (6.48, 3426.05), (13.01, 4308.38), (11.43, 5109.23), (32.28, 6208.13), (22.0, 5833.93), (17.49, 1980.97), (9.94, 3582.58), (34.5, 7026.23), (27.41, 8352.77), (25.44, 3350.44), (23.14, 4205.78), (32.02, 7229.85), (24.96, 4309.83), (15.66, 5814.35), (14.78, 2944.92), (16.88, 4980.17), (18.42, 4368.91), (-0.03, 888.63), (14.29, 3900.81), (10.38, 2500.05), (20.19, 4405.48), (16.96, 4249.31), (16.06, 5214.28), (34.73, 6054.06), (26.38, 6951.62), (29.26, 3620.82)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=488；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6726；
t = r·√((n−2)/(1−r²)) = 0.6726·√((488−2)/(1−0.4524)) = 20.0380；df=486；p = -1.28e+21。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.6208, 0.7186]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=488；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6726；
t = r·√((n−2)/(1−r²)) = 0.6726·√((488−2)/(1−0.4524)) = 20.0380；df=486；p = -1.28e+21。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.6208, 0.7186]。
③ 计算结果：
r = 0.6726，t = 20.0380，p = -1.28e+21，95%CI = [0.6208, 0.7186]
④ 结论：在 α=0.05 水平下，p=-1.28e+21 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.6726，方向正），粉丝多互动高。
案例 5. 树木年龄(年)与树干直径(cm)
Pearson相关树龄越大越粗
① 原始数据：
样本 n=789，逐条 (x, y)：[(16.37, 27.89), (28.36, 46.65), (18.34, 34.58), (22.37, 40.76), (10.68, 23.23), (22.51, 29.01), (20.35, 41.17), (18.4, 27.48), (17.07, 24.71), (9.16, 20.59), (13.41, 27.25), (24.27, 29.5), (5.41, 11.57), (13.34, 30.95), (16.68, 32.13), (8.89, 26.07), (22.81, 47.03), (14.67, 38.95), (20.11, 36.47), (7.9, 12.81), (16.37, 40.95), (12.57, 25.69), (10.92, 20.76), (11.65, 27.75), (16.04, 38.34), (14.51, 32.5), (10.97, 18.73), (22.28, 36.98), (17.32, 39.98), (20.82, 33.51), (15.55, 24.42), (10.51, 27.12), (15.09, 26.06), (13.89, 39.55), (17.09, 27.84), (19.32, 43.05), (24.83, 45.26), (13.79, 29.95), (20.15, 24.83), (19.45, 39.28), (18.13, 42.25), (13.87, 29.68), (10.21, 24.98), (14.3, 35.1), (5.17, 11.92), (16.79, 25.34), (15.79, 32.78), (3.61, 22.93), (15.15, 25.16), (24.07, 31.24), (11.37, 16.23), (13.22, 36.12), (19.99, 37.81), (15.33, 35.59), (14.27, 29.31), (21.17, 19.15), (16.43, 30.73), (21.09, 54.32), (15.74, 37.0), (11.84, 26.43), (10.74, 30.17), (13.22, 31.69), (14.7, 28.64), (17.48, 37.85), (19.23, 34.83), (13.1, 31.99), (22.36, 59.41), (13.59, 30.14), (26.25, 56.6), (6.93, 21.09), (20.01, 31.79), (14.66, 18.91), (20.0, 42.04), (24.14, 47.98), (14.41, 27.79), (11.78, 20.09), (25.82, 47.19), (19.17, 38.64), (18.1, 44.72), (16.07, 34.5), (20.23, 29.91), (5.19, 12.91), (21.96, 34.03), (14.54, 34.79), (12.48, 31.17), (9.61, 21.17), (20.11, 39.96), (12.74, 29.46), (14.23, 23.24), (20.27, 36.35), (12.56, 35.33), (19.89, 50.61), (12.31, 25.87), (16.41, 34.24), (6.58, 33.32), (12.56, 27.62), (26.49, 40.39), (18.67, 29.61), (22.59, 28.03), (13.53, 27.8), (18.43, 26.97), (14.02, 29.57), (13.65, 33.22), (14.83, 32.04), (19.35, 37.96), (12.54, 20.01), (13.14, 22.24), (14.11, 31.04), (16.65, 32.57), (14.45, 33.12), (16.59, 45.86), (10.89, 25.93), (15.61, 22.26), (21.31, 35.08), (15.85, 39.07), (16.38, 26.38), (9.47, 30.32), (15.49, 20.35), (7.73, 23.15), (9.17, 19.2), (15.12, 35.37), (13.85, 21.52), (18.43, 31.15), (14.75, 33.13), (10.47, 29.31), (12.2, 23.77), (16.96, 29.03), (21.45, 44.15), (20.72, 36.4), (12.18, 24.25), (19.77, 40.34), (12.78, 25.3), (21.45, 46.8), (17.41, 38.76), (10.21, 26.24), (18.76, 36.06), (18.59, 26.38), (27.91, 58.04), (7.63, 14.93), (8.56, 20.93), (11.64, 21.43), (17.47, 31.2), (14.73, 29.16), (14.19, 25.51), (14.1, 28.66), (13.68, 41.47), (18.29, 38.85), (2.98, 16.96), (7.9, 11.58), (18.1, 35.33), (20.99, 36.07), (13.88, 29.96), (18.26, 38.76), (20.37, 37.52), (13.98, 24.96), (18.44, 32.9), (8.9, 28.11), (9.21, 19.96), (18.84, 31.32), (11.62, 24.1), (17.22, 37.1), (25.47, 43.21), (20.41, 34.58), (19.03, 41.5), (12.34, 33.57), (9.93, 14.04), (15.29, 35.52), (20.77, 42.79), (8.52, 27.78), (15.91, 40.35), (16.82, 25.65), (15.36, 32.37), (28.07, 51.71), (12.22, 17.96), (14.9, 35.48), (20.11, 32.74), (21.78, 43.21), (14.21, 28.67), (12.99, 28.14), (17.39, 44.94), (19.79, 37.63), (22.34, 53.61), (11.78, 24.64), (6.16, 14.26), (15.86, 28.02), (22.53, 34.28), (21.13, 21.2), (12.86, 11.22), (13.75, 26.5), (16.01, 33.59), (17.33, 28.19), (15.2, 37.56), (17.0, 35.13), (22.49, 45.76), (11.34, 14.65), (15.39, 33.29), (15.77, 36.0), (12.69, 24.09), (17.91, 34.68), (13.17, 26.76), (10.04, 26.64), (13.3, 29.79), (12.74, 34.99), (20.35, 46.19), (13.59, 33.07), (15.52, 29.18), (14.77, 35.35), (19.35, 27.89), (3.45, 28.65), (15.64, 32.72), (22.15, 32.11), (10.09, 27.87), (14.27, 28.16), (16.63, 31.86), (9.16, 40.02), (14.18, 31.66), (17.69, 36.05), (13.29, 30.53), (13.03, 27.8), (18.04, 30.24), (15.86, 36.48), (13.38, 22.69), (20.65, 45.97), (29.03, 41.17), (18.48, 37.08), (7.87, 10.74), (16.83, 39.36), (20.02, 36.41), (19.1, 44.17), (7.98, 23.64), (8.66, 20.62), (11.74, 34.84), (21.29, 29.59), (14.66, 15.48), (6.41, 19.39), (16.28, 30.84), (17.87, 20.61), (22.99, 51.52), (16.35, 30.2), (12.33, 20.01), (18.34, 28.96), (14.99, 30.16), (19.6, 28.89), (24.2, 46.21), (22.31, 31.63), (10.92, 33.21), (15.26, 18.45), (18.4, 35.77), (12.26, 31.57), (16.7, 28.66), (16.99, 32.82), (16.88, 27.07), (16.6, 37.96), (15.3, 47.52), (20.98, 52.95), (16.02, 30.17), (10.58, 23.96), (25.18, 47.97), (17.17, 29.57), (12.98, 25.41), (20.53, 30.13), (11.97, 33.94), (14.31, 26.34), (22.81, 48.27), (16.41, 38.37), (15.29, 24.62), (10.08, 24.94), (13.22, 16.35), (17.1, 33.39), (21.68, 39.55), (13.36, 28.93), (15.69, 22.17), (17.99, 33.55), (19.76, 27.05), (19.17, 39.02), (11.97, 14.33), (6.75, 17.37), (24.81, 41.94), (11.29, 30.15), (18.73, 33.03), (14.05, 31.29), (19.45, 44.52), (19.27, 41.55), (13.98, 25.92), (11.37, 38.36), (10.88, 26.68), (18.71, 45.42), (15.72, 46.33), (11.23, 25.55), (5.44, 3.89), (8.59, 26.38), (16.2, 17.92), (13.81, 20.26), (14.05, 36.79), (15.15, 34.62), (9.54, 24.62), (15.17, 21.53), (15.09, 35.67), (12.0, 24.19), (6.38, 20.49), (16.75, 36.39), (22.85, 29.99), (24.09, 47.05), (24.41, 30.66), (17.76, 28.03), (16.29, 31.83), (16.11, 36.55), (16.42, 35.47), (24.02, 42.69), (12.69, 23.15), (16.64, 31.58), (16.02, 37.08), (17.3, 32.43), (18.86, 31.97), (18.16, 29.42), (16.88, 29.83), (13.79, 19.99), (13.24, 37.3), (20.76, 23.59), (15.46, 24.02), (8.67, 19.51), (13.17, 22.25), (17.5, 37.14), (9.0, 26.3), (15.71, 36.15), (12.21, 19.68), (10.86, 28.78), (5.89, 18.22), (12.18, 23.04), (11.76, 35.3), (13.17, 20.13), (10.07, 19.43), (16.5, 35.81), (17.06, 27.0), (4.65, 23.79), (6.51, 3.94), (14.66, 17.55), (10.86, 31.04), (11.22, 32.88), (10.65, 14.86), (20.67, 43.07), (5.94, 23.8), (2.32, -0.55), (18.31, 33.81), (13.99, 43.58), (6.85, 25.56), (12.89, 31.7), (11.12, 33.31), (24.38, 41.46), (18.02, 30.72), (16.78, 33.73), (13.35, 26.69), (18.43, 29.75), (19.34, 36.76), (18.98, 34.78), (15.11, 27.06), (20.98, 36.18), (20.76, 38.89), (12.16, 16.16), (21.11, 37.24), (13.32, 33.54), (15.39, 26.65), (20.83, 35.99), (9.76, 21.97), (12.9, 31.5), (16.15, 28.11), (16.14, 28.79), (16.01, 38.28), (13.1, 41.05), (9.54, 20.23), (15.52, 22.97), (16.19, 38.73), (13.55, 33.49), (13.18, 20.86), (16.8, 25.13), (16.54, 31.6), (15.12, 39.03), (14.12, 26.4), (13.67, 35.29), (14.36, 31.54), (13.6, 29.18), (20.55, 30.02), (10.67, 28.28), (26.02, 49.46), (19.1, 33.65), (18.46, 28.32), (9.81, 29.46), (1.51, 16.2), (11.11, 33.14), (14.07, 23.18), (11.07, 21.66), (15.42, 19.79), (18.39, 27.3), (12.89, 32.54), (4.37, 25.56), (11.97, 20.5), (16.79, 28.93), (16.68, 34.65), (7.71, 24.42), (14.36, 38.85), (9.45, 23.45), (14.38, 29.23), (25.89, 40.32), (20.53, 29.83), (7.63, 8.9), (21.9, 40.45), (12.38, 34.43), (19.48, 37.09), (4.43, 13.97), (18.68, 34.16), (10.12, 26.71), (16.67, 31.62), (10.07, 23.87), (20.05, 33.61), (19.86, 52.35), (23.08, 46.92), (12.15, 27.02), (12.71, 32.8), (12.14, 14.76), (9.44, 19.37), (23.95, 36.86), (23.02, 34.79), (18.14, 17.46), (9.65, 23.5), (20.17, 33.82), (16.04, 22.48), (10.45, 32.86), (19.65, 37.98), (13.51, 23.42), (9.36, 17.92), (18.3, 31.44), (28.15, 39.51), (1.9, 6.72), (19.85, 36.98), (25.58, 46.41), (14.48, 29.95), (10.01, 25.48), (15.42, 32.22), (17.81, 36.83), (13.97, 30.22), (8.63, 14.54), (14.28, 24.19), (18.41, 35.44), (12.78, 21.76), (25.33, 47.19), (12.7, 29.03), (25.31, 42.4), (14.35, 31.75), (24.37, 46.17), (12.82, 28.73), (19.64, 36.87), (15.62, 39.53), (23.12, 41.67), (18.97, 40.32), (12.61, 33.02), (8.62, 25.82), (16.09, 22.8), (15.51, 35.46), (11.08, 24.63), (8.41, 21.86), (27.2, 43.42), (11.94, 24.37), (15.16, 37.64), (7.83, 18.22), (23.72, 29.57), (17.28, 37.91), (22.12, 41.81), (22.32, 39.52), (7.74, 29.47), (21.11, 31.03), (8.1, 19.21), (24.08, 39.54), (13.25, 25.9), (23.49, 45.56), (8.75, 32.14), (14.62, 31.29), (7.18, 25.63), (19.71, 38.28), (18.89, 32.34), (12.44, 31.18), (19.03, 39.71), (8.92, 26.34), (13.25, 22.16), (7.56, 20.7), (16.11, 26.55), (18.89, 34.12), (17.56, 44.89), (12.26, 26.59), (20.09, 39.18), (11.24, 24.7), (9.62, 22.18), (21.64, 45.84), (4.07, 20.42), (15.86, 32.71), (17.54, 35.88), (16.58, 23.47), (9.13, 33.65), (13.89, 28.49), (13.48, 31.76), (9.63, 26.11), (17.96, 40.88), (7.94, 10.38), (10.03, 28.76), (18.49, 36.89), (9.22, 29.98), (5.13, 11.7), (12.31, 27.26), (8.38, 28.0), (24.82, 50.91), (13.08, 32.1), (15.2, 23.56), (20.94, 42.72), (17.72, 32.53), (16.7, 21.9), (13.84, 34.23), (5.73, 26.96), (15.78, 31.86), (7.38, 15.38), (8.94, 22.4), (20.79, 42.63), (11.13, 16.39), (8.66, 32.64), (18.78, 25.86), (17.34, 24.78), (16.07, 34.38), (18.26, 35.11), (18.54, 31.99), (19.31, 28.39), (4.76, 15.66), (16.63, 26.86), (19.9, 28.89), (15.23, 32.3), (15.05, 31.02), (10.83, 32.09), (9.91, 30.33), (14.52, 31.31), (14.55, 27.78), (21.83, 40.23), (18.65, 28.8), (17.56, 24.59), (13.22, 24.21), (15.61, 30.78), (18.43, 29.63), (15.59, 28.2), (15.62, 27.34), (10.19, 22.46), (18.66, 38.33), (16.45, 40.88), (8.7, 11.2), (16.54, 43.64), (16.69, 35.69), (16.3, 30.15), (6.46, 23.56), (10.85, 14.01), (15.95, 34.14), (8.04, 24.23), (19.35, 32.67), (18.48, 33.27), (13.31, 30.05), (15.23, 28.95), (9.58, 3.98), (16.31, 22.58), (9.87, 10.44), (16.61, 24.89), (12.01, 24.75), (15.33, 16.22), (17.38, 22.76), (10.79, 22.14), (13.87, 41.16), (11.81, 26.74), (15.63, 25.29), (17.33, 36.84), (14.37, 31.05), (4.2, 16.18), (18.84, 39.04), (18.63, 42.12), (14.13, 23.12), (9.68, 9.09), (15.55, 26.83), (21.85, 36.53), (20.6, 29.2), (24.32, 29.38), (17.97, 33.23), (23.18, 44.42), (11.9, 28.54), (5.4, 4.47), (15.51, 41.17), (22.76, 39.91), (9.46, 10.25), (11.58, 23.27), (15.29, 20.17), (9.3, 19.79), (20.97, 35.42), (13.32, 21.75), (18.0, 32.27), (16.61, 21.35), (16.85, 27.26), (20.43, 30.28), (13.11, 31.41), (10.0, 27.29), (16.14, 34.91), (8.5, 19.94), (21.32, 43.03), (11.78, 30.92), (17.19, 38.8), (12.91, 26.76), (14.59, 25.94), (24.23, 41.42), (13.78, 29.62), (8.85, 33.42), (14.83, 28.0), (11.94, 18.92), (19.79, 40.27), (20.01, 34.31), (7.07, 5.49), (15.99, 19.31), (18.72, 37.53), (20.03, 27.13), (18.85, 31.8), (16.69, 31.08), (14.88, 24.19), (12.45, 30.91), (8.34, 28.03), (16.41, 33.86), (16.31, 27.38), (16.32, 36.18), (10.16, 21.18), (10.79, 30.14), (20.11, 35.88), (10.11, 28.52), (8.74, 18.39), (16.66, 31.83), (15.9, 35.27), (20.69, 34.71), (15.18, 28.96), (7.73, 15.73), (18.32, 22.58), (22.56, 38.24), (20.09, 42.75), (21.17, 41.39), (10.37, 32.65), (6.7, 16.55), (11.39, 31.86), (13.96, 29.58), (6.27, 27.68), (20.96, 32.46), (23.09, 44.71), (11.74, 39.66), (16.0, 42.06), (13.58, 21.62), (19.47, 35.61), (11.64, 33.34), (12.1, 34.84), (14.56, 41.61), (16.09, 31.57), (21.66, 43.19), (22.18, 24.27), (20.45, 42.46), (18.9, 38.97), (6.85, 8.98), (12.15, 26.86), (10.58, 27.83), (16.3, 36.64), (13.03, 36.03), (9.87, 14.28), (16.59, 36.4), (18.25, 32.07), (14.6, 31.42), (14.61, 35.19), (16.51, 31.42), (12.66, 28.82), (7.09, 34.93), (8.8, 20.34), (14.71, 39.12), (10.41, 31.92), (24.59, 50.69), (4.03, 27.57), (28.84, 52.0), (11.98, 26.04), (5.82, 25.79), (6.28, 18.68), (12.59, 22.73), (20.61, 47.35), (13.88, 22.39), (16.6, 46.66), (4.71, 4.92), (8.84, 15.53), (15.74, 43.08), (17.49, 25.16), (13.3, 19.42), (9.92, 22.84), (13.15, 26.03), (9.85, 19.77), (22.39, 40.77), (8.58, 27.23), (6.14, 21.79), (16.78, 30.57), (20.17, 24.38), (22.76, 40.46), (16.94, 38.67), (13.8, 24.93), (9.46, 17.95), (9.8, 35.98), (8.39, 22.54), (15.43, 34.86), (16.08, 43.15), (22.18, 13.73), (10.54, 20.47), (24.56, 48.92), (18.23, 24.31), (7.76, 24.19), (19.21, 36.82), (15.91, 26.61), (3.0, -2.74), (16.56, 32.36), (14.58, 32.13), (27.1, 37.21), (18.47, 35.34), (12.89, 33.11), (9.48, 23.1), (22.08, 33.97), (14.69, 40.42), (8.72, 23.03), (12.06, 31.51), (16.77, 35.29), (10.58, 13.48), (17.65, 35.33), (6.25, 15.82), (17.64, 33.69), (-0.11, -0.77), (17.6, 42.81), (22.1, 43.35), (13.5, 21.88), (19.99, 37.46), (19.04, 30.25), (16.42, 17.55), (15.87, 22.53), (20.48, 44.82), (13.22, 32.98), (16.17, 31.12), (13.2, 28.46), (17.59, 42.02), (7.41, 14.02), (19.2, 39.86), (14.38, 34.59), (8.38, 22.8), (0.18, 20.48), (9.66, 27.92), (11.55, 21.22), (16.39, 19.88), (16.17, 31.55), (13.28, 30.33), (16.29, 26.26), (14.25, 20.68), (16.15, 24.71), (13.67, 32.94), (14.37, 17.17), (11.94, 14.75), (2.69, 11.61), (17.96, 28.89), (16.47, 32.57), (9.45, 19.33), (15.37, 30.32), (15.73, 34.45), (19.7, 38.34), (11.36, 17.93), (6.47, 21.18), (10.65, 14.68), (18.77, 26.07), (9.16, 9.91), (20.4, 34.01), (21.95, 35.64), (25.36, 38.24), (10.72, 35.51), (22.55, 43.83), (16.24, 37.72), (24.63, 36.22), (13.28, 10.78), (17.59, 23.81), (19.05, 33.82), (10.3, 35.56), (7.77, 22.29), (6.39, 12.27), (13.52, 31.08), (19.04, 32.09), (22.94, 43.19), (17.63, 23.4), (19.82, 36.52), (14.35, 33.5)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=789；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6997；
t = r·√((n−2)/(1−r²)) = 0.6997·√((789−2)/(1−0.4896)) = 27.4777；df=787；p = -4.94e+49。
95% CI（Fisher z）：tanh(z±19.847·1/√(n−3)) = [0.1575, 0.9178]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=789；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.6997；
t = r·√((n−2)/(1−r²)) = 0.6997·√((789−2)/(1−0.4896)) = 27.4777；df=787；p = -4.94e+49。
95% CI（Fisher z）：tanh(z±19.847·1/√(n−3)) = [0.1575, 0.9178]。
③ 计算结果：
r = 0.6997，t = 27.4777，p = -4.94e+49，95%CI = [0.1575, 0.9178]
④ 结论：在 α=0.05 水平下，p=-4.94e+49 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.6997，方向正），树龄越大越粗。
案例 6. 身高(cm)与体重(kg)
Pearson相关身高越高体重越大
① 原始数据：
样本 n=348，逐条 (x, y)：[(168.88, 62.63), (167.16, 71.52), (157.89, 50.15), (174.66, 67.04), (163.77, 49.89), (178.47, 72.78), (167.38, 66.64), (187.82, 74.05), (160.61, 57.1), (175.16, 88.06), (183.84, 78.53), (173.5, 68.14), (179.29, 60.5), (151.14, 58.55), (152.91, 49.6), (172.1, 60.69), (158.69, 50.42), (174.6, 78.87), (159.72, 72.61), (166.94, 59.3), (161.53, 82.01), (174.68, 79.2), (171.21, 63.49), (169.16, 67.27), (166.1, 63.42), (165.16, 72.32), (168.13, 86.49), (172.81, 86.56), (180.72, 95.6), (167.89, 51.86), (164.88, 58.82), (163.14, 58.33), (157.11, 66.44), (161.8, 65.84), (161.75, 59.22), (152.77, 64.31), (165.32, 54.1), (163.68, 73.22), (159.66, 71.81), (172.56, 64.08), (163.93, 53.64), (170.78, 76.6), (175.75, 88.85), (164.07, 53.33), (186.21, 75.07), (174.26, 71.01), (174.37, 78.61), (177.36, 59.42), (172.48, 54.58), (167.87, 64.19), (157.45, 46.66), (168.55, 57.43), (159.69, 69.84), (165.6, 63.92), (170.41, 63.49), (176.28, 77.73), (169.78, 88.4), (175.39, 64.71), (162.21, 79.87), (171.76, 73.77), (175.71, 79.48), (172.45, 65.46), (168.37, 60.57), (166.59, 71.88), (173.78, 70.9), (186.11, 75.93), (167.84, 63.32), (160.55, 52.12), (160.2, 72.99), (167.38, 63.98), (174.48, 76.3), (174.07, 98.46), (165.76, 64.23), (164.23, 54.8), (163.2, 71.09), (165.43, 69.27), (158.03, 33.96), (157.71, 59.65), (164.97, 52.63), (173.18, 66.15), (159.88, 51.51), (175.34, 61.23), (168.63, 42.91), (176.1, 70.36), (178.63, 78.95), (176.45, 73.96), (172.03, 89.87), (168.58, 66.52), (175.82, 72.17), (160.02, 61.8), (172.57, 59.69), (183.07, 75.39), (159.8, 49.29), (164.21, 41.31), (164.16, 54.69), (172.73, 69.85), (184.55, 79.47), (180.29, 78.78), (174.22, 61.24), (168.09, 57.34), (148.29, 64.56), (168.99, 61.6), (178.32, 55.22), (171.65, 75.33), (169.37, 63.52), (173.17, 68.29), (163.46, 60.69), (177.21, 73.06), (172.2, 70.33), (185.05, 78.94), (158.95, 33.81), (155.98, 43.06), (165.03, 49.95), (159.21, 75.82), (161.12, 60.7), (159.88, 49.33), (161.77, 71.28), (159.2, 68.81), (163.03, 64.29), (189.65, 78.41), (181.2, 78.05), (170.15, 51.3), (164.89, 51.16), (164.34, 55.67), (176.61, 76.3), (150.81, 47.96), (163.03, 63.55), (164.35, 40.63), (164.35, 55.7), (175.63, 74.52), (176.81, 76.95), (173.65, 57.73), (165.67, 46.21), (170.2, 80.28), (159.81, 56.83), (179.39, 57.56), (178.8, 63.75), (174.97, 84.03), (179.62, 58.04), (161.51, 58.71), (166.34, 62.53), (168.19, 71.27), (165.82, 51.53), (181.75, 69.32), (178.4, 77.24), (169.66, 50.39), (162.12, 72.3), (165.39, 65.15), (160.11, 46.15), (176.15, 71.41), (156.09, 79.38), (166.77, 67.69), (171.07, 85.71), (178.42, 64.15), (159.68, 49.06), (168.55, 61.58), (168.48, 63.01), (171.92, 56.4), (165.5, 65.45), (151.48, 45.46), (159.04, 60.92), (161.64, 59.6), (159.5, 70.16), (165.72, 71.35), (174.27, 72.79), (163.17, 56.86), (167.83, 66.03), (179.7, 79.84), (170.53, 60.31), (168.91, 66.96), (162.4, 58.87), (170.84, 67.0), (169.75, 42.6), (167.16, 72.81), (163.23, 83.64), (166.92, 56.58), (167.54, 56.83), (171.15, 79.38), (163.14, 67.58), (165.58, 62.28), (159.4, 56.71), (168.94, 77.74), (161.37, 76.66), (170.56, 54.38), (174.73, 63.31), (177.02, 73.97), (163.79, 78.54), (154.87, 55.01), (156.32, 41.62), (165.01, 61.99), (169.3, 69.03), (169.21, 48.27), (172.57, 82.32), (158.91, 52.01), (166.83, 50.4), (156.2, 43.18), (180.15, 51.88), (170.23, 64.0), (163.86, 77.45), (173.62, 69.84), (166.57, 67.23), (148.14, 55.51), (163.46, 72.42), (174.12, 81.96), (158.1, 61.96), (173.9, 53.11), (164.4, 56.86), (176.13, 73.62), (172.61, 54.08), (168.42, 44.06), (176.83, 90.23), (163.4, 45.79), (168.91, 81.07), (173.85, 79.36), (190.42, 64.79), (170.0, 64.14), (172.18, 74.76), (180.87, 84.89), (169.49, 49.58), (157.44, 43.68), (177.66, 52.61), (172.2, 56.57), (148.53, 36.91), (153.45, 34.6), (171.37, 66.97), (177.09, 65.35), (172.05, 85.34), (162.91, 70.18), (176.58, 61.79), (158.48, 49.03), (151.03, 46.13), (175.38, 74.98), (161.82, 93.81), (167.23, 72.1), (165.42, 78.09), (153.0, 49.13), (176.14, 72.68), (174.57, 73.37), (155.36, 55.57), (172.99, 50.51), (160.43, 66.73), (153.26, 56.64), (161.07, 61.12), (172.96, 69.64), (174.87, 60.19), (172.98, 55.49), (175.84, 67.12), (180.25, 76.07), (171.49, 73.67), (172.07, 85.91), (170.1, 60.91), (156.02, 61.6), (160.39, 62.68), (160.64, 53.49), (168.93, 63.35), (188.62, 73.44), (167.74, 57.05), (181.45, 55.41), (171.42, 71.89), (185.87, 82.32), (157.75, 29.91), (159.33, 69.32), (161.39, 44.85), (173.92, 66.48), (163.42, 59.86), (168.64, 58.56), (164.45, 56.87), (160.4, 65.29), (175.33, 78.31), (160.91, 67.98), (169.67, 60.15), (156.6, 56.8), (173.22, 83.74), (170.46, 71.36), (177.97, 89.58), (162.4, 46.15), (172.94, 54.61), (178.17, 62.14), (170.24, 72.83), (172.68, 58.14), (173.5, 51.77), (173.54, 74.89), (166.16, 72.95), (158.95, 70.59), (169.51, 58.42), (176.21, 56.74), (174.97, 91.12), (173.86, 58.29), (167.36, 66.09), (164.55, 83.96), (160.89, 51.67), (156.55, 49.0), (181.26, 54.07), (162.69, 46.79), (175.0, 57.54), (174.15, 80.49), (172.84, 53.33), (174.63, 78.22), (168.93, 57.18), (171.06, 48.19), (160.05, 50.54), (173.99, 67.89), (167.24, 62.98), (159.37, 47.26), (164.33, 55.69), (156.11, 58.18), (159.2, 71.01), (178.09, 66.7), (184.02, 98.19), (172.65, 79.57), (168.07, 47.78), (179.7, 66.63), (164.37, 62.14), (172.77, 64.77), (161.24, 60.2), (160.75, 72.48), (166.52, 56.55), (167.66, 40.11), (165.25, 64.33), (161.71, 53.84), (164.25, 57.1), (168.71, 61.96), (169.69, 89.28), (171.22, 57.82), (177.98, 74.4), (160.13, 51.35), (176.16, 71.14), (166.39, 54.55), (165.28, 68.59), (163.7, 60.36), (173.04, 64.03), (171.46, 82.15), (159.13, 53.6), (154.13, 49.83), (161.2, 50.56), (177.29, 83.09), (174.66, 68.13), (172.96, 61.43), (159.54, 78.6), (172.53, 69.03), (157.14, 48.73), (155.25, 51.51), (163.87, 62.96), (180.8, 85.21), (160.25, 63.86), (188.32, 66.97), (170.97, 59.71), (182.93, 77.7)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=348；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.4803；
t = r·√((n−2)/(1−r²)) = 0.4803·√((348−2)/(1−0.2307)) = 10.1864；df=346；p = -1.30e-03。
95% CI（Fisher z）：tanh(z±18.591·1/√(n−3)) = [-0.4442, 0.9094]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=348；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.4803；
t = r·√((n−2)/(1−r²)) = 0.4803·√((348−2)/(1−0.2307)) = 10.1864；df=346；p = -1.30e-03。
95% CI（Fisher z）：tanh(z±18.591·1/√(n−3)) = [-0.4442, 0.9094]。
③ 计算结果：
r = 0.4803，t = 10.1864，p = -1.30e-03，95%CI = [-0.4442, 0.9094]
④ 结论：在 α=0.05 水平下，p=-1.30e-03 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.4803，方向正），身高越高体重越大。
案例 7. 睡眠时间(h)与次日专注度(分)
Pearson相关睡得好更专注
① 原始数据：
样本 n=979，逐条 (x, y)：[(8.0, 71.48), (6.88, 69.69), (6.91, 65.73), (6.21, 76.96), (6.13, 74.58), (7.11, 74.35), (6.0, 80.26), (5.68, 72.24), (7.27, 77.76), (4.74, 70.52), (8.1, 90.71), (7.23, 94.67), (6.43, 89.87), (7.74, 54.66), (8.33, 78.48), (7.84, 80.51), (7.46, 89.72), (4.78, 70.51), (5.49, 81.79), (6.23, 65.44), (4.89, 75.02), (7.7, 71.16), (7.44, 70.2), (4.83, 89.55), (6.64, 85.94), (6.02, 65.32), (7.11, 60.34), (7.25, 84.3), (6.92, 72.71), (5.87, 55.73), (7.64, 76.55), (5.31, 72.03), (6.59, 76.55), (6.89, 80.47), (7.6, 74.39), (5.07, 73.46), (6.73, 76.47), (6.19, 66.08), (6.11, 75.13), (6.44, 77.36), (8.88, 95.78), (7.45, 57.71), (6.59, 63.78), (5.88, 64.89), (7.9, 81.8), (4.98, 70.08), (7.88, 73.82), (7.47, 82.44), (5.49, 72.04), (6.97, 79.66), (4.71, 69.59), (6.34, 58.17), (6.32, 71.26), (7.19, 81.07), (5.71, 71.3), (8.36, 88.29), (6.0, 58.9), (7.57, 96.08), (7.75, 95.49), (5.7, 76.57), (8.55, 91.18), (6.42, 84.13), (5.97, 63.08), (7.15, 73.62), (7.49, 81.06), (6.33, 74.15), (6.16, 71.01), (6.48, 76.65), (6.09, 73.83), (5.45, 63.28), (6.56, 70.8), (5.83, 57.65), (8.44, 80.95), (6.35, 76.18), (5.69, 64.77), (6.34, 64.48), (5.11, 70.32), (6.67, 73.15), (7.24, 78.35), (6.4, 75.01), (8.6, 85.2), (7.48, 79.38), (7.58, 78.69), (6.72, 67.12), (7.07, 97.08), (6.16, 78.98), (8.75, 76.69), (7.08, 81.42), (7.56, 71.47), (6.18, 64.09), (7.46, 75.53), (7.58, 77.21), (7.44, 81.76), (5.7, 59.44), (6.54, 74.1), (6.09, 61.5), (7.72, 77.43), (6.54, 81.26), (5.01, 62.97), (6.16, 70.94), (5.96, 78.51), (6.29, 79.35), (6.61, 67.63), (6.4, 65.63), (7.38, 77.43), (6.93, 64.52), (7.21, 79.52), (7.28, 78.86), (6.63, 76.86), (7.76, 81.1), (6.3, 64.49), (7.02, 75.48), (6.73, 69.45), (7.21, 77.98), (7.31, 95.47), (5.45, 58.9), (6.34, 75.98), (6.32, 72.95), (9.02, 94.45), (7.67, 73.4), (7.62, 85.15), (8.1, 87.3), (8.12, 82.14), (7.15, 84.67), (6.8, 78.37), (7.13, 80.6), (6.57, 52.29), (5.42, 68.99), (8.54, 86.15), (8.26, 76.55), (7.02, 79.64), (6.62, 78.76), (5.25, 62.79), (6.4, 83.92), (7.03, 71.06), (8.54, 91.49), (7.18, 57.57), (8.14, 66.44), (7.84, 72.58), (7.72, 76.67), (7.89, 83.67), (6.13, 66.79), (7.09, 84.82), (7.09, 73.21), (5.41, 61.79), (6.84, 77.08), (8.6, 81.35), (5.96, 81.73), (5.48, 73.97), (7.53, 73.01), (7.43, 73.7), (5.78, 44.75), (5.01, 76.86), (5.35, 61.68), (8.23, 77.81), (8.07, 99.64), (6.96, 69.63), (6.22, 64.94), (6.43, 79.59), (5.83, 59.83), (7.81, 62.18), (9.14, 96.21), (7.58, 73.56), (6.46, 63.37), (7.13, 77.8), (5.41, 68.7), (7.0, 63.45), (6.51, 95.87), (7.08, 75.75), (6.21, 77.85), (7.48, 83.79), (6.29, 67.88), (7.05, 58.23), (6.8, 86.11), (7.28, 76.76), (6.47, 66.64), (4.58, 49.51), (7.74, 67.36), (5.96, 78.67), (7.06, 67.58), (6.73, 82.88), (7.13, 87.24), (5.27, 67.36), (6.02, 78.03), (5.71, 81.84), (7.27, 72.0), (8.2, 89.49), (6.74, 78.07), (6.19, 69.86), (7.66, 76.14), (6.29, 105.56), (7.92, 85.13), (6.57, 87.33), (6.53, 83.81), (6.14, 55.92), (8.29, 77.87), (7.43, 84.84), (8.99, 94.68), (8.05, 79.27), (7.95, 68.2), (7.15, 72.81), (8.28, 69.83), (8.31, 84.48), (7.92, 75.06), (6.53, 66.2), (5.88, 76.6), (7.31, 69.2), (6.09, 70.5), (6.36, 74.14), (8.97, 98.51), (5.54, 76.49), (6.32, 72.79), (8.5, 78.38), (8.3, 77.45), (8.25, 79.23), (8.02, 69.19), (7.47, 76.99), (5.97, 68.99), (6.9, 60.27), (6.16, 80.5), (7.03, 69.71), (6.63, 86.26), (6.77, 65.2), (7.44, 77.74), (7.31, 61.56), (6.08, 61.9), (7.88, 68.13), (8.95, 88.68), (7.58, 85.09), (6.98, 63.62), (6.76, 71.92), (7.36, 88.18), (6.57, 57.34), (8.42, 88.79), (6.75, 81.16), (6.68, 68.54), (6.9, 66.32), (8.63, 98.3), (7.38, 81.79), (5.71, 74.84), (6.66, 65.39), (6.87, 67.96), (7.12, 75.73), (7.86, 87.65), (7.07, 85.88), (6.92, 81.39), (6.4, 79.25), (8.09, 87.57), (7.39, 85.74), (6.78, 73.14), (5.91, 69.99), (6.5, 65.81), (7.03, 71.6), (7.79, 77.66), (7.7, 88.53), (7.25, 61.14), (6.88, 85.39), (8.08, 76.91), (6.8, 69.59), (7.54, 87.71), (7.05, 69.01), (8.58, 80.57), (9.63, 95.32), (8.07, 75.14), (7.67, 78.47), (7.38, 87.23), (7.07, 99.54), (8.18, 85.53), (5.87, 64.9), (8.07, 79.31), (6.84, 79.07), (6.52, 76.65), (6.91, 74.58), (7.82, 78.35), (5.61, 68.4), (8.64, 86.2), (6.27, 70.65), (7.44, 71.96), (7.75, 83.98), (6.38, 76.36), (7.99, 79.48), (7.93, 88.1), (7.04, 71.62), (8.87, 76.79), (6.4, 63.56), (8.9, 85.44), (5.58, 89.58), (8.4, 86.32), (7.39, 66.72), (6.89, 62.39), (7.4, 86.96), (7.06, 74.19), (7.42, 70.7), (6.67, 74.97), (6.45, 70.66), (5.73, 86.26), (7.67, 78.5), (6.36, 67.49), (7.78, 88.71), (6.49, 83.42), (6.83, 81.4), (8.24, 80.85), (6.66, 60.05), (7.25, 69.22), (7.02, 86.49), (7.61, 60.4), (8.18, 81.04), (7.75, 94.85), (7.85, 86.63), (4.42, 56.44), (6.94, 65.78), (6.56, 76.33), (6.54, 72.76), (5.15, 62.42), (7.15, 67.54), (5.81, 61.79), (6.58, 72.56), (8.61, 84.29), (4.65, 42.25), (7.74, 69.29), (7.03, 82.86), (7.78, 79.15), (3.29, 55.3), (7.33, 78.62), (8.56, 104.05), (7.74, 86.51), (5.24, 72.08), (7.75, 90.91), (6.72, 79.5), (4.32, 66.11), (7.14, 75.25), (6.41, 67.08), (8.91, 86.43), (8.32, 83.52), (6.27, 75.98), (6.44, 66.52), (6.53, 65.21), (8.32, 65.62), (7.22, 61.8), (7.86, 84.69), (8.0, 72.56), (6.0, 59.57), (7.92, 61.7), (8.68, 92.7), (6.56, 79.04), (5.35, 73.59), (6.08, 79.13), (6.76, 65.49), (7.39, 86.48), (9.47, 92.09), (7.02, 71.0), (5.81, 71.34), (7.4, 79.39), (6.59, 74.52), (6.67, 61.61), (6.61, 58.9), (7.68, 74.19), (7.18, 73.89), (7.9, 84.99), (7.99, 80.49), (6.94, 72.49), (5.93, 80.63), (6.96, 81.78), (5.96, 65.4), (8.29, 80.74), (6.19, 61.21), (4.61, 68.4), (7.87, 86.35), (5.74, 70.95), (6.36, 69.02), (7.28, 85.13), (6.23, 73.4), (6.56, 83.57), (7.09, 78.82), (8.83, 105.83), (7.09, 82.14), (5.95, 69.52), (6.83, 79.19), (5.98, 66.23), (5.65, 59.55), (8.2, 87.12), (7.32, 75.55), (7.39, 74.17), (6.42, 63.51), (5.95, 79.57), (7.23, 59.34), (5.86, 69.46), (8.86, 81.32), (7.95, 84.65), (8.01, 85.2), (6.79, 71.64), (7.05, 76.11), (6.94, 83.42), (6.84, 78.53), (6.38, 67.35), (6.97, 71.72), (7.65, 68.01), (5.9, 78.88), (6.5, 72.03), (6.22, 75.69), (7.1, 73.25), (7.33, 81.62), (7.2, 56.95), (8.64, 96.11), (4.4, 64.36), (8.51, 97.39), (7.24, 83.9), (9.61, 87.75), (5.51, 50.94), (6.31, 67.46), (5.24, 72.33), (7.42, 85.92), (10.39, 87.04), (7.27, 81.77), (7.55, 77.96), (6.29, 63.05), (5.43, 69.44), (6.5, 59.07), (6.39, 66.59), (6.79, 71.85), (8.73, 90.72), (9.18, 85.16), (7.38, 69.8), (7.1, 80.58), (6.98, 74.32), (6.38, 65.11), (5.97, 81.04), (9.17, 75.29), (8.76, 76.17), (8.4, 95.18), (6.78, 78.35), (8.65, 72.2), (6.93, 84.92), (6.9, 81.43), (6.33, 79.5), (6.91, 92.04), (8.38, 80.0), (6.34, 62.5), (5.85, 77.14), (6.43, 83.71), (7.02, 61.37), (6.22, 65.12), (6.06, 60.17), (6.11, 69.12), (5.67, 52.55), (7.29, 66.1), (6.85, 84.26), (7.27, 65.77), (8.77, 74.45), (6.9, 73.52), (7.4, 82.8), (8.94, 85.46), (7.13, 81.63), (6.04, 58.02), (7.74, 66.5), (6.23, 72.53), (8.01, 72.94), (5.98, 60.75), (7.38, 69.07), (9.19, 82.7), (6.36, 57.39), (5.67, 73.34), (5.42, 66.58), (6.34, 75.65), (6.88, 54.69), (7.04, 84.22), (5.87, 67.97), (7.93, 92.29), (6.7, 74.3), (6.49, 67.11), (6.89, 64.21), (7.37, 73.74), (6.7, 73.08), (6.87, 68.92), (6.06, 66.87), (7.14, 71.45), (6.97, 86.4), (4.76, 71.79), (8.71, 94.02), (7.55, 86.3), (6.61, 92.98), (8.04, 84.6), (6.17, 70.25), (8.46, 74.19), (5.37, 72.96), (7.35, 77.93), (6.55, 75.92), (6.59, 66.21), (8.6, 82.27), (4.86, 68.65), (7.75, 75.87), (4.73, 49.88), (8.63, 87.23), (7.8, 76.84), (7.82, 81.57), (8.88, 99.12), (7.82, 77.59), (6.77, 67.28), (7.83, 85.65), (6.58, 72.59), (7.03, 65.3), (5.83, 63.33), (5.56, 83.8), (7.93, 82.72), (6.44, 66.05), (6.42, 78.76), (5.87, 54.28), (6.71, 77.69), (8.26, 80.34), (7.25, 82.28), (6.76, 66.13), (8.21, 89.58), (6.39, 71.25), (8.38, 91.33), (7.05, 77.97), (6.74, 81.11), (7.71, 70.27), (7.88, 79.79), (7.72, 73.76), (8.36, 76.92), (3.94, 76.98), (8.41, 73.53), (8.62, 95.84), (8.54, 92.94), (5.36, 67.56), (5.03, 67.15), (6.34, 60.93), (8.32, 84.41), (6.78, 53.57), (5.88, 67.72), (8.96, 94.28), (7.9, 74.35), (6.87, 67.93), (7.86, 90.0), (8.11, 82.52), (6.91, 68.68), (6.01, 73.19), (7.48, 79.04), (7.89, 92.2), (7.25, 89.27), (6.12, 69.15), (8.11, 80.88), (6.26, 72.87), (7.13, 92.59), (7.34, 71.62), (6.55, 61.49), (7.43, 83.33), (8.7, 99.15), (7.67, 62.53), (5.62, 86.96), (7.51, 77.59), (7.34, 84.96), (7.14, 66.42), (6.84, 72.44), (8.14, 93.15), (6.69, 79.17), (7.11, 81.78), (7.88, 65.49), (6.49, 83.39), (4.48, 54.39), (5.33, 65.46), (7.08, 79.35), (6.36, 65.37), (6.63, 62.82), (6.37, 77.65), (6.4, 75.2), (8.18, 88.2), (7.23, 84.3), (6.46, 83.17), (7.56, 71.29), (4.65, 51.14), (7.45, 83.53), (5.94, 66.64), (8.26, 76.08), (4.67, 78.56), (7.74, 77.87), (7.3, 77.26), (5.9, 64.36), (5.77, 71.94), (6.03, 82.72), (7.8, 62.0), (8.49, 80.02), (8.33, 85.1), (7.78, 81.52), (7.75, 74.23), (6.12, 78.53), (7.4, 81.14), (7.79, 78.18), (5.57, 66.6), (7.92, 72.88), (7.59, 65.39), (7.73, 79.78), (9.1, 108.72), (6.68, 63.59), (6.36, 65.18), (6.61, 72.1), (8.52, 84.44), (6.65, 67.19), (7.31, 77.04), (6.77, 79.61), (8.86, 83.03), (5.7, 58.41), (8.52, 60.69), (7.1, 71.54), (5.3, 51.54), (8.66, 77.75), (4.68, 58.78), (7.05, 64.24), (6.66, 66.23), (7.57, 70.61), (7.45, 76.67), (6.96, 70.82), (8.57, 75.01), (7.42, 86.49), (6.6, 64.87), (8.49, 88.68), (5.79, 73.12), (7.38, 82.19), (5.85, 63.99), (6.97, 58.34), (7.41, 77.99), (5.78, 54.69), (7.63, 77.83), (6.28, 63.13), (5.43, 64.85), (7.42, 62.46), (5.95, 55.87), (4.84, 54.36), (6.73, 61.36), (7.29, 78.31), (6.66, 71.46), (7.43, 70.89), (8.05, 92.11), (6.08, 77.66), (6.62, 70.2), (5.62, 57.74), (7.84, 83.94), (6.84, 70.2), (6.93, 72.1), (8.54, 82.55), (7.07, 83.82), (6.16, 73.72), (7.04, 77.19), (6.68, 64.92), (6.56, 74.4), (7.74, 84.95), (7.83, 88.09), (7.33, 59.27), (5.42, 71.3), (7.42, 77.37), (5.79, 59.38), (5.92, 59.46), (6.16, 76.03), (7.53, 80.15), (7.12, 90.46), (7.13, 54.81), (7.21, 76.25), (7.21, 66.48), (5.74, 58.82), (6.73, 77.76), (6.29, 75.91), (6.54, 57.43), (6.04, 65.97), (5.65, 77.24), (7.55, 87.82), (5.75, 64.77), (7.45, 66.19), (6.94, 60.0), (6.79, 86.19), (7.17, 75.26), (5.58, 76.26), (6.46, 62.27), (8.55, 93.54), (6.68, 65.04), (6.64, 79.64), (6.82, 77.62), (5.04, 53.14), (6.59, 89.82), (6.24, 72.93), (5.88, 62.47), (7.74, 78.65), (6.5, 72.31), (5.46, 63.92), (6.24, 53.52), (7.87, 78.17), (7.2, 79.38), (8.76, 81.55), (7.6, 65.03), (6.96, 69.58), (6.55, 65.78), (6.59, 57.18), (8.88, 81.67), (7.97, 69.22), (6.75, 78.7), (7.02, 74.09), (5.56, 61.25), (5.5, 81.87), (5.32, 58.46), (8.46, 94.02), (5.88, 60.04), (6.85, 74.07), (7.75, 76.16), (5.67, 61.46), (4.97, 48.37), (7.65, 70.96), (6.65, 69.35), (6.51, 76.21), (6.25, 76.25), (8.62, 77.38), (6.45, 77.36), (7.17, 70.45), (6.53, 73.05), (5.48, 64.04), (8.22, 84.11), (7.39, 66.14), (6.84, 89.77), (7.41, 74.73), (6.3, 77.22), (6.36, 65.84), (6.71, 77.89), (6.42, 74.81), (6.59, 62.05), (6.11, 67.88), (8.76, 81.12), (8.34, 91.61), (6.87, 76.98), (7.08, 74.37), (7.11, 65.61), (6.55, 66.34), (7.98, 87.8), (7.52, 70.57), (8.61, 93.44), (8.77, 87.17), (6.93, 81.15), (6.06, 65.87), (7.97, 82.37), (5.87, 70.21), (6.58, 87.88), (5.08, 56.44), (8.87, 99.51), (3.96, 55.74), (7.09, 60.02), (7.29, 87.26), (7.15, 69.22), (7.78, 64.52), (7.32, 71.35), (6.63, 90.91), (7.59, 68.1), (8.52, 81.77), (6.64, 60.65), (6.08, 82.92), (5.61, 63.25), (7.42, 79.96), (8.5, 75.13), (5.81, 67.71), (6.42, 70.44), (6.52, 72.14), (6.64, 79.08), (5.76, 53.55), (6.82, 67.09), (6.05, 73.03), (8.74, 85.99), (6.78, 75.7), (6.82, 72.3), (7.23, 74.97), (7.62, 81.26), (6.23, 67.64), (7.84, 76.09), (7.08, 67.67), (8.57, 91.06), (6.43, 68.27), (6.82, 82.69), (8.46, 81.79), (8.23, 85.67), (6.34, 80.01), (5.08, 65.44), (9.23, 90.3), (7.04, 80.07), (6.88, 63.24), (5.78, 74.14), (6.92, 78.28), (9.06, 73.94), (5.24, 71.17), (7.06, 71.17), (7.09, 73.23), (7.94, 80.37), (7.49, 69.85), (6.07, 53.83), (6.81, 69.8), (7.84, 84.78), (6.67, 83.33), (5.62, 58.18), (6.6, 72.74), (5.96, 76.48), (9.08, 88.29), (5.99, 81.92), (9.18, 87.43), (6.53, 66.99), (9.21, 81.4), (7.57, 74.81), (6.17, 80.93), (5.32, 71.36), (7.31, 79.68), (7.66, 78.38), (6.72, 70.18), (6.64, 60.55), (7.29, 83.2), (8.36, 93.9), (7.57, 73.39), (6.74, 85.87), (6.21, 66.53), (7.25, 67.22), (7.93, 90.13), (7.56, 87.77), (8.13, 78.52), (8.09, 70.58), (6.72, 64.92), (6.28, 88.77), (5.72, 68.62), (6.69, 69.19), (5.15, 67.63), (5.81, 86.83), (6.0, 78.67), (7.33, 78.48), (6.13, 63.73), (7.98, 79.31), (6.36, 73.81), (7.65, 83.03), (6.61, 82.87), (6.12, 89.24), (8.04, 73.02), (4.91, 64.14), (7.41, 78.87), (7.03, 75.42), (7.03, 76.58), (8.79, 77.42), (5.71, 57.91), (6.41, 86.64), (7.43, 60.34), (6.85, 66.05), (4.77, 58.95), (7.45, 88.33), (6.97, 74.6), (4.72, 57.54), (8.51, 91.71), (7.82, 76.26), (6.11, 82.33), (8.69, 90.05), (6.19, 72.91), (7.96, 98.0), (6.6, 62.06), (6.23, 78.82), (8.37, 83.3), (5.25, 81.64), (6.85, 70.96), (5.67, 59.79), (6.88, 75.77), (7.8, 88.99), (5.1, 64.76), (6.82, 77.21), (5.86, 75.85), (6.08, 62.39), (7.16, 78.0), (5.91, 67.17), (8.14, 66.15), (7.29, 77.24), (5.93, 64.54), (7.38, 88.74), (6.76, 69.5), (6.26, 71.1), (9.19, 86.47), (6.97, 75.71), (7.95, 71.69), (4.59, 58.26), (8.45, 88.83), (5.29, 60.46), (5.63, 81.23), (7.36, 66.92), (6.69, 80.88), (6.98, 67.0), (8.19, 89.87), (6.58, 69.4), (7.07, 88.28), (7.62, 72.33), (6.84, 81.72), (6.15, 55.46), (6.68, 65.84), (6.36, 80.4), (6.67, 72.36), (6.72, 79.78), (5.87, 73.98), (6.97, 76.26), (6.52, 74.7), (7.43, 73.09), (6.91, 87.25), (7.02, 71.32), (8.46, 91.86), (8.35, 81.98), (7.68, 69.84), (7.2, 68.31), (7.09, 81.28), (8.44, 75.71), (6.89, 62.26), (7.62, 75.81), (6.91, 77.73), (7.98, 83.01), (5.53, 57.87), (7.52, 81.6), (6.6, 63.1), (6.17, 76.38), (7.45, 73.72), (6.5, 81.19), (7.21, 74.97), (6.66, 71.41), (7.35, 83.75), (6.89, 70.26), (7.05, 95.07), (7.26, 72.0), (8.22, 89.81), (8.32, 108.73), (7.43, 76.58), (6.69, 81.05), (6.63, 71.74), (7.55, 94.17), (5.35, 73.01), (6.59, 75.69), (7.72, 74.29), (6.44, 85.56), (8.55, 74.56), (7.97, 82.69), (7.69, 85.72), (6.81, 64.72), (6.53, 86.63), (7.13, 70.19), (6.75, 69.3), (6.64, 86.63), (7.57, 81.74), (6.35, 63.77), (6.15, 64.63), (7.04, 92.65), (5.64, 71.45), (7.32, 74.28), (5.05, 63.12), (6.51, 79.32), (7.0, 70.23), (8.52, 86.15), (6.09, 77.36), (5.78, 64.88), (7.74, 80.55), (5.98, 70.59), (7.41, 72.42), (7.85, 87.76), (5.59, 70.83), (8.91, 84.58), (7.32, 86.98), (8.22, 89.89), (6.3, 72.71), (5.42, 78.46), (7.8, 76.71), (5.92, 69.14), (5.9, 67.12), (6.96, 48.3), (8.23, 74.85), (5.44, 73.39), (7.84, 74.61), (5.99, 58.14), (5.62, 64.08), (7.47, 72.19), (6.61, 70.83), (6.31, 77.37), (7.28, 77.25), (7.01, 73.8), (7.51, 85.68), (8.7, 79.52), (5.82, 73.39), (6.86, 67.47), (8.26, 85.55), (7.78, 97.48), (7.92, 76.64)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=979；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.5507；
t = r·√((n−2)/(1−r²)) = 0.5507·√((979−2)/(1−0.3032)) = 20.6200；df=977；p = -9.54e+36。
95% CI（Fisher z）：tanh(z±31.255·1/√(n−3)) = [-0.3637, 0.9246]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=979；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.5507；
t = r·√((n−2)/(1−r²)) = 0.5507·√((979−2)/(1−0.3032)) = 20.6200；df=977；p = -9.54e+36。
95% CI（Fisher z）：tanh(z±31.255·1/√(n−3)) = [-0.3637, 0.9246]。
③ 计算结果：
r = 0.5507，t = 20.6200，p = -9.54e+36，95%CI = [-0.3637, 0.9246]
④ 结论：在 α=0.05 水平下，p=-9.54e+36 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.5507，方向正），睡得好更专注。
案例 8. 店铺面积(㎡)与月营业额(万元)
Pearson相关店大营业额高
① 原始数据：
样本 n=450，逐条 (x, y)：[(179.62, 48.42), (182.11, 39.85), (162.17, 50.66), (145.19, 38.84), (123.0, 42.08), (145.84, 49.65), (129.31, 48.11), (87.96, 47.51), (135.51, 32.49), (135.96, 28.67), (91.76, 19.33), (153.66, 35.34), (156.05, 52.81), (184.17, 63.23), (91.04, 30.01), (102.48, 22.92), (154.56, 41.86), (113.45, 34.06), (144.13, 50.45), (167.38, 38.31), (108.81, 50.82), (59.72, 15.87), (115.08, 52.11), (124.2, 50.27), (102.88, 63.84), (183.95, 57.75), (82.35, 17.91), (68.46, 28.15), (133.54, 39.18), (137.76, 31.18), (89.38, 46.79), (172.16, 58.1), (147.04, 55.29), (138.16, 49.73), (139.27, 52.12), (111.96, 38.62), (88.98, 39.78), (60.78, 28.48), (160.05, 45.97), (36.64, 41.88), (96.85, 41.27), (137.44, 36.82), (127.8, 57.87), (44.78, 18.36), (40.48, 25.52), (129.35, 43.77), (64.46, 26.97), (137.27, 33.99), (135.78, 33.13), (104.43, 38.16), (92.35, 35.96), (218.65, 67.64), (74.11, 32.53), (-5.88, 23.11), (117.75, 44.29), (83.24, 36.63), (164.98, 48.69), (117.46, 47.55), (133.43, 36.56), (81.29, 38.72), (63.37, 42.77), (69.92, 21.18), (102.49, 42.87), (139.28, 37.9), (115.74, 38.12), (71.55, 35.72), (210.69, 53.67), (112.96, 21.67), (40.55, 27.25), (86.6, 21.45), (81.55, 41.78), (134.3, 47.14), (190.88, 66.22), (105.7, 54.45), (21.35, 25.94), (116.78, 30.23), (113.79, 33.56), (87.49, 33.21), (156.47, 59.99), (77.19, 35.88), (122.12, 49.06), (125.37, 39.41), (170.62, 49.88), (106.07, 49.1), (131.99, 27.8), (121.89, 48.91), (135.68, 41.89), (61.85, 41.99), (116.01, 30.59), (168.91, 60.3), (113.05, 54.77), (56.03, 43.79), (43.56, 21.7), (69.12, 1.88), (81.67, 40.11), (177.16, 43.41), (100.7, 50.27), (128.77, 20.85), (197.34, 37.44), (149.99, 39.56), (168.72, 71.27), (117.08, 40.14), (121.08, 32.55), (68.15, 28.37), (57.15, 29.46), (61.37, 33.62), (78.53, 51.93), (24.92, 22.74), (123.0, 52.24), (64.99, 46.26), (104.18, 28.04), (128.73, 44.41), (120.89, 56.45), (110.32, 66.64), (165.57, 35.63), (109.63, 42.14), (55.79, 31.12), (122.54, 45.26), (60.7, 19.97), (80.56, 39.71), (84.91, 48.19), (109.85, 38.37), (127.07, 22.25), (139.99, 42.81), (90.07, 26.69), (121.55, 56.36), (47.46, 22.82), (215.68, 79.92), (90.57, 18.61), (162.87, 38.13), (51.68, 33.94), (114.87, 20.08), (177.61, 33.39), (141.36, 35.39), (109.69, 24.51), (97.34, 51.66), (123.46, 34.85), (116.96, 43.19), (86.17, 36.84), (123.77, 30.5), (140.71, 45.38), (104.09, 24.0), (46.87, 45.1), (142.36, 41.88), (223.34, 53.87), (103.02, 37.36), (108.02, 33.97), (191.21, 56.43), (151.96, 35.24), (81.38, 44.42), (114.13, 37.57), (90.82, 25.06), (55.53, 36.66), (68.81, 36.43), (130.54, 29.15), (69.12, 36.35), (76.35, 32.39), (112.7, 47.41), (138.73, 27.9), (124.45, 48.15), (122.0, 38.44), (154.97, 53.66), (122.93, 42.95), (54.64, 43.62), (157.34, 56.03), (149.57, 39.04), (162.28, 46.53), (87.59, 42.29), (131.38, 29.07), (116.79, 54.54), (153.93, 48.43), (71.53, 35.06), (65.39, 38.79), (90.79, 42.37), (141.18, 50.5), (177.36, 41.92), (42.88, 15.1), (123.01, 18.68), (119.13, 27.6), (81.98, 44.68), (136.6, 40.45), (235.85, 62.17), (168.22, 41.23), (205.04, 75.05), (187.68, 34.61), (32.78, 23.23), (97.81, 35.91), (182.32, 36.93), (204.92, 60.27), (164.78, 54.18), (71.81, 25.38), (173.92, 44.22), (78.19, 40.48), (120.21, 50.86), (67.43, 29.71), (192.27, 77.44), (92.43, 54.02), (121.5, 49.42), (132.36, 38.31), (31.24, 32.0), (87.41, 45.32), (70.8, 23.47), (209.74, 68.34), (142.28, 42.24), (140.92, 39.29), (101.56, 27.02), (179.33, 40.8), (108.02, 52.38), (44.58, 13.32), (151.15, 42.26), (49.78, 32.57), (108.39, 36.91), (100.34, 13.02), (184.97, 27.76), (116.83, 22.01), (127.2, 44.57), (78.51, 37.52), (102.73, 33.51), (134.23, 42.51), (139.36, 51.9), (67.51, 31.88), (101.45, 31.51), (117.68, 53.25), (134.67, 39.06), (174.34, 56.59), (111.05, 39.83), (109.97, 34.08), (170.06, 50.31), (38.79, 21.09), (116.44, 33.76), (89.02, 38.21), (127.78, 44.2), (99.77, 37.82), (56.65, 23.68), (141.31, 31.64), (150.78, 49.98), (91.14, 35.89), (127.73, 23.93), (218.36, 27.95), (139.05, 53.09), (154.35, 32.4), (115.15, 43.69), (97.72, 50.53), (123.35, 46.52), (98.1, 23.46), (172.33, 36.35), (158.79, 36.74), (99.79, 41.16), (156.79, 33.75), (164.87, 46.62), (114.02, 34.32), (128.03, 25.14), (78.81, 27.53), (38.39, -5.06), (66.52, 28.6), (127.42, 40.11), (95.55, 41.32), (79.14, 31.44), (104.65, 45.06), (117.99, 37.6), (76.7, 37.01), (121.71, 50.25), (130.02, 32.17), (130.21, 51.47), (181.78, 27.2), (114.59, 36.45), (77.39, 30.87), (74.83, 56.47), (112.41, 37.67), (149.71, 41.14), (125.96, 21.17), (55.39, 18.7), (105.56, 35.66), (155.15, 51.94), (95.72, 45.64), (18.96, 34.95), (54.97, 25.2), (118.56, 34.26), (86.35, 42.67), (143.11, 54.64), (81.69, 42.51), (156.13, 33.38), (142.81, 53.49), (136.24, 31.74), (73.09, 54.72), (137.19, 25.5), (80.46, 33.04), (136.27, 64.83), (62.2, 43.21), (69.03, 36.86), (144.45, 52.88), (129.31, 29.25), (89.59, 41.57), (108.46, 44.62), (92.82, 45.81), (99.27, 28.62), (142.42, 49.93), (88.52, 33.98), (127.91, 58.59), (124.92, 37.89), (134.19, 45.14), (119.94, 54.36), (170.72, 41.27), (132.04, 39.66), (125.51, 18.07), (156.68, 40.12), (86.05, 30.56), (113.16, 18.68), (156.64, 44.42), (115.03, 23.71), (142.53, 41.01), (123.81, 43.19), (146.69, 46.31), (170.7, 46.21), (87.38, 45.77), (145.81, 45.71), (141.44, 38.59), (95.19, 10.79), (131.45, 42.08), (118.34, 44.06), (71.59, 40.18), (203.07, 39.19), (130.78, 37.91), (103.83, 25.64), (95.51, 40.32), (106.81, 25.35), (70.79, 26.59), (46.94, 26.23), (97.09, 46.92), (73.17, 24.81), (174.11, 35.57), (154.84, 37.22), (79.3, 32.0), (138.16, 30.18), (96.34, 43.64), (184.92, 53.69), (96.5, 43.56), (46.57, 30.85), (95.9, 42.91), (91.73, 42.98), (74.0, 44.76), (141.08, 42.22), (96.94, 36.86), (73.36, 39.25), (54.57, 22.34), (154.92, 63.7), (92.93, 33.07), (102.64, 46.55), (122.47, 53.34), (78.5, 25.45), (167.41, 57.65), (62.73, 23.48), (91.98, 48.05), (43.81, 33.76), (132.16, 35.92), (138.7, 40.47), (114.05, 32.61), (216.29, 55.15), (47.2, 32.21), (47.8, 40.69), (72.62, 48.03), (78.64, 29.44), (109.04, 54.15), (173.36, 69.13), (77.56, -3.29), (162.95, 43.67), (107.5, 23.49), (42.73, 23.95), (104.88, 46.93), (72.95, 26.18), (149.45, 40.32), (77.61, 41.17), (150.66, 41.45), (126.8, 43.35), (129.44, 26.92), (62.88, 34.81), (133.63, 36.72), (107.05, 30.46), (75.6, 52.01), (103.73, 26.28), (171.59, 52.57), (192.52, 46.78), (131.96, 52.75), (123.48, 34.47), (94.36, 32.4), (50.74, 9.59), (130.47, 38.13), (59.06, 21.86), (97.77, 39.26), (147.99, 36.98), (129.29, 22.39), (78.83, 31.4), (113.93, 58.6), (74.42, 46.88), (113.68, 52.75), (127.3, 22.57), (163.66, 49.02), (88.36, 38.58), (102.69, 47.17), (66.43, 18.88), (121.74, 52.04), (63.5, 24.24), (110.66, 45.69), (72.65, 39.57), (134.55, 52.44), (85.77, 44.35), (176.56, 48.24), (125.95, 46.36), (168.85, 31.78), (136.1, 39.62), (74.14, 44.38), (152.07, 27.25), (111.63, 36.72), (104.8, 35.01), (154.46, 36.53), (183.76, 66.23), (83.97, 50.26), (159.41, 54.15), (122.7, 53.13), (122.52, 44.28), (109.5, 41.0), (106.88, 37.87), (220.33, 43.39), (108.83, 38.59), (105.55, 48.99), (103.82, 46.3), (89.47, 33.47), (85.69, 21.95), (71.4, 60.98), (157.0, 57.68), (222.8, 70.65), (94.98, 42.96), (56.14, 34.58), (85.02, 48.33), (133.25, 39.15), (146.59, 33.68), (164.64, 42.87), (118.28, 31.53), (147.06, 46.8), (172.38, 52.39), (105.73, 36.31), (180.21, 50.18), (84.75, 28.1), (173.74, 49.13), (188.29, 39.05), (147.88, 27.35), (103.21, 37.15), (66.31, 39.71), (66.64, 49.27), (142.15, 45.98)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=450；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.4889；
t = r·√((n−2)/(1−r²)) = 0.4889·√((450−2)/(1−0.2390)) = 11.8608；df=448；p = -3.30e+02。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.4150, 0.5563]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=450；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.4889；
t = r·√((n−2)/(1−r²)) = 0.4889·√((450−2)/(1−0.2390)) = 11.8608；df=448；p = -3.30e+02。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.4150, 0.5563]。
③ 计算结果：
r = 0.4889，t = 11.8608，p = -3.30e+02，95%CI = [0.4150, 0.5563]
④ 结论：在 α=0.05 水平下，p=-3.30e+02 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.4889，方向正），店大营业额高。
案例 9. 咖啡摄入量(杯)与工作效率(分)
Pearson相关弱正相关
① 原始数据：
样本 n=503，逐条 (x, y)：[(1.55, 65.05), (1.37, 39.46), (2.19, 90.7), (2.89, 88.73), (1.92, 82.94), (2.33, 67.25), (3.03, 54.75), (2.56, 85.67), (1.6, 71.04), (1.74, 75.34), (2.69, 86.47), (1.55, 73.24), (2.32, 52.67), (3.63, 66.64), (3.65, 58.31), (2.5, 85.77), (0.96, 40.82), (3.29, 84.53), (2.4, 81.94), (0.92, 80.32), (1.43, 56.44), (3.12, 80.31), (2.9, 82.32), (2.17, 79.81), (2.34, 86.82), (4.2, 82.14), (2.05, 55.81), (1.65, 68.07), (3.46, 77.51), (4.27, 85.46), (2.38, 26.62), (1.76, 72.29), (2.75, 90.09), (2.25, 78.91), (3.28, 68.01), (1.06, 65.45), (1.9, 59.85), (4.02, 50.73), (1.48, 72.59), (2.87, 60.9), (2.05, 77.7), (2.49, 86.32), (2.59, 54.11), (2.44, 117.28), (2.64, 55.64), (0.46, 82.86), (3.5, 91.42), (0.96, 59.99), (1.53, 62.24), (1.03, 73.31), (0.94, 49.44), (2.81, 70.91), (1.73, 88.63), (0.83, 70.26), (1.96, 43.85), (1.07, 82.22), (-1.41, 79.83), (1.39, 83.79), (2.15, 71.02), (1.32, 59.69), (0.43, 55.15), (0.34, 67.68), (4.64, 95.3), (1.45, 86.99), (2.7, 58.71), (5.51, 79.96), (1.74, 49.36), (-0.06, 24.83), (3.97, 96.47), (1.66, 49.86), (2.34, 93.9), (3.9, 102.79), (3.15, 100.2), (4.23, 87.23), (1.37, 64.04), (2.9, 55.71), (0.49, 57.08), (2.6, 83.71), (1.53, 67.18), (3.65, 73.97), (3.21, 104.01), (2.01, 69.76), (-0.13, 57.02), (3.15, 69.76), (1.11, 60.39), (1.4, 62.09), (1.0, 52.87), (2.06, 71.28), (3.33, 91.36), (0.61, 87.3), (1.98, 80.77), (1.42, 64.78), (2.38, 75.11), (1.91, 68.91), (4.83, 90.72), (1.95, 71.44), (1.66, 53.44), (2.37, 69.67), (1.32, 59.91), (1.36, 93.22), (1.28, 69.28), (2.93, 70.37), (2.09, 68.62), (2.58, 92.34), (1.37, 67.86), (1.34, 61.65), (2.17, 53.48), (2.17, 84.03), (0.3, 66.69), (3.2, 80.14), (2.17, 73.81), (1.45, 64.46), (2.68, 64.37), (0.18, 72.84), (1.02, 55.67), (0.44, 54.32), (1.25, 78.62), (2.8, 71.88), (2.68, 67.8), (2.31, 66.98), (2.26, 75.25), (3.34, 66.01), (3.22, 78.02), (3.01, 78.02), (2.34, 81.75), (2.98, 61.93), (2.06, 86.8), (0.62, 87.59), (2.8, 94.02), (2.32, 80.92), (2.94, 53.68), (5.04, 73.27), (0.96, 55.44), (2.18, 72.3), (2.12, 100.06), (1.78, 67.03), (1.34, 67.5), (0.65, 61.18), (1.84, 87.6), (3.13, 99.99), (2.95, 60.3), (2.27, 67.6), (0.99, 54.14), (3.3, 83.56), (1.32, 68.72), (2.66, 80.28), (3.59, 74.49), (0.87, 64.54), (2.98, 82.95), (2.02, 77.88), (2.47, 79.77), (1.46, 69.37), (1.99, 68.75), (2.33, 81.27), (1.87, 64.88), (2.06, 46.03), (2.46, 68.91), (1.85, 49.73), (3.52, 87.37), (2.29, 43.31), (1.39, 71.66), (1.51, 76.64), (0.02, 46.96), (1.73, 66.41), (2.01, 88.59), (3.17, 74.69), (2.31, 66.05), (0.0, 58.65), (3.89, 68.73), (3.71, 48.11), (2.67, 56.25), (2.72, 72.11), (1.97, 73.9), (3.22, 97.35), (3.46, 60.11), (2.43, 60.97), (1.94, 64.25), (2.33, 75.85), (2.13, 79.98), (1.47, 71.77), (-0.08, 71.8), (3.4, 86.99), (1.19, 69.33), (3.63, 82.09), (2.05, 73.08), (2.2, 77.53), (2.26, 79.99), (0.66, 68.53), (3.7, 63.33), (2.76, 93.64), (0.57, 42.12), (1.36, 40.93), (3.24, 66.55), (1.97, 88.88), (3.07, 73.71), (2.24, 65.2), (3.3, 68.58), (3.25, 54.88), (-0.68, 38.3), (2.25, 89.53), (0.98, 62.22), (1.07, 81.81), (3.45, 86.65), (2.54, 73.6), (2.31, 66.34), (3.42, 84.22), (1.45, 43.55), (2.63, 60.84), (2.85, 91.89), (1.01, 53.51), (1.37, 102.99), (2.7, 79.59), (3.16, 56.93), (2.21, 61.75), (1.26, 58.59), (3.42, 86.68), (0.97, 71.03), (2.83, 77.16), (0.71, 66.8), (1.81, 49.38), (4.81, 73.38), (-0.67, 73.96), (2.06, 81.77), (3.04, 47.64), (2.19, 60.0), (2.08, 100.39), (0.9, 74.24), (3.17, 79.69), (2.13, 70.75), (2.77, 63.9), (2.15, 64.54), (2.14, 73.87), (1.81, 65.78), (2.67, 37.61), (2.6, 74.38), (1.93, 62.38), (4.1, 79.47), (1.7, 92.92), (2.54, 69.8), (1.79, 75.16), (3.9, 70.36), (2.43, 92.39), (2.02, 42.41), (-0.24, 55.18), (1.48, 83.57), (3.41, 72.77), (1.22, 63.15), (2.92, 64.72), (0.86, 83.33), (0.32, 65.51), (3.64, 81.52), (1.99, 48.35), (1.86, 56.34), (1.72, 94.37), (3.25, 81.23), (2.0, 86.37), (0.6, 68.68), (2.17, 78.63), (1.15, 35.28), (3.55, 72.14), (3.84, 81.87), (3.25, 76.47), (2.87, 66.35), (3.48, 65.37), (0.69, 65.1), (1.61, 66.59), (3.0, 65.76), (0.85, 84.29), (2.3, 98.03), (3.08, 77.81), (1.92, 59.89), (1.7, 80.83), (2.23, 59.13), (1.47, 60.79), (0.89, 69.99), (2.49, 83.86), (2.93, 68.5), (2.25, 89.91), (0.7, 60.87), (2.82, 46.99), (1.43, 57.92), (2.99, 101.7), (1.72, 71.9), (0.96, 62.46), (3.09, 93.3), (2.49, 80.21), (1.2, 49.21), (2.18, 49.53), (1.7, 75.47), (1.83, 57.75), (1.27, 64.87), (0.48, 54.15), (2.82, 79.18), (3.58, 83.19), (3.42, 71.98), (2.9, 73.65), (1.15, 69.81), (1.07, 28.24), (1.96, 82.88), (0.68, 66.09), (1.86, 77.59), (0.01, 47.37), (3.0, 79.36), (0.43, 43.38), (2.27, 97.43), (1.5, 82.88), (3.44, 96.36), (2.65, 125.84), (0.24, 55.19), (0.86, 66.2), (0.9, 71.23), (3.27, 63.64), (1.6, 74.59), (3.28, 105.89), (1.0, 63.32), (0.65, 59.06), (3.63, 62.44), (2.99, 75.9), (2.29, 62.5), (2.11, 51.99), (1.06, 61.94), (1.61, 68.28), (2.53, 82.3), (2.11, 81.16), (0.41, 54.36), (1.9, 72.34), (3.77, 81.5), (1.66, 67.76), (1.68, 65.44), (1.32, 35.09), (2.16, 76.92), (1.71, 72.69), (2.56, 76.13), (1.02, 83.16), (2.57, 65.13), (3.6, 112.39), (2.08, 65.18), (3.62, 71.58), (2.64, 86.17), (0.88, 83.97), (1.1, 48.42), (3.17, 87.83), (3.23, 80.71), (1.68, 85.93), (1.75, 62.63), (2.97, 57.46), (2.38, 76.48), (4.44, 62.71), (1.68, 86.58), (2.07, 67.9), (2.57, 64.99), (1.82, 49.55), (1.74, 80.93), (2.5, 62.58), (1.05, 63.11), (-0.07, 63.68), (2.03, 105.32), (-0.2, 38.51), (1.26, 57.15), (2.74, 71.16), (2.75, 61.05), (1.76, 49.55), (2.32, 76.9), (1.38, 71.56), (2.04, 82.08), (1.4, 55.31), (1.59, 65.03), (2.42, 63.08), (2.0, 75.83), (0.63, 51.41), (1.94, 81.66), (1.7, 71.14), (1.32, 74.22), (2.4, 46.56), (3.91, 73.09), (2.1, 92.47), (2.82, 67.74), (1.16, 84.79), (2.15, 63.94), (2.12, 63.8), (1.28, 68.8), (1.19, 57.55), (1.72, 56.08), (3.05, 80.82), (1.68, 59.86), (1.12, 87.1), (3.01, 82.7), (2.69, 89.49), (1.04, 80.79), (1.63, 62.12), (1.98, 97.19), (2.41, 56.62), (2.34, 64.39), (0.09, 44.88), (2.01, 57.65), (2.1, 88.13), (2.2, 71.47), (1.62, 79.5), (2.6, 53.13), (-0.08, 58.13), (1.01, 58.2), (0.35, 89.46), (1.91, 68.46), (3.91, 71.04), (1.99, 75.58), (2.4, 102.34), (-1.02, 75.88), (2.05, 44.0), (3.71, 68.34), (0.99, 39.23), (2.91, 29.46), (2.98, 71.01), (0.33, 86.38), (2.57, 64.62), (2.01, 55.5), (3.16, 40.18), (1.92, 79.21), (3.81, 73.43), (2.01, 96.73), (3.65, 83.4), (2.86, 61.4), (4.3, 86.58), (4.29, 56.25), (4.05, 78.05), (1.2, 56.75), (1.95, 78.35), (3.44, 82.85), (2.16, 91.16), (0.88, 77.94), (1.48, 64.54), (1.22, 91.17), (0.66, 83.73), (3.36, 69.34), (2.77, 86.27), (1.48, 78.66), (3.4, 62.07), (1.59, 81.69), (1.32, 53.6), (-0.01, 58.72), (3.45, 96.11), (2.12, 61.79), (3.25, 78.23), (2.03, 66.27), (1.53, 84.79), (2.64, 52.64), (2.97, 48.71), (3.89, 94.59), (2.94, 66.81), (1.07, 76.02), (2.19, 67.42), (3.15, 58.35), (0.57, 62.35), (1.52, 66.99), (1.62, 48.7), (0.66, 67.22), (0.83, 74.2), (0.61, 47.89), (2.11, 106.63), (3.34, 64.93), (3.84, 54.03), (2.49, 58.84), (2.47, 75.58), (3.14, 81.26), (2.47, 81.01), (2.05, 74.37), (1.67, 89.04), (2.41, 35.19), (2.78, 66.02), (1.77, 78.48), (2.28, 79.95), (1.73, 82.38), (1.1, 64.62), (2.6, 79.75), (3.54, 67.75), (3.79, 64.82), (1.78, 65.91), (2.94, 35.38), (3.37, 53.48), (3.93, 77.13), (2.34, 112.16), (3.16, 71.24), (1.89, 68.41), (1.72, 63.55), (1.7, 33.84), (3.38, 71.62), (2.8, 77.42), (2.26, 60.91), (3.15, 90.06), (2.11, 66.37), (2.79, 83.43), (2.16, 69.97), (1.89, 88.24), (2.16, 38.62), (2.05, 71.26), (2.09, 68.24), (1.61, 74.19), (2.13, 93.68), (1.06, 55.13), (1.2, 78.43), (1.14, 73.58), (1.71, 67.06), (0.67, 79.38), (3.3, 58.63)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=503；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2660；
t = r·√((n−2)/(1−r²)) = 0.2660·√((503−2)/(1−0.0708)) = 6.1764；df=501；p = 1.37e-09。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.1826, 0.3456]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=503；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2660；
t = r·√((n−2)/(1−r²)) = 0.2660·√((503−2)/(1−0.0708)) = 6.1764；df=501；p = 1.37e-09。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.1826, 0.3456]。
③ 计算结果：
r = 0.2660，t = 6.1764，p = 1.37e-09，95%CI = [0.1826, 0.3456]
④ 结论：在 α=0.05 水平下，p=1.37e-09 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.2660，方向正），弱正相关。
案例 10. 通勤时间(分)与心情指数(分)
Pearson相关弱相关
① 原始数据：
样本 n=694，逐条 (x, y)：[(57.77, 57.19), (37.1, 58.52), (47.84, 53.92), (18.74, 54.25), (25.23, 51.54), (44.99, 68.95), (67.13, 59.39), (28.9, 71.27), (53.42, 67.95), (53.44, 81.1), (49.77, 34.95), (37.41, 55.65), (41.57, 80.0), (30.12, 55.73), (35.3, 50.6), (45.19, 79.25), (63.45, 75.95), (58.95, 67.57), (16.59, 41.13), (54.72, 66.02), (37.85, 56.08), (28.03, 38.73), (46.29, 30.3), (37.43, 73.65), (12.31, 45.73), (43.98, 76.3), (43.03, 35.69), (38.62, 50.6), (19.42, 38.04), (40.13, 59.5), (24.01, 51.6), (55.55, 83.44), (29.33, 76.1), (59.75, 49.98), (25.45, 70.16), (32.55, 63.46), (32.03, 66.63), (51.01, 76.33), (35.75, 61.59), (32.08, 56.29), (44.52, 51.96), (46.97, 76.32), (63.54, 58.31), (38.45, 60.63), (40.38, 72.86), (28.73, 64.49), (29.44, 59.17), (15.17, 67.58), (43.58, 76.29), (17.51, 56.92), (43.48, 60.78), (41.0, 37.79), (48.33, 77.98), (56.69, 65.27), (21.14, 58.74), (62.95, 62.76), (70.3, 58.78), (10.44, 62.47), (33.68, 56.68), (13.68, 54.76), (32.98, 51.86), (49.91, 48.25), (48.84, 70.99), (38.14, 46.46), (47.61, 58.66), (55.59, 72.06), (35.16, 48.52), (18.1, 42.52), (48.22, 62.04), (65.74, 57.78), (20.11, 44.32), (26.71, 58.5), (65.12, 54.77), (33.23, 55.72), (46.14, 73.5), (54.1, 75.95), (35.45, 58.1), (50.43, 58.21), (49.88, 63.51), (26.4, 55.57), (54.77, 64.23), (53.68, 43.58), (55.14, 41.7), (38.13, 58.68), (41.63, 73.47), (45.19, 60.68), (36.46, 65.93), (26.07, 81.74), (55.7, 36.56), (37.28, 47.23), (25.49, 61.88), (23.22, 42.68), (61.22, 63.77), (60.85, 61.37), (63.14, 62.79), (28.77, 74.73), (27.94, 49.47), (40.87, 65.1), (26.73, 48.33), (51.57, 67.1), (38.3, 72.29), (34.05, 56.71), (28.65, 36.56), (16.31, 56.57), (54.78, 51.65), (44.77, 54.3), (46.88, 57.57), (22.21, 26.73), (19.94, 57.28), (36.43, 37.2), (19.61, 49.59), (11.5, 59.68), (35.27, 72.42), (39.69, 50.68), (43.85, 39.82), (10.94, 62.51), (31.99, 43.86), (63.45, 57.65), (31.32, 65.05), (51.73, 52.64), (19.9, 48.02), (27.98, 50.62), (2.56, 47.79), (26.76, 58.52), (29.24, 67.57), (39.74, 52.14), (51.42, 66.05), (34.01, 66.64), (57.61, 70.35), (37.06, 60.24), (39.1, 59.4), (36.79, 64.02), (15.14, 43.11), (39.57, 72.71), (14.04, 51.38), (50.91, 58.97), (12.86, 71.85), (33.95, 42.89), (41.76, 61.9), (38.9, 79.34), (45.46, 58.03), (32.21, 42.39), (24.71, 42.27), (42.17, 60.73), (29.12, 59.76), (34.48, 58.4), (46.66, 58.13), (23.51, 59.69), (30.17, 54.08), (39.41, 47.8), (31.75, 48.86), (52.57, 76.84), (53.31, 58.39), (27.02, 38.5), (33.23, 73.22), (49.37, 71.7), (45.4, 39.24), (49.03, 55.72), (50.64, 70.97), (69.96, 53.87), (43.39, 72.35), (55.29, 68.03), (61.62, 75.96), (73.13, 68.41), (65.44, 74.44), (51.66, 53.18), (24.04, 62.21), (26.08, 45.4), (41.39, 62.08), (14.28, 47.87), (52.79, 66.61), (34.47, 52.23), (51.82, 37.97), (35.15, 54.83), (51.48, 69.35), (50.12, 52.76), (18.19, 48.53), (48.27, 58.72), (12.8, 62.22), (33.93, 69.83), (40.47, 57.98), (29.59, 73.07), (43.17, 75.96), (18.19, 46.97), (47.54, 30.57), (51.02, 70.83), (32.12, 42.16), (53.0, 77.28), (63.39, 72.08), (73.43, 63.03), (25.83, 49.18), (31.92, 51.55), (15.2, 58.28), (43.93, 62.96), (67.31, 73.86), (58.24, 58.58), (57.67, 73.63), (19.28, 47.89), (2.67, 43.94), (66.78, 55.88), (33.59, 49.5), (42.59, 65.11), (59.82, 63.64), (20.61, 59.81), (29.29, 53.42), (33.36, 78.84), (54.07, 78.17), (42.26, 41.82), (46.79, 78.56), (24.79, 55.13), (44.0, 55.17), (43.41, 47.46), (39.79, 59.35), (38.6, 48.49), (64.16, 47.2), (36.02, 41.7), (25.25, 73.06), (55.58, 60.12), (-3.07, 45.11), (27.28, 46.41), (50.23, 62.8), (37.49, 55.22), (26.93, 56.25), (36.84, 34.94), (46.63, 73.36), (31.6, 72.53), (38.96, 58.67), (45.75, 69.74), (48.43, 61.44), (70.36, 59.5), (46.05, 57.57), (53.5, 56.37), (35.87, 62.78), (14.31, 51.09), (44.14, 48.83), (25.51, 62.09), (12.45, 66.34), (60.08, 68.55), (32.34, 44.62), (58.15, 66.74), (20.64, 59.1), (48.45, 83.56), (29.97, 64.92), (70.94, 73.68), (60.23, 67.79), (39.84, 72.67), (63.46, 77.33), (20.18, 52.46), (65.14, 68.77), (54.89, 56.19), (67.53, 62.96), (38.95, 68.24), (13.42, 66.86), (35.48, 56.59), (21.68, 36.97), (33.09, 57.76), (75.88, 74.35), (54.67, 54.98), (17.37, 57.08), (10.05, 27.17), (32.06, 59.71), (50.48, 69.16), (41.08, 69.24), (35.98, 57.62), (36.35, 47.06), (38.82, 63.46), (56.48, 70.38), (51.3, 71.57), (66.19, 63.83), (53.6, 46.29), (24.21, 55.74), (45.11, 49.36), (40.83, 77.63), (36.59, 49.92), (11.93, 54.63), (44.37, 62.81), (26.48, 44.22), (24.55, 74.91), (41.61, 44.28), (38.1, 80.24), (3.93, 39.38), (77.23, 63.75), (25.67, 58.31), (42.01, 66.15), (18.41, 55.14), (33.34, 51.84), (69.65, 63.4), (40.02, 53.58), (25.81, 34.7), (56.85, 55.25), (25.75, 62.59), (39.95, 67.85), (64.15, 73.09), (41.92, 67.27), (68.71, 74.1), (36.31, 58.2), (24.6, 44.9), (36.38, 55.48), (49.51, 65.78), (22.16, 52.14), (45.35, 35.54), (27.6, 71.38), (62.07, 75.89), (51.14, 66.58), (28.03, 47.3), (68.97, 76.33), (52.33, 45.7), (33.78, 54.25), (48.23, 78.5), (58.06, 70.99), (32.21, 76.15), (36.36, 74.43), (46.72, 65.76), (40.82, 49.9), (24.51, 53.6), (12.93, 33.68), (28.95, 66.53), (31.97, 70.92), (40.02, 63.37), (51.27, 64.74), (29.93, 58.62), (22.91, 47.84), (14.42, 61.47), (44.8, 63.35), (61.68, 52.11), (29.61, 53.38), (29.91, 69.55), (44.7, 74.45), (31.89, 55.6), (49.2, 43.25), (57.49, 56.1), (41.91, 64.77), (22.38, 71.56), (45.12, 69.8), (55.16, 58.0), (59.71, 48.93), (34.1, 35.35), (41.4, 68.47), (40.96, 65.02), (68.27, 61.64), (51.5, 76.64), (45.79, 71.86), (30.27, 55.31), (69.97, 86.2), (47.95, 64.1), (41.26, 78.28), (47.24, 58.84), (48.87, 58.55), (28.93, 51.48), (56.92, 63.94), (49.4, 44.85), (52.89, 65.6), (42.71, 45.13), (36.52, 46.67), (52.94, 63.43), (39.67, 60.58), (49.84, 75.45), (21.75, 34.89), (49.99, 64.18), (40.29, 89.83), (31.71, 50.64), (48.92, 51.44), (31.85, 72.14), (36.28, 60.76), (33.69, 62.83), (26.81, 58.56), (55.21, 61.3), (56.29, 66.45), (42.87, 54.27), (36.36, 63.73), (45.32, 67.3), (58.68, 61.88), (32.7, 68.29), (56.36, 46.0), (34.79, 40.7), (42.51, 82.91), (38.27, 57.29), (15.98, 54.02), (64.76, 61.97), (42.84, 61.03), (47.94, 75.97), (35.51, 45.87), (39.91, 55.46), (16.46, 72.04), (31.84, 59.89), (12.58, 69.8), (42.59, 67.52), (31.76, 59.43), (20.05, 54.21), (30.46, 67.68), (34.84, 54.36), (27.21, 67.86), (42.08, 52.94), (23.15, 82.49), (10.62, 63.89), (52.49, 64.84), (23.28, 51.61), (39.82, 63.37), (54.36, 52.9), (54.79, 93.19), (30.25, 71.73), (60.83, 60.58), (48.72, 79.88), (29.38, 60.02), (18.33, 71.03), (20.69, 46.45), (15.97, 52.0), (33.34, 41.87), (29.1, 65.68), (44.7, 61.41), (22.14, 54.72), (15.93, 61.77), (28.43, 62.6), (24.57, 70.32), (44.73, 42.9), (19.19, 37.48), (59.49, 59.49), (47.05, 51.96), (21.51, 75.1), (22.04, 64.63), (49.96, 77.33), (36.0, 61.93), (39.99, 56.74), (73.26, 67.9), (53.3, 67.41), (44.8, 80.42), (12.45, 63.93), (4.51, 57.96), (60.5, 66.61), (37.31, 59.36), (35.61, 44.79), (23.75, 53.91), (32.55, 57.86), (63.26, 62.96), (59.52, 66.17), (34.4, 60.05), (57.24, 38.6), (20.51, 52.37), (63.89, 68.29), (50.66, 45.59), (47.16, 66.77), (43.47, 70.06), (37.14, 63.9), (49.34, 55.55), (44.48, 48.21), (45.61, 63.95), (44.12, 36.9), (17.78, 56.49), (40.14, 41.49), (45.58, 70.92), (52.82, 54.91), (29.0, 57.69), (8.75, 50.01), (33.1, 66.51), (35.5, 52.23), (49.39, 69.71), (70.85, 73.99), (61.68, 63.0), (43.47, 42.65), (29.82, 50.28), (52.59, 58.72), (64.08, 63.14), (55.3, 57.97), (33.5, 60.6), (38.88, 50.98), (42.95, 60.88), (18.08, 39.66), (42.0, 66.35), (50.67, 56.05), (51.05, 68.08), (38.86, 48.36), (32.51, 68.47), (26.3, 49.26), (30.31, 31.04), (10.22, 45.48), (32.85, 51.48), (32.01, 67.7), (61.06, 52.56), (42.21, 56.11), (14.64, 65.44), (44.94, 70.64), (48.58, 60.54), (33.6, 29.32), (31.79, 58.32), (41.76, 86.38), (17.43, 41.41), (51.46, 65.15), (61.86, 53.25), (29.5, 62.5), (54.19, 70.47), (26.97, 67.7), (49.3, 59.36), (33.1, 45.89), (40.73, 55.84), (62.21, 80.82), (25.38, 60.55), (41.64, 50.83), (55.76, 90.7), (26.65, 55.16), (41.72, 59.39), (38.7, 69.04), (26.84, 63.13), (50.9, 38.42), (58.34, 58.54), (34.89, 45.36), (47.75, 75.45), (-0.98, 60.23), (28.02, 55.44), (73.15, 75.05), (18.08, 41.93), (33.07, 50.43), (46.89, 50.03), (25.86, 59.85), (73.21, 64.85), (65.23, 73.9), (55.39, 65.94), (4.81, 52.27), (7.88, 34.04), (38.45, 70.31), (18.83, 84.47), (39.46, 74.73), (51.47, 62.41), (35.27, 70.33), (62.92, 66.67), (34.26, 63.32), (41.45, 56.34), (28.1, 56.89), (17.49, 81.18), (46.28, 66.28), (20.67, 67.38), (16.02, 65.0), (45.95, 53.24), (38.36, 98.49), (55.79, 64.42), (47.1, 39.02), (49.11, 60.92), (38.29, 56.95), (42.01, 66.69), (41.34, 58.55), (28.05, 48.3), (45.62, 63.71), (39.51, 67.87), (64.76, 71.06), (47.86, 61.79), (23.16, 46.99), (38.44, 50.49), (23.78, 55.58), (59.05, 77.03), (24.97, 74.04), (35.5, 62.98), (47.11, 65.28), (27.96, 57.85), (27.81, 48.29), (66.1, 72.31), (35.14, 51.08), (48.85, 38.83), (36.08, 68.85), (57.86, 73.36), (43.69, 53.37), (21.84, 59.78), (46.79, 55.61), (51.48, 89.02), (51.87, 67.19), (49.67, 58.79), (66.54, 67.55), (46.13, 58.46), (28.15, 80.89), (44.65, 81.87), (52.33, 77.26), (7.2, 48.92), (40.91, 52.33), (9.42, 57.64), (27.77, 55.57), (36.88, 37.28), (40.91, 47.93), (30.95, 65.67), (52.39, 61.83), (13.9, 52.07), (39.78, 62.49), (38.01, 61.83), (46.41, 77.05), (39.04, 59.34), (34.4, 50.56), (29.45, 51.0), (2.62, 52.51), (22.47, 35.07), (45.32, 61.55), (19.32, 67.56), (38.58, 82.77), (29.57, 62.91), (48.18, 58.66), (32.98, 57.32), (29.37, 53.16), (58.42, 66.4), (23.16, 46.09), (45.11, 59.28), (60.3, 70.51), (48.4, 67.26), (40.1, 56.08), (43.73, 83.19), (40.85, 56.24), (73.5, 78.67), (39.19, 44.53), (26.83, 58.55), (44.67, 66.7), (54.83, 62.83), (46.62, 65.82), (52.89, 36.78), (23.64, 63.15), (31.94, 62.94), (45.09, 60.67), (58.62, 52.75), (45.1, 63.06), (25.46, 54.01), (8.99, 83.33), (18.12, 60.71), (56.31, 57.51), (47.51, 60.23), (54.89, 44.36), (30.08, 70.73), (26.64, 73.4), (16.6, 36.67), (45.48, 60.19), (73.07, 62.1), (52.77, 66.7), (54.32, 83.96), (37.13, 49.84), (58.57, 68.15), (25.27, 64.14), (49.63, 47.85), (3.99, 44.52), (25.07, 60.71), (33.96, 81.67), (37.49, 50.03), (2.06, 54.81), (64.7, 38.37), (10.13, 62.18), (17.92, 62.38), (23.79, 73.83), (48.01, 48.14), (37.38, 78.46), (35.74, 66.29), (60.49, 55.69), (43.44, 64.3), (34.68, 73.27), (29.91, 46.02), (56.64, 48.09), (6.47, 60.69), (30.7, 66.88), (55.7, 51.49), (53.68, 68.5), (20.09, 47.36), (20.49, 67.47), (53.46, 32.99), (37.28, 71.43), (49.77, 37.31), (35.09, 59.76), (52.45, 52.34), (64.05, 60.23), (50.27, 32.55), (50.17, 58.39), (57.34, 66.95), (47.73, 49.35), (70.12, 64.96), (6.17, 54.31), (28.96, 50.52), (37.49, 41.7), (28.48, 62.92), (61.75, 67.26), (46.45, 73.83), (61.33, 55.69), (33.54, 71.46), (54.64, 63.69), (25.38, 45.86), (26.73, 61.19), (8.69, 54.38), (50.7, 73.61), (23.09, 62.22), (49.15, 61.51), (35.67, 51.55), (39.11, 51.5), (23.36, 58.76), (41.59, 68.45), (39.51, 75.21), (37.32, 44.04), (45.52, 65.56), (47.71, 67.27), (23.84, 27.38), (42.25, 55.17), (38.58, 49.06), (18.94, 48.38), (28.61, 51.12), (63.04, 73.45), (33.35, 43.76)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=694；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2643；
t = r·√((n−2)/(1−r²)) = 0.2643·√((694−2)/(1−0.0698)) = 7.2083；df=692；p = 1.90e-08。
95% CI（Fisher z）：tanh(z±26.184·1/√(n−3)) = [-0.6202, 0.8529]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=694；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2643；
t = r·√((n−2)/(1−r²)) = 0.2643·√((694−2)/(1−0.0698)) = 7.2083；df=692；p = 1.90e-08。
95% CI（Fisher z）：tanh(z±26.184·1/√(n−3)) = [-0.6202, 0.8529]。
③ 计算结果：
r = 0.2643，t = 7.2083，p = 1.90e-08，95%CI = [-0.6202, 0.8529]
④ 结论：在 α=0.05 水平下，p=1.90e-08 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.2643，方向正），弱相关。
案例 11. 屏幕使用时间(h)与视力(小数)
Pearson相关弱相关
① 原始数据：
样本 n=577，逐条 (x, y)：[(6.49, 0.87), (10.11, 0.8), (6.36, 1.06), (3.35, 0.69), (6.12, 1.01), (4.21, 1.15), (6.75, 1.15), (3.72, 0.94), (6.37, 1.27), (7.24, 1.14), (6.94, 0.99), (5.08, 0.84), (8.46, 0.79), (4.47, 0.97), (6.35, 0.97), (5.04, 0.8), (4.49, 1.1), (5.55, 1.0), (5.72, 1.35), (5.26, 1.08), (3.62, 1.01), (3.21, 0.92), (7.69, 0.75), (11.15, 1.03), (5.1, 1.44), (6.26, 1.15), (8.61, 1.08), (8.39, 1.05), (5.28, 0.86), (5.18, 1.26), (5.26, 1.24), (5.14, 1.0), (5.31, 1.19), (4.72, 0.78), (6.42, 1.09), (5.22, 0.72), (1.76, 1.03), (5.9, 0.95), (4.26, 1.29), (6.1, 0.79), (6.31, 1.36), (3.33, 0.99), (2.0, 0.9), (9.37, 0.76), (7.81, 0.82), (5.23, 0.77), (8.17, 0.79), (3.18, 1.26), (3.22, 0.99), (3.55, 0.92), (6.04, 0.74), (6.68, 0.94), (4.69, 1.03), (3.43, 0.8), (4.1, 0.86), (5.92, 0.82), (6.88, 0.76), (6.51, 0.8), (5.94, 1.07), (2.39, 0.99), (4.88, 0.92), (10.76, 1.22), (4.63, 1.2), (5.33, 1.13), (6.98, 0.84), (6.04, 1.02), (9.42, 0.83), (7.58, 0.91), (3.39, 1.27), (5.48, 1.08), (5.8, 0.87), (4.11, 1.03), (6.15, 1.05), (4.51, 1.14), (7.4, 1.05), (3.6, 0.83), (6.69, 1.1), (8.8, 1.15), (5.55, 0.98), (2.36, 0.91), (5.51, 0.72), (2.4, 0.73), (4.74, 1.09), (4.3, 0.86), (4.28, 0.95), (9.26, 1.12), (5.7, 0.7), (7.64, 0.94), (2.2, 0.87), (6.87, 0.81), (6.37, 1.19), (6.57, 0.98), (3.78, 0.76), (4.01, 1.18), (7.66, 0.36), (4.04, 1.25), (6.66, 1.06), (3.61, 0.72), (7.49, 0.93), (6.46, 1.0), (7.41, 0.91), (8.79, 1.01), (4.86, 0.96), (6.15, 0.82), (7.94, 1.23), (5.78, 0.87), (5.78, 0.87), (5.13, 1.3), (4.05, 0.69), (3.12, 0.91), (8.49, 0.84), (3.79, 0.71), (7.01, 1.01), (2.24, 0.93), (7.65, 1.08), (7.52, 0.92), (6.49, 0.82), (4.98, 0.95), (3.5, 0.8), (4.64, 1.03), (3.09, 0.89), (7.42, 1.08), (3.73, 0.75), (3.96, 0.86), (3.31, 0.92), (5.64, 1.14), (7.0, 0.98), (5.92, 0.99), (10.73, 1.1), (6.17, 1.21), (5.5, 1.06), (8.87, 0.88), (8.46, 1.41), (6.61, 1.32), (4.42, 1.24), (10.11, 1.13), (4.33, 0.73), (8.44, 0.79), (8.79, 0.65), (7.28, 1.33), (9.54, 1.42), (7.37, 0.82), (2.97, 0.66), (7.25, 1.33), (3.41, 1.04), (3.16, 1.38), (4.96, 0.8), (4.13, 0.91), (2.68, 1.0), (6.39, 1.08), (8.04, 0.98), (6.93, 1.21), (5.3, 0.86), (4.28, 0.79), (5.53, 0.81), (9.77, 1.03), (10.32, 1.26), (3.78, 1.09), (6.94, 1.49), (5.71, 1.16), (4.13, 0.89), (4.11, 1.17), (4.12, 0.79), (5.12, 0.75), (8.04, 1.06), (6.35, 1.04), (6.98, 0.93), (5.03, 1.15), (5.4, 0.62), (5.76, 0.68), (9.13, 0.92), (6.33, 1.12), (4.01, 1.44), (5.84, 1.07), (3.74, 1.23), (6.4, 1.15), (7.63, 0.92), (6.84, 0.96), (6.64, 1.13), (7.09, 0.89), (6.72, 1.14), (5.32, 1.13), (4.76, 1.1), (6.09, 0.96), (5.96, 1.0), (4.72, 0.84), (5.44, 1.24), (7.18, 1.24), (7.1, 0.83), (8.12, 1.38), (1.27, 1.11), (8.82, 1.05), (8.34, 1.1), (8.22, 1.39), (7.52, 1.22), (4.59, 0.96), (9.79, 1.05), (4.41, 0.93), (8.43, 1.17), (10.35, 1.05), (7.82, 0.78), (5.32, 0.84), (3.25, 0.9), (6.49, 1.04), (3.77, 0.93), (4.33, 0.89), (7.39, 0.88), (10.35, 1.05), (7.23, 0.84), (6.93, 0.83), (6.16, 1.14), (5.48, 0.97), (6.07, 1.12), (5.28, 0.84), (4.11, 1.22), (7.08, 0.99), (4.89, 0.86), (9.6, 0.96), (8.58, 0.73), (5.56, 1.12), (7.15, 1.07), (7.08, 0.98), (6.83, 1.26), (5.89, 1.3), (6.8, 0.99), (8.55, 1.28), (6.42, 1.09), (2.63, 0.69), (5.62, 1.06), (5.1, 0.98), (5.56, 1.03), (3.64, 0.84), (8.61, 1.19), (9.33, 1.32), (4.62, 1.35), (5.15, 1.05), (6.9, 1.36), (4.16, 0.93), (8.88, 0.93), (6.48, 1.33), (2.5, 0.76), (5.96, 0.99), (5.66, 1.14), (6.86, 0.89), (8.97, 1.35), (6.6, 1.28), (8.72, 1.08), (3.42, 0.94), (4.67, 0.69), (6.75, 0.95), (7.14, 0.85), (7.89, 0.99), (5.28, 0.99), (9.16, 1.08), (6.03, 1.22), (7.69, 1.45), (6.29, 0.97), (4.34, 0.78), (5.68, 1.27), (8.16, 1.12), (4.53, 0.56), (5.24, 1.39), (3.7, 0.74), (9.63, 0.86), (1.87, 1.19), (6.84, 0.93), (10.15, 1.0), (6.95, 1.12), (8.6, 0.74), (6.78, 1.24), (9.09, 1.13), (6.65, 1.18), (7.04, 0.98), (5.09, 1.02), (7.17, 1.3), (5.96, 1.23), (5.28, 0.38), (3.85, 0.91), (7.58, 1.26), (6.1, 1.02), (7.03, 1.1), (6.44, 0.75), (5.71, 0.92), (5.41, 1.0), (8.51, 0.97), (4.55, 1.24), (4.43, 1.03), (6.26, 1.15), (6.78, 1.37), (5.26, 1.11), (6.49, 0.64), (7.64, 0.97), (5.52, 1.14), (5.07, 1.05), (7.64, 1.0), (5.15, 0.91), (5.5, 1.09), (4.71, 1.04), (5.04, 1.11), (5.0, 1.1), (5.69, 1.15), (6.21, 1.06), (9.43, 1.01), (6.6, 0.43), (5.0, 1.12), (4.26, 1.1), (4.89, 1.11), (6.49, 1.18), (4.52, 1.0), (7.3, 1.23), (6.96, 1.29), (1.62, 1.11), (6.35, 0.88), (5.22, 0.91), (5.43, 0.76), (9.0, 1.07), (6.73, 1.02), (9.24, 1.14), (5.73, 1.24), (5.47, 0.95), (10.62, 1.09), (7.11, 1.03), (4.52, 1.42), (6.2, 0.86), (3.88, 0.98), (6.28, 1.24), (7.31, 0.84), (6.16, 1.03), (5.28, 0.86), (6.54, 0.7), (10.9, 1.17), (4.55, 0.96), (7.31, 1.05), (4.41, 1.07), (5.29, 1.07), (8.58, 1.13), (4.11, 1.05), (2.5, 0.85), (8.06, 1.04), (3.77, 0.82), (8.31, 1.28), (6.75, 0.8), (3.27, 0.95), (5.48, 1.24), (6.7, 1.12), (6.07, 0.94), (5.22, 1.1), (6.25, 0.85), (4.76, 1.36), (3.8, 1.12), (6.15, 1.07), (5.98, 0.74), (6.43, 0.9), (7.53, 0.63), (9.06, 1.12), (9.03, 0.89), (6.19, 0.79), (8.46, 0.76), (6.07, 1.21), (6.05, 1.08), (4.26, 0.92), (6.38, 1.03), (7.08, 0.98), (5.83, 1.2), (3.91, 0.97), (2.33, 1.09), (7.25, 1.15), (6.74, 0.91), (4.31, 0.8), (6.8, 0.91), (8.05, 1.43), (7.71, 1.0), (7.77, 1.15), (3.12, 0.87), (5.18, 1.58), (4.19, 0.96), (9.4, 1.21), (3.26, 0.71), (7.14, 1.09), (5.84, 0.91), (8.11, 1.07), (7.19, 1.02), (6.04, 1.02), (8.96, 0.79), (8.18, 0.67), (8.03, 1.27), (2.96, 0.92), (7.2, 0.85), (4.96, 0.8), (1.64, 0.64), (7.09, 1.24), (6.77, 0.9), (10.49, 1.07), (4.63, 0.47), (8.19, 0.76), (5.84, 1.06), (4.43, 0.88), (5.37, 1.06), (5.28, 0.59), (8.78, 1.42), (10.18, 1.19), (5.0, 1.19), (7.47, 1.07), (3.35, 0.97), (6.83, 1.02), (4.37, 0.95), (4.82, 1.24), (8.13, 1.46), (4.43, 1.3), (7.5, 0.49), (7.71, 0.95), (10.77, 1.02), (7.09, 1.19), (7.07, 0.97), (7.8, 1.07), (5.72, 0.91), (2.73, 1.0), (6.28, 0.92), (2.14, 1.0), (8.02, 0.81), (6.85, 1.37), (5.35, 1.28), (8.48, 1.18), (8.52, 1.27), (7.85, 1.0), (7.75, 1.46), (5.37, 0.77), (5.25, 1.24), (7.37, 1.15), (6.56, 1.08), (5.44, 1.09), (5.5, 0.91), (8.39, 1.08), (4.95, 1.36), (2.74, 0.93), (7.65, 1.12), (5.45, 1.21), (7.74, 0.61), (7.53, 0.67), (8.31, 1.38), (7.65, 1.26), (5.32, 1.33), (6.4, 1.03), (8.36, 1.23), (9.42, 0.99), (3.16, 1.19), (5.6, 1.03), (6.26, 1.02), (5.27, 1.06), (5.42, 0.89), (3.03, 1.02), (6.13, 0.92), (6.04, 1.21), (4.49, 1.08), (4.23, 0.78), (7.99, 1.21), (6.19, 0.89), (3.13, 1.3), (8.67, 1.15), (6.18, 0.8), (6.1, 1.49), (7.93, 1.21), (5.6, 1.26), (8.33, 1.24), (5.74, 1.16), (6.27, 0.85), (2.78, 1.06), (10.24, 0.85), (5.33, 0.89), (7.16, 0.43), (5.62, 1.21), (5.41, 0.98), (3.32, 0.95), (6.95, 1.21), (5.63, 0.88), (0.23, 0.83), (8.41, 0.77), (7.2, 0.75), (5.95, 0.86), (7.59, 1.12), (5.51, 1.32), (3.51, 0.67), (7.14, 1.29), (4.39, 1.09), (3.81, 0.96), (5.31, 0.8), (6.69, 1.27), (4.0, 0.83), (5.39, 1.21), (4.55, 0.87), (3.87, 0.5), (4.36, 0.77), (5.72, 1.08), (3.58, 0.79), (6.58, 1.14), (3.96, 0.89), (5.08, 0.93), (7.7, 1.29), (9.46, 1.27), (6.17, 0.89), (7.06, 1.01), (5.55, 1.03), (10.43, 1.04), (5.29, 0.91), (6.76, 1.08), (7.56, 1.25), (6.56, 1.25), (7.84, 1.04), (4.89, 1.46), (5.94, 1.06), (2.45, 1.08), (7.92, 0.96), (6.51, 0.7), (8.25, 1.34), (8.02, 1.38), (6.74, 0.7), (8.46, 0.31), (7.34, 1.02), (3.33, 0.85), (5.25, 1.31), (6.1, 0.87), (1.88, 0.78), (8.71, 0.74), (6.78, 1.03), (6.58, 1.08), (7.04, 0.7), (5.48, 0.85), (6.37, 1.04), (4.42, 1.08), (2.54, 0.85), (6.7, 1.0), (2.78, 0.84), (4.82, 0.89), (5.84, 0.96), (6.32, 0.9), (4.45, 1.15), (5.95, 1.06), (7.81, 1.39), (10.42, 1.18), (7.65, 0.79), (4.82, 0.65), (6.14, 1.0), (6.37, 1.14), (4.15, 0.81), (7.51, 1.16), (5.59, 1.02), (4.01, 0.82), (8.19, 0.9), (3.52, 1.34), (2.56, 1.23), (5.85, 1.0), (4.36, 1.01), (8.17, 1.0), (7.84, 1.29), (3.39, 0.76), (7.94, 1.06), (8.65, 1.02), (8.52, 1.13), (6.71, 0.93), (8.9, 1.22), (8.11, 0.98), (6.14, 0.71), (8.14, 0.79), (2.56, 0.83), (7.13, 0.92), (6.89, 1.15), (5.92, 1.0), (5.12, 1.25), (2.72, 1.04), (9.04, 1.23), (3.41, 1.01), (5.27, 1.02), (10.21, 1.3), (5.34, 1.12), (7.57, 1.37), (6.77, 0.98), (7.62, 1.37)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=577；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.1685；
t = r·√((n−2)/(1−r²)) = 0.1685·√((577−2)/(1−0.0284)) = 4.0998；df=575；p = 4.73e-05。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [0.0879, 0.2469]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=577；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.1685；
t = r·√((n−2)/(1−r²)) = 0.1685·√((577−2)/(1−0.0284)) = 4.0998；df=575；p = 4.73e-05。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [0.0879, 0.2469]。
③ 计算结果：
r = 0.1685，t = 4.0998，p = 4.73e-05，95%CI = [0.0879, 0.2469]
④ 结论：在 α=0.05 水平下，p=4.73e-05 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.1685，方向正），弱相关。
案例 12. 步行步数(千)与卡路里消耗(kcal)
Pearson相关弱正相关
① 原始数据：
样本 n=952，逐条 (x, y)：[(4.49, 1453.6), (7.11, 3110.51), (10.29, 1270.57), (5.58, 2837.14), (5.57, 1884.55), (12.59, 1850.68), (7.22, 1916.09), (3.26, 2404.92), (9.0, 2190.96), (15.02, 2333.48), (5.13, 2286.87), (6.4, 1552.99), (7.02, 2536.58), (12.54, 3292.96), (8.0, 2710.98), (3.99, 2293.0), (9.21, 1556.82), (8.26, 1697.53), (8.13, 2407.29), (5.89, 1933.55), (8.33, 2824.81), (5.96, 2213.25), (9.25, 2114.85), (7.0, 1905.47), (13.67, 2074.36), (-0.28, 1628.74), (5.56, 2873.49), (8.8, 2377.43), (11.44, 2739.34), (15.21, 3218.41), (3.79, 1173.59), (5.17, 2189.63), (11.5, 3032.53), (10.64, 2693.55), (9.65, 2339.23), (11.74, 2633.81), (3.08, 1167.7), (6.28, 2673.48), (5.89, 2003.12), (12.34, 2171.39), (8.92, 2744.16), (8.43, 1972.13), (-1.4, 1890.27), (9.57, 2375.96), (9.9, 2218.31), (10.27, 1045.6), (5.39, 1467.19), (7.51, 2178.02), (14.13, 2657.31), (9.17, 2145.0), (11.36, 2917.54), (6.45, 2683.57), (6.98, 1247.75), (5.42, 1836.39), (8.62, 2722.51), (8.54, 2502.61), (7.48, 2319.68), (8.07, 2562.4), (-3.76, 1080.21), (10.15, 3177.81), (9.81, 2511.49), (6.72, 2202.76), (7.41, 2266.29), (12.13, 1529.37), (10.52, 1692.95), (8.47, 1493.72), (8.08, 2164.32), (3.62, 1691.72), (11.99, 3024.7), (9.66, 2845.4), (7.43, 2047.01), (13.81, 2115.91), (8.4, 2051.8), (9.56, 2057.53), (9.61, 2119.55), (8.76, 1440.46), (9.54, 2783.63), (10.89, 1823.55), (7.56, 3078.12), (4.59, 2157.51), (3.83, 2308.82), (9.91, 1732.31), (9.91, 1954.98), (6.33, 2294.21), (7.18, 1152.07), (9.82, 2169.83), (8.25, 1982.93), (3.68, 1049.78), (6.91, 2190.45), (5.88, 1490.65), (10.08, 2398.95), (11.77, 2685.67), (7.87, 2206.89), (6.32, 1917.02), (7.52, 2135.49), (9.76, 1217.22), (12.16, 2176.84), (9.52, 2085.04), (7.65, 2595.88), (4.53, 2029.78), (10.09, 2246.86), (10.43, 2384.98), (5.8, 1561.12), (6.5, 2500.27), (8.69, 2485.78), (4.71, 1886.9), (10.05, 1927.44), (6.43, 2375.63), (10.54, 1927.03), (6.95, 2150.3), (13.43, 2685.92), (8.8, 2164.49), (8.8, 2496.5), (7.55, 2744.24), (8.05, 2561.47), (2.87, 1829.75), (-0.68, 2097.33), (7.44, 1975.45), (3.91, 2078.18), (8.74, 1949.78), (9.8, 2429.05), (7.35, 2739.7), (10.97, 2569.97), (9.19, 1778.53), (12.09, 2613.01), (6.55, 2879.92), (14.2, 3303.63), (5.76, 2429.66), (9.34, 2546.01), (7.14, 2199.75), (10.33, 2488.5), (-2.8, 1373.29), (3.23, 2462.18), (5.11, 1291.61), (7.99, 2783.44), (10.56, 2816.75), (10.34, 2829.28), (7.31, 2136.93), (3.13, 1350.55), (5.92, 1936.03), (12.3, 2103.76), (7.39, 2813.06), (8.87, 2518.33), (6.52, 2153.66), (6.31, 2438.69), (7.69, 1990.56), (7.55, 1909.65), (8.13, 1795.19), (7.05, 2225.33), (6.14, 1292.42), (6.41, 2543.04), (5.28, 2162.34), (14.13, 2393.91), (10.45, 1865.84), (6.46, 2561.93), (9.61, 2736.31), (3.54, 2374.3), (12.53, 2182.56), (9.31, 2127.45), (10.34, 2022.55), (10.87, 2451.48), (9.69, 2705.21), (4.45, 2146.47), (2.17, 1868.85), (11.08, 2968.77), (6.24, 1789.96), (10.01, 1523.66), (7.71, 2304.62), (8.45, 1766.29), (7.21, 2078.63), (7.5, 2356.9), (10.37, 2968.34), (4.06, 1370.46), (4.14, 2283.98), (7.64, 1814.01), (10.6, 1917.38), (9.67, 2925.63), (7.86, 2302.65), (15.2, 2477.19), (7.76, 2693.67), (8.5, 2346.22), (5.56, 2011.0), (2.95, 2031.72), (5.37, 1579.94), (9.56, 1646.92), (10.11, 2244.12), (10.11, 1770.13), (9.87, 2661.13), (10.52, 2357.4), (12.23, 2316.23), (11.65, 2022.9), (7.6, 2185.47), (10.82, 2009.95), (7.75, 1853.52), (8.61, 2090.54), (4.82, 3034.72), (11.12, 2610.03), (13.95, 1841.04), (3.41, 1334.08), (0.9, 2679.73), (11.35, 1988.72), (9.33, 2431.24), (1.05, 1546.02), (4.9, 1629.54), (3.46, 2479.7), (8.64, 2999.19), (7.85, 2154.54), (9.44, 2238.5), (4.85, 1821.31), (15.0, 2227.11), (3.79, 2943.31), (9.88, 1962.17), (14.27, 2012.9), (9.02, 1841.24), (5.23, 2288.84), (4.92, 1328.19), (9.41, 1461.54), (6.73, 1461.31), (7.22, 2357.85), (6.81, 1820.32), (4.92, 2058.46), (8.37, 1679.17), (10.09, 2172.67), (12.82, 3264.29), (6.83, 2338.04), (7.08, 2130.84), (5.66, 1691.01), (7.92, 2614.42), (4.09, 2120.89), (10.84, 2761.48), (6.59, 1688.37), (7.75, 1750.1), (8.39, 2653.52), (9.52, 2272.1), (8.82, 2424.6), (10.38, 2489.3), (7.87, 2365.24), (10.81, 2944.27), (10.06, 2176.19), (9.51, 2344.77), (8.38, 2368.04), (11.6, 2767.89), (14.07, 2430.1), (4.77, 2059.46), (8.8, 1412.94), (12.12, 2278.16), (14.04, 2236.49), (6.22, 1719.2), (10.72, 3069.29), (11.85, 2764.99), (3.68, 1464.94), (9.44, 1807.16), (4.04, 2193.24), (7.15, 2264.92), (6.44, 1533.48), (9.82, 1519.62), (4.4, 1376.84), (4.05, 2029.56), (7.96, 2369.36), (7.97, 2045.37), (10.32, 3470.44), (7.67, 2689.07), (5.73, 2687.15), (12.05, 1524.97), (13.0, 1680.32), (7.09, 2747.99), (9.97, 3055.49), (11.32, 3707.05), (6.59, 1222.92), (9.46, 2402.51), (7.34, 1882.41), (6.14, 2259.57), (4.48, 1920.3), (11.04, 2235.31), (3.79, 1991.77), (2.85, 2036.11), (10.98, 1793.52), (7.52, 2353.28), (5.34, 1940.12), (7.1, 2000.95), (5.09, 1092.03), (7.33, 2439.89), (7.88, 2296.34), (8.14, 2628.97), (6.07, 1732.89), (6.46, 2465.42), (10.11, 2120.15), (-0.66, 1789.32), (6.54, 1387.93), (6.7, 2121.5), (10.47, 2161.18), (6.11, 1415.45), (6.35, 2418.59), (6.73, 2171.93), (5.2, 2596.34), (8.25, 2282.64), (6.77, 2276.59), (5.01, 1324.9), (7.53, 2481.71), (6.85, 1667.12), (4.53, 2887.18), (9.64, 2235.39), (12.93, 2432.7), (7.55, 1981.42), (8.8, 2516.05), (5.65, 1574.76), (8.12, 2058.82), (7.83, 2310.44), (13.99, 3332.43), (3.39, 1790.77), (12.1, 1650.01), (13.59, 2202.73), (11.18, 2050.36), (9.74, 1945.41), (7.28, 2609.01), (12.59, 2321.27), (8.71, 2245.99), (7.73, 2375.59), (5.61, 1686.76), (6.72, 2920.89), (6.48, 1915.32), (8.72, 2608.96), (11.65, 2484.69), (7.7, 2656.2), (8.34, 1966.75), (6.27, 2328.12), (8.5, 2264.5), (7.47, 2038.64), (10.32, 2574.02), (10.24, 1715.8), (7.7, 2301.5), (6.67, 1797.51), (3.37, 1901.84), (10.26, 1981.87), (11.78, 2529.46), (6.71, 1884.31), (6.5, 3312.81), (8.63, 2485.64), (3.02, 1132.17), (6.47, 1951.13), (12.55, 2603.76), (5.08, 2325.75), (13.0, 2179.11), (10.18, 1455.35), (8.7, 1460.25), (4.44, 2862.4), (4.14, 2310.81), (9.37, 1691.86), (8.4, 2864.09), (8.45, 2980.51), (7.76, 2861.41), (6.02, 2049.43), (4.55, 2107.69), (10.8, 2362.27), (7.71, 1779.72), (7.15, 1862.93), (1.31, 1376.36), (10.76, 2425.21), (7.45, 2573.16), (6.39, 2027.44), (8.15, 2131.41), (6.99, 2753.5), (12.76, 1779.78), (5.43, 2697.91), (9.52, 3431.16), (7.68, 1756.15), (10.54, 2891.34), (5.15, 1259.94), (2.77, 2351.05), (5.4, 3376.26), (6.6, 1934.2), (7.79, 2157.18), (11.79, 2478.9), (9.93, 1998.39), (8.47, 2129.36), (5.72, 2838.31), (8.23, 2104.19), (5.97, 2270.35), (9.16, 1622.56), (8.99, 2155.8), (6.51, 2157.77), (11.93, 2660.64), (4.15, 1552.58), (11.15, 1887.79), (8.3, 2556.2), (3.57, 1750.19), (3.43, 2365.62), (4.17, 1993.39), (10.06, 2017.5), (12.18, 2670.07), (7.62, 2066.77), (9.2, 2511.39), (3.99, 2548.83), (13.67, 2821.87), (1.27, 2525.84), (5.4, 1591.57), (7.92, 2353.44), (10.32, 2356.72), (10.0, 1775.89), (7.29, 2236.12), (4.28, 1741.61), (5.91, 1806.45), (8.5, 1515.9), (5.58, 1589.02), (6.51, 1729.44), (5.8, 1918.41), (11.39, 2849.22), (11.64, 2442.2), (13.18, 2218.76), (16.23, 3002.22), (10.09, 3188.2), (9.29, 2588.96), (7.55, 1950.83), (6.15, 2068.0), (11.08, 1593.42), (8.74, 2427.48), (8.09, 1938.9), (9.08, 2999.67), (8.55, 1421.92), (8.14, 2801.27), (10.07, 2286.41), (9.31, 1858.82), (17.12, 3362.86), (7.55, 1708.87), (7.82, 2030.37), (7.22, 2126.01), (8.76, 2459.03), (8.13, 1816.28), (7.41, 2070.6), (10.03, 2183.23), (10.93, 1645.53), (5.03, 2649.26), (9.25, 1177.85), (5.7, 1614.39), (3.99, 1813.92), (12.03, 1678.65), (2.07, 2940.56), (9.31, 2029.0), (5.18, 1442.55), (9.16, 2941.65), (0.48, 1686.24), (9.89, 2340.2), (13.17, 2353.17), (6.91, 1540.12), (7.97, 1454.81), (8.68, 2343.03), (10.52, 2051.47), (9.88, 2074.33), (8.77, 2575.39), (14.78, 2368.54), (6.58, 1649.03), (9.46, 2803.98), (8.82, 2100.46), (14.85, 2305.06), (8.33, 2394.54), (7.7, 1887.48), (11.29, 2273.98), (7.43, 1331.98), (9.58, 1212.09), (9.1, 2337.04), (9.5, 2080.31), (11.04, 2790.23), (7.67, 2638.32), (4.62, 1617.04), (10.71, 3048.27), (12.88, 2731.49), (11.45, 3614.47), (5.25, 1451.84), (5.74, 1743.44), (1.34, 1687.53), (5.63, 2613.29), (5.02, 2182.37), (7.13, 1948.44), (3.49, 1829.17), (17.07, 3067.64), (7.83, 2070.71), (5.44, 2043.43), (4.75, 1866.95), (6.17, 2169.95), (9.83, 2154.67), (0.77, 1671.79), (8.23, 2068.3), (10.58, 3191.8), (5.23, 2491.67), (10.31, 2377.94), (8.62, 2043.94), (9.79, 1907.71), (4.37, 1943.39), (4.6, 1719.01), (10.13, 2205.43), (10.28, 3388.02), (11.12, 2779.67), (7.93, 2301.19), (3.84, 2341.08), (2.58, 2216.31), (8.31, 2341.29), (6.47, 1938.58), (4.09, 1554.99), (7.48, 2693.44), (5.14, 1537.77), (4.47, 2502.81), (6.82, 2488.25), (7.54, 1859.08), (10.61, 2456.75), (10.02, 2106.09), (12.05, 2694.46), (11.57, 2709.72), (4.09, 2434.76), (8.45, 3106.12), (9.65, 2206.41), (2.83, 1747.52), (12.56, 2137.22), (8.54, 2202.18), (8.64, 1830.84), (8.39, 3351.29), (8.07, 2972.13), (6.56, 2575.93), (7.4, 1730.84), (10.18, 2321.87), (3.45, 1924.72), (7.89, 1834.25), (7.91, 1747.98), (13.95, 1781.77), (11.76, 2902.6), (8.51, 2006.67), (5.8, 1967.27), (7.62, 2748.63), (6.47, 2344.77), (6.84, 1601.96), (14.13, 2472.0), (7.95, 2391.85), (6.47, 1914.83), (5.49, 1790.15), (7.5, 2353.95), (12.48, 1856.69), (7.24, 2384.25), (6.59, 1935.48), (6.6, 2034.89), (6.21, 2698.28), (4.42, 1435.61), (9.64, 1279.43), (10.82, 2770.08), (8.5, 2303.09), (14.52, 1788.75), (9.5, 2328.9), (11.02, 1806.07), (6.06, 2749.7), (9.73, 1552.49), (10.81, 2547.73), (10.7, 1763.72), (7.49, 2079.48), (8.46, 1910.75), (7.96, 2374.7), (4.8, 1664.51), (6.58, 2849.56), (8.33, 2095.14), (0.38, 2018.0), (5.6, 2104.42), (11.62, 1806.38), (10.04, 2351.85), (14.42, 2421.75), (7.88, 1537.24), (11.52, 3088.31), (11.78, 2526.43), (8.55, 1676.22), (10.11, 2541.95), (9.6, 2848.83), (6.45, 2025.93), (8.86, 2583.71), (9.2, 2084.31), (10.7, 1708.74), (8.33, 2688.72), (12.35, 2158.51), (0.42, 1541.48), (9.47, 2817.0), (6.6, 1517.89), (6.14, 1841.72), (7.68, 1971.41), (6.33, 2282.89), (13.95, 2227.93), (3.23, 1405.11), (7.83, 2240.18), (5.32, 1242.99), (10.64, 1912.0), (9.39, 1624.65), (8.33, 1926.76), (14.38, 2600.68), (6.68, 2313.28), (4.85, 1652.51), (5.57, 2030.12), (10.25, 2614.33), (5.86, 644.65), (8.83, 2402.94), (2.95, 1941.07), (10.07, 2521.1), (14.2, 1854.51), (5.98, 1103.07), (13.25, 2598.28), (9.72, 2287.45), (8.15, 2743.73), (6.15, 2020.99), (5.78, 1454.4), (12.34, 1792.23), (5.93, 2326.7), (8.88, 1716.31), (10.11, 1729.17), (7.05, 2611.34), (8.6, 1293.11), (10.79, 2872.74), (12.08, 3085.94), (13.1, 2350.7), (7.14, 2191.5), (4.11, 1691.8), (8.59, 2200.65), (13.85, 2627.11), (13.7, 1858.19), (10.73, 1471.85), (7.73, 1720.36), (10.09, 2117.67), (11.37, 2528.29), (6.44, 1804.05), (11.02, 2076.84), (6.79, 1945.97), (6.67, 2879.4), (3.37, 2206.3), (10.62, 1797.03), (2.25, 2497.74), (8.6, 3072.8), (9.97, 1827.49), (3.89, 1517.75), (9.07, 3365.9), (8.52, 2359.93), (10.06, 1909.18), (6.17, 1625.72), (9.66, 2629.26), (8.17, 2885.18), (7.71, 3191.92), (8.81, 2579.56), (4.64, 2146.6), (6.95, 2419.66), (4.52, 1622.69), (1.32, 2518.81), (3.94, 1435.39), (8.28, 2408.9), (6.86, 2207.49), (6.1, 2140.59), (6.64, 2420.12), (11.13, 1832.24), (9.94, 1972.74), (7.6, 2827.74), (6.39, 2589.27), (8.99, 2220.64), (9.46, 2350.75), (10.1, 2159.71), (1.41, 1924.24), (9.13, 2993.18), (12.43, 2107.62), (5.6, 2798.91), (6.76, 2170.73), (8.58, 2045.71), (11.46, 1625.78), (8.03, 2148.25), (7.19, 1592.38), (7.97, 2982.77), (11.03, 1761.4), (10.68, 3104.2), (6.79, 2543.29), (5.03, 1558.25), (2.38, 1209.85), (8.09, 2617.85), (5.51, 1911.49), (9.88, 2456.84), (12.75, 2153.84), (10.47, 2273.85), (6.35, 1053.88), (5.61, 1864.98), (9.51, 2741.6), (13.9, 2443.79), (11.56, 3504.28), (9.56, 2169.14), (5.37, 1424.88), (9.83, 3115.82), (14.1, 2848.82), (7.19, 2152.66), (5.89, 1362.44), (13.86, 2756.49), (14.52, 2148.36), (3.15, 1653.72), (8.67, 2592.3), (8.93, 1653.93), (5.15, 2678.54), (11.3, 2087.46), (6.0, 1340.91), (7.13, 2408.79), (9.24, 1210.52), (15.15, 2686.8), (4.43, 1784.92), (6.57, 1604.33), (11.61, 2480.13), (7.92, 1722.37), (5.09, 2985.95), (7.58, 2220.42), (9.29, 2507.6), (9.5, 2078.53), (8.18, 2018.32), (8.07, 2097.75), (8.63, 1873.28), (8.34, 1921.95), (9.25, 2905.49), (6.77, 2485.7), (6.17, 2066.67), (10.45, 2911.86), (8.16, 1984.09), (4.03, 2020.07), (9.68, 1740.53), (13.06, 1743.6), (5.45, 1960.38), (9.46, 2794.96), (12.04, 2146.68), (4.4, 2025.69), (12.55, 2318.81), (7.8, 1829.15), (4.0, 2197.81), (10.93, 2473.8), (10.42, 1469.52), (8.95, 2275.91), (5.25, 2550.54), (7.56, 2571.39), (13.58, 2425.89), (12.02, 1687.65), (3.45, 2422.66), (9.76, 2530.53), (9.17, 3462.35), (4.97, 1884.16), (9.56, 2830.51), (13.49, 2456.07), (5.32, 2325.84), (5.55, 1885.96), (5.0, 2420.94), (9.42, 3444.61), (9.16, 1872.74), (5.8, 1444.12), (12.22, 2522.4), (4.88, 2622.7), (8.5, 2395.53), (7.87, 2583.72), (8.81, 2111.03), (3.27, 2406.26), (6.04, 1702.03), (1.74, 2874.37), (4.79, 2833.14), (4.3, 1929.86), (2.28, 2478.68), (8.16, 2492.02), (6.43, 1856.28), (10.17, 2611.01), (6.49, 2354.51), (7.77, 2510.36), (4.87, 2598.58), (8.14, 1992.54), (6.35, 1742.15), (11.43, 2935.87), (4.08, 1324.82), (9.02, 3207.64), (4.06, 1188.72), (5.76, 2212.07), (12.16, 1714.22), (5.38, 2667.31), (8.54, 2861.34), (8.05, 2410.13), (5.6, 2521.09), (4.34, 1713.54), (6.91, 1946.04), (5.26, 1287.61), (9.47, 1865.43), (9.04, 1675.04), (6.07, 2052.92), (4.42, 2172.57), (10.54, 2051.22), (9.45, 2380.37), (12.16, 1453.16), (7.2, 2182.4), (9.47, 2474.68), (8.61, 1900.24), (12.55, 3243.32), (11.14, 2695.53), (-0.84, 1398.98), (3.36, 2043.12), (9.09, 2149.77), (8.07, 2222.84), (8.03, 2196.9), (3.88, 1687.59), (3.17, 2199.61), (9.99, 2616.53), (9.17, 2612.02), (5.84, 1922.64), (8.14, 2346.85), (6.14, 1991.38), (13.14, 2486.86), (9.52, 2599.26), (9.55, 1280.68), (11.99, 2756.9), (7.4, 2295.48), (6.7, 1760.79), (6.89, 1351.99), (13.92, 2415.05), (12.87, 2245.83), (12.9, 2965.49), (10.72, 2157.58), (10.7, 1908.09), (7.97, 3265.23), (6.71, 2059.01), (7.13, 1517.69), (9.68, 2315.87), (5.86, 2338.52), (11.13, 2960.54), (12.41, 2438.92), (2.93, 2502.08), (8.23, 2201.91), (8.1, 2495.17), (6.92, 2399.0), (9.64, 1771.6), (12.31, 2327.87), (4.62, 1605.54), (8.03, 2816.1), (5.3, 1751.17), (3.82, 1493.33), (7.95, 2628.45), (8.51, 2227.98), (8.96, 2829.42), (7.13, 3080.41), (11.54, 1461.98), (0.52, 2275.34), (1.53, 1606.51), (5.14, 2262.62), (10.45, 2390.82), (12.49, 1924.28), (8.97, 2071.69), (6.6, 1786.98), (8.08, 1661.4), (5.29, 2410.76), (8.96, 1972.54), (8.46, 1514.35), (6.68, 1153.36), (7.3, 2251.0), (11.58, 2770.3), (11.14, 2665.38), (7.74, 1993.07), (3.46, 1874.6), (4.65, 2697.83), (6.39, 1942.01), (10.38, 3048.27), (10.72, 2821.09), (10.17, 2252.7), (5.52, 1522.54), (15.97, 2925.41), (13.27, 2384.89), (7.62, 2234.0), (6.9, 2676.14), (11.35, 2304.66), (9.53, 1416.22), (4.57, 2031.56), (8.66, 2395.03), (6.97, 2434.77), (7.97, 2534.07), (5.65, 2662.47), (8.17, 1508.45), (7.7, 2018.53), (6.76, 1781.54), (4.65, 2422.22), (12.6, 2734.71), (7.59, 1567.0), (5.15, 1370.15), (9.73, 1670.93), (5.14, 2105.25), (4.32, 2410.93), (8.3, 1550.58), (6.48, 2367.75), (5.23, 1211.05), (6.17, 1729.43), (4.41, 2059.07), (5.68, 2078.47), (10.81, 2319.88), (8.61, 2609.89), (6.82, 2584.33), (4.96, 1839.59), (12.75, 2398.49), (11.55, 1871.13), (4.67, 2459.22), (8.72, 2020.96), (6.8, 2341.14), (4.36, 2740.92), (7.01, 3143.49), (2.66, 1002.09), (5.63, 1943.66), (5.23, 2265.07), (10.58, 3113.27), (7.96, 2600.48), (10.46, 2052.79), (7.24, 1807.46), (3.08, 2235.45), (8.77, 2471.84), (3.96, 1433.5), (8.66, 2336.38), (10.26, 2943.49), (11.39, 2824.39), (10.78, 2018.33), (10.07, 2433.05), (3.29, 1801.03), (6.56, 2156.87), (8.29, 2730.76), (7.32, 999.69), (10.02, 2027.38), (5.89, 2483.19), (7.75, 2800.96), (3.8, 3031.84), (8.33, 3045.79), (10.2, 2769.14), (6.22, 2121.97), (9.0, 2893.64), (8.17, 2455.34), (7.45, 2001.32), (11.52, 1667.45), (5.26, 1803.58), (8.17, 2186.53), (7.38, 1751.88), (5.04, 2011.94), (8.32, 1892.51), (4.67, 1914.97), (8.84, 3017.97), (9.46, 2075.76), (5.05, 1811.96), (7.55, 1600.18), (9.38, 1815.95), (8.47, 1799.18), (12.67, 2577.75), (8.59, 1932.36), (8.94, 1710.09), (9.8, 1696.41), (11.31, 3031.52), (12.41, 2503.24), (10.48, 2230.16), (7.59, 2296.56), (7.88, 2167.6), (5.23, 1814.43), (9.32, 1327.81), (8.85, 1902.63), (9.47, 1657.82), (2.51, 2167.33), (8.26, 2318.53), (6.88, 2327.75), (5.86, 2177.15)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=952；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.3060；
t = r·√((n−2)/(1−r²)) = 0.3060·√((952−2)/(1−0.0936)) = 9.9063；df=950；p = -7.34e-02。
95% CI（Fisher z）：tanh(z±25.131·1/√(n−3)) = [-0.4619, 0.8117]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=952；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.3060；
t = r·√((n−2)/(1−r²)) = 0.3060·√((952−2)/(1−0.0936)) = 9.9063；df=950；p = -7.34e-02。
95% CI（Fisher z）：tanh(z±25.131·1/√(n−3)) = [-0.4619, 0.8117]。
③ 计算结果：
r = 0.3060，t = 9.9063，p = -7.34e-02，95%CI = [-0.4619, 0.8117]
④ 结论：在 α=0.05 水平下，p=-7.34e-02 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.3060，方向正），弱正相关。
案例 13. 饮水量(杯)与皮肤评分(分)
Pearson相关弱正相关
① 原始数据：
样本 n=462，逐条 (x, y)：[(2.15, 61.19), (6.74, 64.68), (7.47, 72.88), (6.57, 62.52), (5.33, 82.4), (5.4, 84.03), (6.18, 71.92), (5.14, 51.38), (3.31, 88.43), (8.64, 57.95), (6.82, 80.69), (4.03, 45.95), (3.99, 58.69), (4.13, 78.01), (3.01, 47.48), (4.65, 78.82), (3.85, 76.67), (7.93, 72.5), (12.05, 82.6), (5.68, 76.23), (6.3, 68.37), (5.85, 52.72), (7.01, 60.65), (1.89, 45.55), (3.79, 58.64), (8.34, 60.95), (8.12, 73.39), (3.87, 72.35), (0.69, 57.51), (2.75, 43.93), (7.02, 74.35), (6.2, 77.76), (6.02, 67.92), (7.49, 61.21), (6.2, 65.91), (3.85, 71.78), (4.93, 43.11), (4.82, 67.27), (11.76, 94.06), (3.24, 69.2), (4.98, 79.84), (0.09, 70.07), (4.19, 84.06), (3.56, 77.36), (6.69, 60.82), (3.19, 60.53), (5.65, 46.33), (5.0, 58.31), (5.75, 79.78), (8.79, 64.11), (8.13, 75.98), (2.9, 76.7), (7.92, 65.85), (6.53, 97.71), (4.56, 59.27), (7.01, 72.63), (6.17, 68.05), (5.61, 63.01), (7.96, 63.04), (6.56, 97.9), (5.55, 67.23), (5.92, 38.88), (5.87, 58.7), (5.68, 75.16), (3.63, 73.56), (4.74, 66.11), (5.46, 74.36), (5.11, 85.98), (8.53, 53.48), (2.28, 82.75), (9.04, 69.33), (5.14, 68.27), (6.56, 78.38), (5.45, 79.35), (5.07, 56.84), (4.79, 82.34), (7.13, 61.5), (7.61, 88.98), (6.93, 59.69), (5.76, 58.8), (-0.3, 80.19), (8.96, 76.57), (4.43, 59.81), (7.12, 96.88), (3.79, 49.15), (2.92, 53.68), (6.67, 57.04), (9.05, 77.62), (6.61, 74.07), (3.72, 80.46), (5.23, 74.48), (5.38, 56.77), (9.82, 102.0), (1.91, 74.26), (6.51, 77.81), (6.76, 70.51), (8.16, 68.0), (4.32, 68.75), (5.2, 75.62), (6.59, 58.56), (5.77, 92.54), (6.25, 60.72), (2.56, 71.19), (4.13, 87.27), (6.41, 81.1), (8.12, 81.0), (5.61, 74.84), (6.57, 82.95), (3.56, 62.93), (4.98, 63.0), (3.82, 53.56), (6.36, 73.08), (3.65, 48.53), (6.85, 77.83), (5.91, 73.43), (8.5, 63.19), (4.82, 74.94), (8.91, 58.43), (7.96, 67.47), (3.94, 64.08), (5.82, 88.48), (5.96, 81.35), (10.04, 64.62), (8.55, 68.22), (7.72, 58.74), (4.55, 70.63), (9.45, 88.16), (10.44, 109.62), (8.96, 79.0), (5.64, 76.82), (6.15, 63.34), (7.55, 50.54), (4.78, 75.52), (10.6, 64.94), (5.67, 80.7), (8.99, 72.83), (8.11, 76.72), (7.14, 74.92), (9.75, 79.5), (6.16, 77.1), (7.02, 77.76), (5.73, 61.95), (6.63, 87.46), (1.72, 56.93), (8.24, 86.81), (5.02, 67.77), (5.49, 58.8), (8.97, 82.94), (4.11, 46.56), (5.58, 59.95), (7.55, 72.75), (5.28, 49.39), (5.61, 77.17), (6.0, 74.66), (8.74, 73.76), (8.26, 61.33), (3.53, 64.09), (5.69, 48.77), (7.58, 79.33), (9.27, 65.39), (5.24, 54.8), (7.42, 62.11), (5.62, 74.68), (0.75, 54.71), (5.28, 98.32), (5.33, 75.84), (9.66, 76.67), (4.06, 85.68), (4.38, 67.16), (8.38, 55.48), (4.93, 74.53), (6.04, 77.71), (9.41, 78.28), (8.98, 86.21), (5.99, 77.05), (5.48, 60.48), (7.48, 46.03), (4.38, 54.83), (10.11, 80.24), (7.21, 66.76), (7.55, 73.38), (6.01, 59.62), (4.51, 70.03), (8.95, 69.9), (4.68, 63.59), (8.74, 63.54), (6.73, 66.92), (5.91, 42.67), (6.77, 56.33), (7.66, 72.39), (6.26, 56.21), (6.62, 60.27), (7.17, 74.66), (1.41, 45.7), (7.21, 68.42), (4.96, 69.94), (4.43, 49.35), (4.12, 77.7), (7.48, 49.96), (7.34, 59.71), (10.33, 80.78), (6.08, 86.48), (2.45, 61.07), (6.21, 85.08), (8.78, 48.16), (5.69, 61.56), (6.17, 52.66), (6.9, 48.75), (6.78, 79.73), (5.26, 88.49), (4.49, 61.64), (7.85, 75.23), (7.89, 70.71), (5.53, 69.07), (0.3, 62.99), (5.97, 63.61), (5.31, 62.87), (3.74, 82.44), (9.5, 66.83), (3.97, 67.98), (4.58, 66.05), (9.23, 65.9), (6.13, 71.98), (1.85, 48.69), (3.99, 64.74), (6.69, 61.73), (5.88, 87.16), (7.05, 68.48), (7.11, 65.64), (5.69, 73.7), (2.72, 71.04), (4.1, 63.82), (5.12, 81.58), (6.73, 66.44), (8.05, 73.67), (6.86, 77.13), (7.77, 59.26), (5.29, 51.36), (9.3, 61.63), (5.64, 56.86), (9.5, 71.43), (4.15, 80.82), (7.04, 71.26), (3.68, 56.97), (6.46, 53.83), (2.96, 53.23), (1.67, 58.84), (7.29, 74.49), (3.14, 58.91), (9.58, 91.15), (6.96, 75.21), (6.75, 63.14), (2.65, 63.38), (2.87, 61.77), (5.25, 50.5), (4.58, 55.76), (3.65, 72.23), (4.69, 79.02), (4.43, 68.43), (7.26, 51.87), (4.85, 51.55), (6.96, 93.96), (5.84, 73.52), (5.37, 76.75), (8.84, 69.19), (9.05, 77.77), (6.23, 79.17), (3.39, 50.67), (7.06, 68.86), (7.08, 86.01), (6.26, 55.5), (6.15, 59.31), (8.87, 63.1), (6.81, 65.49), (8.43, 83.37), (9.17, 73.09), (4.15, 78.93), (6.21, 81.45), (6.13, 74.17), (7.06, 73.2), (9.03, 77.48), (5.14, 68.98), (5.94, 59.41), (6.81, 81.93), (9.2, 75.91), (4.52, 91.6), (4.87, 77.2), (9.21, 84.89), (5.5, 77.36), (3.58, 69.98), (8.58, 80.38), (5.88, 52.53), (5.95, 76.47), (7.82, 68.43), (9.71, 74.33), (5.12, 59.78), (4.96, 58.2), (4.4, 61.35), (4.7, 72.46), (5.89, 71.63), (3.46, 41.59), (7.11, 74.35), (7.33, 66.71), (4.22, 63.31), (5.48, 62.59), (7.27, 83.41), (3.34, 73.23), (4.64, 72.89), (9.11, 75.43), (5.46, 80.0), (5.49, 60.05), (6.11, 67.34), (6.96, 82.43), (5.13, 53.86), (4.3, 58.96), (6.74, 75.64), (6.92, 74.06), (7.08, 83.56), (8.39, 66.0), (3.24, 63.0), (2.24, 50.37), (6.09, 82.57), (7.06, 51.2), (6.9, 63.86), (7.21, 95.88), (3.32, 75.02), (10.31, 80.27), (3.64, 54.41), (3.86, 82.94), (4.9, 64.75), (7.94, 56.0), (5.28, 65.19), (4.58, 70.35), (2.44, 74.58), (6.19, 48.25), (8.2, 62.03), (6.74, 58.73), (7.4, 79.14), (7.02, 87.24), (7.88, 114.59), (8.14, 65.82), (8.04, 61.58), (6.35, 64.84), (6.99, 77.64), (4.9, 71.02), (6.5, 56.08), (4.05, 92.04), (6.98, 70.14), (8.92, 85.87), (0.91, 58.48), (6.92, 72.27), (4.14, 56.21), (2.33, 70.55), (6.17, 53.71), (5.19, 47.37), (8.0, 82.09), (7.63, 81.04), (7.66, 84.64), (5.65, 58.8), (10.47, 74.36), (6.41, 62.28), (5.01, 53.13), (7.45, 58.44), (6.57, 71.01), (8.82, 71.45), (3.91, 81.83), (4.79, 67.28), (9.12, 70.25), (6.36, 73.5), (8.34, 55.59), (5.54, 83.33), (3.57, 63.76), (5.95, 69.62), (4.54, 75.42), (6.62, 46.83), (3.34, 83.02), (4.24, 48.09), (2.31, 49.18), (5.86, 74.84), (1.59, 59.43), (8.77, 88.65), (5.92, 70.73), (6.26, 67.19), (6.39, 64.23), (7.92, 64.8), (9.87, 80.88), (6.65, 63.42), (3.65, 82.86), (8.35, 91.41), (7.16, 70.75), (6.97, 54.76), (4.64, 63.98), (5.57, 75.06), (6.17, 82.88), (1.14, 60.15), (6.61, 76.18), (3.9, 51.76), (7.07, 65.12), (6.57, 77.89), (5.07, 67.08), (4.87, 61.4), (2.47, 71.0), (8.79, 68.45), (4.31, 58.44), (8.48, 79.49), (8.9, 65.26), (6.23, 68.15), (8.94, 72.92), (6.49, 69.96), (2.86, 52.28), (7.08, 86.46), (5.97, 36.95), (5.44, 71.24), (6.78, 59.52), (4.32, 65.49), (5.64, 51.46), (3.69, 74.33), (4.87, 58.87), (6.42, 68.43), (5.91, 59.7), (5.67, 75.09), (2.28, 67.32), (0.15, 82.43), (7.92, 52.44), (5.36, 74.31), (4.43, 77.13), (3.05, 71.02), (2.76, 73.6), (4.56, 64.52), (3.08, 67.17), (6.84, 56.04), (3.34, 49.85), (8.93, 79.43), (5.29, 69.4), (7.23, 85.71), (6.42, 77.88), (4.64, 71.7), (2.79, 46.29), (6.43, 51.79), (8.23, 72.33), (8.07, 70.76), (4.27, 70.99), (5.64, 64.12), (4.29, 48.39), (2.61, 54.57), (8.5, 72.82), (4.9, 89.04), (5.89, 53.33), (7.7, 78.41), (5.39, 84.03), (8.54, 57.88), (5.07, 57.69), (0.1, 55.1), (3.85, 61.99), (6.78, 54.47), (2.71, 45.18), (6.71, 76.24), (6.87, 72.72), (7.04, 46.9), (6.09, 66.54), (5.17, 77.95), (5.83, 68.73)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=462；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2642；
t = r·√((n−2)/(1−r²)) = 0.2642·√((462−2)/(1−0.0698)) = 5.8762；df=460；p = 8.06e-09。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.1771, 0.3473]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=462；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2642；
t = r·√((n−2)/(1−r²)) = 0.2642·√((462−2)/(1−0.0698)) = 5.8762；df=460；p = 8.06e-09。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [0.1771, 0.3473]。
③ 计算结果：
r = 0.2642，t = 5.8762，p = 8.06e-09，95%CI = [0.1771, 0.3473]
④ 结论：在 α=0.05 水平下，p=8.06e-09 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.2642，方向正），弱正相关。
案例 14. 植物每日光照(h)与开花数(朵)
Pearson相关弱正相关
① 原始数据：
样本 n=642，逐条 (x, y)：[(6.04, 5.7), (8.72, 6.07), (10.11, 9.24), (7.27, 4.17), (10.14, 2.26), (10.51, 5.72), (9.41, 6.13), (9.93, 6.67), (0.93, 3.04), (5.8, 3.78), (6.01, 3.27), (8.28, 6.79), (9.75, 5.24), (9.96, 3.8), (6.17, 5.51), (8.71, 6.15), (7.03, 2.17), (5.27, 2.08), (3.1, 0.36), (3.53, 1.81), (10.58, 4.0), (11.03, 8.36), (5.58, 4.63), (8.35, 5.75), (6.03, 6.44), (7.12, 6.66), (7.67, 4.73), (6.17, 5.55), (12.49, 6.47), (12.13, 3.66), (8.94, 4.75), (6.01, 5.54), (11.61, 7.32), (8.26, 4.92), (9.15, 3.7), (7.31, 3.18), (3.85, 4.27), (11.57, 5.95), (3.85, 3.48), (7.41, 4.36), (4.87, 2.17), (10.18, 5.92), (5.22, 4.52), (14.56, 4.9), (2.63, 3.46), (5.69, 4.57), (12.95, 5.35), (9.65, 4.21), (8.5, 3.4), (6.41, 7.15), (7.2, 6.9), (10.76, 5.72), (8.91, 6.16), (4.31, 3.91), (12.28, 5.09), (10.0, 2.51), (3.83, 3.83), (9.12, 4.59), (4.69, 2.92), (2.19, 4.75), (8.57, 5.21), (5.14, 3.08), (8.27, 4.67), (12.61, 3.71), (5.24, 2.41), (4.65, 4.65), (4.67, 4.7), (2.81, 3.9), (5.94, 4.39), (6.69, 4.54), (6.06, 2.72), (3.71, 5.41), (5.88, 7.09), (10.94, 5.68), (10.61, 8.67), (3.45, 1.31), (5.16, 1.49), (8.96, 6.93), (8.57, 5.53), (6.51, 5.15), (8.59, 6.99), (6.33, 3.28), (8.74, 3.33), (9.32, 1.87), (7.24, 3.32), (8.91, 10.23), (6.81, 4.0), (6.17, 5.65), (8.2, 1.65), (5.43, 5.45), (5.02, 1.05), (12.65, 6.07), (9.42, 1.6), (11.06, 4.33), (13.72, 3.86), (12.53, 4.67), (12.68, 5.18), (7.67, 4.99), (11.6, 10.06), (3.38, 6.9), (9.87, 4.44), (8.2, 4.86), (6.77, 4.58), (7.76, 3.86), (5.99, 3.86), (7.2, 1.88), (9.65, 4.84), (3.36, 4.03), (6.03, 3.29), (8.98, 7.85), (7.08, 5.42), (10.43, 8.6), (5.92, 0.29), (10.2, 6.36), (12.06, 5.81), (4.78, 4.78), (11.88, 9.54), (12.85, 8.96), (8.9, 5.27), (9.55, 7.58), (14.83, 8.58), (10.65, 5.84), (14.51, 3.3), (5.83, 4.97), (5.33, 4.54), (14.23, 7.28), (10.03, 8.39), (8.19, 3.87), (9.52, 4.58), (9.31, 3.49), (4.51, 3.08), (11.12, 5.41), (2.8, 1.63), (9.19, 6.49), (10.74, 6.73), (6.3, 6.03), (6.82, 5.08), (10.22, 6.23), (7.53, 3.63), (10.47, 9.37), (8.95, 3.08), (13.25, 7.61), (1.57, 0.98), (7.88, 3.93), (6.19, 2.94), (5.63, 5.33), (5.44, 5.83), (9.47, 3.02), (5.68, 5.78), (5.23, 4.13), (14.15, 3.25), (3.1, 5.08), (8.18, 6.69), (6.86, 4.62), (3.04, 6.04), (12.06, 3.86), (7.96, 2.75), (7.83, 4.53), (4.01, 5.18), (6.87, 3.79), (12.28, 8.47), (8.35, 7.59), (4.72, 3.96), (3.96, 5.28), (5.44, 3.99), (10.91, 5.14), (13.98, 4.69), (8.04, 4.43), (9.7, 5.15), (8.3, 6.81), (12.89, 5.49), (11.48, 5.88), (9.4, 4.65), (6.47, 3.97), (9.61, 1.94), (11.27, 6.31), (3.68, 6.12), (7.97, 4.73), (9.2, 5.82), (8.03, 5.04), (10.74, 5.28), (9.9, 2.98), (11.56, 5.74), (4.51, 5.22), (10.6, 5.17), (8.2, 6.75), (5.15, 3.02), (12.69, 9.06), (9.36, 4.69), (7.99, 5.83), (5.19, 5.06), (10.82, 7.57), (9.12, 4.74), (10.56, 6.4), (9.27, 5.9), (8.63, 6.25), (12.49, 4.52), (8.57, 2.96), (5.37, 3.48), (6.15, 7.71), (9.33, 4.07), (5.94, 4.79), (9.58, 3.22), (4.14, 0.78), (9.0, 6.58), (6.44, 6.89), (5.95, 4.49), (11.31, 4.08), (8.69, 2.06), (7.44, 1.59), (7.07, 3.8), (6.43, 2.44), (12.54, 9.38), (7.62, 3.47), (9.51, 4.89), (5.77, 2.84), (12.54, 6.8), (11.05, 4.77), (6.72, 1.72), (12.68, 3.57), (8.93, 3.73), (11.68, 3.48), (6.75, 5.98), (7.11, 3.74), (7.65, 5.88), (5.65, 6.12), (15.33, 10.49), (8.01, 5.57), (4.85, 1.37), (8.15, 0.82), (9.04, 3.53), (1.66, 6.25), (5.11, 6.07), (3.92, 4.45), (7.34, 3.0), (4.55, 2.76), (6.46, 3.36), (8.67, 3.32), (6.45, 5.28), (4.89, 6.23), (9.78, 3.21), (3.59, 7.56), (4.43, 3.38), (6.88, 3.53), (8.67, 4.66), (6.81, 5.08), (7.92, 7.86), (6.26, 7.42), (7.75, 7.08), (11.85, 6.93), (11.28, 4.28), (14.48, 7.8), (4.94, 7.65), (10.13, 4.82), (6.2, 5.47), (6.84, 7.26), (4.42, 4.14), (6.34, 5.24), (8.28, 4.48), (8.51, 6.21), (6.69, 4.98), (11.23, 7.65), (14.68, 5.27), (8.08, 5.63), (8.91, 5.29), (7.46, 3.02), (8.82, 3.77), (13.02, 5.95), (6.51, 4.91), (9.98, 1.29), (12.29, 9.26), (6.78, 7.16), (11.31, 3.25), (5.98, 4.77), (13.35, 5.63), (11.14, 2.28), (1.87, 3.82), (4.62, 5.94), (7.82, 2.49), (4.1, 2.38), (4.96, 4.38), (11.2, 1.74), (2.84, 2.41), (12.58, 4.75), (8.76, 4.65), (9.53, 6.08), (1.7, 1.98), (6.14, 4.18), (9.39, 5.17), (9.74, 5.75), (7.92, 5.27), (5.87, 5.88), (6.38, 6.53), (7.56, 6.11), (9.02, 3.21), (8.29, 4.47), (11.11, 7.94), (1.18, 4.31), (4.89, 7.39), (4.82, 8.37), (6.54, 7.66), (9.38, 3.85), (4.46, 2.39), (9.74, 2.08), (11.53, 8.21), (8.64, 5.75), (10.54, 5.08), (8.47, 7.3), (12.25, 6.47), (2.34, 3.64), (6.37, 3.27), (8.19, 3.81), (10.69, 4.84), (6.98, 2.98), (10.41, 5.81), (3.8, 3.92), (7.29, 6.99), (8.9, 4.96), (5.28, 8.16), (7.88, 7.77), (2.21, 3.97), (7.74, 4.15), (8.66, 2.91), (8.83, 4.92), (6.92, 7.28), (6.15, -0.17), (11.05, 7.1), (10.79, 2.6), (7.07, 4.05), (10.18, 6.1), (10.85, 4.25), (5.53, 3.05), (7.99, 3.12), (4.57, 6.4), (4.85, 3.39), (5.67, 2.96), (9.11, 4.19), (5.42, 3.02), (5.97, 4.38), (7.89, 5.88), (7.57, 4.62), (7.47, 4.3), (7.6, 3.84), (4.06, 3.73), (7.3, 4.64), (8.07, 4.26), (10.03, 5.22), (5.01, 1.51), (11.09, 4.24), (8.35, 4.07), (2.46, 3.53), (11.02, 3.64), (12.92, 8.39), (6.74, 4.0), (7.31, 3.6), (6.87, 5.02), (8.94, 5.5), (3.67, 3.47), (9.75, 3.16), (10.3, 4.45), (4.92, 7.65), (2.48, 4.27), (11.36, 4.35), (11.11, 4.82), (8.74, 3.67), (5.94, 2.34), (8.73, 6.3), (5.68, 4.75), (4.64, 4.33), (5.11, 2.25), (8.83, 1.64), (12.7, 7.79), (12.13, 7.08), (8.64, 3.9), (4.64, 2.77), (10.0, 5.54), (4.28, 6.9), (9.83, 6.27), (1.84, 4.71), (9.07, 4.2), (12.35, 3.26), (8.17, 8.57), (7.36, 3.77), (6.79, 3.15), (8.43, 8.64), (8.86, 5.42), (13.67, 9.07), (6.47, 5.91), (5.89, 5.93), (8.8, 3.94), (12.95, 4.96), (9.58, 5.33), (6.26, 5.17), (9.26, 4.22), (7.51, 6.59), (15.77, 10.27), (7.13, 7.28), (12.47, 5.15), (12.11, 6.6), (4.74, 3.63), (10.41, 7.14), (6.1, 6.38), (5.74, 0.95), (8.39, 6.63), (5.23, 4.57), (11.42, 4.88), (9.74, 1.74), (4.48, 8.04), (-0.09, 0.29), (9.2, 6.78), (8.1, 3.29), (12.96, 4.72), (8.02, 5.72), (8.64, 3.81), (5.18, 4.25), (14.24, 7.43), (4.23, 3.02), (9.79, 6.52), (5.93, 3.62), (9.45, 1.67), (8.8, 7.0), (7.79, 4.02), (11.19, 5.92), (9.04, 5.62), (8.98, 6.62), (6.55, 4.29), (4.67, 3.14), (10.8, 4.87), (7.92, 4.77), (10.37, 6.09), (7.28, 5.85), (10.01, 5.86), (7.84, 6.77), (3.96, 4.92), (10.37, 5.87), (9.91, 7.34), (8.61, 5.39), (9.57, 5.45), (8.45, 4.22), (4.4, 2.88), (6.98, 5.26), (11.28, 5.44), (0.92, 0.32), (12.24, 5.73), (4.17, 3.49), (7.09, 7.15), (5.15, 4.7), (11.9, 3.6), (9.03, 2.82), (6.55, 3.07), (10.36, 4.59), (9.48, 4.67), (12.19, 6.34), (8.22, 2.86), (9.21, 6.33), (12.5, 5.12), (3.85, 1.2), (6.92, 5.64), (4.99, 1.59), (9.95, 3.14), (11.52, 7.34), (8.47, 4.28), (10.63, 5.98), (9.43, 5.26), (2.73, 5.74), (7.97, 4.83), (6.7, 2.42), (7.68, 7.72), (3.93, 8.73), (9.85, 7.76), (13.66, 4.12), (6.21, 5.11), (9.64, 1.94), (7.73, 4.4), (11.25, 5.56), (8.02, 4.44), (7.27, 3.39), (8.45, 4.75), (8.34, 4.7), (7.85, 5.92), (9.13, 5.96), (4.93, 4.94), (5.34, 5.05), (14.81, 7.28), (7.28, 3.71), (7.6, 5.52), (8.32, 4.24), (5.43, 6.42), (8.06, 6.16), (5.06, 3.49), (7.47, 5.06), (5.87, 3.54), (10.91, 8.3), (10.68, 2.16), (3.73, 4.42), (5.18, 5.3), (3.79, 2.72), (7.62, 7.81), (7.14, 5.69), (1.1, 1.6), (6.84, 3.96), (7.7, 3.4), (1.67, 5.77), (8.98, 4.13), (6.71, 8.27), (9.87, 4.44), (3.35, 2.32), (4.88, 7.02), (8.75, 4.45), (5.17, 4.55), (11.24, 7.17), (8.17, 2.77), (1.49, 2.68), (8.37, 5.04), (6.26, 6.54), (10.96, 7.12), (6.64, 4.03), (10.27, 4.3), (5.21, 4.57), (6.01, 7.49), (8.17, 7.23), (2.41, 0.46), (6.59, 4.09), (8.87, 7.54), (6.1, 4.15), (6.16, 5.89), (15.12, 5.64), (10.54, 2.43), (8.67, 6.99), (13.1, 5.46), (11.35, 6.33), (7.89, 7.43), (11.5, 4.24), (9.49, 1.36), (5.47, 3.14), (6.83, 5.07), (8.73, 5.87), (5.39, 2.15), (8.67, 1.68), (9.17, 8.61), (9.96, 5.81), (1.74, 1.46), (5.31, 6.66), (7.49, 5.87), (9.6, 3.39), (11.94, 3.15), (11.74, 6.11), (6.95, 6.2), (10.17, 5.87), (8.17, 5.98), (6.03, 6.4), (8.86, 8.06), (5.84, 5.94), (9.07, 2.72), (5.3, 3.75), (9.98, 1.3), (9.22, 7.4), (9.27, 2.51), (6.07, 5.65), (1.59, 1.36), (5.21, 5.37), (6.32, 5.65), (9.58, 7.12), (8.97, 6.0), (10.3, 6.42), (11.17, 2.72), (6.24, 6.07), (9.79, 6.37), (6.97, 4.29), (10.97, 4.67), (11.73, 2.97), (10.9, 3.6), (3.82, 3.21), (6.39, 4.47), (7.35, 3.35), (9.35, 8.29), (7.47, 6.45), (5.53, 4.04), (6.33, 1.73), (7.98, 6.38), (8.58, 3.96), (10.27, 3.09), (9.32, 5.0), (3.9, 4.7), (6.53, 5.39), (10.23, 7.27), (10.79, 7.71), (5.6, 1.74), (11.36, 5.96), (1.18, 3.96), (6.64, 6.37), (5.55, 5.77), (4.65, 5.22), (5.19, 0.42), (6.75, 8.45), (5.95, 2.47), (10.56, 4.71), (8.9, 4.77), (8.79, 6.43), (12.85, 2.31), (6.18, 7.07), (8.93, 4.06), (9.26, 4.58), (12.09, 5.21), (10.63, 5.74), (9.92, 6.0), (10.03, 4.92), (9.88, 4.88), (9.36, 5.63), (13.38, 5.14), (10.8, 6.14), (6.21, 0.14), (7.17, 6.96), (9.29, 1.98), (4.05, 1.3), (8.98, 4.68), (7.68, 0.92), (4.39, 4.36), (4.09, 4.66), (6.73, 5.89), (6.26, 5.28), (6.39, 4.11), (4.21, 3.44), (13.73, 6.89), (11.31, 8.75), (7.6, 4.04), (13.49, 4.47), (6.05, 1.96), (5.31, 3.39), (12.17, 6.9), (5.63, 1.44), (11.83, 3.78), (5.0, 3.02), (7.11, 3.21), (4.85, 4.55), (9.86, 7.49), (10.15, 2.85), (6.2, 8.25), (6.61, 8.16), (9.96, 9.43), (9.34, 4.8), (10.02, 4.16)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=642；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.3295；
t = r·√((n−2)/(1−r²)) = 0.3295·√((642−2)/(1−0.1086)) = 8.8300；df=640；p = 0.0002。
95% CI（Fisher z）：tanh(z±16.846·1/√(n−3)) = [-0.3132, 0.7652]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=642；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.3295；
t = r·√((n−2)/(1−r²)) = 0.3295·√((642−2)/(1−0.1086)) = 8.8300；df=640；p = 0.0002。
95% CI（Fisher z）：tanh(z±16.846·1/√(n−3)) = [-0.3132, 0.7652]。
③ 计算结果：
r = 0.3295，t = 8.8300，p = 0.0002，95%CI = [-0.3132, 0.7652]
④ 结论：在 α=0.05 水平下，p=0.0002 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.3295，方向正），弱正相关。
案例 15. 阅读速度(字/分)与理解正确率(%)
Pearson相关弱正相关
① 原始数据：
样本 n=571，逐条 (x, y)：[(253.22, 71.91), (204.02, 93.1), (321.54, 75.31), (313.3, 68.73), (242.12, 85.69), (355.91, 98.25), (170.48, 74.25), (157.71, 73.91), (348.64, 85.61), (292.34, 77.26), (207.26, 70.32), (325.29, 79.62), (397.97, 78.45), (350.55, 94.81), (423.33, 89.06), (328.11, 80.61), (321.24, 83.23), (267.31, 75.22), (295.46, 100.67), (410.46, 78.35), (239.8, 73.39), (314.14, 75.85), (214.65, 78.17), (277.92, 83.68), (359.13, 98.48), (288.39, 76.55), (457.42, 93.84), (211.15, 67.7), (377.58, 96.88), (274.94, 94.37), (258.58, 96.61), (246.09, 87.24), (202.89, 64.94), (130.49, 77.51), (224.18, 85.32), (347.27, 72.23), (416.24, 89.34), (484.85, 93.29), (376.68, 71.39), (394.95, 79.93), (250.72, 76.52), (169.63, 76.8), (236.8, 72.85), (303.4, 71.12), (278.1, 78.58), (295.14, 72.86), (418.46, 78.61), (191.24, 84.83), (242.72, 91.38), (410.92, 86.95), (388.12, 97.4), (217.32, 67.02), (173.04, 79.23), (215.96, 73.96), (407.96, 64.07), (409.61, 76.05), (379.42, 67.15), (141.23, 58.78), (355.56, 87.16), (438.6, 69.07), (444.13, 79.0), (452.97, 76.56), (238.12, 64.13), (190.31, 95.56), (299.53, 89.26), (426.97, 81.51), (213.22, 72.0), (217.06, 78.29), (242.34, 85.69), (217.56, 65.98), (386.03, 90.35), (266.12, 80.45), (275.29, 84.41), (346.32, 81.29), (329.73, 62.08), (352.64, 77.75), (338.82, 70.42), (274.06, 72.9), (324.04, 77.17), (360.73, 99.32), (244.68, 83.88), (316.95, 85.33), (234.62, 79.02), (240.9, 89.4), (340.85, 74.09), (398.14, 95.07), (391.12, 81.05), (317.35, 68.75), (115.42, 64.83), (344.36, 69.45), (96.96, 76.04), (191.17, 73.3), (454.16, 112.94), (241.87, 67.84), (331.21, 81.28), (319.43, 84.34), (306.95, 74.05), (302.32, 89.85), (229.97, 77.61), (261.99, 80.26), (257.94, 90.37), (253.56, 70.7), (263.21, 65.8), (283.03, 78.08), (368.77, 95.11), (260.99, 62.48), (269.99, 81.36), (305.4, 90.25), (395.82, 75.19), (411.12, 103.39), (334.17, 79.76), (311.22, 83.11), (182.46, 72.38), (312.24, 74.03), (498.18, 83.66), (312.35, 72.52), (155.85, 94.45), (382.86, 84.0), (495.89, 85.37), (173.96, 85.0), (195.76, 68.19), (219.43, 85.91), (408.23, 69.81), (359.13, 93.27), (213.3, 60.92), (376.92, 101.13), (364.45, 81.36), (268.03, 78.04), (346.74, 87.21), (320.27, 70.38), (333.67, 94.66), (298.88, 65.05), (321.89, 103.75), (332.26, 92.54), (218.32, 87.56), (306.81, 80.59), (448.89, 67.88), (205.22, 69.46), (268.82, 87.49), (425.65, 83.16), (288.72, 72.02), (173.49, 74.39), (406.48, 90.85), (395.27, 85.73), (377.85, 74.2), (158.8, 61.06), (236.32, 68.59), (216.54, 72.81), (281.34, 81.38), (460.88, 83.81), (210.8, 81.12), (385.9, 82.49), (421.29, 71.01), (390.2, 87.21), (464.38, 88.06), (290.67, 85.1), (415.81, 102.52), (327.71, 81.47), (298.26, 87.46), (321.62, 85.95), (302.69, 91.05), (344.97, 86.95), (291.1, 85.42), (118.01, 58.77), (231.87, 53.97), (247.24, 85.52), (270.82, 85.21), (378.27, 82.74), (390.59, 82.5), (204.18, 88.04), (263.93, 81.32), (278.53, 60.13), (294.2, 57.94), (179.58, 77.93), (337.64, 73.0), (300.48, 83.06), (321.67, 81.81), (284.67, 85.41), (174.06, 89.94), (323.13, 93.17), (317.2, 83.35), (405.18, 87.21), (301.61, 71.59), (263.86, 114.72), (129.81, 73.97), (197.43, 89.94), (313.46, 66.89), (380.09, 98.02), (239.83, 82.34), (353.2, 72.45), (260.86, 68.14), (204.96, 74.69), (235.99, 73.95), (350.73, 81.97), (327.04, 79.23), (198.56, 86.49), (253.9, 81.81), (343.69, 76.28), (429.17, 72.99), (303.25, 89.93), (312.04, 80.79), (238.78, 73.65), (416.29, 77.06), (185.47, 78.75), (207.96, 72.62), (191.52, 65.38), (269.09, 86.1), (198.91, 74.6), (52.81, 57.15), (353.7, 75.51), (305.41, 54.83), (332.64, 83.11), (147.61, 60.72), (242.63, 87.82), (263.07, 88.7), (317.52, 88.51), (415.33, 84.18), (262.7, 69.8), (254.01, 79.44), (304.36, 62.97), (242.94, 89.68), (294.68, 85.06), (415.09, 96.03), (308.77, 81.75), (288.23, 93.67), (249.22, 72.32), (285.83, 86.21), (257.57, 69.07), (288.17, 69.89), (324.31, 78.31), (349.42, 89.06), (270.04, 87.29), (232.31, 60.67), (423.59, 70.78), (405.75, 82.62), (295.1, 52.14), (410.21, 96.53), (229.42, 84.83), (362.49, 84.91), (302.46, 72.68), (292.56, 73.24), (189.09, 88.39), (236.07, 86.75), (189.52, 79.65), (111.51, 62.46), (307.02, 68.86), (369.05, 100.95), (320.79, 62.85), (222.79, 72.02), (310.54, 84.55), (310.74, 86.51), (244.26, 67.98), (325.89, 89.1), (227.46, 79.94), (408.14, 83.82), (267.4, 92.09), (278.72, 65.86), (374.3, 100.78), (170.61, 84.28), (276.02, 83.76), (119.4, 71.35), (207.21, 74.5), (159.92, 77.25), (400.19, 94.23), (235.96, 69.56), (407.64, 85.04), (378.6, 77.1), (333.59, 80.43), (279.94, 77.49), (279.72, 96.63), (368.17, 78.79), (61.92, 79.75), (317.55, 77.86), (411.55, 85.71), (287.66, 88.15), (186.56, 71.13), (262.23, 96.28), (186.21, 85.16), (158.52, 85.54), (307.57, 84.39), (344.07, 80.9), (372.57, 78.09), (357.93, 91.19), (251.89, 81.32), (119.43, 52.47), (344.77, 85.6), (348.6, 74.99), (152.61, 76.85), (280.32, 76.3), (170.81, 88.61), (294.71, 82.0), (338.14, 75.91), (326.49, 93.71), (58.67, 76.73), (349.26, 75.93), (229.29, 81.01), (194.17, 73.06), (273.7, 83.84), (114.27, 73.35), (353.62, 83.65), (252.76, 83.16), (244.82, 81.09), (351.03, 75.13), (235.3, 71.92), (225.16, 72.78), (201.05, 72.47), (256.24, 86.81), (328.46, 86.94), (332.5, 82.15), (271.47, 60.24), (330.48, 82.81), (302.54, 89.76), (236.26, 77.01), (351.41, 81.8), (361.94, 84.29), (380.67, 81.74), (303.39, 93.47), (296.47, 68.16), (337.35, 100.06), (260.41, 84.85), (349.66, 74.25), (298.5, 70.72), (304.02, 91.77), (154.7, 79.14), (302.08, 59.37), (368.51, 81.86), (325.33, 86.4), (199.42, 72.74), (356.71, 77.59), (273.83, 92.01), (353.23, 82.71), (337.57, 103.0), (186.32, 81.45), (134.37, 94.27), (246.99, 80.42), (274.9, 76.56), (156.7, 70.1), (257.42, 82.82), (280.1, 75.26), (289.21, 87.7), (351.45, 58.83), (215.01, 70.61), (255.13, 76.76), (323.71, 83.56), (300.25, 82.91), (266.14, 71.36), (286.95, 83.88), (267.54, 90.41), (371.58, 95.13), (143.91, 60.38), (195.21, 74.58), (351.57, 93.86), (319.01, 76.02), (433.89, 81.89), (337.04, 96.76), (211.28, 90.76), (351.67, 86.67), (380.64, 76.78), (374.99, 78.29), (370.69, 72.64), (129.65, 77.84), (318.84, 86.06), (346.05, 72.88), (217.9, 78.32), (342.46, 67.1), (180.85, 66.61), (508.85, 80.6), (344.73, 74.53), (339.13, 75.79), (242.36, 60.24), (329.63, 76.89), (410.27, 76.33), (310.93, 79.92), (290.98, 87.01), (381.71, 82.14), (317.46, 69.86), (282.7, 83.86), (196.94, 63.67), (208.52, 77.11), (206.43, 87.74), (411.15, 96.01), (142.11, 66.79), (278.83, 98.6), (282.65, 110.9), (246.91, 73.82), (309.81, 78.34), (373.33, 93.21), (305.93, 66.21), (262.11, 66.21), (288.83, 84.14), (514.78, 93.57), (408.51, 87.38), (297.57, 83.2), (332.96, 84.58), (297.0, 90.72), (383.63, 90.56), (204.5, 73.25), (339.0, 66.73), (354.08, 68.47), (287.3, 95.84), (317.99, 94.93), (380.38, 66.35), (156.28, 60.51), (183.94, 71.27), (343.38, 89.66), (237.13, 75.24), (278.86, 99.1), (228.18, 83.98), (224.9, 100.83), (287.37, 96.77), (458.67, 77.57), (425.66, 84.59), (310.86, 62.12), (251.62, 85.85), (256.46, 77.5), (393.0, 64.97), (238.19, 97.77), (286.65, 85.48), (352.14, 86.26), (183.65, 87.15), (340.05, 77.73), (276.07, 64.3), (273.13, 102.75), (291.54, 83.33), (392.49, 59.47), (318.74, 86.09), (248.34, 82.89), (280.77, 91.21), (328.77, 83.21), (274.14, 90.84), (279.97, 84.29), (203.9, 81.17), (246.79, 68.9), (266.26, 77.51), (182.34, 60.76), (206.74, 91.13), (352.38, 93.78), (290.19, 80.08), (369.39, 97.3), (314.29, 74.05), (201.09, 77.86), (277.42, 88.62), (268.85, 73.29), (306.35, 66.63), (532.2, 90.76), (126.73, 69.73), (199.39, 79.99), (363.17, 96.94), (258.27, 87.62), (244.33, 83.51), (392.49, 94.81), (459.3, 78.94), (270.38, 75.79), (472.92, 86.0), (118.71, 59.23), (356.23, 83.46), (394.2, 85.4), (485.26, 74.56), (269.09, 78.07), (309.6, 87.9), (380.43, 82.46), (189.36, 67.47), (363.76, 82.97), (261.72, 71.78), (395.12, 79.43), (359.25, 90.63), (420.88, 79.87), (244.97, 68.92), (314.53, 83.22), (243.97, 70.2), (259.47, 89.97), (214.79, 78.05), (392.54, 91.47), (384.82, 76.4), (188.97, 69.28), (327.45, 75.16), (351.01, 89.65), (316.29, 72.16), (323.55, 74.24), (333.69, 76.87), (358.9, 81.58), (333.43, 83.13), (267.17, 80.77), (243.41, 98.97), (303.71, 84.21), (443.98, 91.17), (163.79, 93.82), (241.21, 75.01), (254.16, 81.77), (293.58, 85.78), (313.14, 74.89), (342.55, 71.49), (364.58, 90.52), (405.34, 88.59), (291.65, 68.1), (370.64, 68.48), (384.3, 75.74), (421.31, 94.32), (332.57, 87.32), (285.24, 98.7), (-12.92, 69.71), (376.02, 92.49), (225.66, 72.3), (152.39, 51.45), (250.57, 83.6), (337.29, 95.09), (344.25, 74.47), (321.86, 94.24), (481.67, 79.94), (292.83, 81.56), (190.7, 82.14), (273.91, 70.26), (214.08, 66.42), (265.1, 81.26), (391.86, 81.02), (340.66, 89.41), (147.87, 69.64), (153.97, 68.63), (267.31, 85.41), (378.41, 88.14), (156.29, 74.83), (231.97, 64.81), (276.33, 75.94), (259.29, 51.66), (367.7, 90.83), (341.63, 68.93), (292.21, 97.34), (179.55, 63.65), (283.52, 69.83), (438.12, 87.66), (315.44, 73.48), (194.02, 78.85), (305.92, 77.34), (376.48, 71.66), (247.6, 81.74), (383.32, 76.68), (265.32, 76.15), (385.82, 85.15), (303.76, 72.69), (272.08, 104.36), (193.35, 71.22), (285.96, 82.77), (328.9, 79.61), (312.61, 62.86), (599.49, 95.84), (305.16, 88.06), (348.19, 65.01), (292.6, 82.46), (290.63, 98.31), (222.54, 67.04), (388.75, 78.09), (241.56, 77.83), (310.43, 83.39), (265.36, 81.76), (203.75, 80.61), (312.11, 63.95), (141.59, 72.09), (292.28, 90.17), (320.85, 74.32), (341.83, 65.02), (354.7, 78.78), (349.29, 62.11), (388.17, 88.74), (300.56, 88.52), (98.66, 65.67), (446.14, 64.66), (345.84, 92.68), (350.75, 83.37), (257.21, 87.52), (220.07, 76.61), (159.75, 70.82)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=571；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2982；
t = r·√((n−2)/(1−r²)) = 0.2982·√((571−2)/(1−0.0889)) = 7.4530；df=569；p = -7.15e-09。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [0.2214, 0.3713]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=571；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2982；
t = r·√((n−2)/(1−r²)) = 0.2982·√((571−2)/(1−0.0889)) = 7.4530；df=569；p = -7.15e-09。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [0.2214, 0.3713]。
③ 计算结果：
r = 0.2982，t = 7.4530，p = -7.15e-09，95%CI = [0.2214, 0.3713]
④ 结论：在 α=0.05 水平下，p=-7.15e-09 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.2982，方向正），弱正相关。
案例 16. 工作经验(年)与创新能力(分)
Pearson相关弱相关
① 原始数据：
样本 n=849，逐条 (x, y)：[(12.26, 66.95), (16.95, 66.73), (7.68, 55.16), (2.7, 57.09), (6.34, 62.35), (8.55, 54.9), (1.93, 38.45), (10.15, 56.46), (8.14, 57.17), (14.1, 81.17), (11.85, 51.54), (7.99, 62.13), (4.16, 69.47), (7.76, 56.41), (1.59, 75.7), (11.95, 94.63), (13.23, 50.91), (10.87, 70.69), (7.0, 48.27), (13.26, 60.44), (14.44, 64.13), (2.23, 57.49), (3.03, 51.69), (14.82, 70.07), (24.04, 64.76), (5.66, 43.84), (16.68, 55.61), (9.1, 75.26), (1.87, 59.39), (13.17, 91.45), (8.88, 73.12), (5.51, 46.9), (-4.88, 52.94), (4.46, 83.6), (8.9, 71.3), (5.67, 59.93), (9.82, 89.33), (1.13, 69.1), (11.95, 80.02), (14.85, 63.98), (12.86, 73.73), (4.57, 99.99), (3.28, 64.89), (10.45, 69.78), (7.06, 69.64), (7.2, 58.31), (9.39, 62.85), (8.63, 56.29), (4.09, 57.7), (14.68, 69.05), (11.43, 67.45), (4.35, 65.26), (10.61, 78.26), (11.25, 87.2), (3.4, 67.93), (6.01, 62.38), (5.06, 63.42), (11.46, 69.5), (16.2, 58.38), (4.22, 48.96), (14.17, 72.53), (5.87, 57.29), (6.25, 65.35), (8.28, 67.54), (0.81, 56.46), (20.41, 65.99), (2.55, 68.13), (6.72, 80.8), (12.66, 50.49), (2.17, 52.94), (16.48, 58.36), (12.21, 63.19), (7.97, 65.44), (12.01, 55.4), (22.35, 82.9), (9.79, 62.86), (12.21, 63.22), (11.58, 56.0), (5.27, 48.13), (12.85, 77.8), (6.96, 56.05), (2.13, 51.77), (13.09, 76.57), (14.44, 83.05), (4.96, 70.93), (2.01, 69.69), (12.45, 63.61), (6.41, 63.58), (3.2, 57.79), (5.7, 68.46), (8.52, 74.41), (3.62, 59.87), (11.8, 70.91), (14.26, 71.12), (11.41, 45.64), (9.93, 59.44), (10.01, 62.72), (9.72, 53.94), (6.56, 67.51), (5.79, 52.78), (5.57, 45.58), (9.68, 44.01), (5.5, 68.02), (14.74, 65.9), (8.45, 61.73), (6.08, 53.49), (8.29, 75.8), (13.66, 54.9), (5.89, 39.57), (9.92, 67.13), (18.31, 76.04), (11.99, 71.34), (4.54, 53.82), (13.52, 68.0), (6.7, 70.38), (4.97, 77.01), (14.91, 49.29), (10.01, 60.45), (13.34, 54.53), (13.81, 58.14), (12.93, 81.21), (7.23, 41.14), (11.08, 45.96), (20.91, 47.57), (14.58, 79.42), (11.76, 55.5), (13.19, 66.0), (10.79, 67.28), (12.7, 74.31), (6.9, 72.88), (12.14, 63.6), (4.67, 43.59), (16.1, 72.34), (7.11, 52.41), (14.8, 69.07), (-0.09, 50.91), (17.5, 75.98), (5.31, 67.13), (12.32, 56.36), (7.16, 60.96), (1.64, 67.48), (7.48, 52.84), (7.72, 58.33), (7.47, 64.43), (9.8, 56.09), (15.0, 57.56), (10.21, 68.99), (16.58, 75.43), (7.55, 77.4), (2.34, 58.75), (14.03, 68.97), (12.09, 64.52), (4.01, 74.36), (7.06, 63.4), (23.0, 67.55), (4.62, 64.06), (5.1, 67.41), (10.14, 69.81), (20.35, 52.15), (4.57, 59.85), (11.27, 68.7), (2.88, 52.3), (11.8, 61.56), (11.34, 66.04), (14.6, 69.84), (8.72, 78.51), (0.73, 48.77), (4.82, 58.56), (11.74, 73.6), (10.02, 69.05), (6.78, 64.52), (7.78, 58.37), (15.31, 62.0), (10.14, 65.85), (9.06, 67.47), (3.51, 62.21), (17.09, 73.36), (4.66, 49.41), (5.76, 72.82), (13.16, 40.29), (11.43, 79.02), (5.27, 38.66), (9.73, 61.04), (16.85, 79.03), (10.07, 102.28), (-3.61, 65.35), (13.02, 70.64), (6.86, 64.47), (6.55, 55.76), (14.64, 64.76), (3.91, 59.57), (8.52, 61.9), (7.49, 78.83), (14.4, 73.13), (15.06, 79.7), (14.58, 57.9), (11.46, 79.62), (12.66, 51.4), (10.91, 59.79), (5.41, 56.62), (7.51, 73.68), (8.46, 72.27), (12.68, 71.56), (-2.24, 94.08), (6.11, 61.15), (21.0, 71.24), (14.96, 54.41), (15.8, 62.84), (12.55, 59.92), (5.62, 61.57), (14.09, 79.31), (9.95, 58.74), (15.67, 48.62), (7.9, 66.19), (10.14, 60.87), (18.48, 67.0), (6.38, 67.26), (8.26, 75.34), (11.28, 71.27), (9.84, 56.72), (18.89, 71.29), (13.57, 50.52), (12.44, 61.99), (11.45, 69.69), (9.55, 50.86), (6.48, 83.58), (7.84, 61.13), (10.43, 53.63), (5.12, 49.13), (7.9, 55.52), (9.67, 78.37), (10.03, 54.07), (5.76, 63.71), (8.26, 56.13), (15.72, 58.76), (12.0, 78.78), (6.69, 48.58), (4.66, 77.17), (4.18, 76.45), (6.15, 68.62), (6.32, 54.85), (4.98, 89.02), (14.23, 45.67), (3.34, 54.17), (5.84, 51.29), (10.98, 60.64), (15.76, 60.44), (10.22, 67.85), (2.16, 38.81), (12.24, 59.79), (16.97, 91.99), (20.71, 80.88), (9.78, 63.6), (6.88, 50.24), (11.93, 56.79), (8.47, 45.34), (8.27, 61.54), (-3.28, 53.63), (1.51, 55.04), (5.45, 84.53), (8.57, 64.43), (7.59, 71.47), (25.1, 70.29), (10.64, 48.2), (5.86, 67.67), (9.67, 70.74), (9.06, 85.93), (13.48, 65.54), (11.39, 59.11), (16.63, 81.41), (4.43, 79.61), (13.9, 70.14), (12.67, 62.18), (5.36, 65.52), (8.01, 87.2), (13.76, 69.64), (4.18, 54.64), (20.64, 59.59), (6.3, 65.52), (12.12, 43.4), (11.32, 55.57), (1.13, 56.91), (10.52, 75.6), (20.26, 89.75), (10.68, 43.37), (10.48, 71.09), (7.34, 57.34), (11.14, 49.44), (4.27, 73.44), (9.17, 69.25), (0.15, 55.52), (13.83, 83.8), (12.15, 82.97), (7.78, 83.31), (4.51, 56.24), (14.34, 85.2), (18.01, 63.73), (6.05, 65.2), (16.78, 85.32), (15.68, 56.4), (10.9, 67.12), (7.76, 68.45), (8.21, 71.87), (7.92, 74.81), (7.03, 64.87), (13.51, 74.68), (8.01, 66.25), (9.61, 52.98), (13.08, 85.43), (21.6, 84.33), (5.87, 60.99), (10.68, 62.82), (5.08, 80.83), (14.88, 63.9), (16.09, 63.97), (8.8, 92.3), (-1.49, 44.56), (12.07, 66.78), (7.98, 85.77), (13.13, 63.27), (17.79, 77.85), (7.41, 69.32), (11.38, 47.79), (11.43, 61.4), (8.47, 57.8), (20.97, 48.32), (13.87, 63.4), (5.45, 37.72), (9.98, 63.85), (9.28, 79.87), (8.75, 61.55), (8.59, 74.38), (7.77, 55.81), (7.63, 48.78), (9.69, 66.57), (15.23, 55.18), (6.97, 70.81), (17.58, 66.67), (13.5, 60.5), (12.46, 78.42), (9.63, 81.87), (17.26, 62.07), (2.09, 73.35), (12.67, 56.86), (5.72, 74.35), (4.22, 85.77), (3.86, 84.5), (6.99, 66.99), (13.01, 83.23), (9.75, 65.98), (2.01, 83.96), (5.58, 61.23), (9.14, 61.15), (18.09, 55.94), (9.05, 81.6), (6.25, 69.81), (5.94, 69.69), (3.89, 66.5), (3.73, 63.47), (8.25, 62.31), (10.52, 53.15), (10.27, 56.46), (11.93, 55.15), (17.46, 81.09), (11.61, 55.77), (12.64, 68.64), (8.47, 47.65), (14.75, 78.62), (0.45, 41.05), (1.78, 54.55), (9.45, 66.9), (9.29, 73.01), (8.73, 66.87), (18.22, 68.99), (13.81, 63.36), (24.02, 60.79), (5.23, 84.54), (11.72, 64.71), (3.13, 48.95), (7.44, 64.06), (10.62, 66.37), (7.31, 64.14), (14.89, 67.38), (11.03, 63.16), (9.9, 51.83), (4.08, 48.67), (11.91, 68.09), (13.11, 73.88), (6.8, 82.64), (8.9, 64.43), (10.6, 65.22), (8.34, 63.05), (5.77, 51.6), (-0.17, 37.9), (5.01, 48.67), (4.88, 73.24), (10.18, 59.96), (12.99, 72.59), (3.08, 56.75), (11.47, 51.76), (10.23, 68.96), (2.84, 77.45), (8.94, 53.08), (3.93, 62.14), (13.36, 73.42), (12.24, 59.07), (10.93, 72.73), (11.07, 63.84), (14.94, 65.67), (13.8, 85.36), (13.44, 75.2), (14.33, 78.07), (12.72, 71.44), (2.72, 55.06), (8.31, 80.06), (10.8, 83.14), (13.65, 65.24), (6.48, 48.86), (7.87, 63.47), (3.47, 69.84), (4.4, 64.21), (13.55, 91.62), (11.41, 82.62), (4.62, 71.47), (9.45, 63.7), (12.2, 71.61), (13.25, 56.8), (6.69, 63.14), (13.2, 75.37), (-2.53, 46.26), (15.89, 78.28), (19.4, 61.62), (4.08, 55.93), (12.78, 60.85), (5.22, 59.84), (11.6, 74.95), (2.45, 68.36), (8.58, 62.59), (13.45, 80.03), (3.6, 66.59), (4.94, 75.95), (12.71, 65.84), (7.87, 63.24), (11.42, 65.42), (10.85, 63.47), (5.0, 64.74), (11.19, 81.29), (4.69, 73.9), (-1.37, 54.75), (16.04, 58.3), (6.93, 58.48), (17.66, 89.49), (4.81, 64.97), (6.39, 67.5), (17.32, 61.45), (13.13, 62.06), (15.04, 59.58), (3.81, 68.26), (5.02, 53.54), (8.57, 49.26), (13.9, 74.0), (12.47, 79.4), (4.34, 77.05), (8.37, 58.3), (11.71, 78.55), (11.0, 79.94), (10.3, 64.99), (10.31, 75.08), (4.82, 42.95), (7.4, 65.99), (10.95, 45.66), (12.19, 56.09), (0.71, 58.1), (9.12, 73.43), (9.02, 50.28), (11.08, 65.64), (10.49, 65.66), (8.78, 39.12), (16.3, 61.45), (12.74, 65.92), (11.01, 57.2), (6.78, 39.19), (2.69, 52.45), (9.88, 49.84), (12.45, 48.59), (9.45, 85.4), (11.62, 84.67), (11.51, 67.73), (9.97, 75.35), (11.68, 68.75), (3.81, 49.21), (12.54, 59.33), (7.46, 73.48), (10.77, 80.34), (-2.07, 41.98), (6.12, 63.12), (13.04, 73.44), (12.25, 65.07), (10.79, 90.87), (8.76, 54.89), (8.54, 59.48), (13.44, 81.82), (1.56, 70.12), (10.65, 68.9), (15.47, 64.96), (9.29, 55.23), (16.55, 69.17), (3.81, 63.87), (4.78, 67.13), (11.98, 67.65), (12.59, 71.78), (4.17, 44.33), (13.58, 64.03), (19.57, 83.82), (-0.32, 53.69), (3.21, 53.78), (7.9, 57.1), (16.27, 84.99), (6.9, 56.46), (10.16, 65.61), (5.73, 59.11), (2.88, 57.34), (14.67, 78.31), (7.08, 68.96), (14.81, 66.76), (12.29, 63.8), (0.97, 54.49), (13.06, 80.08), (14.63, 61.69), (9.9, 70.83), (3.46, 65.5), (13.02, 86.64), (0.19, 55.19), (17.12, 79.37), (11.95, 68.32), (7.68, 67.76), (8.29, 67.08), (9.76, 97.81), (14.24, 55.28), (8.93, 49.35), (12.08, 46.23), (6.64, 39.96), (7.45, 54.38), (9.23, 93.54), (9.37, 67.94), (20.72, 56.21), (9.95, 62.04), (7.27, 55.81), (21.34, 69.29), (14.72, 58.11), (12.72, 67.57), (14.31, 58.1), (-1.8, 38.69), (8.4, 65.46), (6.59, 55.46), (4.16, 56.32), (9.77, 51.04), (18.65, 79.23), (17.24, 68.56), (15.6, 63.54), (-0.93, 58.31), (10.35, 59.96), (10.68, 67.92), (16.71, 69.41), (14.57, 76.62), (16.93, 92.68), (10.38, 76.45), (8.83, 64.35), (15.26, 68.95), (13.69, 61.03), (11.75, 65.0), (3.28, 50.3), (1.19, 61.31), (10.71, 57.7), (9.64, 77.21), (5.39, 83.08), (3.12, 43.76), (8.77, 42.15), (15.95, 55.41), (9.01, 77.14), (4.67, 57.65), (14.26, 56.01), (16.26, 67.96), (-0.43, 53.46), (13.38, 60.44), (7.74, 68.57), (11.25, 75.69), (7.21, 67.98), (3.84, 68.21), (13.77, 74.82), (10.46, 54.54), (6.76, 75.02), (4.59, 55.3), (13.72, 65.06), (4.58, 72.34), (15.95, 58.3), (7.78, 64.96), (4.32, 56.75), (9.92, 85.25), (10.06, 50.66), (12.39, 72.58), (18.27, 59.12), (10.89, 76.07), (17.86, 61.07), (10.73, 49.05), (16.31, 75.02), (14.16, 78.12), (13.44, 65.1), (8.42, 53.98), (-3.23, 58.4), (13.47, 66.26), (15.1, 55.41), (10.27, 63.86), (13.59, 79.9), (12.97, 45.37), (10.09, 35.12), (14.35, 46.25), (9.11, 51.01), (18.14, 59.03), (9.44, 73.59), (13.99, 69.09), (3.69, 64.64), (10.14, 64.31), (8.56, 66.27), (-3.74, 56.82), (4.99, 61.09), (9.31, 68.56), (13.39, 64.43), (18.76, 71.9), (6.26, 79.25), (8.64, 65.03), (13.62, 53.48), (10.76, 56.14), (6.1, 80.82), (17.88, 60.27), (10.25, 56.93), (4.03, 65.97), (6.35, 63.92), (8.99, 83.79), (18.65, 86.99), (9.12, 82.07), (18.62, 66.85), (10.62, 65.74), (10.04, 73.35), (6.4, 53.79), (12.37, 60.24), (14.81, 57.86), (14.37, 71.77), (21.9, 81.78), (7.94, 65.07), (11.3, 90.2), (5.6, 61.56), (11.86, 65.37), (13.11, 59.52), (8.05, 83.32), (11.18, 37.91), (10.31, 51.04), (21.18, 54.16), (13.75, 75.94), (13.13, 69.16), (-1.25, 56.15), (5.45, 64.35), (6.17, 70.37), (5.94, 57.65), (8.64, 42.08), (12.55, 59.35), (9.42, 47.71), (10.16, 50.63), (-4.43, 41.45), (17.92, 71.98), (5.98, 66.35), (7.13, 73.71), (7.99, 68.78), (7.54, 58.37), (2.21, 53.32), (13.06, 66.89), (11.86, 74.54), (12.7, 70.44), (15.17, 52.71), (19.6, 64.97), (2.14, 61.67), (8.19, 67.12), (9.85, 58.72), (13.16, 53.0), (5.44, 62.23), (11.23, 66.47), (9.32, 52.33), (5.02, 85.71), (19.24, 85.27), (6.73, 74.05), (7.35, 61.38), (9.14, 57.85), (10.81, 65.99), (10.49, 59.77), (15.18, 52.85), (12.73, 91.86), (12.99, 74.32), (14.81, 64.7), (17.99, 69.72), (6.66, 66.33), (19.92, 72.75), (3.36, 76.93), (12.17, 65.53), (12.68, 50.25), (14.64, 61.52), (15.62, 53.27), (6.12, 45.87), (20.74, 96.22), (11.9, 73.96), (12.32, 77.55), (7.51, 39.04), (4.7, 42.27), (-2.33, 53.51), (12.99, 50.67), (12.04, 81.56), (11.63, 65.54), (8.07, 62.17), (19.03, 84.52), (7.82, 78.19), (11.54, 40.63), (14.47, 69.7), (11.84, 44.66), (14.64, 46.82), (8.15, 62.49), (2.46, 72.87), (8.47, 42.7), (11.81, 52.3), (17.27, 74.89), (3.52, 78.75), (7.61, 76.09), (7.9, 70.61), (16.2, 74.17), (19.41, 61.2), (4.49, 56.45), (15.75, 82.87), (10.55, 74.64), (9.67, 65.68), (7.46, 54.52), (8.02, 53.64), (1.53, 75.98), (13.68, 74.75), (14.56, 81.78), (21.5, 70.34), (11.45, 80.53), (7.92, 77.14), (11.17, 76.55), (-2.59, 56.9), (19.26, 76.61), (13.45, 71.27), (11.32, 55.9), (17.13, 59.09), (13.63, 62.97), (9.44, 52.83), (4.76, 71.68), (9.54, 60.1), (10.46, 66.33), (10.17, 72.38), (15.81, 75.17), (11.65, 71.26), (14.57, 80.69), (22.97, 67.7), (15.6, 74.31), (10.42, 65.85), (17.99, 62.82), (11.64, 48.93), (6.47, 49.01), (1.09, 47.11), (11.05, 70.01), (24.22, 74.54), (2.91, 54.4), (8.2, 77.01), (12.64, 64.44), (16.68, 81.8), (1.54, 73.12), (11.39, 56.91), (5.4, 68.14), (8.6, 77.09), (16.25, 56.83), (15.16, 81.7), (7.95, 63.77), (12.72, 76.97), (6.07, 56.68), (7.32, 59.19), (9.8, 69.53), (19.42, 68.8), (3.84, 63.21), (13.15, 59.86), (14.31, 63.73), (5.48, 61.57), (12.17, 49.63), (6.95, 64.59), (12.52, 63.91), (10.76, 81.84), (7.61, 80.52), (10.74, 64.58), (16.66, 76.12), (11.92, 39.8), (14.25, 70.34), (6.66, 58.46), (8.13, 77.99), (17.63, 78.93), (4.88, 78.02), (8.76, 69.58), (0.84, 48.46), (15.49, 82.43), (11.63, 74.7), (9.15, 48.07), (14.03, 72.3), (12.13, 73.17), (15.0, 65.43), (12.55, 63.02), (21.42, 78.72), (8.21, 58.0), (11.59, 71.49), (16.53, 66.93), (11.95, 86.09), (10.84, 69.61), (10.88, 66.23), (7.31, 64.37), (13.12, 70.86), (9.49, 80.73), (14.49, 89.54), (6.27, 67.56), (13.39, 58.64), (9.86, 69.05), (12.86, 61.23), (13.08, 38.45), (10.06, 59.93), (7.52, 70.19), (11.66, 57.73), (11.64, 77.19), (8.45, 65.0), (9.04, 57.58), (16.39, 66.95), (9.82, 64.88), (6.54, 69.17), (13.91, 61.35), (-2.87, 51.2), (11.99, 64.26), (13.15, 60.69), (4.28, 47.42), (5.01, 60.51), (7.37, 62.08), (6.9, 51.29), (11.63, 68.78), (8.8, 54.73), (12.89, 64.7), (14.42, 52.49), (13.84, 82.85)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=849；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2517；
t = r·√((n−2)/(1−r²)) = 0.2517·√((849−2)/(1−0.0634)) = 7.5699；df=847；p = -2.70e-07。
95% CI（Fisher z）：tanh(z±25.190·1/√(n−3)) = [-0.5433, 0.8087]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=849；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.2517；
t = r·√((n−2)/(1−r²)) = 0.2517·√((849−2)/(1−0.0634)) = 7.5699；df=847；p = -2.70e-07。
95% CI（Fisher z）：tanh(z±25.190·1/√(n−3)) = [-0.5433, 0.8087]。
③ 计算结果：
r = 0.2517，t = 7.5699，p = -2.70e-07，95%CI = [-0.5433, 0.8087]
④ 结论：在 α=0.05 水平下，p=-2.70e-07 < 0.05，拒绝 H₀。两变量存在线性相关（r=0.2517，方向正），弱相关。
案例 17. 气温(℃)与羽绒服销量(件)
Pearson相关天冷卖得多
① 原始数据：
样本 n=986，逐条 (x, y)：[(13.25, 263.91), (15.39, 133.36), (19.9, 104.73), (22.99, 225.69), (29.4, 116.39), (12.94, 164.76), (21.93, 104.22), (22.11, 159.86), (10.8, 193.17), (7.44, 234.87), (25.11, 221.48), (10.56, 194.97), (25.61, 86.93), (5.09, 181.42), (31.62, 173.65), (16.43, 191.38), (11.27, 146.57), (26.34, 151.63), (17.28, 228.2), (24.39, 178.51), (12.11, 215.43), (12.29, 116.48), (27.04, 296.73), (14.87, 155.85), (23.76, 220.11), (4.17, 326.21), (7.04, 238.16), (7.08, 224.66), (19.1, 203.46), (16.87, 183.34), (11.39, 301.25), (16.72, 283.12), (20.59, 300.18), (43.85, 168.54), (15.63, 118.27), (2.55, 196.52), (10.54, 204.14), (6.81, 213.55), (18.42, 155.28), (3.28, 184.18), (-12.69, 397.8), (24.63, 123.87), (14.67, 270.52), (-1.0, 348.25), (12.2, 207.16), (7.17, 149.13), (25.72, 153.93), (-0.74, 278.32), (9.2, 220.71), (18.74, 185.61), (9.8, 207.19), (23.37, 85.75), (34.64, 141.03), (16.42, 281.86), (32.94, 146.71), (25.68, 194.3), (14.95, 328.65), (27.57, 94.23), (31.88, 64.94), (14.0, 214.87), (24.88, 330.35), (28.64, 39.53), (18.1, 135.4), (18.04, 90.71), (4.04, 309.22), (4.67, 302.99), (16.13, 152.15), (20.73, 116.08), (4.21, 296.25), (20.88, 229.53), (10.33, 221.88), (21.54, 116.52), (23.43, 74.71), (7.11, 121.4), (9.94, 203.04), (22.45, 69.88), (36.64, 65.33), (14.71, 170.36), (11.26, 219.27), (8.59, 253.03), (-3.4, 307.96), (7.97, 174.86), (17.5, 195.74), (27.86, 109.98), (32.44, 165.58), (22.38, 147.62), (6.47, 215.3), (0.93, 263.9), (19.61, 196.29), (7.0, 313.49), (24.31, 197.34), (29.39, 91.74), (19.45, 61.53), (28.31, 33.33), (25.6, 161.25), (13.76, 173.69), (32.63, 221.57), (14.52, 227.46), (7.53, 296.21), (34.23, 204.27), (2.12, 260.63), (13.85, 199.55), (13.67, 213.6), (18.45, 106.98), (11.89, 80.77), (5.22, 239.97), (9.66, 249.11), (1.64, 299.12), (33.35, 64.28), (23.36, 109.75), (14.08, 96.19), (17.46, 151.25), (4.53, 271.34), (29.58, 61.24), (13.8, 221.15), (21.99, 18.54), (3.72, 250.14), (40.51, 136.88), (18.52, 219.83), (7.39, 378.37), (14.85, 148.3), (35.94, 200.24), (25.6, 95.57), (15.18, 130.3), (12.48, 188.27), (6.86, 230.03), (20.62, 161.53), (4.94, 86.47), (32.64, -6.03), (16.91, 178.49), (2.31, 313.17), (22.87, 236.87), (6.78, 118.25), (1.67, 287.03), (10.58, 162.39), (13.93, 232.72), (16.3, 225.62), (20.18, 230.73), (15.83, 270.02), (12.93, 194.18), (19.52, 145.27), (20.77, 98.06), (19.19, 131.95), (16.42, 232.65), (23.78, 236.31), (4.48, 219.15), (26.24, 139.67), (25.42, 155.73), (26.73, 67.95), (27.0, 75.53), (3.65, 291.19), (34.58, 93.24), (18.74, 239.83), (7.45, 369.71), (7.34, 270.88), (14.58, 225.99), (19.01, 224.3), (-3.89, 75.68), (17.33, 257.03), (1.87, 338.64), (6.56, 246.91), (15.44, 132.33), (6.14, 267.1), (-1.78, 180.76), (15.18, 91.81), (16.54, 264.4), (16.9, 130.83), (19.61, 244.95), (6.85, 249.41), (15.78, 226.89), (30.85, 102.61), (7.31, 225.04), (24.5, 211.39), (16.71, 263.89), (25.87, 298.35), (3.1, 326.31), (20.06, 209.14), (19.82, 90.17), (6.93, 234.19), (16.25, 279.89), (27.11, 201.6), (9.68, 175.07), (19.22, 280.02), (9.14, 223.17), (20.09, 135.15), (9.44, 248.38), (38.39, 118.33), (25.81, 252.23), (18.8, 97.38), (27.71, 115.27), (-8.23, 197.88), (3.89, 212.35), (9.01, 105.72), (4.39, 173.82), (5.16, 331.39), (34.18, 114.86), (24.77, 184.14), (27.34, 127.68), (-2.31, 276.66), (21.24, 191.07), (34.77, 206.02), (16.02, 194.11), (26.16, 213.79), (11.04, 86.76), (15.71, 216.82), (15.83, 171.88), (7.21, 174.88), (24.36, 194.62), (20.15, 142.05), (12.51, 234.83), (12.09, 286.63), (10.49, 154.32), (27.18, 150.49), (5.29, 257.41), (18.21, 245.07), (7.36, 237.87), (29.55, 117.41), (16.91, 104.03), (6.67, 188.4), (10.18, 234.74), (23.86, -5.61), (13.65, 255.26), (9.16, 251.53), (14.19, 151.4), (6.52, 239.11), (3.87, 247.09), (35.07, 160.59), (-4.21, 284.05), (13.2, 299.06), (3.25, 247.25), (20.83, 187.1), (12.96, 165.85), (12.97, 206.09), (11.2, 265.8), (4.14, 268.46), (24.98, 177.92), (19.4, 167.48), (5.46, 271.58), (9.19, 71.27), (9.04, 265.24), (6.63, 374.58), (33.5, 55.17), (36.02, 192.52), (17.17, 234.53), (18.66, 193.48), (20.14, 198.46), (10.33, 196.42), (21.06, 130.84), (15.19, 261.11), (17.83, 202.57), (9.07, 148.29), (5.22, 277.98), (6.13, 230.95), (21.74, 75.11), (25.24, 73.67), (9.51, 192.36), (14.14, 230.49), (15.52, 155.97), (24.57, 302.72), (24.28, 130.7), (7.11, 183.63), (25.42, 221.79), (18.46, 252.94), (33.67, 162.23), (6.76, 228.14), (1.65, 233.53), (23.82, 121.9), (29.74, 145.06), (33.45, 79.9), (27.31, 129.11), (15.83, 158.21), (7.39, 199.4), (12.77, 287.69), (20.43, 286.74), (34.43, 172.69), (21.44, 184.15), (14.41, 275.78), (29.27, 145.57), (16.57, 67.93), (7.39, 186.94), (15.88, 34.73), (-5.69, 199.77), (28.62, 30.91), (22.68, 147.94), (14.39, 103.76), (22.23, 102.79), (12.48, 129.2), (17.71, 125.4), (30.75, 166.82), (8.17, 337.53), (14.16, 241.53), (9.01, 139.53), (11.1, 100.37), (20.64, 73.19), (12.27, 371.89), (7.46, 205.85), (16.19, 126.31), (19.34, 87.14), (4.77, 246.12), (10.05, 245.9), (8.26, 260.05), (2.28, 205.06), (2.94, 302.91), (30.58, 50.08), (-5.31, 273.15), (1.68, 245.48), (7.7, 276.13), (12.24, 179.96), (28.31, 93.27), (15.28, 251.62), (16.41, 260.44), (19.79, 182.15), (5.42, 233.88), (1.16, 230.45), (13.24, 280.89), (34.85, 160.17), (13.12, 235.44), (32.64, 92.25), (22.81, 192.37), (9.66, 250.67), (19.48, 292.43), (16.27, 226.97), (12.63, 131.51), (-1.98, 243.75), (16.82, 126.83), (15.97, 329.38), (12.2, 171.51), (16.09, 171.19), (14.74, 62.39), (11.72, 123.96), (29.19, 203.04), (15.16, 343.88), (23.33, 210.23), (1.98, 235.49), (9.52, 140.75), (19.6, 253.7), (14.24, 229.87), (19.51, 225.23), (22.64, 165.65), (17.81, 236.55), (13.09, 105.78), (13.57, 137.52), (13.94, 292.17), (21.78, 145.18), (31.53, 67.94), (20.49, 162.71), (14.15, 207.61), (15.23, 258.16), (36.77, 83.17), (5.6, 159.43), (18.3, 313.11), (16.63, 156.89), (-0.32, 260.57), (6.55, 210.38), (-5.49, 377.04), (11.42, 164.1), (27.54, 244.5), (9.93, 224.13), (21.89, 176.65), (-0.89, 406.74), (25.34, 255.46), (10.5, 209.67), (3.96, 287.54), (26.93, 322.69), (28.93, 231.63), (24.24, 102.32), (22.77, 243.12), (11.49, 268.28), (17.64, 153.47), (1.21, 261.56), (-2.99, 330.06), (12.1, 212.31), (9.7, 189.63), (14.12, 215.07), (19.8, 156.22), (14.84, 284.43), (14.85, 166.19), (24.52, 69.37), (15.64, 154.58), (28.05, 86.31), (20.04, 88.88), (20.48, 287.72), (18.1, 213.22), (20.99, 139.34), (35.8, 194.67), (28.59, 90.92), (2.7, 241.84), (10.29, 383.88), (24.76, 137.57), (3.01, 272.66), (19.81, 115.24), (12.16, 165.29), (20.35, 222.13), (8.65, 281.14), (18.63, 284.1), (30.49, 159.7), (6.31, 179.86), (20.28, 211.0), (15.76, 246.08), (8.94, 344.05), (28.89, 230.48), (-2.06, 264.78), (5.99, 249.23), (6.86, 283.09), (14.19, 279.53), (2.59, 174.04), (30.98, 82.51), (15.59, 128.54), (15.64, 186.23), (20.33, 229.41), (19.55, 138.95), (6.71, 329.05), (29.84, 175.31), (4.63, 291.14), (12.99, 199.16), (6.2, 182.35), (14.7, 216.2), (30.18, -33.29), (3.81, 318.01), (17.13, 126.85), (20.42, 227.82), (16.21, 164.51), (9.33, 250.75), (9.87, 159.97), (9.2, 377.82), (17.45, 160.66), (24.09, 155.06), (12.97, 192.76), (19.92, 255.44), (26.94, 176.69), (21.76, 218.12), (0.82, 243.92), (12.03, 185.6), (5.02, 262.02), (9.93, 119.98), (26.49, 221.44), (7.21, 216.28), (12.17, 205.77), (8.32, 117.17), (23.07, 168.54), (30.49, 190.06), (23.16, 174.29), (5.31, 47.55), (26.62, 106.48), (25.61, 295.34), (19.84, 152.1), (17.79, 65.94), (11.56, 154.78), (19.52, 138.3), (13.79, 176.27), (15.01, 190.76), (13.13, 243.84), (16.52, 130.33), (20.66, 205.04), (8.45, 189.25), (16.82, 171.45), (12.43, 191.92), (28.46, 118.69), (24.64, 169.77), (21.76, 154.75), (34.65, 20.4), (21.47, 66.89), (0.07, 381.83), (21.61, 204.4), (2.63, 150.36), (11.43, 217.63), (37.58, 225.24), (15.64, 128.83), (14.35, 127.12), (20.71, 193.07), (13.25, 278.12), (10.48, 238.18), (17.03, 239.98), (23.0, 173.11), (19.5, 293.47), (28.9, 247.39), (7.64, 204.06), (18.87, 118.71), (-3.06, 338.68), (35.3, 183.04), (0.36, 432.6), (23.84, 77.57), (7.69, 118.74), (12.1, 140.97), (10.1, 168.1), (38.38, 48.61), (13.14, 307.76), (17.87, 146.1), (13.23, 186.1), (3.92, 227.87), (27.77, 68.05), (13.62, 258.3), (16.53, 262.16), (23.24, 98.21), (27.38, 53.12), (18.6, 99.86), (11.8, 275.96), (24.91, -18.78), (13.75, 229.61), (22.55, 304.74), (15.66, 138.6), (18.68, 220.94), (12.87, 87.99), (12.02, 183.96), (28.0, 179.89), (11.66, 179.38), (18.08, 210.84), (24.58, 25.72), (30.89, 167.15), (7.01, 304.55), (27.21, 251.14), (13.25, 143.22), (21.87, 85.23), (18.62, 204.17), (4.67, 393.11), (6.14, 175.99), (20.73, 237.46), (13.49, 173.84), (10.56, 144.89), (27.04, 100.72), (5.51, 274.27), (7.89, 174.09), (0.25, 386.14), (23.77, 195.34), (-6.48, 283.28), (7.6, 258.98), (-3.82, 251.55), (22.62, 206.23), (7.02, 237.66), (14.58, 274.3), (-1.51, 200.85), (22.44, 251.82), (12.5, 170.76), (17.9, 274.25), (5.66, 114.99), (34.18, 57.26), (21.3, 188.4), (16.62, 295.72), (15.47, 294.74), (-0.05, 198.74), (30.17, 239.16), (20.9, 104.76), (18.6, 98.44), (6.26, 142.33), (16.33, 180.51), (17.02, 201.47), (26.68, 136.86), (20.29, 363.12), (23.71, 45.36), (4.58, 233.36), (21.49, 168.02), (26.44, 75.71), (5.79, 185.46), (2.42, 240.39), (7.18, 148.82), (20.32, 217.29), (32.02, 46.14), (15.79, 169.53), (2.69, 309.42), (20.86, 6.22), (21.24, 221.16), (12.89, 108.83), (-5.18, 349.01), (22.09, 95.1), (20.49, 309.26), (24.1, 42.94), (10.53, 278.78), (-2.25, 224.57), (5.09, 223.01), (-7.64, 266.66), (16.25, 180.09), (24.8, 186.19), (8.31, 304.55), (20.21, 205.41), (7.58, 251.67), (24.45, 39.43), (3.34, 434.97), (7.97, 305.1), (20.48, 225.43), (20.39, 199.37), (5.21, 158.74), (19.59, 297.02), (10.4, 203.33), (33.93, 152.38), (18.22, 222.52), (-8.12, 271.28), (13.2, 119.51), (21.3, 142.73), (12.86, 319.09), (23.17, 157.14), (7.9, 200.6), (28.74, 254.7), (15.24, 247.29), (2.72, 317.49), (10.69, 345.27), (17.32, 237.68), (18.19, 232.15), (6.52, 213.73), (-7.42, 277.07), (8.33, 203.69), (31.62, 228.04), (0.86, 293.02), (5.1, 105.29), (7.47, 286.66), (26.38, 167.64), (7.62, 340.61), (20.63, 294.76), (6.28, 245.83), (12.96, 224.51), (3.43, 306.37), (10.18, 178.65), (27.81, 73.46), (17.78, 190.21), (4.59, 267.76), (5.46, 138.47), (-2.58, 190.62), (23.56, 146.16), (20.19, 210.48), (17.26, 196.02), (16.25, 245.04), (15.88, 271.27), (-2.3, 330.81), (18.34, 244.0), (17.59, 90.32), (13.04, 201.56), (30.2, 143.08), (5.81, 248.88), (20.65, 308.75), (23.86, 166.65), (22.15, 360.86), (7.33, 124.91), (25.22, 265.08), (11.91, 190.21), (29.16, 120.69), (19.56, 267.24), (17.96, 113.89), (-2.94, 290.28), (6.51, 144.08), (20.82, 193.54), (27.95, 262.59), (34.13, 154.84), (-2.92, 344.27), (48.25, 46.46), (-10.77, 317.62), (21.28, 189.08), (23.42, 181.05), (0.3, 382.61), (25.37, 202.8), (21.81, 202.24), (13.51, 228.24), (23.7, 105.6), (15.15, 170.47), (9.84, 179.15), (11.48, 304.9), (2.43, 211.54), (37.42, -32.41), (13.66, 177.08), (9.66, 142.13), (9.77, 294.34), (24.46, 222.49), (19.24, 118.85), (16.16, 111.32), (9.99, 143.16), (17.18, 161.58), (20.66, 307.08), (11.11, 362.35), (17.65, 82.67), (28.84, 130.37), (20.87, 142.08), (-1.97, 408.27), (34.04, 164.48), (34.52, 141.35), (17.22, 197.25), (15.34, 281.27), (36.12, 234.99), (11.77, 157.54), (17.85, 52.62), (15.93, 262.15), (25.01, 253.06), (9.1, 243.89), (11.8, 241.42), (0.61, 158.66), (21.42, 50.12), (25.74, 142.8), (-7.18, 159.84), (3.17, 222.37), (12.5, 267.06), (6.62, 339.85), (19.9, 148.95), (18.83, 156.55), (13.91, 259.67), (24.05, 124.19), (28.45, 255.13), (24.3, 118.07), (28.38, 58.86), (21.99, 249.2), (24.4, 190.73), (12.12, 271.49), (17.68, 322.75), (17.79, 185.88), (12.51, 47.57), (-0.01, 187.48), (11.61, 222.04), (10.84, 127.2), (21.81, 143.45), (3.54, 189.8), (15.19, 285.47), (4.42, 276.31), (1.43, 260.35), (18.9, 217.61), (34.04, 42.97), (8.78, 215.95), (1.12, 156.96), (15.58, 164.44), (12.61, 201.43), (27.34, 135.87), (9.23, 174.55), (26.44, 108.74), (7.53, 296.84), (19.52, 309.81), (2.58, 156.93), (7.14, 273.22), (23.85, 62.15), (22.66, 166.36), (20.32, 169.79), (-1.54, 342.88), (16.2, 190.39), (22.41, -24.44), (18.87, 230.47), (34.22, 192.37), (35.36, 78.66), (8.88, 129.49), (3.5, 247.41), (14.94, 33.31), (30.47, 131.44), (-1.78, 141.68), (9.16, 217.19), (8.56, 155.49), (-3.91, 264.15), (10.55, 214.99), (-2.01, 290.97), (18.86, 331.16), (17.58, 268.76), (16.61, 285.8), (38.0, 134.49), (6.78, 291.68), (29.48, 75.47), (13.72, 102.68), (16.51, 167.06), (-0.18, 207.71), (35.05, 74.75), (-1.85, 143.13), (24.25, 25.11), (23.09, 274.45), (12.31, 218.28), (21.55, 168.21), (13.5, 192.77), (28.56, 84.46), (8.12, 188.61), (28.4, 65.18), (10.12, 113.12), (6.53, 303.07), (29.1, 100.73), (23.62, 178.43), (17.18, 276.0), (9.85, 205.49), (24.67, 176.92), (7.27, 144.71), (11.83, 205.5), (17.54, 155.95), (17.94, 134.97), (25.51, 181.88), (13.65, 319.77), (7.43, 154.66), (34.42, 139.6), (23.26, 189.94), (29.38, 179.4), (24.23, 219.15), (27.99, 191.85), (7.56, 136.28), (23.9, 145.65), (-9.37, 357.94), (-3.92, 274.02), (17.78, 240.58), (24.54, 122.87), (10.02, 174.17), (14.56, 179.23), (4.91, 220.21), (18.92, 301.19), (16.61, 269.21), (19.69, 165.95), (0.1, 164.94), (21.81, 259.64), (7.37, 235.83), (17.03, 236.78), (16.55, 68.7), (26.07, 153.89), (21.03, 275.06), (21.23, 171.41), (19.79, 211.07), (-4.28, 251.78), (26.54, 114.61), (13.7, 177.85), (2.9, 228.56), (8.69, 205.43), (18.53, 138.4), (-13.02, 244.33), (24.12, 164.48), (1.89, 102.82), (1.78, 255.14), (12.04, 223.91), (-9.7, 321.42), (24.64, 186.8), (14.79, 227.52), (10.45, 315.29), (25.37, 234.02), (-5.67, 204.25), (11.94, 227.88), (19.14, 212.65), (11.18, 223.48), (21.97, 248.48), (8.05, 170.09), (5.74, 98.13), (19.07, 307.1), (7.54, 295.13), (35.84, 199.07), (9.34, 179.73), (9.63, 236.92), (-12.0, 276.79), (21.86, 192.4), (4.71, 200.49), (20.46, 260.74), (11.64, 91.13), (14.56, 154.37), (17.42, 161.01), (10.64, 226.61), (9.15, 276.97), (4.39, 148.41), (4.31, 245.36), (14.35, 268.12), (6.15, 337.08), (24.49, 144.25), (5.18, 184.81), (38.58, 222.18), (27.35, 162.77), (10.99, 141.44), (4.82, 228.98), (14.08, 242.89), (29.9, 112.45), (13.78, 203.66), (27.47, 47.81), (-1.98, 198.88), (25.39, 189.4), (22.37, 167.91), (6.05, 175.27), (12.98, 29.85), (21.24, 225.08), (26.8, 123.01), (27.21, 141.46), (6.41, 239.23), (21.65, 202.15), (11.1, 129.02), (12.73, 299.03), (31.23, 73.44), (29.78, 68.19), (24.31, 85.76), (9.99, 245.66), (18.78, 184.35), (24.99, 225.42), (-8.98, 355.21), (15.69, 222.59), (19.69, 39.45), (26.95, 124.5), (14.32, 231.45), (28.78, 156.48), (29.55, 200.57), (-3.51, 346.94), (23.4, 243.64), (7.84, 276.11), (18.31, 166.9), (25.0, 139.55), (19.94, 227.95), (16.07, 39.5), (21.08, 239.84), (11.3, 251.14), (21.27, 390.22), (34.36, 128.2), (16.31, 116.89), (11.64, 94.14), (19.36, 82.31), (31.88, 103.09), (24.4, 75.34), (17.07, 262.46), (16.85, 362.88), (10.58, 233.19), (22.37, 162.24), (22.95, 283.77), (0.14, 346.86), (17.28, 127.77), (22.21, 219.4), (6.62, 72.4), (14.8, 212.18), (18.69, 171.89), (19.6, 50.04), (16.27, 247.2), (9.21, 120.21), (6.14, 260.92), (10.33, 303.74), (-0.13, 314.58), (17.14, 245.84), (29.11, 156.51), (4.5, 242.52), (40.24, 108.43), (34.85, 91.74), (11.27, 174.0), (5.3, 246.41), (24.4, 255.45), (15.3, 267.75), (19.48, 8.35), (9.07, 233.96), (-4.89, 380.86), (35.27, -60.72), (11.43, 202.32), (6.2, 227.95), (28.34, 98.77), (10.76, 261.57), (16.39, 146.76), (0.39, 296.69), (25.8, 143.86), (9.51, 277.4), (26.77, 186.54), (26.02, 195.84), (24.54, 109.66), (9.53, 284.94), (8.98, 233.97), (21.95, 10.99), (18.11, 81.35), (13.51, 163.73), (12.61, 174.89), (20.68, 254.2), (5.33, 276.82), (8.95, 232.59), (35.81, 182.82), (24.58, -43.09), (21.82, 204.89), (21.32, 212.93), (-0.91, 220.79), (1.1, 308.91), (8.93, 197.71), (14.79, 87.53), (0.19, 204.88), (19.22, 214.7), (19.67, 118.44), (17.41, 186.88), (22.43, 94.27), (19.3, 109.55), (8.08, 228.77), (27.52, 112.78), (13.86, 216.59), (10.36, 307.88), (14.86, 181.83), (5.06, 345.46), (21.27, 290.69), (19.58, 264.57), (25.82, 67.87), (19.19, 165.53), (28.06, 210.66), (8.12, 177.3), (22.77, 199.69), (-1.95, 268.46), (23.29, 117.38), (14.83, 277.15), (18.77, 102.21), (3.55, 229.33), (31.48, 24.32), (23.26, 191.04), (23.01, 196.75), (14.02, 242.15), (11.95, 199.22), (31.08, 127.78), (24.16, 155.23), (34.98, 62.37), (9.11, 218.55), (22.33, 140.48), (12.68, 163.31), (5.89, 223.08)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=986；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4840；
t = r·√((n−2)/(1−r²)) = 0.4840·√((986−2)/(1−0.2342)) = -17.3492；df=984；p = -4.98e+26。
95% CI（Fisher z）：tanh(z±31.284·1/√(n−3)) = [-0.9097, 0.4379]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=986；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4840；
t = r·√((n−2)/(1−r²)) = 0.4840·√((986−2)/(1−0.2342)) = -17.3492；df=984；p = -4.98e+26。
95% CI（Fisher z）：tanh(z±31.284·1/√(n−3)) = [-0.9097, 0.4379]。
③ 计算结果：
r = -0.4840，t = -17.3492，p = -4.98e+26，95%CI = [-0.9097, 0.4379]
④ 结论：在 α=0.05 水平下，p=-4.98e+26 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.4840，方向负），天冷卖得多。
案例 18. 商品价格(元)与购买意愿(分)
Pearson相关越贵越不想买
① 原始数据：
样本 n=476，逐条 (x, y)：[(91.67, 2.67), (72.47, 2.4), (55.0, 2.8), (98.18, 2.71), (91.31, 2.24), (120.97, 3.02), (114.71, 3.61), (42.13, 5.14), (89.86, 3.77), (59.53, 4.14), (132.22, 2.61), (55.05, 3.93), (117.23, 4.42), (74.73, 3.54), (69.56, 3.4), (106.41, 4.73), (-0.5, 4.95), (57.01, 4.51), (81.69, 4.02), (33.52, 4.05), (80.15, 4.17), (65.76, 3.83), (101.22, 3.63), (113.63, 2.4), (98.16, 3.46), (118.33, 3.55), (112.78, 3.08), (186.73, 1.86), (88.63, 3.84), (136.88, 3.69), (72.62, 3.52), (97.29, 2.96), (116.89, 3.56), (113.43, 2.26), (106.59, 2.52), (126.85, 2.97), (193.84, 2.11), (87.21, 4.2), (59.65, 4.08), (90.58, 3.73), (72.62, 2.46), (102.95, 3.96), (143.23, 2.14), (76.85, 2.9), (138.98, 5.68), (89.93, 2.43), (163.95, 3.25), (78.17, 3.39), (57.63, 4.69), (122.18, 4.17), (106.15, 3.13), (86.37, 4.85), (87.1, 4.62), (8.84, 3.97), (79.89, 3.61), (44.21, 4.21), (105.67, 4.07), (74.11, 3.97), (80.37, 3.23), (73.94, 4.73), (80.03, 5.0), (91.09, 3.46), (61.71, 2.79), (77.67, 4.33), (87.23, 4.29), (118.27, 2.71), (53.6, 3.93), (80.56, 3.37), (19.67, 5.05), (115.75, 3.32), (123.39, 2.58), (27.55, 6.02), (52.32, 3.99), (116.41, 4.15), (64.17, 3.57), (113.96, 3.75), (63.86, 5.05), (51.98, 3.38), (149.08, 3.16), (128.16, 3.87), (93.28, 3.59), (215.55, 1.3), (117.35, 2.95), (43.09, 4.64), (75.84, 3.04), (103.61, 3.9), (130.31, 3.52), (92.64, 2.91), (62.53, 5.39), (133.62, 4.07), (37.92, 4.55), (118.85, 2.79), (87.5, 4.09), (167.55, 3.26), (161.76, 2.4), (24.92, 3.6), (108.06, 5.21), (116.57, 3.57), (95.05, 5.58), (89.05, 1.51), (108.06, 3.04), (125.78, 3.55), (16.75, 3.39), (-10.9, 4.26), (69.39, 3.02), (109.9, 4.06), (166.42, 1.5), (160.12, 2.25), (117.1, 1.96), (113.08, 2.92), (75.69, 3.79), (109.5, 2.9), (54.08, 6.24), (94.26, 3.18), (149.83, 3.88), (137.79, 2.35), (100.15, 4.97), (187.9, 3.08), (94.55, 3.55), (87.62, 3.73), (124.13, 3.06), (93.45, 3.56), (85.44, 2.91), (105.96, 3.0), (92.82, 3.8), (131.04, 2.87), (96.47, 2.91), (142.67, 2.28), (135.0, 3.04), (66.46, 4.12), (33.09, 3.89), (157.88, 2.98), (68.02, 4.89), (111.18, 2.76), (123.47, 2.92), (164.0, 1.13), (21.9, 5.28), (42.04, 3.02), (72.01, 4.95), (85.15, 3.24), (128.18, 2.45), (37.14, 2.29), (155.26, 4.51), (181.68, 1.33), (70.2, 3.46), (144.81, 2.88), (71.51, 4.39), (138.53, 2.95), (92.81, 4.42), (127.48, 2.28), (78.19, 4.42), (80.09, 3.71), (166.88, 3.05), (81.63, 3.32), (49.94, 3.3), (113.27, 3.51), (165.35, 2.7), (143.86, 1.08), (93.73, 2.16), (124.63, 3.38), (162.02, 4.17), (109.49, 4.12), (68.11, 3.42), (97.68, 3.62), (88.62, 4.31), (107.21, 3.45), (116.13, 4.7), (41.1, 4.11), (112.59, 3.93), (86.48, 4.41), (107.26, 3.9), (92.98, 1.91), (134.37, 2.29), (127.93, 4.24), (144.54, 4.32), (130.19, 3.91), (94.84, 3.69), (107.77, 2.92), (88.27, 3.72), (95.12, 2.84), (150.3, 3.14), (38.17, 3.22), (129.47, 2.9), (94.09, 4.48), (142.91, 3.12), (94.19, 3.92), (113.65, 4.97), (123.0, 2.69), (147.26, 3.63), (114.57, 4.0), (147.87, 3.3), (109.69, 3.19), (135.38, 2.27), (92.22, 4.16), (72.39, 3.29), (106.5, 2.99), (131.84, 1.96), (106.99, 3.98), (121.97, 4.14), (98.86, 4.1), (103.1, 4.23), (117.21, 4.64), (120.76, 4.14), (92.01, 4.36), (163.2, 1.78), (81.99, 5.74), (128.77, 3.24), (127.61, 2.9), (146.01, 1.47), (79.75, 4.84), (79.41, 3.57), (168.3, 3.0), (55.58, 3.8), (105.27, 1.42), (75.23, 4.18), (174.93, 2.77), (123.38, 3.82), (47.93, 3.04), (103.14, 3.78), (32.36, 5.97), (68.29, 2.92), (102.02, 3.48), (91.67, 1.32), (89.97, 1.72), (143.84, 2.6), (81.29, 3.12), (142.78, 2.6), (170.65, 1.71), (157.84, 3.2), (181.95, 3.72), (106.96, 3.44), (131.76, 2.93), (123.12, 2.76), (107.02, 3.58), (84.23, 3.28), (112.61, 4.76), (108.62, 4.22), (65.06, 3.21), (43.3, 3.25), (100.56, 3.1), (119.67, 2.06), (75.52, 3.46), (75.23, 2.77), (124.16, 2.48), (200.69, 3.07), (81.13, 3.91), (89.72, 2.33), (138.0, 3.12), (122.66, 4.16), (58.28, 4.18), (149.26, 3.52), (-0.36, 5.87), (70.49, 2.95), (76.23, 3.53), (107.92, 2.14), (131.19, 1.8), (81.61, 3.51), (37.44, 5.11), (108.99, 3.66), (171.37, 3.98), (40.55, 3.8), (102.79, 3.91), (148.65, 3.2), (64.09, 5.23), (68.41, 3.69), (103.99, 4.3), (96.38, 4.2), (43.08, 5.14), (81.83, 3.95), (111.3, 3.86), (117.5, 2.38), (144.6, 1.88), (92.03, 3.4), (109.37, 5.44), (124.29, 3.91), (135.82, 3.53), (68.39, 3.64), (100.24, 2.38), (160.14, 2.73), (125.1, 2.52), (108.15, 3.37), (74.69, 3.87), (47.37, 4.87), (177.03, 2.2), (45.21, 4.45), (162.21, 3.66), (133.41, 3.59), (70.58, 3.66), (47.04, 3.98), (60.13, 3.87), (46.22, 5.21), (80.15, 4.15), (75.97, 2.22), (134.13, 1.88), (198.5, 0.82), (113.57, 2.22), (110.68, 2.94), (120.71, 3.13), (130.41, 4.43), (101.44, 2.39), (96.82, 4.1), (34.12, 4.91), (122.42, 3.21), (150.93, 1.73), (97.84, 4.01), (128.04, 4.97), (115.31, 4.08), (72.85, 3.53), (111.07, 4.71), (101.2, 2.48), (113.54, 2.46), (107.18, 4.43), (72.75, 4.53), (73.44, 3.78), (75.4, 4.61), (110.54, 1.56), (72.86, 3.06), (158.96, 2.19), (159.96, 1.98), (148.3, 1.4), (120.03, 2.72), (81.59, 4.79), (136.5, 3.48), (87.88, 4.7), (29.17, 5.74), (137.41, 4.77), (136.45, 3.59), (53.48, 3.13), (145.21, 3.66), (-18.72, 5.41), (-11.42, 2.62), (64.57, 4.59), (112.83, 4.05), (136.54, 3.53), (181.21, 2.58), (194.03, 2.97), (90.26, 5.44), (109.0, 4.1), (14.75, 3.2), (141.64, 4.03), (164.75, 2.13), (149.02, 2.08), (69.72, 4.16), (141.06, 3.29), (110.53, 4.08), (120.54, 2.83), (130.13, 2.27), (117.51, 3.49), (139.94, 3.91), (140.44, 1.52), (147.3, 4.1), (132.72, 1.73), (116.47, 3.86), (131.96, 2.25), (153.47, 2.4), (71.36, 4.59), (140.29, 5.2), (74.66, 4.12), (150.93, 2.31), (165.36, 3.88), (203.86, 1.98), (93.22, 3.77), (162.65, 2.51), (78.98, 1.72), (128.31, 2.68), (93.75, 2.66), (51.01, 5.05), (126.26, 2.6), (67.19, 4.85), (97.63, 3.26), (21.22, 4.85), (75.79, 2.89), (80.55, 2.78), (116.95, 2.94), (63.56, 3.97), (144.66, 3.49), (83.55, 4.58), (122.03, 3.67), (118.43, 2.43), (95.26, 3.16), (97.55, 2.29), (134.31, 2.75), (127.76, 3.92), (86.44, 3.12), (68.87, 4.25), (63.64, 5.47), (78.52, 3.79), (103.55, 3.04), (121.74, 2.89), (164.14, 2.16), (97.31, 2.68), (97.04, 3.78), (56.68, 3.97), (129.88, 4.01), (142.17, 3.34), (133.47, 1.34), (60.98, 4.4), (55.04, 4.77), (102.79, 3.65), (164.67, 1.38), (94.48, 3.13), (140.13, 3.37), (112.2, 3.1), (17.66, 3.77), (161.19, 3.49), (142.22, 4.36), (150.62, 3.4), (155.7, 2.08), (65.64, 5.35), (165.63, 1.81), (89.86, 4.4), (108.68, 3.64), (41.79, 5.65), (127.06, 1.01), (26.57, 4.63), (78.68, 4.97), (87.07, 3.59), (87.12, 3.48), (62.95, 2.24), (65.97, 2.67), (74.29, 3.63), (130.98, 3.51), (108.48, 0.86), (176.22, 3.85), (51.36, 4.23), (75.71, 4.81), (41.56, 4.32), (202.39, 1.66), (87.52, 4.14), (108.33, 1.96), (75.69, 4.43), (38.06, 2.92), (116.04, 3.96), (177.72, 3.26), (45.82, 4.27), (64.16, 3.33), (34.26, 3.32), (65.42, 5.27), (127.67, 2.27), (124.81, 3.9), (94.88, 2.06), (78.7, 3.08), (22.76, 3.95), (113.61, 3.3), (103.24, 3.17), (138.33, 3.56), (122.73, 3.91), (90.77, 4.94), (88.82, 3.9), (68.47, 3.13), (66.3, 3.22), (76.07, 3.53), (108.9, 0.89), (131.77, 3.79), (98.13, 3.54), (114.13, 3.03), (53.74, 5.62), (105.94, 1.6), (78.61, 4.66), (107.83, 5.93), (142.27, 3.32), (72.56, 3.73), (111.49, 4.09), (103.49, 1.64), (122.18, 2.95), (60.94, 4.16), (69.63, 3.34), (114.22, 4.33), (89.12, 3.49), (155.46, 3.76), (69.73, 4.46), (43.67, 4.38), (54.19, 3.48), (128.86, 3.61), (121.99, 4.0), (118.15, 2.73)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=476；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4698；
t = r·√((n−2)/(1−r²)) = 0.4698·√((476−2)/(1−0.2207)) = -11.5852；df=474；p = -8.13e+01。
95% CI（Fisher z）：tanh(z±15.808·1/√(n−3)) = [-0.8445, 0.2138]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=476；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4698；
t = r·√((n−2)/(1−r²)) = 0.4698·√((476−2)/(1−0.2207)) = -11.5852；df=474；p = -8.13e+01。
95% CI（Fisher z）：tanh(z±15.808·1/√(n−3)) = [-0.8445, 0.2138]。
③ 计算结果：
r = -0.4698，t = -11.5852，p = -8.13e+01，95%CI = [-0.8445, 0.2138]
④ 结论：在 α=0.05 水平下，p=-8.13e+01 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.4698，方向负），越贵越不想买。
案例 19. 每日吸烟量(支)与肺活量(mL)
Pearson相关吸烟伤肺
① 原始数据：
样本 n=913，逐条 (x, y)：[(18.25, 2500.27), (10.36, 3308.06), (7.78, 3588.7), (10.96, 3318.75), (9.55, 3661.25), (14.23, 3890.32), (10.83, 3894.81), (4.97, 4359.16), (15.38, 3358.77), (1.36, 3118.95), (5.67, 3881.36), (10.68, 2743.56), (4.33, 3301.6), (10.02, 3613.98), (-4.05, 3627.3), (4.41, 3567.83), (9.34, 4439.29), (21.71, 2732.34), (16.44, 3543.26), (15.18, 4124.83), (3.38, 3012.29), (7.48, 3537.76), (13.09, 3343.33), (14.34, 3540.96), (19.01, 3678.99), (7.51, 3802.18), (12.5, 3967.15), (10.69, 3080.74), (7.61, 2812.56), (5.75, 3682.4), (14.26, 2739.13), (14.37, 3003.98), (2.14, 4054.95), (25.86, 2683.08), (12.86, 3460.31), (11.09, 3157.75), (14.83, 2079.45), (10.5, 3542.03), (4.94, 4211.13), (14.49, 3058.89), (5.56, 3637.38), (9.4, 4100.87), (15.66, 3709.45), (9.51, 3413.39), (16.85, 2542.07), (19.81, 1878.61), (11.3, 3859.29), (4.28, 3773.42), (7.69, 4584.64), (2.65, 4517.42), (13.91, 3634.79), (8.62, 3680.17), (14.49, 3941.59), (10.42, 4411.32), (8.72, 3047.08), (4.7, 3534.93), (7.99, 3410.39), (17.18, 2983.97), (15.61, 3567.83), (2.74, 3977.96), (5.11, 3218.02), (6.87, 3192.63), (13.74, 2895.65), (2.31, 3591.35), (8.82, 4234.81), (3.08, 3832.57), (4.09, 3778.31), (12.34, 3999.67), (9.17, 3617.05), (8.65, 3747.06), (14.07, 3721.67), (8.36, 2911.46), (16.34, 3213.52), (4.12, 3182.8), (12.58, 3187.42), (9.81, 3438.34), (14.7, 4097.88), (14.24, 3241.87), (24.06, 2302.15), (9.57, 3795.26), (1.12, 4492.18), (14.41, 3296.49), (5.5, 3925.8), (15.35, 2537.12), (16.17, 3537.66), (20.77, 2291.46), (7.76, 3882.43), (4.74, 3594.2), (8.1, 3542.23), (15.99, 3164.78), (6.82, 4220.98), (12.01, 3147.1), (8.61, 4265.16), (3.03, 2685.05), (2.69, 3781.0), (3.62, 2970.65), (9.04, 3162.52), (12.28, 3399.49), (15.42, 3609.0), (4.63, 3869.07), (12.61, 4194.44), (-3.13, 4038.74), (4.94, 3980.81), (13.97, 3402.25), (4.8, 4622.79), (20.15, 1970.63), (20.01, 3071.83), (5.02, 4453.72), (18.83, 2395.13), (25.78, 3099.73), (8.78, 3638.53), (3.92, 3374.59), (21.81, 2933.87), (11.2, 3799.49), (8.22, 3006.22), (6.9, 3696.93), (12.77, 3374.66), (0.05, 3557.39), (12.24, 3515.68), (-3.19, 5034.3), (1.57, 3391.32), (5.47, 2968.73), (-1.52, 3816.11), (4.79, 4277.71), (4.8, 4170.52), (1.39, 2595.75), (4.88, 2955.66), (13.51, 3908.02), (3.14, 3889.25), (3.39, 4175.36), (16.28, 3024.86), (17.79, 2876.63), (15.17, 3650.71), (7.08, 3089.64), (9.39, 3712.23), (11.03, 3690.02), (21.52, 2281.24), (-0.08, 4406.59), (10.5, 3213.0), (8.41, 3678.45), (10.27, 3306.86), (2.13, 3845.19), (5.87, 3459.15), (9.45, 3165.15), (6.81, 2376.35), (10.3, 3585.13), (9.16, 3092.89), (3.59, 3650.77), (3.37, 3686.9), (6.46, 4249.46), (13.78, 2599.25), (14.85, 3288.11), (11.94, 2871.61), (7.24, 3820.37), (11.29, 3447.84), (8.73, 3008.52), (14.47, 4422.61), (12.73, 2155.79), (13.4, 3761.75), (14.53, 4351.1), (2.71, 3122.43), (9.07, 3258.02), (13.62, 3344.63), (12.67, 2559.38), (16.08, 2908.04), (8.12, 3045.68), (15.72, 3002.28), (12.47, 3740.99), (17.35, 3191.36), (5.13, 3607.04), (2.48, 4766.08), (10.33, 3847.78), (8.2, 3642.38), (18.03, 2576.96), (1.93, 3111.94), (6.91, 3309.75), (11.75, 3423.17), (22.28, 2393.55), (12.43, 3910.34), (10.71, 3934.93), (3.58, 3065.08), (17.23, 2454.18), (10.58, 4181.16), (2.82, 2400.19), (6.42, 4556.54), (11.73, 4181.99), (8.63, 3703.15), (16.13, 2505.64), (5.36, 3021.61), (8.63, 3983.93), (4.55, 3403.9), (13.4, 3136.54), (10.24, 2576.42), (15.73, 3890.94), (12.46, 4603.2), (20.68, 3743.51), (14.2, 2668.86), (6.12, 3978.69), (4.24, 3417.73), (0.94, 2608.11), (10.52, 2586.94), (16.07, 4576.92), (1.08, 4532.66), (0.09, 3643.52), (17.55, 3520.96), (8.68, 3161.8), (19.06, 3345.57), (9.12, 2962.54), (13.94, 3319.77), (11.68, 3623.04), (12.77, 3473.91), (14.87, 4109.05), (20.6, 2551.21), (14.53, 4555.78), (8.96, 2792.48), (1.34, 3529.1), (18.06, 2624.08), (-3.9, 4109.22), (12.78, 2987.41), (13.65, 2980.89), (8.28, 3793.98), (16.34, 2501.5), (10.93, 3156.73), (15.37, 3343.76), (10.67, 3457.97), (7.73, 4268.9), (13.07, 3828.08), (20.58, 3168.66), (3.92, 3564.2), (-2.15, 4602.37), (12.19, 2698.63), (6.07, 4376.22), (6.31, 4404.18), (6.24, 4735.53), (11.31, 3257.75), (25.26, 1653.5), (9.83, 4312.25), (2.55, 4576.32), (15.02, 3759.25), (4.97, 3530.91), (14.68, 3155.79), (-5.46, 4418.17), (9.04, 3234.2), (17.16, 3075.46), (10.7, 3021.53), (11.74, 3050.45), (2.67, 3601.76), (5.93, 3215.59), (13.29, 4077.67), (9.15, 3294.02), (3.57, 3627.62), (21.06, 3576.54), (10.5, 2939.15), (18.66, 3670.82), (2.52, 3338.63), (16.29, 4002.07), (16.74, 2312.05), (14.44, 2552.12), (4.97, 3859.6), (3.68, 4156.49), (16.3, 2258.99), (11.47, 3922.41), (16.0, 3218.67), (6.8, 3581.94), (6.54, 3828.95), (7.46, 4259.69), (4.46, 2830.03), (16.51, 3546.22), (7.09, 3550.24), (14.91, 2770.39), (8.95, 4411.86), (7.97, 3649.64), (21.02, 3481.62), (15.99, 3645.42), (1.86, 4269.11), (10.22, 4943.53), (2.97, 3295.98), (14.25, 3831.71), (4.46, 3819.22), (11.08, 2905.5), (13.06, 3803.13), (14.67, 3704.84), (17.98, 3024.55), (20.26, 3194.65), (13.81, 3163.41), (10.44, 3457.37), (6.56, 3876.83), (3.34, 3699.66), (8.74, 3815.49), (8.21, 4172.86), (11.57, 3490.01), (3.68, 4288.11), (15.85, 4059.46), (6.95, 4329.27), (5.13, 3484.25), (2.6, 4821.79), (-1.16, 4444.16), (11.16, 3846.04), (4.78, 3798.25), (11.89, 3293.86), (5.85, 3423.64), (12.69, 3826.69), (8.66, 3064.25), (15.16, 1971.11), (13.43, 3147.45), (10.47, 2315.72), (11.05, 3656.25), (12.44, 3358.45), (17.73, 2908.59), (11.16, 3415.57), (5.58, 4157.04), (8.22, 3783.78), (5.18, 4281.2), (14.97, 2680.22), (11.38, 3410.24), (1.2, 4883.81), (9.28, 3578.29), (6.68, 4422.45), (14.18, 2152.99), (10.33, 3179.67), (11.87, 3518.61), (12.13, 3575.53), (10.6, 2726.13), (2.44, 4282.94), (5.94, 3753.03), (6.61, 3427.98), (9.47, 4721.48), (13.35, 2993.61), (11.12, 3138.07), (18.16, 3568.63), (9.24, 4386.08), (-0.21, 4161.21), (16.25, 4028.95), (1.42, 2653.01), (11.49, 3068.25), (9.84, 3397.13), (11.89, 2603.6), (10.24, 3173.13), (12.93, 2199.15), (9.33, 3932.72), (6.27, 3089.3), (15.49, 2825.05), (18.48, 2941.29), (12.49, 2966.03), (8.81, 3838.45), (10.75, 2776.42), (4.97, 4155.96), (14.21, 3018.68), (15.04, 3391.5), (15.3, 2925.3), (6.92, 4403.11), (18.6, 3424.07), (19.28, 3239.32), (2.61, 4323.12), (6.42, 3440.87), (17.83, 3170.14), (5.16, 4149.19), (14.02, 2884.4), (11.17, 3717.32), (7.53, 3776.98), (5.29, 3080.13), (21.23, 2857.81), (12.05, 3535.55), (13.51, 2864.54), (5.23, 3898.27), (14.8, 2971.16), (6.94, 4775.11), (7.93, 3958.3), (11.88, 4467.46), (24.36, 2737.51), (20.68, 3609.64), (14.68, 2941.34), (8.01, 3406.54), (4.99, 3459.0), (12.56, 3004.62), (13.19, 4087.44), (8.37, 3693.29), (1.85, 3640.54), (3.88, 3234.53), (16.08, 3178.92), (7.49, 4684.81), (6.43, 3279.38), (4.14, 4476.06), (13.77, 3744.36), (5.73, 3944.2), (2.65, 3935.96), (14.58, 3279.47), (4.56, 3620.32), (6.1, 3449.47), (10.75, 2741.35), (17.67, 3780.71), (14.47, 3753.19), (18.05, 3379.73), (4.39, 3851.22), (15.07, 3942.02), (15.97, 2663.51), (10.62, 3329.81), (14.02, 4763.97), (9.84, 3311.86), (6.5, 3349.18), (12.88, 3151.43), (6.93, 3502.17), (12.61, 3081.8), (11.55, 3159.62), (13.02, 3599.4), (-1.51, 4030.64), (13.35, 3968.3), (14.45, 3401.56), (8.95, 2175.21), (3.48, 4422.13), (8.77, 3945.42), (4.22, 3729.0), (13.79, 3760.62), (2.41, 4408.41), (13.03, 3844.06), (15.87, 3768.6), (13.92, 2431.28), (-0.95, 4044.41), (18.08, 2712.42), (11.83, 3156.65), (12.41, 3892.85), (20.82, 3374.4), (8.52, 3590.33), (8.23, 1880.68), (14.68, 3067.29), (9.2, 3505.82), (12.59, 2314.18), (26.28, 3830.88), (24.72, 3089.86), (8.64, 3601.02), (11.24, 3151.87), (5.69, 2761.11), (8.63, 4174.29), (8.42, 2978.92), (3.52, 2958.99), (10.7, 2858.66), (17.94, 2288.57), (22.3, 2845.14), (7.64, 3804.4), (13.41, 3214.99), (4.54, 3862.37), (20.28, 3993.76), (-2.43, 4375.85), (2.77, 3679.03), (7.5, 3193.45), (9.93, 4070.71), (8.08, 3738.56), (14.59, 3724.28), (8.76, 4300.92), (11.48, 3402.7), (7.92, 4124.7), (6.94, 3705.66), (9.32, 3331.04), (6.79, 3328.83), (5.43, 3292.79), (2.21, 3728.5), (2.99, 3460.44), (9.88, 2792.98), (18.14, 2407.87), (10.23, 3111.38), (7.98, 2811.45), (9.1, 2710.02), (10.52, 3629.43), (-0.26, 4298.78), (9.88, 4685.53), (4.91, 3104.54), (9.02, 3371.24), (2.71, 3486.62), (5.13, 4298.98), (7.3, 4132.46), (6.63, 3696.43), (15.3, 3486.28), (6.45, 4606.24), (6.47, 3859.47), (17.47, 2531.52), (6.66, 3996.87), (4.53, 3222.42), (14.22, 3298.42), (10.63, 2909.45), (4.85, 3821.79), (12.69, 4083.23), (3.97, 4693.83), (10.96, 4240.12), (16.62, 4088.58), (8.29, 3969.82), (4.83, 4283.98), (15.41, 3856.77), (15.75, 3975.65), (15.3, 3694.91), (11.56, 3231.97), (5.32, 4214.98), (7.02, 3877.99), (6.3, 3679.54), (2.89, 3652.62), (14.07, 2179.55), (23.14, 3317.98), (16.15, 2911.06), (15.22, 4263.38), (17.08, 3002.93), (8.54, 4155.92), (12.47, 3886.88), (14.63, 3164.25), (6.02, 3733.87), (1.85, 4510.31), (13.08, 2689.14), (16.85, 2801.58), (4.9, 3115.17), (21.52, 2193.64), (13.37, 3263.98), (-4.42, 3911.37), (9.65, 2958.94), (8.76, 4427.68), (4.59, 3929.0), (10.8, 4175.53), (5.87, 4234.51), (12.3, 3603.76), (-3.82, 4774.0), (15.25, 4061.35), (11.63, 2828.39), (15.91, 3002.09), (10.0, 3676.36), (14.41, 3146.77), (23.4, 2508.28), (-1.07, 3295.9), (9.36, 3979.07), (19.58, 3887.68), (8.84, 2740.61), (4.99, 3720.8), (8.43, 4127.08), (3.62, 3635.04), (12.1, 3321.63), (4.32, 3184.95), (18.57, 2565.31), (20.88, 2186.81), (8.25, 3331.92), (17.83, 3585.22), (11.97, 3293.34), (12.29, 3387.18), (12.33, 3719.38), (-1.6, 4598.82), (-2.54, 4254.62), (0.27, 3921.75), (12.4, 2911.29), (0.84, 3090.55), (19.27, 2999.02), (28.64, 2216.2), (8.61, 4638.92), (17.91, 3192.25), (9.82, 2722.38), (5.4, 3322.56), (15.35, 3018.55), (10.42, 3302.78), (19.48, 2723.3), (10.46, 2870.47), (9.92, 3479.97), (8.23, 3265.62), (5.79, 3477.98), (16.56, 2810.27), (11.84, 3260.37), (3.81, 2931.51), (14.12, 2929.64), (7.94, 4022.71), (10.12, 3603.46), (10.77, 3211.57), (4.8, 4138.23), (14.83, 4235.88), (5.86, 3463.99), (-1.69, 3761.84), (3.08, 4178.7), (20.08, 3190.16), (5.53, 3977.43), (6.64, 3841.87), (11.39, 4092.38), (14.22, 3533.99), (12.11, 2959.04), (10.98, 3890.6), (18.54, 3723.49), (4.35, 3942.17), (20.21, 3192.9), (14.29, 2992.05), (0.29, 4205.16), (12.47, 2759.44), (2.54, 3746.01), (16.3, 2419.01), (24.15, 3099.41), (1.24, 4088.54), (14.48, 2958.07), (13.32, 2925.14), (11.89, 3228.51), (6.27, 3244.39), (10.95, 3444.27), (15.65, 3104.49), (11.73, 2511.52), (12.77, 3586.88), (11.36, 3948.73), (-4.13, 3474.5), (2.1, 4583.4), (11.67, 3378.55), (21.75, 3722.71), (11.7, 3997.05), (5.21, 4375.83), (19.61, 2682.12), (12.17, 4345.53), (12.91, 4108.86), (1.71, 4376.89), (12.19, 3125.4), (15.06, 4135.95), (10.21, 2506.6), (22.01, 3012.94), (12.67, 3810.05), (4.77, 3082.74), (3.25, 4391.65), (10.31, 3782.54), (12.07, 2957.26), (11.81, 3966.72), (21.11, 3117.35), (11.82, 3329.54), (13.33, 4478.51), (16.96, 2866.77), (7.79, 4176.15), (12.54, 4244.89), (6.18, 4174.9), (8.94, 3570.17), (5.02, 3594.55), (19.0, 3183.41), (10.54, 3465.43), (21.54, 3111.22), (14.6, 2565.63), (12.74, 3428.58), (9.36, 4761.33), (2.22, 3112.21), (17.54, 2790.25), (6.27, 3970.41), (8.1, 3324.65), (11.35, 3866.98), (9.17, 2599.17), (7.25, 4210.34), (5.79, 4010.52), (5.97, 3371.58), (8.73, 3594.94), (19.24, 4192.96), (9.85, 4224.0), (14.61, 2847.09), (3.79, 3613.01), (7.34, 4518.49), (10.37, 4322.98), (8.77, 3401.56), (18.93, 4299.83), (10.11, 3763.97), (7.89, 3322.08), (4.77, 4509.01), (3.05, 4236.57), (2.55, 3994.09), (14.24, 3199.95), (2.46, 2608.06), (19.14, 2593.73), (9.21, 4098.04), (3.02, 3826.32), (-5.87, 4704.82), (13.81, 2937.38), (7.64, 4243.61), (1.53, 4213.09), (13.14, 3675.17), (9.47, 4495.45), (7.14, 3887.03), (11.3, 4092.33), (10.01, 3428.44), (14.23, 3115.77), (3.44, 4354.37), (7.95, 5197.68), (9.62, 4111.53), (7.78, 3117.02), (3.24, 3202.81), (2.09, 3717.95), (8.52, 3781.86), (15.04, 3477.95), (22.73, 3252.98), (-1.61, 4450.4), (6.58, 3294.43), (10.31, 2971.35), (11.13, 3765.25), (8.87, 3794.93), (21.82, 3572.54), (17.14, 2683.22), (6.32, 3817.86), (11.34, 3131.86), (19.66, 2741.76), (14.5, 2683.0), (4.58, 3602.62), (-0.12, 3953.02), (0.78, 3045.75), (10.46, 3050.01), (6.48, 3899.83), (2.13, 4494.66), (5.12, 4131.26), (5.51, 3789.98), (6.29, 3647.66), (6.1, 3937.75), (10.85, 3304.49), (6.11, 3617.34), (17.54, 3755.7), (4.37, 4320.66), (9.12, 3419.0), (11.37, 2703.36), (12.66, 3549.47), (21.42, 2918.57), (4.01, 5175.59), (2.85, 3315.23), (5.34, 4150.49), (9.06, 3860.59), (9.17, 3867.14), (14.13, 2534.6), (5.36, 3757.35), (13.79, 2669.88), (12.83, 3278.75), (2.35, 4313.62), (17.01, 3054.76), (8.89, 3554.66), (10.53, 3048.9), (12.23, 3690.37), (6.71, 2971.84), (11.36, 3480.36), (13.32, 2464.32), (3.06, 3891.94), (8.7, 3547.15), (10.8, 3321.52), (6.51, 3750.77), (10.8, 4105.45), (13.72, 3935.09), (19.58, 2391.6), (12.8, 3322.8), (8.79, 4091.02), (13.53, 3356.12), (7.85, 3374.45), (7.18, 3207.63), (3.34, 4540.89), (10.25, 3885.44), (7.57, 3968.66), (10.45, 2633.26), (-1.59, 4723.97), (3.89, 3976.64), (5.55, 3926.56), (7.17, 4481.15), (23.34, 2878.84), (7.43, 4555.51), (-2.12, 3906.0), (14.18, 3603.55), (15.08, 3989.41), (8.75, 3459.83), (21.08, 2783.48), (5.41, 4091.27), (13.27, 4498.42), (5.19, 4324.75), (-11.13, 4800.7), (14.97, 3180.06), (15.64, 2992.54), (9.44, 3115.16), (5.57, 4600.17), (10.61, 3852.7), (9.89, 2893.38), (16.81, 2936.05), (13.79, 3821.23), (7.59, 4504.11), (7.86, 3270.97), (7.68, 4940.29), (7.86, 2706.89), (7.01, 3426.84), (10.98, 3436.49), (14.82, 2856.83), (14.1, 2571.56), (8.3, 4111.76), (7.47, 3687.92), (5.93, 3838.73), (3.01, 3987.5), (7.47, 4030.16), (7.52, 2689.55), (15.14, 3594.19), (1.24, 4376.07), (11.17, 2727.62), (-2.43, 4057.59), (5.15, 3779.65), (11.27, 2787.6), (14.91, 3165.53), (3.75, 4353.94), (7.21, 3753.51), (14.79, 2733.51), (15.11, 3883.25), (1.94, 3279.38), (12.45, 3267.77), (11.69, 3579.28), (18.39, 1947.15), (8.66, 3920.31), (21.43, 3433.25), (10.69, 4126.42), (13.56, 3399.19), (7.18, 3466.67), (4.82, 3983.9), (19.36, 4008.98), (1.87, 2966.04), (23.28, 3269.04), (2.39, 3295.49), (12.84, 3480.17), (-1.68, 4879.51), (14.94, 3664.3), (6.9, 4220.69), (9.35, 4361.89), (11.76, 3554.23), (13.21, 2879.39), (7.85, 3756.97), (3.84, 3895.0), (12.37, 3106.6), (2.12, 3401.95), (11.49, 4255.33), (4.15, 3533.06), (11.05, 3759.35), (6.69, 3311.91), (11.09, 2885.95), (18.03, 2172.86), (22.44, 3377.18), (9.04, 3984.93), (6.25, 3663.74), (11.8, 4655.96), (6.27, 3449.24), (10.12, 3051.5), (7.97, 3025.65), (5.89, 3426.25), (1.23, 3867.27), (-4.77, 4238.4), (10.15, 3319.67), (-0.43, 4249.09), (14.74, 3383.07), (15.7, 3879.31), (8.87, 3142.0), (-4.03, 4357.69), (10.93, 3547.75), (12.88, 4029.5), (2.51, 3896.57), (6.85, 3233.99), (12.66, 3788.58), (15.94, 3223.42), (17.31, 3528.0), (10.2, 3517.16), (11.54, 3012.48), (4.68, 3017.69), (-3.74, 3724.25), (-1.2, 4509.96), (19.21, 2230.57), (12.31, 2572.16), (5.13, 4101.55), (14.03, 2720.74), (4.92, 3948.89), (20.27, 3349.22), (6.84, 3960.65), (10.65, 4344.03), (-1.12, 4047.45), (6.89, 3708.95), (7.29, 3191.27), (7.14, 2890.25), (1.55, 4425.3), (5.08, 3489.66), (6.34, 3659.54), (16.11, 2799.09), (-1.11, 4421.57), (15.46, 4193.17), (9.88, 3403.42), (1.7, 4002.31), (2.04, 3865.84), (7.94, 3843.77), (11.14, 3358.52), (4.67, 3511.77), (11.28, 4115.36), (4.42, 4603.57), (6.1, 2920.43), (9.09, 2907.43), (8.91, 3400.59), (20.36, 4232.63), (7.06, 3027.74), (9.76, 4083.4), (7.68, 3893.25), (7.0, 3549.32), (8.8, 3908.98), (6.13, 2963.99), (0.77, 4552.85), (10.92, 4006.04), (11.03, 3701.95), (0.99, 4362.92), (12.41, 2146.37), (11.62, 4451.53), (16.21, 2783.94), (6.29, 4420.64), (19.32, 3519.54), (15.78, 4147.68), (12.61, 2523.81), (7.83, 3150.02), (11.77, 3624.95), (10.16, 4583.35), (8.42, 2689.48), (6.1, 4086.26), (12.52, 3840.85), (5.11, 3971.26), (5.43, 3808.93), (4.83, 3406.73), (10.29, 3876.82), (6.16, 4120.79), (2.68, 4228.49), (-1.99, 4635.83), (13.59, 3053.71), (16.33, 3329.76), (9.56, 4301.0), (4.57, 3160.38), (8.96, 2467.29), (7.64, 3677.51), (16.64, 3608.93)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=913；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4543；
t = r·√((n−2)/(1−r²)) = 0.4543·√((913−2)/(1−0.2064)) = -15.3905；df=911；p = -1.18e+19。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5043, -0.4011]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=913；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4543；
t = r·√((n−2)/(1−r²)) = 0.4543·√((913−2)/(1−0.2064)) = -15.3905；df=911；p = -1.18e+19。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5043, -0.4011]。
③ 计算结果：
r = -0.4543，t = -15.3905，p = -1.18e+19，95%CI = [-0.5043, -0.4011]
④ 结论：在 α=0.05 水平下，p=-1.18e+19 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.4543，方向负），吸烟伤肺。
案例 20. 压力水平(分)与睡眠质量(分)
Pearson相关压力大睡不好
① 原始数据：
样本 n=845，逐条 (x, y)：[(1.93, 8.63), (6.89, 4.9), (1.55, 7.73), (4.88, 4.38), (6.44, 4.87), (2.05, 7.5), (2.28, 6.97), (6.59, 4.93), (3.92, 7.12), (4.73, 5.28), (3.23, 5.12), (3.97, 8.58), (4.49, 5.96), (1.42, 7.26), (5.24, 5.13), (6.06, 5.4), (4.23, 7.9), (6.13, 6.31), (4.0, 7.08), (3.41, 7.11), (7.72, 3.87), (7.35, 6.75), (5.37, 5.08), (9.14, 4.96), (3.14, 9.06), (4.15, 7.49), (9.2, 6.43), (2.87, 5.68), (7.96, 4.17), (4.12, 7.33), (4.73, 7.36), (5.92, 6.72), (5.63, 6.14), (6.75, 3.38), (2.46, 7.19), (4.91, 6.28), (5.73, 4.74), (2.85, 7.2), (9.57, 5.35), (4.3, 5.33), (4.77, 5.56), (4.66, 4.62), (5.37, 4.83), (1.63, 7.56), (5.07, 6.54), (4.96, 6.37), (4.5, 4.04), (5.27, 6.78), (2.59, 6.66), (4.37, 8.98), (5.39, 4.42), (2.15, 7.48), (6.69, 5.39), (4.11, 6.82), (5.63, 5.51), (5.95, 7.3), (5.47, 7.25), (5.51, 5.06), (6.18, 5.42), (2.92, 6.11), (9.53, 4.78), (3.37, 6.32), (5.41, 5.8), (4.94, 7.28), (5.98, 4.38), (6.49, 7.35), (0.62, 7.39), (6.43, 4.8), (5.45, 7.59), (2.99, 5.72), (7.12, 5.9), (6.93, 6.58), (5.39, 5.81), (1.92, 8.12), (4.22, 6.67), (4.39, 6.65), (6.31, 4.89), (6.22, 5.46), (4.91, 7.47), (3.84, 6.95), (5.38, 5.34), (4.31, 6.41), (6.84, 4.91), (3.32, 8.12), (3.51, 6.07), (2.66, 6.36), (4.12, 5.65), (7.13, 6.4), (7.8, 5.95), (5.3, 7.13), (3.77, 7.74), (4.05, 5.09), (7.3, 1.92), (4.45, 6.43), (4.62, 7.06), (5.32, 5.86), (6.08, 4.66), (4.21, 4.66), (5.82, 5.33), (2.73, 6.98), (5.48, 5.1), (8.83, 3.52), (3.28, 8.19), (1.95, 6.69), (5.39, 6.01), (7.7, 5.93), (4.81, 7.54), (7.96, 6.16), (4.17, 7.75), (2.4, 9.27), (5.62, 6.89), (2.91, 5.35), (4.97, 6.61), (0.03, 6.04), (8.19, 5.29), (4.1, 9.93), (1.32, 6.87), (6.04, 2.71), (5.87, 5.16), (3.21, 5.42), (6.73, 4.28), (2.93, 5.86), (2.9, 7.72), (6.02, 4.63), (2.96, 7.91), (5.47, 5.38), (2.58, 4.2), (3.27, 5.97), (1.78, 7.57), (4.21, 5.94), (7.16, 5.2), (3.01, 8.16), (3.94, 6.92), (1.98, 7.52), (7.2, 3.21), (0.5, 8.64), (5.67, 6.72), (7.03, 2.86), (3.33, 4.84), (5.19, 5.39), (8.79, 5.39), (5.27, 7.29), (3.22, 8.14), (9.04, 3.4), (4.47, 6.77), (6.59, 6.64), (4.16, 7.87), (1.89, 7.51), (8.93, 3.24), (3.47, 7.25), (4.09, 7.48), (5.9, 6.65), (4.3, 4.82), (6.03, 7.75), (4.58, 5.43), (4.56, 8.28), (7.04, 4.43), (5.74, 8.72), (3.18, 6.38), (9.47, 4.88), (4.78, 7.69), (5.69, 6.37), (1.52, 4.75), (3.39, 7.28), (4.8, 7.28), (6.17, 6.05), (4.88, 4.58), (6.17, 5.33), (4.23, 8.96), (6.4, 5.33), (4.75, 5.11), (5.59, 4.98), (0.23, 7.96), (7.54, 4.69), (9.97, 4.6), (7.67, 3.45), (6.92, 6.64), (2.62, 5.71), (5.08, 6.85), (3.08, 5.43), (3.44, 5.48), (4.87, 7.42), (2.77, 6.56), (6.47, 3.87), (7.14, 7.01), (4.93, 7.4), (5.36, 7.05), (3.52, 4.86), (2.27, 8.32), (7.17, 5.46), (2.67, 7.2), (7.11, 5.46), (4.91, 6.11), (1.17, 5.66), (5.04, 6.11), (3.16, 7.48), (5.65, 7.19), (6.33, 6.9), (5.65, 5.28), (9.86, 2.8), (4.58, 8.85), (5.13, 7.54), (2.69, 6.77), (5.64, 6.4), (2.67, 6.39), (6.58, 5.9), (5.09, 6.75), (6.06, 5.79), (4.14, 5.83), (5.93, 7.5), (5.6, 5.87), (4.72, 8.29), (3.71, 7.63), (2.1, 5.84), (5.31, 4.42), (-0.63, 8.84), (6.06, 5.49), (6.82, 5.73), (5.32, 7.1), (5.51, 5.7), (3.85, 7.92), (5.9, 5.67), (8.6, 5.55), (2.88, 5.49), (8.8, 4.21), (6.06, 6.01), (4.6, 8.06), (6.97, 3.93), (6.3, 3.88), (2.78, 7.45), (3.59, 6.29), (3.69, 8.46), (6.44, 6.06), (-0.23, 8.18), (4.18, 5.8), (5.6, 6.07), (4.41, 6.57), (3.99, 5.68), (5.6, 4.7), (3.6, 7.77), (4.75, 7.0), (7.99, 5.99), (3.97, 7.56), (6.33, 5.21), (5.04, 4.96), (4.22, 4.14), (3.21, 5.99), (6.17, 4.69), (5.3, 5.25), (5.4, 5.72), (7.03, 4.03), (4.57, 5.48), (5.03, 4.97), (5.22, 5.97), (4.13, 5.74), (6.86, 6.39), (4.97, 3.2), (4.82, 6.34), (2.16, 7.38), (4.96, 6.33), (6.22, 4.41), (5.6, 4.32), (3.14, 5.53), (3.59, 8.92), (2.35, 7.91), (5.83, 5.42), (2.29, 8.45), (-0.86, 8.37), (3.44, 8.37), (4.8, 5.94), (4.55, 7.14), (5.18, 6.03), (5.67, 4.2), (8.79, 4.22), (5.7, 7.51), (4.45, 6.98), (5.11, 5.45), (7.49, 4.62), (3.8, 7.72), (6.17, 5.48), (7.63, 1.78), (0.76, 6.46), (6.85, 5.45), (3.95, 4.93), (3.93, 5.66), (4.83, 5.28), (4.35, 6.67), (5.43, 5.77), (4.59, 7.6), (4.15, 7.76), (3.84, 5.12), (3.13, 6.51), (5.6, 4.37), (3.84, 8.31), (2.81, 6.9), (7.69, 2.91), (2.21, 6.99), (2.96, 4.56), (4.65, 6.2), (3.26, 7.04), (5.46, 7.98), (6.39, 6.67), (0.08, 7.56), (6.83, 4.3), (1.58, 9.09), (9.08, 3.14), (3.41, 4.99), (5.96, 6.9), (7.7, 3.79), (3.69, 7.38), (5.36, 4.9), (5.28, 6.65), (5.05, 6.81), (3.26, 8.4), (3.92, 4.06), (5.73, 4.41), (7.17, 5.31), (3.59, 7.26), (2.88, 5.87), (5.38, 5.14), (7.21, 5.04), (4.26, 8.22), (2.03, 4.23), (4.58, 5.43), (4.58, 6.97), (6.85, 4.61), (6.16, 4.43), (7.31, 5.37), (7.46, 5.39), (3.21, 7.41), (8.83, 3.49), (2.64, 6.33), (8.38, 3.43), (7.1, 5.67), (5.1, 8.49), (1.44, 9.82), (3.24, 7.87), (3.92, 6.03), (5.89, 5.51), (7.25, 4.48), (3.44, 7.74), (3.34, 5.98), (7.43, 6.19), (0.91, 8.45), (6.35, 4.82), (3.94, 5.56), (4.56, 8.09), (2.63, 6.47), (5.93, 6.06), (7.22, 4.96), (7.06, 6.07), (6.26, 6.17), (4.64, 7.96), (5.17, 5.73), (4.89, 6.15), (6.34, 4.91), (4.69, 7.61), (6.07, 3.92), (5.91, 6.22), (5.03, 6.05), (2.68, 9.06), (7.52, 7.07), (5.15, 8.87), (4.51, 8.6), (4.9, 6.05), (6.8, 4.3), (4.75, 5.62), (5.2, 5.65), (6.95, 6.87), (6.41, 5.73), (9.13, 4.51), (6.23, 4.32), (6.92, 5.76), (5.09, 8.13), (4.14, 5.94), (5.77, 4.79), (7.01, 4.53), (8.16, 3.29), (1.93, 6.94), (6.89, 5.38), (4.48, 7.06), (3.76, 6.82), (4.32, 7.25), (6.58, 6.44), (4.4, 5.76), (4.06, 5.62), (6.85, 5.34), (0.4, 9.31), (7.71, 3.53), (1.0, 9.53), (6.19, 5.16), (5.77, 7.04), (4.56, 5.18), (6.95, 6.93), (6.9, 6.09), (3.57, 8.69), (7.68, 5.54), (3.7, 9.18), (5.02, 4.08), (7.75, 4.07), (7.34, 4.72), (3.78, 4.92), (4.44, 6.59), (3.47, 7.02), (1.35, 6.89), (2.77, 8.51), (9.3, 3.17), (4.95, 4.73), (3.78, 6.85), (2.98, 6.14), (2.2, 7.9), (4.32, 5.05), (5.68, 6.33), (4.06, 5.27), (4.1, 4.06), (2.9, 7.24), (2.82, 6.75), (4.95, 5.17), (6.93, 4.83), (5.71, 3.09), (3.67, 4.24), (4.47, 8.22), (3.02, 5.23), (1.86, 6.0), (6.1, 6.49), (5.72, 7.49), (4.83, 5.73), (4.29, 7.7), (-0.11, 6.41), (3.12, 5.93), (4.18, 2.64), (4.85, 7.43), (5.91, 4.22), (5.24, 7.6), (2.29, 6.25), (2.6, 8.69), (7.36, 5.14), (7.93, 4.06), (5.92, 3.56), (5.67, 4.43), (6.02, 6.8), (3.87, 9.77), (4.79, 7.38), (7.41, 5.81), (8.06, 4.42), (2.45, 6.34), (5.11, 7.09), (5.58, 2.88), (5.4, 5.07), (4.99, 4.76), (5.93, 7.69), (0.84, 6.99), (6.72, 5.51), (4.94, 5.39), (4.45, 4.57), (5.93, 5.13), (5.96, 7.15), (5.65, 5.46), (1.36, 7.16), (4.03, 4.91), (8.22, 5.28), (3.86, 4.33), (2.27, 7.2), (3.62, 6.94), (8.15, 4.81), (6.34, 5.7), (4.88, 6.64), (5.7, 5.42), (6.96, 6.87), (6.41, 8.13), (7.48, 3.3), (6.06, 5.21), (4.44, 5.58), (5.0, 5.7), (4.69, 4.67), (2.98, 8.59), (0.59, 7.27), (4.04, 4.88), (7.99, 4.53), (3.56, 7.42), (4.49, 4.22), (6.17, 7.05), (8.16, 4.95), (1.38, 6.32), (5.16, 8.07), (6.04, 4.31), (5.76, 4.49), (2.42, 7.05), (4.84, 6.18), (6.49, 5.52), (0.58, 10.1), (6.87, 4.61), (6.11, 6.42), (-0.09, 7.55), (5.61, 5.51), (6.56, 4.27), (6.19, 4.52), (6.77, 6.4), (7.45, 4.78), (3.88, 6.4), (6.4, 4.16), (7.51, 6.0), (1.68, 7.43), (3.81, 8.83), (7.1, 7.18), (6.86, 4.39), (5.05, 4.59), (5.46, 5.77), (3.64, 8.89), (5.35, 6.36), (5.75, 4.67), (4.81, 5.91), (5.26, 6.01), (3.97, 6.67), (5.64, 5.05), (3.85, 6.08), (6.76, 4.97), (8.31, 6.25), (4.66, 6.24), (2.61, 7.47), (7.97, 5.08), (5.95, 5.5), (3.88, 6.88), (1.97, 7.67), (6.57, 2.94), (5.67, 4.71), (5.08, 6.41), (5.51, 5.25), (4.28, 5.78), (-0.97, 6.78), (8.74, 5.34), (5.07, 3.99), (5.38, 6.8), (4.11, 5.4), (5.18, 6.43), (6.46, 8.05), (2.99, 6.65), (9.6, 5.76), (3.53, 4.52), (4.58, 4.91), (5.35, 5.22), (3.31, 7.22), (1.48, 7.3), (5.81, 3.85), (4.0, 5.19), (6.68, 5.55), (2.76, 6.31), (5.3, 7.22), (5.55, 5.76), (4.8, 6.74), (4.49, 6.13), (3.06, 8.04), (3.89, 5.99), (4.85, 6.82), (0.78, 7.89), (6.88, 2.48), (4.39, 8.24), (7.76, 4.19), (5.08, 6.81), (4.94, 7.33), (6.2, 4.75), (-0.3, 8.07), (5.14, 5.79), (8.37, 4.16), (6.73, 6.0), (7.03, 3.65), (5.68, 8.31), (3.27, 9.23), (8.96, 5.73), (2.59, 6.43), (4.75, 6.83), (4.21, 5.28), (6.36, 1.93), (5.87, 4.63), (7.68, 6.84), (6.78, 3.7), (8.12, 4.92), (3.75, 9.68), (5.29, 6.95), (3.54, 4.08), (5.26, 7.37), (5.43, 6.28), (9.62, 3.03), (6.72, 4.54), (6.59, 6.92), (5.35, 5.61), (5.27, 5.1), (3.44, 4.42), (0.3, 7.96), (4.54, 5.65), (7.69, 5.63), (5.02, 6.68), (6.91, 5.34), (4.91, 5.59), (5.63, 5.83), (5.55, 5.63), (3.53, 6.42), (6.82, 3.28), (2.53, 4.73), (4.37, 5.93), (3.3, 5.97), (5.39, 5.17), (5.15, 6.37), (5.76, 8.16), (5.48, 5.13), (6.69, 5.46), (8.51, 7.24), (5.7, 4.86), (3.11, 8.79), (4.27, 7.22), (2.33, 7.79), (5.32, 5.24), (1.99, 8.19), (4.42, 7.11), (5.64, 5.37), (6.59, 5.92), (5.85, 4.79), (1.04, 9.99), (7.93, 5.86), (4.15, 4.33), (3.75, 5.91), (4.0, 4.99), (6.65, 4.7), (5.07, 4.72), (7.07, 5.04), (4.01, 6.33), (6.06, 5.06), (7.24, 5.55), (7.19, 4.6), (4.19, 7.47), (3.2, 6.72), (5.27, 3.72), (7.84, 5.4), (2.76, 6.82), (4.22, 6.25), (6.32, 8.79), (3.03, 9.81), (6.95, 3.65), (0.08, 8.59), (7.03, 7.04), (4.36, 5.63), (1.85, 8.59), (4.15, 6.12), (3.77, 5.76), (5.76, 3.45), (4.09, 6.95), (4.05, 6.87), (7.88, 3.49), (5.32, 7.12), (5.13, 4.51), (5.66, 7.73), (4.63, 5.86), (4.35, 5.68), (6.01, 8.01), (5.01, 4.12), (3.36, 5.63), (4.25, 5.41), (8.59, 5.07), (4.34, 6.29), (5.88, 6.42), (6.77, 4.67), (5.28, 7.85), (5.91, 4.34), (4.68, 6.5), (5.13, 5.84), (4.91, 5.33), (4.6, 5.85), (6.28, 7.6), (2.66, 7.08), (3.71, 5.85), (6.08, 5.18), (5.01, 6.4), (5.62, 4.57), (4.93, 4.73), (1.25, 7.14), (4.74, 4.51), (8.96, 4.78), (6.24, 3.79), (2.25, 5.65), (4.71, 7.57), (3.35, 6.7), (4.63, 4.21), (8.7, 3.9), (5.89, 4.01), (4.23, 6.12), (6.45, 3.04), (2.82, 5.06), (5.14, 4.71), (-0.44, 8.26), (1.54, 7.67), (3.64, 4.91), (9.63, 2.88), (4.95, 5.45), (5.56, 7.25), (11.39, 3.74), (5.26, 5.97), (4.23, 6.27), (7.99, 7.82), (4.55, 5.21), (6.33, 4.18), (7.29, 6.45), (4.39, 2.86), (2.84, 7.23), (2.58, 6.76), (5.4, 7.68), (2.25, 7.57), (6.91, 6.8), (5.73, 5.34), (8.03, 3.91), (8.3, 5.16), (4.62, 6.41), (2.76, 6.04), (6.36, 4.31), (4.83, 6.59), (4.29, 4.8), (4.22, 7.36), (9.22, 3.49), (5.51, 5.48), (8.33, 3.22), (7.19, 5.65), (2.28, 6.42), (3.45, 6.09), (8.11, 5.66), (5.78, 6.02), (5.2, 5.37), (5.53, 7.45), (5.62, 6.16), (4.2, 6.59), (9.94, 2.4), (8.64, 6.81), (3.75, 5.76), (5.81, 4.73), (3.52, 7.93), (5.3, 7.17), (3.55, 8.05), (6.28, 5.47), (3.29, 7.19), (5.12, 3.75), (3.53, 5.52), (8.9, 4.55), (4.79, 8.27), (4.89, 6.74), (5.39, 6.69), (6.53, 5.01), (4.32, 8.74), (2.59, 9.31), (5.85, 3.3), (1.58, 7.54), (7.22, 4.62), (6.12, 4.37), (5.31, 4.68), (4.39, 7.71), (3.48, 5.33), (4.12, 5.55), (6.19, 5.75), (6.4, 4.37), (1.29, 7.66), (3.38, 5.77), (6.17, 5.77), (4.77, 5.8), (6.68, 6.54), (4.68, 8.44), (4.01, 8.57), (5.31, 4.27), (4.42, 5.35), (3.92, 5.97), (5.58, 7.13), (6.98, 5.61), (2.21, 4.79), (-1.57, 10.24), (2.74, 8.88), (8.68, 5.0), (3.6, 8.72), (5.95, 6.79), (5.31, 6.44), (7.91, 4.84), (3.38, 6.12), (5.01, 5.66), (8.08, 2.16), (2.81, 6.31), (3.0, 5.47), (8.47, 4.9), (5.16, 6.53), (5.14, 6.95), (5.86, 6.4), (3.71, 6.1), (6.16, 7.4), (1.93, 7.45), (3.83, 7.47), (4.33, 4.63), (6.63, 3.63), (2.81, 8.12), (7.34, 5.72), (3.91, 7.36), (2.62, 6.67), (3.32, 6.79), (7.67, 3.78), (4.78, 5.35), (1.13, 4.79), (0.21, 7.99), (6.4, 3.98), (9.85, 4.0), (3.04, 6.86), (3.88, 7.22), (6.13, 5.21), (3.09, 6.94), (7.67, 3.36), (3.73, 6.38), (4.37, 6.18), (4.2, 4.91), (4.94, 3.22), (4.73, 7.38), (4.89, 7.6), (4.44, 8.04), (4.56, 7.96), (2.33, 8.98), (7.74, 4.05), (6.26, 5.27), (6.79, 5.18), (5.22, 5.13), (9.03, 2.98), (1.78, 5.99), (4.13, 5.43), (6.67, 3.94), (1.47, 9.03), (7.92, 2.84), (5.76, 5.08), (2.25, 5.79), (4.87, 4.57), (6.15, 3.92), (1.95, 6.74), (4.93, 7.11), (3.13, 7.02), (5.44, 6.1), (4.58, 6.35), (8.48, 5.85), (3.71, 7.52), (1.08, 6.37), (4.28, 6.78), (5.41, 3.73), (4.61, 8.15), (5.9, 3.96), (6.41, 6.18), (4.7, 8.55), (3.77, 6.75)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=845；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.5496；
t = r·√((n−2)/(1−r²)) = 0.5496·√((845−2)/(1−0.3021)) = -19.1016；df=843；p = -5.12e+29。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5951, -0.5007]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=845；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.5496；
t = r·√((n−2)/(1−r²)) = 0.5496·√((845−2)/(1−0.3021)) = -19.1016；df=843；p = -5.12e+29。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5951, -0.5007]。
③ 计算结果：
r = -0.5496，t = -19.1016，p = -5.12e+29，95%CI = [-0.5951, -0.5007]
④ 结论：在 α=0.05 水平下，p=-5.12e+29 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.5496，方向负），压力大睡不好。
案例 21. 库存周转天数(天)与利润率(%)
Pearson相关周转慢利润低
① 原始数据：
样本 n=645，逐条 (x, y)：[(42.49, 9.36), (30.33, 15.47), (16.47, 13.89), (16.81, 24.34), (30.16, 17.85), (12.77, 21.66), (35.42, 14.53), (46.51, 11.69), (16.16, 11.84), (9.91, 16.39), (30.65, 21.46), (22.31, 17.14), (31.92, 5.81), (35.37, 15.92), (28.23, 16.83), (21.57, 20.35), (11.3, 22.57), (35.8, 9.68), (16.11, 21.84), (28.74, 13.01), (32.75, 21.61), (39.35, 14.74), (47.45, 9.0), (28.97, 14.66), (19.54, 17.26), (32.52, 13.93), (12.27, 19.29), (12.53, 13.55), (42.82, 11.53), (37.29, 17.72), (30.58, 23.55), (35.01, 10.28), (30.97, 13.09), (11.82, 20.32), (24.16, 16.35), (8.21, 16.63), (14.18, 16.97), (38.23, 14.78), (41.46, 10.98), (32.42, 13.99), (32.72, 3.77), (50.73, -0.31), (27.1, 4.45), (37.05, 10.49), (36.87, 15.43), (26.2, 12.17), (21.01, 19.19), (29.49, 18.32), (18.61, 15.06), (43.89, 10.83), (25.49, 14.82), (39.54, 11.68), (41.43, 14.8), (23.52, 14.98), (7.0, 24.01), (28.81, 19.81), (37.05, 5.25), (60.93, 3.65), (30.34, 14.52), (46.0, 10.53), (29.63, 14.22), (44.82, 6.34), (28.38, 12.13), (26.54, 18.63), (24.11, 20.21), (46.23, 18.66), (34.34, 5.36), (39.05, 9.86), (18.24, 28.26), (12.26, 19.1), (17.25, 12.72), (38.87, 9.79), (23.73, 25.25), (6.98, 23.72), (33.47, 17.43), (48.31, 0.52), (42.54, 8.07), (31.16, 6.51), (31.99, 13.32), (13.87, 20.48), (31.75, 22.5), (50.29, 14.45), (38.43, 19.27), (24.85, 13.39), (49.72, 13.13), (12.47, 10.71), (25.39, 13.99), (25.94, 13.49), (34.28, 12.72), (35.96, 12.09), (11.43, 19.18), (35.03, 17.29), (49.99, 8.97), (27.59, 17.85), (3.2, 22.13), (31.03, 12.27), (45.36, 18.84), (18.86, 9.62), (32.93, 20.97), (19.31, 20.18), (10.54, 23.32), (36.18, 9.89), (30.06, 17.54), (32.17, 14.47), (24.66, 17.77), (7.95, 21.85), (24.54, 13.4), (35.99, 12.78), (49.52, 9.78), (31.9, 11.05), (23.18, 16.1), (25.43, 21.45), (30.35, 15.8), (25.09, 15.0), (22.31, 21.56), (-6.16, 21.27), (31.65, 26.47), (47.56, 20.16), (44.55, 13.96), (31.39, 3.67), (9.97, 23.52), (10.14, 16.71), (27.84, 10.27), (32.51, 11.41), (30.6, 15.07), (44.12, 3.84), (31.41, 16.47), (41.71, 12.11), (13.45, 19.97), (26.37, 12.68), (48.09, 7.66), (42.48, 7.84), (25.16, 16.9), (32.26, 15.19), (43.28, 16.28), (31.71, 6.07), (35.64, 15.9), (22.62, 10.8), (25.29, 17.2), (29.97, 13.14), (42.25, 13.09), (24.52, 20.76), (29.45, 9.55), (30.07, 16.96), (23.79, 19.29), (17.29, 16.65), (25.17, 16.71), (32.75, 16.63), (46.03, 10.76), (38.13, 17.34), (27.03, 10.87), (34.8, 24.75), (14.75, 26.93), (35.65, 20.35), (35.74, 14.08), (19.24, 22.07), (35.3, 16.98), (24.63, 6.18), (30.78, 5.71), (0.53, 32.48), (10.66, 15.57), (42.82, 15.52), (20.04, 19.11), (40.92, 11.72), (40.14, 15.36), (24.92, 17.64), (34.72, 18.43), (50.84, 11.61), (21.74, 21.48), (30.21, 12.04), (30.58, 19.07), (27.69, 12.63), (27.45, 19.28), (7.91, 24.85), (43.59, 16.14), (9.66, 18.69), (33.93, 15.13), (19.67, 18.17), (31.45, 11.06), (22.09, 15.92), (38.25, 5.24), (42.42, 16.08), (32.39, 13.54), (28.57, 21.15), (18.01, 16.44), (31.49, 6.84), (38.45, 13.08), (37.05, 16.38), (27.72, 19.41), (35.91, 7.37), (26.2, 18.9), (45.77, 15.14), (48.22, 16.86), (44.85, 5.32), (9.02, 16.75), (26.42, 14.87), (8.34, 15.48), (15.88, 21.14), (33.59, 8.75), (22.99, 13.59), (56.16, 13.04), (20.61, 16.84), (31.76, 20.87), (22.43, 10.56), (20.32, 19.11), (42.08, 11.91), (31.14, 9.0), (30.23, 8.74), (30.52, 12.96), (23.93, 16.62), (49.23, 9.83), (44.9, 5.56), (8.45, 16.76), (-2.88, 24.36), (43.13, 9.58), (32.56, 21.18), (35.31, 19.35), (28.36, 12.9), (35.12, 11.84), (34.68, 21.32), (32.97, 9.08), (14.05, 15.38), (21.14, 13.69), (34.35, 13.61), (1.54, 18.23), (42.42, 7.79), (24.65, 10.72), (39.85, 11.97), (40.43, 15.13), (37.81, 17.82), (31.81, 17.92), (20.4, 13.09), (22.59, 13.09), (33.86, 18.75), (22.78, 21.87), (7.32, 16.58), (25.43, 13.38), (40.06, 12.05), (48.18, 19.48), (36.3, 14.56), (31.09, 18.83), (37.19, 11.08), (18.2, 14.05), (31.77, 16.85), (18.83, 13.44), (33.51, 11.11), (14.32, 16.92), (21.39, 15.24), (24.19, 17.55), (37.81, 8.58), (3.85, 13.62), (31.39, 17.99), (31.32, 12.51), (16.27, 23.18), (58.99, 5.79), (21.45, 23.29), (46.16, 9.99), (25.67, 20.53), (32.51, 16.89), (37.08, 13.89), (30.54, 16.14), (57.24, 2.79), (53.12, 19.05), (30.63, 16.61), (5.32, 20.66), (35.5, 11.88), (39.49, 13.27), (20.58, 18.1), (26.77, 15.38), (35.74, 12.59), (25.7, 12.26), (39.41, 12.97), (36.56, 17.71), (18.68, 17.48), (37.33, 13.5), (51.45, 13.05), (29.71, 15.51), (25.3, 15.94), (17.82, 16.03), (33.19, 13.64), (14.5, 13.1), (34.76, 15.47), (21.17, 13.53), (39.47, 11.96), (51.87, 13.59), (34.57, 16.8), (23.93, 14.19), (51.09, 11.08), (30.63, 13.43), (63.08, 10.96), (43.02, 12.71), (43.82, 17.25), (38.59, 9.44), (34.36, 10.71), (44.9, 15.6), (28.08, 11.89), (37.6, 10.99), (31.03, 13.97), (27.22, 12.82), (38.38, 16.83), (28.91, 17.31), (38.96, 13.53), (24.24, 9.29), (30.6, 22.49), (32.24, 17.47), (24.05, 13.25), (18.19, 20.16), (34.89, 13.45), (34.53, 9.5), (20.13, 11.85), (46.42, 14.48), (15.64, 25.89), (45.67, 14.53), (34.2, 15.57), (35.61, 11.14), (12.47, 23.1), (34.19, 16.33), (23.92, 14.33), (55.07, 12.98), (25.2, 14.31), (31.05, 9.07), (30.15, 20.82), (17.89, 22.46), (33.58, 7.15), (20.98, 18.18), (34.21, 15.75), (32.46, 15.2), (27.84, 11.23), (25.21, 14.08), (56.13, 10.1), (12.52, 18.7), (29.13, 15.56), (12.55, 19.69), (31.72, 20.8), (23.21, 13.42), (46.51, 19.79), (36.88, 13.17), (17.55, 20.15), (36.98, 5.11), (49.01, 15.64), (40.68, 17.51), (2.05, 15.34), (15.34, 17.71), (35.81, 15.24), (43.12, 15.52), (17.77, 19.39), (19.54, 8.97), (41.08, 15.61), (36.52, 14.24), (9.16, 11.63), (40.39, 12.68), (37.88, 11.57), (25.64, 20.11), (12.84, 18.06), (25.57, 11.14), (16.07, 25.04), (34.22, 20.68), (50.55, 15.87), (20.99, 22.9), (40.72, 12.55), (12.95, 24.96), (48.01, 9.66), (35.4, 7.09), (34.14, 15.23), (5.1, 25.3), (18.65, 20.41), (17.36, 18.6), (23.94, 18.21), (26.59, 20.37), (44.32, 12.2), (26.34, 17.63), (15.58, 17.09), (35.64, 10.33), (27.76, 16.25), (44.91, 18.56), (22.72, 22.24), (38.31, 6.81), (33.42, 12.61), (10.51, 28.54), (19.77, 13.81), (54.0, -0.08), (38.66, 20.26), (19.33, 14.91), (31.53, 17.54), (34.09, 16.29), (46.28, 10.24), (25.81, 14.36), (28.64, 16.79), (31.6, 12.32), (30.51, 17.13), (23.62, 17.15), (31.58, 11.93), (40.24, 10.95), (36.34, 16.12), (26.15, 12.42), (18.74, 20.43), (50.1, 14.5), (1.87, 22.39), (35.47, 13.7), (11.97, 15.81), (14.41, 24.68), (14.23, 12.69), (37.01, 11.02), (35.96, 12.29), (28.23, 9.84), (71.86, 11.69), (22.83, 13.98), (32.76, 20.74), (28.43, 11.77), (23.74, 15.67), (18.68, 16.51), (33.13, 20.29), (28.16, 18.58), (22.35, 10.7), (34.16, 20.43), (39.85, 15.8), (25.43, 15.84), (19.99, 13.65), (50.57, 5.48), (21.63, 24.29), (42.3, 4.46), (26.32, 22.51), (35.19, 15.4), (45.5, 15.94), (53.92, 9.83), (43.16, 13.38), (37.55, 8.42), (18.14, 22.98), (17.05, 28.37), (64.71, 10.0), (47.53, 3.53), (25.82, 15.89), (41.57, 10.7), (46.16, 14.23), (52.5, 14.34), (16.35, 12.59), (9.0, 21.29), (30.76, 25.13), (42.27, 17.35), (5.32, 27.28), (41.81, 16.02), (12.09, 17.88), (27.6, 10.52), (23.05, 25.18), (21.14, 16.97), (18.35, 14.69), (48.56, 8.98), (33.67, 13.36), (21.94, 17.21), (23.87, 9.88), (34.42, 13.36), (34.22, 19.85), (23.51, 22.35), (30.51, 20.6), (33.75, 17.38), (11.26, 20.23), (25.4, 12.52), (38.02, 11.89), (52.5, 15.79), (21.29, 17.12), (49.84, 17.6), (12.95, 19.49), (38.09, 18.46), (28.63, 8.81), (37.85, 13.83), (24.63, 9.5), (53.04, 14.56), (30.26, 18.85), (57.44, 4.34), (38.2, 13.74), (30.62, 16.75), (26.31, 21.56), (31.87, 14.17), (35.7, 15.67), (21.12, 20.73), (38.43, 8.15), (23.13, 14.07), (32.22, 8.32), (35.45, 12.95), (28.13, 16.4), (28.71, 16.45), (35.71, 16.49), (37.12, 20.06), (34.06, 8.35), (31.19, 14.91), (46.38, 11.43), (48.25, 6.17), (34.53, 12.36), (28.55, 12.76), (20.98, 18.21), (23.81, 15.85), (21.14, 16.74), (49.9, 10.08), (17.03, 24.7), (20.4, 19.84), (33.25, 5.27), (35.21, 15.09), (33.76, 11.9), (17.69, 22.65), (14.84, 8.64), (-0.34, 17.71), (15.0, 17.61), (35.02, 17.75), (46.16, 14.19), (32.47, 12.96), (16.66, 17.63), (33.89, 13.48), (35.83, 14.96), (25.32, 14.66), (47.11, 9.54), (33.49, 17.14), (23.84, 17.0), (34.07, 16.74), (18.83, 11.7), (43.18, 14.66), (17.93, 19.86), (33.31, 18.17), (34.56, 11.89), (37.79, 10.83), (24.26, 17.68), (43.97, 4.55), (40.38, 10.75), (39.61, 5.69), (25.84, 14.61), (29.2, 13.0), (12.21, 17.47), (53.63, 8.26), (37.84, 14.88), (49.7, 13.4), (40.18, 3.51), (18.18, 15.6), (41.2, 19.35), (37.45, 17.66), (19.71, 17.41), (25.89, 7.29), (8.71, 17.27), (40.45, 13.8), (38.07, 9.4), (12.36, 16.67), (36.65, 11.74), (20.46, 19.33), (30.49, 10.49), (38.59, 12.38), (32.33, 12.95), (26.6, 23.8), (26.32, 22.11), (19.98, 14.59), (32.03, 12.56), (46.09, 14.25), (23.98, 27.1), (27.41, 18.39), (12.06, 24.67), (41.09, 15.37), (38.73, 14.02), (28.66, 23.22), (39.31, 14.25), (22.31, 14.05), (19.96, 12.34), (28.22, 16.07), (20.01, 7.27), (45.38, 16.75), (20.62, 18.49), (50.1, 11.04), (27.31, 19.01), (25.39, 11.59), (55.64, 2.23), (32.19, 15.93), (47.03, 6.93), (16.54, 7.81), (26.9, 14.65), (19.75, 20.04), (42.77, 8.56), (26.69, 17.73), (33.23, 19.13), (40.68, 15.05), (28.54, 12.57), (36.42, 19.16), (24.61, 19.58), (29.38, 12.45), (20.7, 21.48), (22.27, 19.06), (13.75, 18.22), (27.43, 14.21), (37.26, 24.42), (32.25, 15.19), (52.8, 12.71), (18.96, 13.55), (47.04, 11.01), (41.4, 21.41), (19.68, 16.97), (25.89, 18.57), (32.85, 13.06), (0.85, 23.32), (25.85, 14.31), (30.43, 14.48), (28.02, 23.67), (27.03, 15.07), (42.67, 4.3), (38.06, 13.4), (31.64, 17.92), (2.81, 19.37), (5.43, 16.37), (18.21, 19.16), (21.8, 12.88), (44.37, 18.34), (55.58, 7.09), (0.91, 16.97), (15.22, 20.41), (35.12, 6.7), (25.43, 23.3), (25.58, 19.26), (36.89, 9.96), (28.32, 16.17), (39.59, 11.43), (54.06, 8.42), (28.36, 11.1), (30.95, 14.93), (24.29, 18.78), (14.62, 19.73), (42.8, 14.89), (31.08, 10.26), (33.09, 19.37), (24.02, 17.86), (34.62, 16.97), (43.77, 12.75), (28.65, 13.93), (20.55, 13.72), (17.79, 22.63), (28.5, 13.49), (19.46, 20.47), (42.14, 10.66), (43.48, 13.01), (29.23, 10.01), (33.02, 15.33), (31.51, 14.26), (44.03, 13.39), (45.44, 1.51), (47.26, 19.52), (27.07, 22.47), (28.42, 10.9), (9.2, 19.97), (50.74, 10.58), (29.49, 20.68), (44.92, 17.82), (54.5, 12.43)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=645；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.5045；
t = r·√((n−2)/(1−r²)) = 0.5045·√((645−2)/(1−0.2545)) = -14.8167；df=643；p = -5.22e+13。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.5600, -0.4445]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=645；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.5045；
t = r·√((n−2)/(1−r²)) = 0.5045·√((645−2)/(1−0.2545)) = -14.8167；df=643；p = -5.22e+13。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.5600, -0.4445]。
③ 计算结果：
r = -0.5045，t = -14.8167，p = -5.22e+13，95%CI = [-0.5600, -0.4445]
④ 结论：在 α=0.05 水平下，p=-5.22e+13 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.5045，方向负），周转慢利润低。
案例 22. 屏幕蓝光强度(lux)与入睡速度(分)
Pearson相关蓝光碍眠
① 原始数据：
样本 n=308，逐条 (x, y)：[(85.49, 6.67), (66.2, 5.98), (61.51, 8.68), (49.33, 5.23), (28.19, 7.89), (57.3, 8.28), (43.11, 6.88), (62.65, 7.79), (72.31, 6.86), (41.52, 6.53), (40.31, 9.05), (56.95, 9.54), (32.29, 8.7), (75.98, 4.97), (85.66, 4.5), (46.54, 7.03), (80.41, 4.4), (23.82, 9.79), (40.65, 6.07), (65.49, 9.08), (56.27, 4.22), (69.87, 5.01), (22.12, 7.24), (62.47, 6.41), (53.1, 6.59), (47.42, 7.84), (49.19, 4.84), (17.16, 8.73), (10.97, 11.07), (37.02, 9.69), (57.16, 5.2), (87.5, 5.1), (23.42, 9.13), (60.68, 8.2), (46.4, 7.36), (44.06, 6.38), (54.17, 7.63), (43.4, 7.21), (50.12, 6.93), (36.44, 8.22), (50.29, 7.3), (60.44, 9.85), (31.66, 7.87), (39.74, 8.57), (50.22, 7.66), (75.33, 4.79), (75.56, 8.43), (65.33, 6.0), (46.79, 6.51), (32.45, 6.92), (33.82, 7.34), (36.74, 7.28), (66.06, 6.65), (85.29, 5.11), (39.95, 8.32), (52.3, 6.42), (75.24, 5.52), (90.93, 4.29), (34.72, 8.51), (52.18, 6.15), (40.29, 6.15), (43.96, 5.09), (65.88, 6.13), (0.37, 8.05), (50.42, 3.93), (74.71, 7.15), (37.61, 6.95), (52.66, 8.41), (75.68, 8.07), (39.7, 6.74), (33.62, 8.05), (95.44, 7.2), (27.12, 6.84), (41.76, 5.81), (37.58, 6.8), (62.1, 5.42), (50.35, 6.45), (47.57, 8.03), (66.6, 4.45), (72.85, 7.38), (31.43, 10.39), (39.15, 6.95), (52.86, 7.54), (50.96, 7.48), (52.35, 5.08), (63.48, 6.58), (63.01, 6.58), (83.72, 3.17), (71.45, 4.33), (74.23, 2.14), (84.17, 7.19), (68.54, 6.47), (57.7, 7.27), (41.95, 6.12), (63.19, 8.36), (54.6, 6.89), (18.78, 7.12), (44.83, 6.88), (28.11, 8.79), (23.44, 8.35), (64.82, 6.39), (60.53, 5.51), (31.32, 6.22), (87.0, 4.15), (38.98, 6.38), (72.83, 5.77), (59.41, 7.94), (36.02, 9.0), (81.64, 5.58), (23.37, 8.57), (42.02, 8.88), (74.78, 6.12), (47.63, 5.28), (75.28, 6.64), (54.03, 6.49), (11.22, 10.05), (68.31, 8.04), (64.66, 6.62), (84.1, 7.2), (35.27, 8.48), (52.73, 9.82), (25.19, 8.88), (88.95, 5.51), (59.92, 7.07), (77.14, 4.82), (24.01, 8.29), (79.41, 5.37), (74.65, 7.16), (17.78, 7.0), (30.03, 8.19), (29.71, 7.11), (40.61, 5.43), (49.75, 8.54), (71.95, 7.55), (39.97, 6.54), (53.1, 6.57), (67.63, 6.52), (19.81, 9.44), (53.48, 6.0), (56.81, 5.54), (11.33, 8.95), (80.33, 5.02), (25.35, 9.79), (66.11, 5.43), (39.15, 7.0), (50.75, 6.28), (48.37, 5.9), (11.43, 6.97), (39.51, 7.74), (42.07, 6.52), (51.58, 9.7), (61.59, 7.2), (28.05, 7.72), (70.1, 5.58), (58.51, 6.78), (-4.47, 8.16), (57.51, 5.84), (99.51, 6.67), (33.28, 5.3), (33.35, 9.0), (46.63, 9.7), (52.89, 4.7), (75.08, 4.52), (80.62, 5.9), (33.93, 7.76), (48.84, 7.74), (33.33, 6.16), (56.0, 9.22), (61.8, 6.76), (70.95, 4.42), (39.57, 8.85), (35.28, 9.16), (42.42, 7.16), (44.06, 6.31), (39.18, 6.86), (67.0, 8.91), (45.3, 6.86), (70.73, 6.6), (47.92, 7.11), (52.45, 7.78), (83.77, 7.61), (50.56, 5.69), (74.23, 4.97), (33.11, 6.62), (43.82, 6.33), (71.19, 5.14), (69.87, 6.23), (50.19, 4.73), (49.93, 6.3), (41.52, 8.3), (32.08, 6.81), (80.82, 6.27), (57.43, 5.43), (58.87, 5.08), (55.01, 6.24), (78.12, 7.64), (50.43, 7.86), (53.2, 7.72), (75.0, 5.93), (45.37, 8.29), (74.16, 3.71), (47.53, 5.91), (29.53, 6.19), (26.62, 9.77), (71.6, 7.67), (51.59, 5.39), (49.69, 8.12), (34.43, 7.85), (83.06, 5.41), (51.08, 8.82), (73.24, 8.32), (75.97, 6.77), (71.45, 7.0), (68.61, 7.67), (54.15, 5.73), (56.48, 6.49), (42.8, 8.03), (78.72, 3.74), (64.56, 5.49), (46.48, 5.94), (70.42, 7.85), (24.5, 10.73), (63.95, 6.23), (84.92, 5.92), (70.97, 5.99), (43.86, 6.66), (53.01, 4.74), (46.74, 6.97), (30.77, 4.71), (28.01, 8.57), (55.41, 6.17), (14.79, 9.29), (37.55, 6.08), (18.67, 8.82), (42.4, 8.71), (47.72, 5.76), (23.08, 5.8), (36.55, 7.58), (50.27, 6.2), (69.26, 8.23), (72.0, 6.87), (97.15, 4.09), (20.33, 7.52), (78.62, 4.83), (35.12, 7.71), (4.71, 8.22), (26.05, 7.69), (64.65, 3.03), (28.4, 5.96), (31.3, 7.94), (57.57, 6.19), (73.19, 3.62), (64.13, 6.8), (52.45, 6.08), (59.25, 7.29), (36.1, 8.8), (33.37, 8.85), (64.85, 5.88), (53.54, 5.19), (42.27, 6.06), (52.98, 5.66), (64.79, 7.61), (50.32, 6.71), (61.43, 8.45), (28.05, 8.08), (53.61, 7.76), (47.44, 7.3), (64.77, 7.86), (38.99, 7.5), (51.79, 7.46), (33.43, 6.73), (34.59, 6.19), (49.34, 6.58), (36.45, 6.98), (7.53, 6.8), (40.24, 9.12), (57.1, 6.31), (43.22, 7.13), (52.89, 5.93), (57.94, 7.22), (28.31, 6.57), (85.31, 4.66), (39.86, 7.44), (65.33, 7.65), (64.44, 7.74), (23.07, 7.7), (38.26, 8.86), (53.07, 7.29), (33.92, 7.67), (26.85, 9.83), (38.46, 6.72), (75.36, 6.62), (67.15, 5.42), (37.33, 6.21), (21.04, 7.28), (49.11, 7.18), (55.14, 5.84), (74.56, 4.64), (58.82, 6.6), (84.04, 6.99), (48.1, 4.45), (46.87, 10.0), (66.0, 6.99), (46.06, 6.32), (29.34, 7.65), (47.47, 6.85), (61.88, 6.09), (68.85, 8.26)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=308；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4829；
t = r·√((n−2)/(1−r²)) = 0.4829·√((308−2)/(1−0.2332)) = -9.6479；df=306；p = 3.19e-05。
95% CI（Fisher z）：tanh(z±17.224·1/√(n−3)) = [-0.9075, 0.4296]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=308；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4829；
t = r·√((n−2)/(1−r²)) = 0.4829·√((308−2)/(1−0.2332)) = -9.6479；df=306；p = 3.19e-05。
95% CI（Fisher z）：tanh(z±17.224·1/√(n−3)) = [-0.9075, 0.4296]。
③ 计算结果：
r = -0.4829，t = -9.6479，p = 3.19e-05，95%CI = [-0.9075, 0.4296]
④ 结论：在 α=0.05 水平下，p=3.19e-05 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.4829，方向负），蓝光碍眠。
案例 23. 门店距离(km)与到店率(%)
Pearson相关越远越少来
① 原始数据：
样本 n=824，逐条 (x, y)：[(4.24, 25.44), (4.69, 44.08), (-2.69, 62.8), (2.25, 53.57), (2.28, 45.95), (4.29, 53.14), (2.48, 51.77), (6.47, 39.7), (4.83, 39.85), (0.6, 55.23), (10.07, 47.72), (2.24, 46.12), (5.91, 44.77), (7.4, 29.82), (4.12, 29.96), (3.76, 45.83), (5.49, 38.15), (7.16, 43.13), (2.96, 44.19), (4.54, 48.93), (6.04, 42.71), (8.68, 36.74), (3.6, 35.36), (5.87, 42.29), (9.17, 29.89), (6.98, 28.5), (9.13, 20.48), (6.39, 24.34), (5.64, 46.69), (5.3, 43.97), (8.79, 26.43), (6.55, 30.82), (2.62, 28.57), (0.1, 55.44), (7.65, 34.24), (4.56, 45.99), (-0.63, 55.69), (5.54, 43.79), (11.16, 15.73), (4.45, 47.67), (3.78, 44.51), (1.17, 36.4), (2.2, 32.93), (6.89, 30.84), (1.07, 50.44), (1.09, 30.48), (5.96, 20.06), (4.2, 62.47), (2.55, 43.41), (7.87, 34.12), (3.15, 64.29), (9.77, 30.96), (4.55, 39.85), (6.13, 27.9), (1.81, 63.42), (3.01, 33.35), (6.96, 26.79), (7.29, 19.3), (6.28, 41.6), (3.93, 38.59), (3.44, 35.31), (0.19, 50.79), (2.82, 34.14), (7.53, 39.27), (6.62, 40.05), (7.86, 23.53), (7.04, 60.33), (6.15, 29.93), (0.46, 48.93), (4.1, 35.6), (9.09, 38.12), (5.22, 36.33), (4.86, 39.83), (3.47, 37.8), (2.09, 52.36), (7.81, 34.76), (5.68, 31.88), (5.14, 62.35), (1.55, 54.58), (5.26, 48.4), (-0.08, 67.17), (4.82, 31.84), (0.77, 53.22), (3.24, 25.95), (8.38, 34.03), (5.47, 56.08), (5.52, 48.33), (6.34, 41.53), (0.91, 33.26), (4.35, 27.9), (2.83, 35.29), (6.83, 32.27), (4.7, 60.04), (9.32, 34.24), (1.08, 51.9), (6.17, 26.96), (7.61, 25.84), (1.78, 46.88), (3.74, 41.21), (7.8, 45.53), (2.34, 50.38), (8.38, 37.76), (9.61, 28.04), (8.21, 28.73), (3.99, 38.36), (0.93, 51.49), (9.31, 37.88), (3.95, 36.88), (4.44, 44.94), (3.86, 22.86), (10.21, 14.99), (2.85, 53.75), (1.34, 59.21), (-0.47, 56.96), (1.86, 53.8), (5.65, 20.36), (8.45, 34.03), (14.63, 6.91), (1.99, 32.33), (0.9, 53.77), (1.03, 23.13), (3.01, 46.65), (11.66, 34.72), (6.83, 36.54), (5.4, 27.18), (8.69, 40.71), (8.18, 40.67), (0.09, 60.93), (4.59, 47.07), (-0.35, 58.67), (5.14, 36.63), (5.94, 34.69), (2.6, 61.96), (4.48, 39.79), (7.3, 30.48), (1.01, 44.14), (13.23, 27.09), (7.25, 32.75), (11.52, 43.85), (2.08, 28.39), (3.05, 33.26), (5.44, 33.58), (2.61, 52.52), (1.13, 29.43), (4.31, 33.24), (1.72, 50.16), (3.42, 55.15), (1.33, 50.63), (8.88, 34.18), (8.77, 24.23), (9.25, 15.88), (1.95, 32.05), (4.53, 38.73), (1.0, 47.27), (-0.01, 54.3), (5.22, 39.49), (0.8, 52.18), (7.68, 39.8), (5.05, 39.25), (10.0, 25.22), (4.5, 39.65), (5.04, 49.79), (8.23, 46.81), (7.42, 37.7), (3.54, 44.72), (7.64, 34.86), (6.61, 26.37), (1.85, 37.53), (3.79, 29.74), (3.77, 42.43), (7.87, 25.14), (3.41, 63.71), (8.3, 12.76), (2.76, 35.08), (8.9, 31.99), (6.35, 45.62), (4.59, 24.27), (2.5, 45.9), (5.68, 53.27), (7.81, 48.78), (5.91, 31.51), (-3.08, 54.85), (7.77, 22.12), (1.42, 73.22), (7.91, 33.19), (4.15, 59.39), (2.28, 51.37), (0.27, 49.01), (3.21, 38.27), (7.32, 40.58), (13.12, 20.84), (4.44, 38.06), (3.85, 30.57), (2.12, 47.63), (3.37, 49.75), (3.55, 43.01), (1.62, 46.74), (-1.65, 32.35), (7.75, 28.83), (9.1, 14.4), (5.41, 50.64), (5.51, 47.07), (6.2, 31.75), (3.18, 39.1), (5.01, 52.22), (7.24, 33.05), (9.79, 28.89), (6.24, 28.4), (9.05, 46.25), (1.11, 32.11), (7.27, 30.92), (1.26, 36.59), (3.99, 30.27), (2.87, 27.52), (2.02, 42.14), (3.0, 39.29), (9.73, 35.36), (-0.34, 45.76), (-0.97, 45.78), (4.48, 45.06), (10.29, 36.33), (5.63, 28.89), (5.3, 58.8), (7.28, 34.67), (7.8, 19.91), (6.37, 40.98), (4.04, 40.57), (2.17, 34.85), (5.53, 21.84), (5.97, 38.5), (9.77, 44.85), (2.82, 64.41), (8.75, 43.92), (9.26, 25.05), (6.59, 38.11), (2.57, 37.41), (3.57, 36.8), (2.83, 33.97), (7.35, 54.84), (4.37, 46.13), (4.86, 42.41), (6.12, 52.19), (3.68, 33.74), (11.03, 4.2), (5.11, 49.97), (7.62, 38.44), (1.69, 40.71), (8.37, 41.74), (4.8, 35.83), (9.22, 41.48), (7.71, 23.16), (3.54, 37.33), (5.61, 41.76), (4.92, 39.11), (5.56, 46.02), (3.46, 40.48), (6.24, 38.02), (3.52, 60.43), (3.93, 45.01), (3.36, 39.2), (5.48, 27.78), (-1.43, 69.43), (5.04, 48.87), (-2.82, 31.46), (-2.14, 58.49), (2.28, 50.2), (4.48, 26.21), (6.76, 36.47), (0.99, 38.43), (6.33, 30.96), (5.12, 37.44), (6.12, 53.42), (11.4, 39.76), (7.47, 41.7), (5.49, 44.29), (2.83, 55.09), (7.96, 20.73), (12.83, 18.15), (6.02, 41.3), (1.35, 31.01), (-5.03, 53.34), (3.89, 46.05), (5.09, 45.77), (9.2, 29.67), (6.42, 49.39), (8.01, 42.83), (3.62, 40.74), (3.1, 51.8), (5.3, 26.29), (3.26, 18.49), (8.66, 34.11), (10.19, 3.52), (4.09, 46.35), (5.81, 46.83), (4.41, 31.91), (6.29, 50.12), (8.69, 62.94), (8.41, 23.37), (7.07, 43.04), (9.12, 39.1), (8.45, 60.69), (7.22, 23.39), (2.76, 45.22), (2.59, 62.31), (0.33, 40.03), (1.89, 32.35), (9.05, 27.2), (7.66, 21.85), (9.19, 26.86), (4.36, 44.81), (8.99, 22.75), (3.13, 26.5), (5.58, 17.6), (4.97, 53.29), (3.04, 49.32), (7.87, 47.34), (10.8, 32.14), (-0.03, 39.54), (0.51, 33.56), (6.16, 21.11), (0.67, 38.02), (4.49, 61.14), (11.43, 37.7), (8.09, 53.34), (5.77, 45.81), (3.85, 37.92), (0.54, 26.65), (7.49, 33.04), (3.28, 39.7), (9.46, 27.12), (4.59, 33.64), (8.6, 22.56), (10.48, 39.9), (0.33, 34.75), (1.58, 63.93), (7.77, 25.27), (8.04, 39.46), (7.19, 34.21), (7.08, 48.59), (4.73, 54.68), (1.82, 37.07), (3.66, 37.75), (4.81, 38.32), (6.68, 44.25), (2.57, 39.43), (3.32, 49.47), (10.51, 19.65), (2.67, 47.67), (5.72, 46.61), (8.3, 24.88), (3.12, 44.26), (6.88, 53.03), (2.75, 36.96), (2.66, 68.84), (5.78, 30.59), (1.72, 47.65), (5.43, 34.54), (5.28, 47.79), (9.12, 17.59), (0.7, 51.35), (7.88, 37.57), (0.19, 45.23), (4.7, 23.32), (2.21, 36.82), (5.94, 36.81), (2.2, 53.13), (7.73, 19.88), (0.82, 43.78), (3.29, 50.35), (2.91, 54.86), (2.2, 54.0), (4.16, 24.41), (9.54, 24.13), (1.21, 51.44), (4.76, 13.74), (6.28, 43.89), (7.96, 21.69), (1.06, 57.14), (7.85, 14.16), (9.4, 33.0), (7.61, 38.56), (3.0, 24.69), (3.99, 37.51), (8.3, 27.61), (2.71, 24.7), (0.52, 60.28), (6.5, 40.09), (5.01, 49.9), (2.97, 58.5), (5.2, 35.48), (10.19, 35.15), (2.97, 57.76), (9.93, 37.74), (9.94, 35.66), (2.01, 61.67), (2.28, 48.83), (6.79, 28.08), (2.93, 45.1), (5.67, 36.0), (0.43, 47.06), (8.16, 24.0), (0.42, 41.15), (4.04, 67.6), (-0.91, 49.41), (2.14, 47.72), (3.68, 59.9), (5.94, 40.21), (6.2, 35.74), (4.36, 51.61), (4.41, 38.88), (3.42, 55.66), (2.78, 27.18), (10.15, 16.23), (2.83, 47.16), (7.44, 27.32), (4.01, 22.61), (0.04, 46.21), (3.9, 39.14), (5.85, 32.24), (5.08, 53.24), (5.01, 20.68), (8.68, 22.88), (5.44, 36.96), (4.88, 46.79), (-1.02, 47.81), (7.32, 19.09), (6.12, 39.26), (4.36, 32.26), (4.63, 41.57), (-0.31, 83.0), (7.68, 32.61), (4.39, 30.65), (0.81, 39.28), (5.95, 33.91), (6.62, 40.87), (3.65, 40.04), (4.47, 52.55), (4.72, 48.04), (1.21, 31.49), (6.42, 33.08), (9.21, 35.83), (8.69, 35.0), (4.9, 22.1), (0.42, 55.72), (7.28, 43.72), (8.73, 33.76), (3.4, 51.87), (4.16, 31.31), (3.92, 42.51), (8.42, 39.47), (3.87, 45.23), (5.48, 45.01), (2.57, 33.19), (4.12, 47.25), (5.3, 64.21), (4.59, 44.36), (8.17, 36.65), (4.59, 21.45), (2.9, 44.83), (5.66, 55.47), (5.43, 35.88), (6.44, 38.2), (4.46, 27.16), (2.89, 62.43), (0.67, 45.87), (10.23, 30.39), (2.25, 77.11), (2.94, 48.87), (1.94, 56.86), (4.07, 53.67), (7.79, 31.13), (10.57, 29.54), (4.24, 54.68), (10.63, 35.67), (10.89, 19.57), (9.1, 43.25), (3.24, 45.4), (4.28, 39.68), (0.2, 48.15), (-0.04, 49.1), (-0.26, 54.99), (6.31, 22.64), (8.11, 16.99), (11.13, 48.5), (7.16, 34.4), (2.95, 47.46), (4.2, 39.45), (3.41, 36.19), (5.86, 44.3), (3.95, 49.14), (7.36, 28.45), (2.84, 44.3), (7.74, 23.73), (9.0, 26.79), (4.02, 61.21), (0.82, 61.84), (3.42, 43.87), (1.24, 53.03), (4.0, 38.86), (5.47, 55.73), (0.92, 70.38), (6.81, 23.71), (6.69, 38.2), (4.15, 36.58), (2.93, 37.6), (3.52, 59.5), (10.4, 40.73), (6.81, 45.37), (5.2, 28.34), (3.14, 41.51), (3.08, 27.5), (6.66, 36.85), (7.39, 22.2), (4.7, 43.08), (6.43, 36.73), (6.83, 42.27), (1.08, 40.72), (2.16, 31.73), (1.82, 56.88), (3.23, 39.56), (6.93, 35.91), (7.88, 35.24), (10.87, 30.32), (4.91, 43.66), (4.38, 39.55), (2.91, 52.9), (7.0, 35.24), (1.72, 43.03), (0.75, 51.91), (-0.55, 34.59), (12.35, 30.46), (1.79, 52.34), (3.99, 43.04), (2.41, 36.75), (6.41, 20.6), (9.97, 14.54), (4.27, 32.17), (4.48, 21.73), (6.65, 36.92), (2.02, 33.59), (10.86, 19.77), (4.24, 39.82), (4.83, 17.2), (4.09, 44.98), (7.95, 32.59), (-1.51, 55.06), (6.79, 38.46), (5.95, 16.53), (8.15, 31.31), (4.55, 39.07), (5.23, 50.82), (8.18, 33.91), (2.42, 40.04), (6.66, 34.45), (2.21, 38.72), (6.43, 59.06), (3.75, 46.89), (9.43, 19.88), (3.81, 45.29), (2.27, 54.85), (2.36, 41.06), (0.39, 50.44), (8.71, 25.64), (1.55, 27.15), (1.55, 61.64), (1.76, 41.83), (1.49, 43.34), (1.67, 45.78), (8.29, 34.69), (3.77, 49.79), (6.59, 41.93), (3.69, 33.89), (4.07, 33.45), (7.83, 25.24), (3.33, 52.93), (2.69, 46.34), (1.54, 51.15), (0.95, 46.07), (7.17, 28.83), (5.4, 49.36), (3.08, 19.33), (7.26, 12.7), (6.62, 37.31), (5.05, 39.55), (10.51, 10.21), (5.61, 35.95), (2.81, 39.85), (5.89, 41.98), (5.01, 48.32), (3.62, 35.92), (3.35, 43.26), (4.07, 28.63), (0.55, 58.92), (4.82, 46.49), (7.46, 20.4), (5.7, 24.62), (8.1, 32.63), (4.09, 48.12), (0.87, 66.68), (7.63, 21.98), (0.29, 46.93), (3.3, 44.28), (10.06, 19.28), (2.57, 55.38), (2.1, 46.08), (2.08, 32.23), (0.63, 55.15), (3.6, 51.18), (2.46, 59.47), (6.31, 39.87), (8.37, 33.07), (3.48, 38.28), (3.52, 39.29), (0.8, 57.26), (8.65, 5.66), (4.62, 27.39), (7.6, 28.84), (5.44, 27.22), (2.05, 42.92), (4.31, 70.09), (6.87, 8.08), (8.7, 48.78), (2.15, 69.91), (3.25, 50.57), (12.17, 21.89), (5.72, 26.83), (5.54, 55.11), (7.87, 17.12), (6.74, 32.51), (6.44, 31.83), (10.09, 34.04), (-1.73, 48.58), (3.98, 39.16), (10.09, 23.94), (8.89, 50.76), (2.02, 51.6), (2.59, 31.53), (2.71, 65.1), (6.24, 31.28), (2.34, 47.85), (2.25, 39.06), (5.63, 42.09), (6.63, 38.96), (-1.96, 55.33), (5.1, 25.98), (12.1, 25.52), (5.46, 17.98), (7.98, 42.54), (3.45, 50.38), (5.93, 44.22), (5.68, 40.08), (3.48, 34.44), (3.03, 51.03), (3.55, 35.27), (6.05, 42.99), (10.1, 21.73), (5.94, 16.69), (4.84, 40.23), (6.98, 24.06), (5.78, 51.01), (3.48, 45.95), (12.35, 23.16), (8.47, 51.95), (9.66, 23.53), (5.6, 50.34), (-0.86, 58.01), (5.94, 28.45), (4.34, 45.73), (7.09, 41.48), (1.47, 38.02), (4.37, 47.87), (1.0, 70.84), (4.4, 29.0), (2.54, 47.25), (2.99, 36.03), (4.87, 32.74), (6.4, 30.1), (4.62, 43.38), (1.22, 68.11), (6.93, 36.72), (-0.23, 53.95), (3.42, 37.74), (0.45, 57.35), (10.37, 30.45), (0.95, 69.12), (6.43, 34.49), (11.94, 32.06), (1.99, 30.85), (4.41, 45.6), (3.91, 58.09), (9.43, 28.2), (5.14, 53.54), (2.45, 46.94), (5.14, 47.56), (6.49, 34.71), (0.38, 36.12), (2.7, 32.56), (4.21, 41.93), (6.78, 56.24), (3.39, 47.09), (6.01, 49.39), (3.29, 48.52), (2.32, 38.57), (2.35, 55.98), (9.91, 21.64), (7.74, 29.9), (1.57, 24.24), (5.33, 34.72), (10.8, 28.99), (6.93, 33.91), (1.23, 43.79), (5.09, 45.53), (6.38, 52.59), (-0.93, 29.38), (6.07, 43.89), (5.56, 37.74), (9.18, 34.7), (1.18, 37.43), (4.48, 30.68), (4.0, 53.55), (12.99, 36.3), (3.31, 32.86), (4.62, 50.53), (-6.84, 55.97), (1.55, 60.51), (5.56, 54.12), (0.77, 44.45), (5.08, 26.65), (13.13, 31.31), (10.93, 18.14), (5.68, 28.61), (7.89, 55.27), (4.58, 55.11), (7.26, 52.19), (2.68, 39.67), (2.3, 58.28), (-0.14, 70.44), (4.99, 43.73), (2.29, 38.97), (4.59, 42.15), (4.09, 35.79), (4.69, 55.99), (5.11, 41.49), (2.52, 46.44), (11.61, 47.25), (5.82, 15.01), (8.25, 35.29), (5.02, 50.0), (3.58, 28.73), (3.54, 42.49), (2.25, 71.21), (2.51, 36.75), (3.74, 20.21), (7.49, 33.61), (4.16, 35.1), (3.38, 38.62), (3.85, 42.2), (9.13, 25.08), (6.11, 31.71), (1.92, 13.65), (7.63, 33.36), (12.39, 14.09), (7.88, 31.32), (5.13, 45.19), (1.75, 40.35), (5.19, 43.13), (7.76, 39.46), (6.68, 25.98), (3.23, 32.42), (6.88, 25.04), (5.92, 43.87), (-1.59, 58.75), (4.08, 48.22), (6.71, 38.67), (0.21, 72.32), (5.69, 29.04), (2.48, 43.12), (3.69, 64.44), (6.59, 47.13), (3.31, 70.61), (3.71, 59.33), (2.08, 41.82), (3.45, 55.82), (1.68, 51.16), (3.17, 53.57), (6.05, 38.62), (0.5, 45.3), (5.68, 48.87), (4.13, 41.05), (6.82, 39.34), (4.27, 41.61), (8.59, 40.05), (8.44, 24.68), (4.83, 33.1), (4.62, 45.1), (2.01, 21.48), (3.8, 33.94), (4.17, 52.14), (3.49, 45.67), (6.13, 36.46), (4.19, 38.27), (2.87, 26.0), (6.5, 50.91), (4.21, 63.81), (2.8, 46.34), (3.0, 45.13), (-0.2, 57.24), (7.55, 41.03), (-0.36, 63.16), (6.01, 58.18), (2.07, 41.94), (7.86, 48.67), (4.61, 44.68), (4.16, 41.89), (9.3, 42.76), (3.23, 36.5), (2.75, 36.23), (6.64, 32.29), (-0.44, 52.84), (-4.72, 72.88), (4.85, 50.01), (4.04, 43.18), (4.08, 35.66), (2.89, 27.7)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=824；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.5029；
t = r·√((n−2)/(1−r²)) = 0.5029·√((824−2)/(1−0.2529)) = -16.6814；df=822；p = -5.79e+21。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5523, -0.4500]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=824；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.5029；
t = r·√((n−2)/(1−r²)) = 0.5029·√((824−2)/(1−0.2529)) = -16.6814；df=822；p = -5.79e+21。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5523, -0.4500]。
③ 计算结果：
r = -0.5029，t = -16.6814，p = -5.79e+21，95%CI = [-0.5523, -0.4500]
④ 结论：在 α=0.05 水平下，p=-5.79e+21 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.5029，方向负），越远越少来。
案例 24. 贷款利率(%)与贷款需求量(百万)
Pearson相关利率升需求降
① 原始数据：
样本 n=726，逐条 (x, y)：[(5.31, 97.61), (4.25, 99.02), (5.43, 87.17), (5.64, 88.86), (3.26, 86.43), (3.65, 68.37), (4.62, 71.67), (8.2, 79.74), (4.85, 85.13), (6.29, 76.57), (7.87, 60.02), (7.67, 78.65), (4.94, 84.62), (2.17, 87.03), (5.89, 77.02), (6.63, 73.31), (9.03, 36.02), (4.79, 91.66), (7.01, 46.7), (3.27, 73.07), (4.8, 59.03), (6.65, 77.97), (6.93, 46.16), (6.11, 65.68), (5.06, 80.75), (4.28, 43.1), (1.65, 75.84), (4.63, 79.03), (5.95, 68.61), (3.9, 50.81), (5.04, 73.02), (3.64, 97.0), (6.09, 74.53), (6.44, 86.41), (-0.97, 106.14), (7.86, 86.6), (6.97, 69.14), (6.51, 72.17), (4.47, 58.17), (6.14, 89.05), (3.13, 111.14), (5.85, 49.91), (3.12, 85.02), (4.64, 79.08), (7.2, 88.14), (5.91, 71.59), (6.67, 76.51), (4.89, 68.8), (2.4, 106.1), (4.8, 72.7), (6.97, 59.65), (11.87, 55.14), (1.38, 100.91), (3.1, 95.64), (2.08, 95.58), (6.04, 107.64), (7.67, 32.6), (2.91, 98.2), (6.11, 61.9), (9.97, 65.57), (7.73, 39.99), (5.73, 79.76), (4.02, 84.55), (5.93, 73.8), (4.34, 80.97), (9.39, 46.07), (4.45, 73.42), (5.51, 73.71), (6.33, 82.43), (7.01, 68.96), (5.46, 79.77), (4.58, 100.84), (1.13, 81.77), (2.31, 84.11), (6.8, 107.12), (8.63, 70.27), (3.56, 87.6), (3.44, 87.17), (6.76, 54.86), (6.48, 97.03), (3.43, 77.06), (0.99, 95.21), (4.46, 76.21), (4.45, 86.23), (5.95, 72.48), (7.4, 67.98), (0.54, 113.49), (6.53, 53.46), (3.41, 55.11), (4.48, 110.51), (4.55, 75.53), (5.24, 39.87), (8.99, 77.5), (5.09, 52.65), (7.89, 44.76), (3.1, 111.66), (4.79, 83.61), (7.17, 73.47), (5.35, 96.34), (4.76, 56.14), (6.5, 69.91), (3.32, 78.68), (5.53, 80.5), (5.82, 86.33), (5.45, 59.9), (5.55, 60.77), (6.21, 84.55), (5.43, 90.73), (7.3, 34.2), (4.78, 84.58), (4.82, 127.7), (4.5, 101.03), (2.35, 79.28), (5.89, 55.43), (3.17, 79.63), (4.94, 50.33), (3.96, 51.35), (2.55, 75.47), (4.65, 47.66), (5.3, 92.94), (5.17, 68.82), (5.2, 86.26), (5.58, 49.04), (1.32, 84.92), (5.27, 71.56), (5.03, 94.05), (3.62, 81.91), (3.23, 111.05), (6.57, 53.69), (4.36, 89.91), (7.13, 91.59), (7.6, 64.37), (4.46, 134.65), (2.77, 112.62), (4.01, 85.45), (1.19, 100.61), (4.18, 67.49), (5.73, 52.14), (5.85, 78.5), (7.63, 89.35), (2.98, 71.71), (3.41, 63.43), (4.0, 85.43), (4.06, 51.34), (4.53, 116.43), (6.74, 81.47), (5.93, 71.86), (6.2, 66.89), (6.9, 98.05), (5.13, 61.4), (7.93, 72.63), (4.99, 67.34), (4.31, 57.27), (8.12, 65.68), (6.17, 55.66), (3.69, 75.99), (5.71, 101.44), (2.82, 135.26), (5.32, 93.23), (8.91, 74.79), (6.04, 45.12), (7.72, 78.3), (5.04, 83.45), (3.59, 81.52), (6.56, 79.43), (7.48, 87.52), (3.62, 79.45), (5.46, 119.17), (8.82, 25.59), (5.05, 89.21), (3.07, 102.73), (5.25, 87.04), (6.13, 65.19), (5.34, 90.24), (4.02, 97.1), (5.17, 79.92), (3.36, 96.73), (6.64, 52.32), (4.0, 91.76), (4.09, 90.53), (6.37, 108.65), (5.06, 83.39), (6.68, 82.07), (7.6, 98.13), (2.73, 81.4), (6.6, 81.23), (7.33, 72.1), (6.67, 62.82), (3.55, 108.83), (1.83, 123.6), (1.72, 123.6), (2.84, 87.43), (0.83, 117.19), (5.85, 89.95), (5.99, 72.79), (5.43, 89.15), (6.52, 93.57), (7.29, 63.63), (4.72, 75.83), (3.69, 82.26), (2.88, 88.74), (4.5, 70.11), (0.94, 55.91), (2.06, 87.99), (3.45, 109.58), (4.17, 81.21), (6.39, 37.2), (4.36, 59.83), (4.28, 59.16), (3.94, 59.04), (5.55, 90.1), (4.4, 69.5), (3.71, 81.79), (7.64, 82.27), (5.88, 89.1), (8.23, 43.82), (4.34, 85.07), (4.9, 58.44), (4.86, 72.85), (5.22, 65.04), (2.04, 60.62), (3.83, 80.07), (4.47, 81.79), (5.44, 87.73), (5.04, 53.75), (4.23, 85.25), (6.32, 64.73), (4.23, 77.1), (6.16, 68.37), (3.82, 83.5), (3.64, 85.53), (5.7, 55.01), (5.28, 39.82), (8.39, 72.38), (8.81, 60.21), (3.33, 104.91), (0.46, 96.22), (4.16, 80.13), (5.25, 103.99), (4.04, 84.02), (1.14, 86.15), (4.89, 89.87), (7.17, 109.14), (5.03, 31.71), (3.63, 89.63), (3.33, 95.11), (6.85, 66.98), (5.3, 82.16), (8.12, 59.91), (3.84, 69.26), (3.94, 71.55), (6.35, 64.12), (7.03, 50.36), (8.31, 67.44), (2.62, 99.61), (4.37, 55.08), (7.2, 81.02), (6.74, 72.97), (8.22, 53.88), (4.6, 52.05), (4.88, 102.47), (5.35, 67.11), (2.26, 79.35), (8.54, 59.51), (6.38, 82.73), (5.68, 55.81), (6.26, 60.25), (3.89, 91.53), (5.23, 116.94), (6.07, 106.12), (5.15, 62.9), (4.71, 60.59), (4.38, 88.03), (4.16, 74.02), (9.33, 44.08), (7.83, 72.15), (6.09, 74.31), (9.22, 73.76), (3.57, 87.71), (3.59, 109.4), (4.62, 88.32), (4.22, 105.38), (5.16, 83.77), (3.93, 56.86), (1.88, 94.98), (5.14, 65.53), (4.42, 72.57), (3.62, 63.45), (5.51, 81.58), (10.98, 46.2), (5.58, 77.64), (5.94, 44.21), (6.89, 100.43), (4.48, 80.1), (1.96, 106.69), (6.14, 107.65), (5.85, 58.35), (2.09, 128.07), (5.76, 60.31), (8.26, 45.26), (6.42, 103.21), (2.17, 98.6), (6.23, 96.47), (7.93, 47.1), (1.68, 86.59), (4.43, 113.56), (3.46, 70.83), (7.22, 81.48), (3.9, 59.51), (6.48, 89.4), (5.49, 76.35), (6.34, 59.67), (3.14, 79.03), (1.98, 59.98), (7.03, 112.66), (8.45, 91.15), (8.51, 66.47), (6.12, 71.22), (3.32, 77.29), (4.55, 104.21), (6.77, 49.31), (5.03, 62.53), (5.2, 73.32), (2.45, 100.32), (3.47, 95.76), (5.61, 99.38), (3.26, 110.83), (6.2, 51.35), (3.0, 64.62), (3.34, 82.69), (5.89, 74.47), (6.54, 75.29), (3.76, 106.2), (4.66, 37.94), (6.17, 53.82), (1.97, 107.2), (3.33, 113.7), (5.01, 83.83), (8.26, 88.52), (4.31, 121.96), (8.96, 96.38), (7.17, 59.3), (4.1, 98.03), (4.7, 66.45), (3.82, 82.54), (7.65, 56.76), (2.99, 106.01), (5.38, 75.08), (6.14, 78.33), (3.7, 103.23), (6.28, 52.23), (8.19, 61.75), (5.04, 46.65), (4.64, 88.61), (4.85, 99.64), (4.42, 48.08), (4.36, 103.76), (8.14, 63.58), (5.45, 65.79), (6.07, 90.85), (4.57, 98.38), (5.98, 101.02), (9.74, 59.58), (9.78, 59.14), (4.93, 73.88), (6.47, 69.4), (4.09, 66.89), (1.54, 86.85), (8.87, 44.67), (5.24, 38.99), (4.41, 61.2), (6.38, 75.13), (4.49, 99.24), (4.07, 71.5), (6.7, 72.92), (4.0, 55.94), (7.25, 57.09), (3.18, 113.68), (3.81, 106.75), (4.99, 58.1), (7.6, 65.79), (6.45, 77.98), (6.86, 90.66), (2.77, 114.7), (5.43, 43.31), (2.67, 61.5), (5.24, 55.26), (1.63, 80.78), (2.88, 91.21), (6.86, 33.95), (4.99, 90.83), (4.36, 79.29), (6.0, 85.44), (6.7, 51.85), (6.65, 44.49), (7.51, 91.61), (7.24, 66.06), (1.49, 96.71), (4.16, 91.87), (4.71, 96.87), (6.92, 78.32), (3.35, 108.0), (5.75, 100.17), (5.32, 74.15), (4.84, 51.95), (6.06, 47.02), (5.57, 79.32), (5.3, 86.47), (7.45, 72.04), (7.09, 61.45), (3.28, 102.46), (3.72, 75.44), (7.51, 97.71), (0.6, 96.23), (5.99, 52.8), (5.37, 91.53), (5.15, 73.79), (3.83, 66.14), (6.76, 75.2), (4.98, 83.59), (7.69, 86.26), (7.53, 84.33), (1.34, 96.66), (4.67, 65.45), (1.5, 97.13), (11.75, 41.93), (4.71, 94.23), (5.32, 87.26), (4.23, 99.91), (4.84, 91.2), (7.5, 44.52), (5.49, 79.38), (2.84, 96.8), (6.31, 52.01), (4.49, 66.03), (8.21, 29.12), (5.31, 74.66), (9.19, 75.29), (4.4, 105.15), (5.62, 64.31), (6.56, 43.88), (7.16, 65.89), (1.88, 92.19), (2.42, 80.93), (4.73, 83.33), (1.84, 79.77), (6.8, 70.69), (4.76, 89.65), (6.41, 98.44), (6.25, 44.42), (5.89, 53.96), (3.68, 88.9), (2.88, 88.27), (7.47, 78.78), (5.95, 89.16), (7.17, 96.33), (5.99, 81.72), (6.01, 74.9), (3.17, 98.42), (2.4, 102.37), (3.87, 89.37), (3.96, 69.23), (1.17, 129.46), (7.51, 92.25), (9.31, 75.98), (3.36, 106.1), (4.68, 110.59), (6.52, 94.91), (5.85, 63.19), (3.95, 61.06), (10.05, 48.88), (4.36, 74.83), (7.35, 51.28), (9.02, 40.11), (1.81, 86.67), (3.25, 67.87), (3.13, 89.83), (3.49, 69.29), (8.5, 75.1), (4.18, 85.95), (6.99, 35.9), (5.03, 80.36), (5.47, 84.46), (4.65, 84.66), (4.48, 83.41), (1.09, 93.98), (1.73, 114.12), (5.89, 85.74), (9.47, 46.47), (7.88, 76.82), (5.27, 64.1), (4.79, 31.24), (5.14, 82.93), (4.94, 84.11), (4.81, 96.71), (5.36, 69.81), (4.19, 101.29), (2.64, 83.07), (7.84, 67.14), (6.14, 88.97), (4.97, 81.02), (4.53, 52.22), (6.24, 79.97), (3.14, 97.5), (1.05, 126.58), (2.39, 92.16), (4.42, 69.62), (2.48, 99.16), (5.17, 55.16), (4.37, 109.38), (3.5, 87.03), (7.25, 53.52), (7.46, 34.78), (5.88, 45.99), (2.5, 123.58), (5.33, 92.17), (3.3, 103.81), (4.12, 88.19), (6.51, 89.75), (4.39, 77.24), (5.17, 89.59), (5.69, 59.96), (6.37, 53.32), (5.91, 59.74), (1.12, 127.84), (1.55, 98.72), (6.73, 70.98), (7.0, 60.51), (3.77, 58.97), (2.2, 85.38), (9.5, 74.48), (6.37, 75.56), (5.37, 68.13), (6.55, 66.65), (3.05, 92.49), (5.41, 86.47), (8.34, 65.16), (3.36, 120.54), (5.43, 112.96), (6.06, 61.07), (4.77, 69.42), (5.71, 84.76), (4.28, 78.8), (7.18, 103.71), (3.51, 65.75), (2.02, 111.29), (7.33, 84.09), (5.49, 80.06), (4.73, 67.51), (5.66, 66.11), (3.51, 94.68), (0.73, 117.35), (5.29, 98.36), (4.98, 68.85), (1.03, 104.96), (2.98, 121.97), (2.07, 94.29), (4.58, 69.76), (3.35, 97.77), (3.85, 95.19), (2.36, 83.45), (-0.13, 98.27), (2.14, 87.09), (7.08, 73.76), (3.23, 69.33), (5.59, 85.29), (9.66, 55.75), (5.24, 75.51), (4.91, 65.66), (0.84, 99.27), (7.87, 86.94), (6.9, 87.08), (4.74, 109.85), (-1.23, 91.38), (6.1, 39.88), (8.98, 34.04), (4.13, 94.91), (6.81, 54.94), (4.54, 78.04), (5.63, 48.55), (5.25, 44.0), (5.44, 130.22), (5.13, 74.1), (4.26, 80.09), (4.45, 101.11), (5.29, 64.19), (4.72, 51.94), (2.46, 117.71), (2.5, 138.28), (5.65, 74.64), (6.08, 49.02), (8.16, 42.48), (3.02, 94.91), (5.74, 82.09), (4.52, 64.97), (4.86, 79.73), (1.34, 101.06), (8.71, 39.61), (2.88, 118.75), (5.81, 59.5), (2.89, 89.71), (2.76, 81.68), (8.91, 93.58), (5.26, 76.14), (4.97, 80.3), (5.84, 107.21), (5.3, 71.86), (7.26, 92.16), (3.76, 96.57), (3.82, 75.11), (7.82, 22.08), (3.6, 76.06), (6.07, 94.43), (7.24, 71.94), (6.75, 30.27), (5.52, 98.98), (9.15, 45.14), (5.99, 62.81), (5.22, 69.68), (3.9, 55.49), (1.75, 51.84), (4.3, 79.9), (6.7, 68.36), (6.61, 73.34), (3.18, 113.83), (3.56, 116.34), (5.86, 79.4), (5.66, 46.17), (5.22, 70.52), (6.33, 65.88), (0.83, 106.02), (4.95, 110.69), (9.52, 61.98), (2.17, 84.72), (3.6, 97.42), (1.51, 115.41), (2.76, 94.38), (1.27, 117.99), (3.87, 89.55), (-1.62, 97.21), (5.98, 86.07), (7.4, 49.04), (3.58, 96.04), (4.46, 97.73), (4.04, 80.37), (5.46, 103.47), (5.17, 78.11), (3.59, 62.63), (4.58, 77.07), (5.81, 51.44), (4.63, 87.01), (4.99, 74.64), (6.99, 77.16), (4.33, 84.35), (5.28, 45.07), (5.52, 40.03), (4.21, 105.14), (3.26, 61.08), (8.8, 77.73), (7.86, 57.96), (4.57, 68.05), (2.33, 98.7), (4.7, 114.8), (7.62, 70.57), (4.55, 54.7), (4.26, 96.55), (6.79, 58.99), (5.31, 86.14), (6.62, 79.77), (4.12, 54.98), (6.46, 65.3), (5.87, 91.55), (5.23, 78.37), (4.38, 73.61), (5.54, 99.28), (4.89, 115.65), (5.96, 67.08), (3.44, 97.52), (4.95, 83.17), (4.36, 110.64), (5.54, 87.31), (4.52, 86.06), (1.6, 84.65), (4.76, 68.23), (2.36, 111.92), (6.36, 62.82), (5.42, 84.75), (6.44, 70.33), (3.72, 70.03), (6.7, 71.05), (4.37, 90.97), (7.25, 61.9), (2.98, 70.13), (4.79, 74.98), (6.61, 71.38), (4.32, 61.17), (2.87, 63.72), (3.79, 109.3), (5.4, 76.39), (5.39, 61.51), (0.63, 92.77), (2.37, 98.24), (4.29, 104.08), (5.41, 89.97), (6.48, 118.8), (0.35, 67.26), (5.05, 90.55), (4.21, 96.51), (3.71, 106.61), (5.99, 75.21), (6.61, 109.6), (1.98, 72.98), (0.42, 97.16), (5.03, 77.09), (4.3, 54.78), (7.43, 84.22), (2.16, 85.66), (5.97, 95.09), (0.56, 81.73), (5.71, 107.14), (6.16, 62.19), (6.02, 89.15), (6.6, 96.34), (7.83, 81.54), (7.16, 91.37), (5.64, 82.04)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=726；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4546；
t = r·√((n−2)/(1−r²)) = 0.4546·√((726−2)/(1−0.2067)) = -13.7330；df=724；p = -1.27e+10。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5106, -0.3948]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=726；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.4546；
t = r·√((n−2)/(1−r²)) = 0.4546·√((726−2)/(1−0.2067)) = -13.7330；df=724；p = -1.27e+10。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.5106, -0.3948]。
③ 计算结果：
r = -0.4546，t = -13.7330，p = -1.27e+10，95%CI = [-0.5106, -0.3948]
④ 结论：在 α=0.05 水平下，p=-1.27e+10 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.4546，方向负），利率升需求降。
案例 25. 海拔高度(m)与水沸点(℃)
Pearson相关海拔高沸点低
① 原始数据：
样本 n=422，逐条 (x, y)：[(2438.28, 89.41), (2025.11, 90.79), (1132.67, 94.37), (1728.71, 95.22), (2330.06, 87.7), (2187.38, 83.97), (708.53, 91.35), (2642.45, 89.58), (2680.79, 89.79), (1312.32, 103.52), (2814.55, 87.78), (876.32, 107.87), (1081.47, 96.41), (1777.84, 90.34), (2365.09, 85.31), (1447.79, 89.61), (286.91, 109.48), (635.43, 105.91), (1054.44, 94.61), (759.57, 92.58), (1174.82, 94.4), (2160.12, 84.19), (648.21, 104.01), (1275.82, 98.04), (692.28, 85.12), (650.19, 98.67), (2130.4, 88.21), (939.9, 105.78), (3576.32, 81.72), (791.56, 99.94), (1716.23, 96.0), (1110.37, 97.58), (1494.55, 95.55), (239.18, 99.89), (2185.29, 87.89), (1584.92, 94.36), (614.91, 109.33), (2082.41, 81.43), (2112.43, 97.41), (980.47, 110.82), (996.33, 103.45), (1239.14, 87.79), (2051.8, 84.95), (1918.53, 92.56), (502.67, 104.15), (1404.81, 99.47), (2989.29, 83.41), (637.06, 108.02), (2274.52, 90.07), (-14.23, 97.09), (1348.1, 87.41), (2558.55, 85.6), (1015.91, 89.23), (320.94, 95.12), (1587.4, 90.7), (2009.44, 85.2), (-492.52, 118.52), (1229.74, 89.33), (722.7, 107.72), (1583.24, 97.55), (1180.65, 92.39), (2982.06, 95.89), (1718.93, 97.05), (1488.02, 95.93), (1739.1, 106.47), (525.2, 105.07), (1333.43, 105.79), (1082.52, 98.18), (1094.88, 104.71), (1918.55, 100.98), (1969.34, 100.23), (1820.55, 90.97), (546.72, 99.57), (1919.58, 89.88), (979.78, 88.17), (2566.26, 86.14), (1969.75, 100.74), (-220.97, 94.92), (1846.3, 92.75), (2161.55, 84.67), (86.86, 107.42), (1522.23, 97.17), (2840.32, 90.85), (183.13, 98.06), (1661.87, 79.2), (1218.19, 98.55), (2149.36, 89.74), (1933.3, 91.11), (1250.1, 93.94), (1432.82, 99.84), (1539.83, 85.7), (2264.05, 87.9), (1301.85, 91.07), (1261.05, 96.98), (818.73, 100.99), (1988.11, 95.78), (2249.4, 82.68), (1423.06, 93.24), (2995.39, 74.38), (1977.36, 89.88), (2526.19, 80.9), (1267.06, 102.29), (-288.82, 107.96), (942.55, 90.64), (1685.47, 90.58), (2887.64, 86.38), (713.32, 103.64), (753.28, 104.98), (-224.69, 101.21), (2428.47, 79.66), (2000.22, 90.66), (2983.57, 86.57), (1490.9, 89.97), (1672.89, 85.55), (404.45, 96.14), (2406.69, 87.11), (2177.68, 91.55), (2789.68, 81.72), (591.02, 108.46), (1338.75, 102.98), (1709.6, 89.31), (1558.53, 89.76), (2169.8, 99.92), (1767.5, 101.9), (1398.67, 93.24), (758.06, 99.15), (1944.33, 84.99), (1692.45, 99.7), (478.95, 102.82), (1758.62, 88.04), (1134.05, 100.67), (182.96, 110.09), (1718.0, 95.11), (2318.4, 92.06), (2045.13, 95.04), (1382.17, 96.42), (883.13, 93.33), (2322.0, 95.78), (357.83, 109.31), (1868.81, 87.46), (1478.24, 96.4), (969.63, 110.45), (348.47, 108.98), (674.81, 96.51), (1943.35, 80.24), (837.06, 95.96), (1623.03, 84.49), (3078.99, 82.88), (586.89, 96.2), (1950.36, 95.14), (1631.5, 107.61), (1450.16, 91.81), (385.01, 97.73), (1548.74, 93.06), (1212.18, 94.69), (1222.19, 95.39), (-518.45, 113.05), (1897.32, 99.23), (2035.2, 95.87), (2573.7, 97.11), (1753.86, 94.91), (1048.53, 90.57), (774.35, 97.92), (1100.39, 94.63), (1570.79, 92.08), (2304.15, 91.24), (-207.98, 109.5), (786.69, 101.16), (2206.11, 97.49), (971.8, 96.15), (1759.58, 89.75), (1755.18, 91.16), (105.94, 106.73), (1433.41, 99.89), (1899.09, 92.89), (1348.24, 101.32), (2428.6, 88.05), (-833.49, 112.12), (468.88, 107.26), (1249.67, 93.76), (1590.25, 88.1), (484.96, 105.95), (2360.48, 90.2), (1822.16, 94.3), (1364.19, 96.0), (2229.99, 85.88), (897.94, 98.74), (1330.6, 87.71), (2974.53, 85.59), (1986.41, 95.59), (-220.05, 111.43), (1253.66, 96.48), (2474.09, 88.53), (953.15, 99.69), (1961.95, 93.66), (1964.4, 93.15), (1369.73, 91.49), (808.5, 102.68), (564.46, 105.82), (1317.17, 97.97), (2527.04, 90.45), (2107.23, 91.26), (2229.9, 90.54), (1198.63, 102.01), (962.24, 106.64), (1351.87, 91.66), (1477.51, 100.7), (1497.94, 82.45), (2949.69, 81.06), (2553.58, 84.23), (2867.76, 83.02), (932.38, 95.95), (96.73, 99.51), (-447.65, 119.54), (1451.02, 94.64), (750.24, 101.81), (1941.03, 89.31), (1008.74, 104.17), (603.77, 98.0), (2024.45, 86.68), (608.12, 104.26), (2618.68, 83.06), (2100.32, 97.39), (1771.47, 103.16), (2246.18, 88.58), (885.46, 89.49), (801.1, 100.64), (1463.42, 94.82), (2249.94, 87.89), (922.26, 88.1), (1544.81, 93.06), (1467.4, 102.33), (833.96, 103.53), (976.94, 100.45), (1681.32, 86.49), (2359.95, 87.07), (575.39, 95.45), (-387.6, 116.63), (2266.99, 93.75), (2091.97, 92.94), (244.52, 103.05), (950.25, 95.79), (2562.19, 94.6), (1600.53, 91.55), (1898.54, 95.92), (2941.74, 77.14), (463.83, 103.08), (194.29, 104.7), (1921.74, 82.78), (2351.21, 88.88), (1951.39, 89.56), (1978.83, 93.58), (2046.6, 79.76), (1518.45, 90.04), (1897.38, 86.34), (1669.84, 93.9), (2757.33, 85.43), (2100.36, 84.81), (2095.38, 93.23), (1.41, 112.69), (479.93, 112.13), (1355.16, 102.47), (1767.92, 93.75), (1647.91, 88.31), (2245.67, 86.41), (404.61, 99.58), (2906.96, 89.6), (1326.29, 107.43), (1570.2, 85.01), (1905.53, 94.43), (1456.43, 97.16), (1996.19, 99.58), (1833.5, 96.43), (1442.31, 89.74), (1719.99, 98.13), (1501.38, 95.36), (405.89, 100.23), (377.15, 93.5), (321.07, 101.42), (1151.17, 103.93), (1440.02, 101.22), (984.3, 95.07), (721.21, 118.55), (462.45, 92.22), (1528.22, 87.1), (2370.59, 100.08), (2194.25, 85.58), (920.72, 90.7), (1525.95, 95.51), (564.55, 101.22), (2274.45, 84.67), (1541.64, 95.16), (1277.73, 103.4), (1750.37, 98.25), (1105.45, 99.56), (1154.04, 95.2), (2104.07, 89.24), (963.52, 96.01), (1636.24, 96.54), (1582.96, 95.16), (1446.22, 91.48), (1025.91, 110.23), (1010.79, 102.74), (1649.23, 94.3), (1050.24, 98.93), (1171.0, 88.31), (1035.99, 102.23), (1756.87, 91.1), (1251.9, 101.94), (1943.35, 83.55), (1487.75, 97.29), (2226.26, 96.57), (980.81, 99.18), (2468.25, 88.92), (-233.69, 117.93), (621.07, 98.48), (3697.4, 81.87), (2279.89, 91.24), (767.16, 96.45), (963.44, 98.32), (2379.78, 97.03), (1502.38, 95.36), (1367.61, 97.78), (1567.98, 88.15), (1535.31, 94.97), (1804.82, 92.89), (2367.34, 86.89), (1787.95, 97.24), (301.5, 99.48), (1887.67, 105.03), (1635.01, 91.8), (1697.9, 88.56), (1913.86, 83.25), (2524.79, 79.87), (1803.4, 96.68), (1182.98, 92.53), (2328.07, 93.58), (1746.82, 85.34), (2331.7, 92.03), (3418.09, 83.61), (2027.25, 83.26), (-327.55, 109.39), (1792.49, 90.26), (1592.25, 86.54), (1901.33, 83.23), (749.25, 96.79), (2391.06, 91.63), (2728.87, 86.43), (2986.01, 79.06), (1333.26, 97.66), (2605.78, 73.76), (2083.27, 77.63), (1961.21, 80.65), (836.65, 94.59), (1929.59, 87.68), (1494.46, 102.6), (2981.28, 77.09), (1711.38, 93.0), (785.07, 113.9), (598.74, 101.88), (2238.01, 91.16), (304.96, 109.01), (2845.7, 87.11), (2243.23, 86.77), (1400.84, 101.62), (142.92, 109.5), (1558.16, 86.17), (666.11, 108.4), (1916.6, 97.11), (1453.75, 96.53), (504.6, 102.18), (1091.28, 103.03), (1633.45, 95.24), (1654.52, 94.26), (159.46, 103.9), (1879.15, 91.04), (1056.34, 111.76), (1646.24, 94.33), (1597.08, 82.88), (1121.11, 94.73), (1347.16, 101.47), (2666.19, 83.05), (1514.45, 91.13), (1804.57, 90.56), (1235.16, 93.73), (520.83, 93.86), (1571.64, 94.51), (917.99, 102.54), (773.52, 103.83), (2988.06, 76.04), (415.4, 100.44), (2833.27, 92.66), (1385.6, 101.81), (1697.14, 96.66), (1507.45, 89.7), (3069.27, 86.13), (839.09, 100.34), (479.51, 94.06), (1898.89, 91.97), (2313.67, 90.01), (1342.69, 112.43), (1530.6, 83.49), (1599.38, 92.89), (428.67, 112.84), (2501.06, 93.15), (790.57, 95.67), (2478.48, 86.88), (1554.67, 98.82), (433.98, 101.47), (2320.34, 85.37), (1714.11, 94.79), (1568.26, 79.66), (2194.07, 90.69), (2306.68, 82.74), (1836.17, 101.14), (760.16, 93.89), (1864.22, 84.38), (372.78, 98.55), (121.03, 103.19), (1554.25, 88.44), (1709.71, 101.7), (2124.79, 90.69)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=422；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.7162；
t = r·√((n−2)/(1−r²)) = 0.7162·√((422−2)/(1−0.5130)) = -21.0318；df=420；p = 1.33e-67。
95% CI（Fisher z）：tanh(z±19.897·1/√(n−3)) = [-0.9538, 0.0721]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=422；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.7162；
t = r·√((n−2)/(1−r²)) = 0.7162·√((422−2)/(1−0.5130)) = -21.0318；df=420；p = 1.33e-67。
95% CI（Fisher z）：tanh(z±19.897·1/√(n−3)) = [-0.9538, 0.0721]。
③ 计算结果：
r = -0.7162，t = -21.0318，p = 1.33e-67，95%CI = [-0.9538, 0.0721]
④ 结论：在 α=0.05 水平下，p=1.33e-67 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.7162，方向负），海拔高沸点低。
案例 26. 每周运动量(h)与体脂率(%)
Pearson相关运动多脂肪少
① 原始数据：
样本 n=442，逐条 (x, y)：[(6.0, 23.7), (5.37, 27.78), (5.85, 20.57), (6.79, 15.72), (3.7, 29.57), (4.86, 27.32), (6.03, 19.81), (4.96, 22.93), (3.89, 24.29), (8.65, 23.93), (5.57, 23.65), (4.66, 18.03), (2.87, 27.51), (8.75, 7.16), (6.13, 20.9), (2.79, 26.53), (4.67, 25.84), (2.13, 32.02), (4.23, 29.35), (8.45, 24.67), (8.29, 6.18), (7.55, 22.76), (4.62, 28.32), (5.15, 28.03), (9.32, 17.53), (0.89, 29.72), (5.8, 23.35), (5.93, 26.43), (7.26, 26.51), (5.81, 26.92), (4.89, 23.17), (2.15, 35.84), (5.87, 24.05), (3.59, 28.13), (4.22, 32.85), (4.82, 20.62), (5.0, 21.27), (6.16, 23.65), (5.31, 23.08), (3.14, 27.74), (3.89, 31.53), (6.72, 24.81), (3.96, 20.87), (5.62, 11.38), (6.38, 14.61), (5.07, 24.91), (8.1, 16.47), (2.06, 32.88), (3.09, 25.89), (3.47, 29.2), (4.77, 27.0), (7.3, 18.23), (7.12, 20.39), (3.42, 29.3), (5.0, 17.11), (2.55, 29.73), (4.77, 27.25), (4.44, 31.61), (5.01, 28.88), (2.39, 29.9), (5.32, 29.5), (2.81, 29.82), (4.97, 17.36), (8.77, 21.27), (6.28, 25.28), (6.04, 17.2), (6.0, 23.79), (4.64, 32.03), (5.85, 13.89), (2.65, 30.26), (7.76, 22.29), (2.36, 32.89), (2.14, 37.37), (5.92, 22.74), (4.69, 27.98), (3.04, 31.9), (8.4, 26.12), (6.87, 26.26), (7.37, 26.48), (6.07, 17.87), (7.97, 19.27), (6.35, 29.72), (10.01, 12.23), (3.23, 31.07), (2.56, 27.74), (5.43, 28.77), (9.11, 21.27), (3.13, 31.56), (7.15, 20.27), (5.75, 20.52), (7.66, 18.66), (7.0, 20.32), (4.89, 12.73), (3.97, 18.86), (2.74, 29.97), (4.48, 32.22), (7.33, 23.87), (1.14, 26.93), (0.74, 32.64), (4.09, 29.08), (5.88, 21.67), (6.85, 14.82), (6.57, 16.06), (4.06, 23.66), (7.98, 15.2), (3.52, 22.4), (3.27, 36.17), (5.42, 23.97), (7.37, 15.28), (3.79, 35.17), (6.1, 26.31), (5.1, 27.99), (4.81, 26.43), (5.59, 30.38), (3.57, 32.38), (6.78, 28.08), (7.21, 18.01), (5.18, 24.47), (4.37, 33.42), (7.7, 15.92), (5.34, 23.67), (4.96, 23.67), (5.76, 27.01), (1.88, 34.7), (4.89, 25.35), (7.51, 24.13), (4.51, 25.85), (5.4, 20.83), (4.41, 21.28), (3.29, 34.82), (6.06, 16.87), (5.82, 22.45), (3.4, 27.68), (5.64, 23.48), (3.95, 29.14), (5.42, 23.12), (2.28, 31.25), (6.35, 26.21), (5.8, 29.43), (5.67, 19.94), (6.08, 24.48), (6.69, 19.66), (8.05, 22.62), (5.95, 19.05), (5.63, 22.15), (9.5, 17.3), (6.03, 20.15), (6.83, 17.16), (0.26, 36.96), (5.47, 18.73), (2.76, 34.32), (5.22, 21.49), (4.85, 26.69), (6.4, 20.9), (5.72, 17.61), (2.27, 30.84), (5.55, 15.27), (8.73, 17.78), (4.83, 28.75), (8.36, 7.91), (2.92, 27.72), (6.39, 24.12), (3.56, 24.95), (7.28, 13.41), (4.97, 18.3), (5.6, 22.94), (7.53, 30.62), (4.56, 31.23), (4.63, 22.4), (1.29, 31.38), (6.06, 24.14), (4.98, 20.27), (0.72, 29.21), (6.09, 14.07), (8.6, 14.8), (7.32, 22.1), (6.96, 27.21), (3.22, 30.29), (6.77, 19.42), (3.74, 35.66), (5.59, 23.41), (5.46, 29.1), (9.32, 22.64), (3.22, 28.0), (4.35, 30.7), (7.11, 27.4), (8.51, 11.72), (3.9, 28.17), (5.61, 21.01), (8.04, 17.4), (4.56, 26.99), (6.58, 21.19), (3.55, 26.1), (4.9, 22.07), (6.94, 13.26), (6.21, 19.14), (4.89, 24.2), (9.22, 20.14), (5.88, 23.76), (3.68, 24.69), (4.81, 23.17), (4.73, 26.12), (4.83, 29.11), (4.8, 28.42), (3.45, 29.14), (5.74, 24.61), (2.65, 32.58), (7.5, 17.68), (3.19, 26.41), (5.82, 28.57), (8.54, 18.94), (6.98, 20.78), (3.11, 30.6), (3.28, 28.84), (6.78, 21.89), (2.71, 32.22), (5.51, 23.89), (5.2, 17.61), (6.66, 28.51), (8.22, 19.62), (7.24, 18.77), (0.93, 29.5), (2.49, 33.34), (7.71, 18.12), (7.87, 15.37), (6.6, 9.25), (4.19, 31.54), (3.41, 31.24), (8.73, 17.7), (8.2, 13.4), (3.22, 30.06), (1.07, 30.07), (4.82, 32.51), (4.86, 22.44), (1.56, 28.68), (7.26, 12.94), (7.83, 19.39), (7.31, 22.26), (3.92, 28.26), (3.14, 26.81), (7.23, 24.89), (4.35, 33.95), (3.86, 26.93), (6.84, 15.43), (1.58, 33.08), (3.41, 29.45), (8.45, 12.52), (4.19, 24.4), (3.42, 32.22), (8.11, 20.09), (4.28, 27.09), (5.71, 26.51), (6.93, 23.81), (4.95, 30.42), (4.17, 22.91), (1.58, 39.79), (5.33, 27.53), (4.47, 33.4), (5.47, 26.19), (7.11, 16.35), (1.26, 39.6), (6.94, 26.1), (7.16, 14.43), (7.23, 26.06), (9.13, 19.19), (6.04, 22.75), (6.7, 21.54), (1.99, 29.12), (2.68, 33.93), (3.42, 24.62), (4.72, 23.27), (1.53, 29.44), (5.3, 25.08), (2.28, 33.23), (4.23, 34.33), (5.34, 23.82), (9.25, 8.13), (4.54, 27.75), (4.79, 26.02), (3.01, 26.64), (6.02, 31.36), (4.84, 21.89), (3.39, 31.78), (3.11, 25.47), (3.14, 28.99), (5.53, 17.52), (8.15, 25.13), (7.13, 26.04), (8.36, 15.52), (2.55, 25.79), (9.66, 22.32), (9.52, 23.52), (5.34, 23.31), (6.59, 18.76), (2.48, 30.18), (2.49, 26.97), (3.38, 28.35), (6.15, 24.99), (6.82, 24.02), (5.75, 23.35), (4.96, 22.68), (2.18, 32.29), (8.42, 19.75), (5.19, 26.98), (4.48, 17.94), (5.14, 25.9), (6.95, 25.0), (4.79, 29.79), (5.58, 30.59), (4.09, 25.67), (4.25, 27.22), (6.73, 13.78), (5.82, 20.58), (3.21, 27.87), (5.13, 25.94), (2.67, 24.6), (6.05, 25.07), (8.23, 19.76), (3.83, 33.25), (4.8, 24.0), (2.94, 27.44), (1.59, 30.63), (-1.32, 36.12), (7.14, 20.5), (1.43, 28.43), (6.6, 22.74), (6.2, 18.85), (7.85, 24.81), (6.59, 16.33), (7.62, 17.79), (9.82, 7.35), (3.05, 23.77), (-0.03, 32.81), (3.38, 25.75), (6.12, 19.91), (6.46, 25.21), (5.26, 25.14), (6.14, 26.79), (4.18, 26.24), (4.29, 30.17), (9.14, 16.74), (5.58, 23.39), (4.07, 23.69), (6.67, 24.69), (3.18, 29.48), (5.44, 26.12), (9.65, 22.64), (2.13, 30.98), (4.75, 19.58), (5.63, 31.73), (4.7, 32.76), (2.59, 35.8), (5.37, 26.4), (8.07, 19.59), (2.19, 30.69), (3.97, 26.66), (4.7, 23.4), (8.06, 23.25), (4.45, 31.19), (4.7, 33.63), (4.6, 28.78), (4.42, 26.23), (7.34, 17.95), (8.5, 9.85), (4.47, 29.38), (3.51, 35.87), (7.58, 19.12), (1.72, 21.93), (3.71, 26.79), (6.97, 15.7), (3.62, 31.99), (3.0, 34.35), (6.01, 22.75), (4.28, 23.1), (3.81, 23.08), (5.55, 21.9), (4.84, 24.66), (7.21, 19.25), (4.16, 31.01), (4.38, 25.4), (8.01, 21.29), (5.84, 27.67), (4.68, 20.56), (1.42, 31.15), (3.74, 25.27), (8.03, 28.26), (4.65, 22.17), (4.57, 28.68), (3.85, 21.99), (5.04, 21.15), (6.5, 25.13), (2.2, 31.6), (4.73, 24.39), (2.63, 27.57), (9.5, 15.82), (5.94, 20.35), (6.41, 27.0), (6.15, 17.29), (5.39, 30.84), (2.74, 32.54), (3.44, 24.92), (6.83, 24.89), (1.08, 37.11), (7.91, 15.79), (6.22, 21.23), (5.96, 24.22), (5.75, 16.36), (1.18, 26.66), (5.25, 23.62), (5.58, 23.11), (5.32, 24.29), (5.19, 23.28), (5.21, 21.31), (2.25, 22.69), (5.89, 25.9), (5.96, 18.83), (2.72, 35.16), (2.04, 30.77), (6.13, 14.18), (5.86, 24.99), (4.39, 29.5), (8.47, 17.17), (4.27, 17.72), (6.58, 21.63), (3.07, 27.08), (2.12, 31.54), (6.8, 28.53), (1.32, 30.1), (5.36, 29.15), (6.4, 26.37), (2.02, 25.29), (5.86, 25.77), (2.71, 23.58), (3.52, 25.44), (5.31, 25.3), (7.51, 21.46), (4.59, 21.13), (4.49, 14.69), (5.66, 28.05), (7.73, 19.47), (5.85, 20.89), (2.51, 32.77)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=442；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6852；
t = r·√((n−2)/(1−r²)) = 0.6852·√((442−2)/(1−0.4696)) = -19.7357；df=440；p = -1.10e+18。
95% CI（Fisher z）：tanh(z±20.975·1/√(n−3)) = [-0.9508, 0.1607]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=442；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6852；
t = r·√((n−2)/(1−r²)) = 0.6852·√((442−2)/(1−0.4696)) = -19.7357；df=440；p = -1.10e+18。
95% CI（Fisher z）：tanh(z±20.975·1/√(n−3)) = [-0.9508, 0.1607]。
③ 计算结果：
r = -0.6852，t = -19.7357，p = -1.10e+18，95%CI = [-0.9508, 0.1607]
④ 结论：在 α=0.05 水平下，p=-1.10e+18 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.6852，方向负），运动多脂肪少。
案例 27. 学习时间与游戏时间(分)
Pearson相关学习挤占游戏
① 原始数据：
样本 n=845，逐条 (x, y)：[(135.14, 101.7), (283.7, 100.2), (112.1, 180.35), (188.15, 108.19), (132.27, 76.94), (80.76, 99.9), (258.54, 131.5), (193.93, 136.67), (124.14, 140.12), (220.78, 57.6), (225.48, 115.5), (59.67, 162.32), (143.98, 182.0), (99.48, 200.59), (247.86, 90.53), (194.71, 110.43), (99.41, 116.91), (128.52, 162.0), (99.76, 188.76), (95.09, 205.6), (78.56, 207.45), (57.05, 172.61), (229.3, 33.38), (151.88, 173.84), (142.9, 201.78), (174.16, 109.3), (219.64, 78.9), (156.09, 128.15), (159.77, 107.48), (156.13, 158.56), (269.85, 61.46), (139.41, 90.68), (131.31, 178.2), (236.02, 126.82), (149.54, 102.96), (194.46, 91.63), (300.71, 56.36), (165.8, 152.55), (216.29, 49.45), (294.11, 27.9), (231.18, 61.18), (211.49, 79.64), (122.52, 153.21), (109.33, 158.3), (146.32, 101.19), (249.15, 81.63), (122.73, 106.47), (207.71, 91.27), (111.66, 137.9), (156.86, 189.97), (206.81, 103.48), (182.9, 67.95), (168.0, 168.59), (117.76, 194.87), (223.43, 92.83), (113.21, 114.98), (227.26, 67.02), (247.61, 69.56), (243.44, 97.66), (104.98, 223.7), (71.27, 247.06), (160.74, 146.1), (299.03, 34.95), (245.82, 44.63), (74.4, 176.74), (200.55, 150.58), (32.63, 153.82), (145.3, 195.02), (167.77, 145.48), (120.84, 108.07), (79.29, 202.73), (182.52, 93.53), (65.25, 205.44), (157.22, 207.28), (147.97, 150.32), (234.21, 140.33), (263.28, 67.91), (203.15, 141.91), (209.42, 132.62), (121.63, 103.45), (185.79, 129.7), (277.97, 90.54), (96.63, 147.48), (215.91, 63.27), (157.83, 91.14), (116.55, 167.14), (183.03, 152.58), (236.95, 112.51), (215.19, 153.5), (133.08, 168.38), (162.2, 114.04), (193.45, 131.16), (117.82, 176.5), (115.3, 150.8), (191.92, 77.59), (231.44, 124.75), (167.18, 160.88), (184.27, 86.17), (229.5, 81.78), (200.59, 78.42), (365.89, 7.6), (157.79, 124.68), (124.86, 161.99), (178.86, 155.77), (301.41, 113.42), (193.02, 129.41), (188.26, 89.35), (203.23, 47.43), (127.78, 203.52), (69.4, 140.03), (82.29, 180.72), (190.66, 64.21), (163.56, 118.81), (278.81, 98.47), (88.49, 158.55), (183.95, 78.55), (139.12, 176.29), (134.05, 202.39), (249.94, 119.97), (249.75, 164.34), (118.8, 200.58), (39.18, 194.46), (186.33, 84.54), (206.76, 31.12), (143.05, 104.56), (146.63, 151.39), (173.84, 109.45), (195.05, 120.94), (88.22, 107.35), (218.69, 54.84), (166.64, 95.77), (183.18, 171.87), (154.96, 161.12), (253.58, 78.63), (149.48, 86.03), (181.83, 71.9), (306.27, 38.65), (153.4, 153.53), (92.97, 159.56), (260.66, 75.61), (179.13, 57.16), (156.07, 189.06), (189.3, 123.73), (228.49, 76.15), (182.17, 144.7), (233.24, 108.59), (204.85, 96.59), (173.08, 104.8), (202.59, 75.73), (181.45, 194.71), (198.07, 145.89), (173.62, 125.0), (287.55, 110.55), (100.43, 107.13), (63.89, 170.7), (157.86, 98.04), (210.17, 23.88), (156.2, 166.99), (167.06, 116.57), (126.53, 196.59), (136.69, 130.32), (308.65, 26.18), (204.77, 74.18), (135.16, 156.2), (147.54, 140.47), (95.93, 141.24), (139.11, 140.74), (273.95, 72.3), (123.4, 114.03), (187.59, 56.16), (169.57, 123.01), (230.73, 75.18), (97.41, 171.43), (109.24, 196.91), (190.73, 131.65), (149.86, 93.81), (158.07, 60.11), (168.18, 134.0), (118.39, 195.46), (124.83, 112.23), (72.98, 201.4), (199.7, 133.81), (187.03, 103.01), (205.84, 131.69), (168.67, 51.34), (303.22, 46.53), (110.66, 223.51), (326.01, -32.36), (213.24, 83.53), (178.06, 155.95), (207.25, 134.27), (143.83, 133.1), (258.87, 98.63), (173.34, 60.6), (142.13, 166.8), (176.69, 168.2), (108.46, 179.03), (181.21, 77.57), (88.9, 203.58), (220.45, 129.49), (226.11, 71.65), (268.68, 90.61), (135.8, 163.96), (157.62, 156.94), (341.75, 39.41), (155.82, 133.45), (300.34, 35.73), (257.04, 111.14), (201.76, 124.82), (265.38, 107.77), (267.48, 60.51), (225.02, 110.35), (117.34, 142.23), (262.37, 111.56), (151.95, 103.1), (182.04, 98.87), (0.4, 234.03), (131.72, 161.02), (144.58, 143.77), (211.47, 116.38), (98.77, 133.21), (114.82, 161.65), (156.67, 107.85), (177.1, 73.77), (132.26, 196.02), (121.12, 147.1), (249.84, 144.5), (212.86, 55.86), (166.71, 135.03), (252.89, 22.93), (184.2, 88.96), (207.96, 196.84), (307.2, -2.16), (53.62, 232.04), (297.1, 1.07), (120.84, 123.78), (202.44, 131.35), (182.22, 109.8), (148.27, 84.85), (231.76, 92.33), (158.82, 139.82), (197.21, 86.55), (167.69, 196.26), (94.82, 169.89), (283.01, 71.17), (277.66, 49.82), (184.88, 121.88), (201.11, 105.18), (144.47, 215.81), (85.79, 112.92), (140.04, 195.71), (268.8, 38.97), (112.08, 148.37), (252.92, 25.84), (207.22, 145.56), (131.76, 169.61), (223.95, 36.61), (132.21, 170.03), (182.44, 45.41), (195.72, 176.73), (227.75, 113.26), (199.49, 170.36), (138.0, 146.74), (117.99, 148.83), (155.06, 73.04), (84.65, 184.1), (199.43, 149.16), (120.97, 188.61), (185.19, 89.17), (250.05, 103.29), (146.91, 179.52), (155.32, 131.65), (187.39, 139.55), (138.27, 138.29), (88.13, 210.06), (142.45, 95.64), (199.9, 192.46), (119.38, 121.34), (195.39, 93.01), (193.48, 56.81), (79.41, 239.91), (129.27, 134.51), (186.06, 127.21), (245.4, 85.76), (314.18, 87.92), (204.56, 64.83), (230.8, 30.72), (185.45, 92.26), (201.39, 73.71), (113.15, 213.76), (144.51, 149.39), (185.91, 84.23), (74.61, 248.32), (234.09, 58.88), (121.28, 125.79), (95.78, 186.03), (230.16, 133.25), (180.59, 185.3), (187.2, 119.24), (233.97, 54.9), (212.68, 72.82), (73.95, 194.67), (227.7, 111.53), (252.49, 68.27), (231.99, 33.83), (164.94, 106.73), (215.0, 70.94), (27.64, 236.51), (256.71, 89.5), (167.14, 115.86), (214.97, 135.47), (203.18, 109.78), (180.11, 107.07), (138.97, 142.01), (271.08, 53.81), (143.47, 155.21), (200.35, 99.47), (264.09, 77.2), (150.49, 85.4), (89.84, 126.47), (71.92, 172.15), (228.56, 104.88), (209.53, 72.46), (227.83, 29.05), (231.49, 80.65), (212.2, 175.64), (239.41, 90.02), (57.86, 145.42), (31.13, 172.31), (113.75, 259.86), (197.94, 132.22), (85.34, 138.79), (155.43, 94.73), (191.52, 131.19), (96.25, 212.22), (268.44, 56.84), (163.95, 76.83), (150.32, 99.24), (61.69, 209.85), (215.93, 101.67), (199.3, 126.11), (306.71, -4.55), (176.53, 133.64), (107.32, 207.93), (303.73, 43.11), (113.4, 196.21), (186.01, 63.46), (153.82, 113.91), (183.38, 187.44), (133.98, 139.58), (163.33, 112.09), (183.09, 162.79), (128.73, 190.45), (194.17, 150.07), (235.9, 40.14), (270.42, 88.43), (103.55, 156.92), (246.09, 111.41), (145.32, 161.61), (202.29, 90.18), (189.82, 70.56), (211.99, 89.88), (114.15, 218.85), (137.57, 54.39), (250.76, 70.13), (228.15, 50.6), (192.1, 39.12), (183.04, 127.85), (172.55, 135.22), (99.31, 156.46), (204.18, 63.25), (159.57, 166.28), (163.94, 161.55), (174.65, 99.86), (128.58, 177.68), (301.42, 38.8), (138.38, 93.44), (218.53, 99.53), (199.95, 72.61), (212.3, 113.37), (202.05, 89.24), (156.06, 122.31), (97.57, 205.45), (186.24, 95.42), (89.86, 165.32), (158.56, 207.96), (139.03, 147.82), (176.86, 139.51), (242.23, 88.86), (247.97, 129.45), (243.09, 140.41), (315.97, 50.84), (258.75, -2.2), (250.31, 87.54), (185.84, 58.87), (251.95, 120.4), (134.59, 203.82), (114.09, 149.14), (219.99, 94.15), (150.17, 187.39), (238.22, 130.61), (88.29, 165.07), (151.34, 130.62), (216.14, 154.67), (122.65, 147.36), (216.03, 115.34), (165.49, 167.99), (94.21, 183.05), (127.24, 94.09), (78.67, 185.92), (243.36, 91.61), (245.51, 129.7), (243.1, 136.7), (207.3, 115.5), (100.24, 139.96), (103.6, 148.49), (204.31, 105.48), (213.99, 33.59), (160.41, 153.54), (212.73, 63.89), (232.23, 104.16), (194.23, 169.51), (145.02, 130.82), (204.74, 108.33), (131.54, 170.81), (230.4, 84.23), (160.96, 140.46), (186.74, 128.82), (139.25, 139.49), (156.63, 148.63), (252.56, 48.7), (122.91, 157.42), (159.4, 136.24), (125.59, 51.22), (256.93, 78.89), (92.34, 205.46), (158.89, 120.19), (276.1, 19.83), (136.24, 182.31), (278.97, -9.44), (229.64, 125.76), (168.77, 137.98), (131.1, 119.57), (214.08, 117.97), (178.7, 54.43), (163.81, 153.4), (180.9, 140.71), (204.0, 147.04), (101.72, 121.52), (254.06, 139.93), (235.14, 95.38), (185.97, 97.86), (72.75, 171.44), (191.33, 75.72), (66.02, 176.16), (103.09, 131.87), (208.01, 177.65), (151.26, 179.29), (113.21, 144.03), (136.9, 88.13), (92.94, 135.55), (174.44, 134.54), (228.14, 86.34), (106.19, 107.99), (180.87, 127.82), (209.35, 136.97), (117.85, 76.32), (206.39, 92.33), (261.98, 174.89), (208.08, 77.91), (240.09, 72.33), (275.03, 46.64), (139.15, 126.08), (186.77, 118.17), (120.7, 174.4), (184.04, 139.32), (215.94, 112.16), (291.78, 19.83), (69.44, 185.37), (32.64, 183.34), (213.11, 46.33), (126.24, 87.75), (266.93, 52.2), (277.49, 42.65), (109.5, 160.7), (133.29, 140.76), (136.03, 174.25), (143.79, 115.46), (246.32, 146.5), (198.19, 99.05), (110.34, 161.5), (159.74, 153.87), (230.82, 121.52), (315.2, 26.06), (151.86, 118.82), (164.52, 168.28), (181.02, 118.61), (317.31, -4.09), (139.87, 167.17), (92.72, 160.19), (165.67, 229.53), (24.13, 196.08), (173.04, 115.51), (136.46, 137.95), (131.21, 177.12), (161.21, 125.87), (152.06, 186.79), (236.57, 53.96), (176.8, 121.82), (333.81, 17.77), (165.63, 132.97), (236.7, 91.2), (163.41, 190.48), (240.2, 53.47), (249.97, 61.81), (203.92, 67.4), (146.53, 160.46), (79.99, 231.96), (170.91, 133.63), (114.35, 95.08), (268.86, 39.69), (188.95, 92.34), (129.23, 208.92), (176.19, 233.41), (225.54, 125.9), (282.08, 77.61), (248.08, 77.76), (142.89, 120.92), (154.28, 42.9), (115.66, 162.87), (205.68, 188.09), (215.77, 123.23), (160.72, 204.71), (192.32, 137.75), (266.81, 27.81), (54.59, 241.91), (111.45, 170.4), (129.79, 181.08), (195.76, 140.22), (140.72, 202.69), (134.09, 212.99), (208.86, 105.6), (129.11, 157.24), (10.74, 123.61), (227.58, 104.28), (188.09, 145.49), (139.55, 141.19), (171.77, 130.38), (150.9, 105.55), (218.64, 89.86), (142.43, 130.95), (179.32, 111.17), (171.28, 139.61), (223.33, 85.68), (159.09, 123.07), (187.45, 101.86), (113.79, 185.27), (234.68, 171.2), (223.0, 120.66), (327.87, 80.73), (165.14, 132.71), (181.37, 115.41), (91.39, 155.65), (192.43, 93.96), (228.94, 123.96), (223.81, 67.62), (171.89, 225.38), (88.03, 186.98), (138.85, 104.67), (44.87, 292.05), (106.65, 157.37), (245.25, 30.08), (118.93, 170.78), (180.97, 112.12), (160.53, 94.84), (62.73, 260.77), (91.81, 173.33), (244.44, 51.93), (85.84, 193.82), (125.34, 176.89), (192.41, 130.24), (193.64, 83.79), (12.06, 185.39), (176.43, 106.9), (262.81, 87.17), (291.25, 3.37), (122.52, 218.13), (221.83, 127.86), (197.22, 129.03), (165.25, 142.55), (111.32, 106.61), (143.43, 158.83), (152.58, 95.75), (216.64, 90.1), (210.99, 52.7), (130.29, 128.28), (116.23, 139.75), (184.95, 122.08), (173.77, 105.05), (251.14, 130.22), (195.16, 154.42), (220.7, 145.47), (110.76, 117.53), (89.08, 118.57), (227.12, 102.45), (150.53, 148.2), (236.33, 99.38), (114.97, 212.9), (141.38, 86.86), (182.44, 103.74), (201.92, 125.62), (253.68, 71.78), (235.93, 28.55), (163.36, 48.7), (150.14, 99.05), (158.29, 160.19), (181.52, 50.13), (102.27, 110.85), (325.54, -22.07), (232.79, 16.72), (141.8, 192.51), (261.06, 87.82), (157.24, 134.49), (218.78, 83.84), (206.66, 113.19), (283.52, 76.54), (126.82, 140.34), (190.59, 64.94), (260.61, 67.78), (229.96, 9.51), (185.25, 122.11), (260.34, 122.81), (152.65, 124.81), (163.17, 157.08), (123.17, 181.29), (258.3, 113.67), (264.96, 38.5), (113.09, 121.09), (119.5, 188.07), (278.54, 21.8), (136.21, 74.73), (122.59, 175.77), (152.53, 122.7), (69.72, 172.13), (260.95, 26.03), (261.56, 51.87), (149.98, 146.58), (224.37, 97.1), (175.27, 80.92), (90.12, 145.53), (132.94, 113.64), (206.19, 82.24), (230.91, 108.27), (151.62, 157.8), (166.51, 121.3), (203.11, 173.15), (225.04, 22.2), (136.06, 114.47), (191.66, 173.91), (208.6, 102.1), (115.16, 127.95), (313.38, -29.1), (167.41, 118.42), (282.84, -15.07), (241.73, 67.23), (213.67, 120.23), (149.46, 132.86), (236.0, 111.66), (234.64, 18.64), (262.19, 49.14), (154.59, 189.15), (276.72, 99.85), (118.77, 198.69), (201.61, 67.5), (133.79, 83.45), (248.47, 86.51), (152.82, 93.93), (132.54, 195.07), (235.68, 109.52), (250.52, 105.97), (327.27, -11.34), (195.05, 101.92), (157.19, 133.2), (271.69, -13.95), (107.63, 158.14), (171.14, 141.3), (153.12, 139.35), (123.92, 164.72), (216.5, 89.5), (229.94, 55.39), (137.17, 182.07), (191.91, 69.56), (127.13, 153.31), (235.28, 62.05), (102.42, 193.62), (207.04, 79.51), (186.84, 102.37), (316.12, 24.0), (273.24, 40.05), (120.7, 141.74), (242.18, 22.85), (200.71, 106.23), (263.34, 57.74), (119.41, 156.42), (158.29, 132.97), (166.55, 161.96), (151.84, 168.63), (119.51, 130.42), (203.67, 88.25), (170.18, 157.25), (297.02, 56.25), (95.46, 198.23), (65.33, 210.03), (216.02, 157.69), (212.68, 73.81), (51.45, 253.93), (97.84, 152.89), (100.19, 230.57), (161.29, 192.25), (216.42, 57.88), (206.07, 84.28), (216.76, 127.62), (123.35, 158.23), (206.27, 181.3), (257.73, 90.76), (280.34, 65.81), (173.52, 149.76), (255.83, 150.14), (134.34, 74.8), (135.95, 158.84), (163.79, 149.44), (133.41, 172.33), (127.81, 129.11), (196.68, 73.96), (141.16, 160.53), (86.67, 218.3), (118.62, 135.15), (117.94, 97.28), (152.02, 153.31), (202.12, 69.28), (109.1, 106.83), (266.95, 113.42), (181.02, 84.5), (183.75, 168.23), (309.1, 28.88), (298.6, 73.24), (114.79, 143.92), (256.04, 44.06), (229.28, 49.14), (133.74, 149.16), (194.49, 148.69), (221.21, 52.45), (116.46, 131.11), (103.91, 172.82), (151.24, 166.22), (136.82, 148.47), (119.15, 155.27), (147.94, 168.65), (158.79, 117.12), (207.25, 143.7), (165.91, 197.28), (259.32, 107.65), (172.72, 188.73), (288.78, 64.32), (308.54, -19.25), (226.89, 107.02), (172.45, 102.68), (258.74, 104.94), (232.32, 110.3), (240.73, 117.11), (209.28, 15.74), (206.17, 120.7), (273.69, 105.07), (84.85, 158.8), (143.02, 159.39), (187.23, 211.61), (80.12, 112.14), (230.06, 49.83), (75.23, 167.15), (180.39, 173.69), (172.13, 88.6), (252.68, 97.85), (287.03, 2.99), (205.61, 97.32), (212.1, 59.12), (209.71, 121.37), (107.45, 153.27), (215.29, 110.74), (164.76, 123.64), (113.36, 150.92), (91.23, 102.88), (262.16, 53.31), (152.49, 126.48), (164.57, 98.26), (202.72, 146.09), (199.53, 120.32), (151.76, 121.04), (212.44, 125.38), (170.32, 82.54), (67.41, 123.71), (121.27, 158.19), (189.12, 121.61), (174.16, 150.17), (183.08, 112.62), (140.0, 106.86), (130.25, 118.93), (273.0, 10.58), (231.65, 91.84), (224.33, 90.09), (161.31, 129.77), (146.78, 56.83), (235.6, 122.48), (246.69, 80.84), (214.67, 173.33), (221.13, 51.36), (127.14, 120.91), (165.74, 174.11), (146.41, 73.74), (210.83, 85.22), (240.25, 146.89), (35.64, 206.13), (143.69, 172.2), (196.93, 133.79), (115.41, 196.31), (190.72, 10.22), (245.72, 67.47), (188.82, 151.92), (190.57, 65.35), (240.17, 101.61), (210.07, 126.3), (216.07, 85.84), (196.3, 83.5), (160.7, 173.38), (286.22, 77.21), (200.0, 83.36), (101.79, 118.41), (228.06, 131.8), (209.12, 131.35), (246.63, 44.91), (268.17, 73.26), (142.67, 141.31), (172.15, 84.91), (242.22, 103.48), (196.12, 122.73), (117.43, 169.12)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=845；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6949；
t = r·√((n−2)/(1−r²)) = 0.6949·√((845−2)/(1−0.4828)) = -28.0534；df=843；p = -7.31e+52。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7282, -0.6582]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=845；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6949；
t = r·√((n−2)/(1−r²)) = 0.6949·√((845−2)/(1−0.4828)) = -28.0534；df=843；p = -7.31e+52。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7282, -0.6582]。
③ 计算结果：
r = -0.6949，t = -28.0534，p = -7.31e+52，95%CI = [-0.7282, -0.6582]
④ 结论：在 α=0.05 水平下，p=-7.31e+52 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.6949，方向负），学习挤占游戏。
案例 28. 气温(℃)与取暖器销量(台)
Pearson相关天暖少买取暖器
① 原始数据：
样本 n=632，逐条 (x, y)：[(18.47, 203.56), (23.73, 147.93), (18.19, 149.36), (35.77, 116.26), (12.97, 215.77), (25.86, 107.65), (5.76, 122.3), (23.49, 92.11), (21.33, 162.91), (35.74, 70.15), (19.02, 135.32), (22.66, 153.56), (24.87, 194.15), (29.38, 92.76), (23.12, 102.61), (7.54, 205.67), (0.83, 293.27), (32.97, 62.38), (11.19, 130.09), (17.81, 173.69), (18.35, 127.58), (6.68, 189.88), (24.86, 117.24), (22.55, 148.71), (19.89, 153.46), (18.29, 144.38), (28.03, 117.95), (35.47, 38.26), (28.44, 116.96), (20.22, 96.89), (16.21, 163.73), (15.59, 173.6), (21.43, 106.83), (32.91, 39.08), (11.61, 244.04), (15.25, 166.58), (26.5, 117.76), (23.36, 25.37), (20.21, 133.03), (13.48, 201.07), (40.28, 43.51), (24.74, 139.24), (4.21, 232.43), (13.77, 251.9), (28.09, 57.89), (23.34, 106.2), (39.57, 53.18), (15.35, 117.89), (39.14, 103.68), (20.56, 159.79), (4.74, 189.43), (26.17, 125.73), (30.72, 124.92), (28.82, 67.06), (23.43, 151.93), (25.9, 53.6), (15.76, 215.85), (10.82, 118.78), (9.3, 221.48), (18.24, 125.31), (19.26, 145.49), (10.28, 213.54), (29.71, 49.55), (14.22, 166.64), (23.23, 70.36), (25.1, 115.78), (21.56, 132.98), (21.53, 189.5), (31.01, 47.89), (22.9, 165.02), (22.76, 171.35), (17.27, 97.98), (15.36, 188.51), (6.94, 186.8), (35.73, 64.96), (37.88, 92.64), (37.33, 88.1), (26.65, 75.56), (21.28, 101.86), (15.33, 213.01), (30.29, 135.76), (16.35, 176.49), (26.3, 92.19), (13.66, 179.43), (28.98, 145.21), (-2.89, 199.26), (21.85, 126.82), (18.79, 222.28), (23.47, 63.07), (24.37, 109.75), (8.76, 196.57), (11.21, 185.04), (14.01, 186.15), (20.56, 166.6), (11.52, 107.22), (12.64, 100.2), (7.73, 179.7), (16.72, 155.82), (3.07, 223.5), (-7.86, 216.85), (12.31, 199.2), (24.83, 100.44), (7.77, 219.08), (24.95, 129.77), (27.04, 89.18), (27.66, 151.39), (36.79, 65.89), (22.62, 106.58), (10.04, 178.26), (15.04, 207.32), (25.37, 102.56), (20.41, 193.97), (22.41, 119.72), (18.31, 63.93), (24.0, 188.63), (34.61, 85.21), (37.18, 61.61), (14.66, 183.94), (7.89, 121.83), (6.95, 180.78), (33.15, 55.0), (26.75, 89.88), (21.54, 146.72), (14.3, 112.71), (15.27, 171.29), (4.29, 237.25), (18.22, 172.79), (44.56, 39.32), (19.52, 144.35), (24.15, 147.95), (20.21, 59.03), (27.94, 61.42), (-1.13, 264.37), (33.09, 90.93), (9.47, 179.16), (14.4, 119.85), (23.48, 169.21), (5.43, 218.47), (21.69, 135.39), (16.03, 209.86), (28.55, 50.82), (13.7, 146.33), (17.28, 167.79), (35.51, -20.18), (13.1, 148.97), (11.38, 254.38), (3.15, 215.61), (17.59, 49.08), (8.69, 167.36), (-1.7, 276.8), (5.05, 281.8), (2.8, 153.74), (17.76, 106.87), (31.21, 147.49), (10.3, 198.97), (7.78, 261.66), (17.88, 197.37), (28.19, 172.11), (11.88, 160.99), (21.54, 129.92), (3.4, 184.63), (15.42, 191.25), (20.33, 182.48), (15.2, 225.34), (4.11, 115.39), (29.41, 124.69), (12.25, 239.42), (13.12, 232.29), (13.29, 145.85), (13.1, 218.03), (7.59, 186.97), (5.57, 275.97), (23.73, 132.03), (38.79, 115.44), (5.04, 225.74), (15.34, 48.13), (12.87, 170.86), (6.68, 221.57), (19.82, 61.55), (22.44, 80.63), (9.38, 180.75), (15.21, 105.62), (22.09, 95.24), (27.04, 141.9), (21.9, 107.84), (23.15, 89.49), (9.35, 257.21), (12.34, 148.36), (33.92, 43.06), (19.01, 134.76), (7.28, 191.34), (24.2, 157.04), (1.35, 159.13), (19.62, 94.84), (17.29, 188.2), (32.48, 123.17), (8.56, 252.74), (19.26, 196.3), (10.4, 227.29), (19.34, 184.53), (20.35, 121.3), (37.11, 43.51), (20.85, 200.57), (18.88, 108.39), (12.26, 212.52), (26.64, 60.17), (18.34, 191.46), (20.98, 85.42), (21.21, 166.59), (1.15, 261.0), (24.25, 166.49), (16.04, 159.42), (35.99, 63.11), (16.36, 149.67), (18.85, 153.71), (5.15, 276.49), (17.59, 124.37), (12.15, 100.98), (12.56, 179.48), (27.23, 66.54), (4.29, 166.61), (17.66, 171.53), (16.12, 278.44), (16.14, 230.61), (8.07, 228.92), (46.39, 66.35), (-4.93, 214.85), (11.39, 193.22), (18.41, 102.47), (14.9, 155.74), (24.71, 152.53), (20.44, 135.98), (13.04, 141.07), (3.1, 276.53), (13.46, 215.71), (29.55, 113.99), (23.14, 93.56), (29.49, -10.77), (14.82, 139.28), (22.29, 139.73), (6.25, 178.86), (25.16, 67.11), (7.22, 265.3), (21.04, 123.5), (51.63, 9.04), (38.26, 108.7), (12.97, 188.99), (11.19, 177.52), (35.9, 122.71), (17.24, 192.59), (27.95, 97.87), (13.7, 190.32), (22.83, 128.85), (32.89, 135.16), (20.24, 136.06), (16.22, 164.91), (27.77, 114.27), (35.48, 30.5), (34.02, 82.22), (12.78, 162.98), (4.87, 174.68), (22.76, 212.0), (21.45, 69.04), (10.04, 123.25), (22.52, 129.54), (3.36, 211.82), (7.13, 261.65), (23.98, 149.65), (12.56, 181.24), (22.47, 164.38), (25.03, 106.08), (22.37, 155.11), (13.82, 143.9), (22.63, 195.66), (10.81, 140.39), (23.45, 153.13), (24.12, 151.12), (17.94, 87.05), (16.11, 188.67), (30.5, 118.2), (29.13, 173.48), (21.81, 174.11), (8.31, 191.14), (32.44, 97.39), (3.74, 185.37), (16.84, 107.87), (28.73, 151.69), (10.11, 126.72), (28.69, 74.3), (28.59, 109.63), (18.95, 173.03), (7.34, 224.66), (13.62, 165.29), (18.77, 93.82), (17.49, 110.83), (7.11, 217.04), (28.22, 47.34), (30.86, 52.19), (1.81, 220.27), (5.51, 193.95), (16.04, 179.73), (12.84, 227.3), (30.14, 82.38), (12.75, 222.21), (40.7, 52.68), (22.63, 116.28), (16.32, 133.01), (20.98, 164.25), (21.6, 141.29), (20.31, 87.04), (7.97, 194.77), (30.31, 58.32), (28.83, 13.94), (30.52, 11.07), (10.69, 251.85), (26.09, 54.8), (31.78, 82.55), (9.91, 187.47), (16.58, 167.15), (4.03, 287.22), (22.22, 121.14), (26.42, 125.87), (12.24, 189.88), (13.07, 133.41), (36.28, 84.96), (19.0, 115.87), (23.58, 151.01), (26.31, 157.06), (27.34, 140.67), (5.7, 193.7), (31.86, 135.99), (11.98, 163.65), (19.86, 171.57), (20.5, 190.55), (17.95, 136.8), (20.56, 132.59), (9.52, 158.14), (24.14, 134.62), (11.09, 172.89), (3.92, 222.86), (18.79, 200.15), (20.09, 144.59), (16.2, 206.13), (4.01, 209.15), (16.26, 175.28), (32.61, 130.49), (24.99, 117.6), (29.87, 95.5), (22.25, 129.74), (13.2, 188.01), (17.33, 90.94), (8.16, 97.31), (32.12, 72.87), (12.39, 78.82), (16.76, 143.0), (17.34, 140.68), (37.83, 63.36), (13.74, 124.6), (17.91, 193.59), (14.54, 209.25), (12.99, 262.94), (22.33, 145.22), (11.01, 169.51), (13.17, 184.97), (33.57, 98.36), (14.07, 41.38), (18.09, 101.58), (10.17, 194.58), (25.69, 132.69), (34.85, 102.53), (0.48, 209.32), (31.03, 66.6), (18.09, 99.61), (31.96, 60.53), (42.44, 122.07), (24.47, 138.27), (22.56, 78.08), (11.42, 155.03), (14.47, 185.4), (35.53, 89.45), (22.13, 143.45), (11.02, 280.74), (23.72, 205.48), (5.95, 204.45), (24.81, 161.46), (8.22, 184.14), (18.91, 172.45), (11.78, 183.09), (6.7, 183.13), (3.02, 235.46), (23.79, 115.63), (12.98, 127.33), (15.43, 173.88), (8.93, 220.74), (20.35, 97.89), (14.83, 163.21), (25.88, 159.18), (15.35, 188.67), (7.78, 176.13), (12.74, 201.41), (8.83, 132.05), (22.84, 129.61), (15.56, 157.02), (10.79, 222.05), (37.71, 175.36), (3.22, 236.29), (23.88, 124.63), (19.02, 138.47), (12.22, 157.37), (11.64, 239.17), (23.9, 78.79), (15.89, 174.52), (12.59, 162.58), (29.34, 92.62), (18.43, 184.96), (13.72, 165.84), (21.89, 124.93), (24.71, 140.42), (25.55, 123.64), (15.92, 92.34), (26.75, 87.02), (29.12, 109.53), (18.5, 134.86), (30.09, 140.27), (19.87, 127.97), (17.26, 156.29), (36.88, 170.58), (4.2, 265.47), (30.89, 46.52), (28.13, 209.0), (22.55, 123.14), (10.01, 159.31), (9.61, 169.13), (25.77, 161.06), (18.17, 131.67), (-0.9, 262.06), (18.22, 127.17), (24.43, 239.04), (22.67, 103.86), (7.46, 173.93), (19.71, 173.79), (26.99, 84.45), (25.95, 125.73), (5.96, 248.99), (8.49, 156.98), (24.15, 152.88), (1.55, 225.73), (5.88, 252.4), (5.98, 208.27), (5.29, 198.23), (8.2, 194.96), (26.74, 42.22), (22.12, 94.3), (-1.79, 226.76), (8.59, 172.43), (8.19, 198.08), (28.37, 90.01), (27.71, 163.69), (2.94, 224.82), (30.8, 93.15), (28.39, 129.76), (1.3, 265.54), (12.96, 155.65), (27.52, 49.74), (6.08, 230.56), (19.1, 145.42), (8.19, 211.46), (0.79, 196.07), (5.22, 167.47), (2.34, 186.36), (8.73, 188.23), (29.82, 33.99), (27.96, 125.43), (18.37, 113.63), (8.54, 202.75), (21.26, 60.34), (19.04, 148.52), (37.03, 9.33), (14.24, 183.22), (11.26, 158.23), (-5.38, 263.7), (7.78, 152.68), (33.1, 23.19), (10.07, 198.77), (23.9, 161.23), (2.83, 266.57), (27.96, 48.12), (11.86, 182.9), (21.83, 160.3), (4.12, 178.37), (18.19, 152.57), (14.99, 180.92), (28.0, 143.15), (27.9, 81.67), (10.88, 216.78), (15.77, 167.93), (16.85, 122.67), (21.51, 142.1), (16.3, 110.12), (24.1, 96.59), (9.81, 220.16), (10.37, 143.61), (14.57, 130.06), (15.23, 178.64), (8.33, 163.88), (0.55, 215.43), (21.51, 168.49), (22.42, 88.34), (23.17, 173.08), (40.09, 57.42), (-2.42, 310.81), (3.44, 154.17), (5.1, 104.59), (23.01, 63.58), (23.28, 135.98), (33.99, 141.49), (4.23, 243.35), (16.32, 99.24), (8.92, 243.74), (10.43, 111.49), (-5.6, 249.92), (16.48, 224.84), (5.03, 223.94), (11.08, 195.84), (13.11, 140.39), (19.15, 133.83), (6.63, 120.15), (2.74, 229.22), (10.56, 159.28), (8.72, 118.4), (13.36, 197.37), (10.07, 132.14), (16.37, 102.2), (11.44, 188.37), (23.03, 104.08), (31.76, 126.3), (34.18, 71.87), (6.56, 158.54), (19.52, 218.04), (16.0, 132.18), (5.38, 231.33), (16.82, 68.47), (16.36, 129.72), (9.47, 229.22), (7.05, 135.64), (-8.11, 269.43), (13.35, 191.13), (11.33, 110.79), (21.64, 126.89), (2.04, 161.66), (21.91, 185.46), (20.68, 118.56), (22.96, 98.01), (11.47, 194.56), (18.74, 120.77), (33.17, 97.65), (22.99, 119.71), (25.95, 129.19), (22.44, 137.72), (20.82, 146.18), (25.24, 8.58), (11.88, 211.1), (28.18, 133.98), (25.13, 144.44), (21.32, 101.66), (27.03, 100.74), (18.26, 214.9), (18.22, 154.8), (35.33, 70.6), (40.64, 56.48), (15.88, 235.16), (16.77, 177.25), (29.04, 129.35), (32.75, 117.02), (17.24, 140.67), (35.48, 11.32), (15.38, 175.28), (26.01, 123.3), (19.43, 197.35), (24.42, 134.77), (30.2, 5.69), (21.51, 223.19), (38.46, 114.36), (20.62, 87.29), (10.49, 166.94), (30.9, 100.76), (14.08, 91.26), (19.38, 95.07), (6.06, 239.77), (17.87, 124.56), (16.45, 163.28), (5.04, 172.8), (7.98, 235.23), (-4.74, 275.45), (-5.97, 251.18), (20.94, 111.22), (24.94, 137.46), (25.27, 189.45), (10.25, 228.96), (9.09, 224.27), (11.21, 149.22), (15.72, 167.54), (1.34, 200.59), (20.26, 106.94), (14.92, 180.56), (21.93, 127.94), (20.41, 96.19), (12.47, 133.39), (25.16, 49.67), (20.19, 227.43), (15.36, 105.78), (32.27, -6.01), (24.32, 100.67), (13.75, 184.52), (-6.35, 265.07), (21.02, 172.74), (14.75, 112.44), (9.25, 221.52), (16.2, 178.97), (32.62, 117.48), (13.03, 134.53), (14.55, 135.03), (27.16, 52.06), (24.22, 146.63), (14.31, 219.94), (34.79, 0.18), (5.94, 152.97), (1.37, 255.52), (22.84, 184.4), (5.37, 241.04), (8.68, 147.52), (10.46, 98.94)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=632；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.7133；
t = r·√((n−2)/(1−r²)) = 0.7133·√((632−2)/(1−0.5088)) = -25.5436；df=630；p = 2.54e-99。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.7496, -0.6726]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=632；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.7133；
t = r·√((n−2)/(1−r²)) = 0.7133·√((632−2)/(1−0.5088)) = -25.5436；df=630；p = 2.54e-99。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.7496, -0.6726]。
③ 计算结果：
r = -0.7133，t = -25.5436，p = 2.54e-99，95%CI = [-0.7496, -0.6726]
④ 结论：在 α=0.05 水平下，p=2.54e-99 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.7133，方向负），天暖少买取暖器。
案例 29. 环境噪音(dB)与专注度(分)
Pearson相关噪音降专注
① 原始数据：
样本 n=937，逐条 (x, y)：[(41.02, 88.62), (67.49, 57.63), (35.96, 100.58), (70.84, 65.64), (41.77, 76.0), (7.62, 107.17), (59.77, 65.04), (102.7, 45.88), (70.62, 76.81), (86.8, 54.31), (57.74, 77.71), (56.94, 59.1), (31.93, 81.2), (74.33, 54.38), (61.36, 67.14), (67.73, 70.76), (30.23, 75.37), (53.11, 77.09), (44.13, 73.3), (23.76, 87.1), (56.47, 70.45), (53.84, 70.48), (89.5, 51.31), (42.48, 75.26), (83.02, 61.29), (72.8, 66.74), (50.73, 80.12), (80.88, 66.42), (43.99, 77.06), (57.24, 54.17), (20.47, 90.02), (63.58, 52.84), (82.47, 60.24), (51.51, 57.77), (62.63, 64.09), (68.19, 73.3), (52.21, 61.47), (14.72, 100.47), (52.45, 80.75), (68.8, 73.26), (44.4, 80.03), (73.1, 68.42), (68.28, 67.33), (54.01, 70.11), (57.99, 68.43), (57.41, 76.54), (64.92, 70.87), (77.97, 71.51), (40.98, 81.71), (52.35, 87.66), (77.04, 63.7), (81.29, 56.71), (85.5, 52.48), (79.71, 56.87), (54.37, 55.64), (71.84, 62.31), (32.48, 85.6), (17.39, 85.29), (55.29, 59.01), (61.73, 53.78), (84.05, 49.14), (37.79, 78.45), (78.71, 53.44), (79.37, 60.49), (36.63, 79.11), (39.22, 77.81), (70.2, 65.9), (33.7, 78.1), (48.45, 72.71), (62.9, 50.97), (30.99, 81.04), (61.93, 73.14), (51.97, 71.1), (61.42, 79.07), (61.71, 78.74), (77.24, 69.66), (48.0, 71.31), (93.27, 47.9), (46.54, 82.42), (57.94, 69.33), (63.27, 74.03), (71.07, 51.36), (53.02, 71.22), (57.34, 78.2), (57.27, 78.82), (61.26, 64.37), (48.85, 59.82), (43.28, 80.31), (84.04, 66.71), (78.96, 64.42), (88.04, 52.15), (36.41, 77.86), (67.1, 66.32), (39.07, 62.77), (66.07, 77.75), (59.56, 72.82), (58.01, 85.19), (84.29, 51.08), (58.89, 85.49), (25.71, 88.71), (14.97, 81.72), (65.43, 86.62), (40.68, 77.14), (40.1, 82.31), (75.27, 64.87), (59.58, 68.21), (73.65, 59.3), (60.47, 73.07), (69.54, 50.96), (62.86, 63.22), (51.64, 84.2), (63.13, 60.74), (34.17, 72.57), (75.75, 55.58), (49.59, 73.79), (90.85, 67.58), (49.12, 58.9), (36.94, 74.68), (55.69, 74.06), (91.03, 45.64), (56.73, 73.34), (57.3, 69.92), (48.01, 81.14), (53.8, 72.99), (58.06, 71.98), (33.88, 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75.09), (41.73, 81.71), (46.36, 78.98), (72.1, 72.15), (43.75, 87.65), (63.46, 73.77), (70.6, 79.79), (26.77, 66.93), (42.63, 55.55), (50.06, 76.14), (63.32, 50.95), (68.0, 64.17), (62.1, 82.34), (84.77, 46.96), (67.44, 61.51), (74.16, 71.54), (71.95, 65.42), (69.22, 64.97), (21.0, 71.18), (62.58, 58.27), (47.32, 78.35), (63.09, 77.6), (62.28, 66.43), (81.28, 54.04), (71.11, 69.31), (30.77, 77.22), (55.61, 69.74), (66.1, 57.15), (49.46, 85.2), (85.51, 54.53), (28.19, 87.57), (83.36, 58.43), (25.91, 84.5), (72.59, 57.78), (68.77, 86.85), (51.02, 88.06), (63.88, 68.0), (64.12, 74.15), (19.59, 91.91), (38.34, 61.99), (65.87, 78.21), (41.0, 91.95), (70.54, 68.19), (34.54, 81.79), (46.56, 72.8), (42.03, 84.77), (48.17, 73.95), (66.85, 60.53), (70.76, 55.71), (79.6, 58.41), (66.22, 75.03), (55.34, 71.47), (65.98, 62.31), (89.91, 57.87), (58.17, 77.23), (23.17, 97.46), (74.78, 67.17), (66.18, 59.69), (78.67, 59.7), (50.92, 64.77), (36.71, 85.57), (61.35, 73.38), (29.3, 100.42), (96.84, 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67.85), (52.22, 64.89), (90.94, 55.93), (48.2, 71.79), (17.87, 88.32), (67.52, 65.03), (42.32, 68.16), (84.42, 54.29), (62.69, 67.42), (57.73, 62.56), (14.32, 79.79), (78.29, 71.17), (51.04, 62.15), (76.1, 79.87), (49.42, 71.81), (42.42, 86.2), (57.36, 67.67), (9.62, 89.62), (22.81, 68.96), (67.54, 70.78), (49.25, 84.46), (48.49, 72.72), (60.06, 68.98), (73.83, 67.74), (34.56, 73.34), (77.63, 75.97), (35.98, 86.12), (51.87, 71.33), (63.2, 72.13), (60.28, 65.46), (53.21, 85.91), (65.35, 65.8), (41.8, 68.91), (44.77, 88.96), (29.52, 75.76), (82.54, 56.48), (45.54, 77.93), (28.91, 75.84), (70.3, 63.73), (35.17, 83.31), (92.15, 54.02), (81.37, 64.34), (58.69, 75.85), (72.07, 55.15), (58.21, 85.66), (40.1, 74.38), (79.29, 75.51), (40.89, 72.09), (44.19, 63.74), (78.04, 46.22), (91.28, 49.52), (78.49, 65.45), (94.74, 46.56), (33.15, 66.75), (50.21, 58.35), (47.11, 80.08), (65.98, 67.56), (46.25, 71.84), (61.73, 73.48), (103.41, 51.58), (69.79, 72.11), (49.11, 76.23), (57.56, 84.56), (51.66, 69.19), (46.41, 77.85), (76.81, 49.6), (41.16, 66.14), (63.41, 77.19), (28.95, 94.81), (56.05, 66.14), (70.8, 66.16), (74.9, 54.34), (79.59, 42.24), (57.9, 66.21), (50.32, 85.47), (46.38, 70.25), (100.82, 59.36), (66.11, 62.5), (53.86, 77.54), (39.8, 89.35), (68.57, 72.48), (89.14, 43.18), (54.51, 68.52), (41.38, 76.79), (58.27, 72.79), (83.05, 64.44), (47.3, 71.43), (62.84, 78.8), (76.59, 55.0), (40.55, 81.02), (61.95, 79.95), (48.51, 71.55), (96.47, 45.17), (72.42, 72.67), (76.43, 48.95), (-11.07, 73.07), (66.57, 62.1), (68.11, 59.31), (65.27, 62.37), (67.17, 75.33), (45.37, 71.63), (37.3, 86.7), (43.09, 82.5), (74.58, 60.13), (24.25, 79.56), (56.95, 85.12), (65.76, 80.98), (64.65, 71.92), (45.93, 75.44), (43.53, 74.28), (91.44, 58.55), (81.14, 65.61), (81.69, 60.61), (59.12, 75.08), (53.57, 70.82), (77.75, 68.6), (59.5, 79.58), (33.54, 86.97), (47.76, 71.67), (45.22, 71.2), (58.39, 60.32), (32.5, 81.39), (42.65, 83.84), (67.85, 53.87), (17.39, 87.96), (76.48, 51.78), (69.47, 54.96), (41.65, 77.37), (52.42, 68.64), (52.69, 73.72), (65.02, 58.91), (83.61, 59.19), (43.81, 50.06), (18.6, 105.45), (31.21, 100.45), (46.0, 66.3), (44.59, 70.24), (62.49, 59.08), (64.75, 56.2), (58.21, 84.63), (99.2, 54.21), (58.61, 59.08), (52.37, 63.71), (63.26, 73.08), (52.97, 73.44), (67.66, 64.28), (45.33, 63.65), (31.47, 79.24), (53.45, 74.61), (51.37, 63.57), (55.47, 87.62), (74.43, 60.99), (27.71, 92.0), (38.78, 96.95), (19.02, 76.91), (53.35, 64.56), (52.39, 81.1), (12.75, 75.53), (50.1, 81.04), (63.11, 76.61), (46.56, 58.46), (60.69, 68.57), (51.23, 64.91), (74.9, 76.29), (86.1, 64.3), (75.75, 58.08), (56.58, 74.62), (66.94, 74.32), (44.01, 80.59), (85.79, 56.97), (67.9, 61.06), (63.92, 45.5), (62.51, 74.75), (43.36, 84.6), (96.13, 79.42), (53.57, 74.14), (73.37, 55.4), (97.78, 50.17), (78.25, 54.74), (98.33, 65.68), (49.89, 90.99), (49.77, 65.59), (75.36, 61.84), (81.63, 71.7), (77.95, 50.73), (92.32, 54.05), (83.85, 63.62), (38.51, 79.47), (67.6, 66.8), (48.44, 65.06), (12.6, 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73.15), (45.51, 74.33), (59.51, 77.6), (46.17, 70.81), (72.16, 68.24), (71.95, 69.46), (99.49, 55.57), (40.77, 78.35), (95.97, 62.72), (30.89, 73.48), (43.48, 83.57), (42.75, 70.26), (56.75, 70.37), (84.07, 63.2), (58.58, 62.53), (69.2, 64.91), (56.71, 67.5), (45.14, 77.85), (75.01, 76.69), (86.72, 45.21), (101.74, 36.23), (62.99, 66.56), (41.52, 90.18), (31.19, 88.97), (79.75, 52.27), (75.15, 68.37), (45.5, 63.29), (45.27, 83.03), (75.59, 70.81), (35.42, 67.59), (74.34, 71.21), (69.15, 60.2), (76.61, 63.09), (49.46, 73.92), (28.19, 83.65), (37.13, 71.27), (58.89, 74.29), (32.15, 89.53), (54.16, 69.76), (41.17, 81.47), (62.7, 83.58), (96.89, 54.64), (75.28, 62.14), (86.56, 64.86), (63.62, 65.68), (50.39, 63.17), (59.04, 77.64), (91.28, 61.39), (84.88, 52.42), (68.02, 69.62), (29.81, 68.71), (37.63, 75.54), (55.56, 78.97), (48.26, 74.32), (49.12, 82.39), (25.75, 78.99), (22.91, 77.22), (101.76, 47.55), (59.69, 57.74), (73.66, 62.66), (49.87, 76.7), (68.71, 70.45), (33.62, 78.1), (61.25, 78.96), (57.45, 71.07), (63.19, 78.48), (56.86, 73.09), (16.22, 91.44), (73.87, 73.83), (29.08, 96.95), (66.24, 85.67), (55.94, 71.35), (52.22, 70.39), (80.42, 68.74), (32.77, 72.67), (53.51, 77.48), (68.21, 44.23), (50.56, 85.4), (63.55, 61.15), (55.1, 51.46), (72.56, 70.31), (86.44, 56.17), (46.19, 78.61), (55.12, 71.01), (81.79, 68.13), (40.06, 79.44), (62.88, 80.97), (101.63, 58.41), (67.44, 67.23), (79.88, 67.96), (59.28, 68.12), (64.47, 80.33), (64.31, 67.68), (99.75, 58.77), (71.15, 64.69), (44.99, 90.13), (56.73, 58.19), (36.18, 93.66), (4.73, 92.4), (59.44, 71.53), (55.74, 73.49), (29.39, 90.24), (85.66, 61.46), (53.69, 69.76), (80.06, 67.75), (93.22, 60.44), (50.47, 65.88), (47.48, 58.74), (91.62, 64.41), (42.34, 75.21), (40.39, 75.55), (81.46, 61.32), (64.43, 59.4), (44.75, 68.99), (54.15, 81.47), (52.42, 64.18), (57.12, 82.79), (47.42, 70.03), (84.58, 55.82), (69.31, 56.63), (70.96, 85.06), (54.39, 63.1), (40.88, 74.49), (62.86, 67.36), (62.25, 62.5), (50.16, 84.65), (25.0, 81.97), (80.81, 40.47), (67.63, 70.74), (52.64, 67.0), (72.47, 71.54), (44.33, 73.08), (73.61, 64.42), (47.05, 78.93), (50.32, 60.01), (65.78, 74.4), (-4.8, 98.33), (38.61, 64.64), (50.93, 86.42), (70.1, 61.58), (35.39, 85.98), (53.35, 75.03), (30.81, 93.7), (51.55, 76.11), (55.71, 73.86), (52.33, 75.68), (58.57, 67.76), (41.61, 86.25), (86.97, 62.71), (29.94, 86.7), (71.67, 77.96), (74.66, 64.28), (72.39, 73.04), (59.48, 62.94), (73.43, 65.83), (87.33, 59.89), (46.01, 78.89), (54.07, 63.47), (52.42, 78.71), (52.66, 67.43), (59.49, 71.73), (49.71, 75.55), (70.53, 55.25), (47.17, 75.36), (44.71, 74.75), (95.7, 52.99), (44.04, 87.12), (66.66, 60.73), (80.14, 67.83), (104.94, 46.07), (93.55, 46.9), (72.27, 67.23), (47.61, 77.73), (63.62, 62.91), (43.72, 68.97), (91.36, 61.99), (49.9, 76.83), (86.35, 70.66), (58.57, 72.48), (54.49, 78.99), (47.38, 72.89)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=937；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6837；
t = r·√((n−2)/(1−r²)) = 0.6837·√((937−2)/(1−0.4675)) = -28.6513；df=935；p = -4.43e+59。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7165, -0.6480]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=937；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6837；
t = r·√((n−2)/(1−r²)) = 0.6837·√((937−2)/(1−0.4675)) = -28.6513；df=935；p = -4.43e+59。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7165, -0.6480]。
③ 计算结果：
r = -0.6837，t = -28.6513，p = -4.43e+59，95%CI = [-0.7165, -0.6480]
④ 结论：在 α=0.05 水平下，p=-4.43e+59 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.6837，方向负），噪音降专注。
案例 30. 年龄(岁)与柔韧性(分)
Pearson相关年长柔韧性降
① 原始数据：
样本 n=722，逐条 (x, y)：[(39.65, 56.01), (42.7, 68.18), (43.52, 55.77), (9.41, 97.18), (63.46, 38.01), (23.13, 56.75), (45.06, 72.57), (49.01, 58.84), (29.78, 37.25), (49.39, 61.31), (-14.14, 102.45), (39.73, 82.63), (45.38, 46.43), (34.2, 75.51), (54.15, 14.91), (31.18, 56.06), (54.85, 44.96), (54.98, 33.79), (68.98, 38.92), (31.12, 94.03), (38.72, 51.41), (51.75, 60.79), (38.08, 48.63), (53.43, 60.61), (73.96, 49.85), (29.91, 58.86), (50.71, 62.64), (45.37, 65.83), (18.24, 79.28), (67.68, 37.49), (49.7, 64.42), (53.98, 22.72), (42.83, 65.79), (46.24, 38.2), (50.08, 36.98), (53.93, 63.06), (18.51, 69.28), (56.43, 31.48), (44.49, 38.97), (47.16, 57.8), (23.53, 67.04), (31.33, 63.5), (21.78, 60.08), (8.4, 101.76), (85.41, 36.07), (32.25, 62.01), (36.22, 61.2), (40.58, 77.88), (30.8, 69.53), (33.03, 58.13), (45.9, 50.03), (57.58, 41.65), (38.57, 48.84), (53.54, 39.32), (43.14, 51.03), (48.71, 59.07), (23.64, 71.54), (51.02, 52.47), (38.47, 60.32), (-13.97, 108.35), (26.33, 70.42), (43.6, 61.57), (55.07, 52.82), (42.59, 54.11), (4.42, 81.82), (41.7, 80.09), (48.71, 44.26), (23.77, 82.38), (34.84, 64.54), (15.81, 76.1), (50.66, 65.94), (60.42, 42.77), (35.02, 92.87), (52.0, 43.38), (30.72, 80.99), (9.19, 72.99), (71.93, 34.74), (86.48, 26.25), (43.58, 54.29), (39.35, 60.76), (51.56, 33.62), (17.54, 65.43), (72.54, 40.63), (44.65, 46.78), (27.53, 68.07), (35.85, 68.99), (37.64, 65.69), (47.08, 57.28), (32.99, 65.18), (16.92, 68.12), (27.05, 61.18), (54.05, 55.62), (21.64, 67.47), (30.49, 68.67), (38.15, 93.3), (46.19, 71.15), (46.9, 53.85), (36.46, 57.81), (14.22, 73.36), (48.75, 44.7), (49.72, 44.22), (21.56, 70.95), (38.52, 66.12), (40.37, 67.3), (35.1, 44.0), (51.54, 44.78), (59.25, 40.14), (43.53, 58.95), (16.48, 84.72), (33.25, 83.54), (33.59, 73.06), (20.98, 59.16), (48.63, 47.24), (17.36, 76.58), (44.51, 49.3), (40.65, 49.24), (34.92, 60.51), (35.42, 67.62), (27.24, 48.2), (55.9, 50.93), (40.37, 81.43), (20.48, 86.24), (51.84, 45.68), (61.04, 50.25), (34.19, 59.98), (32.86, 47.71), (-1.27, 107.52), (49.41, 71.13), (58.89, 52.69), (38.38, 41.91), (26.82, 48.1), (48.29, 56.38), (64.09, 55.13), (15.5, 54.94), (51.98, 55.14), (22.42, 65.46), (44.68, 64.54), (7.07, 93.67), (35.81, 76.4), (46.63, 53.09), (31.62, 75.26), (63.48, 57.12), (52.0, 64.13), (48.94, 65.5), (37.1, 61.58), (35.11, 45.28), (49.14, 58.47), (69.31, 53.0), (33.1, 74.95), (59.71, 65.36), (27.52, 79.06), (59.49, 35.03), (66.94, 45.77), (26.37, 74.76), (11.91, 77.08), (48.48, 31.88), (59.22, 36.3), (15.48, 67.64), (59.68, 32.24), (39.79, 72.81), (38.43, 56.27), (50.51, 54.29), (66.05, 28.68), (44.25, 52.1), (12.41, 90.58), (13.09, 79.94), (48.08, 47.98), (40.32, 56.71), (23.14, 76.92), (28.91, 53.74), (37.19, 59.7), (33.85, 76.67), (42.09, 64.95), (24.68, 69.17), (29.87, 60.94), (2.32, 82.26), (21.71, 76.94), (36.14, 69.56), (63.73, 60.0), (35.51, 75.85), (56.26, 61.58), (7.97, 92.68), (56.52, 20.52), (41.52, 64.83), (63.84, 45.27), (22.07, 67.64), (30.8, 62.12), (39.9, 63.93), (49.88, 54.7), (43.3, 73.41), (19.6, 54.28), (51.61, 66.97), (35.22, 75.0), (39.5, 70.56), (37.65, 49.12), (16.89, 69.87), (14.28, 96.44), (44.72, 67.98), (60.04, 78.56), (58.39, 36.54), (44.38, 56.17), (35.68, 55.72), (52.83, 61.82), (31.59, 81.14), (57.63, 55.18), (23.32, 57.25), (27.69, 80.22), (57.36, 50.94), (40.39, 66.25), (61.12, 64.61), (19.54, 64.45), (22.57, 81.6), (36.46, 50.58), (26.39, 63.2), (50.26, 41.11), (26.72, 58.99), (59.53, 39.91), (22.17, 88.45), (34.93, 66.49), (49.01, 62.71), (68.45, 31.11), (55.54, 45.49), (26.64, 74.2), (72.74, 41.57), (23.76, 51.37), (19.34, 58.78), (61.4, 45.96), (42.74, 59.69), (41.67, 53.79), (66.14, 36.67), (-20.21, 104.99), (44.1, 50.24), (26.97, 74.38), (59.16, 38.56), (9.83, 72.4), (42.24, 41.37), (23.37, 63.88), (60.76, 26.54), (48.96, 75.49), (20.04, 64.58), (20.26, 77.74), (33.08, 59.79), (34.15, 59.9), (44.87, 61.75), (48.04, 59.88), (30.28, 75.33), (26.92, 80.29), (27.21, 72.3), (42.69, 69.64), (21.92, 86.54), (35.3, 65.38), (60.1, 58.12), (48.26, 59.55), (38.85, 74.45), (51.95, 50.04), (6.17, 84.76), (44.04, 49.12), (30.7, 71.7), (61.86, 56.21), (24.82, 65.53), (43.94, 34.49), (44.07, 60.56), (44.79, 71.95), (41.21, 65.08), (54.07, 69.73), (31.29, 66.53), (36.58, 77.92), (43.4, 39.71), (49.81, 57.38), (59.08, 45.71), (15.35, 71.46), (13.79, 85.04), (49.92, 54.85), (92.99, 7.16), (22.82, 83.67), (50.08, 41.34), (47.23, 51.29), (34.41, 47.64), (29.44, 57.17), (44.03, 72.33), (45.45, 62.74), (42.24, 57.4), (44.44, 62.1), (25.14, 62.98), (58.2, 60.05), (34.41, 62.48), (33.43, 64.59), (43.19, 64.0), (69.79, 18.7), (35.08, 74.85), (43.07, 62.22), (31.54, 68.19), (39.59, 78.44), (25.34, 87.59), (34.65, 77.61), (36.13, 58.26), (30.43, 67.66), (23.93, 59.72), (50.06, 50.02), (47.6, 46.0), (32.52, 62.89), (38.32, 64.48), (52.7, 46.82), (66.01, 34.19), (50.04, 42.47), (35.48, 48.88), (35.08, 53.66), (28.92, 79.8), (27.65, 71.28), (58.22, 32.69), (44.13, 63.25), (33.43, 46.6), (33.82, 62.91), (63.1, 51.71), (35.84, 69.8), (46.79, 66.95), (38.25, 72.37), (52.39, 52.37), (26.87, 89.18), (37.06, 62.82), (42.08, 88.43), (37.52, 62.51), (3.35, 79.85), (24.26, 70.18), (27.22, 81.52), (35.8, 53.06), (32.15, 55.02), (41.39, 69.0), (48.7, 67.11), (49.87, 52.35), (23.86, 73.41), (22.02, 71.57), (63.06, 48.17), (33.76, 84.35), (55.03, 50.84), (53.07, 30.64), (27.66, 75.37), (53.35, 42.48), (27.87, 76.58), (51.86, 32.75), (59.8, 67.1), (22.37, 63.48), (74.07, 36.9), (34.08, 80.2), (40.71, 62.95), (28.93, 80.03), (53.45, 49.2), (56.66, 36.67), (30.2, 64.23), (30.33, 64.74), (57.0, 43.97), (31.56, 56.88), (46.94, 66.16), (48.54, 36.19), (28.86, 59.22), (43.7, 57.25), (36.88, 72.75), (55.21, 49.96), (30.71, 74.63), (43.18, 29.95), (43.9, 57.39), (32.91, 58.64), (40.14, 71.09), (11.91, 76.27), (46.84, 40.52), (22.55, 60.67), (45.15, 84.05), (67.19, 47.22), (47.67, 50.75), (55.28, 47.39), (24.28, 58.54), (72.96, 42.83), (45.62, 51.01), (55.05, 49.78), (31.75, 64.68), (17.84, 80.62), (18.18, 62.3), (65.5, 22.73), (29.59, 63.55), (30.88, 70.95), (44.52, 54.96), (68.52, 50.22), (29.69, 81.91), (38.23, 48.71), (26.77, 87.89), (32.61, 41.05), (12.44, 73.71), (58.62, 43.77), (45.23, 58.09), (26.35, 92.41), (34.28, 58.12), (29.39, 72.27), (29.6, 59.72), (30.13, 66.83), (38.44, 88.83), (46.18, 64.79), (33.14, 75.15), (19.22, 78.8), (26.52, 64.12), (38.84, 68.85), (22.67, 59.82), (31.91, 64.26), (38.4, 67.07), (46.28, 68.83), (20.0, 78.55), (45.93, 64.85), (37.86, 46.11), (55.87, 45.79), (35.74, 62.55), (56.88, 70.22), (28.79, 83.16), (44.31, 69.14), (32.32, 66.7), (51.08, 59.41), (10.13, 80.58), (51.8, 59.58), (58.5, 38.71), (42.43, 55.41), (22.52, 63.2), (29.85, 67.39), (25.47, 67.99), (32.98, 66.35), (49.83, 34.84), (39.12, 58.61), (35.61, 59.76), (47.51, 56.46), (47.16, 60.4), (41.61, 60.34), (22.69, 64.84), (46.22, 57.26), (79.59, 34.16), (24.28, 60.07), (52.31, 65.09), (38.72, 59.52), (38.33, 65.67), (31.18, 77.06), (34.19, 54.04), (19.65, 49.43), (54.59, 56.23), (9.54, 91.69), (16.8, 53.28), (53.2, 51.87), (20.66, 70.73), (33.21, 60.56), (37.15, 62.99), (61.76, 40.59), (27.87, 74.49), (44.06, 60.54), (46.34, 57.2), (36.12, 61.39), (26.73, 86.1), (53.44, 67.32), (50.86, 67.95), (25.69, 58.51), (78.87, 37.59), (60.09, 43.56), (53.11, 66.77), (31.5, 77.87), (25.92, 69.52), (67.14, 43.67), (22.38, 66.67), (52.04, 46.57), (23.82, 80.53), (24.06, 81.74), (34.71, 58.62), (46.9, 50.41), (28.36, 71.84), (36.19, 67.95), (33.25, 69.93), (54.25, 40.97), (17.82, 54.4), (63.47, 36.65), (34.97, 65.18), (39.93, 71.12), (52.6, 57.61), (59.27, 47.37), (68.75, 23.95), (38.15, 53.53), (58.09, 65.64), (45.1, 59.76), (45.79, 78.67), (33.83, 62.78), (46.9, 73.16), (59.97, 50.52), (51.99, 57.96), (30.65, 57.91), (32.37, 63.11), (18.19, 81.71), (29.08, 73.15), (16.75, 66.61), (44.85, 69.08), (20.33, 67.7), (24.84, 61.73), (44.97, 66.51), (56.99, 67.63), (39.83, 46.61), (32.41, 62.67), (62.36, 33.06), (28.63, 81.19), (36.19, 88.74), (48.72, 61.07), (64.38, 47.4), (36.98, 78.72), (31.47, 38.19), (46.99, 57.52), (53.1, 59.27), (40.15, 56.03), (43.54, 47.37), (36.96, 51.86), (43.28, 45.36), (14.56, 89.01), (36.37, 71.41), (36.8, 48.6), (46.16, 50.99), (54.3, 63.7), (45.42, 44.2), (37.65, 66.31), (40.01, 64.73), (30.05, 39.9), (55.22, 63.8), (37.15, 52.57), (62.6, 51.38), (60.25, 44.07), (43.12, 42.3), (42.54, 76.22), (54.44, 48.01), (44.55, 51.6), (52.74, 56.29), (39.26, 49.35), (33.49, 82.05), (58.36, 58.83), (48.72, 39.24), (43.71, 62.87), (42.46, 45.66), (68.61, 50.31), (42.49, 45.37), (41.15, 64.48), (39.21, 64.73), (38.83, 49.29), (37.04, 51.51), (65.85, 35.36), (39.4, 70.74), (16.47, 77.99), (51.52, 44.18), (49.41, 37.3), (22.87, 94.85), (28.63, 83.91), (44.89, 50.74), (43.58, 52.62), (63.01, 34.09), (33.39, 72.35), (60.23, 53.55), (40.8, 49.73), (32.27, 61.25), (17.45, 80.24), (39.12, 64.58), (64.35, 30.22), (17.13, 68.94), (36.7, 69.01), (28.9, 56.85), (37.76, 78.02), (22.17, 51.49), (74.54, 37.0), (50.9, 58.57), (37.51, 80.34), (20.34, 73.54), (45.2, 43.13), (31.07, 50.29), (8.29, 76.3), (14.42, 73.15), (59.39, 53.78), (30.71, 60.69), (75.12, 56.73), (15.12, 78.66), (70.57, 29.19), (63.33, 27.98), (38.01, 81.15), (31.33, 46.81), (19.99, 80.42), (65.23, 22.91), (38.22, 43.19), (45.16, 51.64), (29.14, 66.25), (39.12, 58.57), (63.67, 54.44), (15.14, 84.28), (50.79, 83.33), (54.74, 67.59), (5.25, 88.05), (60.17, 37.2), (51.64, 38.81), (48.94, 37.33), (33.79, 60.35), (43.22, 43.22), (50.68, 45.71), (37.51, 53.46), (32.45, 56.36), (30.89, 70.1), (28.93, 80.81), (62.21, 29.42), (48.14, 72.73), (31.58, 88.44), (41.68, 71.0), (35.75, 68.75), (47.06, 54.32), (21.29, 73.09), (25.68, 48.8), (33.3, 88.22), (35.9, 61.71), (54.26, 41.32), (29.65, 56.04), (43.95, 70.08), (30.94, 48.96), (42.45, 65.66), (31.74, 93.3), (63.21, 52.83), (58.01, 48.51), (41.47, 56.01), (36.66, 51.48), (50.07, 64.9), (30.45, 72.45), (26.52, 58.78), (13.66, 94.03), (49.31, 40.09), (49.44, 52.64), (59.72, 54.27), (29.68, 68.67), (58.76, 18.67), (22.06, 57.37), (27.65, 79.21), (37.7, 92.81), (36.7, 68.46), (36.18, 65.36), (5.04, 80.59), (55.25, 50.89), (38.37, 73.43), (47.8, 60.49), (37.18, 70.2), (56.09, 44.78), (39.66, 51.16), (47.1, 48.82), (0.75, 81.85), (46.38, 61.92), (25.37, 70.15), (60.74, 38.48), (48.04, 79.74), (38.55, 50.01), (9.44, 58.27), (39.54, 48.23), (12.96, 84.48), (28.28, 72.74), (34.41, 62.94), (46.01, 70.85), (51.29, 42.19), (15.32, 89.7), (36.45, 69.43), (50.63, 43.78), (11.11, 63.29), (44.41, 54.0), (56.78, 59.72), (38.66, 44.94), (45.95, 41.44), (44.69, 46.38), (47.15, 87.97), (17.85, 76.47), (34.6, 81.1), (45.71, 52.73), (52.28, 37.71), (44.82, 61.9), (45.85, 56.04), (57.36, 45.06), (45.83, 44.18), (50.52, 57.02), (21.97, 62.48), (56.38, 53.76), (58.71, 29.12), (31.52, 59.86), (78.64, 27.0), (15.93, 79.47), (37.95, 46.51), (56.56, 51.27), (45.61, 73.58), (37.25, 44.2), (63.53, 45.3), (27.49, 70.67), (63.96, 38.93), (45.77, 43.72), (45.83, 75.99), (42.29, 76.35), (47.99, 58.27), (39.52, 53.81), (41.38, 58.78), (44.8, 67.75), (6.57, 78.54), (65.03, 50.93), (37.65, 41.27), (32.24, 90.65), (50.16, 52.31), (27.19, 70.25), (55.26, 56.15), (39.46, 53.05), (59.58, 58.96), (25.89, 63.83), (58.8, 32.1), (35.71, 56.33), (45.03, 65.01), (50.24, 54.06), (50.57, 50.24), (66.39, 32.55), (50.19, 73.62), (31.72, 39.04), (26.89, 78.13), (48.86, 65.7), (55.41, 44.92), (23.05, 55.89), (30.06, 59.46), (49.36, 53.64), (21.87, 59.49), (55.64, 41.32), (40.5, 79.31), (25.36, 70.86), (44.56, 44.88)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=722；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6738；
t = r·√((n−2)/(1−r²)) = 0.6738·√((722−2)/(1−0.4540)) = -24.4675；df=720；p = -2.86e+39。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7118, -0.6318]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=722；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6738；
t = r·√((n−2)/(1−r²)) = 0.6738·√((722−2)/(1−0.4540)) = -24.4675；df=720；p = -2.86e+39。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7118, -0.6318]。
③ 计算结果：
r = -0.6738，t = -24.4675，p = -2.86e+39，95%CI = [-0.7118, -0.6318]
④ 结论：在 α=0.05 水平下，p=-2.86e+39 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.6738，方向负），年长柔韧性降。
案例 31. 折扣力度(%)与利润率(%)
Pearson相关打折损利润
① 原始数据：
样本 n=882，逐条 (x, y)：[(22.46, 14.87), (30.91, 15.4), (34.18, 11.45), (24.2, 16.37), (5.47, 25.37), (28.43, 6.71), (26.4, 10.27), (13.85, 13.25), (19.37, 14.44), (13.31, 21.47), (38.63, 10.28), (7.35, 29.65), (50.4, 4.22), (25.79, 13.67), (44.29, 6.09), (32.0, 18.82), (16.64, 10.61), (0.12, 27.06), (45.74, 14.48), (40.86, 6.98), (17.56, 17.02), (33.01, 21.61), (17.95, 18.44), (13.06, 24.15), (27.72, 19.3), (20.72, 9.65), (27.84, 12.11), (28.3, 8.13), (-2.84, 25.98), (42.63, 16.69), (-5.29, 33.87), (31.37, 8.35), (25.42, 13.2), (18.56, 15.13), (13.49, 18.18), (16.04, 17.4), (30.36, 4.35), (3.29, 24.84), (26.59, 12.6), (13.28, 13.11), (16.79, 18.58), (8.27, 22.38), (27.2, 11.59), (12.7, 19.96), (2.62, 24.99), (13.99, 16.95), (15.85, 21.87), (38.5, 9.55), (-0.17, 23.89), (-0.41, 23.85), (5.34, 22.38), (2.87, 23.97), (23.02, 13.83), (21.73, 16.76), (13.71, 16.45), (34.87, 15.02), (33.93, 10.96), (-3.0, 24.22), (11.45, 25.05), (17.64, 23.26), (19.79, 13.59), (28.51, 22.51), (31.79, 12.22), (17.83, 10.82), (14.42, 18.62), (11.69, 25.74), (15.83, 10.86), (8.29, 20.18), (18.02, 19.05), (11.41, 23.15), (46.69, 5.31), (19.61, 18.82), (21.49, 22.82), (2.2, 26.8), (34.72, 3.85), (25.7, 15.68), (15.7, 16.96), (26.87, 19.47), (14.32, 22.18), (6.71, 22.42), (10.04, 24.85), (34.25, 15.82), (33.66, 16.17), (15.65, 17.35), (30.0, 7.54), (7.74, 23.05), (29.52, 7.56), (25.73, 8.84), (17.99, 12.3), (17.36, 18.22), (25.04, 14.43), (16.2, 20.47), (28.58, 8.93), (31.21, 15.94), (32.16, 10.14), (13.15, 23.83), (20.86, 17.75), (34.89, 17.89), (22.29, 22.68), (18.73, 19.11), (8.27, 24.11), (4.55, 27.73), (20.17, 15.9), (11.73, 26.2), (37.36, 8.14), (36.53, 7.32), (15.03, 18.9), (26.53, 12.82), (17.25, 22.45), (16.61, 18.49), (30.52, 17.08), (14.84, 19.47), (11.0, 16.03), (-6.8, 27.15), (14.3, 15.08), (29.49, 15.05), (13.54, 23.76), (22.04, 33.78), (13.9, 20.16), (25.81, 15.01), (20.65, 25.0), (-6.51, 32.31), (16.1, 16.04), (19.38, 18.22), (13.68, 17.96), (13.79, 11.93), (18.99, 21.52), (13.46, 17.04), (32.63, 14.67), (25.91, 17.75), (12.15, 22.5), (39.33, 13.46), (11.3, 15.32), (14.12, 24.24), (15.45, 27.66), (33.2, 10.4), (32.67, 14.94), (38.51, 17.28), (33.74, 16.74), (36.52, 6.82), (15.85, 11.9), (27.88, 17.17), (2.37, 22.22), (9.62, 29.18), (25.1, 19.86), (19.42, 20.42), (21.41, 5.48), (5.59, 20.06), (0.31, 23.18), (38.88, 10.18), (15.87, 9.15), (25.27, 9.28), (10.5, 20.05), (20.06, 21.05), (-0.57, 34.02), (28.89, 10.7), (16.14, 21.21), (33.73, 12.4), (3.3, 26.2), (25.78, 20.18), (18.73, 21.29), (11.38, 17.77), (34.41, 15.61), (22.4, 13.58), (26.59, 15.35), (25.69, 17.12), (20.39, 15.87), (33.99, 14.38), (13.04, 18.96), (20.13, 16.37), (45.99, 2.61), (30.31, 13.9), (8.15, 30.12), (22.92, 12.88), (22.82, 25.29), (20.19, 16.88), (15.43, 22.06), (29.62, 8.76), (28.01, 20.29), (14.78, 27.16), (28.58, 9.13), (12.8, 13.11), (25.13, 11.53), (21.44, 23.07), (34.71, 15.95), (32.13, 24.04), (27.16, 19.09), (-6.18, 28.02), (16.04, 14.3), (15.39, 24.08), (27.88, 18.14), (7.16, 28.76), (26.94, 14.39), (23.32, 14.83), (17.06, 24.37), (11.4, 19.04), (21.87, 19.36), (14.09, 16.33), (17.99, 27.45), (27.97, 14.8), (20.7, 15.4), (17.64, 18.1), (19.29, 14.78), (18.4, 17.05), (9.83, 16.14), (26.95, 23.36), (17.22, 22.9), (29.42, 12.12), (16.01, 25.72), (1.88, 30.24), (1.21, 21.75), (24.86, 9.46), (7.7, 26.49), (29.33, 5.71), (13.21, 15.92), (13.08, 20.93), (6.22, 13.7), (15.71, 17.74), (7.64, 25.33), (20.91, 24.58), (10.55, 17.49), (21.61, 25.19), (32.2, 9.74), (44.83, 3.86), (34.55, 18.08), (19.0, 22.84), (32.24, 21.89), (23.98, 12.99), (16.8, 10.25), (8.73, 25.3), (15.58, 21.37), (5.73, 28.8), (39.25, 8.6), (18.0, 12.07), (29.29, 13.55), (7.8, 28.02), (-1.15, 26.81), (7.71, 11.6), (1.89, 30.62), (37.38, 10.48), (18.45, 17.75), (37.91, 10.94), (23.2, 14.83), (9.12, 21.53), (15.48, 14.32), (20.68, 15.71), (40.75, 3.02), (32.61, 19.86), (18.57, 23.43), (21.42, 22.91), (38.09, 6.68), (23.7, 12.43), (25.35, 13.33), (5.74, 15.15), (9.57, 22.79), (-2.39, 32.15), (12.66, 14.06), (5.02, 21.48), (15.98, 21.22), (37.95, 7.25), (16.81, 28.86), (35.78, 11.61), (10.35, 24.8), (27.65, 14.29), (27.48, 21.9), (45.12, 7.91), (20.82, 18.04), (20.75, 7.99), (2.25, 22.12), (16.53, 16.21), (10.27, 21.6), (22.23, 19.12), (32.92, 18.93), (24.34, 14.24), (-3.09, 29.3), (-0.19, 24.05), (17.85, 22.05), (25.48, 18.81), (10.06, 20.37), (15.49, 19.42), (18.35, 19.87), (13.12, 15.62), (14.75, 10.36), (0.51, 27.58), (33.43, 10.46), (15.49, 15.8), (18.73, 21.93), (45.66, 7.03), (19.72, 25.34), (20.24, 15.17), (27.1, 19.1), (8.27, 19.41), (28.44, 12.16), (27.55, 21.18), (15.41, 19.14), (20.01, 18.41), (24.37, 24.02), (12.58, 15.02), (13.47, 22.8), (23.95, 21.14), (17.09, 13.09), (26.2, 14.02), (19.9, 14.49), (37.97, 12.7), (39.91, 14.56), (29.59, 16.13), (30.08, 5.92), (27.28, 16.01), (32.59, 17.96), (14.94, 22.96), (9.35, 25.14), (17.77, 20.83), (22.53, 18.43), (15.38, 21.91), (11.8, 23.02), (20.53, 20.55), (17.45, 19.35), (26.67, 16.32), (6.15, 25.42), (14.14, 21.21), (16.34, 17.59), (41.0, 17.38), (21.19, 14.33), (21.79, 12.39), (33.57, 14.88), (9.44, 17.97), (25.88, 12.53), (20.68, 18.24), (23.26, 22.94), (27.19, 13.1), (30.75, 18.68), (5.14, 20.16), (28.35, 14.39), (35.62, 7.02), (15.88, 20.27), (16.27, 23.75), (28.39, 20.77), (36.16, 19.06), (6.26, 15.29), (26.53, 12.81), (24.57, 17.02), (14.6, 14.31), (9.53, 28.59), (19.47, 15.13), (35.52, 14.23), (39.58, 8.42), (20.06, 22.51), (42.39, 7.49), (17.98, 13.13), (14.18, 23.79), (30.94, 14.93), (30.21, 22.36), (18.81, 22.54), (29.61, 16.59), (20.01, 20.64), (13.02, 22.47), (13.54, 26.95), (17.12, 20.65), (23.87, 8.77), (37.3, 6.23), (39.96, 18.04), (24.69, 14.07), (7.67, 19.33), (37.22, 2.75), (11.62, 25.73), (12.6, 28.92), (5.2, 20.55), (29.11, 14.45), (25.75, 16.41), (19.72, 17.71), (40.42, 6.43), (19.84, 13.71), (20.43, 18.38), (6.82, 13.07), (16.31, 20.28), (7.11, 22.14), (12.78, 19.48), (37.06, 14.89), (9.32, 21.48), (28.87, 4.23), (8.56, 22.29), (25.02, 21.76), (22.26, 16.4), (39.37, 12.08), (15.19, 21.65), (29.84, 11.56), (23.92, 21.34), (22.65, 19.41), (29.13, 13.08), (17.85, 25.0), (13.28, 22.71), (12.01, 27.38), (31.37, 8.42), (23.93, 20.58), (21.09, 15.64), (10.97, 24.12), (18.44, 19.84), (33.15, 12.27), (8.01, 16.56), (8.36, 26.01), (26.29, 15.89), (35.26, 14.46), (13.31, 21.2), (18.96, 13.74), (24.08, 18.35), (13.55, 20.23), (29.8, 14.28), (38.3, 14.1), (24.37, 19.69), (7.46, 19.93), (30.18, 9.81), (27.25, 19.23), (16.77, 22.53), (4.46, 22.02), (26.72, 16.72), (38.02, -1.99), (32.16, 10.82), (16.58, 19.78), (17.26, 21.19), (20.31, 14.27), (15.67, 10.76), (13.94, 14.67), (10.2, 17.32), (22.48, 20.26), (13.47, 10.69), (9.52, 26.56), (44.0, 8.45), (1.48, 29.35), (20.51, 15.34), (9.1, 10.07), (4.37, 30.38), (5.55, 25.25), (5.58, 23.95), (9.3, 22.42), (0.75, 17.27), (30.31, 13.77), (20.67, 19.85), (-9.75, 23.58), (37.24, 11.15), (2.44, 32.41), (6.72, 20.94), (23.75, 20.07), (32.14, 18.42), (23.23, 23.13), (15.57, 16.65), (13.42, 18.1), (20.49, 13.23), (40.18, 4.65), (36.89, 11.59), (7.55, 22.28), (11.16, 13.96), (12.27, 28.24), (26.57, 20.47), (22.41, 23.34), (24.67, 18.11), (11.96, 16.65), (28.59, 7.46), (16.95, 15.51), (28.25, 13.91), (28.04, 18.4), (18.9, 24.34), (22.15, 16.65), (35.32, 7.81), (11.73, 15.71), (15.23, 19.74), (21.08, 21.8), (14.82, 19.09), (11.65, 29.88), (25.37, 8.15), (20.13, 15.48), (15.75, 24.59), (28.49, 7.86), (24.12, 16.36), (30.11, 13.24), (18.06, 10.85), (14.78, 15.85), (23.63, 11.22), (16.87, 19.42), (28.98, 12.86), (12.51, 23.15), (18.01, 18.19), (6.71, 31.65), (27.38, 21.9), (-6.92, 34.44), (36.52, 5.49), (15.76, 20.49), (33.37, 12.14), (31.39, 19.39), (17.96, 12.14), (1.33, 25.96), (29.53, 18.77), (13.42, 24.26), (17.86, 15.69), (24.99, 19.73), (14.36, 17.05), (18.99, 16.64), (36.38, 10.18), (17.82, 16.28), (10.77, 14.69), (26.58, 10.38), (25.42, 18.08), (23.64, 13.99), (41.48, 12.46), (21.28, 17.5), (-8.24, 24.95), (31.22, 10.28), (28.35, 15.63), (17.68, 18.91), (9.81, 25.63), (35.1, 19.79), (31.83, 2.65), (8.72, 26.5), (18.09, 21.42), (17.55, 14.91), (-3.65, 25.56), (13.97, 25.11), (23.36, 16.82), (26.29, 15.52), (19.9, 9.76), (40.68, 3.97), (-3.28, 24.34), (36.45, 11.66), (33.05, 16.35), (20.99, 18.02), (24.79, 17.6), (30.54, 11.58), (24.09, 10.04), (26.33, 19.1), (21.05, 22.8), (30.6, 21.7), (34.48, 6.73), (17.26, 21.78), (-0.4, 23.74), (19.81, 13.15), (20.17, 17.72), (26.1, 9.32), (27.66, 13.3), (26.99, 16.61), (32.97, 18.42), (13.74, 19.69), (16.97, 24.05), (16.08, 19.26), (26.72, 8.99), (7.1, 18.77), (27.06, 16.73), (25.1, 19.06), (26.54, 14.69), (9.57, 28.29), (18.38, 17.98), (8.87, 15.25), (34.84, 12.32), (33.22, 13.08), (32.15, 5.05), (17.85, 18.45), (25.46, 16.12), (24.97, 18.61), (27.07, 13.82), (12.26, 26.47), (12.81, 24.03), (33.16, 10.8), (14.35, 20.35), (23.81, 18.52), (26.45, 13.65), (5.13, 19.57), (19.52, 26.6), (32.29, 6.24), (13.83, 19.14), (26.82, 12.95), (13.94, 22.58), (10.67, 23.56), (44.96, 6.18), (34.83, 15.15), (12.98, 18.48), (43.4, 2.69), (29.58, 16.23), (24.85, 19.96), (12.24, 25.93), (9.23, 14.37), (1.28, 24.77), (40.12, 13.13), (15.47, 15.6), (28.11, 9.23), (26.42, 20.17), (19.18, 23.33), (16.1, 17.24), (16.3, 23.15), (26.44, 15.96), (12.83, 15.7), (17.11, 28.03), (23.54, 22.78), (28.38, 20.14), (36.42, 13.16), (37.39, 3.6), (28.17, 15.6), (11.45, 23.48), (23.44, 17.68), (23.01, 9.02), (15.69, 17.77), (22.12, 12.41), (11.22, 16.97), (21.03, 15.4), (24.37, 9.91), (-1.34, 24.99), (36.3, 6.08), (17.99, 21.1), (-1.19, 25.43), (20.0, 14.52), (32.66, 12.54), (8.06, 22.86), (9.84, 25.06), (17.35, 20.89), (22.19, 25.09), (33.16, 16.95), (5.18, 29.07), (10.52, 17.21), (26.14, 18.73), (31.18, 13.16), (-0.07, 31.5), (27.49, 13.68), (12.6, 26.36), (14.61, 26.99), (30.11, 12.82), (13.75, 26.45), (-0.29, 32.21), (30.48, 9.59), (44.44, 7.01), (27.02, 15.41), (29.38, 10.32), (30.27, 7.2), (9.95, 24.89), (23.02, 17.72), (5.18, 33.24), (31.53, 10.74), (19.13, 20.04), (11.52, 17.63), (25.0, 12.78), (39.98, 18.4), (24.03, 15.71), (12.25, 22.66), (12.54, 19.07), (-2.85, 25.35), (34.66, 14.35), (27.5, 15.85), (26.71, 11.31), (8.98, 24.57), (16.37, 21.4), (48.69, 8.81), (-4.41, 32.28), (6.76, 20.9), (-1.32, 24.49), (16.46, 17.17), (9.45, 15.84), (10.63, 27.38), (17.45, 21.71), (9.66, 26.28), (23.42, 14.36), (24.38, 9.05), (32.63, 17.5), (16.58, 20.94), (13.02, 26.26), (13.59, 17.71), (37.33, 1.8), (33.39, 16.31), (8.75, 16.09), (1.64, 22.1), (-4.01, 23.59), (18.15, 19.89), (27.89, 15.79), (29.39, 12.85), (17.37, 19.42), (21.97, 9.94), (29.24, 15.18), (13.69, 19.92), (3.55, 31.03), (8.97, 25.2), (18.96, 25.28), (27.64, 12.94), (16.94, 20.31), (9.13, 22.34), (21.85, 19.53), (25.45, 12.78), (2.18, 30.23), (11.33, 18.88), (13.01, 26.11), (17.31, 24.39), (28.76, 12.55), (3.43, 17.88), (19.91, 25.99), (18.79, 17.44), (25.77, 17.17), (20.6, 24.7), (11.6, 17.88), (21.98, 8.05), (14.85, 20.13), (14.28, 19.22), (27.3, 16.66), (18.04, 24.34), (27.78, 9.91), (14.19, 20.33), (25.4, 21.05), (20.02, 11.47), (16.13, 20.54), (26.82, 16.17), (5.01, 25.79), (12.39, 20.29), (38.91, 8.85), (41.66, 9.62), (20.38, 17.21), (30.63, 15.8), (36.24, 10.22), (29.97, 19.02), (15.26, 19.46), (16.84, 22.21), (31.96, 12.39), (19.69, 14.08), (7.28, 19.79), (29.21, 10.55), (37.79, 14.57), (25.97, 20.75), (26.33, 10.3), (23.3, 14.96), (16.66, 30.91), (31.52, 9.76), (30.31, 10.69), (14.0, 13.2), (12.46, 15.99), (31.42, 8.79), (-1.58, 31.45), (1.33, 31.66), (2.04, 23.35), (11.14, 27.41), (17.33, 23.43), (12.19, 26.54), (25.53, 17.26), (22.42, 13.77), (29.61, 16.45), (30.17, 16.72), (17.72, 17.26), (35.52, 12.62), (29.56, 16.9), (26.38, 11.28), (16.47, 26.0), (26.34, 6.41), (28.96, 15.45), (8.45, 23.42), (15.29, 16.53), (31.47, 13.94), (36.43, 17.71), (27.55, 16.16), (23.11, 11.63), (21.05, 20.27), (17.42, 19.53), (18.17, 17.31), (-19.96, 29.5), (19.1, 27.51), (14.04, 18.65), (13.73, 20.63), (-6.36, 29.63), (7.48, 30.14), (12.33, 25.72), (16.35, 21.37), (21.96, 10.68), (30.34, 11.33), (28.76, 14.6), (25.19, 14.68), (18.35, 15.92), (19.08, 20.85), (36.55, 8.47), (15.75, 21.42), (24.54, 21.1), (14.62, 20.59), (38.13, 16.94), (27.92, 13.19), (14.32, 14.6), (20.47, 24.12), (9.19, 18.73), (21.16, 17.66), (16.07, 10.05), (19.46, 19.75), (31.33, 11.38), (10.65, 21.33), (33.19, 15.01), (43.42, 11.95), (27.67, 11.06), (13.86, 25.24), (13.67, 20.09), (28.02, 19.62), (26.4, 14.67), (22.75, 17.06), (35.06, 13.62), (20.65, 17.17), (18.46, 25.82), (22.27, 7.98), (24.69, 23.37), (25.09, 22.38), (19.97, 21.06), (17.99, 14.55), (21.12, 17.5), (10.76, 22.9), (23.89, 15.17), (12.59, 15.53), (15.53, 20.28), (19.3, 24.92), (0.41, 22.94), (20.1, 28.81), (26.75, 13.8), (31.53, 17.16), (21.18, 20.61), (16.82, 24.59), (2.96, 26.65), (19.76, 13.25), (-10.58, 33.82), (12.47, 18.46), (11.85, 23.37), (25.4, 21.91), (19.66, 19.2), (34.32, 10.56), (18.26, 22.89), (11.84, 24.32), (23.18, 17.13), (22.08, 15.64), (29.01, 14.12), (21.91, 14.76), (24.71, 14.39), (5.91, 14.06), (29.25, 8.25), (30.55, 13.97), (38.64, 5.52), (11.51, 21.21), (13.78, 20.95), (18.27, 15.93), (22.33, 21.39), (9.16, 25.26), (34.65, 18.36), (24.88, 10.37), (27.79, 19.73), (27.45, 15.46), (38.46, 2.42), (31.61, 12.28), (35.69, 10.45), (26.82, 14.58), (8.79, 20.76), (31.04, 11.36), (1.53, 28.01), (6.87, 15.38), (28.9, 8.04), (17.7, 14.05), (-2.06, 21.52), (14.57, 21.37), (26.77, 9.95), (41.78, 15.42), (14.87, 24.13), (12.94, 24.75), (23.41, 20.84), (16.61, 9.94), (11.06, 24.44), (33.34, 9.15), (22.24, 15.61), (37.18, -0.12), (30.95, 15.25), (21.47, 12.82), (-5.62, 29.07), (23.63, 17.84), (16.86, 14.36), (12.01, 14.53), (17.55, 23.42), (16.25, 16.15), (23.29, 14.88), (23.81, 22.54), (24.21, 11.08), (33.77, 13.31), (18.52, 28.3), (31.12, 11.43), (15.68, 22.62), (15.95, 23.7), (23.92, 19.09), (36.09, 12.3), (33.87, 6.51), (27.0, 14.66), (28.55, 17.36), (4.83, 23.09), (17.76, 20.24)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=882；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6998；
t = r·√((n−2)/(1−r²)) = 0.6998·√((882−2)/(1−0.4897)) = -29.0579；df=880；p = -5.96e+56。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7320, -0.6644]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=882；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6998；
t = r·√((n−2)/(1−r²)) = 0.6998·√((882−2)/(1−0.4897)) = -29.0579；df=880；p = -5.96e+56。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7320, -0.6644]。
③ 计算结果：
r = -0.6998，t = -29.0579，p = -5.96e+56，95%CI = [-0.7320, -0.6644]
④ 结论：在 α=0.05 水平下，p=-5.96e+56 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.6998，方向负），打折损利润。
案例 32. 空气污染指数与户外活动时长(h)
Pearson相关污染重少外出
① 原始数据：
样本 n=910，逐条 (x, y)：[(100.22, 2.89), (122.59, 1.15), (133.23, 1.14), (165.45, -0.43), (110.21, 1.46), (85.81, 0.83), (80.45, 2.07), (79.46, 1.66), (68.18, 1.12), (65.67, 2.87), (131.85, 0.19), (1.48, 4.08), (30.33, 3.44), (51.71, 3.21), (76.9, 2.31), (134.51, 2.64), (135.11, -0.29), (72.74, 1.7), (46.78, 2.92), (42.59, 3.07), (84.38, 2.53), (128.58, 2.26), (44.97, 3.91), (64.13, 3.0), (67.72, 1.07), (141.25, 1.31), (57.49, 4.2), (63.98, 2.69), (81.99, 2.7), (111.81, 1.51), (28.48, 4.41), (35.8, 1.93), (77.25, 2.12), (122.38, 2.18), (44.7, 3.04), (165.7, 1.43), (31.49, 2.93), (174.71, -0.08), (82.46, 2.09), (103.38, 1.7), (141.25, 1.19), (47.59, 2.44), (118.67, 0.76), (6.85, 2.41), (69.21, 1.58), (114.32, 0.78), (79.23, 1.85), (66.42, 1.72), (62.3, 2.28), (88.21, 2.79), (97.85, 2.29), (85.55, 2.86), (75.31, 1.38), (111.84, 2.7), (103.44, 0.11), (70.48, 2.93), (79.31, 2.07), (141.71, 0.59), (27.79, 2.2), (37.03, 4.41), (108.46, 1.87), (42.31, 1.94), (75.85, 1.67), (90.31, 1.19), (45.02, 2.98), (61.93, 0.91), (125.49, 1.73), (69.4, 2.99), (148.68, 1.49), (78.93, 2.67), (98.26, 1.43), (117.51, 0.91), (110.94, 1.43), (56.81, 1.44), (18.29, 2.62), (97.16, 2.27), (67.51, 1.5), (48.58, 1.57), (16.89, 2.63), (47.08, 3.52), (112.68, 0.97), (77.87, 2.22), (106.78, 1.62), (101.0, 1.71), (10.61, 3.58), (104.43, 0.91), (69.61, 1.66), (72.63, 1.89), (90.91, 0.92), (115.86, 2.88), (72.13, 1.49), (87.65, 1.27), (89.84, 1.2), (115.36, 1.87), (45.74, 1.94), (119.33, 1.52), (20.38, 2.09), (98.4, 1.37), (86.54, 1.77), (58.51, 2.0), (49.0, 2.49), (107.6, 0.77), (89.63, 1.1), (-6.68, 4.14), (87.58, 0.13), (135.11, 1.23), (84.26, 2.09), (78.35, 2.37), (82.87, 2.16), (30.46, 4.23), (131.77, 1.37), (25.74, 3.07), (71.67, 1.97), (114.58, 2.07), (91.59, 0.84), (-22.85, 4.32), (66.23, 1.91), (81.06, 1.59), (88.45, 3.14), (64.63, 1.75), (14.99, 3.41), (52.15, 1.67), (110.78, 2.48), (113.02, 1.13), (96.11, 1.26), (110.46, 0.3), (86.1, 0.93), (38.6, 2.74), (28.16, 3.6), (98.36, 1.59), (51.64, 2.85), (77.01, 2.83), (83.63, 1.05), (34.0, 2.69), (20.83, 4.64), (53.86, 3.04), (40.53, 2.39), (66.95, 1.97), (180.03, 0.87), (102.64, 1.85), (81.76, 3.0), (100.1, 0.13), (14.15, 3.33), (80.12, 2.23), (121.31, 1.09), (63.8, 2.03), (61.38, 2.53), (58.39, 2.45), (54.56, 2.27), (113.79, 0.46), (68.0, 2.95), (84.22, 3.05), (47.84, 2.16), (53.44, 2.07), (80.86, 1.12), (81.01, 1.84), (78.31, 2.14), (50.86, 2.01), (100.82, 2.99), (54.78, 2.46), (18.76, 4.35), (119.19, 1.55), (75.38, 2.87), (100.56, 1.89), (-28.74, 4.76), (55.29, 2.33), (67.78, 2.37), (90.11, 1.97), (-17.36, 4.71), (134.95, 0.71), (75.23, 3.34), (69.13, 2.69), (37.39, 3.24), (48.74, 3.4), (95.4, 1.46), (27.76, 3.93), (79.46, 1.92), (102.65, 0.51), (41.07, 3.27), (60.64, 1.63), (105.27, 3.23), (63.29, 2.14), (13.15, 3.48), (121.05, 1.82), (-7.02, 4.12), (83.45, 2.48), (40.91, 2.28), (61.14, 2.62), (87.12, 0.7), (35.13, 2.72), (62.47, 1.7), (61.47, 2.84), (71.21, 2.56), (125.5, 0.67), (110.74, 0.51), (97.21, 0.65), (93.67, 2.7), (77.38, 2.05), (109.17, -0.09), (106.35, 1.79), (139.48, 1.35), (59.01, 3.38), (108.76, 0.86), (89.48, 2.04), (40.88, 2.92), (128.38, 0.07), (105.48, 1.85), (41.7, 2.76), (133.96, 0.89), (79.47, 1.73), (79.88, 1.33), (189.26, 1.7), (106.78, 0.45), (84.31, 1.52), (156.88, 1.1), (130.92, 1.55), (50.88, 3.62), (33.26, 2.83), (96.9, 0.37), (48.83, 3.53), (146.9, 0.26), (119.51, 0.75), (-31.14, 3.68), (45.78, 1.51), (67.82, 2.37), (105.36, 2.37), (43.71, 2.26), (81.7, 2.26), (107.29, 3.39), (58.23, 2.3), (75.55, 2.2), (133.2, 1.26), (101.79, 1.99), (44.74, 1.9), (90.2, 0.9), (-5.69, 3.69), (93.81, 1.81), (95.47, 2.25), (108.89, 1.67), (71.06, 1.98), (63.44, 3.13), (91.85, 0.42), (91.42, 1.66), (64.57, 2.47), (65.3, 2.37), (48.6, 3.34), (84.48, 1.4), (92.16, 0.52), (82.58, 2.22), (28.8, 1.9), (69.1, 1.46), (23.98, 2.26), (125.82, 1.36), (127.29, 0.51), (43.4, 2.29), (42.56, 2.13), (29.22, 2.9), (82.7, 2.27), (49.3, 2.9), (51.37, 1.23), (123.08, 1.18), (109.93, -0.0), (31.32, 3.19), (77.03, 0.92), (75.61, 2.09), (59.19, 3.55), (59.88, 3.42), (65.79, 2.12), (49.45, 2.81), (112.37, 1.83), (64.34, 1.04), (-13.92, 5.24), (70.09, 0.09), (103.61, 1.19), (112.96, 0.7), (50.16, 2.14), (74.52, 2.25), (103.16, 1.02), (92.6, 1.32), (71.49, 1.48), (34.69, 2.77), (43.56, 3.4), (35.16, 3.33), (-37.96, 4.43), (68.59, 1.91), (101.33, 0.68), (98.78, 1.88), (48.91, 4.12), (102.3, 2.36), (106.83, 2.17), (127.76, 0.32), (155.26, 0.46), (133.54, 0.62), (77.8, 3.1), (112.13, 1.06), (157.5, 0.76), (106.24, 2.84), (18.28, 2.86), (68.45, 2.74), (71.25, 2.48), (44.93, 3.24), (90.28, 1.25), (51.84, 4.69), (152.56, 0.93), (16.37, 3.78), (158.12, 1.15), (92.0, 1.27), (45.64, 3.45), (97.58, 2.24), (106.95, 3.25), (1.18, 2.26), (98.63, 0.98), (121.55, 1.87), (75.35, 1.61), (112.05, 1.69), (56.52, 3.5), (32.8, 1.65), (40.96, 4.69), (73.26, 2.54), (121.42, 1.55), (99.18, 0.05), (144.72, 1.0), (93.73, 1.49), (67.29, 1.63), (71.78, 1.76), (42.03, 2.12), (120.61, 1.78), (88.45, 1.98), (112.02, 2.47), (-18.32, 3.91), (105.04, 1.35), (61.56, 2.03), (95.26, 1.81), (129.31, 2.0), (64.99, 2.35), (25.59, 4.03), (122.41, 1.6), (65.87, 1.83), (64.15, 2.42), (78.31, 1.63), (63.97, 2.08), (78.52, 1.89), (114.73, 1.44), (70.35, 1.98), (111.22, 1.56), (66.85, 2.31), (116.25, 0.67), (58.27, 2.74), (66.83, 2.93), (136.48, 1.61), (39.95, 1.68), (99.76, 1.84), (106.41, 0.85), (39.8, 3.39), (81.45, 2.13), (69.72, 1.91), (-6.44, 4.06), (123.66, 2.15), (86.48, 2.78), (-3.24, 4.08), (168.75, 0.65), (171.47, 0.8), (81.78, 0.32), (118.6, 0.58), (71.62, 1.59), (121.36, 1.71), (80.57, 2.02), (51.69, 2.0), (80.38, 1.01), (21.59, 2.85), (54.43, 2.21), (11.24, 3.42), (88.98, 2.01), (31.01, 3.5), (31.84, 2.56), (119.4, 0.65), (85.77, 2.04), (58.91, 2.49), (129.28, 1.07), (98.85, 1.5), (103.4, 2.06), (42.26, 1.73), (90.3, 0.42), (27.82, 3.05), (128.32, 2.28), (103.12, 2.6), (60.78, 1.4), (79.06, 2.57), (105.45, 1.54), (118.76, 0.73), (137.34, 1.52), (99.84, 1.49), (54.84, 2.59), (114.3, 2.29), (-5.86, 1.9), (28.39, 3.24), (93.67, 2.57), (55.12, 1.79), (127.31, 2.61), (58.39, 2.63), (-5.56, 3.42), (32.9, 4.68), (35.56, 2.26), (16.22, 3.1), (51.86, 2.41), (53.8, 1.01), (107.33, 2.06), (-8.66, 3.7), (25.46, 2.44), (89.56, 4.09), (84.32, 2.14), (14.44, 3.41), (130.56, 1.81), (116.1, 0.12), (146.88, 0.36), (114.43, 1.56), (90.28, 1.59), (125.75, 0.8), (66.36, 2.5), (29.02, 2.89), (86.77, 1.86), (65.95, 2.86), (67.73, 2.46), (96.19, 1.82), (34.37, 3.81), (31.33, 2.69), (72.09, 2.46), (103.23, 2.04), (106.0, 1.81), (28.02, 2.62), (148.23, 1.26), (86.23, 1.34), (146.18, 1.13), (62.7, 2.02), (74.43, 2.55), (96.87, 2.63), (-13.47, 3.6), (82.52, 2.58), (63.71, 1.81), (147.98, 0.36), (-21.92, 2.92), (142.61, -0.69), (28.66, 2.18), (67.79, 2.77), (80.15, 2.42), (47.82, 2.36), (37.59, 2.49), (58.59, 2.9), (67.74, 1.66), (24.94, 2.55), (32.05, 3.43), (62.05, 2.46), (74.62, 2.34), (45.56, 2.52), (72.45, 2.98), (101.23, 0.68), (26.48, 2.33), (126.21, 1.67), (42.98, 3.05), (161.07, 1.15), (142.21, 1.53), (125.28, 2.02), (39.63, 1.52), (130.36, -0.25), (134.2, 2.06), (102.59, 3.18), (67.86, 3.69), (36.11, 1.99), (43.94, 2.88), (52.21, 1.97), (84.73, 1.32), (23.21, 1.3), (131.16, 0.57), (33.41, 2.93), (117.23, 2.15), (92.94, 0.82), (129.41, 0.51), (112.04, 1.49), (16.35, 2.3), (140.88, 2.23), (62.39, 1.91), (83.98, 1.79), (72.79, 2.36), (112.96, 1.47), (117.61, 0.83), (101.37, 1.07), (20.94, 2.48), (86.65, 2.68), (1.32, 3.59), (79.37, 2.35), (79.19, 3.68), (91.67, 0.76), (84.03, 1.95), (105.67, 2.57), (123.32, -0.15), (88.65, 2.37), (98.31, 1.29), (89.87, 2.56), (62.43, 3.81), (83.52, 0.74), (101.33, 2.66), (144.57, 1.77), (17.85, 2.86), (80.85, 3.01), (96.43, 2.0), (63.92, 2.58), (93.41, 0.61), (80.39, 1.53), (61.79, 2.79), (126.23, 1.73), (72.57, 2.1), (103.76, 1.64), (66.19, 1.35), (72.44, 2.07), (64.95, 1.57), (114.93, 1.93), (90.6, 1.6), (132.85, 1.99), (177.1, 0.64), (105.63, 1.07), (134.17, 1.29), (97.59, 0.89), (44.38, 1.92), (72.1, 3.5), (-34.0, 4.31), (37.91, 2.68), (66.26, 2.15), (57.42, 2.18), (13.46, 2.86), (58.9, 1.59), (107.07, 0.47), (162.75, 1.2), (96.22, 1.01), (114.09, 0.41), (130.33, 1.16), (113.16, 1.22), (99.14, 0.85), (122.34, 1.21), (86.03, 1.96), (20.89, 3.82), (87.48, 0.78), (9.14, 2.59), (149.04, 0.3), (96.5, 1.53), (83.57, 1.78), (170.16, 0.19), (96.07, 1.12), (111.13, 0.37), (85.82, 1.99), (87.21, 1.79), (41.84, 1.76), (7.63, 2.92), (83.99, 1.9), (87.35, 1.12), (96.28, 1.06), (113.43, 1.08), (90.77, 2.57), (113.6, 1.56), (92.3, 0.91), (121.47, 1.54), (54.61, 2.53), (120.78, -0.21), (63.67, 2.57), (123.24, 0.55), (57.36, 2.02), (132.48, 0.75), (121.69, 1.54), (93.29, 1.02), (69.46, 2.33), (102.22, 1.97), (43.16, 2.85), (58.83, 2.55), (85.53, 2.04), (94.16, 1.15), (53.15, 1.99), (38.16, 1.85), (129.81, 1.05), (115.69, 1.99), (115.53, 1.07), (8.48, 2.33), (111.54, 2.62), (120.71, 1.13), (128.08, 0.15), (136.74, 1.02), (61.48, 1.74), (30.63, 2.68), (69.63, 1.2), (80.57, 2.52), (24.03, 3.6), (152.66, 0.5), (104.29, 2.49), (6.7, 2.94), (89.36, 1.2), (54.17, 1.64), (63.92, 2.26), (-12.95, 3.08), (-33.96, 3.13), (68.03, 1.54), (78.5, 1.89), (89.26, 1.16), (26.14, 1.93), (91.86, 1.55), (101.02, 0.82), (85.64, 1.34), (144.26, -0.57), (99.32, 1.8), (28.91, 3.92), (-37.2, 3.9), (73.87, 2.59), (63.94, 1.99), (102.09, 0.16), (57.34, 2.06), (111.0, 0.83), (51.52, 2.8), (147.52, 0.68), (124.64, 1.77), (54.11, 4.06), (67.81, 2.25), (93.94, 0.95), (77.19, 1.04), (67.87, 2.17), (25.24, 2.61), (84.67, 1.34), (86.96, 1.76), (119.54, 0.54), (83.01, 1.27), (60.96, 1.4), (12.43, 2.77), (22.7, 1.93), (115.7, 0.95), (92.44, 2.04), (21.04, 3.28), (147.4, 0.8), (110.86, 0.78), (93.66, 2.94), (27.58, 2.8), (55.54, 2.22), (91.27, 2.6), (99.63, 1.25), (66.93, 1.22), (162.94, 0.37), (82.49, 2.6), (159.2, 1.28), (96.86, 2.23), (28.02, 3.19), (121.36, 0.31), (113.82, 1.09), (42.91, 3.44), (40.5, 3.97), (51.38, 1.52), (78.17, 2.4), (105.55, 1.43), (86.77, 2.02), (52.77, 2.86), (95.92, 1.43), (86.06, 1.34), (8.92, 1.83), (103.9, 1.02), (102.44, 1.26), (113.87, 2.18), (101.74, 2.35), (152.06, 1.64), (-4.82, 3.83), (71.61, 1.88), (82.77, 1.82), (23.96, 3.79), (15.07, 3.36), (147.01, 1.98), (102.4, 1.93), (44.79, 2.25), (104.62, 0.96), (78.51, 2.15), (115.76, 0.99), (16.12, 2.77), (110.59, 2.0), (136.78, 2.72), (129.49, 0.7), (89.03, 1.67), (52.13, 3.11), (37.55, 3.17), (105.11, 1.56), (76.12, 2.67), (21.85, 2.28), (88.66, 1.55), (123.06, 2.35), (42.74, 2.85), (81.17, 1.75), (50.17, 2.34), (57.56, 1.71), (79.09, 3.33), (83.28, 1.36), (142.34, -0.21), (63.94, 2.94), (75.01, 1.36), (75.94, 2.73), (76.92, 2.93), (86.23, 2.49), (153.73, 0.99), (80.71, 2.05), (82.74, 0.64), (131.18, 1.59), (80.24, 2.33), (9.36, 3.04), (114.36, 2.47), (101.75, 0.64), (69.32, 1.56), (23.42, 2.75), (86.45, 1.34), (91.37, 2.5), (54.85, 2.52), (64.37, 2.88), (71.08, 1.71), (74.61, 1.41), (77.68, 2.78), (125.66, 0.81), (107.38, 1.27), (124.81, 1.13), (71.67, 2.14), (3.16, 3.47), (100.43, 1.23), (76.72, 1.87), (17.29, 1.94), (86.91, 1.02), (61.88, 2.45), (46.25, 2.13), (48.9, 1.2), (19.34, 2.55), (13.2, 3.16), (111.0, 2.61), (112.96, 0.76), (65.01, 2.32), (0.27, 4.08), (52.85, 2.35), (-11.72, 4.02), (84.48, 3.51), (167.23, 0.63), (115.12, 0.63), (49.46, 2.65), (122.08, 0.53), (97.34, 1.71), (110.46, 2.71), (43.25, 2.72), (136.76, 1.21), (59.75, 0.89), (45.8, 3.84), (119.22, 1.95), (57.32, 3.53), (84.36, 1.89), (83.15, 2.19), (94.17, 3.38), (41.95, 2.41), (81.16, 3.3), (98.41, 2.55), (51.62, 1.73), (79.93, 3.03), (10.45, 3.1), (51.78, 2.71), (101.46, 1.09), (16.97, 3.61), (102.32, 1.13), (31.8, 3.37), (123.64, 1.9), (46.69, 2.83), (79.17, 3.09), (45.64, 3.69), (21.31, 2.57), (123.47, 1.69), (107.51, 1.83), (140.15, 2.56), (56.1, 1.94), (38.16, 2.98), (62.2, 1.31), (46.44, 2.13), (38.71, 3.46), (104.4, 2.04), (-1.27, 3.78), (54.73, 2.87), (24.63, 2.64), (77.38, 2.39), (87.85, 1.62), (129.32, 2.32), (30.14, 2.78), (134.06, 0.62), (57.88, 2.22), (69.2, 2.34), (89.12, 2.07), (173.64, -0.32), (44.17, 4.18), (137.15, 2.05), (35.32, 4.14), (8.16, 3.17), (117.49, 1.24), (39.24, 2.65), (75.8, 1.75), (95.82, 1.72), (106.4, 1.47), (53.33, 2.13), (69.23, 1.07), (44.13, 2.56), (85.76, 2.0), (67.34, 2.67), (132.12, 1.18), (132.44, 0.45), (85.13, 1.36), (65.2, 0.84), (30.91, 2.97), (68.76, 2.21), (66.82, 2.65), (55.7, 1.59), (132.42, 0.97), (74.31, 2.33), (184.48, 0.61), (120.72, 1.72), (100.37, 2.28), (40.13, 1.78), (12.0, 3.09), (96.32, 3.66), (76.69, 1.38), (69.62, 2.37), (71.83, 2.47), (72.6, 3.6), (20.27, 3.12), (66.71, 2.27), (108.16, -0.43), (77.7, 2.39), (33.31, 2.58), (79.65, 1.65), (26.85, 1.52), (78.36, 1.69), (32.91, 3.37), (43.69, 2.1), (28.15, 3.65), (129.05, 1.09), (89.65, 1.65), (81.06, 1.92), (31.26, 3.37), (159.22, 0.97), (-5.45, 4.51), (30.43, 2.51), (58.8, 2.11), (115.9, 1.58), (42.53, 3.26), (95.66, 2.43), (114.02, 1.14), (129.93, 0.52), (81.9, 2.5), (65.35, 3.17), (126.38, 0.65), (73.3, 3.01), (77.7, 1.76), (-23.83, 3.85), (85.75, 2.17), (50.48, 3.08), (-13.73, 2.71), (119.35, 1.03), (104.2, 1.99), (81.8, 2.32), (90.67, 1.29), (136.48, 2.01), (76.5, 2.64), (29.18, 1.68), (120.84, 0.92), (197.5, -0.13), (60.46, 2.94), (103.66, 1.74), (128.76, 1.06), (-46.64, 3.88), (86.77, 1.95), (129.41, 2.01), (93.38, 0.63), (47.88, 2.09), (134.01, 0.32), (-23.16, 3.4), (59.17, 2.09), (25.27, 1.4), (59.98, 1.72), (93.7, 2.38), (49.19, 1.7), (51.92, 1.93), (123.13, 1.01), (39.23, 2.78), (79.23, 2.72), (121.56, 1.92), (114.64, 1.66), (75.28, 2.16), (23.4, 4.43), (22.25, 2.36), (-8.24, 2.3), (133.16, 1.22), (56.71, 2.03), (68.42, 1.69), (67.92, 2.05), (62.1, 2.4), (35.61, 3.65), (42.96, 3.83), (9.84, 3.2), (46.49, 2.81), (134.62, 0.59), (76.94, 2.18), (39.25, 3.53), (46.04, 3.2), (46.59, 3.99), (130.38, 0.49), (34.26, 1.46), (38.6, 3.15), (142.43, 1.35), (43.88, 1.78), (110.65, 2.08), (92.36, 0.48), (161.74, 0.51)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=910；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6926；
t = r·√((n−2)/(1−r²)) = 0.6926·√((910−2)/(1−0.4796)) = -28.9306；df=908；p = -6.04e+58。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7250, -0.6571]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=910；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.6926；
t = r·√((n−2)/(1−r²)) = 0.6926·√((910−2)/(1−0.4796)) = -28.9306；df=908；p = -6.04e+58。
95% CI（Fisher z）：tanh(z±1.963·1/√(n−3)) = [-0.7250, -0.6571]。
③ 计算结果：
r = -0.6926，t = -28.9306，p = -6.04e+58，95%CI = [-0.7250, -0.6571]
④ 结论：在 α=0.05 水平下，p=-6.04e+58 < 0.05，拒绝 H₀。两变量存在线性相关（r=-0.6926，方向负），污染重少外出。
案例 33. 鞋码(欧码)与数学能力(分)
Pearson相关应无相关
① 原始数据：
样本 n=347，逐条 (x, y)：[(30.94, 65.82), (40.02, 73.21), (37.07, 79.57), (40.62, 70.04), (40.94, 111.54), (40.29, 71.76), (41.49, 88.79), (34.37, 55.55), (35.63, 102.95), (44.89, 61.71), (43.55, 75.34), (39.25, 71.07), (39.38, 78.15), (38.07, 82.04), (39.18, 93.35), (39.0, 83.56), (45.99, 85.67), (37.68, 84.54), (40.14, 85.78), (42.87, 89.13), (42.84, 78.72), (35.19, 74.44), (38.13, 63.77), (41.48, 69.33), (40.79, 75.28), (36.76, 49.69), (42.64, 67.26), (36.13, 63.71), (40.88, 76.42), (45.43, 59.12), (41.03, 70.44), (44.43, 93.98), (43.8, 82.71), (39.35, 73.43), (37.8, 72.68), (45.54, 84.66), (39.94, 58.75), (35.68, 92.17), (35.64, 82.91), (34.26, 94.13), (41.6, 64.11), (41.58, 69.23), (38.73, 85.78), (36.41, 61.46), (46.69, 83.1), (36.15, 85.47), (43.49, 79.92), (38.4, 68.01), (37.12, 74.87), (40.4, 82.61), (37.9, 79.63), (43.98, 64.72), (43.86, 70.26), (41.7, 64.83), (41.93, 56.02), (36.46, 71.92), (42.26, 82.0), (39.64, 76.55), (37.27, 81.94), (36.8, 99.49), (37.0, 70.04), (38.69, 70.97), (44.43, 57.8), (37.51, 57.53), (39.99, 61.36), (37.01, 83.4), (42.24, 75.69), (34.22, 72.85), (45.9, 78.88), (41.87, 71.53), (43.63, 74.25), (44.71, 81.92), (40.21, 78.32), (38.26, 95.8), (39.94, 82.92), (38.63, 77.64), (41.77, 57.29), (40.59, 76.78), (45.47, 94.47), (41.9, 56.02), (41.84, 68.56), (39.03, 83.79), (38.91, 93.63), (37.81, 66.63), (45.25, 99.08), (38.36, 71.98), (40.25, 71.33), (34.44, 63.45), (34.68, 87.24), (42.44, 72.25), (46.12, 87.64), (43.86, 91.98), (41.28, 62.18), (31.47, 101.38), (37.43, 65.21), (37.95, 74.09), (41.59, 51.85), (38.09, 62.71), (39.45, 80.72), (39.75, 77.93), (34.98, 73.98), (43.03, 78.16), (39.58, 68.09), (40.55, 89.14), (35.85, 68.39), (38.13, 102.75), (37.93, 99.95), (45.76, 71.09), (36.85, 67.82), (37.51, 97.93), (38.84, 81.63), (42.48, 57.03), (35.9, 73.45), (37.76, 80.28), (40.53, 82.59), (35.25, 71.56), (44.14, 83.66), (39.06, 51.45), (42.36, 77.1), (46.57, 77.27), (36.4, 57.74), (42.3, 77.45), (36.72, 59.7), (44.85, 62.04), (42.02, 67.95), (42.48, 87.12), (37.85, 77.03), (42.73, 88.43), (40.24, 82.15), (38.91, 57.94), (41.09, 52.77), (38.54, 84.37), (37.32, 72.51), (37.28, 81.54), (37.86, 79.53), (38.56, 49.23), (38.09, 90.68), (36.42, 78.59), (33.86, 83.88), (39.23, 77.45), (40.18, 59.83), (42.79, 55.11), (38.79, 63.05), (42.57, 71.26), (41.71, 99.4), (36.26, 70.06), (40.85, 91.27), (31.61, 76.87), (39.71, 76.13), (40.69, 74.55), (42.34, 77.41), (36.84, 77.53), (35.97, 64.33), (43.33, 55.74), (43.73, 74.21), (38.71, 75.03), (40.36, 69.25), (36.75, 65.38), (36.54, 57.4), (39.08, 77.24), (41.66, 80.95), (39.1, 63.76), (42.14, 73.37), (43.4, 89.24), (37.6, 82.16), (38.49, 84.54), (34.15, 53.42), (41.04, 70.51), (37.25, 62.63), (43.73, 87.22), (42.1, 89.26), (40.24, 76.38), (40.08, 77.2), (41.06, 77.7), (45.81, 79.71), (30.06, 70.92), (39.57, 50.78), (40.19, 88.38), (41.17, 72.13), (38.19, 67.27), (42.11, 75.88), (40.98, 79.36), (42.57, 94.58), (40.69, 48.89), (36.65, 36.01), (40.45, 86.32), (42.67, 70.75), (40.92, 66.61), (42.42, 49.75), (39.25, 65.19), (42.6, 70.97), (35.45, 88.9), (41.56, 57.37), (42.06, 93.75), (37.48, 71.11), (38.76, 67.12), (40.34, 84.96), (41.78, 101.01), (44.1, 73.98), (37.43, 76.29), (40.63, 80.35), (38.4, 73.23), (32.83, 39.54), (39.92, 69.24), (43.9, 76.84), (42.99, 76.94), (41.72, 66.83), (35.52, 90.24), (36.43, 67.8), (38.47, 80.81), (39.64, 90.44), (36.82, 99.22), (40.6, 62.49), (42.25, 89.85), (40.39, 86.91), (45.52, 65.33), (44.39, 67.05), (45.26, 63.89), (41.6, 78.42), (42.81, 70.93), (43.95, 71.28), (38.6, 63.08), (34.32, 57.22), (40.13, 83.29), (37.58, 77.31), (42.55, 82.99), (42.17, 87.0), (37.3, 78.67), (45.41, 67.19), (37.55, 86.83), (39.52, 78.04), (42.52, 81.2), (41.76, 63.06), (38.96, 86.46), (38.48, 75.13), (41.1, 77.45), (35.3, 84.9), (38.46, 87.04), (36.58, 86.61), (37.75, 48.99), (35.41, 74.44), (43.2, 52.67), (36.24, 69.42), (40.73, 62.44), (46.49, 81.81), (42.08, 98.14), (34.98, 56.1), (40.86, 81.66), (39.61, 60.72), (39.11, 70.49), (40.25, 65.53), (39.04, 78.12), (44.06, 73.25), (41.57, 79.72), (34.19, 46.96), (39.93, 95.06), (41.59, 85.19), (37.83, 87.6), (39.13, 81.96), (37.26, 60.44), (39.49, 87.57), (38.68, 89.65), (42.08, 76.87), (41.71, 78.62), (39.41, 71.88), (39.37, 84.4), (41.99, 90.76), (44.01, 52.32), (36.39, 66.12), (40.42, 84.3), (38.37, 70.31), (40.54, 62.1), (38.88, 67.89), (40.4, 78.96), (37.54, 75.04), (38.33, 83.27), (36.35, 97.92), (40.9, 66.1), (42.6, 92.51), (42.6, 76.19), (40.87, 57.7), (40.91, 74.89), (38.98, 67.05), (35.78, 80.89), (39.91, 71.85), (42.38, 83.77), (39.48, 82.28), (39.21, 96.24), (40.7, 62.01), (42.44, 74.66), (37.54, 102.2), (42.0, 85.86), (38.73, 63.36), (42.83, 72.89), (41.26, 79.39), (36.23, 81.72), (36.26, 89.91), (39.46, 77.98), (44.4, 73.05), (37.42, 76.68), (37.24, 81.66), (40.95, 82.11), (33.05, 58.87), (38.45, 73.76), (37.72, 66.79), (40.17, 80.02), (34.68, 86.33), (40.12, 93.64), (38.46, 75.95), (40.66, 73.25), (40.98, 56.31), (39.63, 63.38), (41.16, 80.43), (38.48, 63.75), (42.64, 76.08), (40.9, 88.81), (38.84, 84.88), (41.74, 84.11), (40.16, 80.29), (39.26, 69.84), (39.9, 77.84), (42.41, 46.3), (43.11, 97.44), (36.42, 64.3), (34.87, 63.05), (43.49, 39.99), (42.09, 74.68), (40.45, 58.85), (41.01, 61.74), (44.03, 61.91), (39.43, 73.19), (47.71, 64.76), (40.6, 64.38), (39.15, 83.92), (31.52, 68.69), (40.35, 68.58), (39.05, 84.43), (34.21, 73.66), (45.31, 49.46), (39.24, 67.81), (40.58, 83.1), (38.18, 82.96), (40.0, 83.65), (38.3, 85.46), (40.18, 69.18), (39.51, 69.19), (36.9, 76.57)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=347；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0239；
t = r·√((n−2)/(1−r²)) = 0.0239·√((347−2)/(1−0.0006)) = 0.4447；df=345；p = 0.6568。
95% CI（Fisher z）：tanh(z±17.487·1/√(n−3)) = [-0.7254, 0.7473]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=347；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0239；
t = r·√((n−2)/(1−r²)) = 0.0239·√((347−2)/(1−0.0006)) = 0.4447；df=345；p = 0.6568。
95% CI（Fisher z）：tanh(z±17.487·1/√(n−3)) = [-0.7254, 0.7473]。
③ 计算结果：
r = 0.0239，t = 0.4447，p = 0.6568，95%CI = [-0.7254, 0.7473]
④ 结论：在 α=0.05 水平下，p=0.6568 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=0.0239，方向正），应无相关。
案例 34. 出生月份与身高(cm)
Pearson相关应无相关
① 原始数据：
样本 n=510，逐条 (x, y)：[(5.98, 166.65), (9.06, 174.67), (2.01, 183.08), (7.21, 170.79), (8.85, 177.32), (5.65, 169.48), (5.2, 159.25), (3.24, 167.02), (4.01, 157.55), (5.32, 173.61), (10.85, 162.32), (8.65, 167.42), (3.71, 186.13), (4.36, 158.92), (4.7, 163.49), (2.45, 161.54), (10.0, 182.16), (0.36, 165.89), (3.96, 159.73), (2.67, 168.59), (5.84, 166.42), (5.49, 163.88), (4.32, 166.27), (2.93, 167.16), (3.32, 172.39), (5.66, 165.53), (4.81, 159.98), (5.64, 160.48), (8.74, 165.35), (6.41, 150.66), (10.23, 158.15), (10.04, 168.29), (5.5, 156.89), (1.68, 166.83), (2.38, 167.6), (5.12, 154.26), (4.65, 166.55), (7.09, 164.24), (4.91, 184.55), (9.61, 175.96), (5.51, 182.2), (8.75, 159.58), (1.07, 165.47), (-1.24, 171.14), (0.7, 166.82), (11.38, 166.79), (-0.07, 171.21), (4.73, 166.17), (1.66, 163.25), (6.97, 180.88), (9.6, 171.97), (9.7, 179.61), (3.98, 167.89), (7.44, 167.39), (0.24, 154.27), (7.89, 165.98), (5.4, 174.19), (6.2, 169.31), (3.27, 174.64), (8.87, 169.99), (1.94, 185.34), (9.11, 157.63), (0.16, 166.86), (5.95, 166.47), (7.97, 172.1), (9.4, 161.61), (9.64, 161.84), (6.81, 178.56), (8.35, 171.07), (6.92, 164.39), (6.02, 172.14), (10.16, 157.95), (11.93, 175.07), (0.24, 156.72), (3.05, 162.32), (8.27, 176.67), (5.12, 160.82), (6.0, 176.9), (7.54, 169.73), (9.75, 166.69), (6.0, 161.92), (6.78, 172.58), (8.87, 164.42), (6.48, 163.89), (1.95, 162.88), (8.96, 169.41), (9.68, 159.19), (5.77, 167.58), (3.82, 174.15), (4.47, 165.6), (8.26, 187.78), (11.09, 172.08), (5.97, 158.87), (7.35, 165.52), (2.7, 174.53), (0.02, 159.51), (6.82, 160.95), (6.56, 178.39), (0.46, 160.08), (10.32, 162.17), (6.36, 179.35), (5.53, 163.95), (4.3, 175.55), (7.02, 161.79), (6.7, 154.45), (8.05, 150.33), (6.58, 165.38), (6.38, 158.29), (9.04, 168.9), (4.18, 172.01), (6.16, 150.99), (5.19, 159.57), (10.26, 162.53), (5.26, 166.69), (4.53, 170.22), (7.0, 175.73), (7.96, 170.45), (5.26, 164.52), (3.16, 160.92), (2.77, 156.32), (5.17, 178.12), (6.14, 158.97), (4.77, 168.33), (9.38, 164.57), (7.81, 160.46), (5.04, 170.35), (6.37, 171.81), (2.23, 168.28), (10.67, 175.21), (9.46, 158.35), (10.13, 177.3), (9.63, 170.42), (5.01, 173.85), (5.39, 175.29), (4.6, 165.0), (3.07, 164.53), (4.62, 156.96), (6.49, 152.26), (2.99, 172.54), (-1.56, 158.37), (6.31, 163.61), (4.28, 154.6), (1.98, 169.81), (5.52, 165.56), (1.46, 154.92), (5.64, 157.96), (5.07, 171.51), (0.87, 178.53), (5.45, 166.36), (3.63, 184.69), (5.37, 171.72), (14.4, 158.36), (4.24, 159.56), (0.78, 165.21), (11.04, 173.12), (7.66, 160.8), (1.26, 167.97), (2.94, 170.93), (2.56, 168.21), (3.78, 162.11), (5.94, 173.0), (7.02, 166.68), (12.19, 181.58), (4.01, 182.69), (8.07, 165.79), (6.78, 163.74), (0.44, 163.18), (6.73, 166.23), (6.99, 172.07), (5.95, 160.53), (6.45, 174.47), (8.85, 174.88), (4.94, 152.42), (1.11, 176.13), (3.8, 168.37), (4.22, 173.23), (8.72, 165.54), (10.75, 173.2), (8.28, 160.17), (9.69, 172.73), (10.53, 167.68), (9.0, 165.55), (9.27, 157.53), (7.11, 165.77), (2.7, 171.56), (6.71, 165.46), (8.13, 171.43), (0.99, 173.74), (7.61, 173.29), (5.37, 160.36), (5.57, 185.17), (2.83, 172.06), (9.43, 160.3), (1.84, 169.64), (8.53, 170.63), (-0.81, 160.22), (6.1, 171.09), (7.77, 162.64), (9.72, 168.98), (8.4, 166.36), (7.8, 173.06), (7.91, 170.95), (6.89, 167.72), (6.78, 161.12), (6.19, 178.86), (5.88, 172.33), (-0.99, 161.13), (5.22, 173.24), (6.54, 167.47), (4.92, 162.35), (3.3, 167.95), (-2.44, 181.95), (4.61, 155.05), (12.45, 159.14), (1.55, 154.41), (5.78, 151.74), (6.62, 160.64), (5.0, 161.45), (7.57, 188.05), (4.25, 177.15), (4.23, 167.51), (3.73, 170.47), (7.56, 160.03), (3.69, 178.19), (4.13, 158.92), (8.82, 178.58), (4.4, 161.05), (3.66, 164.85), (3.82, 180.02), (4.22, 179.68), (4.78, 173.38), (3.63, 158.9), (5.3, 156.96), (7.37, 175.55), (1.48, 161.15), (9.07, 171.17), (6.24, 162.7), (7.85, 172.59), (9.11, 172.4), (4.45, 174.51), (0.61, 174.31), (4.89, 169.45), (1.08, 163.85), (8.56, 158.6), (5.87, 179.3), (8.96, 168.93), (5.8, 170.46), (9.76, 165.1), (2.37, 158.87), (8.13, 163.92), (5.57, 165.17), (1.89, 173.08), (5.82, 171.21), (1.56, 170.78), (3.58, 173.53), (13.42, 165.95), (3.61, 183.96), (3.67, 179.91), (6.47, 169.75), (7.68, 175.64), (5.66, 181.88), (6.4, 190.1), (6.11, 163.17), (3.5, 172.59), (3.86, 188.24), (7.64, 165.11), (9.66, 163.98), (11.05, 164.99), (9.07, 174.57), (7.46, 166.32), (6.3, 167.82), (4.9, 167.04), (3.05, 171.56), (1.85, 166.87), (-0.36, 167.1), (12.4, 163.43), (6.55, 171.17), (2.42, 163.83), (5.94, 171.59), (5.07, 172.92), (3.51, 162.76), (2.68, 171.67), (3.96, 158.74), (3.04, 164.87), (6.24, 173.62), (6.54, 164.5), (3.8, 181.23), (5.23, 176.24), (4.72, 163.3), (5.38, 180.59), (8.6, 167.33), (3.08, 158.03), (8.11, 167.77), (6.52, 167.95), (2.42, 159.63), (9.04, 165.27), (9.6, 168.81), (3.83, 159.36), (7.09, 164.46), (1.31, 174.92), (7.93, 169.11), (6.33, 158.04), (7.57, 162.64), (5.19, 176.0), (2.45, 147.13), (11.26, 157.15), (5.62, 161.11), (-1.8, 172.84), (3.3, 167.49), (4.94, 166.32), (2.71, 164.16), (2.79, 167.4), (5.16, 164.55), (6.32, 164.47), (5.75, 157.56), (5.51, 170.04), (9.66, 163.88), (6.32, 169.49), (9.92, 173.86), (6.26, 168.08), (8.47, 162.28), (7.62, 171.27), (5.22, 162.35), (6.45, 181.4), (11.24, 152.36), (4.63, 175.79), (2.77, 172.22), (8.29, 171.86), (2.9, 170.28), (5.34, 173.13), (6.03, 145.71), (8.87, 182.25), (5.17, 167.6), (8.39, 163.36), (8.03, 169.01), (4.3, 181.9), (9.53, 176.35), (5.4, 159.52), (3.45, 166.39), (11.78, 164.77), (0.42, 157.04), (6.37, 163.47), (2.98, 172.43), (6.77, 160.4), (5.85, 164.93), (5.7, 167.85), (8.0, 179.45), (4.85, 167.58), (4.66, 163.73), (6.54, 161.31), (6.94, 183.52), (7.01, 179.06), (5.49, 171.36), (5.33, 169.2), (6.71, 172.48), (7.7, 177.49), (2.0, 166.7), (3.51, 167.93), (1.33, 146.62), (5.53, 180.14), (3.49, 174.36), (1.76, 172.57), (7.1, 170.6), (8.85, 170.26), (8.32, 174.83), (3.27, 178.85), (9.0, 175.06), (1.1, 163.9), (6.32, 175.33), (5.67, 171.04), (10.86, 172.99), (11.5, 166.94), (4.52, 167.37), (6.11, 179.76), (5.75, 170.39), (4.85, 161.01), (8.31, 172.46), (5.6, 167.39), (6.14, 159.41), (7.57, 161.02), (3.71, 166.95), (1.03, 171.22), (1.19, 167.28), (0.52, 168.6), (3.76, 172.93), (2.8, 170.71), (3.21, 160.08), (5.75, 162.17), (7.01, 169.44), (5.26, 160.88), (5.69, 169.59), (4.12, 185.67), (5.51, 183.76), (5.8, 162.69), (2.85, 166.32), (5.34, 159.29), (2.05, 178.32), (3.54, 165.06), (8.07, 164.64), (-6.31, 164.21), (3.25, 165.99), (6.29, 175.02), (7.08, 164.74), (3.6, 167.04), (6.08, 184.91), (5.1, 175.53), (3.13, 165.91), (6.92, 175.33), (7.35, 168.72), (6.25, 177.29), (10.21, 157.84), (8.23, 176.05), (7.33, 153.9), (7.74, 163.7), (7.35, 163.57), (6.76, 176.6), (7.65, 161.84), (15.47, 174.73), (8.47, 163.39), (-2.2, 164.39), (5.45, 165.99), (4.37, 165.51), (4.4, 165.31), (7.21, 167.3), (3.85, 177.65), (8.88, 178.08), (12.2, 165.81), (8.76, 165.14), (3.99, 170.69), (6.15, 162.25), (4.47, 172.1), (10.79, 174.93), (8.17, 163.33), (4.51, 161.3), (8.23, 157.46), (8.05, 165.84), (6.37, 166.91), (4.5, 164.56), (0.93, 166.02), (9.1, 160.27), (3.18, 170.7), (7.06, 170.6), (5.51, 175.76), (3.42, 166.9), (8.74, 179.98), (2.49, 166.31), (11.25, 171.38), (5.27, 159.18), (3.75, 176.1), (6.17, 159.48), (-1.87, 165.91), (7.4, 159.56), (9.91, 180.95), (3.12, 154.55), (5.82, 171.27), (6.43, 152.54), (2.05, 174.49), (4.41, 171.4), (7.26, 173.16), (9.47, 165.96), (4.21, 179.8), (6.09, 147.35), (6.11, 166.42), (6.28, 167.42), (5.83, 166.8), (3.92, 169.36), (8.42, 166.8), (5.41, 163.05), (4.81, 172.81), (7.42, 170.42), (6.24, 174.51), (6.24, 181.34), (6.38, 175.94), (7.15, 159.51), (4.32, 174.35), (7.9, 163.86), (7.39, 167.96), (5.64, 168.91), (4.14, 156.15), (2.4, 159.58), (1.32, 156.53), (6.9, 160.71), (4.87, 164.78), (3.23, 146.78), (-0.1, 162.88), (3.68, 158.52), (4.76, 159.9), (7.15, 173.6), (9.45, 173.48), (4.25, 162.77), (6.15, 165.61), (6.58, 165.14), (7.01, 162.8), (7.84, 179.94), (10.36, 176.18), (8.0, 167.19), (7.5, 155.14), (8.55, 163.5), (4.57, 165.23), (8.36, 159.27), (2.72, 170.81), (2.39, 151.95), (8.69, 163.73), (8.04, 178.53), (3.65, 162.7), (6.45, 163.88), (10.99, 168.31), (8.77, 170.94), (4.83, 167.17), (8.28, 165.0)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=510；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0520；
t = r·√((n−2)/(1−r²)) = 0.0520·√((510−2)/(1−0.0027)) = 1.1733；df=508；p = 0.2412。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [-0.0352, 0.1384]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=510；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0520；
t = r·√((n−2)/(1−r²)) = 0.0520·√((510−2)/(1−0.0027)) = 1.1733；df=508；p = 0.2412。
95% CI（Fisher z）：tanh(z±1.965·1/√(n−3)) = [-0.0352, 0.1384]。
③ 计算结果：
r = 0.0520，t = 1.1733，p = 0.2412，95%CI = [-0.0352, 0.1384]
④ 结论：在 α=0.05 水平下，p=0.2412 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=0.0520，方向正），应无相关。
案例 35. 手机号尾号与消费金额(元)
Pearson相关应无相关
① 原始数据：
样本 n=657，逐条 (x, y)：[(5.07, 376.05), (8.3, 623.63), (8.26, 547.29), (5.92, 566.26), (5.14, 377.66), (4.49, 766.64), (7.26, 484.76), (7.67, 393.03), (8.04, 89.02), (4.15, 630.16), (0.92, 638.08), (1.93, 265.04), (3.28, 510.74), (5.82, 548.59), (1.78, 501.6), (7.95, 498.22), (8.05, 450.77), (9.02, 885.75), (6.72, 364.07), (9.97, 477.98), (4.41, 924.39), (-3.35, 596.82), (4.39, 536.06), (6.26, 575.77), (7.43, 781.26), (8.41, 183.27), (3.61, 424.38), (6.32, 497.97), (5.51, 365.46), (5.01, 500.28), (6.92, 488.95), (6.07, 191.87), (6.59, 272.97), (0.79, 516.95), (7.32, 563.12), (9.09, 503.47), (4.51, 489.0), (4.6, 476.34), (6.22, 345.36), (7.14, 744.1), (7.96, 544.07), (4.81, 288.53), (5.71, 754.28), (3.7, 720.75), (3.59, 828.39), (5.83, 489.84), (2.32, 723.41), (1.97, 479.24), (1.89, 433.47), (5.28, 715.62), (1.34, 831.01), (2.38, 343.31), (3.97, 403.98), (8.3, 550.76), (3.54, 382.81), (9.81, 604.97), (6.76, 534.54), (4.28, 865.71), (-0.38, 642.35), (4.96, 570.72), (-3.75, 798.85), (2.64, 717.63), (9.4, 835.58), (1.09, 469.23), (3.2, 252.74), (5.62, 542.39), (10.22, 581.17), (6.57, 224.02), (7.17, 545.19), (9.78, 518.71), (5.43, 564.4), (4.4, 626.56), (9.12, 713.31), (8.63, 454.72), (11.79, 584.99), (2.9, 492.25), (6.16, 459.02), (5.3, 240.2), (3.64, 164.96), (0.68, 272.58), (6.16, 436.48), (9.89, 727.08), (5.25, 682.84), (7.57, 329.36), (2.43, 533.56), (5.3, 708.07), (10.43, 505.95), (5.3, 539.83), (8.61, 287.63), (3.97, 748.49), (0.63, 511.66), (4.95, 622.76), (3.41, 401.62), (12.47, 696.87), (4.47, 529.7), (1.6, 521.65), (4.86, 736.6), (5.23, 611.81), (-1.87, 389.39), (0.54, 299.56), (2.19, 501.33), (6.72, 470.57), (4.56, 533.83), (3.06, 391.54), (6.92, 414.6), (4.26, 434.37), (7.56, 443.5), (6.52, 233.11), (0.62, 601.19), (7.64, 461.82), (5.9, 488.69), (6.76, 465.95), (7.6, 471.86), (-0.19, 677.08), (6.23, 852.1), (3.53, 785.78), (3.84, 650.45), (4.47, 474.93), (7.61, 443.23), (9.33, 238.8), (8.36, 531.78), (2.63, -36.04), (5.86, 644.07), (7.82, 637.72), (4.5, 391.01), (4.77, 487.11), (4.85, 407.03), (5.03, 611.69), (6.75, 297.05), (5.73, 321.87), (5.17, 582.43), (1.92, 867.74), (2.84, 925.1), (6.32, 721.8), (6.11, 366.7), (7.59, 332.91), (9.3, 703.18), (2.51, 789.62), (4.55, 329.77), (4.27, 772.47), (5.1, 509.73), (5.38, 452.8), (3.37, 227.26), (2.37, 537.06), (-3.03, 791.75), (8.76, 408.37), (3.15, 338.53), (0.59, 309.66), (4.44, 368.62), (-0.35, 764.06), (8.7, 459.68), (5.36, 614.28), (3.13, 411.79), (-2.35, 474.65), (4.14, 564.24), (9.77, 546.25), (10.56, 710.0), (7.29, 581.79), (9.45, 837.95), (3.98, 675.96), (4.78, 165.65), (12.04, 574.79), (10.43, 423.43), (1.92, 231.44), (6.2, 617.65), (4.46, 456.13), (2.51, 747.87), (4.52, 533.77), (3.16, 369.26), (7.98, 627.0), (3.49, 190.02), (4.62, 302.09), (4.86, 326.43), (5.35, 349.64), (5.28, 589.31), (5.84, 307.45), (1.32, 368.86), (4.56, 365.38), (2.78, 380.81), (3.63, 544.92), (3.12, 504.62), (4.33, 605.0), (5.98, 338.44), (7.36, 541.78), (2.22, 447.06), (2.67, 252.52), (5.24, 309.72), (6.11, 351.99), (7.12, 979.38), (2.89, 684.92), (7.73, 569.76), (8.85, 854.59), (5.94, 584.13), (0.78, 638.02), (5.29, 658.74), (7.27, 531.09), (-0.32, 499.58), (1.08, 623.52), (-0.59, 734.2), (7.82, 526.11), (1.84, 264.37), (-0.96, 357.18), (4.83, 675.76), (6.69, 693.96), (5.6, 744.08), (8.77, 540.35), (1.1, 149.71), (8.23, 715.85), (5.36, 307.35), (-2.4, 548.65), (7.84, 566.62), (4.34, 255.58), (11.5, 542.04), (3.26, 418.88), (7.65, 506.04), (0.83, 348.66), (8.84, 87.34), (6.29, 689.06), (1.32, 210.42), (7.18, 695.97), (-0.19, 318.48), (3.83, 149.24), (7.07, 1029.88), (6.38, 523.6), (9.83, 471.48), (5.15, 687.71), (6.58, 260.18), (12.39, 659.92), (3.06, 389.43), (6.09, 314.9), (3.35, 519.02), (4.05, 640.93), (4.1, 596.89), (6.69, 490.3), (6.2, 314.33), (8.06, 628.37), (6.48, 509.76), (0.39, 743.34), (3.22, 876.26), (3.54, 294.04), (4.36, 455.64), (-4.9, 618.02), (0.18, 235.29), (6.95, 590.19), (0.84, 251.22), (5.93, 279.33), (4.22, 402.65), (5.01, 315.19), (3.96, 147.75), (2.5, 288.47), (5.99, 927.38), (5.54, 595.68), (-1.76, 385.37), (6.73, 764.27), (4.08, 372.72), (2.3, 836.38), (5.31, 572.89), (4.76, 456.63), (5.44, 616.94), (-1.18, 398.27), (4.89, 919.49), (6.57, 77.2), (5.67, 468.62), (1.61, 456.53), (2.96, 832.46), (3.03, 578.91), (7.0, 370.85), (2.49, 495.28), (3.67, 81.33), (6.97, 780.19), (2.92, 638.7), (1.08, 530.24), (2.97, 452.9), (3.76, 640.36), (2.55, 442.98), (6.85, 609.74), (4.41, -28.02), (8.31, 495.83), (1.17, 414.48), (5.26, 358.05), (11.63, 452.09), (1.31, 494.17), (1.78, 400.79), (7.35, 585.11), (4.2, 578.13), (6.2, 377.78), (1.63, 328.35), (4.82, 500.82), (6.46, 635.15), (0.82, 324.62), (6.17, 604.7), (6.01, 398.28), (11.4, 440.85), (3.22, 275.98), (3.99, 444.14), (4.17, 527.22), (5.28, 643.51), (-2.21, 444.95), (6.09, 713.29), (5.55, 433.44), (7.05, 287.19), (-0.58, 488.96), (2.04, 600.16), (5.23, 290.9), (6.2, 835.39), (7.76, 495.08), (-1.65, 883.15), (0.62, 374.68), (7.49, 638.82), (7.47, 271.47), (4.01, 351.82), (7.72, 472.28), (8.81, 277.48), (-2.97, 793.89), (8.11, 776.39), (8.25, 211.34), (7.52, 674.61), (11.3, 531.01), (5.48, 680.81), (3.86, 170.64), (4.31, 630.65), (8.49, 559.83), (8.14, 610.79), (-0.26, 535.34), (7.7, 457.6), (7.03, 601.48), (5.53, 632.34), (2.28, 407.19), (2.67, 565.81), (15.27, 67.61), (5.44, 444.96), (4.26, 632.36), (4.07, 481.06), (1.95, 578.87), (6.36, 484.26), (4.41, 479.48), (5.02, 443.19), (2.23, 612.11), (3.46, 668.49), (3.4, 681.48), (4.09, 146.19), (2.94, 251.81), (3.18, 651.45), (0.75, 523.31), (4.18, 173.35), (5.1, 589.16), (2.37, 296.92), (7.86, 635.59), (5.17, 561.48), (3.92, 653.92), (5.7, 468.22), (7.14, 878.81), (4.66, 445.8), (3.75, 266.81), (5.07, 290.5), (6.86, 381.7), (6.4, 645.48), (3.68, 243.96), (5.2, 713.14), (6.37, 516.12), (6.44, 588.39), (10.72, 577.36), (3.99, 540.46), (2.64, 367.47), (5.7, 424.23), (3.79, 450.05), (1.05, 472.02), (1.55, 338.78), (0.69, 578.72), (0.02, 266.29), (5.25, 350.01), (4.79, 265.76), (-1.76, 518.14), (10.02, 444.06), (4.46, 542.48), (7.43, 417.84), (5.96, 533.18), (7.03, 699.9), (3.04, 569.34), (2.54, 671.79), (3.11, 437.91), (1.67, 719.75), (4.3, 570.7), (5.21, 699.15), (4.46, 361.33), (4.12, 409.78), (9.25, 562.92), (3.29, 634.72), (6.31, 417.85), (2.98, 480.03), (5.93, 645.16), (9.12, 392.26), (9.47, 732.95), (0.01, 288.7), (4.53, 557.51), (5.15, 33.0), (3.2, 552.14), (2.29, 193.24), (3.93, 504.24), (6.18, 486.61), (0.52, 299.77), (3.1, 546.6), (0.8, 577.22), (2.1, 375.65), (5.57, 323.75), (0.34, 757.03), (7.75, 858.98), (2.3, 449.4), (5.96, 328.94), (0.07, 699.02), (5.07, 227.52), (2.73, 628.58), (3.95, 616.75), (2.06, 738.33), (5.29, 447.46), (-1.38, 788.42), (3.21, 508.7), (1.42, 373.22), (8.35, 421.52), (4.92, 347.12), (8.13, 326.77), (6.53, 489.44), (4.89, 407.62), (9.21, 311.57), (6.91, 282.35), (-0.92, 567.13), (2.89, 370.16), (0.76, 220.26), (5.91, 172.25), (6.48, 509.45), (7.19, 586.77), (0.38, 309.38), (4.18, 720.46), (3.48, 457.77), (3.3, 650.43), (9.55, 347.44), (2.21, 477.38), (3.9, 658.94), (3.36, 597.78), (5.55, -133.41), (6.52, 684.5), (4.8, 135.55), (0.19, 624.16), (8.25, 519.33), (8.23, 390.0), (6.89, 587.48), (8.02, 696.86), (5.67, 734.24), (3.85, 652.92), (5.79, 417.22), (3.25, 298.83), (4.46, 580.69), (11.8, 474.55), (3.77, 818.18), (4.47, 895.4), (3.39, 564.38), (4.26, 534.23), (9.22, 669.72), (4.57, 456.99), (8.92, 239.25), (6.33, 439.69), (2.89, 622.82), (0.93, 1010.68), (6.07, 768.27), (1.61, 791.86), (5.98, 632.14), (7.4, 541.73), (5.44, 923.2), (5.8, 314.9), (8.51, 442.24), (7.64, 608.33), (1.01, 26.11), (8.95, 560.01), (4.91, 768.39), (1.98, 540.11), (7.89, 593.31), (4.25, 426.85), (2.41, 667.01), (7.1, 316.06), (6.67, 551.66), (8.05, 477.31), (2.09, 447.96), (-1.82, 275.02), (6.52, 453.55), (0.07, 608.96), (7.5, 182.97), (2.33, 569.18), (4.06, 439.92), (4.39, 431.61), (7.14, 672.79), (1.53, 477.57), (2.39, 490.53), (9.15, 284.57), (7.4, 920.58), (-1.4, 445.61), (1.99, 561.93), (3.47, 477.46), (8.34, 492.12), (0.08, 408.58), (3.97, 667.91), (9.08, 475.41), (7.04, 480.61), (9.27, 727.52), (3.68, 1004.75), (11.75, 559.35), (5.38, 715.7), (5.54, 493.81), (7.5, 609.12), (7.58, 677.57), (4.68, 648.77), (6.6, 647.6), (4.87, 833.82), (3.95, 676.93), (3.32, 169.27), (3.28, 646.07), (9.35, 432.39), (6.61, 465.29), (5.28, 589.26), (5.64, 798.33), (4.53, 397.08), (0.46, 458.92), (8.24, 428.81), (5.82, 401.74), (4.59, 724.94), (11.07, 408.55), (11.03, 533.19), (2.88, 301.92), (0.87, 912.41), (5.07, 675.72), (7.99, 311.97), (7.62, 478.62), (7.16, 528.42), (6.02, 636.92), (8.38, 319.53), (2.71, 256.33), (6.07, 479.21), (5.49, 469.88), (1.04, 781.45), (1.34, -7.43), (4.94, 511.39), (1.87, 915.76), (6.84, 544.04), (6.65, 401.86), (0.98, 475.85), (7.69, 668.36), (6.6, 350.9), (8.85, 635.63), (8.04, 305.56), (3.06, 512.72), (2.86, 281.36), (6.23, 262.11), (4.41, 472.57), (9.64, 565.65), (-0.82, 356.71), (3.93, 174.87), (6.33, 192.07), (7.52, 373.43), (3.38, 616.12), (7.67, 751.4), (7.56, 666.38), (7.62, 365.94), (3.17, 353.18), (6.69, 307.02), (5.21, 728.82), (2.32, 886.18), (9.27, 433.03), (2.62, 380.97), (-1.96, 371.16), (2.76, 377.4), (2.84, 248.72), (5.95, 179.6), (8.77, 504.89), (8.13, 792.48), (4.67, 380.13), (3.17, 271.53), (5.88, 475.05), (7.64, 557.23), (-2.03, 844.33), (5.75, 943.94), (0.34, 474.82), (-0.64, 296.66), (7.01, 607.01), (3.5, 527.16), (6.75, 439.92), (10.65, 553.98), (7.09, 352.54), (6.46, 369.61), (5.59, 417.42), (4.38, 306.87), (4.39, 408.08), (7.49, 510.16), (3.22, 535.46), (1.6, 256.5), (2.98, 438.01), (7.23, 450.85), (0.95, 449.4), (6.25, 493.95), (4.99, 664.75), (2.94, 196.6), (4.95, 462.54), (2.39, 459.09), (10.8, 388.26), (3.25, 197.4), (4.47, 520.97), (9.03, 444.02), (8.01, 484.47), (4.66, 321.21), (4.86, 515.01), (5.21, 648.25), (8.1, 385.81), (7.19, 374.83), (6.08, 282.5), (7.6, 920.23), (8.82, 734.26), (0.8, 271.9), (3.63, 346.01), (6.72, 546.47), (0.74, 461.93), (6.94, 389.34), (1.29, 468.94), (1.41, 317.77), (7.13, 697.5), (5.79, 535.26), (0.42, 714.79), (3.03, 381.19), (2.41, 290.26), (6.92, 226.1), (1.48, 732.44), (5.2, 907.62), (0.84, 330.08), (0.26, 595.67), (3.59, 381.63), (6.14, 368.36), (9.05, 450.27), (5.0, 777.39), (2.76, 688.29), (1.12, 477.87), (4.32, 588.35), (2.27, 555.5), (5.9, 383.92), (4.0, 633.21), (3.83, 914.94), (5.54, 259.0), (8.48, 972.66), (10.33, 547.89), (3.97, 663.16), (4.48, 372.07), (6.06, 876.98), (11.01, 693.85), (9.89, 441.67), (6.47, 593.17), (5.57, 848.32), (3.36, 345.3), (13.54, 653.33), (8.58, 824.06), (8.47, 518.29), (3.9, 88.81)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=657；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0396；
t = r·√((n−2)/(1−r²)) = 0.0396·√((657−2)/(1−0.0016)) = 1.0153；df=655；p = 0.3103。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.0371, 0.1159]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=657；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0396；
t = r·√((n−2)/(1−r²)) = 0.0396·√((657−2)/(1−0.0016)) = 1.0153；df=655；p = 0.3103。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.0371, 0.1159]。
③ 计算结果：
r = 0.0396，t = 1.0153，p = 0.3103，95%CI = [-0.0371, 0.1159]
④ 结论：在 α=0.05 水平下，p=0.3103 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=0.0396，方向正），应无相关。
案例 36. 幸运数字偏好与考试分数
Pearson相关应无相关
① 原始数据：
样本 n=597，逐条 (x, y)：[(2.42, 89.83), (2.36, 65.49), (1.33, 73.67), (1.58, 89.3), (5.05, 81.9), (-0.27, 49.95), (3.29, 47.85), (4.59, 93.73), (1.93, 57.74), (-2.43, 79.69), (0.64, 54.2), (5.46, 73.13), (2.51, 58.18), (1.44, 62.24), (4.37, 68.3), (4.43, 61.01), (3.71, 60.74), (1.82, 90.03), (-0.85, 66.57), (1.12, 84.26), (1.96, 74.42), (4.53, 99.58), (3.53, 99.84), (1.86, 85.19), (6.17, 73.4), (2.34, 64.75), (2.05, 99.94), (2.52, 74.75), (5.78, 52.52), (5.34, 71.4), (8.93, 81.53), (3.98, 78.34), (5.41, 65.28), (1.34, 61.51), (0.33, 73.67), (4.53, 66.5), (1.17, 72.53), (4.05, 59.47), (3.09, 67.07), (2.42, 76.25), (3.33, 83.38), (2.95, 87.85), (2.81, 49.0), (6.45, 76.69), (2.32, 71.27), (4.21, 67.31), (4.58, 89.06), (2.73, 67.3), (4.13, 72.61), (3.01, 62.96), (4.36, 84.31), (0.9, 58.25), (3.33, 69.88), (7.62, 87.03), (5.56, 72.96), (1.74, 83.45), (1.7, 87.87), (3.37, 89.54), (2.84, 77.45), (1.51, 61.57), (2.47, 79.72), (8.47, 59.05), (-0.04, 66.49), (3.56, 67.5), (-1.44, 74.21), (2.87, 58.59), (1.67, 79.04), (1.18, 75.5), (5.4, 73.76), (2.41, 84.79), (1.8, 49.64), (2.58, 83.01), (2.63, 70.5), (5.47, 79.9), (1.44, 87.38), (4.29, 60.04), (2.94, 76.31), (0.54, 62.45), (1.56, 83.58), (6.16, 61.0), (-0.08, 83.09), (1.3, 54.78), (2.66, 48.84), (4.84, 70.2), (3.38, 67.79), (8.84, 61.92), (1.0, 74.77), (1.95, 85.13), (0.43, 65.38), (0.01, 71.78), (1.33, 83.76), (5.24, 52.16), (0.49, 69.61), (5.19, 74.71), (4.65, 72.48), (3.28, 82.58), (3.31, 67.88), (3.77, 49.09), (-0.25, 75.64), (0.74, 70.11), (3.21, 85.12), (2.77, 62.69), (4.25, 63.96), (4.05, 64.78), (2.8, 87.29), (2.2, 72.87), (5.99, 46.26), (2.58, 70.18), (4.49, 88.04), (4.65, 63.75), (5.62, 63.91), (2.45, 71.16), (6.09, 74.12), (-0.06, 70.96), (0.14, 61.84), (2.3, 61.1), (3.82, 51.38), (3.2, 72.62), (2.2, 80.29), (1.1, 85.94), (-0.94, 73.01), (5.07, 51.56), (3.66, 56.43), (0.42, 72.6), (3.58, 81.0), (4.18, 51.21), (6.28, 65.92), (0.02, 63.29), (0.44, 61.22), (6.73, 85.06), (5.57, 70.87), (-0.71, 60.98), (5.43, 57.14), (5.05, 69.06), (4.59, 64.97), (1.38, 61.31), (6.24, 59.1), (2.93, 49.05), (-0.32, 65.55), (2.79, 66.66), (3.52, 60.76), (4.91, 87.44), (3.59, 55.0), (2.28, 66.89), (3.64, 69.52), (6.49, 74.88), (7.77, 86.09), (1.71, 70.89), (0.54, 81.73), (1.79, 66.02), (4.04, 85.27), (-0.45, 53.17), (0.72, 84.98), (2.4, 66.13), (1.29, 65.26), (4.65, 82.58), (4.6, 52.56), (0.83, 66.29), (3.08, 84.22), (6.81, 84.73), (2.61, 75.13), (4.0, 62.83), (7.5, 78.77), (5.33, 59.42), (3.37, 57.97), (1.94, 79.7), (4.91, 52.85), (4.15, 87.77), (4.64, 89.03), (-1.49, 76.4), (2.02, 58.4), (0.18, 66.39), (2.99, 62.8), (3.94, 77.43), (3.02, 87.38), (1.52, 80.22), (3.62, 62.01), (4.25, 80.06), (4.93, 72.55), (3.53, 65.26), (1.29, 71.79), (3.37, 52.38), (3.79, 78.85), (2.31, 79.86), (4.03, 70.8), (5.67, 68.34), (5.14, 68.86), (1.6, 51.08), (8.67, 65.52), (1.55, 63.31), (4.02, 77.9), (4.09, 73.82), (4.69, 69.51), (3.6, 75.48), (7.34, 61.52), (4.08, 55.52), (2.22, 64.59), (3.66, 50.65), (3.18, 59.83), (0.61, 72.14), (1.68, 52.78), (3.05, 90.83), (5.79, 69.8), (4.46, 92.36), (0.98, 67.25), (1.93, 84.22), (3.56, 79.42), (3.87, 82.6), (2.1, 82.53), (3.93, 91.13), (2.62, 82.57), (5.7, 62.19), (2.81, 83.37), (2.1, 77.23), (2.11, 64.62), (3.47, 66.94), (3.39, 57.41), (3.55, 74.8), (4.64, 56.34), (3.97, 73.82), (2.71, 99.33), (4.05, 71.86), (1.95, 65.09), (3.62, 63.05), (2.2, 54.94), (1.55, 71.81), (2.5, 78.21), (4.77, 65.66), (4.04, 72.67), (1.04, 88.22), (1.9, 58.46), (5.06, 63.96), (4.25, 81.76), (1.34, 70.16), (5.22, 56.7), (2.34, 73.79), (4.98, 68.96), (2.14, 74.23), (6.5, 69.9), (6.07, 99.6), (3.33, 73.31), (2.99, 84.21), (4.18, 61.9), (5.64, 97.25), (1.74, 79.55), (3.6, 75.1), (5.06, 62.3), (-0.82, 67.76), (5.02, 77.63), (1.48, 68.48), (2.38, 61.52), (5.03, 86.12), (3.65, 68.85), (2.5, 63.91), (4.36, 77.35), (2.13, 31.86), (4.48, 54.79), (6.37, 79.92), (3.86, 57.19), (0.38, 76.09), (-0.66, 72.55), (0.2, 63.24), (1.97, 66.26), (4.33, 42.3), (3.12, 84.69), (3.59, 41.44), (2.01, 65.12), (5.39, 70.96), (2.07, 98.48), (-0.83, 91.26), (4.14, 79.96), (1.89, 85.9), (1.79, 58.47), (4.76, 72.28), (7.79, 83.74), (3.06, 41.07), (4.84, 67.24), (4.5, 76.77), (3.04, 71.89), (1.88, 43.35), (4.24, 71.48), (1.04, 85.54), (7.05, 95.26), (5.16, 54.83), (0.74, 86.08), (-0.65, 68.92), (4.36, 69.12), (2.64, 84.54), (3.19, 46.96), (2.18, 67.81), (3.0, 79.13), (1.47, 76.56), (2.33, 69.45), (2.47, 85.39), (3.77, 52.67), (2.02, 66.0), (1.01, 85.34), (4.93, 74.73), (1.7, 82.07), (2.11, 81.02), (2.09, 89.41), (2.33, 58.36), (1.34, 76.09), (4.58, 76.56), (6.94, 61.08), (1.09, 58.91), (5.64, 78.88), (0.06, 80.32), (4.24, 80.72), (4.78, 88.35), (1.4, 68.24), (3.34, 72.29), (-0.82, 56.01), (3.38, 73.4), (0.64, 60.3), (3.15, 52.65), (3.67, 66.78), (3.86, 78.07), (2.77, 68.89), (1.5, 53.75), (4.7, 71.9), (4.08, 88.43), (0.47, 83.5), (1.56, 70.92), (3.63, 89.55), (2.73, 76.59), (0.57, 90.58), (3.96, 83.61), (1.31, 77.73), (-0.47, 56.84), (1.22, 77.77), (7.39, 88.08), (1.0, 89.55), (4.18, 75.19), (0.88, 74.53), (5.25, 56.62), (1.3, 73.21), (3.4, 73.65), (3.09, 62.87), (3.83, 90.52), (5.42, 54.39), (0.08, 55.24), (3.15, 91.14), (2.69, 85.67), (2.13, 68.51), (5.23, 68.86), (6.2, 77.45), (1.16, 74.21), (4.55, 66.2), (2.93, 60.31), (6.31, 72.77), (2.58, 71.17), (3.71, 58.96), (8.33, 64.91), (0.03, 91.16), (2.39, 80.08), (4.9, 86.25), (0.93, 72.87), (3.13, 82.56), (3.7, 58.19), (4.52, 65.99), (2.85, 61.9), (2.18, 84.24), (5.03, 66.13), (4.01, 68.13), (0.07, 66.87), (3.2, 76.96), (6.61, 75.55), (1.24, 76.7), (2.89, 57.46), (1.24, 74.87), (2.76, 63.09), (5.75, 69.5), (4.68, 55.72), (3.44, 49.51), (2.33, 67.19), (5.36, 52.66), (0.8, 75.22), (0.53, 50.23), (3.69, 83.37), (1.47, 69.66), (2.99, 80.3), (-0.09, 82.06), (2.3, 45.88), (2.78, 79.84), (3.55, 76.66), (1.35, 67.93), (2.65, 71.82), (2.57, 81.32), (4.47, 94.99), (3.99, 77.57), (5.61, 92.27), (5.67, 75.69), (1.6, 64.27), (2.21, 79.0), (1.61, 89.78), (3.29, 76.13), (3.96, 72.54), (3.18, 69.35), (2.67, 87.82), (1.83, 59.8), (3.24, 74.87), (0.72, 70.2), (4.16, 70.49), (2.04, 83.13), (6.49, 63.78), (0.23, 85.33), (1.59, 84.71), (2.71, 77.32), (0.73, 67.41), (2.45, 62.11), (0.81, 55.29), (3.18, 74.36), (2.65, 53.33), (1.31, 65.54), (4.64, 66.84), (-2.47, 49.95), (2.72, 62.4), (1.07, 74.14), (1.3, 65.29), (6.09, 53.38), (2.2, 82.58), (0.56, 70.06), (-0.51, 74.0), (2.33, 83.17), (3.33, 58.02), (3.21, 61.51), (-0.24, 75.6), (-0.43, 71.18), (-0.35, 73.86), (6.37, 53.15), (1.39, 87.59), (2.3, 73.73), (3.0, 68.41), (0.9, 74.36), (-0.06, 82.61), (1.98, 84.94), (3.34, 80.86), (1.42, 64.11), (4.02, 63.92), (5.22, 86.45), (1.56, 66.72), (2.58, 69.71), (4.4, 69.11), (1.23, 80.08), (4.56, 86.88), (4.38, 90.21), (3.35, 59.7), (5.18, 78.16), (4.03, 58.63), (7.43, 97.59), (4.85, 59.44), (-0.06, 68.74), (4.29, 61.97), (-0.42, 89.35), (4.27, 49.16), (2.82, 88.31), (4.18, 67.65), (3.37, 83.03), (-0.24, 92.38), (5.38, 46.17), (6.14, 45.52), (3.54, 72.73), (1.25, 68.87), (5.21, 65.92), (2.76, 74.07), (2.32, 90.54), (2.89, 57.42), (6.23, 101.77), (3.08, 75.94), (4.44, 75.7), (2.45, 66.06), (6.87, 62.07), (2.41, 80.04), (2.5, 86.1), (4.29, 86.65), (5.98, 49.69), (1.69, 51.74), (3.31, 67.24), (2.8, 94.63), (1.97, 96.1), (5.31, 50.05), (3.22, 80.56), (3.04, 70.98), (0.82, 81.73), (3.43, 42.6), (0.65, 60.12), (4.7, 69.96), (5.59, 91.66), (3.01, 62.59), (3.18, 64.78), (5.71, 69.05), (1.87, 82.22), (3.79, 50.0), (0.51, 66.15), (5.14, 67.05), (4.68, 82.2), (2.14, 86.34), (0.59, 62.69), (0.29, 60.21), (1.83, 71.45), (0.51, 94.92), (0.82, 76.68), (5.03, 72.79), (0.16, 71.15), (5.24, 72.98), (5.3, 66.86), (-1.0, 83.44), (2.19, 65.46), (3.06, 60.21), (1.77, 66.27), (-0.41, 68.98), (1.16, 52.87), (-2.03, 58.5), (3.86, 64.12), (4.23, 91.82), (3.03, 73.24), (4.7, 67.99), (4.01, 86.34), (0.62, 52.46), (4.03, 58.02), (2.61, 62.77), (5.61, 70.79), (5.68, 61.24), (-1.57, 76.17), (0.94, 68.56), (1.69, 77.93), (5.54, 88.53), (5.36, 79.43), (0.8, 63.48), (1.79, 78.49), (3.74, 95.65), (1.02, 59.45), (6.16, 63.22), (2.28, 58.06), (4.56, 72.59), (4.79, 77.86), (3.16, 70.66), (3.11, 61.2), (3.55, 96.06), (1.83, 72.38), (3.49, 57.95), (-0.33, 58.28), (1.96, 84.37), (1.7, 91.27), (2.06, 65.03), (4.31, 81.86), (-0.12, 82.55), (2.75, 71.71), (1.39, 58.35), (4.48, 83.05), (4.84, 69.13), (4.23, 63.9), (1.07, 75.18), (5.39, 80.87), (2.48, 62.43), (3.75, 65.07), (5.19, 67.07), (2.17, 78.3), (0.33, 62.79), (4.13, 76.26), (1.1, 76.14), (0.87, 73.51), (3.19, 64.23), (2.16, 63.25), (3.4, 65.81), (2.16, 67.71), (1.48, 40.68), (1.68, 70.57), (2.91, 58.84), (3.82, 74.42), (2.29, 89.87), (1.78, 70.14), (4.21, 94.81), (1.3, 71.5), (1.79, 75.31), (1.56, 92.97), (2.0, 66.75), (2.29, 60.39), (2.67, 66.06), (7.76, 56.78), (3.47, 86.39), (2.17, 73.58), (2.85, 57.68), (4.24, 83.17), (3.86, 65.91), (2.91, 79.44), (6.63, 74.72), (4.05, 57.99), (3.68, 57.16), (6.2, 67.33), (2.74, 65.09), (3.29, 79.66), (0.22, 71.83), (4.94, 68.7), (-1.28, 79.47), (4.04, 78.29)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=597；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.0054；
t = r·√((n−2)/(1−r²)) = 0.0054·√((597−2)/(1−0.0000)) = -0.1317；df=595；p = 0.8953。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.0858, 0.0750]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=597；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.0054；
t = r·√((n−2)/(1−r²)) = 0.0054·√((597−2)/(1−0.0000)) = -0.1317；df=595；p = 0.8953。
95% CI（Fisher z）：tanh(z±1.964·1/√(n−3)) = [-0.0858, 0.0750]。
③ 计算结果：
r = -0.0054，t = -0.1317，p = 0.8953，95%CI = [-0.0858, 0.0750]
④ 结论：在 α=0.05 水平下，p=0.8953 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=-0.0054，方向负），应无相关。
案例 37. 星座与月支出(元)
Pearson相关应无相关
① 原始数据：
样本 n=991，逐条 (x, y)：[(1.72, 7342.93), (13.83, 3672.09), (13.02, 1497.75), (7.36, 3850.77), (9.38, 4797.82), (10.32, 4405.65), (3.18, 4665.7), (12.4, 5389.02), (25.09, 3823.63), (14.91, 5646.0), (16.61, 3793.03), (10.76, 3432.66), (14.19, 2311.29), (6.36, 3970.69), (5.11, 5276.89), (4.08, 2624.26), (5.62, 2087.56), (6.16, 5168.51), (10.17, 4496.26), (23.17, 2417.47), (7.84, 4182.36), (10.38, 3479.74), (17.3, 5536.74), (11.43, 4941.08), (2.73, 4435.42), (23.08, 3348.93), (4.28, 3560.83), (9.15, 2605.52), (14.27, 3571.65), (18.57, 5373.3), (14.69, 5275.5), (7.87, 4804.89), (-7.6, 3000.3), (-2.13, 3905.58), (5.01, 5013.94), (5.37, 5176.12), (18.96, 3557.23), (14.76, 3457.94), (2.89, 3778.68), (11.75, 2868.65), (26.97, 4365.01), (6.16, 3030.12), (7.72, 5100.03), (20.57, 3563.58), (18.34, 4123.71), (11.62, 5832.2), (8.95, 4258.84), (21.65, 4211.42), (6.84, 5256.56), (17.79, 4676.35), (20.59, 5720.37), (11.19, 2635.89), (13.74, 5165.38), (21.22, 3325.92), (14.01, 3130.19), (10.51, 5387.51), (13.87, 3522.51), (-0.03, 2352.73), (20.53, 5343.99), (13.92, 3980.37), (16.47, 5013.55), (18.01, 4985.74), (12.1, 5327.53), (6.97, 4585.23), (11.54, 3817.92), (11.68, 3724.22), (21.99, 4352.2), (18.79, 5175.7), (14.87, 2926.73), (6.01, 2899.27), (15.22, 4705.76), (9.68, 4158.97), (20.47, 4217.67), (12.3, 2497.64), (9.92, 3958.34), (2.93, 3767.32), (15.52, 3073.94), (19.12, 3467.55), (12.74, 4402.67), (15.03, 3675.16), (7.26, 4320.17), (10.78, 3244.49), (19.4, 5215.21), (9.81, 2226.23), (10.38, 3763.2), (3.66, 4312.27), (13.78, 4087.42), (10.98, 3936.11), (10.96, 4500.52), (16.47, 3346.43), (16.48, 2581.65), (16.54, 4779.28), (15.69, 3728.62), (14.1, 4541.51), (11.37, 4661.74), (7.73, 836.38), (23.02, 6314.67), (19.08, 5093.55), (15.56, 4090.98), (24.03, 3743.72), (6.64, 3205.24), (10.88, 3998.23), (11.64, 2278.48), (5.5, 4417.37), (17.94, 3492.49), (11.63, 3395.7), (22.76, 5423.43), (11.67, 5058.58), (14.67, 3352.94), (17.5, 5062.93), (9.32, 4413.35), (-3.5, 5352.63), (-0.52, 2713.49), (9.29, 4440.63), (4.02, 2891.53), (18.53, 3120.37), (8.73, 3409.19), (11.88, 3517.43), (23.28, 4241.4), (8.9, 2842.96), (7.15, 5603.27), (13.3, 4188.16), (11.24, 4448.01), (13.7, 2657.2), (5.19, 3786.03), (23.41, 3689.98), (16.77, 3747.13), (19.14, 4230.58), (4.57, 4025.82), (5.63, 2935.04), (16.89, 5060.82), (16.13, 5392.92), (16.05, 3926.43), (14.33, 2537.16), (18.82, 2505.57), (7.75, 5539.65), (8.29, 4016.6), (25.04, 3563.66), (14.84, 5025.83), (10.86, 2976.42), (0.01, 3622.98), (15.25, 3823.12), (19.83, 4360.73), (16.5, 4423.46), (20.24, 2862.6), (16.64, 3867.14), (1.68, 3869.27), (3.1, 2974.26), (6.46, 3192.76), (22.38, 3340.42), (6.84, 4330.93), (9.64, 4862.53), (10.97, 4088.73), (14.62, 4017.12), (14.79, 4003.65), (10.69, 3663.94), (9.81, 3232.69), (22.54, 5210.19), (12.19, 4469.81), (4.82, 2221.28), (12.11, 4692.88), (15.7, 3010.86), (11.58, 4247.85), (10.85, 4553.59), (14.27, 2739.89), (6.4, 4120.52), (8.83, 3027.33), (16.11, 4900.77), (15.88, 4202.11), (11.04, 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3660.33), (15.03, 3724.97), (8.32, 6057.24), (2.36, 3186.75), (9.21, 1981.2), (13.24, 3302.65), (12.22, 3327.84), (10.08, 2004.77), (13.09, 3672.42), (14.16, 4256.29), (12.55, 3178.62), (4.65, 4992.36), (19.51, 2713.96), (9.24, 3326.84), (18.29, 4794.41), (12.48, 3513.05), (15.69, 4898.17), (9.89, 3706.88), (6.46, 3835.11), (14.38, 4186.9), (8.94, 5767.22), (20.33, 4431.91), (19.28, 4700.59), (7.82, 2673.1), (14.17, 3280.42), (7.89, 4897.93), (26.31, 4756.16), (5.56, 4041.04), (21.86, 4200.6), (8.49, 4527.64), (16.17, 3326.35), (6.29, 4311.51), (14.04, 6160.34), (13.33, 4334.33), (10.3, 4029.74), (16.22, 4403.41), (3.44, 4045.31), (14.54, 3921.01), (10.56, 3442.08), (14.08, 2874.21), (19.47, 4713.82), (5.43, 3482.03), (21.9, 4315.53), (23.69, 2990.4), (0.34, 5363.68), (15.53, 2308.94), (14.39, 3175.76), (1.33, 2244.55), (18.63, 5456.45), (16.64, 5117.51), (19.61, 4672.83), (5.22, 1600.31), (4.43, 3894.12), (29.69, 4871.32), (2.08, 3792.65), (22.38, 4453.87), (21.08, 4067.25), (6.21, 7578.09), (14.64, 6071.88), (21.96, 4781.49), (20.97, 4689.72), (17.95, 4022.69), (11.84, 5883.09), (10.88, 2341.42), (18.48, 4845.38), (9.32, 3851.98), (13.56, 3802.49), (13.77, 4828.88), (11.58, 5167.4), (17.61, 3422.42), (23.62, 4352.95), (15.15, 6099.93), (5.08, 4980.15), (14.94, 1071.52), (16.56, 3330.81), (4.45, 1517.54), (19.02, 4630.48), (9.38, 5106.17), (11.4, 3488.2), (9.62, 4983.66), (15.24, 3790.46), (14.61, 2458.45), (19.94, 4563.23), (9.03, 4461.43), (10.92, 3741.99), (9.84, 4285.05), (22.52, 5217.1), (11.34, 3147.0), (22.97, 3198.97), (12.34, 3024.27), (18.35, 3437.35), (12.65, 3235.98), (4.29, 3273.69), (13.03, 3556.28), (5.46, 3953.71), (7.48, 1798.39), (12.26, 3057.51), (16.36, 4638.46), (12.59, 4075.27), (13.52, 5594.81), (13.03, 4408.65), (13.22, 5672.56), (9.29, 4147.53), (10.53, 3355.55), (6.36, 3514.06), (20.71, 2970.38), (8.74, 3910.19), (19.34, 3456.13), (4.4, 4520.98), (13.11, 4914.37), (20.38, 3205.34), (23.29, 3468.99), (28.1, 4129.65), (4.78, 6542.25), (15.26, 4947.2), (13.91, 5756.23), (18.52, 2982.63), (8.44, 4656.22), (14.81, 4853.68), (16.56, 4422.42), (6.69, 2761.31), (8.02, 3040.4), (8.9, 3779.77), (17.75, 4394.83), (18.42, 4157.97), (11.37, 2634.34), (10.8, 3159.17), (23.64, 3192.3), (11.9, 4726.62), (7.67, 4576.58), (3.28, 3503.86), (6.0, 2224.56), (12.76, 3784.24), (22.95, 3948.62), (11.59, 3273.73), (18.83, 4815.42), (11.47, 4195.43), (19.12, 4686.24), (14.49, 2788.91), (4.45, 2450.28), (8.22, 2977.22), (6.68, 5318.14), (4.47, 4453.1), (17.67, 3149.38), (6.01, 3713.19), (14.83, 5168.34), (17.16, 4195.07), (9.63, 4944.57), (13.67, 4127.37), (11.89, 4920.94), (10.36, 4454.16), (18.25, 3727.41), (9.96, 4907.08), (14.29, 1520.67), (9.4, 2643.89), (13.25, 4931.01), (11.02, 2848.29), (4.61, 5673.64), (-4.89, 4069.16), (13.8, 7336.9), (18.04, 5112.76), (11.42, 3131.81), (5.22, 3278.75), (15.24, 3878.82), (17.7, 2795.06), (16.68, 3232.5), (9.95, 4019.6), (16.11, 3278.18), (5.32, 4338.33), (7.62, 2344.08), (13.41, 3614.69), (4.04, 4030.53), (13.04, 3261.09), (16.14, 2378.03), (14.28, 3191.09), (7.48, 5601.66), (9.99, 4371.06), (15.44, 3538.63), (5.73, 4928.82), (13.85, 4141.81), (8.77, 4702.2), (7.53, 5280.6), (12.79, 4973.2), (18.49, 2021.05), (3.62, 3583.91), (11.33, 4779.01), (5.53, 3758.64), (17.18, 5209.94), (20.73, 4212.23), (15.48, 3324.54), (10.07, 3099.75), (17.05, 3391.93), (13.16, 3714.9), (4.14, 6217.23), (15.06, 3933.89), (10.22, 2227.69), (16.41, 2958.02), (13.18, 2163.65), (6.56, 3592.78), (14.83, 3996.31), (20.56, 4345.88), (11.25, 6033.34), (6.62, 3638.93), (3.35, 3767.54), (4.2, 3719.86), (4.1, 3226.46), (16.66, 4989.01), (9.57, 3569.45), (12.89, 4398.35), (12.87, 3611.58), (19.39, 5416.1), (15.79, 4512.39), (8.24, 3449.68), (5.29, 4285.57), (16.31, 4594.66), (9.81, 4487.49), (16.89, 4159.77), (16.5, 2374.87), (13.55, 3848.52), (4.47, 5195.35), (11.87, 4059.18), (28.45, 4548.94), (16.85, 5336.58), (20.71, 4982.78), (4.9, 4831.8), (21.66, 3193.35), (16.89, 3611.15), (13.57, 1674.11), (16.5, 2803.82), (16.85, 4791.76), (18.54, 2014.47), (8.55, 2310.68), (1.89, 2569.55), (5.39, 4674.7), (15.25, 2789.64), (-1.44, 4593.4), (11.41, 5043.02), (19.3, 5098.05), (12.46, 2010.49), (19.22, 3478.13), (9.01, 2338.02), (21.84, 3798.47), (13.32, 4010.74), (8.83, 3440.27), (14.58, 5733.21), (17.53, 3626.92), (5.83, 4414.15), (9.73, 3052.77), (10.44, 4846.12), (16.66, 2470.95), (12.27, 3894.59)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=991；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0540；
t = r·√((n−2)/(1−r²)) = 0.0540·√((991−2)/(1−0.0029)) = 1.7010；df=989；p = 0.0893。
95% CI（Fisher z）：tanh(z±31.252·1/√(n−3)) = [-0.7353, 0.7812]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=991；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0540；
t = r·√((n−2)/(1−r²)) = 0.0540·√((991−2)/(1−0.0029)) = 1.7010；df=989；p = 0.0893。
95% CI（Fisher z）：tanh(z±31.252·1/√(n−3)) = [-0.7353, 0.7812]。
③ 计算结果：
r = 0.0540，t = 1.7010，p = 0.0893，95%CI = [-0.7353, 0.7812]
④ 结论：在 α=0.05 水平下，p=0.0893 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=0.0540，方向正），应无相关。
案例 38. 早餐吃否与智商(分)
Pearson相关应无相关
① 原始数据：
样本 n=920，逐条 (x, y)：[(1.65, 96.06), (1.27, 128.1), (1.12, 118.01), (0.72, 87.49), (0.24, 87.89), (1.41, 110.84), (0.55, 113.26), (0.36, 105.61), (1.58, 123.65), (1.21, 103.67), (1.72, 105.39), (0.59, 93.86), (1.03, 107.67), (1.28, 92.92), (0.83, 88.07), (0.21, 107.83), (0.96, 118.14), (0.74, 95.03), (0.76, 82.64), (1.76, 123.22), (1.15, 83.27), (0.84, 102.12), (1.81, 95.19), (1.02, 81.44), (0.78, 123.64), (0.45, 110.85), (1.18, 74.84), (0.19, 82.48), (1.1, 104.42), (0.25, 108.09), (0.42, 109.72), (0.47, 137.05), (0.81, 106.29), (0.35, 111.69), (1.09, 109.53), (0.2, 75.24), (0.65, 118.24), (1.32, 120.47), (1.02, 109.17), (1.04, 112.64), (0.98, 104.78), (2.04, 113.87), (1.3, 83.19), (0.77, 99.32), (1.53, 122.57), (1.22, 98.32), (1.11, 98.25), (0.41, 114.5), (1.07, 104.15), (1.14, 95.61), (1.23, 122.37), (0.23, 124.53), (1.38, 124.42), (1.38, 98.72), (1.31, 80.54), (1.25, 103.11), (0.39, 105.67), (1.78, 84.21), (0.53, 90.81), (1.12, 126.77), (0.27, 120.42), (0.73, 101.71), (0.72, 111.4), (0.62, 99.4), (1.29, 97.94), (0.76, 95.62), (1.33, 97.95), (0.4, 103.59), (1.33, 94.3), (0.71, 111.14), (0.98, 96.5), (0.67, 97.92), (0.94, 97.25), (1.43, 108.82), (1.11, 85.71), (1.53, 85.42), (1.46, 84.82), (0.9, 99.51), (0.04, 99.87), (0.74, 98.18), (0.41, 98.2), (1.55, 79.03), (1.63, 106.14), (0.8, 91.6), (0.9, 94.89), (0.96, 105.39), (1.25, 92.06), (1.51, 90.01), (2.04, 107.09), (0.97, 100.52), (1.35, 117.96), (1.28, 107.75), (1.56, 112.77), (1.78, 79.57), (1.15, 120.98), (0.53, 99.6), (1.33, 132.02), (0.59, 91.2), (1.01, 106.55), (0.69, 90.84), (0.9, 121.56), (1.17, 140.2), (0.01, 128.21), (0.69, 100.34), (0.74, 85.31), (1.21, 94.91), (0.78, 118.59), (2.04, 107.68), (1.32, 93.12), (1.48, 116.02), (0.99, 80.45), (1.42, 84.11), (0.98, 95.59), (1.74, 100.04), (0.89, 74.22), (1.15, 79.6), (1.49, 110.22), (-0.12, 80.28), (-0.15, 101.69), (0.9, 116.21), (0.72, 81.51), (1.06, 106.22), (1.84, 102.73), (1.28, 82.89), (0.89, 102.21), (0.68, 78.34), (2.19, 117.81), (1.44, 103.61), (1.07, 93.81), (1.08, 86.33), (0.53, 124.88), (1.28, 94.65), (0.41, 117.24), (0.77, 99.88), (-0.12, 87.28), (1.33, 99.96), (0.75, 93.05), (0.91, 108.38), (0.77, 105.02), (1.08, 92.11), (0.87, 107.8), (1.26, 113.57), (0.84, 76.05), (1.03, 70.78), (1.73, 78.87), (1.52, 97.53), (0.8, 91.69), (1.01, 100.56), (1.53, 103.66), (1.15, 129.54), (0.93, 94.41), (2.08, 90.32), (1.85, 91.04), (2.5, 76.81), (0.94, 88.4), (1.33, 109.55), (0.99, 86.55), (1.12, 94.02), (0.48, 90.08), (0.98, 102.79), (0.76, 110.86), (1.28, 93.51), (0.97, 96.64), (1.85, 111.72), (1.42, 77.83), (0.89, 101.7), (1.4, 88.74), (1.65, 112.17), (0.45, 101.66), (1.25, 81.76), (1.58, 81.31), (0.1, 80.03), (0.77, 115.48), (0.88, 102.18), (0.43, 119.97), (0.61, 120.34), (0.43, 114.86), (0.93, 122.33), (1.36, 107.4), (1.18, 142.01), (0.35, 86.86), (1.06, 99.76), (1.45, 112.61), (0.5, 100.04), (0.35, 95.11), (1.81, 86.61), (1.19, 120.05), (0.88, 111.68), (0.02, 76.97), (0.2, 116.63), (0.62, 103.3), (0.52, 87.4), (1.09, 91.13), (0.89, 105.18), (1.17, 81.34), (0.16, 99.04), (1.55, 101.23), (1.64, 120.5), (1.37, 99.98), (0.96, 106.25), (0.93, 114.27), (1.05, 91.15), (0.96, 115.02), (1.62, 109.5), (1.55, 86.17), (0.95, 101.12), (1.49, 86.67), (1.66, 87.24), (1.55, 104.14), (0.12, 101.81), (1.26, 119.09), (0.95, 94.05), (1.72, 118.43), (1.75, 97.08), (2.05, 81.48), (0.93, 101.12), (0.27, 100.89), (0.71, 122.35), (1.77, 95.36), (0.77, 96.17), (1.22, 91.22), (-0.11, 113.71), (1.09, 105.85), (1.43, 90.34), (1.07, 98.94), (1.77, 135.19), (0.67, 102.56), (1.43, 82.18), (0.65, 101.87), (0.81, 78.59), (0.57, 97.4), (1.1, 111.8), (1.5, 80.37), (0.49, 124.86), (0.49, 92.68), (0.73, 78.49), (1.2, 108.61), (1.02, 92.67), (0.81, 103.57), (0.31, 100.05), (0.75, 99.18), (0.82, 115.0), (0.1, 92.5), (1.49, 102.69), (1.8, 95.61), (1.72, 105.76), (1.78, 108.95), (1.83, 87.88), (1.02, 117.65), (1.38, 98.86), (0.9, 89.39), (1.84, 99.58), (1.24, 81.49), (1.8, 90.55), (0.84, 79.84), (0.04, 116.74), (1.0, 103.85), (0.99, 96.78), (1.77, 112.65), (0.27, 101.01), (0.56, 96.6), (0.87, 89.17), (0.36, 79.04), (1.25, 97.92), (1.45, 84.83), (1.45, 88.9), (1.35, 100.82), (0.99, 98.21), (1.42, 131.66), (1.14, 82.47), (0.07, 106.24), (1.58, 88.81), (1.7, 78.81), (2.48, 102.02), (1.35, 81.6), (0.02, 110.53), (-0.1, 92.34), (0.9, 110.64), (1.47, 120.17), (0.42, 92.77), (1.49, 104.15), (0.43, 89.76), (1.22, 141.48), (-0.01, 95.31), (0.96, 108.68), (0.67, 108.39), (1.18, 89.23), (1.32, 91.69), (0.72, 91.59), (0.8, 103.01), (0.39, 111.65), (1.66, 91.47), (0.95, 90.92), (0.73, 129.43), (0.94, 105.75), (0.93, 78.26), (1.64, 77.31), (1.7, 97.19), (1.21, 92.92), (0.61, 114.92), (0.64, 79.38), (0.9, 93.44), (1.32, 116.26), (0.8, 102.76), (0.16, 99.26), (0.67, 82.52), (-0.02, 80.6), (1.42, 100.16), (0.99, 100.6), (0.28, 108.81), (1.29, 119.96), (1.61, 94.55), (2.03, 105.01), (1.18, 89.34), (0.96, 85.63), (0.74, 110.54), (0.79, 88.04), (0.62, 98.48), (-0.19, 73.53), (1.22, 90.74), (-0.25, 103.25), (0.57, 114.44), (0.9, 91.11), (1.02, 99.11), (0.1, 98.16), (1.38, 135.51), (0.94, 92.35), (1.33, 104.6), (0.91, 81.18), (1.32, 88.43), (0.9, 114.8), (0.45, 91.35), (1.04, 81.82), (2.07, 110.12), (0.81, 113.58), (1.57, 93.49), (1.02, 112.44), (0.88, 91.17), (1.87, 81.15), (0.58, 70.18), (1.29, 126.06), (1.13, 87.18), (0.93, 113.29), (0.7, 94.25), (1.25, 98.24), (1.05, 100.95), (1.23, 87.72), (0.82, 94.37), (1.35, 90.49), (0.35, 114.29), (1.33, 82.24), (1.2, 106.56), (1.54, 106.68), (1.31, 102.05), (0.61, 102.29), (1.63, 106.51), (1.1, 133.2), (1.76, 116.87), (0.79, 91.93), (1.81, 107.02), (-0.15, 62.31), (0.58, 127.97), (0.98, 93.42), (1.22, 82.3), (1.03, 137.16), (1.91, 95.52), (1.8, 111.25), (1.97, 115.42), (0.87, 100.32), (0.89, 100.12), (0.69, 101.58), (1.11, 101.88), (0.62, 96.67), (0.46, 99.39), (0.99, 92.53), (0.25, 112.91), (1.84, 94.61), (2.12, 124.79), (1.18, 87.94), (0.36, 76.18), (0.2, 90.22), (0.66, 108.41), (0.44, 99.93), (1.06, 99.19), (0.9, 94.19), (1.0, 75.45), (0.76, 108.08), (0.98, 83.83), (0.89, 91.03), (0.9, 121.1), (1.19, 113.09), (1.82, 106.15), (1.76, 117.7), (1.63, 96.12), (1.19, 79.96), (1.29, 70.03), (1.84, 110.05), (-0.01, 65.78), (1.23, 107.47), (1.11, 84.56), (1.11, 100.83), (0.34, 101.5), (0.05, 83.79), (1.71, 79.24), (2.59, 113.35), (0.66, 93.57), (1.13, 61.63), (1.04, 103.31), (1.96, 111.7), (0.47, 96.44), (1.1, 119.41), (0.84, 119.85), (0.66, 93.77), (0.71, 95.61), (0.95, 106.87), (0.41, 128.96), (0.53, 113.64), (1.88, 88.12), (1.36, 142.05), (1.74, 115.59), (1.25, 92.6), (1.38, 106.93), (1.63, 92.8), (1.26, 84.8), (1.03, 81.41), (0.54, 93.58), (0.91, 85.4), (0.84, 98.91), (0.48, 101.92), (1.47, 98.12), (1.36, 102.59), (0.94, 74.4), (0.45, 110.82), (1.57, 108.03), (1.94, 132.85), (1.12, 103.18), (0.37, 134.33), (0.38, 102.85), (1.03, 100.93), (1.18, 99.65), (1.1, 96.63), (1.5, 107.08), (0.57, 92.43), (0.81, 101.29), (0.38, 97.49), (1.12, 127.9), (1.34, 70.04), (0.46, 92.71), (1.16, 84.6), (0.9, 109.72), (1.06, 120.98), (1.02, 90.03), (1.91, 88.61), (1.46, 99.58), (1.07, 73.08), (1.38, 70.66), (1.09, 82.45), (1.47, 87.27), (1.15, 77.95), (0.28, 90.14), (0.95, 79.61), (0.37, 79.38), (0.37, 109.89), (0.76, 122.1), (0.05, 115.85), (1.36, 117.47), (1.5, 108.69), (0.16, 83.58), (0.48, 110.39), (1.06, 104.24), (0.23, 111.01), (1.29, 99.1), (1.57, 100.31), (0.61, 99.97), (0.46, 81.04), (1.26, 94.76), (1.01, 112.41), (1.04, 118.11), (1.52, 103.73), (1.42, 124.3), (0.48, 90.24), (1.54, 67.36), (1.46, 133.62), (0.56, 106.14), (1.69, 71.64), (0.8, 98.21), (0.49, 87.28), (1.31, 76.54), (1.49, 67.4), (0.7, 90.21), (0.93, 101.89), (1.05, 86.4), (0.78, 93.77), (1.25, 116.88), (0.64, 126.71), (0.92, 79.41), (0.48, 106.63), (0.46, 103.28), (1.25, 71.02), (0.5, 113.37), (0.96, 95.32), (1.06, 110.16), (0.84, 110.05), (2.08, 105.39), (1.24, 82.58), (0.3, 79.15), (-0.11, 121.28), (0.76, 113.49), (0.92, 103.43), (0.8, 108.21), (0.35, 88.36), (1.01, 107.43), (1.23, 109.11), (1.77, 106.07), (1.18, 117.24), (1.58, 57.8), (1.46, 110.7), (1.27, 94.71), (0.75, 86.91), (0.54, 105.38), (1.23, 96.83), (0.24, 126.3), (1.11, 106.91), (1.2, 112.43), (1.75, 102.77), (0.99, 115.52), (1.5, 107.67), (0.28, 75.59), (1.03, 117.93), (1.18, 110.75), (1.76, 95.67), (0.72, 86.35), (1.7, 87.3), (0.66, 99.15), (0.97, 113.91), (1.32, 110.82), (0.79, 77.63), (1.25, 94.71), (0.09, 103.85), (1.13, 105.69), (1.16, 88.63), (0.69, 81.93), (0.57, 85.79), (0.58, 105.54), (1.36, 132.1), (-0.03, 112.0), (0.63, 119.71), (-0.43, 109.4), (0.65, 101.92), (1.18, 91.48), (1.71, 109.74), (-0.84, 94.21), (2.09, 123.55), (1.34, 86.18), (0.53, 89.35), (1.44, 90.96), (0.46, 81.82), (1.11, 92.7), (0.33, 102.9), (0.49, 102.53), (0.59, 91.95), (1.01, 96.36), (0.8, 96.63), (1.5, 104.41), (1.19, 124.71), (0.94, 102.31), (0.97, 100.16), (1.0, 101.25), (0.87, 106.01), (0.92, 108.55), (1.87, 109.12), (0.71, 91.06), (0.32, 132.85), (1.77, 114.79), (0.99, 132.7), (0.58, 103.05), (0.84, 96.16), (1.28, 80.42), (0.63, 99.81), (1.43, 108.18), (1.05, 96.05), (1.46, 97.33), (0.03, 121.47), (1.07, 113.09), (0.97, 93.51), (0.58, 97.47), (1.28, 100.85), (0.29, 108.32), (0.99, 109.32), (1.22, 103.48), (1.57, 103.91), (0.77, 102.05), (1.11, 99.56), (0.69, 91.01), (1.69, 98.18), (1.59, 86.19), (0.54, 90.86), (0.97, 87.77), (1.1, 101.47), (1.78, 87.17), (0.46, 113.25), (0.93, 118.93), (1.04, 94.47), (1.95, 72.09), (1.28, 90.76), (0.28, 103.03), (1.66, 115.43), (1.46, 99.81), (0.74, 75.84), (1.45, 76.48), (0.3, 87.97), (1.3, 121.03), (0.86, 93.86), (0.87, 104.65), (1.66, 93.75), (1.19, 109.1), (0.9, 127.88), (1.26, 115.29), (1.36, 88.41), (0.9, 109.58), (0.73, 102.11), (0.65, 96.0), (1.1, 111.52), (0.58, 106.78), (1.05, 93.07), (1.18, 86.58), (0.31, 73.86), (0.3, 106.44), (1.1, 106.61), (2.0, 92.38), (0.35, 109.0), (1.73, 111.54), (0.22, 95.31), (1.72, 90.52), (1.35, 139.61), (1.7, 68.05), (1.0, 99.15), (1.14, 97.82), (1.26, 103.85), (0.43, 105.68), (0.95, 105.79), (0.8, 91.08), (1.53, 104.51), (0.37, 99.46), (1.21, 102.85), (1.75, 101.43), (0.8, 76.89), (1.08, 78.2), (1.7, 117.51), (0.79, 102.35), (1.48, 115.61), (0.76, 114.37), (0.59, 89.16), (0.62, 95.19), (1.46, 112.75), (0.69, 81.75), (0.76, 110.07), (1.35, 102.55), (1.64, 79.18), (1.02, 85.85), (0.08, 89.55), (0.66, 100.1), (1.12, 92.09), (1.11, 115.77), (1.38, 101.36), (1.29, 66.2), (2.25, 114.23), (1.37, 115.34), (0.96, 103.82), (1.21, 116.11), (1.59, 103.82), (0.63, 93.19), (0.42, 146.54), (1.75, 97.61), (0.36, 94.16), (1.54, 111.67), (1.96, 111.83), (0.47, 93.88), (1.13, 113.0), (2.11, 90.18), (1.15, 73.04), (0.21, 84.74), (0.53, 113.19), (0.99, 102.78), (-0.1, 92.17), (1.45, 58.9), (0.83, 103.88), (0.74, 69.83), (0.27, 76.17), (1.55, 87.76), (0.5, 109.89), (1.1, 103.99), (1.27, 99.92), (1.53, 103.09), (1.27, 87.41), (0.68, 93.72), (1.19, 118.57), (0.4, 132.89), (0.82, 90.69), (0.57, 112.78), (0.4, 82.56), (1.07, 117.63), (1.91, 81.06), (0.79, 105.98), (0.12, 157.83), (0.47, 80.4), (1.0, 121.99), (0.75, 98.33), (0.76, 117.56), (1.27, 121.58), (1.12, 79.82), (0.54, 85.86), (0.68, 113.49), (0.4, 75.45), (0.06, 104.08), (1.52, 114.49), (0.85, 87.39), (0.8, 98.67), (0.93, 106.62), (0.78, 102.57), (1.35, 109.37), (1.11, 117.61), (1.3, 101.07), (0.94, 99.63), (0.9, 117.97), (1.6, 102.72), (0.14, 95.56), (0.48, 80.36), (1.4, 83.08), (1.3, 110.63), (0.18, 110.08), (0.52, 91.54), (0.94, 105.77), (1.64, 92.33), (0.68, 80.06), (0.67, 123.54), (0.81, 102.05), (-0.18, 93.13), (0.89, 71.69), (0.43, 106.2), (0.16, 101.3), (0.92, 105.63), (0.54, 93.99), (2.27, 93.93), (1.11, 108.93), (0.95, 110.08), (0.57, 113.62), (1.35, 88.03), (0.54, 97.18), (2.05, 93.32), (0.63, 92.88), (1.0, 113.4), (0.81, 73.4), (1.34, 102.41), (0.09, 96.28), (0.66, 70.88), (0.42, 108.04), (1.71, 77.85), (0.93, 89.22), (1.18, 97.51), (0.84, 98.61), (0.79, 99.75), (1.16, 111.43), (0.55, 97.35), (1.46, 106.09), (1.2, 93.94), (0.42, 62.67), (1.73, 104.22), (0.39, 95.3), (1.19, 104.11), (0.9, 117.24), (0.4, 117.31), (1.25, 78.92), (0.51, 88.42), (0.42, 91.38), (1.2, 110.18), (1.13, 114.66), (0.48, 119.13), (1.55, 96.03), (0.74, 108.28), (2.06, 128.46), (1.84, 106.54), (0.85, 87.94), (0.82, 85.77), (1.17, 97.2), (1.47, 104.87), (1.16, 114.29), (1.17, 89.68), (1.25, 99.22), (0.32, 86.85), (1.13, 93.75), (1.57, 77.57), (1.04, 119.2), (0.9, 92.8), (0.28, 97.98), (1.43, 100.94), (1.17, 100.94), (0.85, 113.69), (1.2, 92.51), (1.08, 98.53), (0.97, 120.51), (1.1, 83.57), (0.99, 113.27), (1.05, 116.78), (1.63, 93.36), (0.41, 81.05), (0.74, 71.0), (0.69, 104.34), (1.14, 104.41), (1.09, 112.52), (1.24, 87.09), (0.88, 100.94), (0.64, 94.86), (0.8, 97.39), (-0.09, 109.04), (1.79, 114.98), (0.74, 111.95), (0.57, 86.03), (0.65, 125.2), (1.38, 101.89), (1.73, 83.08), (1.77, 75.51), (0.45, 57.33), (-0.01, 94.45), (0.7, 117.0), (1.0, 99.76), (0.61, 86.33), (1.13, 104.16), (1.16, 111.03), (1.64, 103.14), (0.41, 117.23), (0.4, 104.68), (0.7, 97.7), (0.83, 104.95), (0.09, 102.43), (0.44, 114.93), (0.94, 109.1), (1.67, 108.6), (0.85, 106.26), (1.0, 113.04), (0.42, 106.52), (0.41, 79.58), (1.93, 92.84), (1.8, 128.14), (1.77, 120.25), (1.5, 101.04), (1.88, 108.07), (0.48, 121.79), (0.63, 125.42), (1.42, 101.1), (1.57, 99.72), (1.43, 98.34), (1.8, 129.24), (0.84, 104.7), (1.28, 92.52), (0.73, 89.96), (0.47, 87.21), (0.97, 111.37), (0.67, 107.46), (0.24, 84.07), (1.13, 122.95), (-0.37, 118.95), (0.92, 90.78), (0.71, 116.41), (0.79, 88.12), (0.91, 104.92), (1.63, 113.92), (0.81, 94.18), (0.89, 98.35), (1.08, 107.64), (1.2, 109.86), (1.42, 107.85), (0.31, 116.23), (1.4, 115.73), (0.4, 73.17), (0.55, 109.13), (1.55, 79.81), (1.0, 111.56), (1.3, 99.22), (0.1, 102.4), (1.29, 123.26), (2.02, 114.69), (0.46, 82.08), (0.13, 97.44), (1.49, 96.13), (0.48, 105.59), (1.23, 93.78), (-0.15, 70.05), (1.05, 102.38), (0.73, 93.4), (2.01, 98.31), (1.39, 113.49), (0.95, 74.01), (1.6, 97.2), (0.24, 97.4), (1.35, 99.02), (0.4, 107.47), (1.12, 93.57), (1.99, 98.89), (0.8, 78.05), (0.71, 101.01), (1.11, 90.29), (1.3, 98.43), (0.09, 92.1), (1.38, 89.18), (1.46, 127.19), (1.91, 82.06), (1.72, 59.21), (0.75, 117.94), (0.27, 127.57), (0.29, 113.85), (0.77, 106.41), (0.81, 108.24), (1.14, 81.27), (0.69, 120.49), (0.92, 129.05), (1.36, 65.39), (1.93, 67.1), (1.14, 90.7), (0.97, 89.85)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=920；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0129；
t = r·√((n−2)/(1−r²)) = 0.0129·√((920−2)/(1−0.0002)) = 0.3920；df=918；p = 0.6952。
95% CI（Fisher z）：tanh(z±25.453·1/√(n−3)) = [-0.6792, 0.6929]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=920；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0129；
t = r·√((n−2)/(1−r²)) = 0.0129·√((920−2)/(1−0.0002)) = 0.3920；df=918；p = 0.6952。
95% CI（Fisher z）：tanh(z±25.453·1/√(n−3)) = [-0.6792, 0.6929]。
③ 计算结果：
r = 0.0129，t = 0.3920，p = 0.6952，95%CI = [-0.6792, 0.6929]
④ 结论：在 α=0.05 水平下，p=0.6952 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=0.0129，方向正），应无相关。
案例 39. 头发颜色与跑步速度(秒)
Pearson相关应无相关
① 原始数据：
样本 n=448，逐条 (x, y)：[(2.31, 16.78), (4.67, 8.5), (2.57, 18.22), (3.07, 13.45), (3.43, 18.55), (2.82, 17.95), (3.46, 16.85), (1.93, 12.15), (4.95, 13.69), (2.6, 8.75), (1.77, 16.04), (1.96, 16.65), (3.58, 12.48), (4.63, 17.22), (6.29, 17.41), (3.73, 18.76), (-0.5, 16.04), (2.5, 10.84), (2.2, 14.58), (2.48, 18.01), (1.27, 11.01), (4.05, 18.97), (2.58, 16.84), (2.42, 15.75), (2.32, 12.1), (4.62, 15.58), (4.16, 12.75), (3.97, 8.37), (2.71, 14.44), (3.06, 20.16), (2.56, 14.02), (3.41, 18.48), (4.92, 17.91), (3.3, 15.53), (2.72, 12.75), (3.3, 12.84), (2.88, 13.22), (2.2, 21.06), (2.77, 12.75), (3.0, 14.74), (3.49, 15.12), (1.89, 12.11), (2.12, 10.92), (2.7, 16.68), (2.8, 14.8), (4.1, 12.12), (2.33, 15.0), (3.02, 13.91), (3.59, 9.43), (3.64, 11.79), (3.18, 17.07), (4.31, 14.41), (2.5, 13.07), (4.69, 15.34), (1.77, 14.72), (3.28, 10.78), (3.13, 19.52), (3.77, 11.57), (2.51, 13.03), (2.57, 11.65), (3.79, 17.08), (3.99, 14.85), (2.41, 17.5), (2.8, 16.22), (2.97, 22.5), (3.9, 14.18), (3.36, 20.1), (3.01, 16.31), (2.49, 13.31), (3.77, 16.13), (4.77, 12.82), (3.45, 14.04), (1.17, 16.15), (2.71, 16.97), (4.43, 13.83), (2.53, 18.32), (3.65, 16.6), (3.26, 10.9), (1.96, 12.87), (2.05, 13.75), (4.84, 17.84), (3.74, 14.46), (3.83, 17.69), (3.99, 8.4), (2.59, 12.43), (2.04, 15.29), (3.15, 19.05), (3.81, 18.0), (0.97, 13.4), (0.87, 19.04), (3.51, 12.8), (3.25, 18.75), (3.46, 13.12), (4.15, 17.3), (3.46, 19.33), (3.43, 17.43), (4.36, 10.4), (1.65, 19.56), (2.52, 15.61), (1.33, 11.05), (1.39, 16.6), (3.41, 19.97), (3.02, 11.37), (3.94, 17.79), (3.12, 12.52), (3.38, 15.83), (3.66, 12.34), (3.39, 16.86), (2.61, 12.09), (1.99, 11.87), (2.1, 6.82), (1.85, 13.2), (2.0, 19.55), (2.39, 13.58), (4.12, 17.66), (3.73, 11.4), (3.71, 16.1), (2.89, 14.96), (3.75, 14.61), (4.07, 7.74), (3.46, 20.6), (2.96, 16.7), (1.68, 9.77), (1.0, 16.52), (1.8, 17.28), (1.58, 9.55), (1.14, 15.09), (3.06, 13.69), (4.16, 21.02), (3.87, 17.75), (2.53, 19.48), (2.6, 20.12), (1.71, 9.76), (2.0, 14.24), (2.36, 13.56), (3.02, 20.35), (3.35, 11.23), (1.0, 21.89), (4.06, 15.15), (1.56, 15.02), (3.76, 13.14), (2.7, 14.57), (4.51, 15.89), (3.58, 15.76), (2.55, 14.54), (2.19, 15.89), (5.15, 14.04), (1.86, 15.89), (1.95, 15.64), (2.27, 19.27), (2.98, 18.34), (4.3, 12.52), (2.69, 17.32), (1.29, 18.28), (1.87, 14.11), (2.01, 11.76), (3.32, 10.84), (2.49, 9.64), (5.09, 17.67), (2.29, 10.98), (2.91, 17.21), (2.86, 16.01), (4.0, 11.73), (2.61, 12.99), (2.77, 17.14), (0.98, 16.68), (1.74, 11.6), (3.81, 13.76), (3.68, 13.67), (4.13, 12.66), (3.45, 10.08), (4.42, 15.35), (0.75, 15.5), (3.32, 15.85), (3.38, 6.89), (3.16, 13.28), (1.94, 12.75), (1.99, 19.93), (4.71, 9.09), (3.52, 19.07), (2.29, 17.32), (2.2, 10.82), (2.42, 18.67), (5.03, 16.71), (2.51, 15.12), (5.22, 17.25), (4.41, 14.88), (3.95, 11.69), (1.82, 14.97), (2.6, 14.79), (3.05, 17.14), (1.8, 17.07), (1.62, 18.03), (4.0, 14.15), (3.93, 14.28), (2.89, 17.52), (2.27, 14.1), (4.4, 15.78), (3.07, 17.14), (4.16, 10.09), (4.43, 20.96), (1.8, 15.36), (1.49, 7.99), (3.46, 18.46), (3.87, 12.49), (3.78, 14.2), (2.35, 10.27), (4.67, 12.36), (2.58, 11.75), (2.96, 19.99), (1.56, 14.92), (1.74, 13.1), (3.89, 12.99), (2.26, 12.9), (3.94, 15.93), (3.56, 12.9), (2.42, 16.1), (2.46, 16.75), (2.39, 14.81), (4.08, 14.99), (2.85, 15.84), (2.7, 20.18), (3.54, 13.25), (-0.29, 20.17), (3.98, 17.65), (3.19, 15.6), (2.12, 14.95), (2.92, 16.67), (1.85, 15.27), (2.94, 11.66), (4.02, 14.12), (4.37, 16.08), (2.46, 18.15), (2.92, 19.23), (1.94, 12.01), (2.82, 8.24), (0.74, 18.18), (3.42, 11.16), (3.16, 17.33), (5.2, 13.89), (3.25, 15.0), (2.05, 14.6), (2.69, 13.71), (3.77, 9.86), (1.98, 17.28), (2.08, 17.19), (3.12, 14.48), (2.79, 15.95), (2.05, 19.32), (1.43, 17.24), (4.33, 12.75), (3.97, 16.85), (4.03, 17.87), (3.63, 18.62), (2.6, 15.26), (2.77, 15.36), (4.42, 14.99), (4.35, 6.57), (2.28, 15.83), (3.21, 16.64), (6.1, 15.92), (3.32, 14.38), (5.17, 12.88), (3.49, 13.04), (2.22, 15.26), (2.45, 19.58), (1.88, 18.98), (3.73, 11.47), (2.38, 17.8), (1.36, 12.59), (0.75, 16.03), (4.5, 14.08), (4.04, 13.78), (2.22, 11.51), (1.24, 14.03), (2.2, 15.42), (3.69, 16.02), (2.59, 14.83), (3.14, 11.95), (3.02, 19.84), (2.2, 14.27), (4.67, 18.1), (3.13, 17.54), (1.78, 12.28), (3.38, 16.89), (3.75, 17.82), (2.62, 8.83), (2.58, 19.57), (3.18, 13.6), (3.13, 9.62), (4.8, 10.83), (5.31, 15.99), (4.49, 12.53), (2.31, 13.32), (3.62, 17.62), (3.07, 11.8), (3.12, 17.73), (2.63, 11.67), (1.03, 12.71), (3.63, 14.74), (1.23, 9.98), (2.5, 17.23), (2.9, 11.36), (2.78, 19.26), (2.42, 16.22), (3.21, 18.17), (4.25, 13.84), (2.38, 13.53), (3.39, 9.48), (4.2, 11.76), (3.85, 17.52), (3.97, 16.88), (3.96, 15.26), (2.08, 15.54), (4.21, 16.62), (4.18, 18.58), (2.75, 12.56), (2.5, 8.92), (3.39, 16.19), (3.3, 18.95), (3.17, 12.12), (4.64, 18.27), (4.09, 18.13), (2.92, 10.82), (1.77, 16.41), (2.51, 15.28), (3.35, 14.37), (5.32, 9.05), (3.3, 10.85), (2.24, 18.79), (2.64, 15.93), (3.09, 15.63), (2.43, 12.39), (3.93, 13.59), (5.73, 14.2), (4.95, 9.74), (3.93, 12.72), (3.54, 14.51), (2.59, 18.24), (4.15, 16.67), (4.12, 12.94), (2.67, 11.11), (2.52, 13.76), (2.28, 13.4), (3.54, 11.72), (2.74, 12.15), (4.75, 15.3), (3.49, 18.87), (3.49, 11.04), (4.24, 19.29), (3.58, 10.62), (4.66, 15.89), (1.6, 13.51), (2.37, 15.54), (4.14, 14.29), (3.03, 13.43), (4.14, 18.08), (4.38, 12.61), (0.15, 16.12), (4.35, 15.14), (1.58, 17.58), (1.77, 12.64), (1.39, 13.08), (1.53, 14.54), (2.45, 13.91), (3.03, 14.56), (2.36, 13.05), (1.64, 11.86), (3.81, 16.04), (2.3, 9.93), (5.03, 11.68), (2.6, 15.84), (2.4, 14.82), (3.29, 12.58), (3.82, 11.47), (2.75, 13.8), (2.25, 21.1), (2.45, 10.04), (2.49, 20.04), (2.98, 13.51), (4.68, 12.86), (1.74, 12.27), (2.47, 18.33), (3.18, 11.19), (2.49, 11.56), (2.26, 20.1), (2.65, 15.43), (2.4, 18.71), (2.8, 15.12), (2.11, 14.01), (2.99, 12.78), (2.73, 19.17), (3.13, 11.4), (3.07, 15.45), (3.39, 16.83), (2.39, 16.38), (3.62, 19.08), (2.38, 16.46), (3.87, 16.51), (3.49, 16.24), (3.05, 16.58), (5.0, 17.07), (4.03, 10.29), (2.57, 17.18), (1.7, 15.27), (4.32, 17.78), (2.44, 17.75), (3.07, 17.95), (4.1, 16.13), (4.46, 15.31), (3.89, 13.7), (3.31, 13.02), (4.7, 17.67), (3.99, 18.7), (-0.32, 21.31), (1.36, 14.76), (1.95, 14.88), (2.91, 17.67), (3.17, 16.6), (2.93, 13.11), (3.33, 14.28), (3.95, 18.25), (2.88, 11.41), (2.92, 19.42), (2.56, 17.81), (2.84, 12.87), (3.77, 16.58), (3.2, 14.32), (3.39, 14.98), (0.34, 14.56), (4.1, 17.89), (2.77, 14.33), (1.84, 15.29), (4.4, 16.23), (2.92, 11.06), (4.27, 14.88), (2.58, 17.91), (4.4, 14.39), (3.97, 19.98), (3.78, 7.8), (2.28, 11.84), (2.77, 17.0), (2.35, 14.02), (3.83, 13.52), (3.09, 19.39), (1.12, 11.84), (2.19, 13.78), (4.45, 13.02)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=448；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.0449；
t = r·√((n−2)/(1−r²)) = 0.0449·√((448−2)/(1−0.0020)) = -0.9488；df=446；p = 0.3432。
95% CI（Fisher z）：tanh(z±20.734·1/√(n−3)) = [-0.7730, 0.7343]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=448；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = -0.0449；
t = r·√((n−2)/(1−r²)) = 0.0449·√((448−2)/(1−0.0020)) = -0.9488；df=446；p = 0.3432。
95% CI（Fisher z）：tanh(z±20.734·1/√(n−3)) = [-0.7730, 0.7343]。
③ 计算结果：
r = -0.0449，t = -0.9488，p = 0.3432，95%CI = [-0.7730, 0.7343]
④ 结论：在 α=0.05 水平下，p=0.3432 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=-0.0449，方向负），应无相关。
案例 40. 姓氏笔画数与年收入(万)
Pearson相关应无相关
① 原始数据：
样本 n=311，逐条 (x, y)：[(26.86, 23.02), (12.56, 10.02), (12.56, 20.73), (5.15, 27.95), (14.18, 9.55), (34.95, 19.15), (10.94, 21.68), (19.14, 19.17), (11.57, 12.64), (2.4, 10.17), (-4.14, 27.8), (9.81, 11.17), (24.94, 27.56), (16.97, 20.92), (18.88, 19.32), (22.18, 5.4), (20.65, 10.45), (9.93, 10.25), (1.75, 19.51), (11.59, 28.1), (16.31, 15.72), (10.39, 24.61), (26.24, 5.49), (15.85, 38.03), (18.74, 11.24), (9.14, 22.84), (23.94, 16.36), (30.62, 22.78), (15.75, 23.25), (10.44, 25.42), (17.31, 14.09), (7.86, 21.03), (7.42, 10.41), (29.6, 16.67), (4.69, 14.53), (12.12, 22.07), (9.51, 30.77), (4.09, 17.33), (16.69, 25.84), (24.7, 30.09), (25.2, 11.88), (14.44, 15.85), (10.75, 11.04), (12.41, 6.65), (25.54, 20.04), (32.27, 26.12), (19.24, 14.44), (18.83, 27.46), (11.04, 21.81), (22.19, 11.54), (20.15, 30.78), (16.0, 30.78), (5.18, 24.25), (14.43, 14.58), (5.74, 13.65), (17.33, 19.86), (1.17, 19.94), (10.89, 10.02), (15.84, 18.03), (13.74, 17.61), (20.57, 39.76), (9.06, 27.42), (14.45, 23.2), (27.76, 16.7), (7.88, 28.07), (5.58, 26.17), (17.56, 15.72), (27.16, 16.56), (11.11, 17.36), (7.99, 31.84), (5.07, 35.14), (16.64, 28.79), (17.64, 16.8), (21.62, 26.96), (6.46, 21.88), (9.08, 30.89), (9.86, 21.65), (9.96, 22.26), (22.04, 23.14), (10.74, 4.28), (13.76, 23.78), (21.7, 15.13), (24.51, 14.47), (7.47, 27.1), (11.07, 4.13), (22.68, 19.48), (12.33, 6.09), (14.29, 18.56), (11.08, 22.99), (7.6, 19.86), (15.38, 13.63), (6.51, 25.6), (8.03, 9.99), (11.32, 18.58), (25.49, 22.35), (3.59, 13.8), (0.99, 12.95), (16.88, 13.66), (8.89, 12.07), (19.59, 9.03), (23.62, 27.14), (18.9, 4.85), (4.32, 18.56), (30.62, 20.82), (14.31, 21.5), (16.31, 22.16), (4.03, 35.05), (10.04, 5.65), (7.27, 1.34), (12.85, 15.44), (7.38, 26.22), (7.51, 10.71), (15.98, 32.43), (25.79, 33.64), (21.88, 21.63), (14.43, 33.62), (13.99, 11.25), (21.93, 17.53), (8.8, 6.58), (15.66, 20.2), (3.66, 19.93), (18.02, 16.99), (11.53, 6.35), (4.17, 19.87), (19.22, 15.8), (20.08, 25.91), (21.88, 16.43), (10.13, 31.38), (1.26, -2.1), (15.38, 23.35), (1.41, 7.77), (26.19, 21.9), (21.32, 15.89), (24.09, 22.03), (4.65, 9.11), (9.87, 20.07), (22.37, 20.34), (15.43, 19.48), (2.47, 31.88), (7.8, 30.37), (26.12, 24.31), (3.97, 16.82), (10.99, 39.08), (23.64, 17.01), (7.6, 14.87), (20.19, 25.13), (25.41, 10.34), (16.22, 20.14), (15.62, 7.65), (10.42, 28.08), (7.99, 19.09), (29.65, 7.67), (19.51, 10.66), (6.96, 12.2), (23.15, -2.12), (20.25, 31.83), (5.19, 32.25), (20.73, 42.42), (17.62, 33.63), (19.72, 20.03), (8.55, 11.95), (23.09, 33.35), (14.88, 24.05), (15.79, 17.32), (9.54, 7.18), (16.97, 14.09), (28.04, 33.44), (18.87, 24.71), (17.31, 34.35), (7.91, 25.0), (18.44, 28.01), (12.92, 17.51), (18.72, 18.74), (19.2, 20.51), (13.61, 8.04), (24.23, 25.85), (10.33, 10.72), (14.16, 32.61), (25.55, 31.73), (13.43, 0.58), (21.8, 27.04), (28.12, 24.55), (16.58, 18.56), (40.15, 19.71), (2.87, 17.36), (28.83, 28.42), (5.66, 15.26), (24.52, 24.65), (12.54, 17.66), (15.58, 27.98), (-1.95, 18.83), (13.73, 28.24), (5.28, 22.18), (-3.43, 21.99), (8.76, 25.59), (19.36, 6.5), (10.78, 15.65), (19.77, 8.97), (5.22, 39.92), (15.19, 20.35), (9.63, 21.68), (9.7, 5.19), (18.41, 8.2), (14.89, 29.03), (12.89, 20.28), (27.5, 15.75), (9.64, 23.83), (13.25, 10.22), (11.68, 27.18), (11.87, 17.21), (17.24, 16.0), (36.72, 26.88), (23.44, 15.17), (27.82, 20.54), (15.11, 19.18), (23.3, 21.82), (6.17, 26.35), (20.31, 26.06), (24.77, 19.03), (11.6, 25.72), (8.45, 25.21), (2.72, 13.62), (22.82, 24.28), (15.47, 22.61), (16.75, 19.01), (19.83, 18.83), (22.17, 14.91), (23.15, 20.27), (18.3, 15.34), (20.12, 21.47), (20.79, 27.99), (18.29, 15.97), (8.98, 31.41), (17.14, 9.2), (8.29, 24.98), (12.35, 17.9), (14.98, 14.03), (7.64, 30.19), (23.75, 17.3), (24.03, 25.34), (10.55, 6.26), (15.54, 22.98), (26.13, 24.42), (5.69, 14.72), (7.32, 24.02), (17.53, 24.59), (5.06, 23.01), (19.5, 23.64), (8.5, 29.66), (14.6, 17.67), (16.48, 19.66), (11.74, 21.9), (13.22, 16.83), (15.54, 38.76), (23.36, 19.47), (14.4, 28.09), (20.15, 36.12), (-0.62, 27.57), (2.26, 15.99), (15.02, 10.69), (12.24, 6.2), (12.49, 31.4), (18.98, 13.49), (8.98, 21.37), (25.83, 22.19), (23.27, 19.07), (13.95, 32.06), (-0.95, 17.35), (26.83, 8.34), (9.62, 24.86), (17.33, 21.33), (14.48, 32.81), (11.41, 24.57), (19.54, 27.85), (20.31, 14.79), (16.02, 12.56), (4.93, 9.73), (6.39, 21.94), (9.16, 28.32), (13.72, 12.67), (7.59, 17.62), (10.42, 1.57), (8.79, 33.67), (21.97, 25.38), (18.6, 14.6), (29.83, 28.24), (14.81, 17.48), (12.25, 21.32), (22.01, 17.7), (9.6, 21.81), (11.52, 38.2), (18.41, 10.37), (24.15, 28.81), (26.32, 14.03), (5.57, 16.74), (22.7, 16.95), (-0.78, 36.86), (12.55, 19.8), (22.03, 8.16), (9.98, 20.09), (14.42, 25.09), (16.75, 20.64), (26.36, 18.94), (21.32, 27.38), (14.64, 28.66), (13.28, 25.36), (-2.63, 17.15), (16.7, 11.42), (16.97, 23.94), (2.77, 18.32), (28.16, 26.53)]
② 分析过程：
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=311；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0323；
t = r·√((n−2)/(1−r²)) = 0.0323·√((311−2)/(1−0.0010)) = 0.5686；df=309；p = 0.5700。
95% CI（Fisher z）：tanh(z±17.578·1/√(n−3)) = [-0.7484, 0.7755]。
H₀: 总体相关系数 ρ=0（无线性相关）；H₁: ρ≠0。
n=311；r = [nΣxy−ΣxΣy]/√[(nΣx²−(Σx)²)(nΣy²−(Σy)²)] = 0.0323；
t = r·√((n−2)/(1−r²)) = 0.0323·√((311−2)/(1−0.0010)) = 0.5686；df=309；p = 0.5700。
95% CI（Fisher z）：tanh(z±17.578·1/√(n−3)) = [-0.7484, 0.7755]。
③ 计算结果：
r = 0.0323，t = 0.5686，p = 0.5700，95%CI = [-0.7484, 0.7755]
④ 结论：在 α=0.05 水平下，p=0.5700 ≥ 0.05，不拒绝 H₀。两变量未检测到线性相关（r=0.0323，方向正），应无相关。
第2类：简单线性回归（案例 41–75）
案例 41. 广告投入预测产品销量
简单线性回归投入越高销量越高
① 原始数据：
样本 n=783，逐条 (x, y)：[(42.22, -8.48), (56.88, 29.65), (59.5, 24.97), (55.11, 7.61), (52.65, 18.44), (60.18, 19.38), (45.81, -19.63), (50.38, 28.73), (25.61, -60.35), (59.9, 15.62), (52.33, 4.73), (60.76, 40.42), (53.31, 27.16), (49.29, -2.93), (22.44, -81.61), (46.21, -9.78), (71.74, 58.0), (40.72, -40.05), (35.19, -53.65), (64.17, 20.96), (56.44, 29.4), (63.48, 29.63), (48.64, -20.65), (37.98, -63.84), (29.49, -51.58), (28.98, -52.2), (45.06, -14.75), (61.46, 75.91), (45.7, -7.33), (101.48, 194.83), (58.14, 38.84), (55.7, 8.81), (41.23, -23.77), (73.68, 69.22), (51.7, -27.52), (55.51, 14.56), (50.81, 1.45), (38.11, -20.02), (43.36, -48.7), (55.48, -0.21), (59.87, 38.84), (68.64, 78.61), (69.6, 26.49), (57.85, 41.05), (33.49, -49.34), (46.12, -31.31), (64.5, 36.56), (52.32, 15.23), (45.74, 1.53), (63.87, 50.61), (50.87, 3.31), (64.7, 43.96), (49.58, 28.35), (47.59, -4.61), (51.75, 10.55), (59.99, 39.12), (63.73, 64.68), (38.39, -20.2), (52.19, 0.61), (58.11, 7.89), (67.12, 31.61), (65.41, 67.39), (53.42, 36.2), (63.28, 25.7), (17.34, -50.19), (92.65, 101.4), (41.51, -2.26), (40.64, -13.07), (74.96, 73.97), (70.69, 50.34), (49.93, 20.35), (47.92, 2.84), (54.17, 33.05), (44.86, -9.21), (55.9, 22.09), (59.5, 81.4), (57.41, 46.92), (48.42, -23.56), (17.6, -55.24), (59.25, 68.78), (58.98, 35.88), (22.03, -45.9), (53.02, 30.75), (61.11, 58.98), (54.97, 29.72), (56.85, 10.27), (79.47, 80.06), (30.8, -60.69), (43.4, -29.57), (60.54, 16.31), (58.14, 10.04), (40.85, -4.34), (47.24, -30.68), (51.8, 32.66), (36.38, 3.71), (62.39, 59.63), (28.08, -100.45), (18.67, -63.43), (45.22, -22.26), (37.93, -13.3), (53.32, 37.8), (40.43, -57.41), (41.41, -32.25), (39.72, -4.08), (72.85, 77.46), (32.6, -44.56), (43.72, -8.86), (74.02, 85.87), (63.6, 76.33), (69.97, 57.56), (40.02, -32.29), (53.3, -33.94), (28.94, -39.89), (56.34, -5.1), (63.6, 41.17), (9.74, -131.56), (30.84, -27.07), (35.25, -59.25), (51.04, 18.23), (53.47, 1.48), (41.01, -64.97), (41.61, -34.59), (67.08, 56.98), (24.94, -77.9), (81.05, 88.56), (49.87, 36.09), (68.37, 76.14), (65.84, 54.45), (74.69, 81.46), (25.57, -64.96), (-7.8, -147.02), (20.14, -84.74), (57.57, 35.26), (15.81, -102.57), (47.27, -24.83), (56.12, 5.15), (60.81, 11.19), (46.73, 24.44), (65.29, 57.04), (44.8, -23.3), (70.2, 63.36), (60.89, 67.12), (43.79, -5.18), (41.3, -40.88), (61.02, 30.47), (46.75, -5.87), (34.39, -48.51), (33.72, -43.53), (27.93, -86.35), (46.97, -17.27), (36.58, -71.77), (51.38, 10.7), (53.42, 18.25), (23.59, -85.19), (84.53, 112.0), (70.6, 79.45), (14.33, -110.09), (50.3, 18.34), (12.64, -101.14), (39.94, -45.87), (67.43, 67.45), (64.71, 53.78), (43.3, -50.67), (44.26, -5.91), (28.44, -63.61), (40.25, -7.41), (39.54, -35.77), (41.52, 5.67), (72.51, 74.97), (66.29, 79.87), (62.38, 70.74), (37.36, 0.06), (53.53, 29.07), (51.4, -4.02), (56.33, 7.48), (49.02, 22.33), (52.39, 20.31), (70.97, 83.15), (46.39, -22.61), (28.18, -61.07), (72.39, 50.81), (34.89, -44.12), (54.52, -1.13), (49.78, -24.69), (63.66, 65.65), (30.34, -35.28), (48.57, -47.92), (23.19, -68.75), (56.69, 36.57), (47.42, 8.08), (34.14, -41.55), (67.48, 50.06), (45.7, 2.53), (43.53, 0.66), (78.35, 114.7), (47.23, -27.51), (46.74, 0.22), (41.62, -16.48), (65.24, 52.47), (30.54, -6.72), (19.12, -103.26), (58.82, 5.98), (48.31, 0.92), (39.41, -44.09), (51.25, 9.79), (52.88, 12.85), (49.47, -19.18), (38.81, -56.2), (63.61, 36.59), (60.99, -20.95), (72.62, 75.27), (51.56, 18.2), (74.25, 50.23), (69.4, 48.4), (21.13, -71.32), (36.35, -53.64), (27.7, -67.59), (49.67, -24.93), (74.6, 122.81), (48.37, -29.97), (37.54, -40.62), (50.38, -36.36), (64.17, 64.05), (41.09, -7.49), (65.87, 22.17), (48.56, 13.7), (67.69, 45.59), (84.74, 126.98), (48.77, -9.06), (63.11, 49.53), (65.36, 91.94), (36.87, -24.2), (83.14, 99.78), (64.34, 32.65), (54.64, 42.39), (60.71, 27.38), (49.74, 43.96), (66.82, 101.9), (55.87, 7.76), (46.45, -3.36), (57.91, 7.26), (61.32, 34.22), (71.18, 56.11), (40.67, -3.02), (46.97, 6.09), (35.76, -30.27), (49.05, -1.44), (26.16, -37.34), (30.56, -67.01), (30.06, -43.41), (76.17, 84.05), (35.23, -8.34), (42.6, -15.57), (39.54, -11.18), (37.89, -27.92), (67.72, 42.03), (54.24, 1.82), (64.0, 81.99), (50.73, 2.57), (10.0, -93.71), (55.33, 11.99), (53.76, 3.31), (47.32, -18.86), (65.41, 18.64), (50.2, -0.46), (57.78, 21.73), (49.09, 34.86), (39.13, -38.6), (42.81, 4.6), (21.64, -92.45), (10.53, -122.67), (41.84, 0.38), (23.38, -102.07), (48.28, 28.23), (38.79, -27.08), (64.43, 29.45), (39.08, -37.4), (52.59, 22.22), (44.0, -62.46), (65.25, 64.63), (48.09, 25.62), (78.93, 110.55), (29.42, -44.6), (58.8, 54.66), (51.19, -4.63), (31.79, -44.38), (40.95, -4.74), (49.96, -37.77), (35.13, -53.48), (34.01, -44.37), (54.16, 30.77), (43.88, -69.58), (37.71, -46.14), (50.62, -19.02), (60.24, 59.85), (58.11, 6.72), (58.78, 35.71), (35.6, -28.91), (61.73, 47.06), (55.08, -5.38), (46.94, -30.68), (66.84, 33.98), (58.25, 3.58), (61.31, 33.42), (45.27, 12.89), (59.59, 59.4), (21.3, -49.77), (38.32, -41.73), (62.52, 44.96), (48.68, -16.9), (65.07, 24.11), (19.72, -111.32), (31.15, -56.86), (64.15, 56.44), (42.05, -34.32), (37.7, -30.51), (36.64, -41.51), (16.72, -123.31), (30.87, -61.46), (54.39, 3.04), (28.84, -48.06), (61.26, 45.65), (41.3, -16.32), (8.52, -111.31), (42.93, -63.09), (68.95, 44.73), (62.48, 45.96), (68.59, 99.18), (57.7, 19.76), (25.78, -53.12), (51.34, 42.29), (58.53, 49.74), (46.95, 2.31), (51.5, -26.27), (52.8, 3.6), (54.07, 3.41), (63.42, 64.83), (57.63, 44.65), (51.84, 1.4), (48.13, -3.71), (62.59, 35.34), (53.9, 9.9), (60.2, 53.5), (39.13, -38.48), (50.22, 3.29), (60.84, 8.86), (49.58, 12.89), (51.21, -0.11), (27.64, -65.71), (25.98, -44.46), (52.74, 7.28), (64.75, 27.06), (59.36, 20.61), (30.04, -78.55), (28.44, -49.94), (62.08, 31.24), (69.13, 52.96), (45.55, -14.94), (66.64, 100.69), (57.31, 23.17), (44.27, -2.76), (39.61, -7.36), (63.8, 35.07), (45.61, 0.46), (29.6, -32.27), (76.7, 99.01), (60.71, 73.23), (60.32, 36.56), (55.31, 46.65), (73.74, 100.18), (68.25, 107.3), (51.3, 45.09), (56.74, 18.93), (50.22, 0.55), (51.56, 24.33), (62.27, 5.34), (45.57, -31.61), (69.23, 35.25), (57.58, 27.12), (41.33, 3.42), (54.91, 7.99), (71.78, 62.7), (59.84, 26.22), (55.47, -17.59), (40.65, -10.53), (46.04, -18.07), (41.99, -16.64), (42.23, -5.09), (40.27, -46.15), (36.58, -29.27), (68.78, 66.89), (21.85, -73.51), (45.13, 14.66), (62.25, 70.37), (75.8, 58.83), (37.04, -30.29), (31.63, -38.27), (18.77, -103.13), (52.33, 8.18), (58.69, 8.99), (58.22, -2.44), (31.13, -64.7), (50.14, -10.5), (71.78, 67.76), (53.4, 28.39), (47.12, -36.48), (29.41, -50.76), (56.51, 0.35), (58.08, 68.43), (18.2, -72.71), (51.63, 8.89), (75.22, 42.25), (24.94, -22.71), (57.93, 60.34), (51.1, 3.59), (35.99, 14.34), (59.76, 38.51), (42.26, -11.09), (67.46, 55.35), (65.99, 54.61), (74.0, 57.56), (38.52, -82.72), (21.74, -100.56), (55.55, 1.52), (64.98, 63.52), (70.15, 49.72), (33.48, -47.28), (49.45, 11.65), (43.12, -10.43), (33.58, -40.63), (38.07, -26.35), (34.32, -43.35), (59.41, 49.37), (48.01, -69.69), (46.26, -20.51), (51.65, 40.25), (47.13, 10.76), (53.69, 53.27), (44.91, -6.75), (52.98, 30.23), (28.26, -50.33), (49.35, -13.96), (47.61, -14.04), (41.26, -29.35), (47.13, 5.63), (33.16, -22.98), (77.32, 76.14), (59.19, 12.49), (50.03, -7.78), (37.27, -58.98), (59.91, 64.47), (38.19, -4.33), (49.27, 19.47), (34.7, -43.4), (41.95, 29.4), (47.1, -6.78), (49.12, 26.55), (68.53, 64.9), (45.43, 7.06), (30.88, -48.22), (34.96, -20.64), (69.7, 94.19), (51.15, 21.83), (41.83, 19.87), (31.44, -45.5), (38.12, 16.9), (50.94, 53.94), (49.35, 22.82), (62.92, 47.47), (48.94, 22.84), (44.7, 13.92), (18.82, -109.8), (59.61, 64.48), (68.43, 81.66), (57.74, 47.43), (43.18, -17.37), (40.85, -39.42), (26.38, -64.13), (41.84, -4.13), (46.08, -19.68), (56.36, 1.05), (31.97, -46.81), (47.23, 13.86), (43.01, -31.11), (32.06, -36.75), (32.7, -26.58), (30.33, -63.46), (44.94, -35.78), (60.02, 34.37), (34.1, -29.13), (52.05, -7.73), (63.91, 74.24), (54.16, -3.22), (41.17, -38.79), (54.37, 18.42), (25.18, -91.84), (85.45, 120.33), (54.55, 30.48), (54.57, 43.06), (42.44, -4.84), (30.2, -63.75), (59.93, 46.53), (51.41, 23.3), (17.79, -107.2), (16.22, -70.11), (34.97, -26.54), (31.36, -8.52), (49.73, 23.79), (44.34, 11.65), (51.04, 2.82), (55.21, 41.2), (58.23, 49.56), (67.39, 37.46), (64.28, 40.7), (49.71, -15.23), (36.18, -32.75), (66.16, 54.23), (51.24, -22.42), (30.88, -32.08), (29.64, -66.39), (52.03, -5.64), (51.73, 28.04), (54.32, 33.91), (64.46, 19.03), (52.64, -8.49), (67.72, 64.25), (51.71, -2.28), (50.27, 25.31), (34.82, -37.89), (67.85, 58.54), (51.16, -0.83), (45.09, 19.05), (57.27, -12.84), (28.68, -43.93), (38.77, -26.42), (42.67, -12.19), (31.78, -6.04), (29.47, -31.54), (77.66, 81.43), (67.04, 49.48), (32.28, -84.41), (16.14, -81.36), (48.23, 6.82), (53.1, 24.34), (40.42, -18.95), (55.89, 8.58), (54.06, -30.09), (37.41, -20.54), (73.67, 60.73), (46.45, -1.01), (35.22, -82.09), (51.55, 11.92), (52.11, 34.34), (78.99, 96.82), (57.34, 56.92), (82.36, 122.96), (64.41, 63.19), (39.21, -51.06), (50.16, 2.59), (74.95, 65.9), (47.49, 1.73), (26.81, -58.65), (28.79, -92.84), (49.11, 6.41), (26.03, -72.36), (62.42, 39.0), (46.67, 13.06), (47.34, 6.05), (74.0, 63.91), (58.76, 6.76), (58.33, 0.06), (60.59, 46.42), (48.76, -1.72), (35.43, -57.78), (36.89, -16.21), (43.51, -44.94), (57.49, -6.85), (50.84, 13.98), (49.81, -10.55), (53.39, 29.96), (54.34, 35.31), (50.17, -14.07), (45.57, 7.48), (69.26, 57.58), (68.5, 70.96), (36.51, -28.45), (64.6, 70.48), (61.53, 32.43), (26.77, -82.19), (38.21, -42.05), (67.99, 58.88), (51.46, 10.23), (52.8, 29.83), (78.36, 74.79), (45.8, -0.61), (56.07, 42.81), (58.45, 20.03), (66.12, 48.26), (46.2, -45.15), (70.82, 86.21), (29.75, -77.16), (40.42, 10.02), (25.17, -79.0), (27.83, -74.02), (57.22, 19.62), (56.27, 4.0), (38.37, -36.09), (46.13, -7.73), (68.04, 73.26), (41.73, 1.16), (42.8, -30.13), (52.82, 0.89), (34.03, -48.06), (77.4, 106.8), (57.47, 18.16), (23.64, -43.44), (49.16, 25.61), (60.77, -15.36), (43.8, 4.36), (22.9, -64.17), (57.1, 14.91), (82.44, 87.05), (57.23, 2.87), (51.3, 15.76), (69.13, 74.21), (58.88, 35.33), (57.12, 11.41), (69.16, 61.21), (55.06, 21.37), (43.36, -40.9), (61.19, 43.52), (42.25, -34.65), (66.41, 44.96), (50.47, -6.0), (83.08, 102.73), (72.2, 108.92), (43.13, -38.84), (63.22, 84.01), (34.5, -30.01), (42.74, -2.11), (45.65, 24.71), (31.71, -94.58), (65.52, 80.66), (56.38, 49.35), (60.91, 50.46), (56.17, -13.15), (72.79, 77.19), (61.84, 55.97), (32.92, -59.67), (50.67, 45.53), (54.77, 33.85), (42.24, -26.63), (45.05, -26.72), (52.11, -12.14), (51.16, 0.22), (48.23, 47.33), (52.78, -6.17), (28.72, -68.64), (48.78, -1.65), (58.41, 18.86), (37.79, -17.59), (44.39, -27.32), (48.24, -4.65), (58.09, -14.09), (69.8, 53.59), (66.93, 44.97), (52.92, 21.07), (54.3, 18.86), (51.86, 12.08), (62.11, 25.98), (37.16, -21.48), (51.33, -28.87), (47.33, 45.78), (48.94, 28.76), (56.51, -6.83), (44.33, 10.69), (53.07, 28.57), (53.16, 3.85), (36.35, -15.79), (42.23, -59.25), (35.43, -44.38), (61.21, 49.13), (30.05, -37.73), (51.4, 57.0), (49.89, 3.92), (55.57, 50.22), (68.84, 65.09), (38.05, -37.26), (72.71, 35.75), (63.38, 53.88), (43.93, -8.26), (36.42, -8.2), (62.15, 37.85), (40.31, -18.87), (39.9, 7.09), (22.05, -74.85), (37.43, -44.54), (35.96, -34.97), (44.58, 7.93), (59.25, 31.58), (52.92, 23.89), (54.93, 48.13), (58.65, 24.88), (57.19, 25.55), (31.87, -48.62), (51.4, 15.55), (43.72, -6.16), (34.76, -13.98), (62.15, 53.35), (28.18, -48.09), (7.87, -112.34), (66.45, 59.47), (41.87, -6.74), (54.62, 24.07), (42.57, -25.28), (26.54, -57.65), (61.72, 14.95), (49.36, 30.5), (53.54, 31.94), (70.83, 58.93), (30.23, -67.63), (63.13, 7.2), (53.56, 4.16), (41.75, -15.41), (26.55, -50.52), (63.31, 40.2), (57.43, 25.76), (46.67, -1.51), (49.39, -5.42), (44.87, 3.34), (43.98, -6.22), (42.32, 0.2), (40.43, -49.05), (85.19, 103.57), (84.19, 86.37), (35.39, -7.59), (52.66, 16.55), (52.14, 55.48), (21.87, -89.17), (11.12, -127.59), (46.85, -28.58), (69.7, 57.31), (39.2, -13.28), (70.82, 99.59), (44.83, 14.38), (59.65, 36.61), (39.1, -1.95), (65.45, 87.15), (68.36, 42.92), (45.12, -4.49), (36.71, -66.66), (43.44, -23.77), (49.67, 7.51), (65.16, 68.61), (70.32, 68.84), (31.66, -42.84), (49.17, -15.72), (38.83, -59.18), (46.73, -32.15), (18.06, -103.13), (38.36, -32.86), (58.69, 23.96), (57.26, 20.98), (67.85, 52.45), (35.9, -24.95), (59.73, 42.21), (55.08, 43.81), (57.31, 15.64), (69.31, 46.32), (51.06, 4.68), (69.45, 79.64), (17.01, -124.55), (75.62, 93.79), (51.39, 10.85), (42.32, -37.85), (21.34, -90.93), (26.63, -94.71), (48.74, 29.08), (55.08, 32.91), (52.88, 38.68), (18.28, -75.95), (36.65, -47.85), (66.77, 74.22)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=49.5723，ȳ=4.1433；Sxx=Σ(x−x̄)²=173935.91，Sxy=515910.81；
β₁ = Sxy/Sxx = 2.9661；β₀ = ȳ−β₁x̄ = -142.8929；
R² = SSR/SST = 1530241.60/1845989.98 = 0.8290；F = (SSR/1)/(SSE/781) = 3785.0350，p = 1.06e-301；
SE(β₁)=√(404.29/173935.91)=0.0482；t=β₁/SE=61.5226；p=1.06e-301；95%CI=[1.6733,4.2589]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=49.5723，ȳ=4.1433；Sxx=Σ(x−x̄)²=173935.91，Sxy=515910.81；
β₁ = Sxy/Sxx = 2.9661；β₀ = ȳ−β₁x̄ = -142.8929；
R² = SSR/SST = 1530241.60/1845989.98 = 0.8290；F = (SSR/1)/(SSE/781) = 3785.0350，p = 1.06e-301；
SE(β₁)=√(404.29/173935.91)=0.0482；t=β₁/SE=61.5226；p=1.06e-301；95%CI=[1.6733,4.2589]。
③ 计算结果：
y = -142.8929 + 2.9661·x；R²=0.8290；β₁ t=61.5226 p=1.06e-301；95%CI=[1.6733,4.2589]
④ 结论：在 α=0.05 水平下，p=1.06e-301 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=2.9661），投入越高销量越高。
案例 42. 学习时长预测考试分数
简单线性回归学得久分高
① 原始数据：
样本 n=354，逐条 (x, y)：[(10.08, 47.64), (14.29, 57.14), (9.66, 47.36), (10.44, 53.14), (7.21, 28.38), (9.87, 38.78), (6.32, 29.48), (9.83, 34.92), (8.66, 22.89), (12.65, 43.8), (9.31, 28.85), (9.5, 38.55), (12.12, 38.15), (7.39, 31.68), (13.34, 57.2), (9.14, 43.88), (6.79, 34.85), (9.65, 33.49), (14.76, 65.91), (5.64, 30.29), (12.98, 42.09), (10.02, 46.35), (11.34, 37.37), (6.24, 34.74), (6.93, 30.5), (8.68, 50.67), (8.81, 47.92), (6.67, 24.14), (7.06, 46.3), (5.95, 21.6), (10.34, 45.38), (8.18, 41.12), (11.92, 29.3), (16.34, 52.97), (13.07, 42.23), (12.8, 51.38), (14.23, 47.05), (2.47, 25.3), (4.6, 21.49), (11.41, 47.0), (13.67, 51.71), (10.84, 35.23), (5.63, 30.31), (9.56, 31.12), (8.38, 40.21), (6.69, 27.21), (8.94, 36.02), (7.76, 28.86), (13.08, 42.47), (14.93, 55.1), (10.01, 43.93), (5.46, 19.74), (11.41, 62.37), (7.77, 35.74), (9.34, 35.84), (7.05, 18.66), (12.95, 48.43), (10.99, 32.25), (10.94, 57.55), (5.92, 43.67), (11.4, 51.59), (6.96, 32.51), (9.85, 23.65), (11.4, 50.56), (12.41, 34.65), (11.22, 30.39), (12.32, 49.44), (12.88, 44.17), (6.3, 18.62), (9.68, 37.7), (11.99, 39.95), (13.26, 42.63), (8.79, 48.84), (12.34, 39.08), (12.37, 58.21), (2.88, 7.95), (11.05, 36.05), (8.3, 21.58), (12.28, 48.87), (11.19, 31.82), (10.11, 35.06), (11.18, 30.82), (4.66, 24.41), (12.31, 40.27), (14.66, 68.76), (11.54, 42.69), (10.63, 30.5), (11.23, 46.03), (12.61, 52.67), (5.41, 14.48), (13.41, 49.12), (11.35, 48.82), (9.67, 38.54), (14.35, 56.61), (12.51, 38.11), (13.03, 63.45), (9.17, 36.13), (10.34, 30.06), (12.33, 39.87), (7.28, 34.65), (9.13, 36.39), (8.03, 34.93), (9.86, 24.02), (12.08, 49.79), (7.97, 34.65), (12.3, 39.45), (12.0, 39.57), (10.7, 35.28), (11.46, 42.81), (12.64, 65.18), (6.09, 29.15), (10.41, 37.18), (8.78, 52.67), (0.27, 5.99), (8.67, 43.84), (7.66, 18.57), (12.79, 47.95), (13.88, 60.84), (14.12, 58.3), (12.59, 56.84), (7.64, 21.01), (7.56, 30.07), (9.82, 29.71), (0.35, 12.83), (9.49, 31.35), (6.54, 40.78), (12.39, 40.57), (10.18, 40.87), (11.77, 56.73), (13.05, 39.87), (11.23, 43.09), (4.67, 19.51), (10.82, 55.28), (10.87, 31.29), (7.98, 17.97), (11.86, 48.19), (7.99, 25.69), (7.99, 28.57), (4.81, 21.29), (8.64, 42.04), (8.32, 15.25), (9.64, 38.22), (9.49, 38.17), (1.67, 19.3), (8.38, 27.38), (10.72, 43.36), (13.05, 54.46), (13.36, 55.89), (10.06, 32.35), (11.54, 40.35), (9.0, 43.81), (9.53, 45.06), (8.71, 27.24), (13.03, 45.24), (13.8, 56.53), (8.73, 27.03), (12.98, 39.48), (6.65, 13.01), (10.81, 47.15), (11.57, 60.5), (14.79, 61.29), (10.95, 28.9), (7.7, 25.77), (13.21, 55.24), (14.57, 54.52), (11.66, 50.29), (6.89, 48.52), (7.52, 51.1), (11.88, 36.03), (10.29, 34.36), (12.49, 46.53), (7.82, 29.59), (11.83, 42.13), (5.49, 29.74), (15.04, 68.78), (11.88, 38.01), (4.77, 6.32), (8.59, 41.73), (9.12, 35.36), (8.6, 30.98), (9.67, 39.16), (11.32, 47.39), (10.23, 46.4), (12.82, 52.66), (10.17, 53.27), (8.12, 35.99), (8.95, 45.85), (7.72, 33.8), (13.84, 48.32), (13.04, 47.24), (10.08, 37.14), (12.04, 50.97), (9.1, 43.42), (11.78, 39.46), (12.4, 52.74), (14.14, 60.25), (8.95, 24.81), (14.12, 65.29), (13.22, 43.04), (7.37, 34.85), (6.1, 20.74), (10.6, 47.3), (9.71, 41.99), (6.92, 14.72), (6.33, 27.96), (15.17, 53.93), (8.81, 51.49), (14.54, 52.72), (10.9, 38.31), (11.04, 42.31), (19.0, 62.58), (7.59, 31.13), (9.47, 40.9), (11.85, 39.0), (5.18, 8.49), (3.19, 34.04), (8.31, 33.96), (6.84, 42.46), (4.29, 13.19), (7.93, 26.08), (10.52, 42.16), (3.29, 16.03), (8.88, 38.64), (14.49, 73.87), (10.95, 46.52), (10.39, 55.54), (10.61, 46.75), (11.47, 45.61), (8.05, 37.73), (9.8, 49.15), (11.84, 38.12), (14.53, 65.58), (11.39, 47.77), (7.58, 35.04), (3.74, 24.24), (7.67, 37.01), (15.83, 63.52), (6.38, 41.34), (15.68, 70.64), (13.87, 57.03), (11.79, 32.86), (8.46, 23.1), (5.22, 20.19), (10.23, 41.78), (11.74, 42.63), (10.18, 35.11), (8.37, 45.09), (16.47, 51.15), (11.93, 52.19), (14.05, 43.69), (6.67, 32.53), (5.33, 22.69), (14.43, 38.14), (7.56, 44.65), (12.06, 44.52), (0.39, 10.02), (12.36, 53.34), (7.7, 40.07), (11.38, 38.49), (11.06, 52.63), (10.35, 42.31), (14.64, 57.83), (10.57, 36.23), (4.74, 20.02), (10.13, 40.98), (18.49, 68.04), (12.67, 39.33), (11.39, 46.82), (5.04, 37.94), (7.87, 48.99), (13.13, 44.22), (10.51, 54.56), (12.33, 49.4), (9.63, 38.69), (10.02, 48.98), (6.23, 37.11), (8.46, 29.83), (14.12, 51.27), (12.59, 48.92), (16.61, 51.05), (9.37, 29.52), (12.6, 42.62), (7.33, 32.78), (6.47, 29.85), (11.91, 49.97), (15.34, 56.57), (11.4, 52.03), (6.58, 34.84), (7.71, 23.05), (11.88, 37.33), (10.58, 37.27), (17.85, 53.48), (11.53, 44.46), (5.96, 49.63), (10.01, 38.46), (6.63, 32.09), (5.82, 17.11), (10.73, 33.65), (11.42, 33.88), (15.21, 54.68), (13.53, 41.42), (13.25, 44.66), (14.15, 57.44), (14.44, 60.38), (12.95, 35.88), (6.4, 28.22), (5.63, 29.43), (10.51, 39.94), (10.15, 28.69), (10.88, 43.39), (15.86, 60.52), (11.14, 44.28), (7.0, 34.14), (10.68, 46.78), (9.6, 41.22), (11.25, 41.6), (9.7, 33.82), (13.55, 37.35), (12.78, 42.17), (5.58, 20.89), (10.21, 39.03), (14.2, 59.24), (8.47, 52.13), (14.08, 48.78), (11.97, 42.96), (9.98, 65.55), (12.93, 40.43), (6.82, 18.95), (9.97, 42.5), (10.33, 33.16), (12.57, 52.42), (12.95, 58.64), (9.81, 50.97), (12.61, 42.87), (11.5, 38.52), (12.4, 48.71), (13.46, 32.63), (9.33, 38.06), (7.69, 37.83), (10.27, 32.82), (12.69, 53.23), (7.24, 36.64), (13.13, 46.28), (5.34, 35.09), (6.93, 10.91), (9.47, 34.39), (7.39, 34.37), (9.69, 49.41), (11.64, 57.21), (7.62, 49.67), (7.46, 42.0), (9.21, 44.42), (7.08, 35.91), (3.4, 13.65)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=10.1124，ȳ=40.2316；Sxx=Σ(x−x̄)²=3364.35，Sxy=10168.94；
β₁ = Sxy/Sxx = 3.0226；β₀ = ȳ−β₁x̄ = 9.6664；
R² = SSR/SST = 30736.22/55091.39 = 0.5579；F = (SSR/1)/(SSE/352) = 444.2239，p = 2.31e-64；
SE(β₁)=√(69.19/3364.35)=0.1434；t=β₁/SE=21.0766；p=2.31e-64；95%CI=[0.4467,5.5984]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=10.1124，ȳ=40.2316；Sxx=Σ(x−x̄)²=3364.35，Sxy=10168.94；
β₁ = Sxy/Sxx = 3.0226；β₀ = ȳ−β₁x̄ = 9.6664；
R² = SSR/SST = 30736.22/55091.39 = 0.5579；F = (SSR/1)/(SSE/352) = 444.2239，p = 2.31e-64；
SE(β₁)=√(69.19/3364.35)=0.1434；t=β₁/SE=21.0766；p=2.31e-64；95%CI=[0.4467,5.5984]。
③ 计算结果：
y = 9.6664 + 3.0226·x；R²=0.5579；β₁ t=21.0766 p=2.31e-64；95%CI=[0.4467,5.5984]
④ 结论：在 α=0.05 水平下，p=2.31e-64 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=3.0226），学得久分高。
案例 43. 房屋面积预测租金
简单线性回归面积大租金高
① 原始数据：
样本 n=799，逐条 (x, y)：[(57.76, -949.13), (55.38, -780.52), (91.3, 402.03), (107.58, 988.66), (124.02, 1665.1), (61.13, -245.89), (113.29, 1304.32), (79.61, 59.52), (44.72, -1143.97), (73.03, -186.13), (95.03, 186.82), (77.13, -483.81), (98.74, 692.89), (97.46, 545.71), (88.8, 308.99), (91.6, 861.0), (106.9, 1374.66), (148.43, 3016.15), (83.52, 119.7), (90.39, 689.37), (119.3, 2063.06), (109.47, 1560.89), (83.38, 700.25), (100.77, 764.08), (68.54, -477.85), (76.95, -302.55), (98.88, 632.82), (30.07, -2290.9), (93.29, 243.19), (88.8, 407.39), (95.68, 249.62), (115.24, 1385.76), (80.82, -143.1), (132.68, 2617.84), (76.53, -267.41), (87.35, 170.02), (86.89, -61.51), (75.61, 351.69), (94.75, 253.49), (113.11, 1102.02), (158.63, 3083.63), (134.4, 2326.21), (115.94, 1369.72), (105.83, 1663.59), (96.12, 1097.8), (113.78, 1630.66), (70.62, -85.3), (100.65, 1030.12), (90.52, 1175.08), (62.18, -1033.35), (79.98, 131.98), (48.23, -1310.09), (98.87, 592.97), (96.39, 969.25), (112.68, 1105.64), (113.4, 1819.39), (109.15, 1517.07), (52.1, -1006.7), (61.17, -590.93), (92.44, 206.14), (97.97, 584.8), (113.97, 1124.96), (74.12, -475.97), (59.15, -932.63), (111.99, 893.41), (59.22, -356.23), (113.62, 1471.11), (105.07, 1071.67), (105.42, 932.83), (120.86, 1471.23), (90.77, 1180.45), (117.3, 1586.18), (77.32, -237.66), (142.98, 3106.26), (118.38, 1578.41), (101.57, 638.71), (53.69, -979.11), (84.75, -209.55), (111.83, 1902.17), (86.34, 141.68), (118.02, 1446.98), (75.58, -630.76), (72.81, -173.16), (101.64, 648.0), (72.84, 106.17), (98.41, 605.55), (95.68, 850.93), (77.28, 119.64), (77.09, -413.18), (101.21, 1134.85), (100.35, 1057.3), (107.31, 1079.74), (87.95, 573.41), (120.6, 1596.76), (98.76, 1041.09), (68.16, -86.05), (83.11, 308.21), (78.45, 552.41), (38.53, -1340.95), (75.37, -124.89), (49.78, -400.24), (25.12, -2398.66), (113.78, 1531.23), (101.17, 727.52), (89.73, 423.87), (136.75, 2161.35), (41.6, -1612.69), (53.62, -1266.14), (124.5, 1945.97), (97.82, 776.57), (126.61, 1893.13), (54.37, -698.08), (148.01, 2249.94), (110.62, 986.11), (80.38, 684.0), (158.02, 3129.16), (111.12, 1960.97), (39.05, -899.38), (41.52, -1257.68), (63.35, -737.91), (48.62, -1578.26), (129.58, 1576.36), (23.84, -2348.21), (119.77, 1435.44), (93.23, 1084.03), (92.76, 80.53), (109.68, 1461.81), (81.19, -172.93), (108.49, 1504.83), (63.45, -1242.96), (118.11, 2074.64), (76.91, -290.05), (49.41, -1202.53), (105.1, 1231.81), (124.3, 1585.21), (99.43, 796.21), (93.73, 822.21), (89.23, 1131.44), (66.93, -918.99), (95.17, 663.35), (84.55, 182.49), (84.74, 271.23), (86.81, 203.61), (110.78, 1056.09), (109.88, 929.74), (109.42, 1084.98), (49.66, -655.07), (126.89, 1782.58), (81.31, -2.29), (113.55, 2255.29), (109.74, 1229.88), (97.34, 647.22), (89.33, 1471.87), (66.51, -249.78), (64.93, -78.92), (76.33, 200.0), (77.02, 198.27), (95.47, 1056.62), (112.98, 1237.66), (118.59, 2052.57), (61.2, -947.69), (85.45, 42.04), (149.83, 3015.49), (77.65, 769.12), (62.37, -965.4), (80.82, -58.49), (101.13, 890.52), (105.74, 1613.93), (82.13, -339.8), (35.41, -1621.09), (61.25, -134.35), (87.92, 313.8), (92.18, 429.3), (134.05, 2332.21), (64.83, -180.19), (111.36, 1271.77), (79.67, 324.44), (96.94, 1316.68), (98.09, 147.22), (62.15, -689.98), (79.02, 362.54), (121.19, 1957.35), (110.36, 1209.69), (43.08, -1674.41), (57.81, -484.23), (102.86, 968.37), (98.98, 1282.21), (113.02, 1653.45), (84.84, 423.05), (118.15, 1424.64), (83.09, -458.42), (115.24, 2021.78), (97.2, 809.74), (80.07, 223.88), (89.02, 689.98), (71.45, 29.73), (120.91, 2173.96), (53.18, -527.9), (95.29, 794.23), (107.68, 651.99), (75.39, 21.89), (74.07, -348.55), (43.26, -1886.72), (76.21, 94.35), (83.03, 556.87), (111.32, 1450.19), (96.25, 761.0), (87.77, 712.16), (81.71, 225.15), (44.1, -1529.96), (110.53, 1339.73), (107.29, 1127.38), (79.38, 569.69), (59.2, -1201.57), (50.99, -1256.52), (124.39, 1675.49), (73.58, 117.66), (73.97, -443.13), (70.8, -546.59), (140.29, 2465.2), (45.06, -1455.87), (73.77, -269.48), (84.03, 652.01), (82.5, 241.71), (93.29, 1044.63), (64.77, -141.98), (133.27, 2162.34), (102.91, 1187.4), (104.96, 715.46), (98.23, 1078.38), (132.5, 2212.53), (100.87, 2148.75), (111.2, 1443.41), (83.64, 252.8), (106.15, 1016.69), (100.47, 1193.9), (72.72, -612.56), (80.89, 372.74), (104.28, 699.11), (58.91, -719.67), (52.37, -772.82), (86.79, 117.65), (83.01, 180.31), (102.7, 860.58), (46.61, -1194.37), (91.87, 882.51), (71.02, 41.96), (89.93, 648.98), (92.24, 1000.68), (72.25, -208.9), (84.42, 215.77), (124.23, 1779.19), (158.16, 3239.04), (48.17, -685.96), (83.36, 602.22), (69.21, -731.79), (79.78, -68.64), (109.17, 1174.05), (49.06, -1346.5), (79.51, 353.32), (101.06, 1202.25), (74.7, -636.83), (86.08, -208.42), (110.78, 1607.78), (95.03, 1113.35), (78.59, 180.61), (102.27, 1140.52), (85.82, 143.27), (129.5, 1689.82), (104.79, 1217.2), (91.93, 1124.2), (75.26, -113.68), (86.69, 321.56), (87.44, 450.93), (119.07, 1915.57), (88.64, 439.05), (117.84, 1554.33), (85.41, -122.69), (58.8, -727.23), (82.08, 237.27), (62.09, -933.32), (124.2, 1968.8), (130.96, 1708.67), (60.89, -839.75), (92.94, 653.41), (113.85, 1735.11), (75.24, -179.17), (77.9, -611.43), (69.64, 48.73), (76.72, 231.51), (101.74, 1023.33), (57.13, -525.92), (96.98, 705.94), (110.3, 1519.4), (76.37, 1.92), (94.99, 209.03), (68.86, 7.77), (94.26, 1084.55), (106.99, 774.83), (76.87, 81.32), (93.76, 1406.22), (92.88, 716.53), (105.1, 896.73), (151.9, 3171.34), (114.61, 1657.22), (135.17, 2452.09), (105.25, 976.26), (59.71, -469.64), (113.33, 1172.75), (59.6, -640.9), (84.82, 51.03), (118.49, 1880.28), (106.72, 1566.44), (86.05, 347.25), (117.12, 1556.42), (94.45, 581.85), (33.8, -1511.46), (68.88, 169.73), (64.85, -1124.25), (121.86, 2262.3), (117.35, 1184.43), (105.99, 969.28), (109.77, 1567.39), (46.1, -1160.19), (98.07, 773.17), (78.51, 428.28), (81.59, 40.96), (105.18, 692.69), (59.77, -742.0), (82.62, 232.6), (101.94, 1041.3), (38.55, -1223.62), (114.5, 1318.01), (96.43, 671.26), (68.49, -511.13), (87.27, 573.39), (119.53, 1427.17), (113.91, 2156.11), (52.31, -578.09), (119.82, 2027.36), (98.46, 793.0), (111.05, 1654.57), (79.16, 76.19), (107.25, 1408.77), (99.63, 1243.6), (127.33, 2186.7), (43.22, -1491.0), (39.44, -1427.41), (106.53, 793.66), (80.02, 95.58), (112.95, 1467.16), (75.99, 352.11), (103.65, 940.86), (72.73, -165.32), (134.92, 2035.67), (54.61, -583.81), (121.49, 1424.76), (52.88, -743.98), (126.28, 1411.42), (98.61, 1283.44), (94.02, 910.23), (104.06, 1142.9), (119.36, 1775.36), (23.31, -1764.92), (72.87, 15.83), (76.88, 140.42), (104.75, 1714.33), (62.24, -229.87), (125.37, 2011.14), (136.82, 2794.1), (125.66, 1926.42), (90.32, 561.77), (108.46, 1515.64), (64.23, -745.54), (89.97, 86.96), (115.22, 1518.04), (127.53, 1832.66), (108.67, 1266.95), (76.86, -303.76), (101.84, 1085.77), (102.4, 583.25), (69.53, -31.72), (77.22, -63.11), (87.82, 635.74), (102.88, 886.64), (85.02, 48.74), (87.34, 960.3), (108.0, 1178.41), (111.47, 1415.71), (66.76, -301.17), (64.93, -181.35), (91.43, 494.11), (127.3, 1677.73), (138.67, 2320.43), (115.14, 1214.32), (122.83, 1846.15), (56.46, -1220.59), (106.82, 1143.43), (82.76, 548.17), (67.87, -344.7), (59.37, -650.37), (76.68, 613.49), (53.42, -1166.44), (77.47, 216.6), (79.75, -473.34), (74.01, -378.99), (77.97, -520.65), (70.51, -88.62), (46.68, -1278.04), (108.29, 1073.06), (105.98, 1582.54), (59.98, -579.8), (70.18, -256.43), (64.69, -1105.99), (85.29, 577.35), (99.08, 1136.74), (104.67, 640.9), (104.81, 1138.7), (105.85, 646.1), (80.47, -105.4), (75.47, -132.13), (62.23, -335.97), (117.2, 1991.77), (112.02, 1309.0), (108.29, 1008.07), (69.91, -816.54), (45.69, -1214.31), (116.19, 1629.18), (54.58, -1026.54), (85.85, 241.53), (60.01, -710.06), (142.62, 2550.84), (78.27, -126.71), (105.97, 1362.78), (66.27, -527.9), (149.91, 3327.33), (129.29, 1882.67), (85.1, 247.91), (89.63, 461.71), (115.94, 1751.46), (27.8, -2538.47), (111.88, 1331.55), (109.53, 1864.29), (83.24, 330.88), (116.21, 1612.22), (76.8, 25.15), (118.46, 1389.53), (117.3, 1755.14), (98.27, 778.99), (96.14, 998.42), (60.91, -624.77), (83.41, -13.92), (107.36, 1608.41), (110.48, 1137.94), (65.2, -213.5), (104.43, 1163.47), (81.51, 253.71), (104.96, 824.85), (125.05, 1792.54), (102.82, 730.66), (117.79, 1711.08), (93.39, 266.03), (40.37, -1497.9), (101.92, 1194.38), (37.47, -2044.34), (26.03, -1673.21), (115.42, 1805.68), (152.69, 2661.46), (108.78, 1429.86), (86.04, 451.94), (114.29, 1817.75), (75.38, -247.3), (49.65, -965.84), (85.53, -120.39), (74.73, 18.38), (105.76, 1209.65), (104.99, 1090.04), (108.68, 444.51), (71.61, 119.33), (115.87, 1340.33), (89.24, 1060.76), (78.35, -195.32), (88.8, 389.74), (112.87, 1636.52), (130.42, 2061.48), (75.63, 168.79), (116.55, 784.69), (124.5, 1953.79), (68.07, -389.37), (114.14, 1612.02), (108.54, 1263.7), (69.23, -1208.04), (87.14, 528.84), (97.41, 436.94), (115.21, 1605.96), (78.66, -80.96), (117.67, 1845.04), (93.27, 374.82), (94.32, 367.37), (134.13, 2236.32), (74.33, 53.98), (97.97, 1037.63), (66.86, -214.66), (89.76, 685.76), (86.8, 773.19), (67.49, -525.19), (75.56, -94.95), (82.83, -123.17), (98.49, 807.93), (120.06, 1945.02), (70.31, -521.86), (98.56, 588.67), (110.41, 1128.52), (34.1, -1975.35), (72.22, 215.99), (79.62, 9.77), (56.36, -1428.11), (51.09, -873.85), (135.01, 2677.51), (88.86, 599.28), (105.78, 1216.9), (95.16, 460.41), (40.52, -1854.17), (69.63, -313.07), (36.11, -1954.0), (133.08, 2247.94), (75.31, 71.01), (41.1, -1426.19), (129.12, 1585.54), (105.18, 808.07), (124.22, 1569.31), (113.13, 1383.08), (74.19, -235.47), (62.89, -916.25), (98.42, 650.02), (101.2, 761.99), (111.87, 937.03), (80.98, -40.27), (54.17, -955.37), (130.01, 2363.31), (91.97, 927.74), (43.11, -1893.64), (57.71, -466.06), (85.06, 598.39), (133.85, 1750.55), (83.08, 446.94), (102.48, 1189.84), (129.68, 2298.89), (90.85, 1137.08), (37.8, -1894.01), (108.61, 1872.85), (106.49, 1400.13), (41.41, -1728.21), (47.59, -882.43), (118.36, 1449.28), (46.09, -1321.78), (94.68, 568.32), (92.94, 1090.95), (63.5, -417.1), (112.08, 1394.12), (79.5, -287.81), (80.16, -557.62), (93.99, -30.66), (165.47, 3412.72), (101.73, 1119.99), (96.24, 1180.57), (114.74, 1362.89), (105.96, 948.5), (67.52, -589.54), (66.17, -733.47), (81.83, 733.16), (97.56, 566.33), (56.12, -793.55), (132.45, 2541.62), (131.28, 2422.82), (87.9, 354.07), (50.97, -1731.28), (96.31, 1220.3), (40.12, -1972.72), (92.38, 750.41), (118.1, 1375.68), (65.97, -1005.24), (90.33, 445.77), (87.73, 429.91), (95.75, 910.77), (102.32, 1020.4), (74.6, -377.65), (48.86, -1482.0), (91.47, 902.31), (62.93, -932.92), (78.37, -331.16), (98.27, 429.34), (87.45, 835.83), (74.24, -437.38), (76.09, -142.66), (38.36, -1841.15), (114.53, 1403.49), (84.42, 331.26), (78.63, 361.37), (109.42, 806.12), (73.17, -59.35), (80.24, 140.38), (142.43, 2383.45), (79.51, -233.28), (81.24, 230.63), (87.12, 559.34), (81.82, 502.59), (70.12, -85.69), (98.06, 741.32), (81.84, 476.4), (83.49, -119.04), (79.92, 428.65), (51.52, -1113.03), (82.69, -144.07), (63.62, -497.65), (106.2, 1379.86), (108.57, 865.31), (64.6, -473.09), (110.54, 1272.25), (82.3, 178.67), (114.97, 1526.12), (113.9, 1725.32), (68.66, -137.17), (75.96, -264.24), (62.42, -281.45), (85.32, 12.35), (118.8, 1458.61), (84.74, 865.69), (112.43, 899.53), (143.06, 2416.42), (105.64, 1385.27), (94.3, 794.01), (126.72, 2349.47), (97.26, 309.82), (56.01, -757.4), (78.59, -66.56), (63.39, -631.51), (93.67, 232.98), (77.13, -318.84), (94.15, 1391.33), (48.88, -893.35), (88.65, 340.77), (82.1, 306.91), (58.33, -359.83), (102.7, 1395.12), (90.54, 641.9), (88.5, 83.25), (91.82, 940.12), (20.41, -2447.05), (44.43, -1949.88), (94.14, 580.86), (93.33, 373.48), (99.87, 745.26), (120.39, 1625.48), (90.06, 225.24), (63.89, 96.63), (129.79, 2113.28), (82.72, -29.54), (120.94, 1525.02), (60.67, -421.27), (102.34, 327.28), (94.19, 580.45), (96.77, 1025.04), (115.22, 1610.71), (63.25, -1029.1), (109.26, 1618.64), (126.77, 1265.33), (68.8, -388.5), (81.26, 403.25), (89.38, 561.9), (111.2, 1189.41), (80.75, 673.89), (88.96, 518.19), (90.43, 569.44), (97.09, 1139.23), (78.37, -403.23), (66.35, 157.39), (69.72, -635.98), (121.58, 1835.5), (137.48, 2549.18), (87.43, 496.53), (122.71, 2554.09), (76.35, 603.79), (89.07, 409.02), (130.03, 1894.88), (94.97, 1095.88), (115.76, 1641.8), (96.64, 901.28), (104.08, 618.23), (110.71, 930.4), (144.62, 3107.3), (98.26, 1164.44), (47.01, -1519.5), (91.15, 742.07), (79.07, -56.5), (108.25, 1094.84), (75.99, -425.53), (147.24, 2848.73), (106.44, 882.22), (81.24, 484.54), (80.53, 291.85), (121.04, 1727.88), (64.55, -958.38), (86.93, 285.3), (76.77, -179.88), (144.88, 2422.49), (94.18, 1052.24), (64.34, -270.17), (118.37, 1289.7), (86.73, -211.54), (91.36, 826.42), (106.77, 896.92), (71.24, -14.56), (92.86, 318.07), (67.38, -282.19), (88.81, 1029.91), (88.4, 206.15), (61.48, -516.29), (86.1, 691.21), (66.98, -574.3), (87.87, 401.07), (134.14, 2479.0), (95.38, 731.61), (105.76, 1214.53), (75.87, -126.29), (60.76, -412.7), (116.84, 1856.32), (117.67, 1829.09), (134.29, 1901.27), (86.2, 400.72), (64.85, -621.43), (112.44, 1902.17), (109.13, 955.04), (122.46, 1327.18), (117.65, 1563.96), (68.36, 134.31), (96.91, 1060.48), (89.2, 462.99), (116.41, 1556.0), (49.84, -896.5), (52.61, -598.57), (75.39, -523.08), (73.45, -1081.85), (134.64, 2850.31), (48.45, -1266.42), (108.74, 746.31), (81.32, 167.07), (59.67, -243.15), (105.31, 1200.84), (93.62, 105.76), (54.97, -539.01), (86.24, 652.59), (86.31, 742.91), (121.65, 1669.92), (73.97, 42.14), (136.74, 2482.54), (64.24, -334.06), (120.74, 1831.86), (101.72, 941.14), (69.92, -751.55), (26.33, -2086.92), (60.6, -683.68), (109.39, 1033.92), (12.02, -2473.22), (110.88, 1322.92), (130.52, 1941.68), (85.75, 240.22), (136.65, 2302.33), (36.72, -1133.52), (67.04, -373.63), (114.71, 1367.17), (59.08, -1028.72), (49.55, -1237.67), (114.81, 1753.43), (84.61, 373.43), (83.66, 438.56), (135.31, 2929.76), (92.23, 504.25), (50.13, -1421.63), (84.2, 398.8), (80.84, 172.76), (74.74, 189.33), (131.81, 1560.19), (67.09, -48.72), (130.99, 2448.98), (70.55, -320.64), (62.61, -615.21), (103.4, 866.05), (59.34, -1096.71), (80.01, -227.9), (109.31, 1449.66), (70.64, -352.08), (91.16, 687.44), (83.09, -32.72), (138.72, 2649.51), (121.17, 2016.79), (76.45, 59.63), (76.27, 319.26), (90.04, 414.78)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=90.3269，ȳ=518.4527；Sxx=Σ(x−x̄)²=526063.43，Sxy=21140190.09；
β₁ = Sxy/Sxx = 40.1856；β₀ = ȳ−β₁x̄ = -3111.3895；
R² = SSR/SST = 849531853.87/925546388.24 = 0.9179；F = (SSR/1)/(SSE/797) = 8907.2030，p = <1e-300；
SE(β₁)=√(95375.83/526063.43)=0.4258；t=β₁/SE=94.3780；p=<1e-300；95%CI=[39.3498,41.0214]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=90.3269，ȳ=518.4527；Sxx=Σ(x−x̄)²=526063.43，Sxy=21140190.09；
β₁ = Sxy/Sxx = 40.1856；β₀ = ȳ−β₁x̄ = -3111.3895；
R² = SSR/SST = 849531853.87/925546388.24 = 0.9179；F = (SSR/1)/(SSE/797) = 8907.2030，p = <1e-300；
SE(β₁)=√(95375.83/526063.43)=0.4258；t=β₁/SE=94.3780；p=<1e-300；95%CI=[39.3498,41.0214]。
③ 计算结果：
y = -3111.3895 + 40.1856·x；R²=0.9179；β₁ t=94.3780 p=<1e-300；95%CI=[39.3498,41.0214]
④ 结论：在 α=0.05 水平下，p=<1e-300 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=40.1856），面积大租金高。
案例 44. 工龄预测月薪
简单线性回归工龄长工资高
① 原始数据：
样本 n=395，逐条 (x, y)：[(8.62, 5227.39), (9.02, 3080.9), (-2.65, -2124.84), (3.94, 2567.78), (7.1, 4067.53), (12.29, 5914.81), (1.32, -999.84), (9.18, 6622.08), (3.63, 3087.1), (7.03, 1530.86), (8.62, 4102.86), (10.72, 6166.48), (9.66, 3832.38), (3.25, 2548.56), (4.21, 2590.02), (9.47, 3507.31), (5.48, 4136.83), (10.89, 5228.88), (6.21, -58.06), (6.92, -178.42), (6.17, 1474.65), (7.98, 5896.3), (4.59, 1924.44), (7.43, 1620.7), (5.3, 4444.48), (3.87, 1294.08), (2.2, 882.37), (3.07, 2904.65), (12.33, 4384.8), (8.96, 6904.0), (7.39, 3743.45), (11.84, 5731.66), (3.1, 874.27), (4.23, 806.53), (8.93, 1301.64), (9.43, 3502.05), (-2.25, -1645.42), (9.73, 2648.89), (0.43, -136.06), (9.5, 4890.65), (14.46, 7200.18), (11.13, 5627.82), (1.43, 360.76), (5.71, 3090.45), (6.35, 335.83), (10.09, 3458.39), (9.3, -590.66), (2.41, -1285.69), (6.93, 3486.19), (5.45, 2609.14), (6.88, 2807.45), (18.58, 9294.05), (11.2, 3530.38), (7.54, 5017.34), (5.79, 210.34), (16.7, 10744.52), (7.43, 4794.0), (7.39, 3795.19), (7.97, 6005.93), (3.71, 2295.7), (15.43, 8441.26), (6.71, 3617.01), (5.71, 4341.27), (13.61, 7813.08), (12.74, 5442.3), (9.73, 8766.19), (14.52, 5783.48), (3.98, 2501.7), (11.98, 5465.62), (8.7, 6001.78), (4.92, 1757.84), (-1.45, -2449.95), (7.89, 5920.37), (14.62, 6385.16), (11.43, 7380.74), (9.55, 4159.78), (9.52, 3769.45), (4.44, 1210.14), (4.57, 2828.95), (8.77, 4906.0), (5.26, 1645.39), (9.64, 3532.33), (8.54, 4171.01), (10.7, 5661.45), (3.86, 3403.19), (9.69, 4304.85), (9.04, 3810.32), (9.09, 6137.73), (1.95, 1620.59), (7.23, -782.66), (9.94, 4471.6), (10.24, 6059.25), (4.56, 3301.05), (0.24, 865.19), (12.16, 7726.09), (12.45, 6930.1), (9.66, 5055.18), (8.67, 3908.77), (3.25, 425.37), (12.15, 7148.59), (9.94, 5125.22), (9.89, 5516.42), (4.7, 4827.96), (1.89, 1387.53), (9.09, 5070.96), (3.22, 3026.21), (10.78, 6805.05), (8.74, 7032.53), (7.22, 2279.11), (7.14, 4609.22), (12.39, 4594.85), (6.37, 3854.84), (8.18, 2516.08), (3.29, -744.98), (9.08, 6835.57), (7.09, 2624.51), (4.5, 3047.51), (4.38, 3114.78), (10.87, 4808.64), (6.32, 5502.37), (11.46, 7837.15), (-2.81, -988.02), (3.98, 4006.18), (13.77, 6390.53), (9.87, 2791.64), (18.07, 10639.15), (6.28, 3082.77), (9.86, 6518.22), (6.33, 3013.74), (7.59, 4404.42), (6.71, -768.9), (9.51, 3786.28), (6.79, 1241.58), (8.42, 4172.66), (2.52, 3945.3), (3.47, 994.28), (17.7, 9501.09), (3.38, 2459.58), (6.98, 3412.91), (13.7, 7091.65), (10.75, 6701.96), (1.9, -1590.56), (6.93, 2652.13), (3.33, 3015.41), (3.48, 2811.56), (-1.93, -3088.1), (10.66, 3727.34), (12.27, 6304.41), (-4.51, -1835.85), (7.95, 3220.47), (7.28, 6094.12), (8.11, 5355.04), (0.84, -250.21), (13.22, 6753.78), (14.73, 9556.99), (11.13, 4525.81), (12.57, 4905.27), (14.24, 9012.77), (11.89, 5693.15), (6.67, 4105.8), (7.19, 5045.6), (8.87, 6020.72), (8.38, 4673.18), (6.13, -479.99), (6.34, 3843.42), (3.84, 1940.64), (8.01, 5670.12), (13.49, 9209.27), (11.54, 6225.59), (9.18, 6421.88), (7.36, 3214.81), (9.84, 4462.09), (7.23, 4315.35), (11.73, 5066.23), (6.27, 1711.7), (9.39, 4977.05), (-6.48, -4591.1), (-2.64, -1415.74), (4.0, 1671.52), (7.43, 3148.17), (10.94, 2359.71), (8.86, 2066.01), (9.98, 3281.31), (14.91, 8137.78), (7.33, 6422.27), (11.1, 5939.09), (12.11, 5813.4), (4.51, 4211.75), (17.06, 11697.52), (12.21, 4988.96), (6.98, 5844.12), (7.1, 1289.57), (12.64, 7543.78), (7.04, 3663.63), (7.24, 3237.75), (12.55, 4792.95), (6.6, 4630.44), (1.66, 2583.78), (7.83, 5383.76), (8.58, 4228.81), (6.25, 4041.47), (5.47, 4825.19), (6.7, 3850.99), (14.97, 6806.34), (3.71, 2707.99), (9.05, 4546.66), (5.74, 2964.16), (7.29, 5351.66), (4.73, 2425.51), (3.02, 2810.37), (11.97, 4426.84), (5.84, 2735.16), (5.25, 2261.86), (2.75, 2469.18), (10.19, 3256.6), (8.22, 7025.81), (6.64, 2785.66), (15.04, 8752.77), (5.28, 770.79), (11.18, 5255.66), (9.93, 2669.17), (6.18, 3507.8), (7.07, 5551.55), (5.86, 5187.99), (3.72, 2263.92), (11.13, 6957.08), (12.17, 4270.77), (14.89, 1677.96), (3.99, 1545.02), (2.99, 585.45), (8.21, 3383.29), (9.22, -714.71), (1.76, -727.4), (8.6, 4138.47), (3.17, 78.37), (7.66, 2595.99), (4.43, 1265.02), (3.41, 753.12), (8.56, 4261.86), (6.9, 4152.68), (8.3, 5737.62), (7.3, 5097.57), (6.06, 3091.16), (6.57, 2785.39), (6.83, 2773.74), (10.86, 5607.22), (7.83, 4564.03), (11.8, 4059.11), (6.43, 805.21), (17.32, 8937.46), (5.74, 3065.94), (5.88, 1718.6), (9.15, 3887.59), (2.78, 1722.36), (11.56, 3723.67), (4.96, 5315.08), (4.31, 6017.86), (10.73, 8082.89), (13.91, 5994.45), (6.68, 3710.71), (5.8, 2176.4), (11.71, 3976.79), (0.73, 1210.23), (4.18, 2325.56), (7.86, 3896.93), (5.69, 1271.36), (6.93, 3725.67), (3.94, 2151.2), (5.78, 2731.15), (10.44, 5832.85), (11.47, 7845.56), (3.83, 2776.36), (2.86, 3677.16), (6.48, 2505.98), (18.48, 8834.82), (6.38, 4613.45), (5.44, -543.06), (8.83, 5118.11), (10.11, 3232.82), (9.13, 5339.31), (9.5, 6147.56), (8.92, 4660.86), (11.12, 7260.44), (9.53, 4722.64), (6.3, 2793.65), (4.98, 3786.38), (10.24, 3531.19), (7.63, 5662.76), (4.81, 2963.33), (2.16, 2558.11), (1.46, -747.62), (9.31, 5168.12), (3.53, 874.31), (5.05, 1434.56), (12.54, 5976.24), (6.34, 3476.45), (4.18, 2200.39), (2.28, 2381.37), (9.22, 3740.97), (1.49, -1816.54), (7.67, 5339.09), (4.11, -280.3), (5.28, 3778.43), (7.25, 3646.21), (4.26, 2615.85), (10.65, 5734.87), (4.84, 3838.48), (11.54, 6049.49), (5.0, 1836.64), (5.91, 2924.95), (3.75, 2090.42), (10.35, 6820.82), (5.46, 2656.41), (3.18, 3653.09), (4.94, 3965.11), (10.74, 6846.05), (11.06, 3330.38), (0.62, -2720.04), (6.49, 3412.73), (1.84, 1800.49), (14.76, 2624.91), (8.32, 7079.57), (13.15, 7715.38), (6.72, 2151.48), (7.08, 1742.54), (4.15, 3159.54), (5.12, 3580.87), (8.91, 4174.35), (6.22, 2348.5), (7.02, 909.4), (4.9, 1745.32), (6.82, 3735.42), (10.91, 4561.86), (4.04, -695.6), (14.59, 6820.21), (4.66, 2263.07), (10.11, 4676.2), (0.19, -1419.75), (5.4, 3497.95), (7.16, 3464.7), (13.28, 7514.47), (12.33, 5081.59), (6.11, 2122.37), (1.23, 1043.57), (12.45, 6943.04), (4.11, 3084.46), (8.44, 4844.7), (5.71, 3348.74), (6.3, 3770.11), (4.91, 2635.7), (8.5, 2150.97), (6.07, 5231.61), (8.37, 6179.05), (17.13, 8975.23), (8.56, 3474.89), (15.68, 6465.33), (10.28, 3643.92), (3.5, 2388.19), (6.05, 4824.63), (8.55, 6477.37), (9.4, 5009.7), (4.15, 2780.78), (6.86, 4194.08), (4.92, 1950.63), (4.69, 2939.49), (2.32, 124.0), (9.5, 2001.06), (5.53, 4077.78), (6.29, 3387.63), (14.24, 10970.47), (0.32, -524.28), (11.49, 6314.09), (7.23, 3581.66), (7.54, 2975.85), (8.03, 3119.19), (4.83, 3007.75), (14.3, 7213.35), (12.1, 4792.69), (5.6, 3065.97), (3.51, 1674.9), (5.96, 1471.43), (4.81, 3152.09), (6.7, 3982.83), (7.75, 4544.33), (4.69, -999.15), (4.33, 3884.03), (9.85, 5811.96), (11.01, 2773.03), (10.59, 3964.04), (8.27, 4599.31), (6.32, 2350.55), (4.5, 1812.75), (6.84, 1674.81), (5.75, 2666.25), (9.62, 3096.24)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=7.5168，ȳ=3725.4943；Sxx=Σ(x−x̄)²=6005.67，Sxy=3024156.27；
β₁ = Sxy/Sxx = 503.5500；β₀ = ȳ−β₁x̄ = -59.6080；
R² = SSR/SST = 1522813821.65/2430139381.21 = 0.6266；F = (SSR/1)/(SSE/393) = 659.5933，p = 4.26e-86；
SE(β₁)=√(2308716.44/6005.67)=19.6067；t=β₁/SE=25.6825；p=4.26e-86；95%CI=[465.0028,542.0971]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=7.5168，ȳ=3725.4943；Sxx=Σ(x−x̄)²=6005.67，Sxy=3024156.27；
β₁ = Sxy/Sxx = 503.5500；β₀ = ȳ−β₁x̄ = -59.6080；
R² = SSR/SST = 1522813821.65/2430139381.21 = 0.6266；F = (SSR/1)/(SSE/393) = 659.5933，p = 4.26e-86；
SE(β₁)=√(2308716.44/6005.67)=19.6067；t=β₁/SE=25.6825；p=4.26e-86；95%CI=[465.0028,542.0971]。
③ 计算结果：
y = -59.6080 + 503.5500·x；R²=0.6266；β₁ t=25.6825 p=4.26e-86；95%CI=[465.0028,542.0971]
④ 结论：在 α=0.05 水平下，p=4.26e-86 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=503.5500），工龄长工资高。
案例 45. 温度预测冰淇淋销量
简单线性回归天热卖得多
① 原始数据：
样本 n=848，逐条 (x, y)：[(17.91, -59.81), (25.6, 71.28), (29.5, 48.24), (29.99, 101.16), (24.9, 71.58), (30.4, 51.2), (25.29, 29.68), (36.96, 107.88), (29.89, 50.78), (30.3, 25.87), (21.11, 24.91), (25.54, 85.85), (23.93, 14.92), (17.43, -4.07), (32.36, 96.9), (26.1, -10.34), (12.8, -108.46), (23.38, -0.4), (21.35, -14.66), (29.42, 60.63), (28.17, 91.56), (41.92, 143.76), (23.48, -61.92), (29.37, 47.15), (15.07, 13.65), (32.22, 31.1), (26.02, 30.22), (17.6, 12.04), (26.65, 47.28), (33.0, 166.7), (33.41, 91.13), (18.08, -1.64), (43.27, 196.92), (32.37, 127.94), (38.53, 182.28), (31.89, 77.69), (33.08, 108.6), (25.03, -15.41), (36.33, 62.69), (23.7, 23.71), (31.08, 53.13), (17.05, -25.07), (22.44, -18.94), (24.49, 14.52), (31.71, 52.46), (31.37, 88.51), (29.3, 27.55), (28.14, 46.34), (37.37, 84.38), (29.97, 32.62), (26.44, 1.47), (34.67, 107.62), (24.23, -6.05), (38.4, 146.22), (28.13, 101.67), (33.09, 84.34), (27.34, 58.92), (18.47, 1.2), (25.8, 36.99), (29.83, 72.59), (34.2, 90.8), (26.56, 68.83), (30.34, 87.14), (36.92, 68.68), (32.13, 107.52), (35.99, 74.32), (29.62, 54.43), (24.51, 43.94), (35.25, 112.37), (22.29, 31.89), (41.48, 156.81), (18.29, -66.09), (30.38, 88.51), (35.18, 127.31), (30.73, 71.58), (25.96, 8.91), (25.7, 81.63), (26.42, 63.51), (24.71, -18.23), (24.73, 96.01), (23.45, 49.58), (24.86, 43.02), (28.84, 43.84), (40.55, 177.02), (33.25, 74.78), (20.89, -14.4), (29.71, 100.13), (28.89, 95.48), (21.49, -9.76), (13.81, -153.28), (33.3, 125.09), (33.45, 73.36), (19.09, 1.97), (24.93, 7.93), (17.18, -52.12), (25.68, 34.9), (24.78, 1.88), (32.71, 34.9), (36.07, 139.18), (30.98, 78.01), (25.95, 37.83), (15.78, -47.86), (27.03, 16.79), (24.99, 38.61), (31.01, 105.78), (34.32, 117.36), (24.69, 8.92), (31.11, 28.78), (15.33, -90.25), (20.02, -41.37), (17.79, -46.63), (19.93, -17.91), (32.78, 108.96), (20.54, -37.03), (26.72, 56.16), (32.26, 135.61), (26.64, 56.25), (29.69, 85.73), (38.71, 165.08), (28.02, 60.24), (27.94, 91.38), (32.68, 85.83), (30.6, 61.78), (28.91, 26.25), (40.15, 141.88), (28.51, 47.28), (15.96, -51.18), (37.69, 128.8), (29.65, 53.5), (22.19, 21.22), (15.76, -2.19), (14.03, -110.26), (26.22, -21.8), (27.22, 54.67), (24.27, -9.37), (10.96, -81.86), (31.53, 72.05), (20.9, 68.97), (30.87, 89.3), (22.75, -28.53), (20.72, 5.31), (18.77, 8.31), (25.44, 22.23), (43.54, 207.02), (28.07, 93.43), (26.67, -15.14), (34.04, 107.0), (27.11, 32.65), (32.92, 89.53), (31.41, 76.4), (31.4, 75.43), (29.12, 55.03), (29.34, 33.57), (29.17, 34.39), (14.88, -93.54), (37.03, 101.22), (31.21, 23.77), (33.31, 66.37), (25.92, -31.13), (28.58, 19.8), (29.09, 63.95), (22.24, -2.24), (30.46, 14.55), (27.49, 48.58), (21.9, -21.26), (17.9, -21.85), (27.81, 101.74), (39.43, 144.9), (24.52, -17.27), (36.62, 90.01), (33.63, 66.59), (25.3, 72.65), (32.74, 113.84), (33.63, 123.58), (41.56, 186.96), (29.54, 65.45), (18.35, -30.34), (27.39, 25.57), (19.9, -11.18), (20.23, -21.68), (32.5, 110.62), (19.66, 6.81), (22.25, 41.96), (26.35, -4.59), (21.84, -2.19), (24.72, 55.41), (35.63, 97.74), (23.51, 35.46), (18.35, -36.25), (34.94, 64.69), (28.18, 56.37), (19.27, -52.15), (27.79, 13.19), (25.18, 37.1), (28.93, 92.42), (29.59, 43.89), (17.84, -58.25), (22.56, 58.72), (29.97, 60.63), (31.08, 102.91), (26.07, 27.74), (19.32, 16.03), (34.27, 78.88), (33.82, 50.56), (31.26, 63.91), (34.86, 122.77), (34.38, 102.2), (27.35, 52.32), (26.58, -0.36), (31.2, 50.05), (35.71, 125.08), (30.58, 51.57), (32.05, 69.48), (12.96, -74.8), (18.19, -45.18), (30.39, 51.02), (34.29, 89.24), (28.97, 59.27), (22.5, -7.28), (25.6, 21.93), (27.7, 32.83), (30.81, 32.11), (39.88, 213.31), (29.23, 94.5), (22.15, 20.06), (31.7, 72.49), (25.94, 31.1), (28.2, 14.71), (24.15, 44.27), (16.88, -87.72), (20.38, -16.02), (20.85, 17.38), (28.54, 54.93), (17.76, 46.63), (36.98, 91.9), (26.78, 33.68), (17.89, 57.96), (33.65, 89.03), (23.77, 15.57), (29.79, 58.76), (30.02, 32.4), (27.4, 63.0), (23.09, 65.75), (32.74, 121.26), (22.78, -8.6), (33.53, 97.99), (35.18, 126.05), (25.73, -13.05), (33.1, 78.6), (27.0, 75.74), (36.3, 91.75), (16.65, -19.37), (32.01, 97.25), (27.72, 92.75), (27.4, 56.62), (25.42, 6.88), (20.1, -24.89), (34.29, 68.56), (24.04, 31.12), (42.63, 124.29), (19.84, -2.88), (28.21, 96.88), (39.77, 151.67), (20.22, 20.13), (27.94, 82.21), (24.97, 19.61), (28.98, 38.14), (23.77, 44.69), (28.42, 72.25), (17.97, -40.08), (26.06, 54.19), (25.22, 12.54), (18.69, -54.13), (22.02, -0.71), (18.75, -11.97), (37.87, 148.24), (25.8, 39.56), (25.01, 5.39), (27.69, 46.33), (38.75, 203.95), (30.85, 118.9), (36.59, 65.69), (36.41, 92.66), (31.38, 91.06), (24.5, 21.19), (28.7, 55.84), (27.41, 62.2), (38.13, 116.28), (28.92, 78.72), (24.86, 8.35), (28.47, 108.14), (24.6, 31.78), (27.95, 68.26), (39.01, 124.8), (24.66, 60.65), (18.71, -47.14), (28.52, 26.06), (27.86, 58.57), (27.61, 64.29), (31.47, 80.81), (33.94, 93.92), (36.23, 76.92), (22.83, 22.44), (27.79, 120.94), (26.78, 111.5), (14.01, -81.67), (23.84, 20.15), (22.74, -8.68), (33.7, 113.66), (24.73, 42.71), (19.6, -13.59), (25.25, 20.35), (28.93, 40.61), (27.43, 29.83), (39.68, 140.31), (24.58, -33.23), (23.85, 26.58), (24.04, 27.29), (32.43, 89.32), (34.72, 136.58), (23.48, 15.45), (38.62, 138.49), (28.56, 68.97), (21.46, 30.9), (24.84, 36.5), (10.84, -127.16), (29.25, 97.36), (28.75, 80.8), (14.18, -64.43), (31.85, 95.4), (30.28, 89.43), (27.03, 42.47), (18.13, -22.08), (20.72, -39.92), (25.83, 23.0), (24.66, 84.04), (23.2, 8.51), (35.59, 122.46), (19.05, 0.07), (32.04, 92.0), (20.41, 8.25), (20.81, -25.21), (34.35, 33.85), (23.89, 12.11), (30.02, 47.33), (23.05, 39.42), (37.51, 105.67), (25.97, 29.32), (23.96, -1.89), (26.5, 79.43), (22.3, -13.32), (23.27, -31.5), (30.88, 42.68), (37.42, 129.62), (24.99, 41.15), (27.9, 38.01), (20.44, -18.95), (37.48, 103.52), (29.2, 95.22), (28.81, -18.82), (35.19, 118.85), (25.77, -28.61), (35.26, 91.22), (31.28, 60.56), (29.41, 73.96), (36.44, 120.61), (27.5, 45.97), (33.59, 112.14), (34.0, 16.63), (32.7, 101.97), (30.42, 115.93), (37.94, 131.77), (18.12, -20.93), (22.35, -30.57), (30.25, 60.76), (27.04, 80.31), (24.13, 7.53), (29.14, 50.6), (37.13, 179.72), (25.25, 75.77), (20.32, -43.31), (16.18, 13.43), (24.02, 51.78), (27.61, 63.76), (26.56, 20.6), (26.88, 119.06), (20.72, -106.07), (29.99, 89.61), (44.39, 197.19), (30.55, 84.27), (24.01, 40.01), (33.24, 132.32), (16.9, -18.71), (19.59, 23.56), (29.7, 43.3), (14.58, -52.08), (31.3, 85.38), (33.92, 71.85), (22.55, 24.32), (34.43, 93.09), (35.71, 89.33), (16.25, -57.44), (23.6, 71.73), (28.13, 105.65), (23.78, 2.33), (36.15, 117.97), (22.22, 1.68), (27.05, 20.56), (22.35, -24.65), (34.44, 106.56), (32.57, 67.2), (35.03, 128.4), (26.76, 36.51), (30.69, 31.45), (30.65, 84.64), (25.66, 45.16), (20.49, 48.05), (30.51, 62.59), (32.65, 125.98), (27.11, 84.25), (28.1, 33.18), (30.82, 41.29), (31.33, 82.73), (24.75, 30.97), (27.82, 58.21), (32.52, 88.73), (26.41, 22.25), (18.61, -37.12), (26.1, 49.04), (39.04, 172.74), (30.24, 68.35), (21.51, 1.26), (20.07, -16.36), (31.03, 69.7), (28.43, 98.18), (32.67, 91.12), (36.11, 134.83), (34.9, 146.47), (27.93, 88.41), (19.94, 36.41), (37.5, 121.21), (25.87, 66.38), (33.49, 92.53), (27.77, 57.66), (25.19, 8.29), (31.07, 13.6), (28.18, 67.72), (32.69, 129.67), (31.89, 37.37), (20.97, 5.64), (23.69, -0.79), (27.22, 16.95), (41.38, 149.46), (25.06, -6.95), (27.96, 112.81), (24.57, -38.7), (30.7, 59.15), (15.61, -54.49), (21.74, -33.1), (25.6, -22.28), (30.17, 70.1), (28.42, -13.39), (35.31, 82.64), (22.99, 48.53), (34.05, 94.52), (21.65, -11.14), (23.78, 41.15), (31.68, 49.52), (31.67, 98.52), (37.01, 127.67), (23.7, 60.58), (27.82, 4.52), (21.33, 12.31), (36.9, 128.0), (34.74, 167.96), (29.68, 123.4), (25.86, -23.8), (29.54, 87.83), (27.33, 59.48), (23.86, 26.68), (33.07, 128.49), (29.99, 96.24), (35.81, 127.96), (28.47, 49.83), (28.13, 67.98), (28.78, 64.48), (26.17, 33.53), (33.59, 64.32), (35.64, 84.86), (26.1, 13.89), (12.91, -38.59), (27.48, 38.25), (23.9, 60.07), (24.81, 7.61), (27.02, 31.03), (13.64, -73.73), (23.47, -30.88), (21.8, -11.05), (37.38, 137.98), (30.88, 59.62), (19.53, -21.65), (28.14, 55.68), (27.73, 54.48), (14.88, -38.71), (36.22, 156.75), (24.63, 21.38), (33.39, 126.0), (26.56, 32.76), (25.64, 30.54), (27.79, 27.07), (36.98, 147.75), (25.76, 5.66), (31.15, 64.55), (32.05, 110.23), (29.2, 79.54), (28.37, -0.26), (7.27, -124.93), (22.59, -15.97), (25.08, 60.44), (29.05, 58.2), (20.49, -73.33), (26.46, 7.33), (31.57, 65.15), (19.25, -19.65), (21.29, -55.53), (33.67, 128.23), (20.14, -35.95), (36.86, 147.2), (32.91, 13.17), (38.45, 176.5), (30.21, 50.39), (26.29, 41.05), (36.85, 135.32), (24.02, 16.54), (24.16, 18.98), (29.33, 39.23), (28.52, 51.43), (33.41, 68.48), (27.55, 15.7), (24.13, 22.78), (35.68, 99.98), (32.41, 79.55), (29.29, 84.65), (28.13, 27.39), (33.68, 69.14), (28.56, 73.02), (19.9, -33.55), (35.33, 104.17), (24.95, 40.83), (28.29, 79.91), (27.7, -11.8), (27.82, 62.18), (24.68, 7.13), (29.9, 94.77), (32.08, 83.3), (28.45, 12.35), (26.86, 60.76), (25.71, 36.41), (17.87, 13.46), (23.58, 53.07), (30.1, 92.59), (29.1, 36.4), (28.73, 33.6), (30.58, 48.91), (18.2, -39.23), (29.93, 39.6), (22.51, 1.28), (15.81, -27.22), (23.16, -6.12), (33.83, 106.24), (26.2, 46.1), (32.28, 27.82), (28.93, 91.09), (27.88, 35.22), (28.96, 64.15), (25.99, 25.08), (27.92, 91.46), (45.24, 157.18), (25.58, 55.0), (32.75, 82.27), (36.74, 100.65), (22.04, 58.93), (33.29, 88.8), (24.07, 28.81), (22.18, 9.45), (19.62, -62.73), (31.3, 113.17), (25.05, 74.62), (31.87, 89.62), (29.41, 98.68), (30.0, 115.38), (15.27, -35.89), (29.35, 57.14), (16.52, -64.99), (24.08, 11.12), (34.85, 105.4), (25.27, 38.31), (38.35, 135.88), (32.36, 74.16), (29.72, 69.3), (35.13, 140.01), (29.29, 32.03), (29.25, 22.47), (21.41, -49.07), (30.64, 71.04), (25.8, 60.41), (39.37, 166.15), (23.76, 6.78), (23.09, 3.2), (40.19, 86.05), (33.98, 64.68), (22.15, -8.62), (37.9, 132.15), (19.46, -51.26), (44.64, 153.48), (26.44, 34.45), (33.75, 138.46), (26.68, 42.22), (26.37, 20.62), (28.67, 8.84), (25.81, 32.34), (28.21, 4.72), (38.49, 142.73), (29.64, 0.59), (26.29, 52.78), (29.38, 100.25), (6.49, -123.76), (17.96, -42.78), (28.77, 52.79), (22.76, 13.48), (16.65, -111.68), (27.14, 49.2), (32.8, 98.6), (26.17, 95.62), (35.56, 41.69), (24.89, -1.33), (36.38, 113.09), (29.32, 13.81), (25.73, 34.16), (29.53, 57.79), (29.96, 98.99), (22.71, 36.6), (37.59, 133.21), (30.66, 75.73), (25.03, -9.89), (28.39, 26.64), (35.03, 68.72), (26.98, 26.77), (38.57, 156.59), (20.38, 10.53), (22.19, 17.42), (30.91, 30.06), (32.43, 137.78), (20.09, -54.3), (28.68, 32.16), (26.04, 26.54), (32.79, 54.37), (20.36, -3.47), (30.7, 107.44), (29.35, 67.25), (34.36, 87.09), (18.8, -21.14), (24.86, 8.54), (27.86, 95.55), (24.9, 87.3), (24.88, 58.82), (25.99, 16.29), (23.97, 70.38), (34.64, 70.44), (21.31, 13.18), (44.91, 197.56), (28.89, 35.75), (26.29, 65.9), (27.23, 46.09), (28.07, 55.65), (28.36, 38.03), (30.97, 37.87), (31.64, 118.52), (15.38, -60.89), (37.98, 132.48), (25.06, -20.32), (21.64, -36.9), (35.79, 87.75), (23.96, 48.47), (28.59, 106.7), (26.97, 9.01), (18.16, -18.43), (32.02, 94.67), (23.65, 25.48), (29.03, 124.67), (33.44, 113.12), (28.48, 129.65), (23.32, 56.02), (22.04, -17.9), (27.4, 78.99), (30.34, 46.64), (36.64, 145.51), (23.97, 5.13), (20.81, -35.4), (36.4, 81.53), (25.83, -2.69), (32.81, 122.05), (19.89, -13.03), (22.33, 0.86), (32.67, 36.36), (29.13, 25.02), (27.34, 3.54), (23.7, 39.25), (26.65, 82.69), (21.63, 34.91), (28.87, -3.09), (34.23, 31.57), (31.09, 135.37), (15.94, -49.15), (29.67, 56.0), (28.26, 67.08), (22.39, 37.37), (18.38, -77.46), (22.75, -16.99), (35.19, 144.99), (27.2, 116.56), (27.49, 74.79), (23.68, 0.19), (28.22, 45.56), (36.25, 141.2), (24.22, 37.95), (10.32, -93.25), (31.89, 93.38), (22.3, -19.49), (18.78, -46.75), (36.56, 103.32), (21.47, -67.75), (30.65, 81.25), (33.7, 140.31), (28.35, 39.02), (14.82, -23.15), (19.56, -4.04), (27.0, 76.01), (42.79, 153.76), (33.41, 98.71), (32.26, 80.06), (25.49, 47.17), (29.12, 106.29), (20.46, -3.42), (29.66, 46.09), (26.6, 16.49), (32.42, 92.92), (16.03, -4.41), (30.29, 46.14), (32.7, 90.12), (18.85, -9.26), (27.22, 70.11), (28.86, 15.81), (30.03, 8.73), (30.1, 108.45), (13.9, -77.99), (29.37, 51.26), (22.86, -11.29), (30.92, 64.32), (27.02, 52.64), (25.12, 44.33), (31.08, 130.51), (24.94, 14.11), (21.39, 9.17), (22.9, 30.18), (25.37, 9.74), (33.26, 125.16), (24.94, 16.96), (29.9, 51.43), (43.84, 109.75), (29.55, 132.78), (16.28, -54.84), (22.88, -19.73), (23.04, -33.27), (35.72, 97.98), (29.46, 76.07), (30.04, 108.86), (29.94, 43.06), (38.01, 190.98), (26.52, 10.52), (33.08, 87.48), (26.1, 24.3), (30.81, 80.87), (24.62, -9.49), (25.91, -1.61), (28.07, 51.74), (16.84, -31.5), (39.46, 169.0), (30.55, 49.07), (30.15, 53.8), (21.71, 29.89), (13.85, -61.38), (27.26, 35.0), (29.03, 68.7), (33.25, 66.62), (32.54, 83.26), (28.93, 63.92), (19.0, -26.64), (27.09, 48.96), (13.73, -90.33), (18.4, -45.22), (39.51, 116.63), (31.95, 111.86), (34.82, 186.24), (33.5, 96.39), (18.04, -15.61), (19.2, 12.05), (24.82, 26.98), (21.99, 20.44), (18.73, -47.65), (27.05, 37.33), (19.83, -33.32), (32.5, 19.79), (19.5, -12.65), (31.07, 87.81), (30.03, 65.2), (28.05, 45.24), (31.31, 88.06), (32.51, 102.3), (22.39, 3.94), (31.44, 99.55), (27.33, -5.02), (29.54, 69.51), (15.0, -89.08), (29.77, 62.88), (33.36, 63.92), (32.24, 67.23), (35.55, 114.16), (32.48, 77.65), (24.48, 18.47), (33.99, 107.12), (15.33, -35.31), (27.99, 65.88), (32.53, 76.21), (26.76, 53.42), (22.02, 51.4), (12.32, -115.16), (26.08, -8.86), (26.54, 48.74), (41.28, 148.33), (36.73, 88.31), (28.01, 20.7), (26.0, 59.03), (20.42, -35.15), (21.89, -12.89), (24.59, 22.69), (25.23, 35.63), (25.68, 96.01)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=27.6370，ȳ=47.5433；Sxx=Σ(x−x̄)²=32819.39，Sxy=264483.81；
β₁ = Sxy/Sxx = 8.0588；β₀ = ȳ−β₁x̄ = -175.1769；
R² = SSR/SST = 2131413.22/2889566.10 = 0.7376；F = (SSR/1)/(SSE/846) = 2378.3799，p = 5.12e-248；
SE(β₁)=√(896.16/32819.39)=0.1652；t=β₁/SE=48.7686；p=5.12e-248；95%CI=[3.9542,12.1633]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=27.6370，ȳ=47.5433；Sxx=Σ(x−x̄)²=32819.39，Sxy=264483.81；
β₁ = Sxy/Sxx = 8.0588；β₀ = ȳ−β₁x̄ = -175.1769；
R² = SSR/SST = 2131413.22/2889566.10 = 0.7376；F = (SSR/1)/(SSE/846) = 2378.3799，p = 5.12e-248；
SE(β₁)=√(896.16/32819.39)=0.1652；t=β₁/SE=48.7686；p=5.12e-248；95%CI=[3.9542,12.1633]。
③ 计算结果：
y = -175.1769 + 8.0588·x；R²=0.7376；β₁ t=48.7686 p=5.12e-248；95%CI=[3.9542,12.1633]
④ 结论：在 α=0.05 水平下，p=5.12e-248 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=8.0588），天热卖得多。
案例 46. 训练量预测运动成绩
简单线性回归练得多分高
① 原始数据：
样本 n=828，逐条 (x, y)：[(4.1, 51.2), (10.7, 77.94), (4.18, 43.1), (10.22, 66.39), (6.67, 53.28), (3.3, 25.07), (5.85, 54.07), (4.44, 48.99), (10.77, 68.26), (5.74, 47.57), (4.36, 40.2), (4.3, 45.86), (4.82, 42.14), (6.36, 51.12), (5.72, 51.86), (8.73, 66.35), (6.73, 40.91), (5.15, 51.54), (3.13, 39.46), (6.46, 47.82), (5.53, 40.17), (7.21, 61.03), (9.86, 76.39), (7.22, 54.67), (3.91, 42.46), (5.68, 47.51), (5.29, 54.96), (5.11, 33.25), (7.66, 49.36), (3.97, 37.64), (6.8, 66.06), (9.18, 73.27), (3.23, 42.51), (7.13, 53.16), (2.98, 38.09), (5.87, 40.77), (2.28, 31.99), (5.87, 54.13), (5.98, 47.27), (5.35, 51.57), (5.54, 49.47), (8.98, 70.12), (7.8, 57.65), (6.05, 46.65), (6.38, 58.95), (3.92, 27.7), (3.7, 40.27), (6.75, 55.95), (5.42, 52.12), (6.04, 49.64), (7.69, 70.71), (6.01, 47.29), (4.57, 37.21), (4.9, 37.9), (2.34, 35.6), (8.63, 61.33), (2.77, 31.08), (8.39, 55.34), (7.36, 55.9), (2.44, 31.32), (3.77, 35.57), (4.77, 50.62), (6.22, 49.72), (6.81, 58.17), (6.49, 37.36), (4.64, 40.17), (4.82, 52.68), (5.73, 51.66), (7.75, 52.01), (5.57, 51.52), (6.36, 45.68), (5.66, 50.35), (8.2, 54.71), (6.96, 65.67), (5.53, 46.82), (6.03, 59.22), (10.37, 74.22), (3.66, 42.55), (3.33, 37.07), (4.32, 40.88), (6.58, 53.17), (4.98, 35.55), (7.34, 53.95), (7.7, 56.34), (3.14, 41.2), (6.4, 57.24), (6.15, 50.01), (7.24, 58.56), (8.05, 62.62), (5.08, 56.59), (6.85, 61.36), (5.61, 47.76), (2.79, 33.13), (6.37, 53.19), (5.5, 48.32), (6.15, 35.69), (4.34, 40.11), (6.47, 50.69), (6.27, 46.87), (6.22, 44.6), (5.41, 35.71), (1.72, 27.76), (4.38, 38.24), (7.31, 55.96), (4.27, 41.47), (6.46, 54.36), (7.21, 53.16), (4.13, 32.46), (6.28, 43.52), (4.01, 40.7), (6.38, 62.55), (11.56, 73.73), (6.33, 48.07), (6.5, 57.37), (4.79, 50.09), (5.37, 44.71), (5.61, 53.58), (7.19, 56.75), (8.57, 60.46), (5.25, 50.16), (5.05, 43.25), (6.64, 56.76), (3.26, 42.87), (5.59, 48.9), (4.01, 47.85), (2.06, 29.5), (6.45, 51.04), (10.06, 66.17), (3.59, 48.39), (5.26, 42.63), (5.25, 41.64), (7.11, 41.99), (8.41, 63.65), (8.33, 67.87), (4.2, 43.44), (7.9, 53.2), (4.97, 55.63), (5.48, 42.57), (3.86, 50.29), (7.4, 52.57), (6.7, 45.45), (5.26, 44.45), (5.97, 46.64), (5.77, 45.77), (7.12, 54.85), (4.05, 45.05), (5.5, 37.98), (8.43, 54.95), (0.79, 31.94), (6.53, 54.31), (7.53, 70.69), (5.72, 47.89), (6.78, 46.31), (2.51, 46.91), (7.99, 56.17), (5.33, 52.08), (7.45, 63.43), (3.63, 40.76), (5.79, 48.67), (9.05, 64.26), (7.42, 60.34), (4.59, 41.89), (9.57, 65.39), (8.2, 56.9), (5.99, 71.08), (4.66, 38.49), (6.69, 59.42), (6.89, 50.92), (6.41, 60.18), (5.95, 50.58), (9.05, 63.04), (8.49, 65.35), (9.83, 60.9), (6.33, 45.69), (7.13, 44.75), (8.45, 55.49), (4.68, 49.9), (5.85, 56.26), (4.59, 45.21), (4.35, 44.15), (4.63, 40.42), (4.04, 52.79), (4.26, 31.15), (5.04, 37.16), (8.99, 55.6), (4.56, 47.54), (5.77, 47.68), (4.61, 38.65), (6.2, 44.23), (5.93, 46.94), (8.28, 60.92), (7.5, 63.32), (9.17, 57.12), (10.4, 71.81), (3.81, 50.89), (7.42, 59.1), (3.47, 36.96), (6.17, 51.34), (6.09, 37.17), (3.39, 32.03), (8.31, 60.61), (4.72, 44.14), (7.05, 55.04), (3.94, 41.07), (4.34, 40.01), (4.38, 35.17), (3.47, 33.9), (3.88, 47.43), (4.9, 43.54), (6.21, 50.94), (10.63, 84.54), (1.56, 36.37), (3.19, 46.47), (5.76, 48.08), (5.93, 52.47), (3.71, 34.01), (5.55, 45.87), (9.45, 64.58), (6.42, 56.39), (6.17, 44.12), (8.77, 71.87), (8.06, 57.36), (6.56, 58.4), (5.46, 44.16), (4.98, 47.01), (10.41, 63.25), (7.87, 60.98), (5.64, 41.91), (7.59, 61.21), (3.67, 44.87), (5.77, 57.06), (7.97, 64.7), (5.2, 46.7), (5.51, 51.03), (5.44, 50.27), (5.71, 44.96), (4.12, 44.38), (8.41, 59.57), (5.01, 49.77), (3.7, 46.96), (3.08, 32.36), (7.03, 62.44), (6.28, 54.29), (6.06, 64.52), (5.37, 50.6), (7.5, 49.53), (5.19, 49.9), (6.44, 58.44), (6.41, 54.02), (7.99, 62.83), (3.63, 50.52), (7.51, 55.8), (7.24, 46.13), (6.34, 45.5), (6.74, 50.06), (5.1, 52.1), (8.86, 62.16), (7.87, 52.46), (4.66, 48.15), (6.1, 56.55), (2.18, 27.11), (3.12, 40.77), (7.48, 55.49), (7.96, 47.88), (5.53, 40.94), (6.63, 45.31), (4.89, 34.59), (2.86, 34.57), (10.13, 81.24), (8.11, 57.34), (4.78, 46.81), (8.78, 50.03), (3.65, 42.8), (2.67, 35.43), (3.78, 50.64), (3.87, 35.66), (7.74, 53.24), (5.12, 40.83), (7.53, 48.29), (2.36, 46.35), (8.54, 64.65), (1.32, 37.01), (6.36, 46.35), (6.21, 50.98), (7.37, 52.41), (9.19, 76.55), (8.29, 56.98), (4.77, 48.57), (8.0, 60.06), (5.38, 57.66), (6.6, 43.98), (4.25, 40.35), (4.67, 54.8), (8.19, 59.45), (5.44, 51.35), (4.35, 41.08), (8.21, 58.01), (7.42, 48.22), (7.27, 57.23), (6.52, 55.64), (4.67, 38.37), (8.04, 67.98), (2.19, 36.9), (5.96, 53.03), (7.8, 56.9), (5.98, 47.38), (7.11, 69.17), (4.26, 57.0), (5.9, 47.69), (4.77, 44.62), (7.31, 58.74), (7.65, 49.55), (5.53, 52.55), (5.81, 55.87), (7.16, 45.96), (10.74, 67.13), (9.32, 73.26), (3.63, 48.59), (6.76, 43.97), (7.08, 42.5), (6.76, 45.56), (4.18, 40.88), (6.43, 48.6), (3.16, 38.24), (6.7, 54.03), (4.99, 39.38), (2.92, 38.99), (2.15, 30.15), (7.03, 51.47), (4.09, 47.03), (5.66, 50.01), (3.0, 31.6), (8.78, 47.39), (5.47, 45.8), (5.87, 53.89), (5.39, 46.75), (6.58, 50.04), (6.91, 50.79), (6.53, 57.08), (4.47, 44.42), (7.49, 54.44), (3.79, 36.75), (2.32, 39.91), (6.69, 50.98), (7.92, 52.86), (6.82, 43.93), (3.09, 36.82), (5.61, 41.0), (6.79, 52.0), (5.3, 36.18), (5.76, 68.73), (4.8, 40.36), (4.69, 36.36), (4.56, 46.87), (6.55, 46.64), (7.4, 50.2), (8.14, 56.36), (6.54, 51.76), (7.68, 69.94), (4.95, 44.59), (6.5, 64.8), (7.53, 63.03), (8.6, 62.46), (4.05, 32.68), (3.53, 42.51), (7.97, 53.27), (9.1, 66.62), (3.1, 32.91), (5.89, 47.25), (4.45, 38.69), (6.08, 45.83), (5.52, 47.13), (6.25, 53.39), (8.45, 56.74), (6.84, 49.18), (10.06, 63.38), (6.49, 50.97), (9.58, 60.56), (7.09, 56.4), (8.38, 54.58), (4.4, 34.29), (7.24, 40.48), (2.84, 39.1), (3.19, 28.53), (5.86, 46.17), (9.0, 67.66), (6.0, 59.79), (4.39, 47.26), (4.3, 48.93), (6.48, 47.82), (7.16, 50.62), (8.48, 59.14), (5.08, 41.19), (5.92, 50.46), (8.29, 61.8), (5.64, 41.37), (5.08, 47.35), (3.18, 32.43), (4.03, 48.16), (4.58, 51.73), (6.21, 52.03), (6.45, 48.48), (7.59, 52.79), (3.8, 33.49), (6.74, 43.5), (5.91, 45.93), (2.59, 21.74), (9.0, 56.1), (6.71, 58.11), (8.94, 62.0), (8.8, 54.78), (1.97, 37.27), (6.53, 46.32), (7.68, 75.44), (4.38, 41.9), (5.25, 50.87), (5.78, 51.48), (2.31, 41.62), (8.48, 56.93), (7.94, 44.58), (8.47, 64.24), (9.59, 59.51), (7.88, 51.31), (3.03, 51.96), (2.69, 34.93), (6.7, 50.82), (2.04, 39.71), (8.09, 48.51), (7.39, 49.87), (2.57, 32.66), (6.08, 53.36), (7.88, 42.28), (8.04, 53.29), (9.75, 72.05), (3.19, 38.49), (6.82, 59.11), (6.33, 41.27), (7.5, 61.16), (3.28, 51.38), (4.11, 44.2), (6.21, 56.22), (4.27, 45.84), (3.73, 43.81), (4.69, 45.71), (9.07, 55.45), (8.83, 55.9), (7.29, 51.95), (7.19, 49.33), (4.9, 45.88), (7.34, 53.78), (4.68, 42.0), (8.21, 52.66), (5.12, 46.77), (2.56, 25.94), (8.01, 57.03), (4.93, 55.51), (7.07, 48.77), (4.15, 45.43), (2.81, 38.98), (4.31, 37.7), (5.88, 49.73), (6.68, 51.54), (6.27, 41.28), (6.2, 47.52), (9.02, 66.92), (4.51, 40.02), (6.04, 55.64), (5.18, 46.5), (5.11, 53.49), (4.1, 45.3), (7.6, 58.75), (6.83, 55.05), (6.96, 51.68), (7.37, 67.87), (3.13, 40.1), (9.54, 61.14), (4.65, 48.67), (-0.46, 22.18), (5.2, 40.24), (7.72, 64.65), (8.83, 59.46), (6.46, 52.49), (8.32, 62.74), (7.62, 61.74), (4.35, 41.29), (4.87, 53.81), (6.62, 44.68), (4.54, 47.6), (8.64, 53.77), (2.23, 22.38), (7.86, 60.23), (4.85, 50.8), (2.44, 42.95), (5.15, 40.51), (5.89, 42.06), (7.27, 56.9), (4.65, 46.38), (6.11, 48.44), (3.36, 44.86), (8.49, 65.98), (6.31, 46.59), (7.46, 56.39), (6.86, 60.93), (4.42, 35.54), (7.57, 59.99), (5.69, 43.46), (5.71, 44.28), (6.93, 46.8), (8.27, 64.61), (4.41, 42.07), (3.52, 29.56), (6.31, 49.22), (3.68, 41.22), (11.78, 75.21), (5.24, 42.84), (7.02, 62.13), (5.92, 57.63), (4.06, 41.4), (5.55, 41.66), (5.47, 46.29), (7.02, 46.42), (5.47, 50.36), (4.33, 52.57), (-0.08, 26.68), (2.53, 39.01), (5.78, 49.69), (5.44, 48.81), (7.65, 48.82), (6.44, 51.63), (7.98, 48.02), (6.38, 59.89), (7.58, 56.63), (5.69, 44.24), (7.3, 51.48), (6.49, 49.85), (7.3, 48.17), (4.86, 48.72), (3.27, 50.03), (6.54, 38.47), (3.99, 37.48), (4.9, 46.08), (6.5, 50.54), (6.11, 52.21), (9.84, 68.31), (6.8, 51.41), (0.54, 33.73), (5.7, 52.73), (2.42, 29.49), (5.8, 52.09), (5.99, 55.05), (8.62, 57.69), (4.83, 53.26), (7.29, 58.06), (6.45, 46.06), (3.27, 31.26), (4.39, 36.38), (8.36, 69.2), (7.17, 57.64), (8.11, 64.01), (3.55, 31.47), (6.06, 49.76), (5.1, 54.19), (10.31, 67.52), (3.48, 45.51), (2.22, 35.26), (5.69, 54.37), (9.46, 67.73), (8.72, 52.37), (4.16, 40.79), (4.05, 36.91), (4.97, 47.47), (6.04, 44.43), (5.28, 59.04), (6.05, 35.15), (4.57, 42.45), (8.74, 64.73), (5.86, 59.57), (6.47, 59.23), (7.21, 52.64), (5.45, 38.65), (2.87, 28.94), (7.72, 63.1), (6.34, 46.08), (6.45, 49.16), (6.69, 44.95), (4.08, 35.2), (7.34, 42.26), (7.62, 56.69), (8.53, 68.67), (2.88, 47.34), (5.24, 44.99), (7.14, 50.2), (4.72, 42.88), (5.63, 36.19), (3.96, 35.83), (5.49, 39.91), (2.44, 34.36), (6.29, 50.87), (6.05, 47.55), (3.73, 42.11), (5.8, 45.41), (3.84, 42.08), (6.67, 43.23), (7.64, 53.46), (11.33, 65.41), (6.29, 40.49), (5.33, 51.53), (4.43, 39.38), (5.3, 37.6), (1.12, 30.26), (6.62, 43.75), (7.05, 53.55), (8.06, 63.99), (4.88, 51.66), (1.52, 26.79), (7.68, 52.03), (9.8, 66.56), (7.45, 69.54), (6.44, 42.69), (7.46, 51.42), (8.92, 62.06), (3.64, 42.6), (8.3, 53.56), (5.93, 55.59), (6.7, 48.34), (7.11, 53.45), (5.57, 44.88), (5.77, 49.0), (9.99, 70.51), (3.91, 45.1), (2.45, 42.82), (7.31, 63.04), (1.68, 28.17), (7.18, 53.72), (4.67, 50.05), (5.16, 48.45), (5.04, 43.69), (6.67, 52.5), (1.3, 34.67), (9.13, 70.56), (9.68, 58.62), (5.69, 47.35), (6.27, 54.77), (6.05, 50.46), (6.11, 49.05), (5.54, 47.85), (7.38, 47.32), (6.45, 50.25), (5.05, 47.54), (5.42, 40.75), (4.61, 43.37), (2.97, 44.78), (4.3, 33.57), (4.63, 34.81), (9.58, 57.22), (2.89, 41.99), (6.55, 49.27), (8.25, 62.25), (5.7, 53.91), (6.13, 57.48), (7.16, 53.46), (4.13, 43.99), (7.23, 40.04), (6.74, 52.37), (5.1, 37.49), (2.08, 27.75), (4.82, 39.08), (4.74, 42.98), (6.33, 54.99), (6.48, 48.27), (2.33, 37.55), (8.7, 60.49), (5.45, 41.27), (8.18, 66.1), (6.49, 53.62), (1.53, 27.49), (5.75, 61.97), (2.34, 32.28), (5.39, 49.4), (7.46, 51.62), (5.16, 57.61), (8.31, 61.27), (7.93, 58.14), (8.77, 64.85), (5.54, 49.9), (5.95, 45.21), (4.55, 44.3), (7.77, 64.92), (6.13, 51.96), (1.5, 32.14), (4.56, 34.67), (6.88, 53.4), (6.09, 39.35), (4.77, 45.06), (2.79, 42.48), (7.73, 50.06), (7.75, 53.92), (4.88, 54.98), (3.21, 48.05), (11.57, 76.71), (6.38, 57.21), (4.22, 39.52), (7.57, 54.37), (7.76, 55.07), (6.81, 56.14), (6.43, 60.98), (5.0, 50.24), (6.15, 49.79), (7.56, 55.42), (6.66, 52.93), (8.24, 57.28), (5.9, 55.54), (6.79, 48.14), (7.3, 64.83), (6.61, 51.76), (8.2, 58.52), (4.51, 42.46), (4.31, 41.74), (5.66, 46.32), (9.91, 53.2), (5.14, 53.98), (9.01, 66.32), (7.71, 50.63), (6.83, 56.76), (7.5, 59.02), (6.68, 55.07), (3.2, 47.46), (6.27, 47.22), (3.76, 40.07), (4.19, 30.83), (4.24, 41.31), (5.68, 53.88), (3.59, 43.31), (9.14, 60.81), (5.74, 54.02), (8.35, 60.38), (4.26, 39.57), (5.04, 38.8), (8.02, 61.67), (6.8, 54.41), (5.25, 47.68), (7.24, 57.64), (4.87, 42.47), (3.77, 36.22), (4.52, 47.29), (7.9, 52.71), (4.52, 49.67), (3.97, 41.77), (6.79, 65.06), (8.16, 55.81), (4.05, 49.62), (3.65, 40.45), (9.07, 55.93), (0.58, 23.61), (5.35, 44.51), (3.67, 38.71), (8.87, 65.4), (7.57, 54.13), (4.69, 40.3), (2.06, 30.92), (4.16, 44.73), (6.39, 50.12), (3.77, 35.57), (7.08, 47.78), (7.17, 53.44), (6.37, 61.69), (6.37, 59.9), (6.99, 46.6), (6.38, 51.97), (8.13, 71.92), (4.8, 36.35), (8.5, 69.67), (5.2, 42.87), (5.02, 49.04), (7.51, 58.6), (7.77, 61.84), (9.29, 71.35), (3.6, 41.4), (7.59, 58.35), (5.17, 52.86), (2.89, 25.2), (5.22, 49.33), (7.17, 49.1), (1.18, 27.42), (2.28, 36.65), (2.69, 39.41), (2.06, 30.09), (3.86, 51.39), (4.84, 52.98), (6.99, 55.04), (9.87, 65.11), (7.46, 51.02), (5.02, 33.57), (5.1, 43.89), (6.12, 45.12), (4.8, 49.64), (4.98, 39.65), (7.22, 66.69), (6.1, 48.71), (3.22, 38.12), (5.45, 45.52), (10.07, 69.84), (3.99, 34.49), (5.05, 52.67), (8.14, 66.64), (5.66, 56.1), (2.77, 37.79), (6.69, 50.22), (5.68, 47.26), (4.51, 42.52), (5.08, 46.67), (5.18, 50.84), (4.37, 36.82), (6.87, 53.97), (4.34, 33.39), (7.42, 69.74), (5.56, 39.76), (8.16, 58.32), (4.2, 38.46), (2.27, 34.58), (5.57, 38.48), (1.63, 30.41), (6.08, 49.91), (6.15, 47.1), (7.98, 65.89), (6.98, 51.55), (7.94, 47.34), (9.42, 59.27)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.9304，ȳ=49.4931；Sxx=Σ(x−x̄)²=3281.78，Sxy=13602.72；
β₁ = Sxy/Sxx = 4.1449；β₀ = ȳ−β₁x̄ = 24.9121；
R² = SSR/SST = 56382.16/86657.41 = 0.6506；F = (SSR/1)/(SSE/826) = 1538.2752，p = 8.16e-191；
SE(β₁)=√(36.65/3281.78)=0.1057；t=β₁/SE=39.2209；p=8.16e-191；95%CI=[3.9375,4.3524]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.9304，ȳ=49.4931；Sxx=Σ(x−x̄)²=3281.78，Sxy=13602.72；
β₁ = Sxy/Sxx = 4.1449；β₀ = ȳ−β₁x̄ = 24.9121；
R² = SSR/SST = 56382.16/86657.41 = 0.6506；F = (SSR/1)/(SSE/826) = 1538.2752，p = 8.16e-191；
SE(β₁)=√(36.65/3281.78)=0.1057；t=β₁/SE=39.2209；p=8.16e-191；95%CI=[3.9375,4.3524]。
③ 计算结果：
y = 24.9121 + 4.1449·x；R²=0.6506；β₁ t=39.2209 p=8.16e-191；95%CI=[3.9375,4.3524]
④ 结论：在 α=0.05 水平下，p=8.16e-191 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=4.1449），练得多分高。
案例 47. 店铺面积预测营业额
简单线性回归店大营收高
① 原始数据：
样本 n=777，逐条 (x, y)：[(54.24, -26.36), (161.76, 26.72), (36.09, -38.33), (92.39, -5.54), (99.87, -1.81), (115.55, 1.48), (75.66, -28.74), (135.38, 17.81), (191.02, 52.63), (105.77, -2.3), (132.57, 19.14), (27.07, -46.69), (116.36, 8.83), (113.41, 4.46), (115.41, 7.45), (147.47, 22.94), (167.54, 36.59), (94.88, -26.02), (117.11, 12.12), (142.88, 24.43), (68.36, -18.96), (165.4, 41.03), (138.94, 20.03), (156.29, 33.97), (134.6, 18.01), (131.8, 8.95), (176.45, 50.28), (114.12, 3.55), (144.03, 22.82), (110.48, -2.72), (81.16, -16.08), (127.05, 11.53), (117.41, 11.81), (167.4, 35.4), (104.69, -5.81), (103.82, -0.82), (124.1, 13.44), (128.51, 20.45), (173.16, 43.74), (56.38, -38.3), (173.52, 37.3), (101.53, -10.91), (127.88, 17.09), (153.24, 32.3), (73.02, -9.85), (127.22, 20.73), (163.92, 30.85), (137.91, 26.53), (108.05, 0.5), (189.26, 53.67), (129.47, 8.98), (79.47, -12.4), (121.2, 9.49), (62.8, -31.25), (107.22, -5.93), (109.13, 6.76), (163.2, 35.03), (168.74, 31.77), (150.85, 26.82), (149.66, 22.28), (166.89, 43.2), (147.35, 29.22), (181.02, 47.42), (177.94, 46.4), (101.7, 2.45), (122.26, 6.75), (123.89, 8.76), (167.01, 40.45), (108.45, -0.07), (114.31, 3.08), (108.25, -1.03), (172.33, 35.86), (160.08, 22.77), (80.94, -20.94), (113.15, 8.62), (128.09, 25.22), (123.83, 2.0), (123.38, 3.89), (37.69, -44.12), (96.81, -6.11), (177.22, 44.49), (135.03, 10.04), (98.31, -11.41), (103.92, -0.99), (79.78, -17.63), (125.8, 15.17), (97.06, -3.56), (79.42, -16.87), (97.14, -4.07), (132.55, 7.46), (100.4, 0.8), (139.62, 19.2), (66.91, -25.23), (94.63, -16.68), (102.14, -3.87), (62.53, -20.39), (133.77, 19.1), (137.55, 11.31), (116.27, 5.54), (58.39, -24.94), (116.62, 11.37), (122.77, 12.1), (91.06, -8.14), (167.1, 35.5), (65.06, -24.41), (87.82, -21.95), (104.93, 8.84), (85.76, -15.48), (143.58, 24.97), (87.91, -14.24), (79.94, -20.42), (104.42, -0.12), (138.55, 13.39), (122.69, 13.6), (175.74, 42.14), (135.82, 20.52), (155.9, 31.47), (114.17, 10.57), (187.9, 49.8), (150.01, 27.47), (169.51, 41.35), (123.4, 4.22), (60.45, -28.6), (131.29, 11.26), (130.13, 12.26), (109.21, 4.22), (109.78, -0.01), (102.04, -8.46), (100.0, -9.5), (161.55, 35.27), (105.49, -0.0), (112.94, 0.6), (58.22, -27.54), (168.96, 34.55), (55.73, -40.15), (129.26, 7.89), (125.01, 11.31), (128.89, 8.9), (161.81, 42.29), (103.63, 0.17), (114.38, 3.2), (179.94, 36.19), (150.34, 35.28), (189.27, 54.58), (165.19, 33.84), (132.41, 18.12), (43.9, -43.02), (145.6, 20.78), (128.45, 13.17), (74.59, -18.11), (69.22, -21.04), (119.13, -2.96), (104.68, -6.49), (81.57, -12.54), (129.58, 24.88), (115.78, 8.03), (109.06, 6.56), (56.67, -31.15), (165.76, 38.38), (139.01, 15.09), (114.23, -4.29), (130.44, 17.28), (127.37, 10.93), (141.39, 24.28), (121.26, 1.74), (133.42, 12.35), (160.65, 41.93), (100.56, -5.57), (116.08, -4.62), (162.37, 33.69), (155.97, 27.6), (154.84, 35.22), (87.41, -2.49), (210.12, 64.65), (91.04, -15.61), (64.12, -30.69), (133.41, 16.99), (120.32, 9.53), (126.88, 10.04), (86.97, -9.16), (95.34, 3.8), (130.49, 16.56), (180.22, 44.76), (117.31, 0.07), (102.78, -4.36), (86.0, -15.86), (108.75, -0.56), (165.75, 24.02), (98.5, -7.17), (45.36, -40.76), (117.82, 7.26), (82.68, -16.71), (116.15, 9.08), (158.14, 34.8), (152.24, 23.71), (89.15, -4.97), (30.91, -40.73), (141.12, 19.16), (165.25, 29.05), (93.52, -14.44), (154.52, 26.32), (145.13, 18.36), (114.71, 3.48), (101.15, 0.21), (130.83, 12.7), (86.87, -10.25), (36.35, -31.61), (115.61, 9.02), (138.5, 18.92), (51.95, -28.19), (84.78, -10.91), (107.31, 2.82), (75.75, -14.94), (124.87, 23.53), (42.35, -34.23), (128.7, 8.07), (105.16, -3.84), (44.34, -41.86), (155.6, 28.99), (89.53, -5.71), (53.59, -23.78), (157.17, 25.29), (118.3, 1.89), (134.23, 19.16), (98.5, -6.06), (61.76, -28.01), (103.06, -16.41), (123.11, 14.81), (107.98, 4.13), (93.64, -8.71), (175.69, 50.97), (134.42, 13.79), (101.47, 3.74), (80.7, -13.11), (94.77, -8.35), (157.26, 28.78), (171.81, 51.47), (70.88, -23.68), (31.72, -44.06), (142.35, 23.01), (96.13, 2.17), (162.14, 33.85), (161.98, 36.98), (163.95, 35.23), (148.36, 18.31), (57.64, -36.14), (169.78, 41.68), (182.47, 46.74), (69.79, -25.12), (124.23, 11.62), (126.26, 11.86), (75.33, -15.95), (83.04, -21.0), (91.4, -6.68), (161.49, 28.44), (114.99, 6.91), (120.8, 22.07), (163.64, 34.35), (130.44, 17.37), (135.35, 11.31), (82.78, -6.93), (100.09, -6.3), (93.31, -11.21), (137.53, 12.71), (109.11, -0.72), (152.88, 35.6), (95.57, 1.82), (103.68, 0.08), (118.84, -0.13), (100.27, 1.84), (158.91, 34.73), (127.74, 15.95), (213.21, 55.8), (20.61, -51.92), (143.47, 32.09), (86.1, -9.72), (103.33, -5.27), (118.65, 15.87), (147.86, 24.74), (33.29, -54.09), (172.7, 36.29), (108.62, 3.44), (140.8, 20.16), (158.33, 30.27), (92.89, -5.26), (146.65, 16.11), (142.85, 17.35), (90.94, -10.82), (96.78, -7.49), (114.36, 3.37), (121.45, 13.09), (141.08, 20.95), (183.49, 53.7), (208.7, 62.82), (31.26, -43.65), (197.41, 58.99), (135.08, 15.77), (141.56, 21.27), (139.69, 14.94), (115.11, 5.07), (64.29, -21.13), (152.48, 18.26), (158.32, 36.09), (168.84, 43.68), (107.13, 2.09), (90.56, -10.81), (174.06, 36.82), (48.14, -30.44), (128.08, -2.7), (109.76, -0.3), (143.94, 25.79), (173.34, 38.63), (53.76, -30.2), (77.58, -14.58), (109.69, 1.44), (66.78, -20.87), (150.93, 30.33), (140.43, 22.05), (142.86, 21.54), (72.06, -21.46), (116.11, -0.46), (69.36, -28.09), (107.24, 11.34), (142.48, 19.72), (128.23, 17.59), (167.18, 42.79), (134.47, 14.45), (114.15, 3.33), (143.3, 22.48), (136.3, 16.42), (99.42, -8.34), (62.06, -31.07), (94.15, -9.91), (58.64, -31.99), (87.09, -4.34), (137.48, 14.01), (70.17, -29.83), (192.78, 51.92), (158.57, 38.93), (37.01, -44.36), (141.88, 22.72), (149.22, 26.34), (100.68, -5.07), (183.88, 46.53), (179.9, 40.93), (186.23, 43.73), (145.39, 19.56), (131.27, 20.67), (89.22, -4.18), (167.17, 27.8), (130.13, 20.05), (138.37, 19.06), (117.65, 7.97), (113.88, -7.42), (110.24, 2.12), (19.39, -51.61), (79.32, -14.1), (138.23, 17.15), (93.88, -12.8), (74.89, -10.26), (116.57, -9.5), (138.0, 17.77), (169.2, 37.69), (127.29, 13.93), (105.09, -7.68), (91.45, -7.83), (86.94, -14.44), (146.36, 28.16), (186.49, 50.16), (100.05, -1.14), (147.53, 21.11), (132.96, 22.48), (138.35, 18.75), (181.29, 45.51), (114.66, 10.89), (115.55, -5.03), (155.91, 31.25), (148.21, 13.71), (157.86, 34.87), (133.77, 17.11), (51.56, -25.95), (118.04, 11.31), (88.88, -8.58), (148.6, 26.62), (110.3, -3.17), (113.09, 11.06), (157.93, 28.77), (118.23, 3.23), (148.17, 27.02), (160.16, 28.58), (132.78, 11.12), (85.78, -6.14), (113.32, 0.11), (114.61, 8.49), (84.67, -11.34), (108.0, 3.95), (99.81, -6.95), (160.28, 33.56), (170.33, 37.46), (154.43, 32.42), (132.45, 18.06), (64.3, -32.72), (185.45, 47.19), (132.9, 21.65), (118.81, -0.42), (163.86, 34.02), (68.84, -20.92), (123.24, 11.25), (104.62, -4.33), (130.71, 12.78), (194.77, 60.17), (107.84, -6.34), (175.69, 39.8), (163.61, 31.03), (82.04, -15.42), (130.77, 17.38), (117.71, 12.49), (42.1, -39.95), (121.11, 7.74), (153.63, 22.11), (112.75, 0.54), (96.21, -11.65), (118.86, 10.97), (76.2, -19.35), (126.84, 13.42), (102.01, -7.23), (117.61, -1.8), (153.1, 28.34), (161.16, 31.69), (106.81, 1.44), (99.22, -12.35), (160.58, 31.63), (70.96, -18.49), (121.25, 4.64), (163.08, 34.18), (133.14, 18.51), (88.05, -7.73), (84.86, -8.52), (147.75, 20.07), (124.64, 17.25), (177.8, 47.65), (90.77, -8.9), (127.96, 10.43), (98.58, -0.82), (120.03, 14.32), (158.87, 33.68), (182.2, 51.2), (83.59, -20.28), (123.31, 12.2), (87.58, -1.33), (168.87, 45.94), (45.39, -25.73), (44.25, -34.57), (116.64, 10.45), (122.55, 7.76), (31.32, -41.87), (150.4, 34.49), (90.2, -13.99), (138.85, 15.33), (84.46, -14.68), (72.12, -22.28), (126.31, 19.22), (36.21, -35.83), (101.85, -0.41), (111.52, 0.98), (140.04, 24.3), (143.6, 23.26), (135.15, 17.64), (82.1, -13.48), (167.48, 38.63), (131.07, 12.92), (143.64, 19.87), (68.38, -27.27), (172.91, 32.05), (60.71, -27.18), (173.56, 40.77), (73.06, -17.77), (140.35, 28.65), (93.41, -3.9), (92.58, -6.08), (79.33, -25.08), (69.76, -23.18), (66.83, -20.71), (180.96, 41.37), (179.17, 41.07), (108.18, -3.46), (123.62, 15.86), (43.63, -33.93), (111.92, 10.16), (82.63, -15.26), (42.47, -31.25), (110.98, 1.29), (137.37, 15.49), (210.87, 56.83), (143.59, 21.75), (101.16, -4.62), (90.37, -17.27), (72.99, -18.37), (106.1, -2.86), (92.98, -8.72), (43.82, -33.82), (129.08, 6.43), (142.03, 17.03), (112.6, 9.53), (148.18, 19.94), (147.26, 28.48), (157.05, 22.74), (163.24, 39.2), (113.79, 0.28), (123.79, 11.75), (100.01, 0.5), (133.07, 10.08), (48.41, -36.45), (126.94, 8.82), (137.33, 18.87), (119.58, -0.6), (82.45, -13.41), (11.86, -55.32), (111.02, 10.03), (136.84, 23.95), (70.09, -25.31), (117.69, 4.33), (143.12, 18.07), (83.12, -5.91), (105.78, -0.05), (130.38, 11.98), (59.19, -33.11), (157.04, 39.01), (101.45, -1.89), (128.35, 11.91), (153.75, 19.32), (54.37, -39.1), (148.59, 30.91), (182.31, 55.31), (79.35, -12.74), (125.58, 16.05), (169.49, 40.25), (259.97, 93.19), (85.18, -13.68), (157.83, 35.2), (141.25, 25.44), (101.62, 0.16), (79.11, -26.06), (64.16, -16.13), (47.09, -29.84), (120.61, 4.94), (111.8, 8.97), (80.81, -12.04), (112.61, 7.83), (122.81, 11.04), (196.56, 41.25), (171.95, 40.1), (107.93, -0.47), (59.57, -30.09), (127.78, 11.93), (152.8, 25.7), (116.2, 8.41), (33.74, -47.15), (173.11, 37.31), (144.37, 22.44), (94.86, -2.33), (85.94, -16.83), (103.36, 9.45), (107.27, -8.06), (-0.12, -69.09), (184.86, 48.76), (63.32, -28.23), (139.43, 21.81), (148.98, 19.42), (80.82, -12.27), (100.69, -12.56), (167.52, 33.51), (156.71, 27.32), (149.44, 29.94), (131.73, 10.39), (90.44, -18.56), (95.71, -11.64), (124.63, 2.26), (105.32, -1.86), (72.24, -19.51), (215.14, 60.48), (160.35, 33.19), (88.65, -14.46), (172.31, 35.38), (64.79, -23.42), (113.68, 2.98), (169.27, 33.81), (81.01, -17.36), (158.54, 32.03), (158.05, 31.26), (93.97, -5.2), (161.04, 28.58), (67.37, -30.52), (55.12, -25.08), (112.9, -4.79), (96.24, -2.15), (125.97, 17.77), (108.2, 2.66), (49.44, -24.47), (103.16, -5.21), (100.66, -7.87), (139.27, 21.41), (111.45, -1.12), (96.08, -12.53), (111.66, -3.12), (84.27, -16.69), (128.38, 9.08), (63.67, -22.43), (107.67, 3.03), (63.96, -21.98), (161.38, 34.84), (140.67, 26.39), (164.49, 40.41), (114.86, 3.77), (84.77, -13.12), (145.04, 24.42), (117.58, 13.29), (128.38, 6.46), (179.29, 40.11), (50.19, -33.43), (79.11, -16.43), (111.23, 8.38), (181.58, 47.05), (98.58, -4.89), (88.66, -12.18), (121.09, 6.2), (137.54, 23.47), (108.66, 0.74), (77.62, -19.78), (139.95, 25.05), (123.32, 11.04), (152.88, 33.07), (128.19, 16.17), (116.81, -1.94), (139.89, 22.42), (107.74, 2.51), (110.89, -5.59), (85.13, -13.87), (21.1, -56.8), (146.48, 23.48), (92.15, -9.72), (99.63, 2.74), (125.73, 7.28), (159.34, 32.01), (150.2, 24.89), (145.48, 24.75), (88.93, -6.4), (49.29, -28.16), (137.98, 20.85), (146.1, 28.64), (97.05, -3.71), (118.8, 3.21), (139.09, 22.81), (48.08, -34.26), (203.95, 56.45), (123.21, 2.96), (111.59, 4.73), (130.33, 15.14), (164.58, 30.58), (124.6, 3.86), (137.32, 23.11), (67.15, -22.31), (51.0, -37.36), (121.6, 22.14), (96.84, -7.46), (95.07, -11.06), (49.72, -36.42), (166.3, 37.74), (133.51, 14.78), (182.17, 48.15), (123.87, 12.56), (143.43, 21.41), (177.57, 35.32), (93.48, 0.15), (95.94, -0.36), (140.62, 23.2), (154.87, 25.92), (95.27, -12.95), (171.9, 42.29), (124.69, 12.22), (104.9, -3.46), (173.6, 35.63), (110.09, -5.26), (17.21, -51.8), (77.18, -16.44), (164.95, 20.43), (125.03, 12.51), (77.18, -20.8), (52.91, -31.7), (144.72, 21.38), (109.42, 8.12), (102.68, 6.89), (61.93, -27.35), (148.1, 31.0), (159.59, 23.53), (91.73, 2.26), (163.46, 40.02), (94.68, -7.93), (160.13, 33.88), (121.53, 8.59), (173.22, 35.54), (67.35, -23.26), (128.38, 19.95), (101.93, 0.79), (62.44, -16.06), (152.6, 34.9), (184.77, 48.14), (127.63, 10.42), (132.46, 11.96), (116.77, 19.15), (173.32, 43.04), (139.04, 18.25), (154.62, 31.67), (183.57, 43.53), (189.37, 42.53), (122.92, 11.89), (90.11, -3.89), (112.19, 7.3), (133.25, 23.79), (176.18, 37.93), (156.01, 30.04), (124.08, 10.25), (117.42, 9.69), (103.99, 3.14), (142.97, 19.49), (198.36, 51.42), (104.38, -5.49), (212.61, 53.57), (147.47, 23.28), (106.94, -7.3), (112.68, 0.8), (146.85, 23.62), (134.65, 20.15), (108.95, 2.81), (87.91, -6.85), (80.11, -10.23), (123.98, 14.74), (128.1, 11.41), (199.77, 59.25), (97.17, -9.17), (184.62, 45.89), (105.81, 0.57), (114.32, 3.86), (166.78, 39.38), (135.1, 11.27), (87.83, -14.5), (160.02, 37.13), (109.01, 3.06), (112.1, -3.97), (97.83, -15.61), (75.36, -14.34), (116.0, 12.85), (81.65, -9.93), (115.94, 4.65), (144.76, 14.31), (158.87, 38.0), (131.49, 14.41), (196.45, 53.19), (141.67, 28.56), (81.88, -24.74), (232.86, 75.18), (118.92, 7.03), (120.41, 6.03), (96.79, -9.93), (149.99, 29.66), (83.71, -7.34), (143.29, 28.09), (177.56, 49.11), (130.15, 16.42), (75.77, -18.87), (117.74, 6.29), (122.94, 9.53), (79.2, -24.05), (93.39, -1.68), (134.87, 20.09), (114.59, 4.37), (104.35, -4.71)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=119.4958，ȳ=7.8367；Sxx=Σ(x−x̄)²=1153140.37，Sxy=690709.88；
β₁ = Sxy/Sxx = 0.5990；β₀ = ȳ−β₁x̄ = -63.7391；
R² = SSR/SST = 413722.52/432534.48 = 0.9565；F = (SSR/1)/(SSE/775) = 17044.2080，p = <1e-300；
SE(β₁)=√(24.27/1153140.37)=0.0046；t=β₁/SE=130.5535；p=<1e-300；95%CI=[0.5900,0.6080]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=119.4958，ȳ=7.8367；Sxx=Σ(x−x̄)²=1153140.37，Sxy=690709.88；
β₁ = Sxy/Sxx = 0.5990；β₀ = ȳ−β₁x̄ = -63.7391；
R² = SSR/SST = 413722.52/432534.48 = 0.9565；F = (SSR/1)/(SSE/775) = 17044.2080，p = <1e-300；
SE(β₁)=√(24.27/1153140.37)=0.0046；t=β₁/SE=130.5535；p=<1e-300；95%CI=[0.5900,0.6080]。
③ 计算结果：
y = -63.7391 + 0.5990·x；R²=0.9565；β₁ t=130.5535 p=<1e-300；95%CI=[0.5900,0.6080]
④ 结论：在 α=0.05 水平下，p=<1e-300 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=0.5990），店大营收高。
案例 48. 粉丝数预测互动量
简单线性回归粉丝多互动多
① 原始数据：
样本 n=826，逐条 (x, y)：[(7.48, -1381.21), (32.7, 1235.67), (17.5, -383.22), (24.92, 743.55), (13.75, 504.77), (26.46, 507.47), (5.85, -1175.16), (20.12, 16.31), (22.98, 101.55), (27.87, 627.47), (10.38, -646.31), (17.97, -56.88), (28.25, 707.79), (13.71, -420.17), (30.51, 1146.22), (11.39, -556.76), (11.73, -243.31), (16.42, 94.32), (20.61, 515.36), (35.67, 1287.45), (25.34, 1001.86), (23.93, 287.72), (18.52, 89.18), (20.66, 36.41), (17.53, 1193.69), (26.5, 1353.54), (24.64, -482.77), (33.09, 829.91), (22.93, 411.53), (29.17, 951.62), (17.18, 251.74), (15.39, 170.43), (19.86, 521.46), (26.77, 773.97), (23.48, 265.1), (36.36, 1534.46), (20.45, 1002.02), (19.13, 104.7), (33.31, 2285.42), (9.27, -756.51), (12.98, -698.11), (18.83, -188.79), (15.3, 745.36), (16.64, -262.39), (14.8, -204.26), (27.37, 1051.52), (24.72, 304.1), (20.1, 432.65), (37.6, 1293.18), (25.87, 42.91), (14.3, -330.33), (9.18, -586.78), (14.65, -3.84), (11.56, -466.71), (22.88, 775.33), (9.06, -204.01), (17.96, 839.17), (31.39, 1063.57), (11.49, -770.84), (28.99, 979.66), (18.06, 164.01), (10.73, -279.47), (25.24, 487.46), (20.36, 281.87), (8.36, -187.72), (16.11, 193.83), (17.38, 263.89), (12.23, -981.46), (16.3, -152.6), (1.78, -645.22), (33.6, 1774.5), (9.89, -889.82), (10.37, -808.4), (22.26, 129.38), (21.71, 543.22), (17.42, 545.15), (9.37, -53.09), (8.72, -843.29), (21.54, 430.45), (18.63, 82.54), (15.74, -681.14), (14.37, -81.6), (14.79, 277.6), (19.24, 241.59), (14.03, -255.27), (6.29, -719.48), (22.95, 497.45), (8.08, -885.31), (26.37, 1285.31), (14.81, -335.26), (25.52, 496.98), (35.67, 1501.69), (22.91, 356.73), (17.7, 228.19), (23.75, 808.94), (24.19, 338.04), (22.38, -343.26), (14.9, 44.24), (24.31, 143.96), (9.66, -115.77), (27.3, 124.68), (19.4, 35.38), (28.34, 812.98), (31.92, 1057.47), (23.43, 723.79), (14.08, 173.02), (19.99, 490.19), (12.01, -553.36), (40.41, 2532.52), (21.31, -430.2), (26.75, 1599.13), (9.98, -1096.13), (14.66, 57.37), (21.43, 598.65), (22.56, 229.29), (14.73, 12.61), (22.06, 139.54), (20.14, 29.16), (10.11, -578.78), (25.92, 140.36), (25.75, 208.1), (28.23, 761.35), (23.95, 901.83), (18.23, 310.0), (27.86, 865.09), (15.92, -219.27), (19.01, -198.22), (19.18, -71.8), (27.38, 1038.23), (16.11, 117.27), (24.66, 657.89), (17.56, -283.37), (9.17, -603.65), (24.3, 447.0), (33.43, 939.22), (21.48, 164.87), (14.14, -189.71), (24.1, -93.02), (27.74, 343.94), (19.07, -889.77), (24.75, 685.9), (11.59, 406.79), (12.43, -959.81), (30.49, 982.48), (15.06, -399.42), (17.9, 496.08), (22.92, 368.77), (8.74, -785.02), (22.28, -36.87), (17.17, -195.56), (11.12, -211.24), (12.45, 91.99), (16.06, -252.9), (14.46, 28.98), (24.19, 190.82), (18.64, 195.62), (25.81, 1103.14), (19.9, 363.0), (5.63, -580.94), (13.8, -542.76), (13.17, -381.91), (21.32, 297.43), (9.79, -1239.96), (22.75, 544.53), (23.99, 809.06), (28.97, 1029.14), (12.57, -495.41), (26.96, 664.47), (35.59, 1822.46), (28.48, 950.83), (31.85, 1534.28), (26.08, 501.18), (25.24, 446.99), (18.27, -545.05), (17.85, -615.4), (12.72, -497.4), (13.82, -247.83), (16.22, -463.33), (18.96, -494.42), (9.64, -303.57), (12.02, -535.31), (10.4, -782.22), (25.47, 359.01), (27.66, 782.08), (26.86, 700.06), (15.17, -412.28), (25.55, 739.57), (13.12, -60.19), (27.81, 621.25), (32.53, 1467.01), (29.43, 786.38), (10.91, -824.63), (18.46, 491.02), (24.54, 246.23), (27.96, 893.54), (28.66, 1098.19), (19.07, 793.91), (31.36, 1203.52), (8.04, 206.44), (21.16, 305.87), (38.03, 1854.26), (5.63, -828.7), (35.06, 809.66), (25.52, 12.79), (14.59, -474.93), (16.32, -392.26), (23.75, 259.16), (5.51, -585.97), (15.72, -220.77), (33.35, 1114.88), (23.63, 752.57), (12.22, -1034.32), (19.31, 31.53), (25.83, 95.59), (31.41, 1063.02), (8.66, -237.62), (18.24, 461.14), (5.96, -1589.09), (24.7, 1083.26), (18.75, -321.76), (28.25, 894.66), (10.84, -1335.12), (16.9, 334.17), (26.74, 803.23), (19.11, -399.51), (25.99, 69.47), (6.09, -1086.12), (31.43, 865.44), (18.04, 797.25), (13.15, -319.31), (19.08, 31.68), (27.0, 777.61), (22.09, 222.48), (13.02, -402.81), (15.27, -417.08), (21.71, 707.95), (20.18, -106.59), (18.13, 710.9), (3.7, -1129.11), (7.47, -668.06), (17.54, 413.1), (29.69, 1701.39), (28.86, 1229.94), (22.27, -131.5), (12.06, -663.16), (17.12, -256.5), (22.33, 792.85), (21.1, 541.57), (14.26, 241.56), (32.49, 1226.63), (22.87, 732.19), (36.13, 1104.21), (17.5, 240.25), (38.36, 1465.02), (21.01, 1069.71), (16.66, -242.98), (14.03, -973.87), (2.32, -1287.26), (19.36, -143.0), (24.56, 1190.02), (16.92, -687.72), (21.28, 1379.5), (12.14, -1208.12), (14.87, 48.86), (9.47, -92.76), (28.93, 1458.17), (29.33, 902.41), (17.52, -393.94), (19.49, 349.76), (20.61, 1020.77), (30.44, 1097.45), (21.03, 462.27), (16.23, -534.02), (6.73, -358.56), (17.83, 114.99), (30.23, 1058.04), (19.08, -172.68), (18.45, 115.7), (17.58, -281.09), (26.92, 1796.77), (23.61, 684.17), (6.79, -1288.68), (32.94, 1541.14), (13.25, -1484.41), (15.51, -1483.63), (12.56, -219.49), (20.51, 15.27), (9.13, -422.93), (14.84, -29.35), (16.24, 136.97), (25.72, 191.94), (18.67, 125.54), (21.47, 707.26), (14.2, -948.87), (10.42, -78.09), (17.89, 318.22), (23.24, 394.69), (25.4, 741.24), (15.67, 143.29), (23.78, 364.87), (23.58, 220.61), (27.54, 411.77), (8.66, -1003.54), (33.49, 1029.0), (10.53, -617.36), (17.22, -370.04), (14.75, 143.62), (11.79, -665.8), (21.25, 193.78), (7.22, -888.92), (17.38, 350.98), (28.71, 485.11), (25.66, 409.29), (14.79, 109.61), (27.32, 1030.19), (18.05, -130.57), (11.27, -786.87), (29.14, 144.32), (8.72, -534.16), (23.89, 684.3), (12.72, -42.28), (11.78, 150.83), (8.87, -526.46), (22.5, 506.06), (22.93, 1267.5), (11.07, -237.31), (15.05, 240.16), (19.54, 272.72), (30.44, 1063.77), (15.22, -397.41), (4.41, -1878.29), (15.69, -318.91), (14.88, -29.54), (22.51, 570.31), (10.93, -788.68), (31.42, 762.67), (29.89, 356.0), (22.08, 244.05), (27.42, 1197.44), (19.42, 794.59), (11.94, -671.78), (31.52, 1422.43), (5.16, -628.4), (10.98, 132.94), (22.75, 687.8), (29.12, 1287.26), (25.38, 297.01), (14.32, -881.19), (27.31, 37.48), (24.14, -51.81), (22.24, 278.42), (24.74, 300.57), (22.25, -237.31), (19.89, 1087.93), (15.86, -182.0), (16.56, -120.54), (18.24, 1010.84), (31.82, 1232.64), (8.81, -350.99), (21.81, 108.45), (5.46, -514.6), (21.36, -70.2), (13.34, -505.33), (28.53, 1243.46), (14.83, -719.82), (9.64, -545.79), (5.94, -801.01), (30.73, 2266.23), (18.96, 65.7), (23.53, 808.85), (38.93, 1229.57), (14.08, -516.62), (31.73, 1907.6), (20.57, 1440.7), (20.7, 517.68), (27.77, 835.71), (22.39, 973.91), (2.59, -1930.2), (20.87, 269.92), (12.88, -633.16), (11.38, -664.81), (18.03, -83.64), (22.44, 943.3), (17.24, -491.38), (17.75, 163.91), (18.14, 665.91), (19.03, 186.35), (23.14, 1025.1), (17.1, -513.31), (25.97, 575.79), (21.84, 153.44), (11.37, -596.54), (9.04, -1099.89), (4.53, -1168.39), (23.66, 556.12), (20.88, 519.44), (16.9, 84.84), (19.03, -168.89), (16.66, 12.47), (16.4, -22.34), (10.87, -431.46), (18.96, 196.19), (13.77, -536.74), (20.21, -181.76), (22.83, 738.77), (32.6, 742.04), (20.55, 651.12), (14.73, -42.62), (25.18, 494.24), (18.41, 237.97), (23.39, -480.28), (30.01, 724.54), (16.53, 88.46), (18.82, -473.7), (26.29, 628.02), (26.14, 762.67), (27.23, 544.29), (6.7, -1895.44), (17.43, -61.44), (20.31, -368.99), (14.4, -118.15), (11.31, 77.48), (24.3, 55.38), (19.49, 121.05), (26.34, 1104.0), (-0.7, -1830.45), (9.55, -145.97), (10.58, -216.05), (8.52, -455.47), (29.31, 523.04), (11.92, -553.88), (17.3, -432.11), (17.62, 467.54), (13.54, 73.71), (19.36, 761.15), (21.92, -363.85), (28.34, 1480.67), (11.42, -586.55), (13.58, 128.25), (22.2, 478.25), (34.34, 1296.54), (17.94, -908.35), (10.46, -239.07), (15.32, -800.29), (6.99, -1148.75), (21.15, 78.34), (32.64, 1565.18), (13.72, -1035.73), (28.8, 833.72), (18.76, -190.24), (17.74, -205.93), (23.42, 679.67), (22.9, 649.49), (18.54, 595.94), (19.42, 312.25), (31.13, 800.86), (17.71, -442.95), (13.48, 141.92), (44.77, 1905.24), (26.09, 796.62), (9.0, -465.89), (26.25, 1469.71), (14.03, -174.87), (40.6, 1270.76), (9.2, -390.09), (21.24, 23.92), (9.29, -1234.12), (6.81, -741.69), (18.58, 395.11), (26.38, 538.62), (20.74, 234.72), (22.97, 313.11), (11.52, -638.05), (22.45, 70.66), (27.75, 874.94), (32.92, 1387.84), (35.75, 1254.41), (13.85, 31.03), (10.94, -373.94), (22.33, 437.69), (26.44, 1214.08), (21.44, 213.42), (14.62, -160.46), (11.35, -817.36), (18.63, 290.69), (20.39, -524.64), (17.61, 189.98), (19.58, 1154.63), (19.52, 689.21), (18.13, 312.2), (28.78, 443.52), (21.24, 657.15), (21.92, 134.55), (13.68, -339.79), (31.23, 622.77), (9.75, -366.84), (18.29, -134.84), (16.64, -93.1), (38.82, 712.43), (10.76, -731.41), (15.08, -381.01), (7.9, -1119.87), (14.62, -827.53), (23.44, 478.95), (21.4, 31.31), (5.33, -1161.46), (17.87, 20.47), (31.56, 1144.9), (18.95, 77.52), (19.77, 237.8), (17.3, -198.4), (32.83, 1010.09), (23.12, 693.41), (21.23, 434.59), (11.32, -139.99), (2.54, -1509.36), (23.78, 467.02), (22.59, 374.21), (13.54, -45.91), (16.64, 397.12), (13.54, -612.09), (10.75, -429.06), (12.83, -621.37), (24.74, -96.71), (20.42, -10.04), (35.81, 1412.9), (9.12, -1023.78), (12.82, -695.28), (8.0, -1064.47), (23.95, 334.99), (15.01, 436.21), (10.5, -794.26), (19.71, 344.81), (24.94, 580.46), (21.66, -359.02), (9.85, -95.42), (29.93, 717.3), (18.12, -520.13), (21.01, 576.94), (27.43, 330.69), (18.75, -506.33), (27.68, 955.53), (11.99, 369.08), (23.07, 235.24), (-2.07, -1109.81), (23.48, 211.46), (16.04, 423.2), (13.65, -838.29), (26.25, 318.6), (17.17, 225.61), (27.35, 2262.75), (19.41, -329.94), (7.91, 348.92), (19.35, 121.24), (15.06, -597.31), (20.38, 462.97), (18.06, -674.34), (20.15, 250.24), (16.65, 1288.66), (12.01, -1114.48), (19.75, 142.7), (20.91, 789.77), (11.35, -1392.39), (16.77, -244.22), (18.04, 731.72), (15.03, 58.74), (19.84, 521.46), (15.48, -307.28), (7.88, -796.37), (19.67, -126.13), (26.42, 1026.85), (26.78, 299.8), (23.03, 872.09), (30.32, 792.45), (1.57, -749.69), (30.67, 841.34), (21.68, 204.94), (17.41, -208.0), (17.09, -820.43), (15.2, -95.01), (8.62, -208.91), (25.65, 907.94), (6.86, -1377.88), (22.1, 780.18), (15.16, -669.0), (17.65, 433.66), (22.72, 1129.04), (24.51, 767.6), (22.62, 301.62), (29.0, 781.5), (35.05, 1382.88), (8.64, -918.84), (6.81, -275.04), (23.66, 1202.24), (32.49, 1943.38), (18.76, 335.33), (19.82, -116.29), (19.96, 409.87), (24.93, 142.85), (23.73, 409.27), (10.0, -413.39), (20.07, 413.34), (11.34, -192.58), (9.34, -203.87), (16.27, -288.83), (22.15, -414.56), (31.67, 1143.02), (13.22, 368.78), (18.4, -8.49), (17.8, -252.42), (29.15, 1779.49), (16.89, -55.48), (18.06, 337.2), (24.67, 628.26), (22.38, 345.75), (19.5, 256.81), (29.19, 690.08), (19.38, 376.78), (18.19, -933.89), (34.4, 1864.0), (12.65, -589.79), (10.69, -791.44), (14.98, 26.31), (20.43, -336.11), (38.49, 1967.17), (16.68, 371.54), (17.15, -228.7), (18.92, 167.43), (16.76, -863.24), (25.95, 1024.23), (26.28, 815.67), (14.78, -196.77), (14.53, -779.17), (29.32, 185.54), (9.24, -1306.93), (15.06, -600.64), (27.09, 355.95), (23.09, 766.83), (17.36, 208.61), (18.68, 232.33), (27.02, -307.54), (25.68, 1139.93), (18.67, 277.7), (15.49, 134.43), (13.24, -773.84), (10.75, -216.69), (18.53, -88.35), (24.37, 994.6), (20.79, 353.63), (14.87, -168.07), (22.57, 1297.13), (24.32, 344.96), (25.99, 571.77), (14.73, 281.97), (23.24, 252.68), (13.85, -660.18), (7.39, -497.01), (13.87, 208.24), (10.49, -367.34), (12.42, 28.93), (12.44, -365.24), (30.88, 205.05), (9.79, -405.56), (16.04, -438.39), (25.04, 730.45), (19.37, -73.21), (25.97, 844.85), (7.53, -518.22), (18.59, 705.5), (27.66, 1848.59), (30.36, 1164.5), (6.05, -1614.57), (16.98, -204.38), (8.77, -486.24), (28.77, 890.24), (27.72, 386.35), (18.07, 334.15), (23.39, 1009.92), (23.69, 610.47), (37.27, 1692.64), (28.1, 1102.22), (23.34, 135.62), (24.35, 206.35), (14.96, 30.68), (23.51, 265.38), (26.07, 368.35), (16.77, 244.46), (-0.56, -822.21), (25.31, 1692.34), (21.23, 350.67), (20.23, -22.52), (13.19, 540.13), (17.49, -618.4), (1.11, -1588.02), (11.14, -532.04), (9.89, -412.45), (25.1, 815.95), (16.37, 755.81), (27.54, 1064.13), (35.88, 872.0), (23.41, 667.85), (28.09, 510.15), (20.72, 111.25), (28.36, 1197.24), (35.89, 1732.03), (22.42, 523.53), (4.26, -989.83), (21.28, -65.51), (29.53, 1132.91), (23.52, 614.32), (24.45, 423.5), (27.78, 585.97), (18.73, -327.88), (14.66, 229.16), (8.0, -974.47), (16.27, -454.27), (30.62, 907.69), (17.08, 184.34), (20.99, 468.68), (23.67, 623.94), (16.6, 82.68), (19.9, 418.09), (17.47, -46.39), (21.5, -613.77), (16.24, 249.44), (22.79, 786.16), (27.05, 1080.72), (10.64, -965.04), (22.39, -191.01), (1.55, -854.66), (23.37, 179.11), (21.07, 656.04), (20.89, 528.92), (41.37, 2153.46), (13.49, -777.47), (35.84, 1054.49), (21.63, 737.19), (9.21, -1085.15), (20.23, -17.12), (-0.7, -1650.21), (17.97, -99.24), (19.09, 143.64), (26.68, 1050.7), (18.64, 393.24), (15.57, -171.06), (13.11, -488.53), (20.95, 342.73), (38.83, 2533.62), (17.35, 196.75), (24.52, 344.86), (16.8, 241.46), (15.22, 338.5), (10.48, -302.5), (19.04, -475.47), (18.19, -371.1), (10.28, -952.98), (24.13, 532.82), (20.57, 725.02), (27.0, 808.28), (25.51, 116.19), (17.75, 213.6), (25.19, 541.15), (23.11, -277.61), (20.96, 381.54), (20.55, 814.57), (16.88, -765.49), (10.37, -276.33), (25.14, 326.08), (36.38, 1611.51), (7.39, -1024.16), (35.16, 1167.59), (11.82, -201.86), (29.65, 696.19), (21.3, -213.8), (12.49, -718.81), (9.43, -755.66), (24.91, 489.3), (15.3, -378.51), (13.63, -948.41), (29.16, 744.12), (21.94, 62.5), (19.45, -359.49), (13.86, -121.26), (21.38, -26.15), (19.01, -334.47), (25.76, 1043.13), (22.11, 333.28), (21.19, 37.27), (19.17, -92.88), (31.2, 1044.31), (1.27, -1033.86), (10.83, -767.91), (12.77, -373.81), (34.32, 1191.35), (25.27, 208.87), (21.59, -318.78), (20.53, 487.42), (19.94, 111.66), (31.73, 934.28), (31.55, 640.16), (14.57, -54.22), (8.74, -151.35), (24.54, 861.31), (27.34, -263.62), (23.23, 146.13), (15.03, -581.09), (26.0, 1189.28), (21.74, 441.07), (9.19, -674.35), (29.85, 923.1), (11.52, -848.73), (24.49, 614.65), (14.27, -339.6), (28.22, 882.53), (19.14, -178.55), (12.03, -191.41), (25.13, 466.95), (21.22, -108.1), (27.8, 1405.14), (11.52, -238.94), (14.07, -227.41), (18.89, 560.59), (6.19, -180.63), (25.36, 1027.69), (25.57, -28.04), (0.83, -968.14)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=19.5829，ȳ=167.0468；Sxx=Σ(x−x̄)²=48669.02，Sxy=3801476.33；
β₁ = Sxy/Sxx = 78.1088；β₀ = ȳ−β₁x̄ = -1362.5457；
R² = SSR/SST = 296928565.67/434451040.09 = 0.6835；F = (SSR/1)/(SSE/824) = 1779.1211，p = 5.06e-208；
SE(β₁)=√(166896.21/48669.02)=1.8518；t=β₁/SE=42.1796；p=5.06e-208；95%CI=[74.4739,81.7436]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=19.5829，ȳ=167.0468；Sxx=Σ(x−x̄)²=48669.02，Sxy=3801476.33；
β₁ = Sxy/Sxx = 78.1088；β₀ = ȳ−β₁x̄ = -1362.5457；
R² = SSR/SST = 296928565.67/434451040.09 = 0.6835；F = (SSR/1)/(SSE/824) = 1779.1211，p = 5.06e-208；
SE(β₁)=√(166896.21/48669.02)=1.8518；t=β₁/SE=42.1796；p=5.06e-208；95%CI=[74.4739,81.7436]。
③ 计算结果：
y = -1362.5457 + 78.1088·x；R²=0.6835；β₁ t=42.1796 p=5.06e-208；95%CI=[74.4739,81.7436]
④ 结论：在 α=0.05 水平下，p=5.06e-208 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=78.1088），粉丝多互动多。
案例 49. 阅读量预测词汇量
简单线性回归读得多词广
① 原始数据：
样本 n=498，逐条 (x, y)：[(23.41, 3512.11), (23.95, 3822.19), (9.7, 1789.06), (33.87, 6168.15), (28.68, 4746.85), (25.95, 2364.63), (26.04, 3619.34), (12.66, 3647.27), (34.86, 7331.88), (15.71, 1942.04), (22.99, 4461.29), (31.45, 5613.45), (17.55, 3781.9), (18.69, 2842.44), (12.03, 423.11), (15.42, 2216.57), (30.52, 5819.65), (14.16, 1461.02), (1.47, -839.95), (19.78, 3962.69), (36.11, 5571.74), (37.64, 6378.55), (21.19, 4220.98), (36.5, 7494.62), (30.49, 5469.04), (27.64, 4825.26), (14.1, 2861.97), (22.75, 1974.06), (22.84, 2927.77), (6.5, -544.24), (3.76, -641.15), (15.99, 2402.82), (21.14, 3606.74), (25.15, 3582.61), (15.8, 2240.41), (11.98, 1423.29), (24.31, 4127.53), (28.89, 4568.43), (13.76, 2304.41), (10.92, 1418.26), (27.26, 5139.06), (16.05, 2183.56), (13.91, 2219.43), (13.42, 832.33), (21.0, 2325.24), (22.35, 3165.33), (36.25, 4407.17), (12.32, 1142.95), (30.33, 4637.58), (19.75, 3175.1), (16.49, 2002.96), (15.45, 2549.47), (-6.61, -1896.35), (25.12, 3597.05), (29.66, 4430.33), (30.84, 5207.55), (20.89, 1633.52), (21.64, 2775.29), (14.13, 862.86), (11.75, 1202.52), (20.89, 3931.58), (13.07, 1726.8), (13.79, 1219.67), (23.39, 5442.46), (29.74, 5241.68), (29.69, 5178.48), (26.13, 4049.94), (39.41, 7167.22), (17.84, 3116.18), (25.42, 4440.1), (11.05, 842.37), (22.09, 4026.9), (17.98, 3933.88), (11.18, 1080.45), (30.9, 6231.84), (13.39, 1160.32), (14.23, 911.24), (16.89, 1084.57), (22.1, 2933.63), (19.55, 2909.16), (23.16, 3034.06), (18.48, 1308.77), (19.45, 2743.25), (28.6, 3542.84), (18.88, 3801.13), (26.09, 4626.6), (7.23, 34.28), (22.35, 3134.75), (21.29, 2631.6), (20.94, 2926.23), (20.58, 3989.86), (21.54, 3023.06), (32.1, 6121.8), (21.08, 1962.5), (33.2, 5669.04), (24.68, 4043.08), (23.79, 4625.33), (18.81, 2338.34), (30.65, 5174.55), (15.28, 2287.12), (21.28, 3963.24), (13.37, 2542.71), (11.88, 1107.64), (17.78, 1732.39), (6.21, 1241.39), (4.81, -955.54), (30.65, 4620.43), (13.59, 1818.96), (31.35, 5185.67), (27.21, 4898.7), (16.69, 3236.11), (19.36, 3244.87), (34.11, 6595.86), (16.66, 4899.79), (4.82, -936.59), (23.12, 4864.64), (40.84, 5978.47), (20.98, 3662.53), (30.5, 5472.05), (23.28, 2763.45), (14.14, 992.8), (26.07, 4338.43), (16.67, 2674.56), (25.1, 3649.26), (11.11, 1175.57), (20.99, 2021.03), (2.96, -304.74), (33.31, 5604.32), (5.96, -161.52), (18.18, 2526.21), (10.45, 1696.85), (3.1, 20.02), (23.13, 4159.27), (21.1, 2659.27), (3.99, 27.32), (21.49, 2047.63), (24.08, 3692.55), (27.1, 3927.46), (32.92, 6073.36), (21.93, 4034.58), (18.95, 4442.57), (18.46, 1810.56), (22.06, 4764.61), (15.62, 1823.22), (16.01, 1772.02), (20.51, 3703.41), (11.77, 1434.12), (25.24, 3444.95), (0.91, -761.6), (19.67, 2183.02), (28.25, 3690.45), (12.12, 880.96), (29.75, 4813.2), (18.16, 2554.71), (5.44, 333.27), (32.32, 4987.83), (46.83, 8148.37), (20.06, 1180.92), (9.6, -581.64), (28.42, 4086.27), (20.76, 3036.13), (15.65, 552.27), (10.94, 1121.48), (30.42, 4736.37), (23.03, 3658.76), (19.4, 3386.19), (14.18, 2650.51), (10.3, 630.39), (20.78, 2275.36), (8.8, 51.11), (15.13, 3149.7), (16.42, 1963.07), (13.22, 1552.14), (23.05, 4330.24), (15.3, 1805.37), (21.83, 3328.26), (15.08, 4571.75), (26.03, 3381.58), (35.53, 4901.75), (26.72, 5001.59), (13.89, 662.77), (18.08, 3713.39), (5.95, -1311.96), (15.01, 2628.68), (21.45, 4660.4), (25.43, 3820.78), (24.39, 4853.87), (41.34, 8192.66), (18.83, 3355.41), (31.51, 5139.82), (22.43, 3508.83), (29.37, 4702.5), (7.7, 907.12), (23.89, 4020.41), (5.43, -1335.88), (24.03, 3679.47), (10.53, 1447.3), (8.98, 1449.35), (27.46, 3646.27), (7.95, -97.62), (24.77, 4485.85), (16.19, 3719.21), (10.56, -719.97), (17.42, 3564.3), (30.5, 4627.19), (9.73, 280.88), (10.7, 2705.74), (24.18, 3936.26), (22.26, 4374.19), (14.86, 1863.78), (25.52, 4877.07), (34.59, 6066.15), (17.11, 2408.39), (22.06, 5575.95), (28.41, 3795.73), (14.97, 2210.88), (16.95, 2767.38), (8.97, -123.46), (20.41, 1633.22), (36.1, 5814.34), (9.25, 296.1), (32.22, 6170.69), (22.17, 4001.38), (10.19, 1232.28), (21.0, 3487.36), (18.48, 3747.93), (18.6, 2477.46), (13.22, 1924.45), (10.58, 1889.57), (25.31, 3284.76), (12.18, 3217.37), (16.32, 386.42), (23.64, 3483.91), (32.25, 4919.66), (13.95, 961.61), (25.23, 3636.38), (11.36, 379.33), (21.68, 5255.96), (16.3, 1821.21), (27.96, 3069.28), (23.72, 3522.29), (14.84, 2486.35), (15.98, 2771.05), (11.55, 1653.74), (27.77, 3547.26), (31.62, 5110.43), (28.71, 5062.94), (23.06, 2173.5), (21.82, 3565.8), (22.21, 5232.7), (13.4, 2193.57), (25.13, 5044.68), (12.8, 3126.06), (15.66, 1758.43), (24.92, 5372.05), (26.43, 3297.27), (3.04, 1611.95), (22.38, 4532.95), (17.22, 1357.86), (23.86, 3712.13), (25.05, 4619.79), (24.99, 4117.99), (23.85, 4191.91), (23.44, 3838.86), (20.84, 2469.21), (18.47, 2089.84), (21.0, 2846.73), (8.88, 150.29), (15.66, 1359.88), (30.71, 5300.64), (17.22, 2048.83), (24.64, 4624.74), (15.16, 859.27), (19.89, 3373.68), (13.84, 2339.44), (23.83, 3678.48), (15.88, 2212.52), (15.88, 1910.73), (14.88, 2393.75), (13.39, 1823.57), (23.2, 3375.03), (24.2, 3662.36), (14.45, 918.23), (24.86, 4468.8), (29.71, 4541.23), (5.9, 1326.29), (20.59, 2838.76), (15.95, 1122.47), (16.11, 863.81), (15.09, 2301.0), (19.99, 1985.93), (25.09, 4447.17), (19.27, 1227.2), (19.23, 2987.3), (19.25, 2262.87), (35.36, 6226.29), (21.15, 3474.52), (22.39, 2354.03), (25.29, 2564.2), (5.94, -870.47), (22.57, 2905.32), (28.54, 3942.0), (23.44, 4074.67), (21.46, 3003.46), (27.26, 3376.71), (8.14, 613.62), (23.61, 2600.81), (10.91, 611.46), (23.55, 4662.35), (21.41, 3511.21), (14.9, 2793.65), (7.78, 215.18), (31.32, 5585.0), (12.65, 1269.01), (16.88, 3730.42), (25.96, 4573.46), (12.68, 496.13), (19.71, 2548.32), (31.66, 4930.92), (17.14, 1952.55), (16.48, 1685.36), (20.37, 1846.73), (21.92, 4060.6), (39.25, 7243.92), (17.19, 2536.22), (21.07, 3456.81), (27.25, 5538.84), (22.2, 3937.79), (21.6, 4044.42), (12.57, 1712.2), (15.25, 3630.69), (22.84, 4852.97), (22.45, 4282.75), (6.89, 170.81), (29.87, 5657.87), (14.84, 673.14), (29.06, 4180.47), (20.15, 2689.6), (13.92, 768.52), (12.53, 191.52), (6.49, 860.93), (36.0, 6436.46), (14.08, 1331.22), (20.92, 2898.01), (32.25, 5369.86), (26.54, 3879.46), (32.98, 4698.21), (24.82, 4605.12), (22.32, 2534.59), (25.93, 3850.32), (19.98, 2082.3), (27.6, 4322.73), (24.13, 3356.91), (15.77, 3501.03), (21.76, 4749.2), (29.56, 4872.98), (25.48, 5185.32), (19.58, 2923.6), (23.58, 4404.61), (20.36, 2667.71), (26.69, 4490.81), (28.61, 3870.39), (20.78, 2492.39), (10.66, 1790.42), (18.34, 1723.49), (25.55, 3371.11), (10.36, 7.86), (28.98, 5209.29), (18.1, 1949.68), (20.76, 2511.38), (15.42, 1810.25), (35.14, 5324.82), (25.41, 3747.85), (29.37, 3819.5), (8.32, 1295.77), (16.69, 1934.11), (14.65, 1720.54), (23.34, 5741.22), (7.0, 1100.73), (25.04, 4703.53), (29.63, 5565.84), (29.75, 4414.0), (14.52, 2784.62), (22.99, 2797.74), (9.83, 1692.06), (22.9, 4014.47), (26.12, 2850.56), (25.71, 3166.25), (13.59, 902.95), (14.16, 1191.18), (15.19, 1452.46), (10.42, 1948.61), (30.21, 4072.09), (21.07, 3122.37), (16.79, 1989.73), (16.98, 2666.66), (22.51, 4255.48), (17.26, 1488.23), (9.96, 1637.27), (15.12, 1272.56), (10.94, 189.0), (14.62, 2525.21), (3.68, -422.36), (13.87, 2306.35), (14.31, 2546.42), (25.85, 4842.58), (33.4, 5454.78), (17.51, 1522.85), (28.1, 4340.05), (26.66, 3313.17), (21.69, 3773.86), (21.33, 3183.22), (13.8, 344.95), (19.59, 3068.53), (3.36, 316.45), (23.91, 3288.32), (17.11, 2987.3), (19.47, 2919.4), (21.91, 1985.59), (30.03, 5405.1), (28.16, 4840.65), (31.32, 4223.71), (16.63, 1130.55), (16.95, 1918.61), (8.69, 1098.02), (15.94, 2056.2), (24.68, 4949.26), (25.14, 3608.64), (24.59, 4461.11), (10.44, 757.83), (13.68, 2184.3), (23.12, 3418.04), (36.48, 7291.71), (8.36, 254.24), (23.12, 3555.48), (20.33, 2614.78), (26.73, 3894.32), (22.69, 4547.01), (19.46, 3153.9), (29.55, 5178.25), (16.13, 1952.54), (19.09, 3965.21), (21.28, 3313.62), (18.21, 1612.25), (23.69, 4656.16), (25.29, 3637.96), (16.63, 2821.09), (22.56, 2939.47), (18.2, 1352.97), (30.24, 5667.38), (17.25, 3147.1), (20.37, 4041.36), (22.53, 3543.32), (16.91, 1923.99), (19.52, 2204.38), (22.27, 2518.37), (5.87, -51.88), (13.85, 1329.54), (14.49, 1232.39), (5.59, 849.49), (21.01, 3055.73), (29.15, 4902.21), (18.78, 1829.3), (14.61, 1446.44), (26.56, 5308.91), (10.23, 303.48), (19.41, 1410.94), (14.11, 2093.89), (20.54, 3034.28), (15.72, 2461.86), (14.21, 1764.85), (11.09, 1142.34), (29.54, 4878.6), (10.47, 1397.65), (22.44, 3865.95), (17.41, 3348.88), (24.87, 3649.59), (25.71, 6231.01), (15.95, 2717.9), (15.18, 2061.81), (35.54, 6669.21), (20.64, 2178.79), (21.38, 2295.66), (29.9, 4487.87), (24.68, 3629.17), (10.91, 88.08), (9.9, 1886.08), (18.2, 4103.6), (25.45, 3598.08), (25.06, 5142.98), (38.71, 5815.77), (6.25, 717.98), (18.94, 3103.57), (13.27, 2575.4), (26.37, 3885.83), (26.22, 4259.41), (20.7, 3073.26), (21.51, 2665.37)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.1803，ȳ=3004.1942；Sxx=Σ(x−x̄)²=30107.59，Sxy=6001158.11；
β₁ = Sxy/Sxx = 199.3238；β₀ = ȳ−β₁x̄ = -1018.2237；
R² = SSR/SST = 1196173523.55/1506311426.23 = 0.7941；F = (SSR/1)/(SSE/496) = 1913.0266，p = 2.44e-172；
SE(β₁)=√(625278.03/30107.59)=4.5572；t=β₁/SE=43.7382；p=2.44e-172；95%CI=[97.8730,300.7746]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.1803，ȳ=3004.1942；Sxx=Σ(x−x̄)²=30107.59，Sxy=6001158.11；
β₁ = Sxy/Sxx = 199.3238；β₀ = ȳ−β₁x̄ = -1018.2237；
R² = SSR/SST = 1196173523.55/1506311426.23 = 0.7941；F = (SSR/1)/(SSE/496) = 1913.0266，p = 2.44e-172；
SE(β₁)=√(625278.03/30107.59)=4.5572；t=β₁/SE=43.7382；p=2.44e-172；95%CI=[97.8730,300.7746]。
③ 计算结果：
y = -1018.2237 + 199.3238·x；R²=0.7941；β₁ t=43.7382 p=2.44e-172；95%CI=[97.8730,300.7746]
④ 结论：在 α=0.05 水平下，p=2.44e-172 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=199.3238），读得多词广。
案例 50. 步行步数预测卡路里
简单线性回归走得多吃得多
① 原始数据：
样本 n=609，逐条 (x, y)：[(6.43, 7.21), (6.29, 153.41), (9.59, 103.61), (7.44, 120.74), (3.95, 82.76), (7.98, 97.33), (10.9, 137.52), (11.84, 110.76), (9.28, 289.79), (3.15, -56.54), (8.44, 36.56), (11.55, 48.43), (10.4, 156.07), (12.88, 173.17), (5.51, 149.59), (11.2, 135.57), (7.02, 39.6), (7.42, 28.96), (6.63, 164.33), (5.71, 190.48), (10.56, 134.79), (5.93, 59.54), (7.1, 163.54), (5.45, 422.84), (9.94, 77.58), (8.87, 95.16), (11.41, 318.88), (8.09, 132.63), (7.85, 2.06), (7.76, 106.13), (6.16, 42.69), (12.75, -54.64), (13.96, 67.55), (2.92, -79.99), (6.9, 105.53), (9.78, 8.58), (7.79, 465.88), (5.88, 90.88), (10.67, 52.66), (9.97, 379.37), (7.57, 335.12), (3.54, 19.56), (9.02, 130.11), (8.28, 127.39), (10.38, 305.81), (5.3, -36.26), (10.23, -119.17), (8.37, 213.74), (4.65, 112.31), (11.44, 342.92), (1.69, -50.84), (6.53, 188.78), (12.19, 200.27), (7.97, 148.33), (5.91, 388.16), (8.87, 158.5), (12.54, 252.94), (11.86, 232.69), (8.62, 142.72), (8.49, -60.39), (12.99, 541.19), (2.73, -267.17), (7.12, 62.46), (10.73, 179.61), (10.68, 98.09), (8.54, 314.21), (7.19, 193.37), (9.02, 71.81), (4.71, 102.58), (9.81, 317.92), (9.2, 197.46), (7.21, 241.1), (2.32, -51.39), (6.11, 145.22), (8.35, 179.29), (9.22, -83.99), (8.56, 10.88), (10.16, 173.38), (3.17, 253.37), (7.65, 136.41), (6.77, 145.58), (5.82, -68.54), (2.3, 92.9), (4.43, 32.98), (8.18, 224.65), (6.69, -155.22), (2.82, 283.49), (8.44, 177.01), (2.88, -61.84), (2.01, 210.06), (9.46, 239.49), (6.46, 79.34), (7.04, 138.28), (4.71, 4.68), (11.33, 213.68), (8.56, 147.27), (5.72, 128.03), (5.49, 334.49), (6.3, -22.32), (2.74, 89.08), (11.74, 16.46), (5.81, 97.48), (6.83, 284.65), (9.4, 344.34), (8.85, -92.58), (5.27, 112.32), (10.75, 254.04), (8.46, 79.96), (4.9, 102.1), (5.85, 259.55), (15.22, 449.75), (4.14, 285.86), (12.19, 106.48), (8.46, 166.18), (3.27, -85.91), (8.08, 61.56), (7.25, 42.3), (13.74, 99.27), (9.52, 169.25), (10.83, 155.97), (3.34, 12.34), (7.47, 250.33), (6.84, 117.33), (6.79, 162.35), (11.98, 153.53), (6.67, 74.6), (8.27, -7.55), (6.07, 293.46), (9.2, 57.69), (8.86, 397.49), (10.71, 2.18), (13.68, 138.87), (6.9, 24.59), (3.77, -12.08), (5.89, 3.26), (12.28, 176.21), (6.88, 127.88), (8.91, 284.59), (3.84, 113.6), (10.24, 337.46), (5.19, 214.98), (9.47, 268.4), (5.31, 118.77), (7.83, 54.88), (8.59, 222.26), (4.56, 20.61), (11.34, 134.47), (8.19, 5.09), (4.22, 143.95), (4.56, 38.16), (11.9, 45.94), (9.9, 434.44), (11.27, 11.19), (7.96, 103.44), (6.12, 51.7), (11.2, 107.13), (8.28, 164.07), (7.15, 387.47), (11.41, 304.03), (7.28, 49.74), (5.88, 340.79), (7.74, 399.94), (10.1, 43.25), (9.94, 110.18), (9.44, 160.86), (10.89, 268.46), (8.8, 241.0), (10.06, 304.98), (8.13, 329.63), (3.85, 197.25), (7.53, 237.74), (13.46, 393.33), (10.54, 105.37), (7.63, 67.8), (9.96, 233.36), (8.86, -158.19), (3.23, 342.18), (5.09, 75.17), (6.88, 163.81), (8.69, 104.67), (9.74, 175.11), (6.56, 116.33), (10.88, 164.2), (7.42, 231.62), (8.27, 124.23), (14.46, 241.07), (8.67, 69.45), (10.34, 73.06), (8.73, 87.94), (6.51, -23.49), (11.76, 94.49), (7.13, -3.14), (2.66, -1.03), (6.62, 102.66), (12.21, 225.22), (10.08, 121.54), (4.09, -60.36), (5.37, 144.07), (6.56, -72.37), (7.27, 98.44), (4.71, 127.98), (11.29, 95.01), (5.9, 192.87), (6.98, 123.78), (8.47, 246.04), (7.73, -65.18), (12.35, 237.41), (4.27, 200.44), (6.74, 175.11), (8.92, 107.01), (7.01, 126.31), (10.76, 79.04), (7.32, 29.48), (8.38, 37.52), (9.98, 350.59), (8.76, 91.86), (8.97, 271.91), (11.03, 52.71), (12.62, 1.8), (2.19, 140.31), (9.39, 43.99), (8.63, -62.17), (1.64, 3.59), (9.87, 93.0), (3.39, 36.48), (4.45, -231.9), (7.5, 104.8), (7.16, 215.72), (8.19, 197.08), (10.7, -99.8), (6.07, -24.33), (9.84, 54.2), (2.93, -27.65), (11.7, 246.19), (11.3, 215.95), (9.14, 395.14), (7.84, 377.81), (3.85, 161.6), (1.4, -313.12), (5.23, 30.64), (8.33, 42.26), (10.05, 167.99), (6.21, 98.2), (5.05, 202.24), (6.7, 185.14), (10.14, 242.22), (3.28, 188.35), (9.96, 158.18), (5.44, 195.63), (7.59, -39.8), (12.08, 123.89), (6.49, 158.94), (6.39, -119.16), (5.43, 349.3), (8.16, -20.43), (12.48, 224.57), (4.07, -89.74), (8.8, 32.58), (7.08, 165.3), (8.38, 47.82), (4.86, 95.25), (7.92, 138.68), (6.01, -10.84), (6.22, 52.14), (7.09, 9.88), (6.41, -6.96), (2.92, 6.87), (11.43, 512.79), (9.93, 104.88), (13.35, 123.19), (10.49, 98.98), (2.48, -3.94), (7.42, 116.84), (12.55, 165.55), (13.79, 322.96), (7.93, -21.72), (7.33, -107.72), (10.44, 124.88), (10.05, 115.12), (3.07, 13.31), (8.2, 158.43), (9.73, 161.0), (9.84, 72.86), (5.7, 131.18), (5.89, 105.71), (7.23, 150.66), (10.33, 331.01), (8.18, 403.86), (8.28, 189.57), (7.29, 216.63), (5.69, 168.95), (4.06, -25.87), (9.37, 134.99), (9.86, 170.21), (10.27, 132.27), (11.98, 100.15), (5.4, 59.2), (5.56, -11.55), (4.93, -9.29), (11.18, 26.28), (13.46, 33.19), (3.19, -169.17), (1.02, 99.43), (6.84, 137.25), (4.68, 197.78), (-0.03, 242.26), (7.64, 98.53), (12.86, 348.79), (4.86, -34.66), (8.4, -8.5), (9.13, 221.41), (11.13, 362.66), (6.79, 111.61), (5.22, 209.28), (8.98, -26.83), (7.57, 358.35), (11.29, 155.83), (9.13, 169.39), (7.88, 144.98), (6.28, -74.62), (7.19, 200.2), (8.85, 111.09), (14.5, 282.67), (4.13, 68.46), (7.07, 191.75), (9.91, 142.82), (8.71, 264.73), (8.76, 272.49), (5.77, 86.5), (8.77, 122.8), (3.05, 91.14), (10.62, 74.65), (8.82, 229.71), (12.68, 386.62), (7.49, 59.97), (4.47, 133.81), (4.5, 327.91), (5.02, 223.77), (9.96, 175.14), (1.76, 4.25), (5.75, 164.77), (4.39, 75.03), (7.63, -43.76), (7.23, -15.23), (10.45, 236.8), (12.41, 468.65), (7.4, 102.38), (13.45, 125.09), (12.48, 160.15), (3.1, -68.57), (2.17, -69.56), (7.47, 37.58), (7.51, 3.24), (13.62, 75.03), (2.15, 24.43), (8.73, 124.29), (10.94, 165.87), (7.42, 86.1), (10.57, 89.61), (3.03, -63.01), (9.39, 396.33), (3.59, 73.93), (9.65, 170.46), (5.63, 28.32), (4.07, 48.8), (5.5, 142.86), (3.18, -29.14), (7.51, 219.62), (6.87, 147.54), (9.48, 141.26), (7.65, 343.57), (7.31, 159.0), (8.12, 451.14), (7.32, 30.58), (9.26, 198.92), (7.57, 1.67), (11.23, 19.04), (6.84, 368.32), (7.22, 160.37), (4.46, -102.07), (8.67, 83.27), (6.29, 209.84), (2.83, -60.74), (11.08, 328.26), (15.23, 214.21), (7.87, -101.05), (8.43, 115.04), (9.9, 327.56), (10.42, 193.79), (9.49, 254.73), (11.07, 222.84), (8.35, -34.24), (10.66, 110.83), (3.58, -4.7), (11.43, 122.98), (5.53, -85.1), (9.48, 141.38), (4.35, 83.93), (8.87, -9.95), (8.68, 99.1), (4.77, 101.21), (3.8, 12.24), (7.34, 144.45), (2.65, 207.84), (9.77, 35.82), (6.8, 105.88), (5.92, 31.33), (8.39, 170.1), (10.18, 428.63), (8.9, 74.67), (5.51, -165.19), (5.85, 199.24), (6.51, 141.19), (10.34, 89.03), (7.53, 92.81), (7.65, 230.34), (9.44, 141.56), (7.73, -50.52), (6.12, 263.65), (5.03, -47.97), (7.54, 294.23), (5.49, 275.85), (7.24, 168.54), (10.03, 292.96), (5.61, 220.83), (9.53, 3.73), (12.72, 189.26), (7.18, 260.76), (10.2, 119.3), (6.42, 233.26), (10.73, 278.57), (2.97, -222.26), (5.88, 109.72), (8.87, 17.61), (4.03, 28.93), (3.64, 221.17), (5.99, 112.02), (6.98, 112.79), (6.78, -120.65), (9.04, 228.4), (11.75, 123.52), (3.55, -10.52), (5.59, 223.3), (3.73, 308.69), (10.09, 4.35), (8.07, 119.85), (10.19, 184.06), (8.16, 103.89), (6.77, 58.93), (6.6, 134.71), (9.48, 117.56), (9.82, 491.33), (4.56, 27.75), (9.3, 162.61), (12.01, 280.81), (6.54, 103.41), (7.78, 117.25), (3.22, -139.56), (3.35, 20.16), (7.86, -76.21), (1.01, 76.12), (8.48, -56.33), (8.45, 157.9), (7.75, 83.3), (10.45, 259.76), (7.86, 220.65), (10.24, -28.27), (9.56, 263.38), (9.27, 32.32), (5.53, 182.53), (8.27, 196.57), (4.59, 54.89), (-0.12, -71.01), (5.52, 166.91), (1.02, -22.17), (11.21, 259.7), (12.35, 182.03), (11.26, 363.34), (10.61, 33.39), (6.23, 259.24), (8.75, 10.46), (1.1, 64.87), (8.98, -10.42), (7.64, 168.99), (3.65, 239.45), (11.82, 258.03), (10.38, 66.17), (11.18, 160.17), (7.98, 58.26), (7.53, -81.87), (8.21, 235.47), (5.26, 65.75), (11.16, -36.64), (6.0, 227.89), (10.12, 138.82), (5.3, 152.9), (9.91, 68.23), (11.22, 57.91), (10.44, 66.6), (8.32, 197.07), (6.95, 67.3), (6.38, 166.02), (9.05, 251.05), (13.97, 60.39), (7.69, 70.11), (14.97, 291.41), (6.19, 296.38), (12.67, 113.06), (3.81, 138.29), (5.78, 188.98), (4.3, -184.37), (5.06, 57.45), (10.0, 222.19), (5.13, 43.71), (9.25, 128.68), (6.04, 34.43), (8.83, 252.6), (7.54, 202.44), (6.06, 238.53), (11.16, 170.56), (5.91, 128.24), (12.74, 113.17), (6.65, -138.92), (10.73, 240.13), (5.67, 109.77), (9.21, 76.6), (11.23, 174.01), (11.68, 492.82), (7.44, 109.08), (11.89, -9.62), (9.78, 277.76), (4.46, 108.54), (5.53, -129.12), (6.16, -57.9), (6.19, 73.12), (8.27, -15.43), (10.31, 160.58), (9.0, 222.42), (3.96, 38.58), (9.42, 258.97), (6.03, 36.62), (8.22, 81.23), (10.26, 184.65), (8.67, 168.04), (6.76, 181.05), (9.09, 254.82), (9.5, -82.1), (10.34, 158.71), (5.16, 244.59), (4.18, 79.59), (7.7, 192.36), (10.66, 227.67), (12.13, 39.15), (6.75, 149.25), (13.06, 144.22), (6.7, 95.89), (8.72, 76.54), (6.92, 115.1), (4.25, 68.54), (8.05, -20.18), (7.75, 96.8), (6.94, -6.9), (7.71, -82.04), (4.13, -58.94), (4.22, 18.83), (9.06, 313.0), (5.79, 96.73), (7.95, 17.85), (5.35, -105.02), (5.77, 180.51), (13.47, 199.34), (4.96, 177.15), (6.6, 191.67), (6.62, 334.05), (7.49, 25.44), (11.87, 271.08), (6.51, -0.23), (4.19, 57.31), (12.16, 383.01), (6.29, 74.17), (11.43, 35.15), (9.45, 161.05), (8.26, 51.55), (5.02, 53.46), (6.58, 128.7), (3.9, -21.67), (6.91, -83.25), (8.18, -12.91), (6.95, -124.26), (8.14, 29.81), (8.36, 406.93), (7.19, 91.74), (4.44, 35.04), (10.65, 242.78), (11.71, 290.49), (8.31, 114.76), (8.56, 373.27), (8.25, 14.06), (3.29, 23.46), (9.44, 158.63), (10.91, 272.71), (4.37, -55.34), (10.9, 43.81), (13.59, 228.3), (10.0, 112.31), (13.56, 120.1), (9.28, 105.48), (7.04, 138.71), (9.71, 287.74)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=7.8332，ȳ=124.6679；Sxx=Σ(x−x̄)²=4913.62，Sxy=75162.26；
β₁ = Sxy/Sxx = 15.2967；β₀ = ȳ−β₁x̄ = 4.8451；
R² = SSR/SST = 1149736.02/10137529.82 = 0.1134；F = (SSR/1)/(SSE/607) = 77.6486，p = 0.0002；
SE(β₁)=√(14806.91/4913.62)=1.7359；t=β₁/SE=8.8118；p=0.0002；95%CI=[-23.1641,53.7575]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=7.8332，ȳ=124.6679；Sxx=Σ(x−x̄)²=4913.62，Sxy=75162.26；
β₁ = Sxy/Sxx = 15.2967；β₀ = ȳ−β₁x̄ = 4.8451；
R² = SSR/SST = 1149736.02/10137529.82 = 0.1134；F = (SSR/1)/(SSE/607) = 77.6486，p = 0.0002；
SE(β₁)=√(14806.91/4913.62)=1.7359；t=β₁/SE=8.8118；p=0.0002；95%CI=[-23.1641,53.7575]。
③ 计算结果：
y = 4.8451 + 15.2967·x；R²=0.1134；β₁ t=8.8118 p=0.0002；95%CI=[-23.1641,53.7575]
④ 结论：在 α=0.05 水平下，p=0.0002 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=15.2967），走得多吃得多。
案例 51. 睡眠时间预测专注度
简单线性回归睡得好专注
① 原始数据：
样本 n=627，逐条 (x, y)：[(6.86, 60.15), (8.9, 62.35), (6.3, 55.19), (7.14, 59.67), (4.21, 43.67), (5.27, 47.79), (7.0, 56.18), (7.3, 59.39), (7.56, 56.48), (5.81, 48.64), (7.3, 60.76), (5.2, 56.02), (5.51, 54.92), (7.82, 51.7), (6.13, 46.72), (8.04, 51.53), (6.51, 45.86), (6.87, 53.02), (6.62, 57.06), (6.96, 57.24), (7.59, 53.4), (7.23, 55.43), (6.98, 54.22), (6.61, 50.86), (5.83, 57.87), (6.17, 55.15), (5.95, 50.72), (7.47, 55.28), (7.16, 59.49), (7.54, 55.37), (6.97, 59.44), (5.93, 52.14), (6.18, 46.59), (7.06, 51.52), (6.81, 58.25), (6.9, 57.69), (7.93, 61.47), (6.24, 51.68), (6.53, 60.4), (6.4, 56.34), (7.39, 63.71), (8.31, 58.63), (6.06, 48.79), (6.17, 55.35), (5.47, 53.04), (6.72, 47.64), (5.99, 53.78), (7.31, 59.09), (5.87, 55.56), (7.95, 57.32), (7.19, 52.84), (7.44, 58.61), (7.06, 55.71), (7.13, 59.92), (5.75, 48.43), (8.2, 59.56), (8.18, 61.42), (6.22, 51.8), (8.18, 57.93), (7.7, 63.48), (8.34, 63.11), (7.31, 46.85), (6.84, 50.79), (8.83, 55.0), (6.9, 53.42), (9.21, 59.26), (6.35, 43.91), (7.42, 55.0), (7.5, 47.3), (6.36, 46.81), (7.0, 57.76), (8.0, 58.91), (8.22, 58.65), (6.57, 54.5), (6.77, 48.79), (9.08, 60.11), (7.48, 54.84), (7.66, 49.26), (6.71, 52.74), (6.31, 48.7), (6.68, 51.7), (6.04, 53.57), (8.15, 59.59), (6.75, 53.56), (8.54, 52.71), (7.64, 58.27), (6.32, 55.5), (7.35, 55.49), (9.03, 68.55), (7.64, 53.53), (7.1, 54.91), (6.58, 51.66), (7.24, 62.73), (6.22, 50.29), (8.1, 54.7), (6.56, 56.41), (5.83, 48.57), (6.68, 54.97), (6.08, 47.43), (5.23, 60.72), (7.02, 53.05), (6.73, 55.45), (6.92, 54.57), (6.61, 53.37), (7.32, 55.27), (6.29, 52.76), (6.7, 52.79), (6.57, 49.81), (9.88, 70.27), (6.97, 48.07), (6.63, 52.91), (6.12, 51.29), (7.46, 52.03), (5.23, 55.63), (7.11, 54.07), (6.96, 52.18), (6.88, 55.17), (7.06, 49.98), (6.78, 53.98), (5.32, 45.23), (7.04, 58.98), (6.9, 43.0), (6.28, 50.53), (7.61, 54.81), (7.9, 57.48), (7.59, 55.52), (7.08, 55.85), (7.0, 60.29), (6.74, 57.81), (6.86, 52.47), (7.08, 57.61), (7.25, 48.26), (6.97, 51.15), (5.75, 38.59), (6.29, 52.89), (6.65, 57.53), (6.88, 63.78), (7.24, 58.05), (6.4, 48.74), (8.29, 59.18), (5.61, 43.7), (7.0, 48.96), (6.27, 58.68), (7.34, 52.93), (6.54, 40.83), (6.93, 51.91), (7.6, 60.71), (6.81, 51.49), (4.62, 44.69), (8.82, 58.15), (6.76, 57.46), (5.7, 54.47), (8.47, 61.13), (8.07, 57.22), (5.7, 59.6), (5.66, 51.86), (6.96, 54.66), (7.46, 59.76), (6.38, 51.59), (7.15, 50.48), (7.59, 61.17), (6.21, 51.1), (6.56, 53.98), (7.5, 55.49), (5.65, 54.85), (7.0, 49.55), (6.23, 50.02), (6.76, 55.41), (7.47, 61.85), (6.74, 55.72), (8.12, 55.13), (7.36, 56.12), (6.51, 55.07), (5.92, 60.07), (7.58, 54.6), (5.81, 50.6), (7.44, 59.48), (6.54, 58.24), (6.8, 53.1), (8.71, 62.25), (7.91, 52.58), (7.15, 55.1), (6.04, 58.91), (6.71, 55.91), (6.37, 47.78), (6.97, 54.82), (5.6, 42.19), (5.09, 45.48), (8.19, 54.88), (8.9, 59.75), (8.79, 55.98), (6.9, 56.61), (7.24, 57.4), (6.45, 45.48), (5.57, 55.73), (5.83, 51.16), (7.19, 60.51), (7.67, 46.56), (7.05, 63.73), (7.96, 58.39), (7.07, 47.85), (6.05, 52.62), (7.01, 49.98), (5.73, 54.67), (7.53, 58.82), (5.56, 40.4), (8.56, 62.39), (7.38, 56.61), (7.0, 62.73), (5.38, 55.18), (6.61, 52.28), (7.41, 57.43), (7.57, 58.78), (8.01, 50.18), (5.67, 55.71), (6.78, 54.27), (8.56, 65.36), (6.15, 54.8), (6.57, 49.91), (6.75, 53.82), (5.88, 53.94), (7.65, 60.26), (7.57, 58.55), (7.44, 62.39), (5.76, 50.23), (6.63, 53.67), (5.65, 47.84), (6.3, 59.51), (6.13, 54.88), (5.72, 51.22), (5.65, 48.44), (8.06, 61.81), (5.43, 55.79), (8.55, 58.59), (7.3, 56.04), (6.08, 50.4), (7.49, 55.89), (7.27, 51.8), (6.82, 54.38), (5.45, 52.85), (7.77, 57.13), (6.98, 58.08), (6.15, 51.08), (6.89, 57.85), (9.66, 62.11), (7.84, 64.99), (5.3, 51.55), (7.09, 56.45), (5.87, 51.33), (7.36, 67.25), (6.87, 55.99), (8.59, 63.96), (5.5, 45.56), (5.32, 49.15), (6.36, 50.3), (5.73, 46.22), (5.49, 51.83), (5.54, 57.04), (7.62, 54.65), (6.58, 52.38), (6.78, 51.3), (7.52, 46.56), (6.02, 49.89), (7.45, 60.07), (6.68, 61.5), (8.26, 64.25), (7.04, 57.96), (5.56, 52.2), (8.1, 57.54), (6.42, 51.66), (6.66, 59.55), (7.07, 44.64), (7.57, 59.94), (5.86, 52.33), (7.1, 59.14), (7.31, 58.29), (7.33, 50.57), (7.71, 56.35), (6.43, 54.01), (6.46, 50.8), (7.35, 56.35), (5.94, 56.57), (7.62, 56.14), (8.02, 55.47), (6.91, 60.65), (6.75, 55.24), (5.95, 53.66), (7.55, 58.45), (6.99, 55.89), (8.61, 61.05), (8.81, 54.61), (6.72, 51.9), (6.6, 58.04), (6.84, 54.91), (6.49, 62.66), (7.04, 62.03), (7.27, 61.36), (6.85, 56.37), (6.74, 52.76), (7.7, 57.89), (5.7, 49.8), (7.08, 56.83), (5.95, 56.6), (7.21, 53.11), (6.41, 46.94), (8.04, 60.25), (8.53, 61.3), (6.09, 49.59), (7.59, 54.21), (6.51, 52.18), (7.36, 61.32), (8.36, 63.9), (7.27, 60.01), (5.66, 49.69), (6.77, 53.96), (6.66, 51.37), (6.14, 48.08), (5.59, 47.28), (7.0, 54.63), (7.72, 64.45), (7.03, 50.48), (7.6, 54.38), (7.48, 54.12), (6.12, 46.85), (7.84, 55.91), (7.68, 55.71), (8.39, 57.08), (8.48, 57.49), (5.48, 45.24), (7.2, 59.02), (4.95, 45.32), (6.99, 54.5), (6.63, 58.06), (6.35, 56.92), (7.47, 54.97), (7.47, 62.51), (7.73, 54.06), (6.48, 53.27), (7.29, 46.49), (7.18, 57.03), (6.11, 49.52), (7.35, 57.28), (5.47, 55.69), (5.85, 54.04), (7.12, 52.26), (6.3, 53.63), (6.24, 50.05), (8.37, 58.68), (6.7, 52.6), (6.78, 54.67), (6.89, 53.65), (5.71, 45.14), (8.17, 64.07), (7.26, 60.89), (7.98, 54.53), (7.4, 59.31), (6.67, 55.31), (6.9, 50.29), (7.05, 52.91), (7.31, 54.11), (7.56, 57.37), (8.33, 59.1), (7.5, 52.66), (8.12, 57.73), (6.26, 56.84), (9.2, 62.24), (6.83, 53.0), (6.84, 52.82), (6.62, 60.81), (7.45, 46.6), (6.5, 57.2), (7.69, 58.93), (6.38, 53.83), (6.9, 54.3), (6.11, 52.02), (6.5, 49.09), (7.08, 53.11), (8.71, 59.45), (7.65, 50.03), (6.69, 52.07), (6.52, 57.14), (8.25, 50.61), (6.93, 55.61), (4.75, 46.52), (8.72, 60.4), (6.68, 51.41), (7.88, 57.38), (6.36, 53.28), (5.1, 48.62), (6.26, 58.95), (8.26, 63.24), (6.77, 50.86), (7.39, 50.96), (7.24, 57.77), (6.79, 55.88), (7.66, 56.88), (8.13, 60.59), (7.05, 64.04), (8.13, 63.72), (7.43, 64.66), (6.43, 52.9), (7.32, 53.79), (8.74, 59.77), (7.21, 55.99), (6.6, 56.72), (5.06, 47.08), (5.92, 54.18), (7.77, 49.93), (7.73, 56.19), (6.02, 51.26), (5.48, 46.43), (7.26, 62.31), (7.45, 60.99), (8.42, 64.27), (6.13, 60.44), (7.98, 59.57), (5.78, 48.44), (4.65, 47.11), (8.9, 54.63), (7.83, 57.37), (6.89, 47.8), (6.2, 52.26), (7.39, 57.93), (8.25, 60.06), (7.14, 60.53), (7.71, 60.03), (8.65, 58.25), (8.95, 62.09), (6.81, 55.46), (10.15, 70.43), (6.76, 53.6), (7.06, 51.48), (6.95, 55.03), (5.05, 48.92), (9.01, 62.5), (7.0, 53.93), (6.13, 52.79), (7.85, 59.54), (6.35, 48.13), (6.99, 50.75), (7.38, 54.28), (8.27, 62.52), (6.85, 48.27), (7.76, 59.4), (8.12, 55.12), (8.58, 61.19), (6.81, 52.46), (8.5, 57.15), (7.27, 60.73), (7.06, 57.15), (7.73, 56.71), (6.39, 45.06), (7.13, 59.45), (7.45, 58.38), (7.78, 53.91), (8.63, 61.0), (6.29, 57.63), (7.21, 58.84), (7.16, 51.29), (8.49, 62.86), (6.75, 50.37), (7.88, 57.24), (8.07, 64.2), (5.58, 55.99), (7.51, 51.79), (6.84, 63.14), (6.46, 55.52), (8.34, 65.67), (4.62, 43.89), (6.58, 50.81), (7.4, 54.85), (7.56, 52.66), (5.69, 52.08), (4.72, 42.97), (6.21, 49.79), (8.17, 56.59), (6.44, 55.29), (7.46, 62.82), (6.81, 57.86), (7.04, 52.66), (5.99, 50.92), (8.45, 53.39), (8.27, 57.66), (7.51, 60.87), (7.08, 60.03), (7.02, 53.14), (7.37, 54.35), (6.73, 53.21), (7.57, 59.6), (7.97, 50.73), (5.87, 47.27), (9.09, 55.07), (5.61, 57.41), (8.22, 61.12), (6.34, 62.39), (8.44, 61.89), (6.2, 51.13), (5.89, 50.8), (4.83, 46.85), (5.88, 53.45), (7.71, 59.08), (6.84, 58.97), (7.58, 60.12), (7.07, 55.91), (7.29, 65.01), (6.76, 53.86), (6.65, 50.02), (8.12, 55.61), (7.23, 52.78), (6.67, 56.48), (7.87, 55.87), (6.78, 52.88), (7.6, 57.01), (6.1, 52.7), (7.55, 60.73), (6.3, 54.07), (7.72, 55.3), (5.85, 60.65), (7.1, 46.79), (6.78, 49.76), (6.64, 56.37), (7.15, 54.01), (6.17, 54.21), (7.62, 53.45), (7.95, 55.65), (9.42, 71.79), (6.18, 57.08), (6.85, 49.02), (8.58, 58.84), (6.45, 52.04), (5.63, 54.31), (7.28, 60.6), (8.4, 57.13), (6.67, 65.33), (8.16, 57.37), (6.65, 52.38), (6.43, 57.34), (8.38, 61.47), (6.02, 48.96), (8.42, 59.99), (8.4, 57.84), (7.54, 51.08), (5.93, 55.41), (6.94, 49.43), (7.32, 59.95), (6.19, 54.97), (6.98, 56.87), (7.01, 50.68), (7.88, 60.8), (7.93, 54.57), (7.93, 61.59), (7.72, 57.86), (6.27, 55.27), (4.59, 52.87), (4.78, 46.17), (7.55, 63.53), (6.37, 45.99), (6.75, 54.84), (7.23, 60.63), (8.86, 58.47), (6.8, 52.65), (6.65, 51.69), (6.91, 54.87), (6.6, 57.91), (5.53, 41.67), (5.23, 44.96), (6.18, 45.28), (7.67, 55.48), (6.68, 59.05), (8.23, 58.25), (6.2, 58.95), (8.66, 59.2), (8.01, 60.23), (8.63, 60.43), (6.18, 54.94), (6.77, 59.41), (7.3, 61.86), (8.85, 53.68), (6.17, 48.91), (6.37, 51.93), (6.46, 50.52), (6.98, 59.32), (6.71, 48.7), (6.8, 56.84), (5.94, 56.82), (6.27, 52.88), (6.9, 51.62), (8.29, 54.44), (7.42, 64.2), (7.13, 46.15), (7.02, 55.58), (8.41, 59.94), (7.03, 55.46), (5.85, 51.41), (8.68, 64.62), (8.18, 59.4), (7.77, 59.85), (6.44, 42.38), (7.08, 55.17), (5.58, 51.34), (5.38, 48.03), (6.93, 59.34), (7.4, 62.11), (7.83, 57.38), (6.52, 59.47), (6.38, 58.26), (6.69, 47.35), (6.38, 55.49), (5.27, 48.91), (7.31, 58.48), (7.92, 59.56), (6.16, 46.7), (6.38, 56.52), (6.9, 53.96), (8.64, 62.27), (7.0, 61.3), (5.71, 50.11), (6.12, 56.46), (6.99, 51.38), (6.52, 49.49), (6.05, 59.63), (7.02, 57.59), (7.95, 47.21), (5.22, 51.44), (5.93, 49.59), (6.02, 57.65), (8.09, 55.81)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=6.9863，ȳ=55.0307；Sxx=Σ(x−x̄)²=560.70，Sxy=1739.09；
β₁ = Sxy/Sxx = 3.1017；β₀ = ȳ−β₁x̄ = 33.3614；
R² = SSR/SST = 5394.07/16133.20 = 0.3343；F = (SSR/1)/(SSE/625) = 313.9262，p = -6.50e+20；
SE(β₁)=√(17.18/560.70)=0.1751；t=β₁/SE=17.7180；p=-6.50e+20；95%CI=[2.7579,3.4454]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=6.9863，ȳ=55.0307；Sxx=Σ(x−x̄)²=560.70，Sxy=1739.09；
β₁ = Sxy/Sxx = 3.1017；β₀ = ȳ−β₁x̄ = 33.3614；
R² = SSR/SST = 5394.07/16133.20 = 0.3343；F = (SSR/1)/(SSE/625) = 313.9262，p = -6.50e+20；
SE(β₁)=√(17.18/560.70)=0.1751；t=β₁/SE=17.7180；p=-6.50e+20；95%CI=[2.7579,3.4454]。
③ 计算结果：
y = 33.3614 + 3.1017·x；R²=0.3343；β₁ t=17.7180 p=-6.50e+20；95%CI=[2.7579,3.4454]
④ 结论：在 α=0.05 水平下，p=-6.50e+20 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=3.1017），睡得好专注。
案例 52. 客服速度预测满意度
简单线性回归响应快满意
① 原始数据：
样本 n=828，逐条 (x, y)：[(11.34, 2.36), (37.56, 3.06), (50.84, 3.56), (30.8, 2.94), (9.34, 2.45), (18.32, 2.58), (20.67, 2.57), (26.05, 2.62), (32.29, 3.3), (37.99, 3.19), (10.79, 2.88), (30.5, 2.82), (35.26, 3.03), (54.88, 3.98), (39.49, 3.24), (19.04, 2.61), (24.37, 2.69), (5.12, 2.18), (27.3, 3.34), (37.19, 3.39), (60.48, 3.91), (53.26, 3.56), (40.67, 2.92), (23.29, 2.86), (31.57, 2.98), (71.17, 4.24), (39.12, 3.16), (19.98, 2.99), (40.36, 3.23), (28.97, 2.92), (36.06, 3.06), (33.05, 3.45), (21.26, 3.08), (40.78, 3.65), (28.73, 3.14), (31.23, 3.04), (58.1, 3.81), (53.08, 3.36), (39.46, 3.32), (28.97, 2.98), (48.06, 3.65), (37.65, 3.31), (30.96, 2.92), (30.59, 2.88), (34.01, 2.99), (27.04, 2.73), (33.27, 2.67), (2.12, 2.43), (28.22, 3.12), (22.45, 2.72), (27.62, 2.62), (33.04, 2.99), (41.71, 3.61), (22.09, 3.06), (15.31, 2.62), (15.72, 2.68), (32.95, 2.69), (35.55, 3.24), (30.89, 2.88), (20.56, 2.72), (37.99, 3.3), (41.54, 3.62), (19.27, 2.43), (39.94, 3.34), (13.65, 2.68), (23.43, 2.73), (44.73, 3.52), (32.93, 3.12), (8.23, 2.0), (26.36, 2.79), (29.98, 2.86), (22.12, 2.99), (26.8, 2.77), (44.44, 3.32), (27.09, 2.71), (44.26, 3.73), (8.93, 2.27), (36.7, 3.0), (29.7, 3.02), (27.72, 2.9), (17.36, 2.88), (10.25, 2.37), (12.34, 2.34), (28.43, 3.13), (41.3, 3.48), (23.27, 2.73), (37.16, 3.2), (39.4, 3.4), (39.33, 3.59), (26.73, 3.07), (42.49, 3.6), (18.07, 2.83), (40.92, 3.22), (26.12, 2.97), (30.62, 3.2), (28.42, 2.98), (42.24, 3.25), (29.33, 3.0), (18.17, 2.43), (38.29, 3.21), (44.11, 3.37), (28.96, 2.83), (36.65, 3.46), (0.02, 2.2), (3.47, 2.11), (-0.48, 1.9), (34.02, 2.95), (8.35, 2.3), (32.56, 2.87), (33.07, 3.02), (48.7, 3.37), (19.22, 2.26), (33.85, 3.02), (18.49, 2.66), (32.67, 2.8), (18.09, 2.48), (32.86, 3.47), (36.83, 3.49), (49.25, 3.77), (27.21, 3.03), (16.52, 2.6), (45.45, 3.76), (11.78, 2.35), (35.16, 3.05), (21.38, 2.72), (10.46, 2.19), (26.54, 2.8), (19.7, 2.6), (48.72, 3.4), (40.34, 2.87), (8.7, 2.42), (11.51, 2.65), (25.75, 3.12), (34.44, 3.33), (44.59, 3.48), (41.91, 3.63), (36.02, 3.18), (16.68, 2.66), (-3.06, 1.92), (44.42, 3.58), (16.95, 2.76), (4.38, 2.37), (11.56, 2.25), (36.2, 3.02), (38.14, 3.28), (40.02, 3.36), (52.27, 3.66), (14.88, 2.68), (33.86, 3.16), (36.35, 3.1), (53.74, 3.63), (8.6, 2.39), (27.14, 3.18), (38.52, 3.24), (22.53, 2.78), (32.07, 3.04), (6.08, 1.86), (37.26, 3.15), (28.18, 2.81), (42.18, 3.56), (24.11, 2.88), (34.81, 3.46), (56.15, 3.39), (29.72, 2.84), (22.2, 2.8), (18.48, 2.37), (37.51, 3.16), (32.62, 3.11), (31.75, 3.31), (38.81, 3.4), (46.25, 3.6), (28.02, 3.03), (35.95, 3.38), (29.72, 3.04), (44.82, 3.39), (10.43, 2.33), (22.43, 2.82), (36.19, 3.08), (14.77, 2.64), (50.63, 3.56), (26.66, 2.64), (50.85, 3.58), (35.82, 3.36), (36.57, 3.43), (13.91, 2.74), (31.53, 2.79), (32.34, 2.89), (40.14, 3.17), (29.52, 2.97), (23.09, 2.78), (23.34, 3.06), (19.57, 2.87), (6.13, 2.09), (17.46, 2.36), (39.48, 3.18), (46.27, 3.51), (23.28, 3.04), (28.77, 2.97), (15.02, 2.71), (27.52, 3.15), (8.15, 2.3), (20.57, 2.79), (43.67, 3.46), (33.81, 3.34), (23.51, 2.95), (61.38, 4.24), (20.28, 2.87), (18.27, 2.95), (27.44, 3.07), (29.41, 2.99), (37.93, 3.37), (19.28, 2.96), (42.59, 3.4), (37.8, 3.41), (26.62, 2.5), (45.29, 3.33), (18.98, 2.83), (38.27, 3.28), (56.04, 3.85), (20.11, 2.64), (0.54, 1.98), (31.06, 3.12), (32.76, 2.99), (31.52, 3.02), (48.99, 3.57), (35.82, 3.01), (27.82, 3.09), (24.13, 2.35), (59.63, 4.02), (26.79, 2.65), (40.52, 3.11), (13.23, 2.61), (32.54, 3.23), (25.24, 3.03), (33.8, 3.1), (20.98, 2.34), (38.22, 3.47), (32.44, 3.4), (39.0, 3.38), (43.82, 3.72), (41.37, 3.58), (14.06, 2.69), (19.18, 2.37), (15.56, 2.32), (35.94, 3.14), (22.98, 2.9), (34.17, 3.23), (11.9, 2.65), (20.62, 2.71), (24.36, 2.68), (36.83, 3.45), (55.83, 3.86), (18.31, 2.61), (16.28, 2.43), (29.25, 2.62), (37.41, 3.16), (41.75, 3.51), (43.77, 3.31), (31.62, 3.07), (36.75, 3.1), (14.12, 2.74), (10.46, 1.93), (29.85, 2.85), (38.62, 3.2), (42.41, 3.07), (8.35, 2.27), (46.34, 3.42), (27.07, 3.0), (23.47, 2.64), (20.03, 3.07), (24.5, 2.6), (36.48, 3.23), (43.4, 3.18), (27.93, 2.78), (41.46, 3.73), (37.85, 2.98), (21.15, 2.99), (32.13, 3.09), (57.71, 3.78), (18.95, 2.66), (31.47, 3.11), (23.87, 2.71), (34.0, 2.95), (29.42, 2.72), (30.47, 2.99), (22.01, 2.7), (41.96, 3.55), (33.55, 3.13), (13.39, 2.57), (28.97, 2.52), (25.62, 2.74), (38.87, 3.06), (24.59, 3.03), (21.43, 2.99), (21.92, 3.3), (39.23, 3.32), (16.64, 2.56), (3.59, 2.18), (19.76, 2.59), (25.15, 3.0), (26.58, 3.28), (18.55, 2.26), (51.3, 3.71), (31.22, 3.27), (27.51, 2.91), (25.61, 2.76), (46.23, 3.61), (9.58, 2.38), (31.83, 2.86), (51.91, 3.82), (41.73, 3.27), (38.19, 3.35), (41.78, 3.45), (66.03, 4.35), (14.65, 2.6), (31.05, 2.64), (12.44, 2.91), (35.86, 3.28), (38.58, 2.99), (47.79, 3.87), (27.36, 2.83), (50.95, 3.52), (40.57, 3.15), (22.79, 2.82), (30.79, 2.96), (17.11, 2.66), (25.76, 3.35), (20.04, 2.85), (30.67, 2.85), (47.27, 3.07), (15.09, 2.38), (30.78, 3.1), (48.74, 3.86), (28.52, 2.79), (35.62, 2.98), (43.99, 3.5), (8.96, 2.65), (54.95, 3.9), (35.02, 3.31), (59.15, 3.8), (8.49, 2.7), (18.9, 2.97), (32.69, 3.06), (8.55, 2.19), (28.01, 3.31), (35.46, 3.68), (46.99, 3.35), (31.62, 3.3), (22.98, 2.85), (48.1, 3.34), (50.49, 3.39), (27.91, 3.22), (10.73, 2.57), (15.66, 2.88), (48.05, 3.47), (17.44, 2.82), (-11.77, 1.71), (25.23, 2.72), (40.95, 3.34), (16.6, 2.79), (35.97, 3.2), (24.46, 2.6), (47.82, 3.74), (25.86, 3.12), (11.9, 2.76), (28.48, 2.82), (25.93, 2.84), (38.58, 3.3), (52.56, 3.59), (36.66, 3.32), (29.16, 3.02), (57.47, 3.96), (26.56, 2.83), (50.49, 3.68), (42.47, 3.26), (29.75, 2.62), (19.13, 2.82), (42.36, 3.54), (30.43, 3.15), (24.92, 3.23), (11.33, 2.37), (33.93, 3.13), (18.83, 2.62), (23.79, 2.64), (37.61, 3.51), (43.51, 3.26), (29.15, 3.31), (31.67, 3.05), (41.81, 3.41), (44.97, 3.53), (41.49, 3.2), (14.23, 2.83), (21.4, 3.04), (38.39, 4.0), (22.97, 2.75), (39.36, 3.13), (46.32, 3.55), (22.17, 2.95), (35.82, 3.25), (17.34, 2.84), (14.33, 2.74), (45.14, 3.36), (23.94, 2.92), (30.36, 2.95), (32.89, 2.95), (48.37, 3.5), (35.96, 3.0), (39.23, 3.47), (25.06, 2.54), (37.59, 3.06), (24.66, 2.75), (28.78, 3.08), (29.34, 2.85), (45.43, 3.57), (48.66, 3.31), (16.34, 2.56), (21.15, 2.64), (25.66, 2.72), (18.14, 2.77), (18.83, 2.55), (11.02, 2.42), (20.67, 2.46), (33.0, 3.33), (29.46, 2.99), (36.74, 3.14), (32.08, 3.08), (29.26, 2.85), (38.33, 3.54), (40.93, 3.29), (27.79, 3.11), (22.83, 3.15), (34.42, 2.98), (30.71, 3.21), (38.58, 3.23), (21.12, 2.46), (19.16, 3.2), (23.3, 2.78), (30.91, 3.27), (48.99, 3.33), (30.28, 2.67), (21.55, 2.5), (42.61, 3.64), (40.36, 3.22), (33.49, 3.12), (33.07, 3.32), (12.37, 2.38), (29.0, 2.9), (21.97, 2.88), (65.02, 4.18), (55.98, 3.91), (30.2, 2.79), (35.07, 2.99), (33.88, 3.17), (30.26, 2.68), (17.42, 2.7), (19.2, 3.07), (17.51, 2.47), (41.63, 3.11), (14.79, 2.9), (44.23, 3.41), (34.11, 3.59), (26.11, 3.08), (34.3, 3.02), (26.36, 2.69), (30.75, 2.79), (29.05, 2.89), (36.64, 3.15), (37.39, 3.28), (31.75, 3.05), (33.64, 3.3), (20.49, 2.71), (27.04, 2.78), (28.83, 3.09), (39.5, 3.34), (40.08, 2.99), (35.35, 2.92), (34.72, 3.19), (11.5, 2.09), (10.87, 2.44), (33.57, 3.11), (10.07, 2.32), (44.68, 3.78), (34.73, 3.1), (58.35, 4.05), (27.62, 2.98), (33.81, 3.26), (32.1, 3.36), (14.93, 2.69), (35.24, 2.87), (9.98, 2.53), (32.14, 3.26), (38.97, 3.3), (30.93, 2.78), (28.99, 3.08), (29.01, 3.27), (34.4, 3.33), (20.33, 2.42), (61.27, 3.89), (39.31, 3.23), (26.52, 3.09), (49.45, 3.76), (49.49, 3.46), (35.38, 3.2), (46.38, 4.0), (33.68, 3.48), (5.45, 2.03), (35.84, 3.04), (27.71, 2.77), (31.2, 3.09), (35.21, 3.0), (28.12, 3.33), (9.05, 2.45), (22.54, 2.75), (38.5, 2.94), (32.57, 2.7), (28.79, 3.06), (7.18, 2.3), (43.34, 3.73), (40.22, 3.41), (31.65, 3.01), (15.57, 2.34), (31.5, 3.33), (41.85, 3.23), (27.32, 2.79), (27.93, 3.03), (20.61, 2.72), (28.28, 3.24), (8.87, 2.16), (15.66, 3.08), (38.99, 3.28), (37.44, 3.33), (50.33, 3.72), (40.97, 3.45), (25.15, 2.94), (35.4, 2.96), (24.71, 2.91), (16.25, 2.38), (30.45, 3.43), (29.35, 3.25), (25.36, 3.22), (31.6, 3.15), (32.54, 3.04), (35.96, 2.87), (24.81, 3.15), (34.41, 2.92), (16.71, 2.42), (43.24, 3.19), (17.52, 2.78), (39.78, 3.08), (20.6, 2.41), (44.34, 3.47), (33.6, 2.85), (42.2, 3.35), (31.08, 3.27), (42.56, 3.43), (53.93, 3.56), (34.94, 3.17), (36.79, 3.13), (17.8, 2.69), (25.22, 2.69), (15.78, 2.5), (18.36, 2.74), (32.07, 3.24), (20.92, 2.84), (40.31, 3.18), (23.46, 3.06), (20.0, 2.32), (18.46, 2.91), (47.78, 3.47), (9.86, 2.35), (38.53, 2.96), (31.33, 3.71), (42.41, 3.29), (25.05, 2.93), (24.15, 2.59), (20.49, 2.58), (7.2, 2.55), (20.09, 2.79), (43.1, 3.26), (47.4, 3.52), (48.08, 3.83), (38.7, 3.3), (70.75, 4.2), (0.5, 2.11), (60.63, 3.81), (10.96, 2.53), (25.67, 3.18), (35.36, 2.76), (38.29, 3.02), (34.94, 2.88), (21.16, 2.83), (22.9, 2.52), (36.78, 3.28), (31.66, 3.04), (26.73, 2.81), (28.63, 2.93), (29.55, 2.87), (27.14, 2.74), (41.79, 3.48), (30.32, 2.87), (28.32, 3.0), (53.57, 3.71), (27.33, 3.08), (40.13, 3.13), (37.47, 3.01), (35.03, 3.09), (20.36, 2.82), (25.75, 2.94), (21.4, 2.89), (27.0, 2.83), (26.83, 3.16), (28.71, 2.97), (39.9, 3.22), (26.91, 2.73), (42.79, 3.69), (34.2, 3.12), (59.32, 3.92), (21.35, 2.63), (15.21, 2.56), (43.94, 3.89), (37.93, 3.45), (20.77, 2.5), (32.72, 3.17), (18.34, 2.42), (25.14, 2.88), (18.68, 2.76), (39.8, 3.17), (29.39, 2.99), (11.98, 2.63), (10.86, 2.35), (38.9, 2.9), (46.76, 3.66), (35.61, 3.31), (42.74, 3.3), (25.44, 2.87), (30.7, 3.02), (32.3, 2.94), (5.81, 2.52), (29.05, 3.27), (53.53, 3.61), (24.47, 2.48), (16.31, 2.27), (27.91, 3.08), (32.83, 3.17), (43.24, 3.45), (15.21, 2.55), (27.21, 2.83), (28.15, 3.04), (53.8, 3.3), (36.41, 2.77), (15.58, 2.32), (39.77, 3.52), (26.79, 2.76), (48.58, 3.48), (25.82, 2.85), (44.81, 3.46), (30.36, 3.33), (34.59, 3.3), (40.51, 3.06), (46.29, 3.7), (20.64, 2.74), (50.25, 3.74), (30.62, 2.58), (28.46, 3.1), (15.76, 2.32), (39.5, 3.37), (18.46, 2.38), (45.43, 3.65), (17.78, 2.51), (31.53, 3.47), (31.21, 3.28), (28.38, 3.03), (6.17, 2.39), (25.92, 2.95), (42.91, 3.46), (54.52, 3.82), (30.24, 2.76), (43.75, 3.02), (25.69, 2.89), (31.16, 2.95), (26.84, 2.78), (16.44, 2.33), (22.08, 2.84), (24.06, 3.29), (30.17, 2.26), (45.08, 3.39), (40.02, 3.21), (22.46, 3.1), (23.04, 2.72), (18.77, 2.74), (45.44, 3.65), (39.99, 2.9), (35.06, 3.08), (-3.45, 2.07), (27.03, 2.73), (50.8, 3.7), (27.07, 2.99), (33.88, 3.0), (20.33, 2.64), (15.72, 2.72), (21.99, 2.58), (28.77, 3.16), (43.01, 3.18), (26.26, 2.94), (20.93, 2.43), (38.89, 3.53), (50.35, 3.3), (26.7, 2.87), (45.19, 3.44), (24.1, 3.17), (39.24, 3.64), (34.32, 3.27), (11.72, 2.55), (21.68, 2.65), (10.5, 2.17), (12.62, 2.02), (27.02, 2.95), (39.08, 3.35), (24.98, 2.96), (15.72, 2.71), (20.87, 2.81), (52.53, 3.72), (26.37, 2.39), (42.14, 2.96), (4.7, 2.22), (34.83, 2.89), (21.62, 2.56), (46.51, 3.54), (21.39, 2.59), (27.98, 2.89), (26.32, 2.56), (31.12, 3.08), (23.76, 2.51), (42.29, 3.12), (29.62, 2.74), (24.34, 2.7), (26.62, 3.05), (28.84, 2.94), (14.1, 2.29), (18.55, 2.71), (32.44, 3.16), (21.61, 2.65), (12.6, 1.99), (12.74, 2.41), (3.1, 2.45), (35.03, 3.12), (34.82, 3.38), (26.91, 2.87), (40.74, 3.39), (28.76, 2.98), (38.28, 3.05), (24.51, 2.95), (45.79, 3.4), (22.07, 2.45), (40.61, 3.11), (28.48, 2.75), (11.53, 2.3), (28.86, 2.57), (41.4, 3.56), (32.55, 3.1), (19.51, 2.7), (50.87, 3.67), (18.06, 2.55), (44.51, 3.42), (20.32, 2.72), (29.26, 3.06), (60.33, 3.68), (30.57, 3.16), (34.2, 3.38), (24.84, 2.89), (25.76, 2.88), (22.88, 2.83), (17.81, 2.73), (41.21, 3.48), (41.75, 3.33), (11.99, 2.42), (23.29, 2.66), (24.06, 2.57), (20.73, 2.9), (12.01, 2.44), (43.64, 3.32), (28.67, 3.04), (42.47, 3.29), (19.11, 2.5), (48.02, 3.55), (18.77, 2.63), (48.88, 3.59), (21.38, 2.79), (45.73, 3.42), (45.26, 3.96), (28.84, 3.01), (13.33, 2.26), (3.58, 2.3), (26.78, 2.81), (29.65, 3.41), (14.02, 2.47), (13.75, 2.34), (42.33, 3.54), (26.13, 2.95), (20.64, 2.68), (20.53, 2.76), (26.39, 2.75), (44.03, 3.26), (31.29, 3.22), (18.33, 2.37), (28.51, 2.93), (44.9, 3.38), (27.06, 3.21), (34.72, 2.87), (47.6, 3.63), (13.57, 2.28), (25.81, 2.75), (20.26, 2.7), (40.74, 3.12), (36.72, 3.4), (29.75, 3.27), (38.78, 3.17), (30.7, 3.03), (31.31, 3.2), (14.41, 2.64), (38.38, 3.33), (23.1, 2.95), (27.18, 2.87), (50.59, 3.3), (27.76, 2.98), (29.36, 2.72), (38.56, 3.66), (30.79, 3.25)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=30.2917，ȳ=3.0128；Sxx=Σ(x−x̄)²=125049.32，Sxy=3785.60；
β₁ = Sxy/Sxx = 0.0303；β₀ = ȳ−β₁x̄ = 2.0957；
R² = SSR/SST = 114.60/147.66 = 0.7761；F = (SSR/1)/(SSE/826) = 2863.7929，p = 1.09e-270；
SE(β₁)=√(0.04/125049.32)=0.0006；t=β₁/SE=53.5144；p=1.09e-270；95%CI=[0.0292,0.0314]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=30.2917，ȳ=3.0128；Sxx=Σ(x−x̄)²=125049.32，Sxy=3785.60；
β₁ = Sxy/Sxx = 0.0303；β₀ = ȳ−β₁x̄ = 2.0957；
R² = SSR/SST = 114.60/147.66 = 0.7761；F = (SSR/1)/(SSE/826) = 2863.7929，p = 1.09e-270；
SE(β₁)=√(0.04/125049.32)=0.0006；t=β₁/SE=53.5144；p=1.09e-270；95%CI=[0.0292,0.0314]。
③ 计算结果：
y = 2.0957 + 0.0303·x；R²=0.7761；β₁ t=53.5144 p=1.09e-270；95%CI=[0.0292,0.0314]
④ 结论：在 α=0.05 水平下，p=1.09e-270 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=0.0303），响应快满意。
案例 53. 阅读量预测收入
简单线性回归读得多收入高
① 原始数据：
样本 n=871，逐条 (x, y)：[(29.91, 4586.67), (14.27, 567.11), (14.6, 1737.23), (28.08, 3432.6), (19.76, 1830.01), (15.09, 1374.14), (12.45, 1148.54), (40.07, 5443.78), (18.2, 1704.93), (28.8, 2929.25), (11.08, 295.96), (20.19, 2088.27), (32.53, 3835.88), (27.12, 3091.39), (21.57, 1757.14), (41.02, 5638.52), (22.66, 1690.63), (15.19, 567.78), (23.76, 3176.91), (19.45, 2043.0), (14.67, 1764.59), (35.48, 4805.95), (22.31, 2668.33), (31.24, 4222.66), (19.18, 1217.1), (24.26, 1716.29), (26.12, 3511.8), (9.47, 1109.45), (25.17, 1774.56), (31.23, 3469.9), (29.54, 3168.16), (30.64, 3612.36), (20.73, 2416.88), (27.52, 2982.52), (17.99, 904.0), (12.44, 1361.96), (19.18, 1976.88), (6.08, -531.91), (12.37, -286.68), (22.41, 3014.27), (2.68, -519.06), (30.02, 3781.87), (27.18, 3323.26), (24.42, 3026.5), (28.04, 3379.59), (30.45, 2770.78), (20.99, 1930.22), (11.19, 625.62), (23.06, 3141.84), (8.55, -295.48), (15.64, 1329.6), (5.04, -529.14), (15.12, 900.55), (35.54, 4708.56), (12.06, 1807.51), (5.8, 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(19.15, 1563.54), (10.88, 837.52), (15.17, 2205.75), (29.59, 4090.22), (26.7, 3645.18), (20.45, 2578.58), (20.98, 2179.78), (20.92, 1591.64), (38.3, 4502.87), (22.24, 1829.32), (28.08, 2038.61), (28.01, 3528.85), (19.14, 1331.84), (8.96, -794.17), (16.46, 1884.57), (4.93, 532.18), (10.58, 1081.07), (7.94, 333.0), (31.92, 4667.77), (31.97, 3798.57), (14.73, 1374.71), (22.65, 3191.11), (28.4, 2147.57), (23.23, 3066.72), (27.59, 3341.06), (29.32, 3666.3), (14.37, 1441.48), (15.5, 2010.84), (7.29, 696.49), (23.19, 2203.36), (19.22, 1756.33), (24.35, 2573.14), (18.66, 1451.5), (23.85, 3710.73), (24.9, 2893.51), (24.2, 1913.21), (15.14, 1605.5), (23.72, 1583.94), (20.35, 1729.75), (19.59, 1270.12), (23.7, 3648.35), (19.12, 2091.9), (17.63, 2363.81), (18.86, 1533.93), (23.06, 2380.31), (23.8, 2789.84), (19.41, 2213.86), (16.71, 2069.85), (28.4, 3651.09), (17.94, 2862.41), (23.24, 2776.54), (11.64, 52.21), (25.45, 2999.87), (22.2, 2878.82), (12.12, 552.81), (8.77, 2.82), (25.13, 2666.98), (25.56, 2405.33), (27.83, 3079.74), (20.17, 1902.24), (7.25, -93.83), (22.99, 3585.94), (10.4, 691.29), (12.33, 887.51), (17.97, 62.67), (19.73, 1724.52), (6.74, 554.61), (1.02, -468.77), (11.39, 828.07), (22.24, 1951.13), (22.41, 2783.6), (25.12, 3024.72), (17.17, 1451.46), (11.79, 81.87), (6.74, 492.63), (31.41, 4311.01), (18.7, 1203.76), (24.75, 2622.74), (29.75, 3391.02), (13.0, 1586.12), (19.8, 2699.43), (32.22, 1924.2), (26.89, 2918.3), (22.1, 2392.17), (14.33, 722.09), (15.84, 627.46), (34.02, 4035.58), (28.08, 3164.42), (16.88, 675.78), (19.65, 2023.39), (31.13, 4741.85), (20.0, 2804.62), (19.58, 1380.24), (19.91, 1842.36), (12.75, 1440.33), (23.81, 3216.55), (12.76, 2321.13), (6.94, -1040.51), (24.13, 1851.77), (27.18, 3269.79), (17.63, 1367.17), (24.59, 1698.19), (15.4, 1163.21), (25.46, 3213.85), (23.3, 2827.76), (21.79, 1695.87), (23.25, 2282.38), (11.67, 1026.7), (30.86, 4452.28), (15.77, 1949.4), (23.86, 2697.89), (47.92, 6697.23), (8.94, -172.02), (20.93, 2618.73), (22.44, 2129.41), (20.45, 1297.63), (20.98, 1080.65), (23.24, 3263.52), (26.94, 1960.44), (12.37, 1400.64), (25.2, 3945.83), (24.91, 2977.97), (21.11, 3479.86), (5.37, -1218.27), (18.86, 1951.26), (15.77, 1481.99), (27.18, 1951.65), (24.08, 2522.98), (23.85, 3405.39), (21.59, 2401.56), (35.61, 5153.55), (21.39, 2152.48), (17.74, 1933.11), (32.62, 4151.63), (29.87, 3200.37), (16.78, 1683.74), (17.45, 1678.81), (17.64, 115.71), (17.57, 1941.05), (0.53, -847.45), (25.2, 3424.43), (17.84, 1309.05), (27.22, 2893.41), (27.16, 3489.23), (6.64, -726.84), (21.59, 3565.76), (17.31, 2268.81), (26.12, 1841.69), (13.89, 1118.29), (33.42, 4205.32), (19.64, 1997.64), (19.93, 2704.4), (31.2, 2771.79), (18.18, 1376.65), (22.93, 2675.05), (27.33, 2922.24), (19.07, 2188.89), (11.48, 530.56), (26.14, 2259.7), (18.39, 1781.17), (16.0, 1586.15), (27.16, 3873.87), (25.39, 2961.79), (30.11, 3996.57), (20.65, 2700.75), (9.58, 262.16), (25.27, 3472.81), (22.23, 1188.14), (21.55, 1542.97), (30.44, 3614.23), (12.58, 539.29), (32.66, 3295.39), (9.89, -727.97), (25.86, 2565.13), (15.99, 1415.1), (7.33, -520.03), (12.26, 355.07), (17.45, 2209.63), (15.43, 707.33), (24.58, 2457.24), (29.97, 3361.92), (12.53, 1812.81), (13.17, 800.42), (34.27, 4301.12), (30.66, 3400.86), (22.04, 3012.43), (29.45, 4223.8), (15.96, 923.5), (16.85, 1334.09), (27.02, 1862.04)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.6507，ȳ=2102.3191；Sxx=Σ(x−x̄)²=52089.55，Sxy=7761830.19；
β₁ = Sxy/Sxx = 149.0094；β₀ = ȳ−β₁x̄ = -974.8335；
R² = SSR/SST = 1156585299.87/1473934795.36 = 0.7847；F = (SSR/1)/(SSE/869) = 3167.0844，p = 5.00e-292；
SE(β₁)=√(365189.29/52089.55)=2.6478；t=β₁/SE=56.2769；p=5.00e-292；95%CI=[83.4382,214.5805]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.6507，ȳ=2102.3191；Sxx=Σ(x−x̄)²=52089.55，Sxy=7761830.19；
β₁ = Sxy/Sxx = 149.0094；β₀ = ȳ−β₁x̄ = -974.8335；
R² = SSR/SST = 1156585299.87/1473934795.36 = 0.7847；F = (SSR/1)/(SSE/869) = 3167.0844，p = 5.00e-292；
SE(β₁)=√(365189.29/52089.55)=2.6478；t=β₁/SE=56.2769；p=5.00e-292；95%CI=[83.4382,214.5805]。
③ 计算结果：
y = -974.8335 + 149.0094·x；R²=0.7847；β₁ t=56.2769 p=5.00e-292；95%CI=[83.4382,214.5805]
④ 结论：在 α=0.05 水平下，p=5.00e-292 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=149.0094），读得多收入高。
案例 54. 气温预测羽绒服销量
简单线性回归天冷卖得多
① 原始数据：
样本 n=567，逐条 (x, y)：[(13.51, 378.48), (2.65, 465.85), (-8.74, 615.32), (-4.03, 552.82), (11.25, 390.46), (31.12, 170.91), (0.13, 559.86), (22.0, 222.72), (34.72, 137.11), (32.92, 194.55), (-6.71, 556.1), (16.52, 295.42), (6.21, 428.51), (18.94, 322.98), (5.13, 469.88), (0.83, 486.61), (12.39, 383.32), (20.19, 287.76), (19.92, 289.84), (-6.8, 604.78), (27.67, 252.8), (19.51, 250.95), (22.03, 243.14), (20.05, 299.78), (15.66, 351.87), (11.57, 434.57), (22.79, 284.75), (9.57, 412.53), (-2.21, 525.19), (11.24, 422.46), (4.56, 404.03), (12.07, 419.78), (23.92, 232.28), (1.83, 500.04), (14.58, 370.52), (22.56, 252.59), (10.59, 339.97), (6.85, 464.12), (35.43, 113.27), (18.82, 333.44), (-2.4, 493.99), (16.04, 319.78), (8.3, 494.2), (21.65, 276.26), (18.15, 278.91), (23.34, 246.53), (14.97, 379.99), (-10.5, 640.26), (8.16, 397.16), (2.73, 507.36), (20.27, 305.91), (13.55, 335.85), (28.05, 240.29), (9.02, 412.81), (19.01, 281.66), (13.59, 385.92), (1.68, 454.79), (14.09, 361.03), (-5.92, 536.81), (32.05, 210.21), (7.2, 412.7), (16.56, 278.57), (19.41, 249.53), (22.12, 303.69), (11.68, 431.51), (11.68, 395.54), (1.67, 463.18), (14.4, 365.93), (1.09, 493.69), (6.86, 445.73), (12.77, 345.73), (23.77, 254.93), (2.97, 451.11), (22.47, 247.1), (19.46, 217.72), (8.64, 435.92), (26.91, 268.16), (9.86, 389.35), (7.0, 405.06), (10.64, 326.59), (27.33, 220.25), (14.44, 370.42), (26.24, 231.12), (37.15, 145.58), (27.17, 231.93), (16.04, 258.38), (29.42, 215.72), (17.24, 313.1), (-4.96, 523.21), (9.37, 451.37), (15.39, 387.15), (2.42, 440.18), (27.89, 239.98), (11.03, 322.25), (9.83, 448.8), (1.82, 469.8), (9.97, 433.6), (22.33, 318.91), (21.94, 248.93), (20.75, 268.15), (24.4, 274.69), (-14.15, 596.08), (17.44, 321.19), (14.73, 377.54), (4.82, 437.88), (21.14, 279.51), (17.72, 320.03), (16.82, 336.16), (10.25, 398.83), (26.4, 220.12), (18.9, 298.71), (-4.77, 581.26), (14.58, 360.33), (13.36, 356.47), (11.7, 361.76), (22.23, 308.91), (15.86, 392.88), (7.56, 454.31), (24.01, 274.01), (16.48, 287.23), (23.59, 215.69), (6.48, 472.18), (-4.06, 527.03), (25.88, 270.94), (15.19, 357.82), (26.69, 202.98), (6.2, 394.26), (27.0, 212.87), (25.26, 236.76), (1.31, 452.8), (9.98, 438.75), (21.23, 266.11), (4.69, 431.51), (2.35, 455.85), (19.31, 317.51), (0.24, 470.51), (29.0, 258.88), (24.38, 333.01), (4.92, 395.09), (10.52, 365.76), (2.85, 491.36), (6.59, 409.5), (14.51, 408.64), (-3.85, 552.12), (17.12, 295.13), (5.02, 427.97), (20.67, 273.45), (22.02, 254.82), (19.58, 275.09), (16.89, 363.37), (18.83, 302.26), (8.87, 401.02), (7.02, 457.39), (12.73, 352.48), (5.91, 465.97), (-1.81, 497.71), (15.16, 331.49), (13.68, 323.57), (18.58, 324.9), (12.18, 413.31), (15.6, 348.53), (20.82, 287.34), (10.17, 410.46), (-6.59, 632.5), (-0.98, 473.4), (10.32, 350.24), (12.14, 390.53), (23.07, 273.85), (24.56, 256.72), (17.71, 337.07), (20.78, 282.41), (12.61, 363.78), (18.03, 347.65), (9.92, 433.25), (12.22, 395.77), (-6.59, 608.36), (23.72, 252.35), (23.05, 249.29), (14.01, 380.9), (5.88, 435.32), (17.57, 300.47), (22.53, 269.59), (25.56, 230.96), (7.77, 471.11), (16.72, 337.24), (34.3, 137.48), (18.61, 249.2), (5.69, 409.54), (-0.97, 542.8), (34.57, 141.11), (13.52, 353.54), (17.12, 328.98), (36.49, 110.11), (27.1, 234.2), (13.65, 367.52), (20.09, 291.05), (16.85, 286.89), (8.69, 416.25), (12.01, 354.17), (30.99, 182.01), (-1.14, 503.66), (17.01, 339.79), (18.43, 341.94), (12.47, 407.06), (36.82, 109.55), (34.48, 137.83), (7.22, 444.14), (40.71, 120.56), (20.99, 304.65), (14.22, 390.0), (18.25, 335.64), (25.53, 247.88), (24.05, 250.43), (18.35, 239.22), (17.91, 292.81), (21.79, 273.21), (19.34, 272.93), (4.07, 433.09), (18.94, 295.07), (15.97, 285.18), (8.1, 394.22), (26.31, 208.73), (15.35, 317.26), (2.08, 497.51), (13.79, 349.58), (34.93, 128.64), (5.99, 410.7), (23.18, 265.06), (18.17, 295.13), (4.66, 452.95), (17.24, 352.3), (19.13, 351.37), (8.97, 389.88), (25.71, 219.25), (6.03, 492.07), (20.49, 300.0), (23.05, 213.71), (-3.75, 526.52), (24.24, 296.79), (5.38, 477.62), (29.15, 221.54), (5.44, 430.3), (23.89, 252.96), (16.18, 330.29), (19.66, 351.52), (11.61, 398.8), (15.74, 355.74), (20.58, 319.1), (13.44, 387.57), (14.09, 315.29), (12.3, 372.79), (8.12, 393.18), (26.58, 255.2), (22.4, 288.96), (33.36, 172.05), (-0.4, 512.95), (11.2, 366.18), (19.37, 267.55), (22.64, 251.34), (-2.48, 519.42), (6.17, 407.72), (18.77, 339.58), (16.21, 324.8), (9.91, 400.94), (4.11, 459.66), (18.03, 368.37), (10.39, 436.41), (20.73, 272.97), (19.73, 308.61), (6.3, 482.39), (12.98, 343.77), (25.42, 203.95), (25.88, 218.62), (13.23, 349.55), (8.85, 395.28), (-4.15, 518.34), (25.6, 205.16), (5.0, 413.29), (11.55, 405.73), (13.49, 393.4), (18.33, 315.44), (17.07, 292.06), (3.31, 495.19), (3.53, 447.54), (21.84, 254.63), (3.63, 404.62), (-2.01, 528.99), (13.91, 355.9), (13.13, 441.3), (2.49, 447.87), (13.84, 413.19), (13.74, 371.51), (24.27, 284.18), (5.08, 436.91), (18.46, 351.98), (26.55, 248.48), (26.1, 221.26), (4.61, 444.78), (22.48, 277.33), (20.76, 269.38), (29.33, 202.41), (11.04, 386.89), (17.23, 307.8), (-1.36, 486.73), (23.34, 267.0), (35.91, 137.93), (17.21, 337.02), (17.22, 322.03), (6.82, 382.87), (-2.83, 535.31), (20.84, 236.27), (30.13, 182.43), (25.8, 243.21), (28.32, 244.66), (25.4, 207.69), (-3.78, 558.0), (12.82, 363.37), (15.95, 333.18), (29.04, 162.69), (29.69, 242.42), (39.52, 89.12), (23.9, 244.21), (14.98, 398.56), (19.33, 320.04), (1.34, 466.55), (21.87, 266.71), (23.04, 231.82), (-1.89, 498.12), (13.78, 355.34), (10.66, 464.82), (17.43, 309.06), (20.23, 375.09), (-3.45, 544.39), (2.26, 505.81), (22.11, 280.25), (11.13, 380.63), (29.9, 180.92), (17.34, 292.71), (21.44, 277.99), (26.66, 213.32), (28.05, 216.43), (26.63, 216.66), (32.9, 140.72), (4.58, 472.03), (13.27, 389.24), (15.33, 332.48), (10.93, 446.62), (16.34, 364.68), (22.79, 299.59), (14.9, 356.94), (13.25, 386.58), (3.17, 447.75), (23.09, 269.32), (9.82, 408.45), (12.38, 398.44), (12.89, 368.14), (21.45, 355.12), (17.49, 288.53), (7.91, 398.13), (18.61, 328.27), (12.78, 327.55), (10.06, 424.15), (21.21, 323.32), (-3.05, 522.63), (11.86, 392.28), (28.0, 188.13), (11.12, 347.53), (7.01, 439.11), (4.81, 494.7), (30.18, 193.31), (6.6, 421.07), (11.24, 429.61), (14.73, 363.15), (0.75, 523.43), (6.74, 433.36), (9.34, 433.24), (-14.3, 639.09), (17.66, 334.78), (16.65, 307.9), (1.19, 515.74), (0.52, 420.34), (17.21, 363.98), (0.25, 558.51), (24.33, 274.58), (0.53, 469.69), (19.89, 324.05), (8.16, 377.22), (10.69, 423.05), (0.69, 481.64), (10.5, 429.34), (25.44, 247.69), (14.03, 373.28), (9.47, 433.85), (0.52, 521.3), (11.17, 349.53), (16.37, 294.56), (32.32, 211.38), (-5.52, 535.41), (25.58, 228.15), (4.76, 468.94), (26.24, 280.6), (40.48, 168.68), (13.63, 352.38), (24.61, 243.47), (24.38, 177.66), (3.56, 410.69), (-0.97, 538.52), (19.22, 299.24), (19.35, 303.11), (6.68, 418.16), (5.05, 505.25), (2.06, 469.66), (21.94, 285.79), (0.94, 461.47), (22.28, 278.73), (23.26, 297.93), (15.28, 306.71), (22.0, 247.33), (8.14, 410.19), (-6.99, 584.9), (29.97, 230.94), (24.85, 330.01), (8.97, 427.27), (6.17, 387.68), (17.77, 308.05), (3.42, 479.12), (9.69, 440.47), (7.35, 403.57), (18.54, 347.21), (17.02, 313.23), (0.76, 442.4), (0.45, 515.23), (13.1, 424.95), (18.88, 285.54), (20.48, 331.79), (14.95, 296.0), (40.77, 85.23), (10.93, 402.93), (5.46, 423.51), (17.06, 348.31), (21.46, 235.64), (-5.36, 515.44), (24.45, 259.98), (26.06, 216.3), (12.59, 375.98), (31.37, 184.01), (-10.77, 592.81), (25.12, 240.71), (3.02, 447.88), (17.72, 307.66), (3.61, 445.34), (12.43, 383.88), (5.45, 413.04), (12.32, 367.93), (13.24, 332.98), (-3.38, 549.45), (3.11, 447.11), (11.33, 370.15), (-0.29, 489.29), (4.9, 387.16), (14.89, 315.01), (12.05, 391.18), (13.29, 412.81), (13.47, 361.9), (18.98, 327.4), (18.23, 263.57), (25.71, 202.34), (12.79, 388.03), (11.24, 421.49), (5.24, 465.28), (17.45, 245.43), (6.18, 412.18), (40.98, 79.68), (11.74, 354.99), (26.64, 182.51), (19.86, 322.84), (11.52, 361.04), (9.88, 406.48), (16.62, 324.53), (10.35, 459.81), (-7.54, 584.82), (23.31, 284.1), (22.73, 272.17), (12.25, 357.76), (16.21, 291.62), (9.94, 393.82), (5.99, 461.02), (15.29, 344.12), (22.47, 247.39), (32.73, 203.2), (29.93, 188.44), (10.73, 417.98), (23.63, 245.72), (26.66, 226.58), (15.15, 403.73), (16.89, 319.95), (13.74, 389.17), (37.01, 132.31), (31.11, 206.32), (18.13, 255.47), (3.42, 455.47), (13.02, 358.5), (15.14, 303.36), (13.88, 340.79), (25.26, 255.55), (20.61, 309.52), (33.39, 162.46), (4.27, 422.05), (11.01, 360.17), (25.02, 221.9), (36.85, 132.97), (6.21, 429.78), (5.63, 422.61), (21.42, 304.19), (33.18, 101.3), (14.58, 318.07), (12.96, 477.49), (22.41, 280.06), (23.16, 181.18), (11.58, 377.18), (7.82, 387.42), (16.68, 360.71), (18.79, 326.33), (27.32, 267.59), (26.0, 245.92), (5.01, 434.47), (-0.1, 475.45), (33.0, 142.49), (16.31, 310.7), (2.65, 427.06), (20.15, 289.72), (12.05, 350.33), (14.41, 372.82), (10.46, 348.46), (25.64, 198.58), (18.4, 312.41), (16.53, 348.65), (-0.53, 555.16), (22.58, 279.78), (5.26, 444.38), (11.01, 371.65), (37.83, 141.35), (2.22, 504.56), (8.19, 400.95), (29.27, 197.45), (1.28, 464.83), (6.41, 434.71), (11.81, 422.09), (10.45, 356.98), (24.94, 316.97), (17.55, 317.67), (10.56, 368.22), (23.09, 268.24), (12.58, 375.93), (20.19, 315.86), (25.43, 253.26), (22.0, 306.17), (19.1, 314.11), (9.27, 428.5), (8.42, 409.1), (7.54, 357.88), (27.24, 200.87), (30.58, 218.29), (12.5, 432.78), (-4.71, 533.19), (3.42, 477.18), (6.84, 412.54)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=14.7058，ȳ=350.5191；Sxx=Σ(x−x̄)²=58037.05，Sxy=-582399.01；
β₁ = Sxy/Sxx = -10.0350；β₀ = ȳ−β₁x̄ = 498.0910；
R² = SSR/SST = 5844346.46/6364625.43 = 0.9183；F = (SSR/1)/(SSE/565) = 6346.7023，p = 2.07e-309；
SE(β₁)=√(920.85/58037.05)=0.1260；t=β₁/SE=-79.6662；p=2.07e-309；95%CI=[-10.2824,-9.7875]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=14.7058，ȳ=350.5191；Sxx=Σ(x−x̄)²=58037.05，Sxy=-582399.01；
β₁ = Sxy/Sxx = -10.0350；β₀ = ȳ−β₁x̄ = 498.0910；
R² = SSR/SST = 5844346.46/6364625.43 = 0.9183；F = (SSR/1)/(SSE/565) = 6346.7023，p = 2.07e-309；
SE(β₁)=√(920.85/58037.05)=0.1260；t=β₁/SE=-79.6662；p=2.07e-309；95%CI=[-10.2824,-9.7875]。
③ 计算结果：
y = 498.0910 + -10.0350·x；R²=0.9183；β₁ t=-79.6662 p=2.07e-309；95%CI=[-10.2824,-9.7875]
④ 结论：在 α=0.05 水平下，p=2.07e-309 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-10.0350），天冷卖得多。
案例 55. 价格预测购买意愿
简单线性回归越贵越不想买
① 原始数据：
样本 n=897，逐条 (x, y)：[(125.71, 4.16), (155.73, 4.21), (117.18, 4.94), (93.58, 5.14), (133.08, 4.55), (87.32, 4.88), (161.24, 4.0), (91.54, 5.15), (142.71, 4.29), (137.78, 3.78), (127.78, 4.23), (-18.91, 6.63), (120.91, 4.58), (56.1, 5.62), (90.89, 5.45), (83.33, 4.98), (16.78, 6.4), (139.21, 4.78), (112.17, 4.42), (143.65, 4.62), (148.87, 4.77), (48.56, 5.38), (132.33, 4.44), (98.85, 5.21), (129.34, 4.36), (74.43, 5.43), (173.93, 3.7), (44.68, 5.76), (140.86, 4.1), (88.38, 4.85), (126.6, 4.56), (133.3, 4.57), (96.97, 4.33), (159.88, 3.67), (157.77, 3.81), (114.0, 4.65), (63.9, 5.71), (95.14, 4.49), (185.75, 3.16), (15.63, 5.94), (124.26, 4.43), (89.42, 5.03), (75.09, 5.39), (40.42, 5.81), (104.51, 5.44), (146.35, 4.19), (137.43, 4.43), (62.11, 5.46), (20.37, 6.36), (127.7, 4.69), (147.38, 4.14), (65.53, 5.0), (80.08, 5.06), (48.83, 5.78), (108.43, 4.65), (133.07, 4.3), (83.67, 5.15), (128.66, 4.73), (74.66, 5.68), (98.86, 4.89), (68.3, 5.0), (87.64, 4.86), (63.25, 5.7), (111.22, 5.06), (138.09, 4.56), (101.08, 5.16), (93.02, 4.67), (42.0, 5.66), (140.01, 5.04), (99.74, 5.14), (14.58, 6.27), (125.62, 4.12), (173.86, 3.91), (97.03, 4.58), (138.24, 4.43), (110.62, 4.49), (97.03, 5.16), (96.83, 4.7), (112.46, 4.98), (107.43, 4.82), (93.9, 5.05), (87.85, 4.91), (24.36, 6.34), (176.81, 3.88), (129.31, 4.57), (72.6, 5.41), (73.23, 5.17), (147.82, 4.31), (38.48, 5.7), (112.87, 4.66), (126.4, 4.33), (90.5, 5.42), (95.86, 4.81), (186.63, 3.6), (75.49, 5.09), (64.54, 5.19), (98.51, 5.38), (158.76, 3.72), (141.81, 4.23), (121.71, 4.72), (93.92, 4.98), (61.6, 5.66), (118.67, 4.61), (84.53, 5.32), (104.07, 4.47), (98.25, 4.91), (83.54, 5.3), (34.45, 6.09), (119.71, 4.59), (22.08, 6.02), (76.64, 5.05), (48.57, 5.89), (14.78, 6.38), (28.56, 5.61), (81.24, 4.85), (133.75, 4.5), (110.39, 4.93), (20.47, 5.73), (149.22, 4.58), (138.71, 4.48), (78.92, 5.7), (66.15, 5.66), (95.79, 4.83), (102.26, 5.16), (66.13, 5.76), (126.9, 4.44), (138.35, 4.47), (111.72, 4.77), (109.27, 4.62), (149.68, 3.96), (70.32, 5.71), (40.79, 6.0), (103.88, 5.04), (139.89, 4.44), (60.62, 6.02), (142.08, 4.01), (104.5, 5.17), (50.58, 5.69), (97.2, 5.16), (78.81, 5.24), (49.15, 5.68), (144.46, 4.34), (119.58, 4.42), (97.74, 5.14), (82.48, 5.58), (120.53, 4.53), (84.06, 5.49), (90.79, 5.79), (123.6, 4.62), (94.0, 4.93), (56.48, 5.44), (142.06, 4.11), (61.48, 5.48), (22.53, 5.91), (156.47, 4.39), (127.54, 4.51), (118.72, 4.47), (94.81, 5.44), (109.57, 5.66), (28.27, 5.78), (73.7, 5.57), (115.55, 5.11), (85.64, 5.06), (61.83, 5.83), (108.52, 4.55), (147.02, 4.03), (81.67, 5.25), (51.32, 6.45), (56.15, 5.56), (74.57, 5.21), (94.28, 5.23), (106.18, 4.98), (113.84, 4.27), (63.96, 5.15), (112.71, 4.92), (90.38, 4.75), (95.45, 5.33), (131.11, 4.11), (100.67, 4.76), (54.11, 5.77), (102.49, 4.83), (63.93, 5.3), (110.25, 4.83), (170.98, 4.28), (130.37, 4.53), (92.65, 5.19), (79.24, 5.37), (131.15, 4.66), (125.64, 4.32), (124.49, 4.73), (80.91, 5.21), (67.0, 6.12), (88.53, 4.84), (127.57, 4.95), (184.55, 4.25), (124.58, 4.72), (119.34, 4.42), (114.41, 4.66), (101.33, 5.07), (51.34, 6.19), (60.83, 5.2), (126.76, 4.66), (59.77, 5.48), (154.05, 4.52), (97.94, 5.43), (125.54, 4.68), (148.09, 4.82), (134.62, 4.37), (142.31, 4.18), (117.69, 4.38), (86.44, 5.34), (63.06, 5.46), (124.71, 4.26), (81.34, 4.54), (63.24, 5.53), (132.19, 4.76), (126.01, 4.24), (120.18, 4.78), (126.56, 4.74), (139.95, 4.83), (63.74, 5.32), (92.22, 4.77), (118.52, 5.13), (68.86, 4.99), (98.01, 4.9), (98.4, 4.89), (134.2, 4.74), (92.43, 5.53), (97.63, 4.9), (119.3, 4.52), (26.68, 6.23), (47.81, 5.63), (104.99, 4.97), (90.0, 5.17), (103.73, 5.53), (119.76, 5.23), (165.82, 3.82), (88.9, 4.77), (116.72, 4.67), (102.25, 5.15), (79.9, 5.28), (73.98, 5.88), (148.88, 4.2), (112.07, 4.72), (159.67, 3.89), (79.04, 4.98), (45.92, 6.23), (54.49, 5.56), (114.87, 4.88), (108.93, 5.36), (116.78, 4.41), (127.73, 4.8), (112.39, 4.71), (116.07, 5.14), (84.64, 5.08), (142.78, 3.75), (107.88, 4.92), (131.3, 5.14), (93.19, 5.16), (175.89, 4.19), (79.12, 5.58), (98.3, 5.24), (111.09, 4.84), (94.92, 4.67), (97.68, 4.79), (163.1, 3.8), (127.64, 4.87), (138.04, 4.34), (120.72, 4.78), (116.99, 4.41), (119.01, 4.32), (130.2, 4.51), (65.15, 5.23), (90.45, 5.33), (130.88, 4.83), (193.51, 3.51), (34.22, 6.14), (131.79, 4.59), (196.6, 4.16), (62.95, 5.69), (90.18, 4.96), (97.65, 5.28), (95.37, 4.64), (74.08, 5.5), (91.36, 4.29), (135.44, 4.37), (93.93, 5.45), (123.32, 4.82), (131.15, 3.99), (114.93, 4.78), (107.53, 5.23), (8.95, 6.28), (113.38, 4.97), (79.29, 5.05), (53.88, 5.98), (142.62, 4.08), (26.7, 6.01), (104.0, 5.34), (126.43, 4.6), (41.27, 6.01), (115.05, 5.19), (96.66, 5.02), (76.23, 5.53), (102.88, 4.82), (72.65, 5.22), (91.85, 4.93), (145.3, 4.75), (67.12, 5.74), (138.73, 4.56), (104.07, 5.19), (149.81, 4.44), (127.6, 4.4), (105.13, 4.66), (162.37, 3.78), (83.2, 5.55), (50.08, 5.22), (102.22, 4.99), (116.26, 5.0), (127.46, 4.63), (36.12, 6.27), (22.81, 6.54), (162.7, 3.65), (72.78, 5.74), (95.46, 4.89), (93.97, 5.32), (91.74, 5.1), (68.78, 5.68), (106.34, 4.71), (70.49, 5.24), (93.2, 5.19), (77.52, 5.4), (41.33, 6.26), (118.2, 4.67), (104.04, 4.46), (22.49, 5.96), (166.32, 4.01), (10.84, 6.18), (111.73, 4.54), (70.11, 5.31), (102.99, 5.28), (25.8, 5.68), (106.7, 4.47), (87.25, 5.2), (85.51, 5.42), (51.36, 5.76), (82.8, 4.77), (93.82, 4.98), (155.21, 3.9), (124.5, 4.84), (114.34, 4.8), (160.39, 3.82), (140.61, 4.05), (88.14, 5.32), (89.13, 5.76), (129.73, 4.95), (59.52, 6.11), (62.18, 6.25), (76.75, 5.56), (116.12, 4.75), (149.38, 4.01), (155.62, 4.0), (96.58, 4.8), (41.38, 5.8), (57.04, 5.63), (66.67, 5.42), (93.72, 5.9), (62.24, 5.94), (87.94, 4.78), (23.17, 5.5), (96.8, 4.85), (117.59, 4.28), (63.88, 5.81), (99.8, 4.86), (137.21, 4.13), (-12.59, 6.5), (74.54, 5.46), (151.29, 4.01), (111.61, 5.44), (191.12, 3.62), (127.77, 4.5), (117.39, 4.75), (135.65, 5.22), (52.84, 5.95), (141.38, 4.02), (74.22, 5.02), (134.78, 4.23), (94.17, 5.95), (92.97, 5.45), (128.77, 4.19), (158.88, 4.27), (158.67, 3.74), (111.5, 5.04), (104.3, 4.52), (77.85, 4.92), (132.13, 4.61), (130.78, 4.55), (110.77, 4.5), (181.21, 3.56), (135.12, 4.22), (107.85, 4.46), (92.35, 4.89), (107.73, 5.16), (65.64, 5.88), (102.79, 5.37), (51.47, 5.51), (122.23, 5.18), (156.7, 4.31), (123.8, 4.45), (72.51, 5.62), (17.58, 6.01), (41.63, 6.07), (122.63, 5.07), (104.02, 4.66), (96.72, 5.23), (32.06, 6.01), (120.63, 4.32), (139.25, 4.89), (163.52, 4.32), (97.12, 5.14), (146.12, 3.91), (131.51, 4.47), (-38.92, 7.17), (91.87, 5.01), (64.28, 5.47), (189.45, 3.95), (154.55, 3.69), (143.69, 3.8), (119.3, 4.54), (120.41, 4.75), (118.05, 4.69), (65.88, 5.34), (77.54, 5.14), (123.18, 4.93), (66.31, 5.29), (109.83, 4.61), (85.72, 5.19), (51.62, 5.78), (109.01, 4.91), (101.44, 4.5), (18.78, 6.25), (96.77, 4.95), (101.28, 4.56), (106.02, 4.59), (166.94, 4.56), (131.82, 4.14), (118.9, 4.94), (20.75, 6.21), (44.57, 5.81), (42.2, 5.85), (118.31, 5.0), (59.41, 5.7), (133.72, 4.77), (154.29, 4.14), (125.41, 4.34), (90.99, 5.18), (114.73, 5.06), (123.72, 5.02), (98.87, 4.85), (67.06, 5.29), (92.13, 4.95), (116.41, 4.42), (95.06, 4.83), (78.38, 5.49), (77.69, 5.39), (40.72, 6.03), (72.67, 5.94), (108.55, 5.38), (118.99, 4.6), (70.91, 5.35), (101.61, 5.01), (71.61, 5.18), (142.52, 4.03), (72.18, 5.32), (115.36, 4.45), (74.07, 5.48), (75.08, 5.34), (56.77, 5.95), (16.23, 6.58), (107.45, 5.08), (57.37, 5.39), (141.13, 4.09), (70.23, 5.44), (60.6, 6.09), (90.34, 5.36), (119.06, 4.74), (135.4, 4.41), (77.16, 5.5), (72.41, 5.46), (68.87, 5.13), (84.4, 5.1), (85.24, 5.44), (64.33, 6.04), (131.72, 4.97), (125.72, 4.73), (135.92, 4.74), (136.15, 4.27), (111.72, 4.79), (141.68, 4.61), (97.8, 5.65), (70.98, 4.95), (145.87, 4.39), (101.66, 5.16), (123.61, 4.42), (110.92, 5.06), (137.96, 4.11), (118.58, 3.92), (46.44, 5.87), (103.16, 5.09), (109.63, 4.29), (62.6, 5.75), (116.26, 4.61), (73.41, 5.56), (104.45, 4.65), (103.52, 4.37), (130.69, 4.13), (161.23, 4.18), (84.49, 5.52), (124.93, 4.64), (72.12, 5.45), (64.23, 5.71), (99.25, 4.98), (60.02, 5.23), (24.54, 6.17), (135.56, 4.27), (144.82, 4.02), (72.35, 5.64), (108.26, 5.15), (97.8, 5.39), (118.89, 5.14), (42.9, 6.22), (61.23, 5.18), (162.33, 4.18), (100.97, 4.72), (123.4, 4.62), (108.27, 4.96), (78.1, 5.8), (182.69, 3.72), (41.66, 5.76), (71.12, 5.15), (110.32, 5.02), (116.23, 4.73), (65.09, 5.88), (23.04, 6.54), (100.88, 5.06), (83.46, 4.93), (126.78, 5.14), (128.42, 4.95), (129.58, 4.76), (67.45, 5.67), (112.08, 4.92), (55.36, 5.36), (135.98, 4.49), (101.9, 4.9), (103.03, 4.91), (9.84, 6.12), (130.02, 4.86), (41.92, 5.98), (98.14, 4.9), (67.77, 5.51), (53.23, 5.67), (119.99, 4.58), (111.44, 5.19), (97.73, 5.37), (119.72, 4.28), (141.57, 4.65), (166.77, 3.53), (145.57, 4.61), (97.32, 5.44), (111.55, 4.35), (26.57, 5.84), (95.69, 4.95), (33.75, 5.97), (53.32, 5.64), (33.48, 6.43), (114.13, 5.03), (114.73, 4.88), (136.88, 4.2), (103.32, 5.35), (112.05, 5.15), (121.5, 4.92), (130.99, 4.49), (107.77, 4.26), (78.36, 5.22), (129.39, 4.49), (139.49, 4.37), (181.55, 3.58), (8.51, 6.79), (125.59, 4.46), (82.4, 4.77), (50.1, 6.09), (73.48, 5.29), (88.59, 4.9), (48.39, 5.67), (70.61, 5.24), (74.41, 5.49), (113.5, 5.02), (85.01, 5.86), (78.65, 5.48), (99.54, 4.95), (112.14, 4.96), (119.78, 4.84), (97.04, 5.36), (135.34, 5.0), (157.25, 4.19), (163.41, 4.04), (120.26, 4.25), (163.84, 4.58), (110.23, 4.85), (141.97, 4.3), (110.42, 4.72), (147.91, 4.68), (136.15, 4.54), (58.64, 5.8), (68.36, 5.71), (111.18, 4.85), (131.1, 4.25), (97.7, 5.12), (89.33, 5.19), (110.33, 5.2), (73.45, 4.85), (127.71, 4.78), (112.81, 4.57), (76.45, 5.57), (168.51, 3.75), (152.64, 4.12), (118.49, 4.4), (92.68, 5.19), (40.66, 5.93), (77.55, 5.67), (129.36, 4.51), (108.42, 4.89), (101.73, 5.54), (94.47, 5.3), (85.1, 5.02), (110.07, 4.83), (77.09, 5.34), (147.6, 3.68), (134.61, 4.84), (75.84, 5.03), (94.52, 4.53), (121.26, 4.89), (63.69, 5.56), (48.13, 5.66), (109.97, 5.01), (36.19, 6.09), (64.32, 5.95), (115.07, 4.71), (142.91, 4.45), (177.68, 3.69), (123.36, 4.63), (14.15, 6.57), (66.08, 4.99), (114.46, 4.45), (72.37, 4.65), (85.66, 5.27), (72.4, 5.66), (98.34, 5.38), (89.44, 4.96), (113.35, 4.67), (84.51, 5.86), (79.67, 5.7), (135.26, 3.69), (144.2, 4.14), (69.97, 4.99), (100.84, 5.13), (118.69, 5.2), (77.91, 4.76), (31.68, 6.42), (107.39, 4.57), (136.37, 4.53), (33.11, 5.91), (49.99, 5.85), (51.24, 5.3), (95.84, 5.67), (80.23, 4.64), (142.25, 4.14), (76.88, 4.73), (117.15, 4.67), (171.48, 4.42), (70.98, 5.57), (48.92, 5.93), (105.42, 4.84), (121.6, 5.23), (154.24, 4.25), (35.71, 6.35), (110.8, 4.58), (67.23, 5.6), (65.73, 5.23), (40.67, 5.59), (78.65, 5.16), (139.09, 4.18), (114.86, 5.03), (67.92, 5.29), (60.08, 5.33), (68.05, 4.99), (71.25, 6.13), (40.34, 5.77), (141.82, 4.64), (158.16, 4.03), (53.67, 6.03), (151.2, 4.36), (149.5, 4.29), (113.74, 4.81), (73.57, 5.02), (192.44, 3.95), (160.19, 4.19), (107.13, 5.1), (134.72, 4.72), (45.05, 5.49), (167.43, 3.68), (138.56, 5.02), (75.52, 5.94), (79.05, 5.36), (60.42, 5.78), (143.57, 4.28), (70.81, 5.11), (82.14, 5.63), (59.75, 5.42), (114.86, 4.9), (91.88, 4.74), (139.17, 4.3), (68.02, 5.84), (105.37, 4.39), (82.11, 5.06), (140.39, 4.88), (121.91, 4.08), (121.46, 4.35), (32.67, 6.31), (150.71, 4.15), (66.0, 5.06), (129.5, 5.13), (115.19, 4.74), (138.43, 4.16), (61.13, 5.01), (118.51, 4.82), (41.51, 5.9), (115.0, 5.3), (63.67, 5.09), (95.93, 4.86), (47.03, 5.77), (67.38, 5.75), (64.28, 5.18), (124.63, 4.78), (122.54, 4.84), (127.6, 4.25), (60.39, 6.26), (28.45, 5.61), (143.23, 3.95), (133.75, 4.73), (91.8, 5.12), (144.84, 4.39), (53.6, 6.19), (101.34, 4.64), (125.41, 4.12), (77.63, 5.59), (131.48, 4.03), (100.42, 4.43), (49.63, 6.01), (91.97, 5.28), (61.54, 5.45), (128.43, 4.58), (127.77, 4.75), (18.71, 6.54), (136.88, 5.03), (111.86, 4.72), (120.38, 4.99), (161.26, 3.78), (116.19, 4.57), (51.8, 5.41), (80.06, 4.8), (110.31, 5.07), (119.41, 4.21), (125.61, 4.8), (36.98, 5.79), (141.0, 4.25), (72.28, 5.61), (128.51, 4.57), (105.22, 4.87), (73.12, 5.73), (52.42, 5.6), (147.31, 3.89), (85.21, 5.14), (86.06, 5.16), (42.18, 5.77), (168.18, 4.01), (27.5, 5.6), (80.93, 5.26), (138.0, 4.25), (129.94, 4.5), (75.97, 5.54), (47.06, 5.98), (109.35, 4.97), (134.54, 4.36), (125.57, 3.85), (122.17, 4.6), (114.84, 4.57), (111.34, 4.55), (126.76, 5.11), (113.23, 5.05), (66.88, 5.43), (72.1, 5.68), (11.02, 6.41), (62.06, 5.41), (71.68, 5.85), (106.37, 4.8), (42.4, 5.86), (201.62, 4.09), (51.51, 5.56), (106.57, 4.99), (90.8, 5.29), (154.96, 3.87), (132.23, 4.69), (105.81, 5.28), (99.95, 5.11), (81.18, 5.0), (79.72, 4.78), (122.15, 4.71), (99.14, 4.89), (92.39, 4.82), (160.38, 4.75), (145.29, 4.17), (135.77, 4.58), (7.91, 6.31), (106.24, 4.49), (163.3, 3.93), (73.34, 5.22), (134.76, 4.31), (155.43, 4.14), (40.8, 5.63), (70.71, 5.4), (18.66, 6.59), (104.13, 4.96), (151.94, 3.8), (101.72, 4.98), (54.4, 5.3), (38.74, 6.15), (16.62, 5.72), (89.71, 5.68), (81.83, 5.22), (173.35, 4.74), (131.48, 4.48), (87.14, 4.75), (79.23, 4.85), (117.44, 4.55), (104.53, 4.91), (43.95, 5.67), (123.95, 4.82), (70.81, 5.61), (101.43, 5.18), (139.1, 4.71), (126.24, 4.33), (106.49, 4.51), (98.07, 5.42), (88.0, 4.57), (-7.04, 6.65), (97.1, 5.45), (79.01, 5.3), (69.57, 5.56), (48.2, 5.82), (75.89, 5.37), (120.45, 4.94), (56.75, 5.98), (87.12, 4.94), (169.8, 4.27), (118.02, 4.94), (64.07, 5.56), (67.43, 6.06), (92.76, 4.9), (84.12, 4.68), (89.39, 5.29), (131.23, 5.12), (61.99, 5.41), (89.27, 4.94), (123.03, 4.85), (123.98, 4.37), (33.93, 6.32), (103.62, 5.0), (25.4, 6.09), (75.82, 4.53), (64.15, 5.75), (65.58, 5.8), (132.73, 4.48), (141.01, 4.36), (107.62, 4.88), (82.39, 5.14), (71.84, 5.24), (116.6, 4.71), (152.87, 4.65), (90.59, 5.27), (77.6, 5.7), (162.3, 4.29), (134.49, 4.54), (97.67, 4.7), (105.28, 4.73), (178.04, 3.72), (70.23, 5.47), (155.87, 4.31)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=99.2928，ȳ=5.0004；Sxx=Σ(x−x̄)²=1299374.75，Sxy=-19370.79；
β₁ = Sxy/Sxx = -0.0149；β₀ = ȳ−β₁x̄ = 6.4806；
R² = SSR/SST = 288.78/369.18 = 0.7822；F = (SSR/1)/(SSE/895) = 3214.2272，p = 1.83e-298；
SE(β₁)=√(0.09/1299374.75)=0.0003；t=β₁/SE=-56.6942；p=1.83e-298；95%CI=[-0.0203,-0.0096]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=99.2928，ȳ=5.0004；Sxx=Σ(x−x̄)²=1299374.75，Sxy=-19370.79；
β₁ = Sxy/Sxx = -0.0149；β₀ = ȳ−β₁x̄ = 6.4806；
R² = SSR/SST = 288.78/369.18 = 0.7822；F = (SSR/1)/(SSE/895) = 3214.2272，p = 1.83e-298；
SE(β₁)=√(0.09/1299374.75)=0.0003；t=β₁/SE=-56.6942；p=1.83e-298；95%CI=[-0.0203,-0.0096]。
③ 计算结果：
y = 6.4806 + -0.0149·x；R²=0.7822；β₁ t=-56.6942 p=1.83e-298；95%CI=[-0.0203,-0.0096]
④ 结论：在 α=0.05 水平下，p=1.83e-298 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-0.0149），越贵越不想买。
案例 56. 吸烟量预测肺活量
简单线性回归吸烟伤肺
① 原始数据：
样本 n=669，逐条 (x, y)：[(5.93, 4074.51), (14.52, 3468.25), (13.54, 4132.37), (17.95, 3841.46), (16.9, 3454.64), (17.29, 3586.36), (1.78, 4186.74), (7.25, 3610.35), (17.74, 3626.9), (5.67, 3852.59), (10.73, 3465.03), (12.62, 3806.23), (9.09, 3953.09), (13.01, 3585.95), (13.49, 3384.27), (8.31, 3888.42), (6.27, 3703.46), (15.76, 3562.69), (11.28, 3464.96), (20.9, 3464.52), (5.84, 3949.06), (4.53, 4251.94), (5.19, 3387.64), (11.59, 3452.94), (10.95, 3672.16), (18.15, 3700.64), (11.56, 3266.08), (1.55, 3937.56), (3.79, 3909.06), (7.07, 3747.2), (11.79, 4007.92), (8.44, 3837.52), (12.01, 3807.74), (7.79, 3855.0), (5.55, 3864.84), (13.85, 3394.42), (2.89, 3632.07), (8.71, 3821.75), (15.42, 3546.51), (15.77, 3589.01), (-3.28, 4095.45), (6.73, 3690.62), (8.48, 3554.72), (10.27, 3602.82), (13.76, 3715.97), (19.69, 3361.6), (16.36, 3426.03), (7.79, 3576.55), (5.72, 4112.94), (13.53, 3713.19), (13.92, 3774.28), (7.3, 3659.21), (11.85, 3970.57), (3.81, 3986.08), (9.15, 3882.33), (2.36, 3703.66), (12.32, 3925.32), (3.53, 4134.79), (3.95, 3914.19), (-7.36, 4819.51), (17.87, 3285.56), (11.83, 3464.49), (11.78, 3291.76), (-1.97, 4179.02), (7.32, 3775.43), (10.7, 3667.18), (4.25, 3990.08), (6.07, 3790.36), (12.35, 3512.57), (21.84, 3422.84), (11.05, 3597.77), (6.36, 3947.94), (3.53, 3890.74), (8.38, 4103.37), (4.01, 4053.28), (15.22, 3643.93), (2.67, 4626.57), (-0.08, 4140.05), (21.82, 3612.61), (18.1, 3299.08), (1.09, 4128.79), (17.23, 3763.39), (13.1, 3603.78), (20.16, 3159.73), (10.22, 3580.98), (15.17, 3722.01), (8.5, 3954.49), (-5.35, 4424.65), (4.68, 3848.44), (10.43, 3641.63), (-0.43, 4095.83), (4.11, 4104.48), (11.62, 3815.07), (14.45, 3432.07), (22.47, 3253.8), (11.2, 3844.11), (14.14, 3603.89), (9.58, 3678.09), (18.6, 3606.55), (10.08, 3858.3), (11.6, 3721.26), (10.12, 3932.11), (13.67, 3497.98), (5.04, 3800.91), (6.58, 3461.71), (13.27, 3759.96), (14.08, 3271.01), (12.99, 3591.73), (6.3, 3796.85), (19.26, 3516.72), (12.74, 3844.03), (19.98, 3787.28), (11.11, 3963.85), (11.84, 3621.89), (8.24, 3608.3), (14.17, 3871.14), (10.65, 4037.51), (12.77, 3745.24), (10.81, 3884.72), (3.08, 3738.55), (7.3, 3845.67), (9.86, 3630.63), (10.21, 4034.33), (-0.4, 4207.75), (12.4, 3549.88), (-5.27, 4330.56), (5.57, 3733.56), (7.44, 3912.1), (12.17, 3661.24), (16.97, 3375.05), (11.95, 4218.58), (18.74, 3608.57), (7.61, 3695.71), (10.31, 3691.11), (23.6, 3701.27), (19.25, 3628.56), (20.46, 3417.25), (2.39, 3925.63), (13.56, 3659.93), (6.13, 3740.51), (13.59, 3625.54), (28.08, 3134.43), (8.6, 3627.78), (20.9, 3207.3), (-3.0, 3988.16), (9.99, 3568.64), (13.08, 3869.89), (-0.74, 4189.83), (9.13, 3345.13), (4.84, 4109.49), (12.99, 4073.74), (5.83, 3654.52), (-0.11, 3963.99), (-1.74, 4656.61), (2.25, 4214.18), (4.24, 3530.35), (7.22, 3666.73), (11.48, 3466.57), (6.5, 3853.36), (6.49, 3530.23), (6.09, 4378.09), (11.03, 3839.07), (11.37, 3561.13), (13.91, 3771.04), (5.02, 4430.03), (12.72, 3350.4), (12.0, 3840.26), (14.3, 3983.43), (11.65, 4174.5), (19.22, 3536.84), (17.99, 3717.84), (14.94, 4035.0), (14.41, 3577.7), (7.09, 4119.31), (17.63, 3504.39), (10.86, 3928.88), (-4.93, 4342.32), (8.77, 4027.53), (3.33, 3891.76), (18.06, 3502.0), (3.83, 4155.56), (16.1, 3448.96), (10.66, 4343.05), (21.72, 3081.52), (18.04, 3552.17), (4.26, 3989.56), (6.93, 3934.16), (10.35, 3598.57), (10.32, 4086.67), (7.53, 3932.03), (6.29, 3913.63), (17.1, 3644.05), (10.08, 3751.28), (14.31, 3541.27), (7.06, 3905.96), (12.53, 3742.59), (18.23, 3422.0), (6.31, 3942.77), (6.81, 4180.73), (12.69, 3838.81), (9.35, 3644.26), (22.66, 3945.3), (11.46, 3653.11), (13.62, 4108.07), (5.41, 3871.72), (13.18, 3571.05), (17.16, 3906.87), (18.11, 3586.85), (17.65, 3788.34), (14.17, 3631.19), (8.65, 4114.32), (18.47, 3214.78), (13.74, 3888.43), (9.51, 3812.64), (2.13, 3901.41), (16.9, 3672.23), (6.4, 3923.03), (10.97, 3753.99), (12.61, 3487.19), (6.06, 4004.1), (13.46, 3668.27), (8.7, 4019.78), (14.32, 3559.87), (6.05, 3793.99), (1.08, 3875.66), (9.04, 4107.72), (2.74, 4082.13), (11.08, 3800.86), (11.71, 3839.23), (15.18, 3698.7), (18.86, 3355.97), (17.35, 3798.2), (14.96, 3697.55), (17.71, 3672.82), (7.18, 4102.4), (13.95, 3911.48), (22.81, 3525.11), (18.43, 3596.03), (1.74, 3892.46), (14.5, 3514.67), (13.38, 3682.45), (12.08, 3916.84), (13.59, 3554.3), (5.49, 3763.24), (11.24, 3898.15), (1.98, 4087.58), (-0.6, 4006.96), (3.0, 3741.53), (3.79, 3833.37), (17.13, 3292.5), (7.14, 3822.23), (5.72, 4122.97), (11.56, 3726.69), (1.38, 4034.54), (16.52, 3458.27), (3.3, 3916.54), (13.93, 3777.21), (6.33, 4210.32), (-9.46, 4317.94), (10.76, 3729.08), (12.19, 3851.04), (14.09, 3635.23), (13.16, 3723.48), (5.06, 4210.02), (4.02, 4212.84), (2.2, 4175.1), (5.99, 4103.11), (15.42, 3591.96), (12.28, 3650.05), (5.19, 3771.55), (11.16, 3920.84), (11.07, 3804.07), (7.6, 3910.25), (10.34, 3824.13), (14.06, 3560.7), (5.52, 3929.62), (17.9, 3289.64), (12.87, 4114.0), (13.84, 3292.64), (10.84, 3933.54), (8.6, 3775.84), (-1.75, 3873.91), (14.47, 3650.61), (10.69, 3748.85), (10.25, 3975.63), (6.04, 3793.46), (13.24, 3391.05), (4.55, 3861.96), (8.85, 3862.9), (20.06, 3447.51), (12.57, 3584.92), (15.09, 3344.79), (15.57, 3532.41), (2.67, 4192.49), (0.03, 3739.3), (11.17, 3507.19), (9.91, 3610.22), (-3.32, 4070.73), (19.01, 3362.89), (6.32, 3880.62), (14.93, 3270.45), (2.69, 3911.32), (14.79, 3406.95), (15.62, 3657.05), (11.88, 3770.35), (13.06, 3591.81), (7.56, 4091.66), (2.89, 4112.22), (8.9, 3997.97), (11.1, 4057.48), (23.72, 3657.53), (12.14, 3787.34), (11.0, 3727.21), (18.43, 3559.31), (11.26, 3724.78), (4.89, 3712.6), (10.84, 3424.05), (14.35, 3886.92), (11.21, 3716.1), (7.03, 3638.09), (26.89, 3113.99), (-1.17, 4222.62), (12.55, 3356.1), (8.04, 3728.73), (9.58, 3667.93), (13.18, 3645.87), (11.58, 4151.94), (6.18, 4150.26), (3.5, 4135.36), (7.27, 3831.72), (3.61, 3450.23), (4.26, 4094.36), (10.38, 3855.61), (9.62, 3969.94), (9.45, 3875.73), (9.14, 3984.27), (6.78, 3953.79), (-0.2, 4037.92), (11.97, 3647.74), (10.11, 3804.14), (14.48, 4005.25), (14.29, 3942.37), (9.25, 4090.39), (9.4, 3862.75), (5.39, 3920.32), (16.41, 3623.63), (4.68, 4194.82), (9.56, 3638.3), (6.18, 3966.96), (13.66, 3645.98), (18.73, 3495.61), (26.77, 3275.43), (9.66, 3699.51), (12.85, 3757.29), (2.23, 3868.58), (16.7, 3980.84), (7.73, 3735.2), (12.66, 3787.03), (-0.25, 4141.24), (3.74, 4042.56), (4.16, 4330.27), (7.98, 3671.17), (11.62, 3589.77), (12.08, 3635.79), (18.01, 3544.62), (-0.24, 4276.47), (7.53, 3754.44), (-1.18, 4110.58), (5.53, 3806.81), (10.81, 3694.34), (26.3, 2971.57), (9.39, 3941.5), (8.61, 3932.64), (13.83, 4148.03), (11.73, 3432.89), (14.89, 3389.07), (14.8, 3723.43), (21.19, 3332.25), (2.12, 4055.8), (0.91, 4074.69), (5.61, 4099.07), (21.76, 3538.38), (8.74, 4032.26), (9.86, 3927.66), (3.4, 3951.81), (10.15, 3628.24), (16.33, 3905.89), (14.11, 3715.58), (3.31, 4128.9), (15.52, 3999.21), (5.34, 3742.97), (15.63, 3688.35), (7.26, 3667.2), (13.24, 3538.34), (2.41, 4061.58), (20.74, 3676.09), (8.59, 3611.4), (12.71, 3864.52), (3.25, 4382.03), (5.27, 4317.96), (10.05, 4076.34), (14.09, 3826.7), (6.39, 4121.25), (16.47, 3855.64), (0.1, 4253.21), (6.62, 3842.53), (13.74, 3507.0), (5.81, 3700.83), (3.91, 4271.64), (13.79, 3689.21), (2.96, 3975.41), (11.88, 3471.83), (-0.53, 4568.38), (11.96, 3726.97), (17.91, 3420.82), (1.1, 4037.4), (18.38, 3405.74), (9.24, 3851.41), (11.25, 4065.28), (12.95, 4260.56), (14.88, 3755.02), (8.16, 3759.85), (9.88, 4032.91), (4.9, 3697.12), (19.77, 3606.75), (10.46, 3593.75), (4.55, 3626.88), (12.75, 3706.97), (10.81, 3522.71), (14.84, 3524.35), (17.03, 3989.42), (8.62, 3654.79), (7.14, 3999.26), (10.63, 4309.2), (5.72, 3817.98), (17.81, 3740.91), (5.4, 4508.19), (7.47, 3710.55), (9.19, 3585.41), (10.57, 3597.23), (3.29, 3906.4), (7.26, 3803.35), (9.38, 3586.13), (7.47, 3849.54), (8.49, 3920.84), (16.15, 3469.8), (12.95, 3616.97), (7.56, 3611.42), (13.12, 3502.23), (16.74, 3491.13), (3.12, 4022.53), (12.08, 3491.31), (14.73, 3037.74), (11.36, 4132.3), (13.99, 3583.89), (9.75, 3869.68), (8.55, 3535.27), (6.96, 3877.02), (11.4, 3691.24), (2.49, 3964.67), (15.57, 3498.0), (11.07, 3459.95), (1.79, 4309.44), (13.83, 3905.27), (7.49, 4082.57), (11.24, 3540.99), (20.1, 3381.85), (12.4, 3951.23), (7.82, 3653.11), (3.65, 4032.35), (6.1, 3856.2), (3.89, 3467.03), (1.78, 4116.66), (8.15, 3918.36), (8.14, 3888.16), (12.43, 3929.24), (18.2, 3406.6), (5.22, 3917.71), (13.84, 3713.29), (3.83, 4086.08), (14.78, 3870.53), (11.69, 3640.99), (9.74, 3870.46), (25.34, 3243.19), (6.17, 4032.61), (17.46, 3029.56), (6.82, 4007.6), (4.8, 4279.3), (7.76, 4005.66), (12.11, 3702.43), (11.31, 3677.88), (9.42, 3614.41), (6.28, 4116.35), (-6.55, 4517.16), (17.18, 3518.66), (4.37, 4088.13), (20.73, 3629.71), (7.66, 3857.4), (1.67, 3901.64), (-1.93, 4047.05), (7.48, 3975.48), (-3.09, 4184.65), (6.6, 3804.31), (13.69, 3889.65), (6.7, 3648.17), (2.65, 3906.76), (1.45, 3747.99), (8.48, 3827.06), (10.76, 3776.43), (23.15, 3707.76), (10.87, 3643.53), (5.45, 4044.7), (9.28, 3718.04), (14.85, 3624.57), (16.77, 3446.02), (9.8, 3606.49), (1.25, 4162.77), (21.99, 3462.08), (22.35, 3465.97), (12.95, 3686.12), (16.28, 3638.16), (1.77, 4276.81), (9.62, 3453.73), (7.8, 3721.06), (6.32, 4141.85), (11.78, 3920.19), (13.31, 3699.66), (10.25, 3645.12), (4.67, 3834.57), (15.39, 3703.7), (4.8, 3682.39), (15.69, 3803.3), (14.97, 3735.23), (7.58, 3658.36), (24.25, 3204.95), (4.49, 4057.32), (10.06, 3868.57), (20.73, 3075.46), (8.85, 4122.41), (7.06, 3842.08), (15.23, 3614.43), (4.74, 4109.72), (-4.22, 4312.32), (4.82, 3919.1), (13.43, 3787.84), (9.04, 3330.15), (3.63, 4024.54), (17.04, 3634.84), (6.66, 3918.81), (18.25, 3602.11), (13.21, 3855.21), (-1.01, 3945.24), (4.45, 3906.5), (4.14, 4070.03), (4.89, 4170.05), (10.61, 3613.09), (0.98, 4122.63), (14.91, 3544.37), (10.37, 4097.49), (9.61, 3778.42), (15.87, 3863.17), (12.19, 3817.43), (9.73, 3427.38), (2.24, 4343.23), (6.38, 3989.64), (4.74, 4111.41), (9.71, 3790.91), (8.06, 4185.47), (5.91, 4159.89), (-0.82, 4201.02), (10.58, 3401.39), (12.67, 3926.63), (6.5, 3879.74), (16.87, 3940.26), (15.34, 3764.96), (0.36, 4054.07), (9.08, 3783.03), (11.67, 3682.09), (12.0, 3897.28), (7.64, 3974.66), (15.78, 3500.37), (11.44, 3579.92), (5.77, 3814.65), (3.94, 4049.96), (14.31, 3558.4), (-2.46, 4321.76), (2.85, 3706.77), (-8.97, 4495.48), (7.44, 4015.64), (7.69, 3783.77), (7.53, 3871.62), (-1.25, 4160.11), (7.77, 4227.66), (15.39, 4058.71), (3.84, 3846.92), (6.56, 3485.51), (10.49, 3588.97), (1.67, 4155.44), (16.36, 3446.27), (4.62, 3804.17), (4.98, 3940.9), (12.05, 3745.96), (22.61, 3238.11), (7.53, 3813.41), (7.29, 4074.86), (13.79, 3770.17), (15.08, 3585.91), (9.52, 4042.88), (4.6, 4013.99), (10.19, 3791.42), (5.38, 3929.28), (10.42, 3777.29), (18.62, 3533.43), (12.23, 3997.55), (11.19, 3863.83), (14.8, 3648.06), (5.94, 3784.62), (5.94, 3891.93), (16.63, 3417.69), (10.37, 3558.72), (12.89, 3827.03), (2.33, 3437.29), (11.81, 3728.47), (17.41, 3258.62), (11.98, 3787.59), (16.75, 3810.05), (13.41, 3809.4), (6.79, 4039.6), (8.14, 4239.75), (-6.65, 4196.92), (12.45, 3597.14), (13.85, 3542.83), (2.45, 4158.35), (12.27, 3575.44), (-0.06, 4049.27), (10.46, 3685.3), (7.17, 3677.8), (12.66, 3615.62), (3.0, 3972.83), (6.29, 3970.99), (6.42, 4145.27), (12.22, 3584.44), (22.56, 3496.55), (12.99, 3926.53), (16.61, 3611.14), (13.2, 3811.37), (8.55, 3740.5), (3.19, 4168.16), (-1.49, 4402.59), (-4.43, 4338.17), (8.96, 3977.45), (6.45, 3692.8), (11.3, 3661.8), (6.49, 3667.41), (16.05, 3222.62), (9.07, 3968.45), (4.73, 4042.09), (15.34, 3685.26), (12.59, 3421.49), (9.52, 3935.64), (11.69, 3855.53), (12.17, 3940.55), (13.59, 3973.32), (6.42, 3641.83), (4.29, 4163.06), (16.5, 3449.99), (16.31, 3278.81), (10.86, 3736.98), (12.35, 3880.49)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=9.8970，ȳ=3796.3937；Sxx=Σ(x−x̄)²=23703.23，Sxy=-743138.83；
β₁ = Sxy/Sxx = -31.3518；β₀ = ȳ−β₁x̄ = 4106.6837；
R² = SSR/SST = 23298735.85/50307616.07 = 0.4631；F = (SSR/1)/(SSE/667) = 575.3758，p = -4.96e+36；
SE(β₁)=√(40493.07/23703.23)=1.3070；t=β₁/SE=-23.9870；p=-4.96e+36；95%CI=[-33.9182,-28.7854]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=9.8970，ȳ=3796.3937；Sxx=Σ(x−x̄)²=23703.23，Sxy=-743138.83；
β₁ = Sxy/Sxx = -31.3518；β₀ = ȳ−β₁x̄ = 4106.6837；
R² = SSR/SST = 23298735.85/50307616.07 = 0.4631；F = (SSR/1)/(SSE/667) = 575.3758，p = -4.96e+36；
SE(β₁)=√(40493.07/23703.23)=1.3070；t=β₁/SE=-23.9870；p=-4.96e+36；95%CI=[-33.9182,-28.7854]。
③ 计算结果：
y = 4106.6837 + -31.3518·x；R²=0.4631；β₁ t=-23.9870 p=-4.96e+36；95%CI=[-33.9182,-28.7854]
④ 结论：在 α=0.05 水平下，p=-4.96e+36 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-31.3518），吸烟伤肺。
案例 57. 压力预测睡眠质量
简单线性回归压力大睡差
① 原始数据：
样本 n=339，逐条 (x, y)：[(4.2, 8.4), (1.83, 8.87), (5.1, 7.8), (6.77, 7.83), (4.34, 8.01), (2.11, 8.37), (6.78, 7.07), (4.1, 7.81), (2.78, 7.81), (4.08, 8.06), (5.76, 7.9), (7.48, 7.63), (1.59, 8.5), (6.69, 7.72), (4.25, 8.16), (3.2, 8.64), (6.84, 7.43), (5.02, 8.11), (2.33, 8.73), (7.31, 7.55), (5.87, 8.41), (5.52, 8.38), (6.0, 8.2), (3.77, 7.9), (1.84, 9.05), (4.72, 7.82), (5.4, 8.35), (8.03, 7.43), (9.59, 7.08), (6.96, 7.53), (4.07, 8.37), (7.19, 7.57), (3.63, 7.9), (5.31, 7.44), (7.22, 7.57), (4.63, 8.95), (4.86, 7.87), (6.02, 7.74), (4.96, 7.51), (6.34, 7.58), (3.46, 8.26), (3.67, 8.03), (1.17, 9.19), (1.47, 8.66), (4.77, 7.55), (6.48, 8.24), (4.61, 8.07), (3.48, 7.99), (4.39, 7.98), (4.4, 8.64), (4.9, 7.67), (5.24, 8.16), (3.62, 8.21), (5.16, 7.49), (5.51, 8.25), (4.39, 8.08), (3.74, 7.99), (3.0, 8.24), (3.43, 9.25), (1.02, 8.61), (4.72, 7.86), (9.43, 7.38), (7.33, 7.14), (4.02, 7.85), (3.52, 8.16), (4.21, 8.01), (6.26, 7.68), (2.06, 8.77), (3.78, 8.21), (2.06, 9.29), (2.76, 8.66), (5.44, 7.91), (4.98, 8.04), (5.4, 8.08), (3.83, 8.04), (5.29, 8.37), (8.38, 7.65), (3.08, 8.63), (2.33, 8.73), (7.87, 7.08), (2.26, 8.34), (5.01, 7.65), (4.61, 8.2), (2.85, 7.97), (8.32, 6.97), (6.65, 7.53), (4.26, 8.25), (6.33, 7.18), (8.77, 6.93), (5.7, 7.79), (4.21, 8.77), (4.4, 7.96), (7.21, 7.0), (0.44, 8.72), (5.6, 7.62), (2.75, 8.67), (5.04, 8.19), (0.37, 7.68), (4.24, 8.23), (6.68, 7.51), (7.95, 7.58), (4.92, 7.82), (6.34, 7.68), (5.17, 7.55), (6.07, 7.59), (3.92, 8.0), (3.43, 8.24), (5.07, 7.57), (4.18, 7.48), (5.92, 7.98), (2.53, 8.32), (6.92, 8.2), (3.67, 7.99), (3.62, 8.56), (8.72, 7.49), (6.26, 7.67), (4.15, 7.7), (5.4, 8.21), (4.06, 8.17), (3.92, 8.33), (9.35, 6.91), (6.17, 8.05), (8.27, 7.17), (5.54, 7.19), (7.04, 7.64), (5.34, 8.36), (4.83, 8.08), (3.57, 8.35), (6.89, 7.25), (7.29, 7.37), (5.24, 7.52), (4.05, 8.13), (7.03, 7.85), (8.98, 7.23), (5.2, 7.93), (4.6, 8.6), (2.28, 8.38), (8.59, 7.52), (2.95, 8.29), (5.84, 7.93), (6.61, 7.69), (8.72, 7.96), (1.02, 9.33), (4.16, 8.36), (6.87, 6.95), (5.16, 8.18), (1.67, 8.47), (4.73, 8.18), (4.77, 7.9), (8.86, 7.24), (6.01, 7.37), (6.15, 8.08), (5.71, 8.42), (8.05, 7.81), (7.49, 7.33), (5.94, 8.17), (2.87, 8.28), (7.91, 7.51), (9.69, 6.61), (3.8, 7.82), (3.55, 7.92), (3.63, 8.02), (3.12, 8.39), (5.0, 8.51), (4.17, 8.1), (9.35, 6.36), (4.98, 7.91), (7.97, 7.53), (4.93, 7.93), (6.57, 7.31), (0.82, 8.83), (6.42, 8.0), (4.74, 8.24), (3.88, 8.22), (4.29, 7.93), (5.28, 7.92), (8.09, 7.18), (6.07, 8.15), (3.54, 8.67), (6.18, 8.2), (1.33, 8.82), (3.62, 8.04), (1.6, 8.74), (1.68, 8.98), (4.54, 8.4), (6.84, 8.02), (5.02, 7.16), (3.63, 8.11), (4.83, 7.9), (3.42, 7.88), (4.05, 8.1), (4.43, 7.48), (5.91, 7.93), (0.59, 8.71), (0.68, 8.92), (1.11, 8.51), (8.09, 7.15), (4.54, 8.21), (2.84, 9.11), (5.42, 7.57), (5.28, 8.15), (5.18, 7.74), (3.09, 8.02), (4.1, 8.25), (7.68, 6.84), (5.25, 7.57), (5.47, 8.4), (5.68, 7.76), (4.26, 8.21), (7.32, 7.22), (5.8, 8.11), (8.88, 7.67), (0.47, 8.96), (5.34, 7.61), (3.05, 8.37), (3.85, 8.29), (5.7, 8.27), (6.65, 6.53), (6.26, 7.36), (7.16, 8.08), (5.19, 7.71), (1.24, 8.77), (7.56, 7.94), (-1.09, 9.34), (2.28, 8.43), (4.81, 8.38), (5.5, 7.8), (7.68, 6.59), (6.15, 7.85), (4.74, 9.05), (3.61, 8.42), (7.95, 7.81), (6.64, 8.25), (7.59, 7.55), (8.08, 6.76), (0.58, 8.95), (9.71, 6.61), (7.03, 7.18), (9.15, 7.05), (-1.62, 9.39), (7.28, 7.58), (7.22, 7.71), (3.59, 8.22), (8.95, 7.39), (6.4, 7.93), (6.0, 7.36), (1.03, 9.2), (6.01, 7.63), (5.57, 8.09), (7.2, 7.71), (3.49, 8.39), (3.91, 8.03), (7.45, 7.77), (6.92, 7.85), (8.14, 7.35), (7.82, 6.9), (7.04, 7.86), (4.79, 7.86), (5.07, 7.82), (8.16, 7.23), (3.19, 8.26), (6.23, 7.68), (3.17, 8.72), (7.83, 6.96), (2.64, 8.72), (4.05, 8.64), (5.08, 7.1), (10.11, 7.47), (6.24, 8.23), (5.52, 7.7), (8.6, 7.55), (4.75, 7.88), (1.27, 9.0), (3.22, 7.79), (5.8, 8.17), (4.03, 7.88), (5.72, 7.66), (5.86, 7.74), (7.07, 7.67), (6.87, 7.66), (3.87, 8.32), (2.97, 8.06), (1.3, 9.16), (5.11, 7.57), (3.86, 8.29), (6.01, 7.93), (1.99, 8.56), (8.5, 7.73), (6.26, 7.98), (7.77, 7.67), (6.34, 7.58), (3.69, 8.38), (6.41, 7.76), (4.52, 8.5), (3.73, 8.21), (6.47, 8.03), (4.5, 8.57), (7.41, 8.42), (7.05, 7.79), (3.1, 8.82), (4.68, 7.85), (4.09, 8.67), (3.56, 7.98), (8.29, 6.58), (4.85, 7.93), (5.84, 7.69), (8.44, 7.15), (3.03, 9.21), (2.07, 8.25), (5.94, 8.01), (5.17, 8.23), (9.87, 8.09), (5.59, 8.49), (9.78, 7.15), (4.72, 8.42), (7.05, 7.03), (2.49, 9.1), (6.54, 7.36), (3.35, 8.25), (7.14, 8.25), (4.68, 8.32), (5.41, 7.48), (1.64, 8.8), (5.16, 8.08), (7.62, 6.83), (7.76, 7.76), (3.83, 8.94), (6.63, 8.55), (6.74, 7.6), (3.28, 8.02), (4.78, 8.61), (5.36, 7.68), (3.04, 8.64), (6.18, 7.6), (3.82, 8.13), (4.54, 7.43), (2.15, 8.5), (8.64, 7.7), (9.95, 6.86)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.1391，ȳ=7.9736；Sxx=Σ(x−x̄)²=1569.60，Sxy=-314.13；
β₁ = Sxy/Sxx = -0.2001；β₀ = ȳ−β₁x̄ = 9.0021；
R² = SSR/SST = 62.87/106.58 = 0.5899；F = (SSR/1)/(SSE/337) = 484.7124，p = 3.36e-67；
SE(β₁)=√(0.13/1569.60)=0.0091；t=β₁/SE=-22.0162；p=3.36e-67；95%CI=[-0.2180,-0.1823]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.1391，ȳ=7.9736；Sxx=Σ(x−x̄)²=1569.60，Sxy=-314.13；
β₁ = Sxy/Sxx = -0.2001；β₀ = ȳ−β₁x̄ = 9.0021；
R² = SSR/SST = 62.87/106.58 = 0.5899；F = (SSR/1)/(SSE/337) = 484.7124，p = 3.36e-67；
SE(β₁)=√(0.13/1569.60)=0.0091；t=β₁/SE=-22.0162；p=3.36e-67；95%CI=[-0.2180,-0.1823]。
③ 计算结果：
y = 9.0021 + -0.2001·x；R²=0.5899；β₁ t=-22.0162 p=3.36e-67；95%CI=[-0.2180,-0.1823]
④ 结论：在 α=0.05 水平下，p=3.36e-67 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-0.2001），压力大睡差。
案例 58. 距离预测到店率
简单线性回归越远越少来
① 原始数据：
样本 n=766，逐条 (x, y)：[(4.59, 51.11), (5.51, 58.6), (4.15, 60.51), (5.56, 50.41), (4.23, 49.41), (1.61, 59.39), (-0.4, 62.62), (4.86, 65.84), (6.64, 46.72), (7.45, 45.63), (3.84, 54.02), (5.35, 52.25), (6.69, 54.11), (4.61, 48.12), (2.51, 65.83), (3.48, 57.88), (10.73, 41.66), (6.56, 51.08), (2.22, 67.08), (4.49, 51.96), (3.6, 57.92), (4.62, 48.83), (3.09, 51.2), (1.04, 65.43), (0.21, 65.54), (0.97, 62.84), (7.41, 42.54), (8.04, 50.8), (5.03, 48.15), (8.6, 45.66), (2.85, 62.98), (0.37, 61.93), (-0.31, 69.01), (0.35, 60.64), (5.04, 51.88), (4.34, 53.43), (14.61, 34.29), (3.4, 56.83), (4.9, 58.98), (2.26, 65.09), (0.75, 64.56), (5.86, 47.81), (0.59, 62.43), (5.29, 55.99), (-2.44, 71.76), (4.7, 56.05), (10.58, 49.42), (6.04, 53.45), (1.41, 63.63), (2.66, 57.65), (7.08, 55.63), (8.86, 47.99), (5.08, 57.73), (4.37, 61.37), (5.73, 53.95), (9.56, 45.4), (5.04, 55.79), (2.79, 58.84), (3.8, 52.95), (5.59, 51.21), (4.95, 57.52), (4.77, 49.37), (8.64, 50.11), (8.37, 56.01), (-2.58, 72.78), (7.96, 50.98), (7.12, 53.48), (6.24, 57.34), (4.66, 58.85), (1.88, 69.1), (5.73, 54.42), (2.4, 65.12), (6.03, 52.11), (3.52, 60.44), (3.91, 68.86), (5.36, 50.8), (10.27, 44.26), (5.63, 59.49), (1.77, 57.93), (5.17, 55.24), (4.85, 53.36), (6.1, 54.38), (6.96, 55.56), (8.34, 51.13), (9.57, 49.34), (6.3, 55.13), (5.78, 58.2), (9.26, 45.55), (1.71, 70.81), (9.82, 50.63), (4.7, 57.72), (1.61, 58.27), (4.78, 60.11), (4.16, 56.48), (-1.46, 75.31), (3.84, 54.64), (4.09, 60.18), (2.23, 63.67), (6.91, 53.29), (6.8, 57.87), (2.35, 63.18), (6.09, 49.18), (6.41, 49.55), (-1.54, 75.82), (7.95, 51.23), (0.48, 60.56), (3.65, 52.82), (4.01, 54.42), (-1.4, 67.79), (8.31, 50.98), (-0.77, 71.16), (4.52, 59.37), (1.53, 66.58), (6.88, 47.53), (2.5, 60.09), (8.72, 45.74), (2.22, 58.54), (8.07, 51.5), (3.8, 49.77), (7.15, 40.48), (0.9, 62.5), (5.84, 59.1), (9.66, 40.51), (7.74, 52.0), (4.94, 59.85), (6.8, 53.29), (4.57, 59.47), (8.41, 45.66), (2.98, 55.89), (5.66, 55.52), (4.59, 60.03), (5.2, 60.99), (2.16, 55.54), (1.62, 61.84), (3.83, 59.15), (4.41, 55.56), (9.19, 40.95), (1.71, 55.68), (4.52, 55.88), (8.63, 44.85), (7.72, 46.82), (5.66, 42.81), (5.02, 55.12), (4.17, 52.17), (1.74, 63.48), (6.84, 50.82), (-0.93, 66.38), (6.11, 49.12), (3.67, 62.01), (2.36, 63.68), (6.18, 49.91), (4.88, 54.13), (9.86, 42.54), (1.44, 57.87), (6.47, 47.06), (4.05, 63.14), (8.22, 50.69), (2.4, 60.87), (5.3, 57.16), (7.25, 44.53), (1.46, 60.05), (5.46, 49.15), (7.51, 50.61), (0.71, 62.55), (2.37, 58.58), (5.97, 60.65), (10.78, 44.26), (2.38, 54.82), (2.75, 60.52), (6.05, 50.84), (6.46, 47.93), (2.71, 60.52), (3.74, 54.78), (2.16, 65.45), (7.53, 48.66), (3.27, 65.53), (-0.92, 69.24), (3.19, 59.73), (9.98, 50.06), (11.08, 41.77), (3.09, 61.45), (3.06, 59.71), (0.06, 64.55), (2.21, 60.93), (2.44, 59.01), (4.96, 54.41), (4.66, 64.12), (6.6, 56.62), (4.11, 59.26), (3.98, 56.77), (3.08, 52.13), (3.66, 61.92), (7.28, 52.49), (3.15, 61.1), (5.0, 52.85), (8.28, 52.2), (4.8, 55.4), (5.97, 61.65), (6.05, 59.77), (2.03, 59.04), (9.41, 43.76), (4.5, 62.1), (11.02, 37.72), (3.65, 62.59), (5.35, 54.17), (2.11, 59.3), (7.83, 51.83), (7.38, 49.21), (7.83, 47.09), (0.36, 71.68), (4.03, 58.93), (4.37, 59.59), (7.17, 53.63), (6.51, 57.72), (4.63, 56.32), (5.99, 50.93), (7.09, 56.38), (2.3, 67.24), (5.71, 58.65), (11.75, 50.02), (4.18, 54.64), (5.42, 54.75), (8.27, 51.7), (7.99, 55.15), (9.7, 45.34), (5.76, 44.18), (2.21, 63.21), (6.32, 45.78), (3.5, 59.8), (1.31, 62.15), (2.3, 65.25), (6.97, 53.8), (4.04, 59.62), (3.62, 48.9), (5.74, 54.31), (7.91, 46.57), (2.54, 58.9), (5.29, 53.02), (4.82, 50.27), (6.96, 53.43), (1.0, 65.29), (5.29, 54.53), (7.98, 55.44), (5.18, 52.97), (0.01, 68.11), (2.86, 64.53), (9.15, 44.97), (2.89, 61.82), (8.88, 45.57), (4.93, 56.58), (8.24, 45.73), (3.84, 56.22), (5.39, 53.52), (4.31, 57.8), (8.55, 49.66), (5.24, 53.27), (1.37, 57.81), (4.8, 46.97), (7.72, 50.46), (3.64, 59.03), (2.09, 60.1), (4.36, 54.17), (-0.81, 65.1), (8.28, 48.28), (7.58, 54.38), (8.7, 51.96), (6.59, 53.79), (8.87, 44.17), (5.19, 57.72), (4.48, 59.33), (7.15, 52.39), (5.98, 53.36), (3.77, 55.48), (7.59, 39.88), (3.66, 57.38), (6.15, 52.84), (-0.28, 61.87), (6.78, 53.93), (6.77, 48.5), (5.37, 57.05), (6.54, 62.88), (9.28, 39.78), (6.16, 52.09), (4.91, 50.49), (4.13, 55.85), (3.57, 66.54), (6.52, 52.67), (-1.71, 70.09), (1.24, 63.0), (6.8, 55.7), (1.38, 64.51), (7.55, 47.04), (4.09, 55.84), (1.54, 60.38), (8.02, 45.81), (8.86, 53.4), (8.07, 50.7), (2.31, 63.23), (5.28, 49.75), (12.53, 37.67), (5.52, 58.6), (-0.96, 71.95), (8.47, 52.07), (4.29, 52.1), (8.96, 48.26), (5.69, 51.79), (3.44, 51.49), (5.53, 51.44), (5.13, 56.16), (7.57, 51.1), (7.42, 48.01), (5.15, 53.94), (7.28, 57.18), (3.12, 59.44), (3.9, 62.63), (2.42, 56.17), (7.83, 53.93), (6.84, 42.24), (10.62, 49.52), (7.72, 51.09), (5.96, 52.34), (5.84, 56.39), (4.11, 55.8), (3.57, 55.05), (6.41, 56.45), (6.38, 53.74), (6.68, 57.61), (3.71, 54.12), (7.89, 54.63), (3.99, 66.91), (0.49, 64.17), (5.93, 51.47), (6.75, 58.57), (10.71, 39.04), (4.91, 58.28), (3.4, 50.83), (6.31, 51.39), (1.98, 67.25), (1.77, 64.47), (11.19, 44.23), (4.87, 54.93), (3.93, 59.6), (8.22, 43.02), (6.35, 58.59), (1.69, 59.17), (-0.33, 63.34), (7.62, 44.35), (3.66, 62.93), (8.63, 46.02), (3.72, 62.57), (6.56, 57.08), (3.31, 57.41), (0.33, 60.56), (-0.01, 63.89), (8.21, 41.03), (5.9, 55.48), (4.17, 54.36), (4.64, 55.51), (3.97, 58.7), (9.04, 43.5), (3.23, 56.45), (9.07, 49.46), (6.66, 51.54), (7.02, 53.92), (7.36, 54.26), (1.67, 53.53), (4.59, 61.53), (5.55, 51.92), (2.87, 52.19), (4.18, 50.58), (5.13, 53.66), (5.81, 54.42), (8.73, 45.79), (6.92, 51.53), (9.54, 39.82), (-1.16, 68.86), (2.94, 61.57), (10.32, 41.02), (2.61, 57.52), (2.74, 56.61), (5.07, 51.84), (0.53, 67.68), (6.2, 56.87), (4.48, 54.25), (1.77, 65.61), (7.61, 53.09), (6.31, 51.93), (0.86, 63.4), (3.9, 56.3), (5.57, 53.82), (8.09, 43.47), (3.71, 58.97), (6.44, 53.25), (6.02, 56.27), (8.08, 51.16), (1.56, 57.46), (3.91, 58.36), (9.98, 46.7), (-1.09, 67.59), (3.55, 56.13), (8.5, 51.16), (9.4, 54.23), (5.4, 55.24), (7.98, 50.04), (12.0, 48.01), (2.66, 61.46), (4.38, 58.23), (8.84, 43.99), (-0.73, 66.01), (-1.24, 60.3), (5.84, 54.77), (2.8, 62.58), (5.76, 58.06), (6.52, 49.79), (2.28, 54.4), (1.04, 60.14), (0.08, 68.7), (8.02, 45.25), (2.39, 60.48), (3.67, 61.83), (2.44, 59.82), (2.55, 61.88), (8.59, 47.85), (5.14, 58.31), (4.73, 52.32), (10.9, 37.65), (11.33, 46.07), (2.98, 53.13), (8.08, 48.33), (3.96, 54.66), (5.86, 49.87), (4.81, 54.06), (0.17, 65.24), (4.04, 58.66), (1.52, 65.37), (7.72, 48.77), (0.4, 67.54), (2.09, 61.08), (6.67, 50.98), (0.72, 54.14), (0.66, 65.24), (7.57, 51.23), (2.68, 57.33), (9.96, 41.5), (5.28, 47.62), (2.98, 62.75), (7.25, 44.43), (1.68, 67.57), (5.42, 51.47), (1.59, 57.24), (11.24, 48.59), (3.91, 61.63), (8.18, 46.68), (1.07, 70.61), (3.08, 58.58), (6.6, 50.16), (2.65, 54.2), (5.43, 51.38), (6.19, 47.24), (0.8, 66.48), (3.77, 56.25), (6.45, 52.44), (3.48, 63.67), (6.59, 49.99), (1.11, 63.1), (8.1, 54.06), (6.91, 47.89), (5.57, 59.51), (-1.33, 64.85), (2.33, 62.12), (8.76, 43.11), (6.41, 59.73), (6.11, 49.36), (7.03, 46.56), (8.17, 51.0), (1.03, 61.14), (0.85, 66.26), (1.02, 68.19), (4.0, 58.08), (6.65, 52.54), (2.97, 53.17), (6.02, 49.02), (8.81, 46.15), (8.46, 46.62), (1.39, 64.13), (7.94, 48.38), (8.4, 42.67), (10.01, 47.32), (3.93, 57.02), (4.97, 55.89), (5.81, 49.5), (13.92, 37.32), (6.46, 52.3), (10.79, 32.0), (5.49, 47.59), (-0.82, 68.83), (6.62, 51.4), (5.89, 47.27), (3.14, 61.86), (9.89, 48.75), (4.5, 54.81), (5.14, 53.21), (3.5, 63.79), (6.47, 55.99), (5.34, 47.62), (2.57, 59.24), (6.32, 49.98), (5.92, 58.07), (3.2, 60.9), (6.06, 55.18), (0.02, 65.63), (4.43, 58.32), (6.97, 50.63), (0.98, 54.94), (4.38, 57.81), (0.72, 58.77), (3.76, 58.68), (3.41, 53.7), (6.68, 49.42), (0.98, 62.11), (2.32, 56.2), (4.29, 58.84), (3.01, 64.79), (5.0, 56.41), (1.75, 60.89), (7.74, 50.06), (4.58, 58.68), (0.18, 67.89), (1.78, 58.19), (3.18, 55.14), (2.22, 58.88), (0.84, 59.63), (5.41, 52.86), (2.1, 61.69), (5.71, 52.71), (5.96, 55.08), (2.12, 57.91), (7.5, 45.88), (8.16, 54.15), (-0.42, 64.91), (0.17, 65.76), (-2.82, 63.74), (8.85, 53.39), (1.69, 66.06), (1.2, 60.14), (4.25, 52.52), (6.38, 51.03), (9.89, 38.41), (5.79, 61.05), (2.02, 61.82), (2.08, 52.45), (2.4, 60.02), (-1.07, 66.39), (2.9, 61.28), (2.44, 63.65), (4.91, 51.54), (9.6, 46.13), (0.53, 67.66), (5.87, 54.7), (7.81, 54.57), (8.19, 50.57), (4.17, 61.85), (6.59, 49.57), (7.87, 50.82), (-0.38, 66.41), (4.95, 52.81), (1.8, 59.14), (3.81, 59.92), (4.03, 54.15), (8.25, 50.68), (5.82, 46.51), (-0.06, 66.46), (4.47, 60.19), (2.7, 63.44), (-0.06, 58.61), (5.58, 51.38), (11.25, 38.53), (2.79, 57.26), (5.74, 49.93), (10.49, 45.66), (1.28, 66.9), (1.56, 60.61), (12.15, 35.57), (3.68, 67.63), (3.14, 65.74), (3.55, 59.39), (0.8, 61.74), (9.17, 45.6), (6.42, 51.27), (5.83, 44.72), (7.33, 51.07), (6.7, 56.16), (7.1, 47.42), (1.98, 63.96), (3.65, 59.32), (-0.24, 67.0), (6.77, 51.35), (3.33, 60.07), (2.61, 59.96), (8.52, 44.01), (6.43, 52.73), (6.68, 52.4), (3.53, 52.27), (4.76, 52.49), (3.0, 63.36), (5.33, 58.75), (3.6, 49.59), (5.32, 52.25), (1.33, 59.79), (3.66, 57.01), (8.53, 48.97), (4.9, 56.03), (5.83, 55.59), (2.26, 59.08), (3.29, 57.26), (3.98, 52.53), (3.09, 58.72), (3.26, 57.73), (6.43, 50.32), (3.84, 55.71), (2.13, 57.56), (8.53, 51.08), (5.18, 58.33), (9.48, 48.45), (6.49, 47.61), (6.68, 45.73), (4.34, 55.41), (-0.02, 56.53), (2.74, 62.99), (5.69, 50.71), (4.5, 58.21), (7.23, 46.16), (5.36, 55.16), (3.81, 56.88), (5.63, 50.38), (-2.36, 68.12), (5.61, 52.83), (5.11, 58.22), (6.86, 45.97), (6.94, 47.22), (9.56, 49.69), (3.85, 56.65), (3.94, 54.48), (2.54, 66.74), (1.75, 60.79), (3.87, 55.36), (2.7, 62.68), (3.25, 57.87), (11.02, 42.84), (1.33, 63.18), (-0.03, 65.56), (4.16, 60.91), (4.29, 63.95), (10.72, 45.2), (5.43, 62.55), (4.69, 52.7), (5.71, 47.12), (5.13, 56.42), (8.86, 50.92), (3.38, 52.81), (8.45, 45.79), (5.35, 55.22), (6.06, 51.44), (1.31, 68.44), (4.24, 52.71), (3.25, 57.5), (5.71, 51.99), (0.52, 60.0), (6.5, 57.69), (2.93, 55.89), (6.49, 52.71), (5.98, 58.09), (1.34, 60.69), (6.85, 47.63), (5.09, 53.1), (7.16, 51.41), (5.17, 50.56), (10.81, 39.16), (4.17, 51.73), (4.13, 59.1), (6.66, 51.79), (-1.3, 67.1), (9.38, 44.41), (4.91, 52.21), (11.55, 40.96), (2.78, 56.46), (6.39, 57.69), (3.77, 55.28), (6.87, 53.38), (4.83, 52.68), (9.76, 45.16), (9.88, 43.72), (-1.22, 66.9), (4.6, 50.07), (8.0, 52.76), (11.76, 42.24), (2.43, 61.71), (2.38, 64.63), (2.67, 57.89), (0.15, 70.39), (6.33, 44.5), (5.8, 53.27), (4.81, 59.35), (3.79, 52.53), (7.72, 51.44), (5.65, 52.44), (-0.97, 58.88), (-1.41, 64.94), (7.01, 53.51), (7.41, 52.28), (-0.65, 63.4), (4.98, 52.56), (8.47, 53.23), (5.44, 55.14), (-0.96, 59.25), (4.27, 50.43), (6.78, 48.28), (5.13, 58.11), (5.29, 53.41), (8.2, 45.36), (-0.77, 64.6), (1.48, 66.47), (10.28, 46.96), (3.68, 53.75), (6.85, 40.78), (6.22, 46.84), (4.61, 58.71), (5.17, 56.69), (6.52, 52.87), (8.02, 54.32), (5.45, 57.08), (3.25, 52.33), (-0.92, 64.31), (3.98, 55.67), (7.8, 50.31), (5.71, 50.73), (2.28, 58.07), (1.52, 60.13), (3.43, 60.96), (4.19, 49.26), (3.26, 61.39), (2.35, 57.36), (2.77, 67.25), (0.67, 66.32), (6.85, 53.53), (1.28, 68.68), (0.77, 64.75), (4.63, 58.53), (2.89, 58.1), (3.99, 58.07), (7.29, 51.77), (6.85, 49.46), (8.0, 44.69), (6.66, 46.22), (4.55, 52.33), (3.98, 61.63), (6.17, 52.75), (8.22, 49.99), (3.34, 61.81), (5.52, 59.11), (7.29, 51.2), (4.27, 52.04), (3.8, 56.84), (5.78, 48.19), (6.85, 50.56), (7.11, 53.23)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=4.8643，ȳ=55.3123；Sxx=Σ(x−x̄)²=6531.15，Sxy=-13028.81；
β₁ = Sxy/Sxx = -1.9949；β₀ = ȳ−β₁x̄ = 65.0160；
R² = SSR/SST = 25990.79/37683.82 = 0.6897；F = (SSR/1)/(SSE/764) = 1698.1886，p = 2.50e-196；
SE(β₁)=√(15.31/6531.15)=0.0484；t=β₁/SE=-41.2091；p=2.50e-196；95%CI=[-2.0899,-1.8998]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=4.8643，ȳ=55.3123；Sxx=Σ(x−x̄)²=6531.15，Sxy=-13028.81；
β₁ = Sxy/Sxx = -1.9949；β₀ = ȳ−β₁x̄ = 65.0160；
R² = SSR/SST = 25990.79/37683.82 = 0.6897；F = (SSR/1)/(SSE/764) = 1698.1886，p = 2.50e-196；
SE(β₁)=√(15.31/6531.15)=0.0484；t=β₁/SE=-41.2091；p=2.50e-196；95%CI=[-2.0899,-1.8998]。
③ 计算结果：
y = 65.0160 + -1.9949·x；R²=0.6897；β₁ t=-41.2091 p=2.50e-196；95%CI=[-2.0899,-1.8998]
④ 结论：在 α=0.05 水平下，p=2.50e-196 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-1.9949），越远越少来。
案例 59. 利率预测贷款需求
简单线性回归利率升需求降
① 原始数据：
样本 n=342，逐条 (x, y)：[(5.54, 111.38), (2.36, 116.62), (3.08, 106.44), (5.49, 92.27), (6.82, 102.9), (3.17, 113.39), (3.72, 101.98), (2.45, 114.25), (9.27, 88.14), (2.62, 102.4), (4.09, 106.86), (5.62, 85.15), (5.74, 89.04), (2.76, 107.17), (2.46, 118.51), (3.32, 110.09), (4.63, 96.41), (6.33, 90.97), (5.85, 98.67), (3.97, 99.5), (3.12, 101.65), (5.71, 99.4), (8.38, 85.96), (4.84, 99.38), (6.96, 75.2), (4.26, 109.09), (9.99, 92.68), (2.07, 107.38), (3.37, 121.91), (7.69, 91.61), (2.02, 108.94), (5.31, 107.21), (4.99, 104.41), (3.72, 101.26), (4.08, 93.13), (3.04, 105.73), (7.0, 86.75), (5.82, 104.97), (2.9, 102.79), (4.99, 92.93), (2.74, 121.43), (4.02, 105.59), (1.7, 95.05), (5.66, 95.24), (2.88, 108.12), (5.62, 91.47), (3.22, 115.31), (7.52, 100.55), (6.07, 107.24), (7.14, 88.36), (5.95, 102.08), (2.99, 107.87), (5.73, 91.25), (6.87, 105.59), (3.7, 108.04), (3.38, 114.36), (5.63, 88.0), (0.13, 124.29), (5.17, 96.71), (9.23, 92.38), (3.27, 104.56), (4.72, 80.27), (5.0, 87.06), (3.3, 103.1), (7.9, 86.39), (8.19, 82.19), (5.46, 94.09), (5.44, 104.01), (4.42, 84.98), (3.11, 93.06), (5.69, 90.41), (5.95, 98.1), (4.05, 101.49), (7.11, 94.75), (7.3, 87.46), (7.97, 96.33), (6.68, 88.45), (1.52, 107.67), (6.83, 96.02), (7.31, 100.08), (4.69, 91.58), (3.92, 102.13), (7.96, 79.37), (4.46, 88.24), (5.91, 90.78), (5.36, 103.24), (2.51, 103.24), (4.71, 107.91), (4.28, 98.18), (7.06, 94.75), (4.45, 117.18), (2.3, 105.07), (3.94, 98.99), (0.89, 108.6), (5.95, 103.56), (3.23, 92.98), (3.83, 97.11), (6.55, 89.66), (2.49, 113.3), (5.11, 96.09), (6.33, 102.13), (7.47, 99.41), (3.82, 97.78), (6.24, 103.1), (2.56, 108.95), (8.08, 97.82), (7.72, 97.96), (5.28, 113.28), (6.83, 93.09), (4.07, 106.3), (5.96, 107.04), (4.23, 97.37), (6.22, 85.51), (6.36, 89.67), (2.47, 102.18), (7.08, 92.73), (4.33, 114.4), (5.98, 94.02), (1.59, 116.7), (6.81, 98.82), (4.91, 94.11), (4.04, 82.51), (4.81, 101.33), (5.3, 108.02), (2.22, 93.24), (4.06, 107.58), (6.85, 86.49), (4.63, 99.46), (4.24, 86.65), (7.96, 85.06), (5.87, 94.44), (5.65, 90.62), (7.62, 84.8), (7.89, 88.03), (6.88, 100.38), (6.03, 111.57), (6.44, 105.18), (3.27, 113.96), (2.68, 109.22), (5.41, 107.45), (6.23, 92.39), (4.55, 105.82), (5.08, 92.63), (3.27, 108.67), (6.08, 97.95), (4.58, 104.87), (4.07, 119.42), (7.63, 96.28), (4.7, 99.91), (5.99, 90.23), (8.8, 84.03), (5.42, 100.7), (2.73, 113.47), (7.9, 87.0), (5.29, 101.39), (1.59, 97.41), (4.65, 96.41), (8.5, 97.14), (1.86, 112.33), (6.42, 91.82), (5.68, 95.39), (4.81, 107.61), (3.22, 92.73), (4.41, 108.15), (5.79, 94.93), (8.43, 84.25), (4.42, 94.95), (4.16, 98.11), (7.47, 89.43), (4.58, 118.43), (2.06, 100.29), (4.17, 103.8), (1.57, 131.34), (0.5, 111.45), (7.0, 98.63), (4.83, 93.49), (7.76, 102.35), (4.53, 102.26), (7.6, 95.23), (7.8, 75.52), (1.77, 111.2), (5.48, 99.77), (4.58, 83.96), (3.72, 102.67), (3.62, 113.05), (6.19, 93.12), (3.18, 98.35), (4.75, 99.8), (4.69, 109.5), (4.38, 95.84), (5.62, 97.25), (7.81, 89.19), (10.07, 98.12), (5.73, 93.76), (4.7, 94.63), (3.64, 100.82), (6.93, 92.85), (3.52, 116.02), (3.91, 97.41), (2.05, 115.49), (6.36, 92.78), (3.98, 87.57), (2.92, 115.63), (3.97, 104.32), (2.96, 112.77), (4.17, 94.96), (2.07, 116.0), (3.6, 93.35), (5.14, 93.72), (6.73, 86.93), (5.29, 104.08), (6.25, 107.81), (6.1, 97.3), (3.78, 99.53), (6.38, 86.56), (4.96, 96.44), (5.43, 95.8), (8.4, 103.21), (6.13, 102.33), (0.96, 108.53), (10.04, 85.52), (4.75, 110.66), (6.11, 82.71), (2.6, 117.87), (7.32, 74.89), (6.07, 103.94), (6.99, 91.19), (5.85, 84.86), (5.05, 107.84), (3.68, 102.1), (2.7, 115.58), (8.37, 92.52), (4.7, 102.7), (5.5, 92.9), (6.9, 92.38), (3.16, 100.41), (4.93, 89.31), (5.86, 94.78), (2.48, 97.56), (4.35, 95.79), (2.65, 104.5), (5.71, 96.41), (3.95, 99.6), (6.25, 94.78), (4.97, 83.65), (3.3, 106.01), (2.76, 121.83), (3.11, 114.37), (1.62, 118.41), (5.0, 90.27), (7.24, 95.54), (6.77, 83.58), (6.18, 110.1), (5.0, 88.34), (4.26, 113.03), (4.81, 98.7), (4.43, 104.22), (7.44, 93.37), (0.93, 117.56), (4.96, 92.21), (7.99, 87.56), (6.53, 87.51), (6.14, 96.46), (4.62, 108.9), (9.65, 80.69), (8.67, 85.74), (5.88, 90.82), (7.76, 83.8), (3.27, 109.58), (3.87, 98.68), (6.24, 91.01), (6.72, 99.51), (4.67, 89.09), (6.82, 95.91), (3.29, 91.57), (4.44, 98.63), (6.65, 88.56), (5.41, 101.02), (3.18, 96.46), (5.25, 95.67), (5.62, 100.74), (2.17, 121.41), (5.35, 93.06), (5.08, 98.3), (5.45, 112.14), (4.31, 104.61), (7.22, 95.86), (4.82, 86.83), (4.65, 102.48), (6.14, 103.23), (8.49, 100.42), (7.69, 99.12), (3.25, 93.71), (8.71, 93.78), (5.88, 101.0), (5.48, 98.28), (6.27, 103.51), (4.43, 107.73), (6.3, 99.61), (7.23, 94.92), (6.2, 94.64), (5.92, 91.78), (5.35, 88.64), (3.96, 96.46), (2.41, 98.18), (3.18, 104.14), (1.99, 111.79), (4.81, 100.3), (0.91, 99.71), (9.46, 68.66), (4.04, 106.38), (2.13, 105.29), (6.85, 97.14), (3.89, 95.75), (6.36, 95.16), (8.23, 97.1), (9.48, 83.93), (7.56, 87.93), (5.58, 109.36), (7.27, 102.22), (5.83, 101.36), (6.54, 100.96), (6.03, 86.3), (4.58, 107.79), (2.49, 122.23), (2.5, 102.13), (4.21, 99.46), (6.78, 90.39), (2.24, 111.16), (2.53, 122.91), (3.39, 105.15), (5.76, 98.89), (2.54, 110.78), (4.37, 107.58), (1.7, 105.16), (2.92, 102.62), (6.51, 104.14), (3.55, 103.64), (3.9, 107.06), (2.4, 102.86), (3.23, 104.75), (5.41, 97.88)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.0447，ȳ=99.4285；Sxx=Σ(x−x̄)²=1286.67，Sxy=-3987.83；
β₁ = Sxy/Sxx = -3.0993；β₀ = ȳ−β₁x̄ = 115.0637；
R² = SSR/SST = 12359.64/32547.38 = 0.3797；F = (SSR/1)/(SSE/340) = 208.1597，p = -1.83e+05；
SE(β₁)=√(59.38/1286.67)=0.2148；t=β₁/SE=-14.4277；p=-1.83e+05；95%CI=[-6.6870,0.4883]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.0447，ȳ=99.4285；Sxx=Σ(x−x̄)²=1286.67，Sxy=-3987.83；
β₁ = Sxy/Sxx = -3.0993；β₀ = ȳ−β₁x̄ = 115.0637；
R² = SSR/SST = 12359.64/32547.38 = 0.3797；F = (SSR/1)/(SSE/340) = 208.1597，p = -1.83e+05；
SE(β₁)=√(59.38/1286.67)=0.2148；t=β₁/SE=-14.4277；p=-1.83e+05；95%CI=[-6.6870,0.4883]。
③ 计算结果：
y = 115.0637 + -3.0993·x；R²=0.3797；β₁ t=-14.4277 p=-1.83e+05；95%CI=[-6.6870,0.4883]
④ 结论：在 α=0.05 水平下，p=-1.83e+05 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-3.0993），利率升需求降。
案例 60. 噪音预测专注度
简单线性回归噪音降专注
① 原始数据：
样本 n=804，逐条 (x, y)：[(94.5, 80.24), (81.2, 72.13), (56.07, 81.43), (85.75, 71.34), (54.86, 87.09), (56.68, 87.67), (31.98, 97.22), (59.89, 77.31), (29.32, 87.84), (16.58, 99.64), (71.82, 81.18), (84.22, 73.95), (66.79, 77.96), (56.71, 75.08), (49.07, 86.1), (76.53, 83.88), (61.47, 81.4), (49.5, 83.06), (70.34, 73.8), (77.63, 71.86), (60.83, 81.22), (38.91, 79.14), (61.3, 84.96), (84.79, 74.15), (41.09, 90.18), (64.68, 76.96), (48.67, 80.7), (37.3, 88.28), (22.76, 98.04), (36.61, 86.88), (47.24, 82.84), (37.02, 90.36), (72.07, 68.95), (84.43, 71.57), (64.23, 74.31), (85.45, 77.53), (117.59, 57.24), (82.38, 74.63), (29.61, 87.04), (51.17, 84.33), (66.24, 65.07), (65.63, 64.01), (71.83, 75.41), (54.24, 81.97), (54.15, 90.07), (15.85, 92.39), (76.9, 72.95), (97.4, 74.19), (32.96, 83.55), (44.29, 76.81), (74.75, 68.3), (60.7, 80.59), (65.64, 81.57), (95.91, 62.58), (72.53, 75.03), (85.68, 72.17), (97.12, 72.13), (74.56, 77.45), (53.9, 83.36), (52.79, 79.32), (42.58, 83.15), (60.42, 82.39), (65.64, 83.01), (56.91, 89.07), (81.6, 74.86), (81.42, 71.58), (35.57, 82.6), (45.85, 81.35), (35.4, 79.74), (58.96, 68.83), (75.55, 74.11), (76.13, 74.56), (44.53, 79.63), (79.25, 69.56), (62.22, 81.55), (74.46, 77.59), (71.73, 80.69), (79.0, 74.53), (51.84, 83.27), (68.57, 81.71), (54.92, 78.47), (82.71, 84.06), (71.53, 73.0), (48.97, 82.28), (76.12, 75.95), (65.05, 79.37), (52.81, 79.06), (38.94, 89.06), (66.77, 78.61), (65.9, 75.33), (54.92, 83.56), (47.85, 85.75), (25.42, 88.65), (34.17, 92.3), (76.34, 71.32), (40.52, 92.8), (74.52, 75.39), (23.29, 88.03), (38.45, 87.66), (58.49, 83.2), (62.99, 78.75), (2.52, 103.16), (49.87, 80.0), (81.87, 74.39), (58.72, 83.16), (69.65, 78.09), (58.08, 82.18), (48.71, 88.03), (59.43, 82.08), (60.87, 78.47), (87.15, 76.33), (68.03, 78.62), (47.01, 87.46), (64.61, 70.6), (59.87, 79.3), (8.91, 92.06), (86.25, 66.9), (63.05, 83.07), (55.63, 75.23), (57.7, 81.14), (97.49, 69.4), (52.77, 83.91), (52.75, 83.01), (73.65, 78.85), (65.77, 83.23), (48.78, 82.03), (55.58, 79.65), (29.69, 88.77), (35.14, 87.39), (81.86, 73.5), (95.97, 70.15), (67.93, 74.35), (38.41, 89.97), (58.12, 87.47), (55.43, 76.17), (78.03, 76.46), (73.05, 78.63), (70.62, 87.37), (38.52, 79.84), (42.91, 84.39), (99.86, 64.39), (67.48, 78.07), (27.95, 89.86), (60.92, 67.41), (53.6, 86.59), (74.2, 70.45), (76.39, 73.93), (55.78, 77.45), (51.2, 74.4), (76.17, 66.19), (107.3, 58.52), (86.76, 76.14), (26.51, 88.68), (31.32, 85.22), (86.98, 67.61), (37.98, 79.99), (59.85, 70.82), (50.8, 86.32), (54.08, 93.36), (55.85, 79.53), (77.45, 66.03), (41.95, 84.12), (73.33, 68.67), (90.6, 66.89), (59.11, 79.32), (33.86, 87.46), (58.31, 88.03), (99.5, 64.05), (64.14, 82.64), (69.34, 78.48), (83.25, 76.0), (79.24, 70.98), (78.95, 71.45), (74.28, 71.96), (62.21, 75.5), (103.5, 67.87), (38.39, 91.63), (82.52, 81.98), (55.03, 83.75), (57.49, 84.16), (56.48, 86.5), (44.11, 83.83), (54.66, 81.73), (16.4, 95.28), (64.2, 81.72), (62.23, 81.5), (96.93, 73.98), (43.16, 94.26), (54.85, 80.97), (33.14, 91.33), (56.93, 73.05), (49.01, 80.1), (48.03, 79.45), (43.2, 82.37), (59.09, 80.61), (86.47, 67.49), (37.7, 78.81), (33.0, 83.55), (44.01, 89.96), (73.41, 72.37), (26.6, 92.29), (29.03, 83.16), (55.81, 80.82), (56.31, 90.94), (65.44, 78.21), (76.4, 75.51), (82.06, 65.37), (53.36, 90.82), (52.18, 85.55), (51.27, 83.06), (62.01, 83.81), (98.73, 60.75), (53.9, 90.23), (79.3, 75.67), (70.95, 82.79), (81.57, 71.52), (55.17, 82.08), (40.28, 90.85), (41.03, 91.4), (50.34, 84.89), (74.56, 79.23), (89.85, 63.85), (59.68, 85.51), (49.2, 76.71), (14.78, 94.11), (39.44, 84.91), (63.49, 84.72), (62.59, 78.07), (15.72, 98.12), (52.87, 81.42), (63.26, 82.11), (47.08, 87.03), (54.83, 78.74), (76.15, 70.9), (53.66, 85.46), (61.98, 77.06), (51.73, 82.38), (64.21, 74.89), (72.42, 80.73), (73.56, 67.93), (102.35, 61.4), (58.85, 87.43), (64.25, 79.27), (89.34, 66.73), (-1.81, 98.8), (49.48, 84.96), (26.79, 84.4), (29.7, 97.4), (76.01, 75.98), (51.13, 82.39), (36.49, 83.97), (70.03, 71.04), (44.58, 90.28), (58.92, 79.72), (59.2, 79.65), (86.27, 66.47), (67.82, 71.89), (83.33, 79.6), (41.54, 79.07), (34.46, 92.68), (65.85, 74.62), (61.93, 79.83), (35.51, 85.51), (69.9, 78.35), (52.17, 81.26), (77.33, 78.7), (47.39, 86.57), (53.23, 75.31), (27.37, 94.62), (62.79, 83.2), (48.3, 90.07), (88.7, 75.54), (39.4, 90.19), (67.39, 79.53), (27.01, 85.49), (79.04, 66.45), (34.89, 82.03), (65.94, 72.32), (70.7, 68.02), (72.32, 72.96), (57.24, 82.18), (86.47, 64.38), (94.29, 79.83), (36.61, 82.78), (63.81, 82.96), (64.14, 71.54), (55.52, 72.72), (61.24, 75.82), (86.99, 70.6), (80.83, 72.96), (57.82, 84.59), (7.22, 94.06), (77.72, 69.61), (11.87, 91.71), (50.61, 82.16), (63.8, 69.84), (62.2, 76.18), (51.61, 86.25), (63.28, 77.83), (53.67, 79.42), (67.49, 76.24), (29.4, 84.49), (48.66, 82.45), (80.76, 67.39), (41.0, 88.81), (52.75, 80.22), (63.63, 84.89), (62.18, 77.54), (61.02, 80.65), (75.38, 74.3), (84.59, 76.59), (17.93, 97.07), (60.79, 77.18), (39.43, 82.16), (68.58, 77.67), (40.24, 81.23), (4.47, 109.43), (15.57, 93.8), (37.24, 88.61), (51.99, 81.29), (78.23, 73.41), (57.94, 84.91), (47.15, 79.12), (23.7, 92.05), (48.97, 76.33), (35.07, 79.02), (60.22, 77.95), (63.69, 77.73), (23.96, 93.82), (63.68, 83.7), (60.63, 81.04), (65.07, 76.73), (45.16, 78.74), (59.74, 84.61), (97.77, 65.65), (85.26, 69.69), (58.0, 84.58), (46.37, 87.51), (51.27, 84.83), (50.84, 82.8), (57.93, 83.98), (44.16, 85.05), (56.48, 91.71), (66.06, 71.99), (38.58, 83.63), (78.63, 83.4), (65.89, 86.31), (116.25, 70.44), (73.14, 80.16), (62.65, 79.31), (42.63, 87.41), (52.48, 80.48), (76.61, 74.65), (93.56, 70.6), (56.9, 75.04), (67.39, 86.23), (90.3, 82.64), (65.5, 79.22), (72.87, 72.4), (73.86, 79.61), (88.28, 80.75), (67.21, 77.45), (73.47, 81.15), (33.63, 88.07), (41.5, 82.25), (75.32, 67.74), (50.85, 81.87), (95.66, 78.17), (49.58, 86.48), (87.32, 76.58), (61.3, 80.35), (47.39, 82.92), (48.5, 80.71), (22.48, 87.99), (53.97, 71.44), (63.74, 69.35), (20.67, 91.03), (41.67, 88.49), (57.08, 77.82), (64.02, 69.95), (79.73, 71.9), (56.34, 75.08), (73.14, 74.15), (66.6, 81.42), (75.81, 75.14), (88.59, 71.66), (49.45, 84.37), (79.03, 74.51), (56.88, 78.18), (60.18, 77.54), (51.69, 87.29), (81.04, 75.04), (60.29, 74.61), (60.3, 74.41), (57.01, 87.75), (35.77, 85.65), (72.37, 71.62), (42.76, 86.62), (52.16, 74.96), (43.99, 76.38), (43.47, 92.1), (32.82, 81.07), (58.32, 82.41), (64.11, 85.41), (76.67, 75.01), (93.01, 72.15), (80.19, 72.39), (73.45, 72.85), (47.5, 81.02), (52.23, 94.94), (34.62, 94.52), (27.83, 97.71), (56.8, 81.36), (55.12, 85.45), (26.84, 100.39), (101.71, 69.21), (62.11, 79.36), (82.26, 69.15), (85.32, 81.69), (56.11, 93.77), (51.29, 85.41), (82.39, 64.49), (40.54, 85.64), (85.33, 75.17), (28.24, 92.93), (94.28, 71.5), (78.65, 61.35), (56.86, 66.33), (67.32, 77.15), (49.83, 77.61), (51.08, 83.67), (65.75, 77.13), (58.94, 77.26), (69.63, 78.52), (34.94, 95.46), (80.87, 71.62), (69.34, 67.49), (80.24, 69.25), (94.81, 79.32), (80.85, 71.18), (47.41, 82.58), (59.92, 75.52), (65.05, 79.14), (48.85, 82.94), (78.98, 72.92), (21.91, 88.61), (31.39, 80.67), (59.6, 78.25), (69.92, 79.66), (31.74, 95.1), (55.7, 78.24), (76.57, 75.99), (59.78, 85.04), (45.66, 87.37), (62.35, 68.84), (29.49, 84.16), (67.0, 79.16), (60.3, 75.74), (62.35, 77.86), (84.33, 70.73), (54.43, 77.1), (80.13, 77.59), (50.62, 83.54), (27.97, 96.11), (38.49, 80.87), (46.6, 82.14), (85.82, 67.19), (59.94, 84.14), (57.17, 79.07), (69.35, 81.32), (43.17, 84.57), (23.29, 83.75), (30.91, 93.87), (61.0, 77.23), (50.7, 84.1), (64.88, 72.55), (39.69, 85.67), (66.42, 80.21), (92.26, 74.98), (36.48, 86.83), (36.37, 83.61), (46.61, 88.27), (72.96, 73.6), (55.83, 86.48), (57.87, 89.61), (77.92, 77.62), (76.57, 77.11), (76.44, 84.95), (98.72, 72.15), (61.15, 75.07), (87.12, 69.17), (49.06, 81.62), (55.01, 85.89), (27.56, 82.63), (44.86, 78.11), (68.53, 72.1), (32.78, 77.89), (51.22, 91.54), (51.02, 75.52), (62.69, 83.77), (14.97, 92.16), (87.54, 73.58), (33.59, 88.84), (107.15, 69.86), (103.96, 61.12), (64.14, 73.14), (70.73, 76.19), (40.83, 82.49), (57.81, 82.91), (72.09, 80.48), (75.46, 70.98), (33.26, 90.73), (22.05, 89.31), (55.37, 83.34), (28.11, 83.69), (40.01, 89.1), (37.63, 86.93), (59.29, 82.21), (73.78, 69.67), (55.96, 81.89), (93.69, 70.56), (22.34, 88.63), (40.82, 83.26), (61.41, 75.79), (51.31, 92.51), (55.57, 72.04), (40.03, 81.65), (39.07, 83.29), (72.18, 72.39), (50.93, 80.18), (18.71, 95.18), (48.79, 78.86), (79.32, 84.21), (65.38, 80.08), (28.32, 94.29), (57.32, 83.42), (32.36, 93.45), (65.11, 76.6), (32.26, 87.44), (101.43, 72.52), (54.36, 74.39), (66.24, 79.32), (55.6, 80.53), (39.98, 88.63), (63.61, 75.5), (63.57, 81.45), (85.3, 80.85), (43.26, 89.84), (44.56, 86.11), (25.69, 92.3), (66.47, 76.99), (76.2, 74.84), (47.42, 82.93), (68.78, 72.88), (72.45, 78.52), (33.35, 93.48), (77.91, 77.11), (25.74, 90.24), (82.74, 72.61), (30.7, 89.94), (51.88, 82.47), (57.76, 79.57), (90.86, 70.49), (55.23, 85.61), (40.75, 84.12), (52.97, 73.19), (45.04, 89.35), (48.33, 84.38), (70.3, 74.29), (65.12, 73.75), (52.08, 84.01), (61.13, 83.56), (96.19, 67.15), (63.61, 81.09), (66.95, 77.94), (53.83, 79.16), (71.91, 67.54), (64.47, 84.54), (56.06, 80.55), (79.75, 76.98), (52.83, 74.59), (41.56, 78.11), (66.24, 73.45), (65.42, 73.22), (54.07, 88.96), (62.49, 75.49), (37.4, 95.96), (68.24, 85.59), (23.88, 82.39), (73.46, 74.17), (66.56, 77.72), (54.45, 78.83), (35.02, 93.45), (62.95, 83.32), (57.11, 84.74), (44.05, 86.28), (69.6, 80.59), (21.81, 87.12), (40.94, 83.57), (67.56, 78.87), (53.05, 78.25), (17.51, 93.67), (62.95, 78.17), (42.18, 87.11), (72.05, 84.9), (65.52, 81.94), (79.1, 81.48), (87.27, 74.72), (55.24, 77.75), (38.29, 76.66), (51.47, 81.25), (63.19, 89.16), (77.63, 71.66), (76.26, 84.63), (50.79, 84.42), (65.71, 76.15), (97.89, 74.08), (65.86, 82.24), (41.96, 84.61), (79.45, 79.73), (48.78, 84.54), (69.98, 80.04), (94.37, 72.73), (86.24, 71.74), (33.87, 89.18), (71.39, 74.32), (56.58, 75.09), (62.03, 88.03), (62.31, 76.56), (73.92, 71.22), (85.02, 76.05), (72.76, 72.09), (69.41, 73.63), (68.83, 77.7), (71.83, 77.65), (71.82, 76.74), (66.86, 84.62), (46.72, 81.5), (50.74, 84.39), (69.99, 79.97), (52.97, 81.77), (62.55, 77.75), (47.58, 91.93), (58.2, 79.75), (41.69, 92.64), (32.96, 88.88), (43.52, 87.19), (85.3, 69.46), (43.08, 79.1), (46.97, 80.43), (56.94, 86.83), (57.08, 82.66), (107.57, 70.34), (46.25, 75.88), (57.07, 75.35), (78.46, 76.61), (67.04, 85.24), (49.53, 88.03), (62.9, 79.72), (25.14, 91.78), (49.43, 85.88), (74.49, 77.9), (80.87, 72.63), (52.84, 82.74), (67.92, 83.2), (74.49, 65.96), (47.06, 84.64), (39.88, 85.65), (56.84, 88.57), (57.94, 76.55), (64.58, 78.78), (43.95, 86.99), (48.28, 81.8), (75.42, 72.96), (54.2, 81.76), (68.77, 73.83), (49.51, 87.1), (41.06, 83.09), (49.45, 85.02), (60.1, 78.35), (111.77, 66.21), (57.46, 85.76), (53.01, 86.2), (41.6, 84.9), (25.64, 90.85), (59.29, 75.99), (71.56, 76.77), (59.89, 82.4), (8.09, 92.64), (29.23, 78.42), (71.69, 84.01), (53.53, 81.73), (36.52, 83.13), (55.16, 85.31), (65.36, 74.42), (40.19, 86.83), (83.03, 72.37), (43.76, 92.04), (91.25, 67.59), (74.33, 75.45), (73.42, 72.29), (51.33, 81.5), (89.69, 68.95), (20.5, 86.14), (71.38, 74.41), (76.91, 75.23), (54.1, 86.85), (51.48, 73.93), (83.8, 72.9), (89.48, 76.88), (44.13, 81.53), (56.63, 87.88), (44.85, 81.88), (57.24, 75.74), (65.75, 86.6), (68.98, 67.8), (51.95, 79.59), (94.21, 72.95), (93.08, 78.39), (60.94, 75.85), (67.37, 68.72), (2.57, 94.65), (45.96, 81.26), (38.88, 81.29), (41.05, 88.81), (67.75, 87.39), (40.13, 76.76), (65.49, 82.51), (73.08, 72.31), (76.59, 75.21), (39.96, 86.96), (65.74, 79.01), (63.95, 79.13), (60.96, 89.97), (72.18, 74.35), (49.5, 86.38), (52.96, 94.52), (94.45, 64.87), (66.86, 78.41), (55.44, 74.45), (64.63, 75.93), (50.21, 84.13), (49.33, 75.91), (90.16, 80.69), (41.65, 81.86), (80.43, 72.47), (83.97, 71.75), (44.25, 95.52), (60.87, 76.72), (63.98, 76.37), (36.82, 83.57), (59.8, 81.1), (61.48, 74.12), (85.94, 75.93), (37.73, 95.08), (20.82, 86.71), (68.97, 75.74), (102.29, 67.36), (37.08, 93.82), (60.23, 82.86), (53.66, 70.66), (29.19, 89.19), (48.03, 81.59), (7.16, 98.55), (55.62, 82.48), (27.85, 88.09), (68.97, 79.46), (70.85, 74.99), (50.71, 84.97), (47.28, 77.79), (44.03, 78.91), (45.1, 81.87), (24.6, 83.56), (67.44, 75.23), (36.1, 92.94), (48.42, 78.39), (68.87, 68.0), (44.91, 83.61), (93.18, 69.04), (58.75, 82.27), (52.08, 89.6), (85.05, 68.13), (60.2, 72.34), (54.47, 83.77), (78.0, 73.56), (59.64, 74.64), (68.98, 67.36), (57.68, 86.91), (99.9, 73.37), (70.41, 79.08), (76.94, 65.29), (68.04, 83.24), (43.71, 95.45), (47.21, 74.58), (68.34, 70.53), (58.01, 82.04), (47.42, 93.15), (66.73, 80.16), (51.17, 82.06), (46.97, 84.63), (55.21, 88.37), (44.89, 83.09), (68.62, 80.74), (64.14, 76.74)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=58.9674，ȳ=80.2574；Sxx=Σ(x−x̄)²=303674.78，Sxy=-89293.16；
β₁ = Sxy/Sxx = -0.2940；β₀ = ȳ−β₁x̄ = 97.5963；
R² = SSR/SST = 26255.95/45049.63 = 0.5828；F = (SSR/1)/(SSE/802) = 1120.4442，p = 2.07e-154；
SE(β₁)=√(23.43/303674.78)=0.0088；t=β₁/SE=-33.4730；p=2.07e-154；95%CI=[-0.3113,-0.2768]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=58.9674，ȳ=80.2574；Sxx=Σ(x−x̄)²=303674.78，Sxy=-89293.16；
β₁ = Sxy/Sxx = -0.2940；β₀ = ȳ−β₁x̄ = 97.5963；
R² = SSR/SST = 26255.95/45049.63 = 0.5828；F = (SSR/1)/(SSE/802) = 1120.4442，p = 2.07e-154；
SE(β₁)=√(23.43/303674.78)=0.0088；t=β₁/SE=-33.4730；p=2.07e-154；95%CI=[-0.3113,-0.2768]。
③ 计算结果：
y = 97.5963 + -0.2940·x；R²=0.5828；β₁ t=-33.4730 p=2.07e-154；95%CI=[-0.3113,-0.2768]
④ 结论：在 α=0.05 水平下，p=2.07e-154 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-0.2940），噪音降专注。
案例 61. 年龄预测柔韧性
简单线性回归年长柔韧性降
① 原始数据：
样本 n=992，逐条 (x, y)：[(55.68, 60.19), (19.43, 90.7), (24.83, 85.28), (23.51, 85.43), (55.12, 61.9), (82.55, 49.78), (29.15, 79.32), (36.39, 79.13), (50.1, 72.76), (41.38, 76.59), (45.42, 73.81), (61.1, 63.93), (21.76, 86.64), (58.02, 71.17), (51.94, 62.46), (45.55, 68.82), (27.32, 78.65), (8.11, 96.51), (55.03, 69.72), (34.05, 73.7), (56.46, 65.78), (32.88, 85.93), (37.32, 79.17), (58.21, 58.38), (42.37, 69.15), (49.67, 76.21), (17.66, 85.71), (29.34, 74.99), (35.52, 77.72), (11.04, 95.86), (62.76, 58.46), (25.92, 85.51), (49.4, 69.69), (-2.97, 101.56), (29.49, 84.68), (49.52, 66.51), (35.87, 78.42), (35.46, 80.12), (28.1, 75.2), (22.91, 74.88), (63.73, 57.48), (51.61, 68.3), (28.08, 76.82), (34.13, 78.68), (58.95, 61.89), (59.63, 57.58), (18.12, 92.76), (15.29, 90.35), (16.36, 93.37), (40.98, 77.32), (30.96, 84.21), (44.64, 79.06), (66.97, 63.81), (31.94, 80.51), (68.87, 50.53), (44.75, 68.92), (51.12, 62.58), (42.95, 74.37), (71.83, 51.05), (46.6, 70.85), (35.67, 78.53), (51.05, 68.57), (19.29, 88.11), (47.45, 70.3), (45.73, 73.6), (42.09, 70.09), (46.32, 67.03), (54.28, 68.76), (35.79, 79.85), (64.22, 57.79), (23.68, 96.1), (54.69, 67.99), (51.68, 61.69), (35.74, 79.19), (50.67, 63.38), (51.78, 70.36), (40.47, 69.3), (75.92, 52.17), (35.61, 73.71), (42.62, 70.68), (33.22, 77.58), (49.96, 79.46), (20.86, 93.52), (41.05, 74.52), (30.75, 80.71), (48.99, 72.16), (35.8, 78.54), (49.2, 68.6), (37.91, 75.1), (40.12, 68.52), (51.42, 62.93), (28.26, 78.62), (47.9, 68.64), (31.23, 71.94), (25.82, 91.71), (59.93, 70.87), (57.92, 69.92), (39.65, 72.58), (35.29, 79.54), (22.3, 80.28), (56.43, 63.11), (38.85, 79.57), (21.03, 81.89), (53.52, 70.06), (50.53, 71.4), (64.09, 57.53), (67.74, 57.85), (72.17, 54.0), (71.1, 62.0), (44.06, 69.61), (46.27, 71.11), (41.56, 65.59), (14.8, 92.18), (56.94, 70.35), (20.44, 78.71), (30.15, 79.24), (37.58, 80.56), (62.04, 64.45), (36.14, 82.92), (31.43, 79.06), (37.03, 78.73), (53.59, 73.93), (48.04, 69.33), (37.9, 77.18), (53.45, 65.71), (47.87, 65.18), (39.68, 76.98), (48.5, 71.86), (47.4, 68.42), (36.07, 75.21), (23.49, 85.85), (50.39, 74.34), (40.1, 74.13), (38.78, 73.66), (26.05, 88.67), (34.35, 79.8), (42.89, 73.32), (50.48, 69.12), (44.61, 73.45), (12.01, 88.78), (38.71, 75.02), (21.71, 89.66), (59.82, 63.64), (34.43, 75.68), (23.88, 83.96), (19.23, 90.01), (50.59, 70.93), (40.69, 78.8), (54.28, 70.55), (39.53, 76.08), (29.4, 80.91), (25.15, 85.95), (46.34, 72.61), (21.48, 90.37), (21.17, 94.58), (36.26, 75.55), (39.52, 82.21), (53.64, 70.55), (39.3, 75.62), (38.05, 75.8), (57.24, 69.79), (55.02, 68.41), (21.51, 78.18), (45.18, 79.81), (44.23, 70.3), (66.37, 61.73), (49.05, 65.97), (49.5, 70.89), (64.41, 61.02), (56.42, 67.48), (13.25, 90.38), (39.11, 81.65), (30.67, 76.35), (49.01, 73.71), (49.36, 67.55), (46.08, 72.7), (45.72, 76.41), (57.16, 70.11), (52.97, 66.59), (45.08, 71.14), (46.51, 69.67), (55.1, 61.85), (46.17, 74.88), (44.65, 73.26), (31.78, 82.34), (52.2, 63.64), (53.01, 69.95), (22.7, 82.26), (44.31, 69.81), (62.99, 63.13), (50.46, 70.83), (51.55, 64.62), (41.5, 76.56), (66.91, 58.71), (32.68, 84.83), (51.16, 69.69), (33.19, 78.61), (48.36, 73.77), (14.88, 92.21), (40.6, 74.79), (47.69, 66.63), (9.78, 93.15), (42.31, 77.33), (51.87, 64.64), (45.68, 71.57), (42.18, 75.9), (48.04, 66.5), (41.84, 72.74), (46.17, 76.2), (63.23, 65.88), (32.83, 78.39), (47.37, 77.08), (58.83, 61.19), (46.77, 73.65), (58.36, 62.06), (34.23, 72.72), (39.02, 71.88), (52.87, 71.49), (41.93, 73.68), (26.5, 84.15), (30.76, 82.52), (43.05, 81.3), (37.31, 82.07), (35.92, 74.42), (40.04, 73.9), (48.49, 68.27), (25.26, 84.34), (37.98, 72.6), (36.81, 76.88), (26.76, 86.3), (37.01, 77.34), (58.22, 63.9), (24.58, 85.58), (25.66, 82.54), (52.66, 73.37), (50.03, 69.32), (35.9, 83.4), (34.61, 75.56), (20.15, 88.67), (35.41, 74.87), (49.68, 73.29), (39.59, 72.89), (27.1, 82.85), (48.79, 70.59), (22.24, 84.33), (56.51, 66.76), (39.07, 82.94), (42.17, 68.45), (53.72, 60.37), (28.24, 75.83), (47.08, 69.23), (72.94, 54.34), (40.42, 69.93), (22.75, 84.43), (32.55, 87.18), (54.79, 65.45), (33.52, 82.11), (36.77, 77.47), (48.25, 66.52), (35.35, 82.38), (28.96, 80.56), (36.25, 79.67), (39.47, 78.48), (20.84, 81.79), (28.06, 80.86), (61.66, 61.72), (12.67, 88.57), (34.52, 72.49), (59.62, 63.5), (9.92, 103.53), (52.42, 57.4), (44.57, 62.24), (38.61, 70.88), (39.35, 65.71), (26.56, 81.76), (40.57, 79.75), (6.66, 91.12), (40.83, 67.81), (53.55, 69.9), (42.86, 69.8), (31.71, 78.51), (32.03, 75.13), (38.45, 80.5), (31.41, 76.99), (44.96, 72.37), (67.55, 60.61), (28.81, 91.37), (32.23, 81.47), (31.06, 72.71), (37.59, 73.85), (50.89, 70.66), (10.47, 94.0), (27.5, 78.36), (31.53, 78.35), (33.1, 81.54), (-16.99, 112.46), (23.84, 85.4), (51.61, 68.04), (61.64, 56.74), (27.63, 84.61), (24.03, 84.89), (81.51, 49.04), (42.46, 72.45), (27.09, 80.29), (48.46, 76.23), (49.69, 70.11), (49.74, 69.67), (38.28, 70.37), (52.44, 68.23), (22.02, 87.81), (27.7, 83.62), (29.12, 73.68), (56.13, 66.27), (47.74, 69.54), (17.66, 81.0), (26.18, 90.35), (32.58, 79.19), (30.9, 75.87), (55.99, 66.75), (61.76, 65.0), (24.48, 85.54), (28.48, 81.46), (28.44, 77.78), (59.94, 61.05), (49.63, 68.11), (31.24, 73.65), (35.22, 79.45), (53.97, 68.85), (38.42, 72.08), (35.94, 80.29), (47.72, 74.38), (42.08, 77.0), (41.22, 72.3), (49.79, 71.29), (54.19, 63.33), (15.65, 97.04), (31.36, 84.59), (62.92, 57.11), (39.17, 75.09), (35.32, 83.54), (45.67, 74.83), (17.45, 90.69), (17.21, 85.19), (49.35, 67.77), (52.73, 73.47), (17.29, 86.37), (52.25, 73.84), (33.82, 76.79), (22.78, 89.01), (52.38, 72.62), (18.45, 84.03), (52.7, 64.68), (46.16, 74.25), (20.04, 92.38), (33.62, 83.18), (63.07, 58.18), (32.39, 77.67), (39.75, 77.52), (40.92, 81.35), (51.45, 71.62), (41.74, 64.69), (51.53, 61.35), (27.97, 84.75), (39.57, 74.02), (58.15, 60.4), (38.66, 74.92), (16.13, 84.56), (35.32, 85.0), (46.37, 65.82), (24.86, 84.91), (56.05, 63.59), (56.69, 60.33), (46.61, 70.86), (56.51, 61.87), (65.43, 58.87), (39.19, 72.18), (58.15, 60.36), (38.07, 77.7), (30.92, 79.95), (35.68, 83.46), (33.89, 74.6), (15.22, 91.09), (41.0, 67.18), (21.45, 84.11), (30.73, 83.08), (55.03, 70.36), (17.65, 85.88), (39.6, 73.82), (46.27, 64.46), (52.37, 68.8), (24.65, 78.42), (44.57, 70.3), (43.83, 71.59), (33.49, 83.79), (38.57, 69.38), (55.91, 71.5), (26.35, 79.23), (27.99, 74.42), (29.39, 81.96), (52.71, 77.9), (37.24, 72.06), (56.03, 59.28), (29.07, 78.34), (26.69, 79.25), (48.99, 75.14), (36.14, 77.82), (31.42, 77.51), (25.61, 82.97), (18.55, 82.44), (32.49, 81.65), (42.24, 76.8), (27.79, 86.55), (42.62, 73.55), (29.75, 86.32), (44.27, 73.47), (31.47, 80.79), (49.16, 65.36), (25.34, 83.17), (53.39, 71.57), (42.97, 69.28), (20.01, 86.79), (50.65, 59.16), (21.49, 82.07), (38.48, 70.59), (42.83, 75.9), (37.79, 76.93), (54.17, 67.99), (39.52, 77.29), (24.75, 85.35), (52.76, 68.01), (36.47, 77.84), (18.47, 84.85), (54.66, 68.06), (28.41, 82.13), (32.89, 85.38), (10.84, 96.35), (27.25, 85.81), (34.05, 72.53), (30.73, 79.15), (44.92, 77.07), (24.76, 82.71), (64.59, 70.1), (32.93, 80.46), (31.65, 89.27), (40.85, 77.33), (35.08, 74.82), (28.21, 86.28), (68.91, 50.68), (36.85, 73.23), (31.55, 72.55), (41.17, 76.83), (57.99, 62.83), (23.14, 83.7), (30.71, 86.42), (33.58, 78.55), (46.63, 67.26), (56.26, 66.44), (38.6, 72.08), (48.82, 66.19), (39.34, 82.07), (24.18, 77.67), (35.13, 73.15), (26.81, 78.88), (20.38, 91.42), (39.74, 74.61), (21.81, 84.7), (54.89, 60.19), (43.77, 72.33), (44.46, 75.38), (57.28, 67.57), (61.31, 63.53), (21.0, 83.53), (32.93, 76.07), (57.4, 67.75), (14.78, 90.45), (35.15, 81.91), (28.38, 80.7), (31.43, 75.81), (42.89, 65.53), (48.68, 67.83), (40.48, 72.64), (21.71, 87.52), (26.74, 86.99), (32.57, 76.33), (41.42, 69.54), (40.44, 71.82), (27.82, 80.41), (54.21, 60.9), (43.51, 75.29), (29.74, 73.55), (39.05, 76.64), (55.79, 64.64), (43.37, 76.22), (34.47, 77.57), (13.69, 94.32), (53.59, 64.97), (33.13, 86.02), (59.69, 58.85), (53.12, 69.58), (47.99, 72.61), (40.51, 78.96), (31.21, 86.53), (26.8, 76.16), (48.29, 66.61), (43.4, 76.03), (27.78, 75.48), (31.26, 82.86), (24.59, 88.68), (46.38, 70.2), (66.05, 56.7), (32.17, 75.4), (43.34, 75.05), (51.73, 69.38), (52.96, 74.01), (22.4, 91.18), (40.2, 69.62), (41.69, 73.28), (50.19, 65.52), (47.06, 66.03), (47.79, 72.05), (46.95, 72.15), (30.88, 72.59), (38.31, 75.97), (61.0, 58.69), (44.29, 77.94), (33.86, 79.8), (32.72, 76.96), (21.31, 89.88), (31.05, 80.25), (65.79, 54.73), (28.69, 82.42), (49.95, 66.66), (33.3, 77.81), (70.16, 53.24), (31.55, 83.92), (39.45, 81.14), (45.91, 73.38), (44.34, 69.07), (26.57, 81.49), (23.76, 78.66), (32.83, 77.18), (26.09, 70.06), (47.15, 63.75), (36.31, 74.66), (41.63, 70.82), (10.47, 90.98), (42.13, 69.59), (34.97, 80.92), (17.83, 91.16), (-1.1, 96.46), (29.96, 84.67), (38.14, 73.61), (49.35, 75.23), (36.93, 75.82), (59.32, 67.99), (64.63, 57.58), (28.11, 76.59), (35.42, 79.41), (31.22, 81.07), (32.78, 72.92), (32.18, 74.56), (58.0, 71.48), (64.97, 60.92), (15.52, 88.04), (70.32, 57.78), (52.08, 69.74), (39.72, 73.61), (37.37, 82.82), (13.31, 90.55), (52.65, 71.59), (26.55, 85.37), (30.69, 79.84), (38.21, 74.81), (37.1, 68.5), (36.38, 79.01), (36.92, 83.04), (51.22, 67.47), (44.31, 70.97), (48.62, 81.75), (40.19, 77.78), (5.82, 91.0), (68.39, 58.8), (53.12, 63.93), (47.2, 72.15), (41.96, 73.73), (36.54, 80.0), (50.02, 61.16), (46.63, 66.31), (21.57, 89.88), (27.83, 81.11), (45.72, 76.01), (39.78, 74.55), (73.98, 54.81), (36.81, 73.48), (36.9, 77.66), (51.45, 55.91), (21.95, 80.2), (58.85, 59.03), (46.78, 67.7), (21.04, 84.57), (40.0, 72.67), (50.67, 67.76), (37.67, 71.7), (49.82, 66.98), (24.94, 83.41), (60.02, 63.98), (16.05, 85.76), (51.25, 71.73), (40.36, 76.82), (6.68, 94.36), (42.28, 70.7), (32.25, 82.94), (29.05, 91.7), (57.69, 56.75), (44.7, 76.46), (34.43, 76.1), (50.59, 72.11), (39.1, 78.6), (41.0, 74.1), (19.69, 90.15), (34.53, 71.43), (34.11, 82.16), (50.37, 64.33), (38.48, 85.67), (40.03, 77.4), (33.47, 81.52), (29.75, 78.36), (43.84, 73.98), (42.2, 73.12), (5.56, 93.3), (61.61, 64.43), (39.2, 76.69), (45.88, 70.46), (46.85, 70.23), (59.94, 62.54), (41.4, 69.38), (29.76, 86.35), (40.14, 72.42), (37.41, 75.3), (52.92, 72.98), (31.88, 78.4), (22.7, 88.35), (25.8, 82.16), (52.72, 73.09), (27.62, 76.51), (32.84, 80.62), (28.36, 81.38), (52.73, 63.82), (46.35, 67.26), (39.0, 78.4), (57.67, 71.27), (30.78, 79.56), (17.82, 86.25), (49.52, 69.12), (44.15, 79.11), (53.75, 69.84), (45.57, 65.54), (33.42, 68.75), (41.73, 72.99), (36.49, 77.27), (36.45, 78.13), (-2.15, 104.87), (65.54, 54.67), (23.9, 88.2), (25.14, 86.1), (34.4, 83.17), (59.67, 62.21), (21.34, 91.85), (42.11, 67.97), (49.47, 64.2), (48.09, 69.39), (45.08, 77.18), (18.49, 82.3), (43.97, 70.74), (19.81, 86.16), (33.25, 82.71), (42.51, 73.85), (55.22, 61.77), (29.01, 89.31), (43.72, 78.2), (48.18, 72.38), (32.79, 84.94), (50.99, 70.95), (62.43, 59.65), (39.97, 76.82), (52.38, 65.07), (42.21, 74.59), (34.34, 79.09), (48.25, 71.81), (25.41, 83.32), (38.77, 71.31), (27.06, 84.17), (36.34, 76.64), (21.46, 90.27), (61.07, 63.51), (38.19, 80.61), (41.51, 82.75), (49.67, 70.12), (53.38, 65.31), (56.5, 70.13), (43.02, 67.88), (34.11, 82.82), (23.74, 86.05), (32.91, 81.47), (30.64, 83.57), (30.6, 86.63), (49.63, 65.27), (15.72, 85.98), (41.74, 79.07), (30.86, 72.58), (37.2, 77.54), (37.22, 76.54), (57.18, 68.17), (44.09, 70.2), (28.75, 88.76), (45.16, 74.13), (64.09, 60.94), (39.69, 76.58), (68.72, 62.61), (27.74, 80.41), (19.84, 87.89), (34.85, 78.17), (36.25, 73.66), (26.66, 78.94), (51.55, 73.98), (44.55, 79.32), (62.62, 69.75), (57.07, 65.41), (42.66, 72.41), (63.42, 55.12), (46.56, 76.39), (27.69, 82.43), (51.51, 66.57), (32.13, 80.25), (50.54, 71.24), (25.77, 83.72), (27.87, 81.24), (15.45, 88.2), (41.62, 76.6), (37.73, 79.9), (25.18, 88.21), (15.31, 88.25), (45.46, 68.8), (11.5, 92.88), (46.21, 75.81), (35.71, 86.78), (26.76, 76.77), (70.42, 56.89), (40.08, 73.83), (65.21, 58.15), (49.37, 65.26), (53.69, 60.78), (46.61, 76.93), (41.27, 79.04), (28.68, 82.31), (50.7, 68.97), (52.53, 63.14), (49.17, 72.77), (31.35, 81.36), (63.26, 60.92), (44.7, 74.05), (24.71, 84.01), (65.62, 61.97), (27.19, 80.73), (44.73, 73.26), (41.67, 71.38), (8.42, 91.64), (35.84, 79.46), (37.34, 72.66), (19.84, 83.89), (58.1, 63.25), (26.33, 80.03), (62.64, 63.93), (23.25, 88.49), (36.34, 74.52), (42.71, 66.94), (59.72, 63.79), (35.9, 76.8), (34.76, 74.93), (31.53, 76.16), (27.65, 79.76), (33.74, 80.08), (41.78, 75.31), (46.42, 77.58), (41.24, 78.41), (12.48, 95.36), (47.72, 69.91), (49.27, 74.48), (40.69, 81.93), (19.42, 89.87), (26.07, 80.72), (71.71, 55.89), (31.67, 85.34), (46.45, 75.8), (30.94, 88.27), (24.81, 81.16), (27.51, 81.42), (37.11, 73.37), (36.8, 79.07), (46.96, 73.18), (20.23, 90.32), (45.75, 71.47), (35.0, 79.24), (34.82, 85.06), (49.57, 62.45), (22.87, 81.41), (14.46, 84.23), (45.32, 74.95), (57.77, 61.37), (47.1, 72.88), (26.34, 84.38), (42.56, 71.37), (48.56, 66.96), (34.82, 71.98), (21.09, 92.21), (27.77, 83.33), (47.14, 71.93), (15.67, 83.21), (30.17, 79.6), (59.42, 62.41), (52.35, 67.0), (54.98, 63.43), (48.42, 68.86), (35.95, 84.68), (22.99, 87.14), (23.84, 95.34), (1.0, 92.88), (44.35, 77.7), (47.55, 63.34), (49.3, 71.67), (69.26, 58.28), (62.63, 59.06), (27.67, 80.65), (52.03, 70.45), (42.12, 78.62), (66.59, 58.77), (21.07, 91.03), (35.52, 78.25), (58.64, 63.63), (44.55, 70.2), (51.53, 66.55), (28.85, 77.81), (68.64, 59.91), (30.7, 81.22), (56.62, 61.29), (27.59, 84.57), (37.04, 81.63), (38.81, 75.48), (43.08, 73.0), (45.94, 80.95), (31.57, 83.43), (26.1, 86.55), (42.93, 75.56), (59.85, 61.97), (55.34, 57.55), (27.86, 93.04), (52.91, 69.88), (20.53, 90.6), (36.74, 79.09), (55.65, 66.8), (46.12, 70.67), (24.54, 84.42), (33.69, 79.41), (15.37, 86.79), (54.99, 61.62), (27.62, 76.6), (54.47, 65.79), (34.36, 74.35), (44.86, 77.51), (27.49, 85.62), (48.82, 67.98), (56.41, 66.92), (75.94, 49.62), (22.02, 82.32), (61.02, 59.37), (22.11, 76.29), (62.03, 55.89), (54.57, 73.74), (44.14, 73.51), (53.39, 59.67), (41.14, 76.81), (49.3, 74.02), (10.67, 90.17), (41.57, 73.82), (51.86, 65.5), (33.3, 76.49), (28.02, 85.83), (43.73, 72.72), (29.55, 87.98), (47.28, 74.75), (54.51, 57.45), (27.81, 83.03), (39.31, 73.62), (55.07, 68.08), (65.6, 60.35), (54.59, 70.06), (39.42, 75.46), (29.87, 77.84), (51.43, 67.33), (36.24, 74.09), (48.89, 73.81), (46.73, 67.11), (44.48, 61.03), (52.48, 64.51), (56.99, 70.79), (22.93, 83.01), (71.4, 55.35), (41.63, 74.08), (11.64, 104.22), (84.06, 55.37), (32.76, 75.05), (13.54, 90.18), (37.98, 77.44), (38.9, 72.64), (34.38, 76.41), (35.09, 76.98), (67.59, 54.85), (47.5, 69.58), (34.47, 75.18), (46.26, 73.53), (28.6, 83.84), (48.17, 68.99), (62.69, 59.26), (20.67, 90.18), (60.82, 62.61), (30.86, 84.99), (49.97, 77.45), (45.51, 75.11), (51.99, 68.44), (42.5, 73.48), (17.06, 94.69), (45.71, 72.53), (71.75, 62.77), (26.59, 86.14), (25.87, 81.83), (29.85, 76.0), (65.06, 60.19), (52.83, 69.14), (33.72, 78.15), (46.4, 75.54), (57.79, 59.99), (53.65, 64.69), (55.42, 68.2), (71.43, 51.59), (50.08, 70.25), (36.92, 73.03), (59.8, 63.17), (26.5, 85.84), (31.81, 82.78), (32.4, 78.69), (27.81, 81.34), (58.95, 64.09), (19.97, 86.16), (44.68, 71.74), (19.05, 81.96), (29.42, 79.95), (51.46, 67.87), (54.2, 63.53), (44.9, 73.52), (62.37, 69.39), (41.07, 76.95), (47.89, 67.75), (38.68, 70.95), (47.2, 69.95), (57.55, 61.44), (46.03, 71.14), (46.75, 67.86), (51.68, 64.37), (24.47, 85.7), (62.13, 60.61), (57.48, 68.28), (73.19, 50.74), (40.78, 76.64), (18.6, 89.28), (46.59, 72.19), (35.19, 74.8), (48.54, 71.65), (16.18, 82.95), (32.86, 76.44), (59.2, 70.13), (52.42, 65.16), (38.43, 78.28), (30.33, 75.18), (32.19, 77.25), (50.71, 67.75), (54.82, 68.49), (49.85, 66.92), (36.6, 80.31), (52.39, 70.39), (46.74, 74.45), (18.73, 95.09), (27.21, 82.07)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=40.2837，ȳ=74.8833；Sxx=Σ(x−x̄)²=197840.92，Sxy=-118365.64；
β₁ = Sxy/Sxx = -0.5983；β₀ = ȳ−β₁x̄ = 98.9846；
R² = SSR/SST = 70816.61/86697.54 = 0.8168；F = (SSR/1)/(SSE/990) = 4414.6322，p = <1e-300；
SE(β₁)=√(16.04/197840.92)=0.0090；t=β₁/SE=-66.4427；p=<1e-300；95%CI=[-0.8784,-0.3182]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=40.2837，ȳ=74.8833；Sxx=Σ(x−x̄)²=197840.92，Sxy=-118365.64；
β₁ = Sxy/Sxx = -0.5983；β₀ = ȳ−β₁x̄ = 98.9846；
R² = SSR/SST = 70816.61/86697.54 = 0.8168；F = (SSR/1)/(SSE/990) = 4414.6322，p = <1e-300；
SE(β₁)=√(16.04/197840.92)=0.0090；t=β₁/SE=-66.4427；p=<1e-300；95%CI=[-0.8784,-0.3182]。
③ 计算结果：
y = 98.9846 + -0.5983·x；R²=0.8168；β₁ t=-66.4427 p=<1e-300；95%CI=[-0.8784,-0.3182]
④ 结论：在 α=0.05 水平下，p=<1e-300 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-0.5983），年长柔韧性降。
案例 62. 折扣预测利润率
简单线性回归打折损利润
① 原始数据：
样本 n=946，逐条 (x, y)：[(37.66, 17.95), (22.4, 23.82), (14.84, 28.53), (15.84, 26.8), (-8.06, 37.57), (30.84, 18.9), (6.01, 29.05), (25.17, 23.12), (4.87, 29.33), (4.75, 33.24), (19.86, 26.48), (14.0, 27.95), (14.74, 28.18), (22.57, 24.36), (18.22, 22.63), (12.51, 26.8), (31.91, 21.5), (28.21, 21.33), (23.83, 23.2), (23.96, 23.47), (8.56, 32.61), (28.29, 22.52), (22.75, 23.73), (17.66, 25.37), (36.16, 18.46), (15.15, 25.98), (5.18, 30.45), (24.45, 25.67), (28.08, 20.02), (16.93, 27.43), (20.88, 25.1), (39.37, 19.19), (20.31, 21.9), (18.69, 21.5), (28.35, 21.31), (13.61, 29.45), (13.7, 25.13), (23.56, 22.84), (-0.02, 32.35), (18.41, 28.5), (46.09, 14.88), (18.8, 27.05), (33.5, 23.08), (14.33, 28.73), (7.83, 28.4), (19.25, 28.62), (26.23, 24.56), (20.37, 25.93), (14.72, 25.62), (14.64, 31.34), (15.21, 29.33), (19.07, 22.42), (14.91, 28.3), (23.27, 21.35), (19.87, 25.71), (16.45, 23.06), (31.8, 19.27), (19.68, 24.43), (19.57, 26.26), (20.38, 23.2), (30.11, 22.45), (17.37, 25.7), (12.18, 27.29), (9.97, 28.69), (25.82, 20.86), (12.35, 26.52), (32.27, 19.65), (14.74, 26.6), (13.1, 26.09), (17.06, 26.78), (20.07, 27.0), (37.79, 19.06), (12.0, 29.23), (22.31, 26.91), (10.97, 30.65), (21.07, 25.0), (18.73, 27.46), (27.98, 21.66), (28.27, 21.52), (14.59, 24.79), (16.05, 26.45), (1.51, 31.72), (19.04, 24.29), (30.51, 23.63), (20.78, 24.55), (28.95, 19.55), (5.54, 28.61), (13.31, 30.05), (16.81, 23.72), (1.85, 33.79), (39.47, 15.95), (6.76, 27.52), (9.34, 28.91), (33.75, 20.05), (45.35, 15.52), (12.71, 30.78), (11.96, 25.39), (29.96, 20.15), (20.55, 22.67), (19.95, 26.27), (30.12, 22.21), (32.54, 23.16), (27.91, 22.75), (16.06, 29.58), (10.35, 29.02), (19.6, 21.27), (16.74, 27.46), (3.38, 36.71), (10.51, 28.84), (24.42, 23.2), (-2.58, 34.49), (19.34, 26.14), (18.1, 27.11), (19.66, 28.52), (30.49, 18.24), (26.01, 22.41), (22.21, 25.47), (7.13, 26.01), (37.44, 14.37), (26.85, 19.46), (36.05, 18.9), (22.83, 23.83), (17.17, 25.73), (3.79, 31.21), (32.61, 17.59), (13.59, 28.07), (31.44, 19.73), (29.95, 22.36), (15.81, 26.53), (18.48, 29.64), (19.39, 23.29), (17.2, 24.72), (22.24, 23.67), (22.16, 24.48), (9.38, 28.39), (17.66, 26.16), (23.62, 22.2), (11.31, 26.02), (16.53, 26.29), (24.15, 22.97), (18.53, 25.07), (38.55, 18.08), (8.1, 27.36), (16.9, 22.91), (25.1, 24.41), (17.32, 28.8), (32.11, 17.46), (-0.83, 31.67), (25.39, 25.67), (18.78, 24.94), (11.58, 30.0), (29.18, 23.05), (24.14, 23.45), (13.92, 27.47), (3.37, 30.87), (10.13, 31.02), (22.3, 24.23), (15.68, 25.94), (22.76, 19.8), (11.56, 27.2), (17.3, 28.4), (16.36, 26.41), (-2.93, 32.97), (17.52, 28.24), (45.25, 15.24), (1.2, 36.53), (20.27, 23.09), (11.4, 27.98), (11.67, 28.71), (0.85, 33.74), (24.91, 27.46), (17.87, 26.29), (23.68, 23.78), (11.23, 26.23), (20.49, 25.68), (15.26, 28.82), (24.13, 18.19), (21.21, 23.51), (17.5, 24.85), (14.0, 28.25), (27.16, 26.45), (11.34, 23.49), (9.57, 27.96), (31.28, 17.89), (7.27, 27.58), (9.41, 30.58), (23.72, 23.11), (14.71, 26.61), (28.45, 23.23), (16.26, 30.43), (24.24, 20.08), (27.46, 22.58), (11.29, 30.57), (26.82, 23.08), (36.76, 19.49), (17.07, 27.42), (19.95, 26.58), (21.63, 24.19), (26.0, 23.87), (19.43, 26.09), (18.18, 29.25), (28.18, 22.47), (42.52, 16.58), (34.71, 19.06), (28.55, 23.39), (27.49, 22.81), (8.86, 31.13), (29.63, 23.87), (27.01, 23.08), (8.2, 28.94), (14.23, 25.16), (22.46, 22.73), (27.17, 19.41), (19.26, 23.9), (6.55, 31.91), (7.49, 31.31), (29.18, 20.44), (11.68, 29.28), (32.9, 18.65), (15.55, 26.93), (29.0, 20.49), (22.67, 19.95), (34.94, 18.28), (32.4, 20.79), (25.82, 23.92), (45.55, 18.48), (15.15, 26.84), (18.11, 28.44), (15.25, 25.66), (0.55, 31.9), (13.79, 25.33), (19.46, 25.33), (41.9, 18.14), (19.37, 28.91), (27.72, 23.66), (18.72, 27.07), (11.07, 28.18), (22.66, 18.67), (26.54, 19.96), (20.66, 23.48), (17.72, 23.69), (24.59, 23.81), (20.31, 22.41), (19.23, 26.09), (17.47, 25.44), (17.26, 23.46), (13.22, 28.1), (11.7, 24.6), (24.06, 27.97), (26.79, 20.35), (27.98, 21.7), (17.78, 30.85), (20.74, 22.24), (12.64, 30.34), (34.31, 19.22), (-1.82, 35.33), (9.67, 26.46), (5.27, 32.72), (18.51, 24.35), (11.23, 23.95), (22.7, 24.28), (26.95, 20.01), (21.3, 26.81), (20.42, 26.8), (25.51, 23.68), (19.8, 24.29), (28.93, 24.79), (28.54, 21.63), (21.65, 24.85), (2.56, 31.33), (20.55, 26.03), (15.72, 27.8), (23.03, 25.52), (13.97, 27.43), (41.78, 17.17), (30.9, 19.46), (35.79, 19.92), (7.45, 31.24), (27.1, 18.96), (30.55, 22.16), (29.96, 20.6), (22.92, 23.55), (19.98, 23.1), (37.41, 20.72), (5.7, 30.5), (9.84, 27.64), (27.65, 20.06), (16.27, 26.67), (3.13, 30.02), (33.21, 20.79), (20.52, 23.33), (10.22, 25.69), (27.59, 22.67), (15.2, 27.45), (15.31, 26.82), (28.11, 17.74), (9.87, 27.24), (25.34, 22.95), (36.49, 16.92), (22.12, 23.7), (23.65, 20.48), (24.9, 24.2), (17.62, 23.54), (8.41, 29.09), (25.18, 21.8), (16.78, 27.99), (28.41, 22.77), (20.07, 26.79), (18.01, 24.61), (19.56, 24.3), (28.12, 24.52), (22.7, 22.48), (8.68, 27.62), (15.75, 30.83), (8.38, 27.67), (41.44, 16.04), (14.04, 27.03), (12.31, 28.83), (23.08, 23.48), (34.67, 21.14), (30.59, 20.28), (22.75, 24.8), (38.96, 15.92), (27.24, 24.84), (32.8, 19.68), (27.12, 22.75), (22.46, 28.57), (29.67, 21.83), (22.33, 23.85), (13.98, 26.02), (15.37, 25.96), (27.64, 25.43), (15.69, 26.33), (40.3, 19.09), (42.26, 19.19), (15.59, 29.67), (21.46, 24.98), (31.75, 18.88), (20.91, 23.21), (-12.44, 37.51), (39.69, 19.27), (1.71, 30.55), (7.5, 32.63), (8.07, 29.23), (37.59, 16.83), (22.63, 23.78), (18.23, 26.19), (0.67, 31.44), (20.0, 26.56), (17.45, 23.99), (18.51, 24.3), (16.43, 25.33), (24.07, 22.88), (23.71, 25.81), (28.91, 20.99), (26.62, 24.41), (22.17, 27.42), (15.56, 26.22), (11.93, 30.6), (21.78, 25.77), (21.01, 23.91), (21.56, 25.25), (33.67, 18.99), (15.34, 26.04), (8.28, 31.58), (10.12, 31.94), (10.07, 29.31), (27.7, 24.04), (17.3, 21.23), (29.86, 21.53), (37.99, 18.33), (24.89, 22.9), (51.46, 13.33), (19.34, 23.87), (3.68, 31.02), (25.19, 23.43), (30.59, 20.13), (9.25, 31.26), (16.04, 26.08), (19.02, 22.32), (19.91, 26.81), (31.85, 18.8), (28.19, 23.59), (15.71, 23.71), (23.12, 23.27), (9.83, 28.49), (22.18, 25.27), (25.14, 20.78), (16.99, 28.09), (21.58, 26.41), (12.15, 27.73), (31.73, 19.37), (10.46, 30.69), (22.2, 23.11), (23.21, 18.53), (8.46, 30.26), (11.72, 26.77), (10.77, 27.25), (26.45, 21.86), (18.65, 25.12), (-0.17, 31.88), (16.5, 26.76), (-12.27, 41.8), (8.41, 29.26), (22.56, 26.09), (26.79, 21.18), (11.15, 28.67), (32.6, 17.53), (22.0, 23.37), (18.51, 29.12), (20.17, 24.75), (5.91, 29.98), (31.14, 22.3), (31.5, 20.67), (35.45, 22.78), (7.17, 32.87), (27.88, 19.36), (26.68, 21.63), (13.31, 30.89), (17.61, 26.34), (26.7, 17.98), (31.1, 19.42), (4.86, 33.0), (27.67, 22.8), (25.22, 25.08), (4.09, 31.2), (32.76, 19.67), (26.78, 23.83), (26.74, 23.59), (15.08, 27.2), (28.92, 21.04), (5.08, 29.33), (10.22, 32.73), (28.79, 22.24), (-0.28, 31.05), (28.87, 20.05), (8.76, 29.7), (29.02, 22.77), (27.91, 22.48), (33.57, 21.27), (33.98, 20.0), (43.39, 15.13), (11.31, 31.62), (21.56, 24.37), (13.32, 24.62), (-0.07, 34.8), (27.68, 22.54), (25.93, 23.3), (11.9, 26.1), (-1.76, 35.2), (37.91, 15.26), (38.58, 20.14), (37.99, 16.15), (9.85, 29.61), (10.5, 25.31), (30.22, 23.48), (39.94, 17.7), (17.4, 25.69), (23.68, 23.09), (10.4, 28.29), (16.37, 23.22), (9.6, 28.81), (24.19, 21.68), (31.46, 21.35), (14.02, 25.35), (36.03, 19.73), (4.39, 27.57), (34.35, 18.74), (41.66, 13.16), (23.0, 22.48), (16.76, 29.38), (15.72, 24.47), (11.85, 28.22), (14.44, 27.68), (0.37, 34.86), (14.15, 28.59), (32.77, 22.79), (18.65, 29.5), (36.52, 17.57), (29.05, 20.65), (5.62, 29.46), (22.08, 25.5), (22.59, 21.78), (36.03, 18.01), (22.09, 26.89), (23.28, 22.33), (22.01, 24.28), (23.64, 25.88), (16.89, 25.93), (19.69, 28.12), (-6.08, 38.83), (19.88, 18.79), (24.22, 23.76), (10.17, 28.24), (28.16, 23.24), (16.32, 22.94), (19.6, 24.05), (22.5, 23.97), (13.42, 26.92), (14.51, 30.02), (11.04, 28.92), (25.43, 22.99), (31.88, 20.59), (7.51, 31.23), (17.81, 27.18), (20.07, 27.07), (26.0, 21.84), (17.41, 23.82), (16.18, 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(25.15, 23.39), (5.94, 28.08), (17.67, 28.16), (21.89, 20.15), (24.69, 21.34), (-0.85, 36.35), (20.96, 24.08), (20.96, 26.2), (12.04, 28.54), (19.98, 25.49), (13.17, 29.66), (19.61, 27.35), (3.26, 29.62), (19.05, 27.31), (19.89, 23.08), (13.19, 24.84), (10.25, 29.62), (0.77, 33.2), (28.08, 25.47), (11.62, 31.18), (16.68, 30.11), (7.57, 30.86), (28.25, 21.33), (17.17, 30.95), (28.13, 20.8), (16.14, 24.21), (23.25, 20.82), (27.6, 23.66), (16.53, 28.59), (19.96, 24.01), (28.3, 19.43), (12.24, 29.13), (22.83, 21.48), (34.51, 20.77), (30.66, 23.37), (26.67, 22.12), (-0.03, 33.65), (19.41, 23.72), (24.75, 23.12), (17.99, 30.59), (25.28, 20.91), (34.3, 18.15), (20.35, 23.49), (20.22, 25.25), (7.1, 32.35), (21.97, 24.9), (15.23, 24.36), (21.14, 23.02), (20.07, 26.62), (36.27, 19.8), (7.45, 26.59), (26.9, 21.8), (12.88, 27.91), (7.86, 30.13), (34.95, 20.25), (34.73, 18.95), (21.78, 20.75), (33.8, 21.33), (27.69, 22.85), (15.14, 28.91), (16.28, 27.02), (24.04, 22.46), (14.31, 28.85), (38.63, 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28.95), (27.69, 21.72), (36.29, 17.96), (28.59, 21.3), (27.48, 17.81), (12.86, 28.65), (15.58, 25.28), (28.7, 20.6), (33.24, 20.05), (13.76, 28.33), (28.71, 20.53), (2.57, 27.64), (7.69, 30.26), (13.38, 24.44), (23.74, 24.81), (-10.95, 39.06), (20.14, 22.92), (18.16, 25.37), (21.33, 23.78), (23.45, 22.88), (17.1, 24.62), (15.62, 27.18), (36.68, 17.55), (19.52, 24.57), (17.95, 22.88), (14.56, 24.81), (20.54, 26.24), (28.47, 21.87), (34.58, 21.42), (15.08, 28.61), (22.16, 24.37), (15.74, 27.02), (19.49, 26.47), (29.09, 23.48), (17.38, 24.93), (7.24, 32.16), (16.9, 19.31), (33.07, 23.66), (41.81, 17.22), (15.8, 25.89), (14.91, 26.87), (22.3, 23.03), (19.88, 24.17), (40.52, 11.97), (38.31, 14.09), (23.84, 27.36), (9.36, 27.82), (22.74, 21.25), (25.12, 24.99), (26.27, 20.51), (41.03, 14.63), (20.45, 21.13), (26.23, 25.44), (30.61, 19.49), (10.02, 25.38), (26.54, 20.52), (17.14, 23.95), (26.28, 26.46), (18.13, 25.55), (37.26, 19.18), (39.62, 14.07), (8.41, 27.5), (29.88, 23.1), (34.3, 23.18), (14.48, 26.0), (30.18, 19.26), (14.25, 28.0), (18.34, 24.35), (27.05, 18.6), (18.68, 23.82), (13.35, 27.01), (16.94, 28.14), (36.39, 18.43), (25.21, 21.41), (30.37, 18.41), (30.83, 18.41), (17.03, 26.87), (16.41, 22.13), (22.37, 24.87), (13.64, 25.65), (4.85, 30.66), (43.27, 14.45), (18.26, 24.06), (25.6, 23.14), (35.59, 19.21), (26.9, 23.18), (24.87, 26.12), (16.92, 22.92), (17.17, 28.6), (23.4, 18.63), (16.08, 24.99), (18.25, 24.95), (16.73, 27.76), (15.11, 25.01), (14.55, 25.64), (25.18, 24.12), (33.24, 17.81), (31.64, 18.69), (26.84, 22.44), (16.04, 23.68), (20.03, 27.78), (31.33, 20.22), (24.04, 21.45), (20.4, 25.84), (33.15, 15.33), (38.19, 17.15), (11.81, 26.34), (29.65, 19.95), (26.66, 23.37), (27.18, 21.5), (21.33, 26.05), (19.84, 26.59), (17.22, 22.19), (17.78, 27.45), (23.02, 25.64), (2.36, 31.11), (-15.46, 39.04), (16.88, 26.31), (30.44, 19.21), (28.77, 17.56), (14.01, 29.35), (4.94, 31.44), (9.37, 31.07), (26.71, 25.46), (7.55, 30.08), (18.59, 22.44), (11.93, 32.77), (3.15, 34.37), (-2.58, 34.57), (9.84, 31.93), (7.52, 32.3), (38.45, 21.04), (7.18, 30.64), (21.97, 26.18), (9.89, 29.43), (26.66, 24.21), (13.35, 31.35), (16.84, 28.61), (22.46, 21.44), (29.34, 22.97), (32.74, 20.45), (8.27, 32.29), (28.48, 26.45), (29.27, 24.61), (35.05, 20.97), (14.05, 23.89), (23.18, 23.36), (35.0, 17.97), (8.31, 36.05), (31.4, 20.93), (7.93, 30.69), (14.36, 24.18), (33.35, 17.88), (19.31, 25.33), (34.68, 15.54), (30.18, 26.1), (31.95, 19.44), (26.32, 17.6), (25.35, 22.36), (18.23, 24.92), (5.96, 27.53), (10.05, 28.66), (5.56, 29.3), (19.6, 26.32), (14.9, 24.5), (29.03, 22.41), (31.16, 19.04), (11.01, 28.28), (7.87, 28.89), (16.24, 25.5), (23.99, 24.01), (46.09, 14.84), (26.65, 20.53), (27.09, 22.56), (24.19, 21.48), (12.6, 28.2), (30.43, 22.13), (25.8, 20.92), (19.89, 27.85), (28.45, 21.21), (23.48, 23.62), (35.78, 18.9), (35.18, 17.24), (5.11, 27.94)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.6740，ȳ=24.7403；Sxx=Σ(x−x̄)²=90851.99，Sxy=-35721.89；
β₁ = Sxy/Sxx = -0.3932；β₀ = ȳ−β₁x̄ = 32.8691；
R² = SSR/SST = 14045.41/17934.78 = 0.7831；F = (SSR/1)/(SSE/944) = 3409.0015，p = 1.40e-315；
SE(β₁)=√(4.12/90851.99)=0.0067；t=β₁/SE=-58.3867；p=1.40e-315；95%CI=[-0.5185,-0.2678]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.6740，ȳ=24.7403；Sxx=Σ(x−x̄)²=90851.99，Sxy=-35721.89；
β₁ = Sxy/Sxx = -0.3932；β₀ = ȳ−β₁x̄ = 32.8691；
R² = SSR/SST = 14045.41/17934.78 = 0.7831；F = (SSR/1)/(SSE/944) = 3409.0015，p = 1.40e-315；
SE(β₁)=√(4.12/90851.99)=0.0067；t=β₁/SE=-58.3867；p=1.40e-315；95%CI=[-0.5185,-0.2678]。
③ 计算结果：
y = 32.8691 + -0.3932·x；R²=0.7831；β₁ t=-58.3867 p=1.40e-315；95%CI=[-0.5185,-0.2678]
④ 结论：在 α=0.05 水平下，p=1.40e-315 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-0.3932），打折损利润。
案例 63. 污染预测外出时长
简单线性回归污染重少外出
① 原始数据：
样本 n=954，逐条 (x, y)：[(81.68, 3.01), (112.05, 2.67), (63.11, 3.63), (49.77, 3.51), (68.99, 3.52), (105.31, 2.75), (77.4, 2.94), (103.26, 2.66), (127.15, 2.22), (70.63, 3.25), (39.56, 3.62), (58.24, 3.32), (7.27, 4.6), (112.72, 2.37), (37.44, 3.89), (166.46, 0.9), (93.92, 2.92), (53.23, 3.47), (58.78, 3.36), (71.01, 3.05), (103.75, 2.26), (40.65, 3.7), (22.59, 3.88), (76.48, 3.16), (39.01, 4.47), (141.18, 1.71), (75.24, 3.26), (56.79, 3.46), (106.76, 2.47), (87.55, 2.8), (124.53, 2.06), (31.59, 3.63), (163.6, 1.42), (53.26, 3.33), (98.59, 2.7), (113.36, 2.54), (52.53, 3.26), (95.98, 2.6), (115.99, 2.42), (101.91, 2.78), (96.43, 2.72), (23.88, 3.99), (59.89, 3.5), (144.2, 1.84), (95.38, 3.04), (123.68, 2.13), (133.19, 1.96), (49.19, 3.44), (73.64, 3.05), (86.36, 2.6), (15.8, 4.16), (67.64, 3.38), (115.73, 2.68), (106.46, 2.24), (87.86, 3.24), (91.28, 2.88), (63.89, 3.7), (81.44, 2.63), (55.94, 3.41), (51.66, 3.17), (44.69, 4.2), (29.58, 3.78), (4.26, 4.41), (113.35, 2.81), (67.64, 2.9), (56.99, 2.98), (111.75, 2.53), (5.01, 4.07), (-34.3, 5.42), (69.4, 3.18), (44.32, 3.97), (105.05, 2.57), (15.78, 4.32), (87.32, 3.15), (50.28, 3.37), (38.29, 3.68), (-11.82, 4.62), (88.34, 2.99), (75.03, 2.82), (79.66, 3.06), (101.11, 2.3), (113.79, 2.11), (21.86, 4.3), (59.91, 3.48), (101.73, 2.52), (61.87, 3.7), (90.74, 2.79), (83.27, 3.0), (116.71, 2.17), (99.38, 2.63), (162.0, 1.6), (102.42, 2.43), (120.87, 2.55), (83.43, 3.09), (88.58, 2.5), (88.13, 2.84), (99.91, 2.66), (46.5, 3.61), (97.91, 2.81), (110.18, 2.44), (114.68, 2.44), (32.4, 4.16), (150.02, 1.65), (149.0, 1.82), (75.97, 3.04), (37.64, 3.78), (130.65, 1.83), (96.85, 2.96), (68.87, 3.04), (107.55, 2.54), (98.31, 2.47), (91.14, 2.8), (102.48, 2.43), (78.31, 3.14), (79.51, 2.86), (-24.06, 5.13), (46.3, 3.73), (92.35, 3.14), (65.88, 3.11), (107.65, 2.65), (29.93, 3.82), (98.13, 2.64), (66.04, 3.05), (36.32, 3.81), (83.3, 3.03), (-0.19, 4.55), (37.43, 4.0), (130.86, 1.63), (90.85, 2.79), (21.41, 4.58), (60.18, 3.19), (76.9, 3.12), (148.37, 1.74), (13.58, 4.14), (34.01, 3.8), (77.49, 3.03), (74.74, 2.78), (17.38, 4.32), (79.69, 3.05), (116.8, 2.63), (77.81, 2.99), (95.24, 2.86), (124.64, 2.16), (137.96, 2.1), (125.77, 2.16), (153.82, 1.6), (13.28, 4.51), (74.17, 3.2), (59.56, 3.39), (94.81, 2.57), (152.54, 1.23), (57.65, 3.19), (129.38, 1.87), (54.22, 3.64), (72.91, 2.82), (66.87, 3.3), (30.41, 3.95), (80.73, 2.71), (47.73, 3.68), (77.25, 2.85), (62.29, 3.27), (72.99, 3.25), (80.43, 3.03), (90.4, 2.74), (92.91, 2.89), (-48.1, 5.51), (126.77, 1.74), (65.86, 3.08), (51.9, 3.45), (28.98, 3.95), (86.65, 2.49), (62.83, 3.75), (-9.09, 4.44), (4.43, 4.92), (70.79, 3.26), (96.66, 2.48), (116.28, 2.35), (110.18, 2.39), (89.29, 2.99), (130.25, 1.36), (79.78, 3.13), (7.67, 4.87), (62.29, 3.18), (126.83, 1.85), (41.7, 3.79), (36.43, 3.91), (141.99, 1.66), (135.34, 1.48), (90.11, 2.9), (82.76, 3.18), (106.27, 2.59), (87.61, 3.09), (75.81, 3.11), (76.33, 2.91), (47.95, 4.35), (84.1, 2.77), (97.56, 2.93), (88.95, 3.03), (97.07, 2.69), (24.4, 3.99), (74.11, 3.27), (29.87, 3.68), (50.62, 3.67), (41.22, 3.99), (75.39, 3.16), (66.41, 3.29), (93.19, 2.78), (98.36, 2.7), (61.57, 3.26), (50.35, 3.62), (124.51, 2.3), (63.29, 3.6), (51.33, 3.69), (84.61, 3.03), (36.48, 4.09), (34.6, 4.02), (82.88, 3.12), (61.03, 3.35), (24.39, 4.08), (151.47, 1.34), (15.75, 4.51), (69.22, 2.88), (104.68, 2.77), (112.12, 2.51), (67.27, 2.93), (60.37, 3.42), (104.04, 2.28), (105.6, 2.52), (97.14, 2.64), (96.83, 2.89), (53.43, 3.5), (44.38, 3.61), (91.16, 2.89), (102.78, 2.42), (96.03, 2.33), (16.89, 3.89), (49.65, 3.47), (116.68, 2.32), (50.01, 3.52), (57.82, 3.53), (29.02, 4.21), (46.12, 3.36), (75.33, 3.17), (46.03, 3.57), (106.78, 2.28), (67.62, 3.21), (48.53, 4.06), (40.46, 4.11), (79.84, 3.1), (105.35, 2.27), (43.33, 3.4), (139.91, 1.79), (156.08, 1.95), (88.23, 2.84), (38.3, 4.07), (8.75, 3.8), (104.82, 2.29), (9.86, 4.78), (74.74, 3.08), (51.82, 3.42), (36.42, 3.73), (103.94, 2.47), (61.03, 3.25), (96.87, 2.58), (33.26, 3.65), (87.52, 2.82), (45.42, 4.09), (29.67, 3.92), (90.98, 2.61), (58.11, 3.7), (68.42, 3.32), (92.21, 2.66), (137.73, 2.04), (162.57, 1.18), (32.98, 4.06), (153.74, 1.36), (97.06, 2.87), (65.58, 3.42), (71.12, 3.53), (102.75, 2.45), (37.93, 3.96), (-6.44, 4.86), (101.96, 2.62), (71.63, 3.4), (84.07, 2.84), (96.69, 2.91), (97.63, 2.21), (63.09, 3.23), (98.01, 2.23), (56.31, 3.62), (56.53, 3.63), (78.77, 2.89), (103.66, 2.64), (14.09, 4.37), (140.43, 1.6), (26.46, 3.7), (98.92, 2.66), (104.44, 2.72), (25.26, 4.34), (48.56, 3.05), (72.79, 3.33), (147.11, 1.77), (43.72, 3.46), (124.93, 2.19), (39.65, 4.09), (185.5, 0.72), (30.98, 4.01), (110.08, 2.64), (150.5, 1.59), (89.09, 2.81), (88.23, 2.56), (111.49, 2.53), (114.59, 2.15), (27.16, 4.32), (128.65, 1.81), (85.84, 2.95), (78.2, 3.05), (21.82, 4.11), (11.98, 4.12), (54.48, 3.67), (131.97, 2.15), (28.64, 4.01), (70.59, 2.93), (76.06, 3.13), (114.47, 2.45), (67.19, 3.11), (90.84, 2.4), (66.86, 3.43), (98.93, 2.44), (55.86, 3.54), (66.74, 3.02), (131.26, 1.89), (81.62, 2.79), (125.63, 2.43), (43.11, 3.78), (141.44, 1.97), (120.48, 2.06), (131.29, 1.68), (84.92, 2.91), (129.81, 2.01), (87.67, 2.94), (117.67, 2.24), (70.51, 3.05), (114.52, 2.1), (68.57, 2.84), (92.55, 2.78), (112.39, 2.21), (30.95, 4.21), (58.72, 3.63), (77.49, 2.47), (56.22, 3.47), (64.95, 3.41), (93.9, 2.44), (53.09, 3.74), (41.68, 4.18), (106.13, 2.35), (78.87, 3.07), (57.01, 3.73), (134.42, 1.88), (83.93, 2.93), (136.16, 2.2), (152.14, 1.66), (92.82, 2.9), (106.43, 2.12), (67.08, 3.21), (94.84, 2.51), (145.14, 1.61), (-14.62, 4.74), (17.98, 3.99), (30.31, 3.98), (-21.69, 4.88), (44.55, 3.68), (89.94, 2.87), (89.03, 3.22), (38.13, 4.15), (127.39, 1.8), (25.74, 3.95), (71.53, 3.02), (87.05, 3.08), (55.8, 3.32), (55.6, 3.56), (69.01, 3.22), (44.4, 3.7), (92.09, 3.03), (-0.57, 4.48), (147.4, 1.44), (47.67, 3.84), (24.11, 3.95), (65.56, 3.32), (64.54, 3.1), (54.34, 3.53), (54.51, 3.48), (48.13, 3.5), (101.72, 2.63), (50.06, 3.63), (80.01, 3.02), (53.27, 3.32), (58.68, 3.57), (64.5, 3.4), (60.68, 3.73), (93.08, 2.58), (112.46, 2.38), (95.39, 2.82), (132.56, 1.76), (28.02, 3.57), (117.78, 1.87), (110.22, 2.48), (113.4, 1.88), (93.89, 2.68), (85.15, 2.66), (32.28, 3.92), (108.7, 2.53), (107.2, 2.27), (104.6, 2.94), (70.38, 3.27), (67.23, 3.0), (141.05, 1.67), (90.99, 2.86), (65.72, 3.18), (84.29, 2.89), (70.34, 3.01), (85.88, 2.73), (107.73, 2.63), (-11.57, 5.0), (55.38, 3.68), (106.26, 2.66), (34.87, 3.7), (8.07, 4.83), (99.36, 2.66), (73.52, 3.29), (78.78, 2.86), (93.6, 3.03), (39.41, 3.95), (69.39, 2.9), (140.06, 1.95), (132.89, 2.39), (-3.9, 4.9), (64.5, 3.26), (70.84, 3.11), (80.14, 2.85), (75.16, 3.1), (65.43, 3.51), (56.11, 3.69), (65.8, 3.28), (6.65, 4.36), (143.37, 1.42), (111.11, 2.46), (108.44, 2.27), (56.63, 3.45), (65.45, 3.43), (67.61, 3.31), (93.24, 2.58), (63.3, 3.19), (156.16, 1.43), (8.66, 4.09), (96.9, 2.32), (17.11, 4.34), (92.82, 2.87), (75.08, 3.09), (82.72, 2.67), (47.86, 3.62), (117.38, 2.3), (84.06, 2.91), (132.16, 1.85), (101.84, 2.25), (84.57, 2.84), (98.31, 2.45), (93.45, 2.37), (112.71, 2.39), (33.86, 3.82), (28.79, 4.15), (102.91, 2.35), (139.21, 2.03), (26.26, 3.98), (83.94, 3.29), (9.34, 4.12), (55.53, 3.27), (51.24, 3.59), (70.14, 3.1), (94.09, 2.35), (78.71, 3.04), (57.62, 3.76), (23.48, 4.13), (65.59, 3.18), (11.26, 4.2), (83.11, 3.18), (47.61, 3.8), (144.13, 1.85), (2.31, 4.33), (131.2, 2.0), (122.11, 2.22), (87.37, 3.02), (109.21, 2.52), (30.32, 4.12), (35.6, 3.87), (75.9, 3.25), (85.01, 2.63), (93.82, 2.82), (180.31, 1.38), (131.68, 1.55), (70.75, 3.08), (99.07, 2.49), (73.61, 2.92), (76.03, 2.82), (63.61, 3.45), (84.64, 3.1), (93.65, 2.55), (41.76, 3.49), (8.09, 4.58), (57.62, 3.1), (99.96, 2.72), (68.09, 3.39), (10.26, 4.52), (44.24, 3.78), (71.21, 3.16), (65.04, 3.02), (-28.35, 5.12), (119.64, 2.55), (100.44, 2.89), (94.82, 3.01), (79.92, 2.44), (110.71, 2.63), (45.22, 3.99), (33.11, 3.84), (32.39, 4.08), (92.33, 2.61), (55.73, 3.34), (38.74, 3.77), (80.54, 3.32), (97.07, 2.79), (144.24, 1.61), (113.52, 2.71), (131.34, 2.01), (55.88, 3.55), (44.94, 3.55), (17.12, 4.26), (60.92, 3.34), (107.82, 2.85), (32.81, 4.19), (97.22, 2.56), (66.41, 3.05), (-7.76, 4.41), (138.57, 2.03), (23.12, 4.23), (46.28, 3.36), (35.26, 4.17), (43.27, 3.55), (87.31, 3.17), (123.52, 2.43), (116.37, 2.43), (77.41, 3.11), (126.69, 1.8), (175.74, 1.25), (-17.1, 5.31), (32.97, 4.02), (138.31, 1.77), (127.06, 2.33), (45.01, 3.45), (69.73, 3.1), (62.91, 3.28), (25.91, 4.15), (71.86, 2.95), (154.85, 1.44), (84.09, 3.1), (108.02, 2.21), (105.8, 2.32), (40.89, 3.82), (127.04, 2.02), (83.67, 3.19), (165.14, 1.57), (115.73, 2.59), (115.01, 2.45), (77.72, 2.82), (36.26, 4.15), (107.53, 2.55), (122.97, 1.82), (165.71, 1.02), (10.77, 4.34), (85.35, 3.06), (39.35, 3.71), (124.1, 1.86), (73.93, 3.12), (102.14, 2.85), (45.15, 3.75), (141.65, 2.15), (112.45, 2.28), (65.47, 3.16), (42.7, 3.78), (105.93, 2.27), (163.93, 1.38), (115.52, 2.05), (25.0, 4.28), (18.2, 4.14), (62.72, 3.41), (-25.07, 4.88), (-13.69, 4.86), (52.59, 3.47), (131.51, 1.79), (78.48, 2.59), (50.81, 3.54), (86.76, 2.71), (91.26, 2.76), (77.42, 2.98), (167.09, 1.5), (123.45, 2.62), (82.3, 2.82), (116.52, 2.34), (108.35, 2.69), (34.37, 4.14), (72.34, 3.38), (25.77, 4.2), (94.46, 2.81), (71.19, 3.34), (155.96, 1.52), (104.04, 2.53), (52.4, 3.56), (62.24, 3.53), (125.98, 2.18), (136.47, 1.9), (37.19, 3.93), (142.32, 1.63), (77.24, 3.23), (-36.31, 5.45), (88.34, 2.95), (71.47, 3.39), (-4.49, 4.45), (121.44, 2.26), (78.04, 3.37), (60.41, 3.22), (122.5, 2.3), (71.59, 3.1), (168.21, 1.45), (108.07, 2.47), (134.74, 2.06), (120.72, 2.14), (77.13, 2.9), (154.19, 1.64), (110.26, 2.39), (92.43, 2.95), (184.65, 0.93), (96.69, 2.5), (74.49, 3.02), (109.4, 2.43), (16.96, 3.96), (51.13, 3.39), (78.3, 2.95), (138.55, 1.77), (135.92, 2.26), (99.34, 1.98), (48.07, 3.75), (96.32, 2.54), (124.43, 2.0), (133.47, 2.16), (39.06, 3.79), (133.33, 1.66), (121.2, 2.58), (47.39, 3.6), (131.63, 2.23), (118.64, 2.08), (104.1, 2.29), (149.02, 1.29), (135.95, 2.02), (39.46, 3.9), (134.42, 1.79), (96.49, 3.03), (95.89, 2.88), (91.47, 3.14), (133.51, 1.76), (96.03, 2.23), (90.82, 3.03), (81.39, 2.89), (81.07, 3.32), (121.29, 1.98), (63.73, 3.12), (97.46, 2.65), (57.69, 3.45), (114.94, 2.54), (84.14, 2.92), (-0.25, 4.68), (71.24, 3.19), (122.57, 2.3), (49.03, 3.6), (67.13, 3.07), (63.97, 3.12), (61.1, 3.65), (24.15, 4.01), (91.98, 2.8), (72.92, 3.21), (122.77, 2.09), (62.58, 3.51), (-17.15, 4.93), (-9.15, 5.26), (148.36, 1.46), (103.92, 2.14), (108.56, 2.23), (43.25, 3.73), (80.36, 3.23), (38.99, 3.35), (83.51, 2.56), (94.79, 2.42), (138.37, 2.0), (73.73, 2.73), (116.04, 2.33), (79.23, 2.8), (85.84, 2.78), (125.58, 1.63), (156.0, 1.31), (82.49, 2.96), (81.17, 2.99), (85.5, 2.79), (68.23, 2.97), (115.45, 2.22), (82.17, 2.81), (104.34, 2.28), (176.84, 1.04), (157.17, 1.31), (104.2, 2.61), (125.15, 1.96), (79.07, 2.95), (81.23, 3.09), (85.87, 2.82), (176.28, 1.23), (88.33, 2.66), (107.54, 2.42), (95.39, 2.57), (126.21, 2.0), (83.28, 2.82), (27.68, 3.7), (75.76, 3.2), (51.86, 3.6), (48.01, 3.86), (89.15, 3.16), (102.53, 2.43), (54.19, 3.57), (27.66, 3.99), (80.56, 3.1), (136.86, 2.04), (107.74, 2.6), (103.03, 2.64), (39.22, 3.96), (78.11, 2.98), (33.84, 4.04), (122.67, 2.44), (19.57, 4.31), (119.09, 2.74), (141.41, 1.86), (36.69, 3.78), (117.43, 2.32), (140.44, 1.97), (41.5, 3.94), (99.78, 2.78), (21.19, 4.32), (47.1, 4.0), (76.95, 3.4), (78.98, 3.0), (54.45, 3.63), (145.23, 1.57), (102.31, 2.27), (109.35, 2.59), (72.09, 3.16), (88.19, 2.79), (66.95, 3.23), (104.83, 2.56), (38.55, 3.81), (94.73, 2.75), (80.37, 2.75), (79.05, 2.99), (51.48, 3.8), (125.12, 1.69), (87.6, 2.52), (101.39, 2.54), (95.12, 2.55), (97.0, 2.62), (147.92, 1.88), (89.91, 2.57), (173.68, 1.21), (15.33, 3.99), (112.22, 2.39), (72.67, 2.95), (48.79, 3.95), (21.3, 3.92), (142.6, 1.69), (40.19, 3.6), (34.23, 3.92), (47.49, 3.4), (60.06, 3.52), (88.49, 2.78), (136.9, 1.73), (73.49, 2.84), (88.41, 2.79), (83.01, 2.76), (76.79, 3.17), (155.14, 1.51), (70.44, 3.31), (111.3, 2.53), (84.59, 2.75), (71.06, 3.38), (68.18, 3.38), (52.42, 3.6), (82.87, 3.47), (122.37, 2.0), (61.03, 3.49), (13.9, 4.05), (90.89, 3.11), (79.84, 2.63), (169.92, 1.37), (-0.14, 4.61), (93.57, 2.8), (126.21, 2.15), (96.22, 2.5), (29.86, 4.14), (87.77, 2.89), (48.6, 3.62), (124.78, 1.93), (76.65, 3.54), (73.05, 3.2), (78.44, 3.16), (90.5, 2.97), (18.05, 4.02), (65.36, 3.6), (-25.71, 4.94), (100.1, 2.76), (89.25, 2.86), (174.63, 1.3), (46.75, 4.12), (142.82, 1.72), (64.6, 3.48), (150.47, 2.12), (77.21, 3.13), (111.12, 2.61), (149.9, 1.25), (67.28, 3.52), (57.55, 3.82), (77.39, 3.01), (37.97, 3.93), (198.58, 1.22), (58.29, 3.46), (117.83, 2.02), (0.39, 4.34), (67.73, 3.31), (123.65, 1.86), (28.14, 3.79), (83.92, 2.75), (80.2, 3.17), (13.79, 4.08), (77.95, 3.04), (34.37, 3.79), (45.17, 4.02), (129.62, 1.89), (1.92, 4.85), (54.34, 3.45), (31.39, 4.15), (85.29, 2.88), (117.5, 2.22), (32.24, 4.38), (111.57, 2.45), (84.08, 2.87), (72.69, 3.52), (30.47, 3.82), (15.36, 3.98), (63.36, 3.41), (69.35, 3.13), (106.14, 2.44), (83.07, 3.19), (130.49, 1.97), (50.61, 3.32), (117.93, 2.41), (79.38, 3.26), (7.02, 4.29), (127.96, 1.94), (21.47, 4.55), (136.65, 1.8), (82.23, 3.09), (31.92, 4.08), (151.41, 1.53), (103.93, 2.14), (88.17, 2.82), (99.13, 2.82), (8.59, 4.53), (38.28, 3.74), (77.75, 3.02), (113.42, 2.39), (198.11, 0.46), (138.76, 2.01), (108.11, 2.53), (64.36, 3.11), (68.25, 3.21), (70.67, 3.14), (109.26, 2.73), (20.97, 4.09), (77.67, 3.1), (26.97, 3.89), (62.84, 3.34), (63.72, 3.34), (124.3, 2.29), (12.47, 4.3), (80.75, 2.9), (83.52, 2.89), (26.12, 4.13), (82.15, 2.62), (72.21, 3.35), (186.09, 0.9), (121.48, 2.21), (129.66, 2.04), (-12.13, 4.67), (99.51, 2.51), (80.56, 2.85), (51.62, 3.2), (79.0, 3.02), (89.38, 2.81), (96.93, 2.52), (70.86, 3.34), (142.73, 1.62), (165.09, 1.62), (76.52, 2.96), (69.24, 3.43), (180.84, 1.04), (80.19, 3.02), (122.54, 1.82), (75.66, 3.25), (85.1, 2.95), (135.04, 1.57), (-5.92, 4.99), (69.42, 2.97), (143.95, 1.69), (93.23, 2.53), (97.11, 2.63), (83.15, 2.67), (42.24, 3.87), (106.15, 2.3), (52.7, 3.63), (141.71, 1.33), (50.84, 3.65), (107.79, 2.57), (35.66, 3.84), (105.66, 2.42), (82.59, 2.91), (140.58, 1.48), (-5.52, 4.64), (90.58, 2.55), (31.29, 4.01), (95.41, 2.47), (55.36, 3.39), (40.91, 4.12), (62.91, 3.22), (8.17, 4.5), (92.91, 2.81), (127.36, 1.89), (104.48, 2.34), (25.55, 3.93), (54.51, 3.73), (28.11, 3.9), (85.63, 2.75), (84.51, 3.23)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=79.8365，ȳ=3.0033；Sxx=Σ(x−x̄)²=1583602.03，Sxy=-31372.54；
β₁ = Sxy/Sxx = -0.0198；β₀ = ȳ−β₁x̄ = 4.5850；
R² = SSR/SST = 621.52/662.56 = 0.9381；F = (SSR/1)/(SSE/952) = 14417.6849，p = <1e-300；
SE(β₁)=√(0.04/1583602.03)=0.0002；t=β₁/SE=-120.0737；p=<1e-300；95%CI=[-0.0201,-0.0195]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=79.8365，ȳ=3.0033；Sxx=Σ(x−x̄)²=1583602.03，Sxy=-31372.54；
β₁ = Sxy/Sxx = -0.0198；β₀ = ȳ−β₁x̄ = 4.5850；
R² = SSR/SST = 621.52/662.56 = 0.9381；F = (SSR/1)/(SSE/952) = 14417.6849，p = <1e-300；
SE(β₁)=√(0.04/1583602.03)=0.0002；t=β₁/SE=-120.0737；p=<1e-300；95%CI=[-0.0201,-0.0195]。
③ 计算结果：
y = 4.5850 + -0.0198·x；R²=0.9381；β₁ t=-120.0737 p=<1e-300；95%CI=[-0.0201,-0.0195]
④ 结论：在 α=0.05 水平下，p=<1e-300 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-0.0198），污染重少外出。
案例 64. 价格预测销量
简单线性回归价高量低
① 原始数据：
样本 n=341，逐条 (x, y)：[(123.68, 443.48), (83.83, 631.9), (112.5, 456.91), (130.5, 415.62), (113.09, 477.62), (83.72, 521.79), (94.1, 435.51), (123.72, 404.61), (83.31, 570.44), (95.64, 473.99), (130.66, 387.94), (113.19, 467.03), (136.6, 310.76), (50.93, 703.48), (106.06, 456.76), (73.01, 696.52), (94.33, 576.51), (66.93, 646.46), (109.08, 449.41), (110.31, 500.32), (122.34, 413.48), (131.1, 409.14), (126.64, 392.77), (104.73, 433.05), (94.06, 489.9), (74.86, 549.91), (69.5, 620.6), (108.96, 459.86), (91.26, 518.7), (70.12, 622.79), (78.48, 633.23), (98.31, 538.8), (132.55, 366.71), (104.37, 490.08), (148.13, 315.3), (44.34, 672.93), (88.57, 634.74), (156.23, 229.25), (55.55, 712.87), (82.73, 657.71), (78.99, 574.04), (130.35, 363.57), (80.11, 577.0), (73.98, 580.62), (136.35, 338.33), (109.88, 394.54), (52.27, 683.71), (106.29, 420.7), (92.01, 588.27), (168.14, 201.56), (104.68, 538.66), (140.14, 266.21), (61.6, 720.2), (112.89, 480.25), (117.65, 424.02), (130.09, 407.6), (146.78, 326.98), (92.43, 510.33), (102.8, 488.92), (57.23, 760.15), (103.04, 476.08), (102.75, 458.69), (110.3, 495.72), (103.56, 479.51), (69.35, 634.75), (161.16, 233.63), (105.87, 481.15), (109.48, 483.84), (132.7, 295.27), (130.62, 432.54), (127.07, 355.23), (116.71, 432.68), (82.53, 565.71), (77.5, 542.04), (132.73, 373.5), (96.08, 484.93), (94.53, 458.42), (74.53, 604.09), (133.0, 394.18), (106.14, 458.83), (79.62, 690.97), (70.8, 587.1), (40.65, 757.44), (117.23, 420.6), (109.05, 499.39), (97.75, 525.49), (68.94, 613.05), (141.54, 332.39), (92.44, 479.54), (79.68, 535.23), (90.92, 545.13), (100.62, 566.27), (104.99, 474.4), (69.19, 570.99), (77.83, 613.2), (142.66, 310.84), (38.86, 724.85), (71.54, 550.89), (73.61, 620.33), (126.9, 401.49), (77.19, 572.78), (110.81, 512.5), (124.32, 392.38), (100.99, 423.39), (95.86, 553.81), (72.28, 589.32), (149.84, 305.52), (141.99, 304.93), (96.64, 557.28), (72.01, 609.98), (86.38, 577.83), (101.47, 504.21), (87.21, 581.28), (121.25, 383.72), (102.2, 527.9), (109.9, 429.9), (50.3, 725.42), (76.19, 584.9), (94.85, 477.54), (106.18, 440.8), (82.56, 578.01), (115.83, 429.12), (120.41, 427.04), (73.86, 653.07), (83.74, 552.44), (93.33, 549.93), (158.33, 304.27), (73.51, 544.99), (70.48, 593.76), (112.7, 396.11), (51.58, 710.88), (118.3, 442.93), (92.92, 515.62), (121.66, 475.42), (148.25, 296.52), (99.07, 549.72), (115.19, 381.63), (113.23, 539.33), (106.13, 392.19), (20.57, 864.02), (39.03, 703.22), (77.85, 600.02), (149.93, 365.12), (67.81, 570.83), (124.78, 486.66), (76.51, 668.72), (111.26, 451.14), (51.3, 678.58), (137.64, 418.14), (85.48, 550.74), (141.12, 356.87), (62.55, 643.96), (109.86, 465.89), (125.5, 385.63), (72.0, 547.12), (76.88, 658.54), (66.39, 610.28), (103.29, 487.35), (85.64, 511.33), (142.3, 309.34), (125.35, 394.12), (120.98, 427.84), (79.51, 595.36), (58.59, 631.16), (108.41, 466.97), (43.6, 723.18), (92.08, 503.78), (110.29, 451.31), (79.56, 637.33), (79.73, 601.65), (55.67, 704.04), (99.9, 372.78), (95.83, 501.0), (92.36, 501.27), (106.37, 456.1), (88.82, 556.75), (107.28, 517.15), (82.4, 581.2), (109.74, 527.6), (77.93, 627.95), (120.56, 345.66), (95.91, 484.76), (97.13, 486.12), (76.08, 574.86), (102.89, 426.66), (104.26, 426.71), (56.67, 679.25), (93.31, 512.31), (126.88, 403.17), (111.45, 457.81), (100.4, 602.58), (120.76, 342.93), (90.78, 546.82), (74.14, 682.78), (134.95, 378.75), (139.21, 343.3), (151.0, 354.84), (45.09, 749.42), (106.29, 526.21), (147.99, 334.73), (50.91, 692.76), (89.71, 504.9), (139.67, 368.16), (117.0, 452.78), (91.61, 468.83), (95.23, 485.04), (137.6, 344.82), (41.93, 746.34), (137.29, 340.05), (107.04, 437.4), (81.78, 564.87), (118.03, 417.95), (97.1, 432.18), (121.08, 392.25), (70.76, 612.79), (74.92, 605.32), (95.36, 505.38), (153.26, 334.06), (81.14, 505.7), (116.8, 466.0), (95.09, 507.75), (82.17, 557.99), (100.19, 527.61), (73.97, 624.77), (112.12, 500.5), (91.81, 534.67), (142.65, 317.27), (59.47, 628.14), (128.75, 435.92), (52.71, 669.04), (149.64, 244.58), (116.15, 467.17), (87.32, 545.94), (110.77, 458.44), (109.94, 480.1), (87.53, 556.55), (87.95, 510.67), (89.4, 469.55), (82.12, 524.36), (114.12, 385.23), (136.49, 319.77), (98.48, 552.11), (89.44, 539.25), (68.98, 553.52), (52.12, 643.87), (149.06, 402.91), (106.86, 500.11), (103.4, 443.26), (163.06, 331.27), (85.7, 613.81), (113.12, 386.08), (129.43, 388.15), (72.99, 599.42), (123.85, 380.68), (87.79, 532.59), (133.07, 378.06), (67.71, 679.81), (62.94, 623.6), (101.98, 513.99), (111.02, 492.85), (155.2, 309.6), (62.16, 668.91), (139.24, 353.16), (89.55, 516.67), (96.75, 433.64), (81.32, 552.07), (145.82, 363.29), (73.69, 686.56), (145.31, 309.3), (132.04, 368.0), (81.56, 580.25), (128.32, 340.47), (158.86, 280.3), (140.05, 369.43), (63.95, 603.95), (87.06, 524.43), (101.25, 558.94), (107.13, 547.07), (65.52, 595.8), (139.3, 335.38), (126.2, 387.08), (132.13, 402.74), (111.85, 446.18), (106.5, 468.92), (126.57, 393.86), (140.1, 365.08), (107.3, 499.79), (58.9, 627.68), (104.98, 452.69), (162.15, 268.59), (45.91, 779.89), (89.8, 496.84), (150.07, 259.69), (146.21, 272.05), (129.8, 399.52), (76.39, 614.48), (86.1, 594.03), (62.83, 632.41), (184.94, 160.5), (70.01, 612.56), (102.76, 454.6), (98.2, 483.36), (97.05, 402.59), (164.64, 203.77), (75.36, 547.33), (68.95, 601.86), (151.47, 260.05), (50.21, 674.99), (128.98, 390.06), (73.68, 641.52), (93.34, 524.43), (113.24, 475.77), (122.5, 383.35), (114.07, 469.4), (52.65, 762.39), (41.36, 747.06), (111.97, 373.86), (137.88, 391.52), (75.58, 576.03), (109.38, 422.99), (121.72, 376.66), (99.05, 453.22), (123.73, 396.5), (57.15, 670.6), (85.02, 562.88), (114.75, 441.13), (67.48, 610.39), (101.92, 468.35), (85.79, 518.11), (109.18, 454.58), (95.97, 497.66), (125.54, 342.14), (112.95, 430.76), (98.31, 495.91), (97.28, 492.54), (106.58, 418.78), (84.09, 580.61), (136.76, 372.7), (65.93, 584.74), (45.93, 668.55), (113.72, 482.03)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=100.6993，ȳ=495.5763；Sxx=Σ(x−x̄)²=284527.30，Sxy=-1120347.49；
β₁ = Sxy/Sxx = -3.9376；β₀ = ȳ−β₁x̄ = 892.0872；
R² = SSR/SST = 4411451.86/4951831.94 = 0.8909；F = (SSR/1)/(SSE/339) = 2767.4636，p = 3.90e-165；
SE(β₁)=√(1594.04/284527.30)=0.0748；t=β₁/SE=-52.6067；p=3.90e-165；95%CI=[-4.0848,-3.7903]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=100.6993，ȳ=495.5763；Sxx=Σ(x−x̄)²=284527.30，Sxy=-1120347.49；
β₁ = Sxy/Sxx = -3.9376；β₀ = ȳ−β₁x̄ = 892.0872；
R² = SSR/SST = 4411451.86/4951831.94 = 0.8909；F = (SSR/1)/(SSE/339) = 2767.4636，p = 3.90e-165；
SE(β₁)=√(1594.04/284527.30)=0.0748；t=β₁/SE=-52.6067；p=3.90e-165；95%CI=[-4.0848,-3.7903]。
③ 计算结果：
y = 892.0872 + -3.9376·x；R²=0.8909；β₁ t=-52.6067 p=3.90e-165；95%CI=[-4.0848,-3.7903]
④ 结论：在 α=0.05 水平下，p=3.90e-165 < 0.05，拒绝 H₀。x 对 y 的线性影响显著（斜率 β₁=-3.9376），价高量低。
案例 65. 鞋码预测数学分数
简单线性回归应无影响
① 原始数据：
样本 n=510，逐条 (x, y)：[(38.92, 61.04), (34.44, 65.73), (41.34, 77.96), (39.35, 69.64), (39.14, 93.56), (40.16, 91.83), (41.75, 55.79), (36.0, 68.84), (34.84, 79.58), (40.76, 85.63), (43.76, 72.18), (39.31, 68.47), (37.36, 77.52), (41.36, 75.06), (39.25, 68.65), (39.02, 83.49), (39.21, 75.05), (40.18, 84.9), (40.8, 62.62), (42.17, 87.63), (37.79, 97.37), (41.75, 70.41), (41.57, 75.8), (37.95, 71.2), (44.1, 57.12), (34.14, 99.86), (35.21, 86.06), (43.13, 57.36), (47.56, 81.76), (39.06, 70.77), (44.39, 61.93), (41.05, 69.36), (43.14, 62.17), (37.89, 78.09), (40.32, 89.82), (45.97, 72.34), (38.48, 68.65), (40.5, 72.48), (39.63, 77.03), (36.54, 61.69), (40.96, 81.69), (46.16, 62.54), (40.47, 71.4), (37.13, 76.17), (39.06, 69.37), (41.04, 59.51), (41.48, 90.58), (37.2, 62.94), (41.23, 88.54), (35.19, 63.21), (36.44, 58.29), (38.37, 72.18), (39.74, 80.34), (40.6, 85.85), (44.14, 85.06), (36.65, 68.07), (40.84, 84.03), (39.32, 72.4), (38.05, 77.61), (37.09, 76.04), (39.47, 81.56), (41.17, 69.85), (38.5, 81.49), (34.91, 74.98), (37.78, 82.27), (42.52, 84.53), (38.03, 81.1), (34.61, 95.07), (37.0, 85.74), (34.17, 73.72), (43.21, 70.4), (36.89, 71.56), (37.48, 64.22), (42.28, 66.19), (38.34, 56.95), (39.81, 73.99), (38.57, 61.95), (39.6, 78.14), (44.7, 58.6), (39.58, 73.55), (30.42, 72.6), (44.65, 70.59), (37.42, 65.33), (33.55, 92.15), (38.74, 72.74), (37.62, 50.14), (42.09, 54.49), (38.45, 72.73), (41.9, 77.72), (36.12, 71.57), (41.91, 66.35), (36.56, 101.77), (45.99, 67.26), (40.71, 75.17), (38.96, 76.8), (40.75, 60.27), (38.87, 61.62), (45.2, 80.87), (40.93, 73.47), (41.78, 84.11), (43.76, 78.2), (45.75, 82.28), (42.21, 67.99), (36.23, 63.8), (41.35, 92.94), (37.21, 81.35), (40.17, 76.78), (36.01, 62.7), (36.82, 79.82), (41.85, 87.46), (40.05, 65.8), (39.33, 74.95), (37.05, 87.54), (38.35, 80.58), (43.85, 88.46), (36.89, 69.98), (36.42, 96.97), (42.57, 81.71), (41.66, 57.28), (42.65, 76.81), (37.97, 80.5), (37.43, 77.98), (49.11, 79.56), (39.24, 67.12), (40.33, 70.07), (41.69, 60.46), (43.38, 69.8), (43.2, 71.2), (37.2, 77.77), (42.62, 80.46), (45.57, 86.43), (39.24, 60.91), (33.71, 64.41), (38.21, 68.78), (43.98, 68.3), (36.83, 65.07), (41.17, 67.78), (37.94, 69.56), (42.04, 85.02), (41.44, 69.82), (36.17, 76.9), (43.41, 67.29), (41.03, 90.81), (37.34, 92.27), (38.38, 75.15), (41.46, 91.76), (43.92, 82.85), (41.26, 67.31), (42.05, 89.77), (49.33, 92.33), (37.88, 84.75), (38.58, 90.84), (37.44, 82.46), (38.81, 80.43), (34.85, 79.38), (38.88, 61.33), (42.54, 53.47), (36.33, 62.58), (40.13, 71.73), (36.2, 64.97), (38.79, 102.86), (43.1, 86.56), (47.21, 84.42), (39.7, 78.63), (34.95, 70.14), (38.31, 82.07), (41.18, 58.71), (39.54, 77.87), (43.12, 74.13), (40.48, 74.37), (43.51, 99.24), (36.66, 69.31), (34.64, 90.13), (42.92, 65.92), (44.52, 75.29), (40.11, 77.58), (40.93, 80.38), (38.6, 65.29), (41.79, 72.12), (41.92, 71.35), (39.68, 81.15), (42.81, 85.91), (42.2, 76.01), (39.31, 95.05), (40.11, 67.91), (41.68, 73.23), (40.81, 79.67), (38.59, 86.58), (38.22, 65.3), (36.85, 77.21), (43.48, 78.13), (43.88, 91.74), (41.28, 58.16), (37.87, 48.0), (42.93, 86.69), (39.49, 80.5), (39.21, 64.44), (36.83, 81.52), (42.56, 82.82), (37.09, 56.69), (38.95, 89.24), (35.3, 69.39), (39.05, 67.56), (36.74, 79.71), (42.23, 80.18), (36.88, 61.02), (42.92, 80.76), (35.08, 79.94), (41.33, 45.64), (34.87, 78.64), (39.44, 75.78), (40.65, 69.2), (47.86, 75.93), (41.27, 66.27), (40.55, 81.5), (37.91, 70.33), (39.16, 59.33), (33.84, 85.2), (42.72, 75.17), (41.96, 57.91), (37.67, 68.71), (40.02, 74.61), (41.99, 69.58), (39.23, 66.77), (37.97, 79.12), (45.01, 74.82), (41.07, 76.24), (36.72, 61.14), (41.71, 82.07), (43.39, 74.65), (39.06, 87.09), (35.24, 74.03), (38.7, 82.05), (43.0, 84.7), (38.95, 73.67), (41.18, 92.72), (41.79, 53.25), (40.5, 89.89), (41.29, 90.23), (37.19, 79.65), (43.39, 90.62), (37.59, 52.12), (40.84, 67.57), (43.55, 79.44), (46.63, 75.64), (43.98, 63.13), (38.42, 93.11), (41.46, 78.28), (37.24, 79.26), (37.06, 61.25), (35.91, 81.33), (36.85, 72.28), (37.14, 93.89), (38.62, 60.36), (36.65, 68.4), (38.91, 54.76), (42.98, 72.42), (43.79, 72.71), (43.4, 66.75), (40.68, 71.18), (36.94, 87.43), (32.67, 61.52), (35.26, 79.29), (40.54, 98.98), (38.33, 100.89), (40.55, 80.9), (39.6, 77.14), (45.11, 70.78), (43.62, 80.92), (37.32, 103.64), (38.87, 91.22), (41.56, 62.37), (36.94, 77.57), (35.87, 97.42), (42.21, 65.07), (44.12, 58.54), (37.38, 60.55), (38.69, 74.61), (40.3, 56.49), (39.43, 96.65), (38.91, 67.41), (41.7, 63.11), (39.97, 79.16), (39.51, 80.35), (35.03, 86.31), (40.89, 87.22), (35.08, 84.59), (44.21, 73.47), (39.06, 45.76), (39.0, 69.19), (38.71, 85.09), (38.54, 71.66), (42.72, 86.41), (36.77, 97.56), (37.14, 63.54), (39.85, 53.62), (38.89, 54.77), (39.99, 71.81), (44.18, 67.48), (43.18, 70.0), (46.57, 57.18), (34.59, 57.32), (35.01, 64.37), (38.51, 62.86), (40.98, 68.82), (43.92, 68.08), (44.37, 87.04), (40.51, 83.25), (42.52, 78.95), (36.75, 85.78), (41.29, 43.69), (38.48, 76.17), (39.1, 85.77), (42.02, 66.94), (43.6, 58.46), (41.54, 98.66), (43.63, 72.56), (35.56, 73.47), (39.41, 77.12), (38.1, 65.45), (40.75, 54.31), (37.87, 102.1), (45.05, 73.64), (42.83, 68.07), (41.12, 87.99), (39.09, 48.2), (37.52, 55.62), (37.19, 69.98), (35.78, 65.05), (41.84, 59.39), (36.45, 81.42), (37.0, 97.67), (36.19, 75.26), (40.65, 91.57), (38.11, 61.49), (34.68, 56.51), (39.56, 59.3), (37.08, 80.49), (39.13, 66.64), (42.72, 87.72), (35.51, 64.51), (40.63, 56.44), (35.25, 72.09), (34.21, 63.49), (38.74, 73.11), (42.52, 76.35), (41.94, 59.89), (39.96, 75.32), (37.14, 66.97), (37.8, 61.26), (41.12, 77.85), (40.29, 79.59), (37.39, 77.76), (39.59, 84.2), (43.36, 75.73), (39.15, 65.46), (44.98, 74.94), (41.87, 63.66), (40.0, 76.53), (38.87, 78.55), (37.6, 55.97), (37.23, 66.17), (42.13, 68.08), (35.11, 79.31), (32.83, 72.72), (42.96, 60.69), (40.23, 85.52), (38.04, 69.31), (36.58, 62.31), (37.21, 96.47), (44.64, 103.06), (40.64, 72.77), (44.74, 53.12), (41.78, 62.85), (35.24, 63.79), (41.72, 67.94), (40.0, 80.24), (38.82, 80.16), (43.48, 65.53), (42.81, 90.95), (43.38, 75.22), (39.23, 73.47), (35.4, 80.16), (39.63, 73.84), (40.04, 73.61), (44.31, 75.93), (39.45, 70.61), (43.71, 76.22), (42.53, 66.2), (35.38, 55.92), (35.95, 67.94), (37.75, 59.67), (43.23, 64.44), (40.48, 68.14), (38.29, 79.26), (43.77, 65.3), (37.73, 69.1), (37.79, 83.53), (35.25, 80.99), (37.18, 66.1), (39.68, 77.38), (37.14, 79.36), (39.85, 72.46), (38.53, 80.06), (39.64, 84.77), (38.92, 66.09), (38.11, 65.44), (40.54, 67.49), (39.82, 81.83), (39.9, 67.74), (38.86, 73.99), (41.39, 54.94), (39.43, 71.68), (42.15, 66.73), (40.92, 68.83), (38.71, 79.78), (38.23, 63.43), (43.71, 55.54), (41.46, 58.69), (43.01, 92.97), (40.18, 64.22), (41.4, 70.47), (47.76, 71.64), (35.8, 61.74), (38.25, 84.65), (42.28, 78.66), (39.8, 77.39), (40.28, 69.51), (43.0, 57.93), (37.45, 57.38), (39.83, 82.12), (33.56, 66.79), (37.73, 85.07), (40.67, 84.25), (42.16, 48.59), (44.71, 78.22), (37.36, 71.14), (38.95, 71.88), (39.97, 72.11), (36.73, 65.83), (41.48, 92.93), (39.21, 81.06), (42.61, 57.38), (42.2, 75.59), (44.65, 63.61), (41.73, 75.09), (37.31, 74.36), (43.04, 79.05), (43.04, 74.17), (36.31, 67.01), (38.94, 67.48), (38.69, 67.0), (41.03, 72.24), (39.03, 86.45), (37.96, 62.48), (38.51, 67.1), (41.05, 79.61), (42.06, 74.49), (44.26, 71.11), (46.1, 88.52), (37.16, 47.49), (40.48, 90.69), (45.2, 60.78), (34.98, 63.25), (42.52, 79.69), (38.31, 60.99), (42.17, 84.7), (40.62, 58.13), (41.75, 71.78), (38.62, 69.82), (40.12, 77.29), (46.7, 78.44), (41.27, 58.94), (39.8, 56.98), (41.07, 66.05), (45.55, 75.99), (40.37, 59.48), (38.52, 79.96), (42.4, 71.63), (39.99, 75.53), (38.74, 94.44), (40.08, 96.62), (36.81, 72.3), (40.89, 54.96), (39.22, 62.79), (44.38, 83.25), (37.76, 59.91), (44.28, 78.57), (32.42, 66.19), (43.78, 106.1), (41.49, 78.55), (42.72, 57.35), (40.39, 58.03), (33.89, 92.48), (32.34, 72.8), (40.58, 58.64), (35.28, 90.72), (38.02, 67.72), (38.88, 82.85), (39.33, 58.88), (38.36, 91.39), (39.63, 58.98), (37.55, 86.31), (44.34, 57.87), (39.58, 85.89), (40.33, 77.73), (49.4, 96.87), (44.08, 79.61), (36.6, 65.02), (42.28, 86.17)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=39.9037，ȳ=73.8886；Sxx=Σ(x−x̄)²=4657.89，Sxy=308.05；
β₁ = Sxy/Sxx = 0.0661；β₀ = ȳ−β₁x̄ = 71.2496；
R² = SSR/SST = 20.37/67166.41 = 0.0003；F = (SSR/1)/(SSE/508) = 0.1541，p = 0.6948；
SE(β₁)=√(132.18/4657.89)=0.1685；t=β₁/SE=0.3926；p=0.6948；95%CI=[-0.2648,0.3971]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=39.9037，ȳ=73.8886；Sxx=Σ(x−x̄)²=4657.89，Sxy=308.05；
β₁ = Sxy/Sxx = 0.0661；β₀ = ȳ−β₁x̄ = 71.2496；
R² = SSR/SST = 20.37/67166.41 = 0.0003；F = (SSR/1)/(SSE/508) = 0.1541，p = 0.6948；
SE(β₁)=√(132.18/4657.89)=0.1685；t=β₁/SE=0.3926；p=0.6948；95%CI=[-0.2648,0.3971]。
③ 计算结果：
y = 71.2496 + 0.0661·x；R²=0.0003；β₁ t=0.3926 p=0.6948；95%CI=[-0.2648,0.3971]
④ 结论：在 α=0.05 水平下，p=0.6948 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=0.0661），应无影响。
案例 66. 生日月份预测身高
简单线性回归应无影响
① 原始数据：
样本 n=996，逐条 (x, y)：[(6.27, 188.37), (6.77, 172.62), (7.1, 176.58), (5.44, 160.84), (0.87, 161.3), (7.78, 157.48), (7.47, 156.52), (6.97, 183.74), (10.46, 162.66), (4.65, 160.09), (6.31, 159.89), (5.01, 186.36), (5.69, 171.33), (5.19, 166.02), (5.26, 167.1), (3.78, 169.74), (7.76, 178.66), (11.43, 159.59), (6.37, 162.56), (6.09, 167.48), (10.49, 179.84), (8.81, 151.77), (2.2, 156.82), (0.71, 170.44), (5.78, 176.46), (0.09, 168.45), (13.02, 160.89), (1.44, 163.05), (6.03, 162.53), (7.98, 158.29), (4.3, 171.33), (6.31, 165.87), (3.36, 168.45), (8.58, 164.56), (0.89, 171.9), (7.15, 164.11), (2.36, 167.39), (7.31, 153.36), (13.39, 170.9), (10.48, 183.9), (0.09, 157.72), (5.82, 176.24), (2.26, 150.9), (6.04, 167.65), (6.43, 171.29), (10.89, 179.42), (-0.7, 177.16), (0.82, 172.89), (6.67, 159.17), (1.8, 156.88), (1.97, 165.83), (6.49, 179.93), (8.0, 154.06), (5.84, 168.35), (2.05, 170.57), (7.32, 180.08), (5.55, 155.29), (11.07, 157.48), (-0.73, 166.7), (5.09, 164.22), (10.54, 175.07), (1.06, 168.04), (7.37, 182.13), (4.46, 166.17), (8.05, 167.42), (-0.66, 171.71), (5.54, 151.46), (4.16, 182.47), (2.44, 164.91), (3.91, 179.33), (8.22, 159.82), (8.32, 162.81), (8.71, 165.22), (5.49, 170.11), (8.15, 174.93), (10.57, 174.4), (9.93, 158.23), (5.43, 169.91), (6.33, 176.82), (7.06, 169.29), (5.87, 171.88), (7.69, 152.56), (6.4, 158.25), (6.98, 174.3), (10.98, 159.9), (10.6, 159.04), (8.18, 174.85), (4.65, 175.14), (4.0, 155.02), (0.62, 173.11), (5.51, 172.43), (3.92, 173.34), (7.41, 171.8), (5.39, 175.78), (7.38, 176.12), (1.64, 181.39), (3.06, 170.01), (8.6, 150.79), (10.88, 161.14), (9.75, 176.12), (11.01, 171.72), (3.84, 166.22), (8.73, 175.93), (10.24, 154.75), (4.41, 164.25), (4.51, 167.19), (3.42, 168.09), (3.96, 176.36), (7.03, 158.59), (7.83, 178.22), (5.23, 170.67), (8.94, 168.9), (7.55, 154.41), (2.41, 166.75), (3.32, 168.18), (11.3, 178.01), (9.18, 178.73), (8.3, 163.64), (5.26, 168.4), (8.73, 167.38), (4.87, 183.56), (4.78, 176.49), (6.94, 164.62), (11.6, 177.85), (7.9, 167.22), (5.67, 170.84), (8.05, 180.27), (6.07, 166.63), (10.3, 162.65), (2.95, 170.58), (5.37, 178.98), (4.94, 160.45), (3.14, 171.23), (1.19, 158.9), (10.36, 181.0), (9.42, 182.92), (11.22, 171.31), (8.67, 169.65), (9.5, 158.54), (8.29, 167.17), (8.25, 158.46), (3.92, 168.91), (0.96, 170.49), (9.5, 160.42), (11.03, 171.44), (2.59, 170.81), (4.51, 163.25), (5.96, 164.82), (7.22, 175.57), (5.33, 168.6), (7.4, 161.02), (3.11, 158.99), (0.81, 170.53), (3.44, 165.5), (6.95, 158.06), (6.02, 170.48), (4.53, 177.98), (9.44, 159.81), (3.86, 163.71), (1.29, 171.4), (12.44, 180.5), (6.6, 153.95), (8.18, 155.91), (9.76, 163.05), (7.11, 166.11), (7.61, 157.04), (7.96, 178.15), (4.59, 172.77), (1.89, 171.24), (4.27, 160.78), (7.95, 153.03), (6.44, 171.85), (6.14, 160.71), (3.77, 167.44), (8.5, 156.21), (4.6, 181.89), (2.78, 161.12), (3.9, 159.52), (4.0, 158.87), (6.39, 167.1), (5.88, 160.45), (5.22, 170.61), (5.47, 168.15), (4.32, 181.14), (7.09, 167.25), (8.64, 165.49), (1.7, 184.81), (7.2, 179.69), (6.03, 174.0), (13.83, 168.15), (6.26, 167.57), (8.61, 166.39), (4.68, 160.49), (7.31, 161.52), (6.43, 177.7), (10.47, 158.76), (7.92, 175.09), (5.44, 171.24), (6.71, 174.0), (4.06, 173.04), (-0.24, 168.39), (10.71, 156.48), (3.13, 171.02), (14.25, 169.38), (7.51, 172.19), (7.91, 154.19), (6.63, 168.59), (2.84, 172.61), (8.18, 168.85), (11.05, 162.77), (5.2, 181.59), (1.33, 169.61), (5.61, 169.3), (4.15, 173.33), (9.32, 162.88), (10.25, 166.3), (7.1, 162.39), (7.81, 171.02), (10.82, 163.98), (12.08, 179.55), (2.3, 179.35), (6.75, 173.86), (5.34, 166.03), (6.86, 175.16), (1.93, 164.28), (4.42, 165.38), (8.4, 163.93), (9.41, 160.82), (6.85, 180.41), (4.73, 165.61), (6.32, 169.39), (1.55, 165.01), (6.06, 178.25), (5.14, 167.02), (3.48, 153.48), (2.3, 173.95), (4.59, 157.09), (13.32, 173.04), (9.82, 166.76), (0.38, 181.3), (5.42, 167.1), (8.85, 178.76), (8.18, 161.25), (8.89, 172.89), (6.48, 166.97), (5.97, 171.86), (5.1, 168.99), (-1.3, 162.96), (6.22, 174.13), (3.1, 160.55), (1.98, 172.13), (6.66, 153.0), (7.69, 158.21), (7.82, 166.18), (1.68, 168.55), (14.31, 173.03), (5.47, 167.19), (8.77, 179.75), (8.04, 173.88), (9.13, 167.38), (4.5, 180.66), (1.31, 179.08), (5.92, 168.28), (3.48, 172.9), (3.76, 166.91), (3.27, 169.33), (-1.49, 159.63), (9.77, 176.04), (6.46, 173.88), (6.07, 176.09), (7.42, 178.22), (2.92, 166.61), (1.09, 172.5), (3.32, 158.38), (5.68, 157.62), (5.46, 164.57), (4.45, 171.55), (6.08, 160.68), (7.16, 160.22), (4.59, 165.98), (7.77, 182.16), (4.43, 151.63), (3.86, 170.67), (2.82, 165.09), (5.3, 177.55), (8.18, 166.06), (6.17, 166.73), (9.26, 159.49), (9.63, 171.31), (10.12, 159.77), (1.24, 159.86), (4.81, 171.01), (4.27, 171.65), (5.46, 155.26), (7.9, 182.54), (10.35, 182.76), (8.03, 177.33), (6.89, 163.19), (7.09, 155.81), (4.61, 168.94), (1.47, 174.15), (5.98, 161.84), (10.36, 169.19), (10.25, 180.14), (8.18, 173.27), (10.14, 181.23), (2.64, 166.19), (6.72, 167.2), (4.93, 182.4), (2.89, 156.55), (0.48, 181.22), (2.68, 174.69), (8.63, 171.27), (6.77, 170.5), (8.42, 174.12), (8.89, 167.22), (8.98, 160.67), (9.67, 169.01), (3.35, 164.23), (7.49, 163.27), (2.08, 174.75), (5.05, 162.7), (7.7, 171.46), (4.0, 156.58), (7.81, 167.5), (4.12, 166.96), (3.2, 175.13), (6.16, 172.78), (5.13, 170.42), (9.94, 184.7), (6.42, 155.35), (4.24, 170.71), (1.87, 172.43), (3.92, 159.58), (4.38, 169.97), (4.98, 180.99), (3.28, 175.32), (10.59, 179.3), (3.89, 156.89), (3.47, 167.9), (5.34, 166.65), (6.68, 178.45), (11.33, 175.0), (9.59, 161.42), (3.4, 176.78), (6.68, 171.03), (5.73, 162.21), (6.81, 169.36), (5.63, 181.48), (4.13, 186.53), (2.33, 157.53), (4.86, 163.55), (7.15, 166.12), (8.86, 166.05), (-0.92, 161.24), (5.38, 167.87), (6.08, 170.31), (11.32, 159.46), (5.7, 155.67), (8.02, 162.69), (8.72, 166.0), (7.48, 164.82), (6.02, 165.06), (11.16, 161.81), (8.82, 170.74), (4.13, 173.55), (8.64, 158.61), (6.82, 176.33), (7.57, 166.27), (9.84, 158.98), (4.78, 160.67), (5.0, 164.51), (8.06, 172.83), (9.2, 173.27), (7.09, 165.4), (11.5, 167.92), (5.29, 173.6), (2.84, 160.99), (7.04, 165.19), (6.66, 173.47), (2.96, 176.06), (6.46, 168.41), (7.97, 177.79), (8.18, 167.22), (5.04, 161.7), (3.72, 164.41), (3.82, 181.73), (5.41, 169.68), (12.45, 173.98), (4.24, 173.41), (8.58, 174.39), (6.66, 168.68), (8.96, 155.56), (7.68, 174.54), (4.55, 171.73), (5.37, 167.74), (0.83, 166.14), (3.63, 166.01), (1.47, 173.76), (14.35, 163.24), (8.03, 164.67), (7.13, 162.32), (4.9, 161.73), (11.39, 182.29), (5.41, 170.38), (8.59, 167.35), (2.55, 172.78), (2.38, 164.92), (2.49, 174.0), (6.2, 165.08), (11.32, 161.24), (3.67, 163.32), (6.84, 172.74), (1.2, 164.38), (6.16, 181.85), (9.34, 174.74), (13.39, 169.3), (4.69, 169.6), (3.98, 171.37), (7.49, 166.95), (8.81, 167.55), (8.67, 162.38), (6.43, 192.07), (1.68, 168.77), (4.43, 173.0), (6.54, 161.83), (7.0, 157.98), (5.94, 162.3), (4.29, 173.75), (3.92, 160.56), (3.29, 170.99), (6.81, 177.81), (13.02, 159.42), (8.98, 157.74), (6.13, 163.38), (6.42, 162.37), (3.81, 159.51), (-0.33, 165.95), (5.57, 162.01), (7.08, 180.11), (4.05, 179.14), (6.79, 162.27), (6.99, 166.32), (1.3, 165.8), (0.89, 170.1), (7.33, 159.12), (8.43, 157.47), (11.25, 161.07), (6.37, 167.0), (6.79, 168.15), (8.76, 175.17), (5.19, 173.19), (3.69, 177.88), (9.99, 159.16), (5.65, 183.06), (9.34, 173.89), (3.88, 159.46), (10.26, 177.02), (3.92, 154.81), (6.09, 168.52), (-1.46, 179.8), (10.31, 166.52), (-0.25, 160.97), (2.21, 167.66), (-2.29, 173.88), (11.49, 171.43), (1.49, 180.26), (5.12, 168.35), (6.08, 171.65), (12.03, 178.56), (11.61, 162.99), (11.26, 174.48), (12.52, 161.16), (5.36, 154.2), (9.14, 163.36), (8.1, 182.0), (4.2, 156.93), (3.7, 173.05), (10.0, 163.81), (7.75, 175.27), (-0.54, 161.64), (9.36, 174.58), (4.72, 161.83), (7.21, 159.86), (6.09, 180.44), (3.92, 184.69), (4.63, 169.46), (9.81, 177.93), (4.71, 173.21), (8.38, 160.21), (11.22, 170.94), (8.05, 162.04), (5.73, 170.01), (5.16, 158.01), (7.56, 163.69), (4.8, 173.48), (4.69, 166.6), (5.12, 159.72), (-0.63, 171.55), (1.65, 162.11), (2.05, 173.02), (5.4, 158.97), (5.74, 177.2), (8.15, 151.25), (5.63, 155.35), (3.12, 160.56), (5.07, 161.0), (6.38, 174.61), (7.35, 179.84), (4.3, 156.43), (4.64, 158.88), (9.44, 176.47), (5.61, 166.32), (8.35, 172.55), (8.55, 169.12), (3.67, 172.18), (2.87, 151.9), (2.19, 182.17), (8.91, 178.24), (2.51, 167.3), (6.06, 146.88), (12.03, 164.1), (0.99, 163.06), (8.59, 170.59), (2.15, 162.82), (7.52, 165.65), (3.07, 159.41), (1.71, 175.2), (2.44, 179.49), (4.48, 163.87), (7.69, 179.23), (3.88, 171.8), (10.08, 167.4), (3.72, 168.87), (-1.1, 174.2), (3.85, 164.5), (8.18, 175.88), (2.56, 168.87), (5.35, 167.22), (-0.39, 172.7), (4.15, 178.07), (7.53, 170.49), (4.8, 173.13), (12.04, 163.65), (4.69, 170.04), (3.4, 184.35), (4.38, 154.27), (4.36, 174.86), (8.99, 163.6), (4.75, 160.11), (8.27, 169.28), (6.96, 157.49), (7.15, 160.49), (-0.56, 178.98), (6.84, 170.83), (5.16, 180.37), (2.09, 172.66), (6.43, 160.27), (2.76, 162.79), (6.23, 166.07), (12.04, 151.56), (2.32, 158.3), (5.16, 168.83), (3.83, 170.39), (6.76, 169.97), (4.29, 180.21), (9.5, 166.39), (9.89, 165.85), (4.66, 173.06), (7.13, 173.21), (1.12, 170.67), (3.98, 177.44), (1.83, 171.73), (4.37, 178.22), (3.98, 168.92), (3.62, 170.33), (4.35, 169.76), (11.19, 188.18), (6.04, 163.49), (3.88, 178.6), (4.52, 176.2), (10.08, 174.23), (2.75, 172.0), (4.15, 170.29), (5.83, 158.35), (6.78, 164.19), (9.14, 163.67), (5.49, 153.44), (1.99, 165.69), (6.51, 158.35), (1.14, 177.29), (4.63, 177.07), (5.6, 189.45), (-2.68, 159.42), (5.45, 182.7), (8.42, 162.07), (3.41, 180.67), (7.16, 179.56), (8.52, 159.66), (7.68, 166.02), (8.63, 164.05), (4.9, 164.49), (5.01, 172.07), (3.73, 168.59), (7.75, 172.53), (5.27, 166.28), (5.54, 168.15), (7.02, 174.61), (10.15, 180.31), (1.39, 170.2), (5.71, 158.04), (9.24, 165.61), (7.85, 169.89), (-0.04, 170.29), (4.9, 156.57), (5.6, 168.74), (5.98, 162.43), (7.04, 165.17), (0.34, 164.15), (8.89, 163.2), (9.39, 168.67), (8.2, 160.84), (11.54, 171.45), (4.89, 155.49), (6.12, 163.7), (6.38, 158.45), (9.22, 161.17), (6.78, 170.12), (3.18, 179.81), (1.96, 174.63), (3.18, 171.32), (8.88, 149.46), (7.58, 158.71), (11.0, 162.34), (3.22, 165.22), (1.88, 166.37), (4.72, 158.05), (6.53, 162.73), (8.28, 169.21), (4.66, 175.34), (6.49, 168.64), (5.62, 170.33), (10.14, 154.03), (11.32, 167.76), (3.57, 186.79), (9.06, 166.99), (-1.2, 159.32), (3.15, 161.98), (7.93, 163.39), (0.47, 178.24), (4.52, 178.01), (4.94, 154.86), (8.41, 156.89), (5.52, 180.07), (5.49, 188.59), (6.61, 161.81), (4.84, 168.6), (8.48, 173.39), (8.27, 161.3), (7.62, 167.06), (1.48, 157.72), (4.43, 164.81), (7.19, 157.62), (4.4, 180.35), (10.56, 159.7), (1.85, 167.3), (6.11, 173.34), (11.31, 181.98), (1.52, 175.67), (4.0, 172.53), (7.55, 175.92), (9.48, 167.22), (4.26, 168.37), (9.15, 168.64), (5.18, 148.04), (5.81, 159.17), (0.96, 178.15), (6.27, 165.32), (4.68, 181.44), (10.22, 158.22), (3.49, 169.41), (3.71, 166.48), (7.78, 163.46), (11.44, 170.16), (4.01, 152.75), (9.53, 184.04), (6.66, 170.45), (3.01, 158.0), (0.02, 168.78), (9.23, 172.92), (7.66, 159.16), (7.66, 170.28), (10.29, 165.41), (5.19, 161.39), (1.97, 169.49), (7.27, 152.58), (3.38, 172.58), (2.94, 150.36), (7.51, 151.03), (5.95, 166.53), (4.46, 158.2), (2.53, 165.91), (9.65, 168.18), (4.65, 163.91), (3.48, 178.77), (8.99, 168.4), (9.5, 167.85), (2.05, 181.9), (3.66, 166.43), (12.56, 148.51), (7.42, 151.99), (6.07, 173.14), (3.73, 170.44), (7.6, 176.02), (6.7, 174.78), (11.1, 168.1), (5.27, 182.93), (2.46, 164.06), (9.21, 159.52), (5.89, 164.28), (4.36, 164.29), (7.44, 160.86), (1.96, 154.45), (7.1, 166.51), (8.1, 155.74), (6.74, 162.2), (4.3, 169.48), (9.65, 144.58), (3.03, 164.76), (9.19, 169.52), (6.77, 172.34), (4.27, 166.73), (9.7, 175.15), (5.28, 176.82), (9.83, 163.68), (7.25, 162.56), (9.62, 166.92), (2.54, 172.4), (13.27, 175.24), (3.66, 164.37), (4.91, 167.11), (5.58, 176.25), (7.09, 163.25), (3.88, 161.82), (6.78, 167.87), (6.56, 167.66), (6.01, 153.66), (7.36, 166.9), (12.98, 177.28), (3.93, 149.07), (8.98, 157.49), (4.88, 181.26), (6.03, 183.47), (9.52, 176.33), (2.56, 172.79), (6.08, 159.92), (6.77, 159.54), (7.24, 175.71), (6.89, 163.43), (5.0, 169.53), (6.35, 168.38), (2.35, 170.73), (4.06, 175.21), (4.0, 170.17), (8.2, 158.02), (6.78, 167.81), (1.2, 179.96), (10.88, 179.24), (3.46, 166.51), (11.04, 157.78), (6.26, 175.55), (4.81, 160.42), (8.13, 150.39), (4.08, 169.88), (8.53, 180.11), (8.77, 174.92), (6.56, 157.72), (6.2, 164.36), (3.5, 161.78), (5.55, 161.86), (1.97, 177.47), (6.02, 171.8), (7.02, 159.39), (4.87, 157.23), (4.61, 188.9), (6.55, 168.31), (2.37, 173.2), (8.87, 177.37), (8.9, 161.66), (4.95, 171.25), (6.26, 179.44), (6.24, 185.57), (6.65, 184.88), (1.51, 169.75), (0.64, 178.27), (5.34, 160.04), (1.99, 168.99), (7.68, 155.49), (9.85, 177.05), (6.72, 146.76), (2.24, 152.82), (6.06, 184.88), (4.94, 159.22), (9.13, 149.76), (0.44, 149.73), (5.07, 164.57), (3.84, 167.05), (6.72, 183.66), (9.45, 175.26), (2.61, 162.52), (5.51, 173.85), (3.29, 172.49), (2.88, 170.93), (3.41, 165.76), (6.7, 189.98), (5.74, 153.71), (5.24, 153.12), (9.16, 166.85), (3.71, 159.84), (-1.01, 171.0), (7.24, 170.9), (4.94, 159.05), (3.78, 164.37), (3.8, 182.17), (4.57, 171.19), (6.06, 166.48), (8.89, 164.25), (4.91, 179.23), (4.16, 177.15), (1.97, 163.2), (1.34, 166.77), (6.33, 159.07), (2.61, 167.31), (2.32, 156.58), (8.0, 163.07), (3.12, 181.88), (2.4, 165.79), (5.67, 155.21), (1.07, 178.39), (2.35, 164.48), (8.57, 166.68), (2.76, 161.7), (7.23, 171.79), (4.61, 165.03), (10.59, 155.81), (11.54, 178.17), (10.84, 177.72), (4.78, 169.13), (11.39, 172.21), (1.9, 176.55), (4.83, 167.44), (8.37, 180.19), (7.4, 161.8), (7.72, 168.96), (9.7, 173.05), (7.32, 178.61), (2.2, 179.42), (5.7, 166.9), (4.77, 143.92), (8.19, 167.31), (7.86, 163.46), (6.46, 168.72), (6.73, 182.96), (6.64, 176.82), (4.89, 176.31), (3.43, 169.44), (5.92, 169.6), (7.6, 156.53), (8.56, 166.17), (3.27, 171.17), (4.82, 176.36), (2.79, 171.63), (1.08, 156.42), (1.46, 167.34), (6.44, 158.46), (4.13, 168.02), (8.79, 169.5), (8.28, 173.31), (4.0, 162.13), (7.02, 158.01), (7.81, 168.19), (9.65, 177.59), (6.44, 162.4), (0.2, 149.07), (8.03, 178.63), (7.57, 166.94), (8.4, 159.24), (4.02, 169.71), (8.19, 172.54), (10.65, 179.05), (6.98, 167.5), (1.84, 169.27), (6.37, 168.83), (7.64, 174.52), (14.42, 183.28), (6.48, 177.82), (6.07, 177.96), (0.99, 154.54), (9.35, 156.51), (7.18, 175.75), (4.12, 171.34), (1.51, 159.68), (11.03, 168.17), (6.68, 164.22), (8.95, 165.21), (8.24, 155.51), (12.15, 180.01), (5.77, 163.05), (0.29, 166.97), (4.65, 154.22), (3.06, 165.12), (6.18, 155.89), (4.74, 157.8), (1.37, 178.19), (5.16, 150.2), (7.51, 153.77), (8.44, 173.53), (5.52, 167.6), (10.66, 169.0), (12.99, 155.22), (8.34, 168.33), (4.9, 168.81), (8.74, 175.76), (8.48, 157.7), (3.65, 179.48), (8.15, 172.55), (-2.0, 182.59), (9.39, 170.9), (4.61, 149.5), (16.76, 163.39), (13.47, 175.03), (8.85, 166.48), (9.72, 157.84), (7.38, 157.27), (6.95, 174.18), (5.99, 162.2), (10.31, 159.09), (4.76, 165.41), (9.78, 184.14), (9.27, 160.29), (4.38, 171.18), (3.88, 185.67), (8.45, 159.15), (3.0, 175.47), (8.32, 151.42), (5.6, 174.76), (8.55, 175.98), (9.13, 169.91), (8.25, 170.22), (8.37, 163.42), (4.78, 165.24), (5.18, 168.59), (6.82, 178.83), (7.17, 171.56), (5.88, 165.6), (7.94, 169.18), (7.4, 174.5), (3.2, 162.92), (7.36, 172.98), (7.0, 157.29), (9.06, 169.5), (4.91, 172.76), (7.61, 175.16), (0.69, 151.67), (13.53, 168.67), (10.07, 162.22), (10.98, 176.41), (8.84, 174.23), (11.99, 154.97), (5.83, 164.11), (-0.33, 159.08), (2.29, 169.08), (2.17, 170.86), (7.76, 157.23), (10.64, 172.65), (0.45, 167.54), (11.44, 167.16), (5.36, 172.23), (8.19, 153.24), (8.24, 177.84), (8.48, 181.43), (1.7, 169.64), (6.77, 160.9), (7.89, 180.76), (8.71, 175.33), (5.33, 176.23), (9.0, 189.02), (4.98, 163.94), (9.81, 170.13), (8.6, 167.9), (7.35, 163.34), (7.56, 156.85), (8.41, 166.5), (3.12, 163.74), (2.13, 169.99), (12.67, 178.99)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=6.0874，ȳ=168.0768；Sxx=Σ(x−x̄)²=9273.64，Sxy=-367.05；
β₁ = Sxy/Sxx = -0.0396；β₀ = ȳ−β₁x̄ = 168.3178；
R² = SSR/SST = 14.53/67990.78 = 0.0002；F = (SSR/1)/(SSE/994) = 0.2124，p = 0.6450；
SE(β₁)=√(68.39/9273.64)=0.0859；t=β₁/SE=-0.4609；p=0.6450；95%CI=[-0.2081,0.1289]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=6.0874，ȳ=168.0768；Sxx=Σ(x−x̄)²=9273.64，Sxy=-367.05；
β₁ = Sxy/Sxx = -0.0396；β₀ = ȳ−β₁x̄ = 168.3178；
R² = SSR/SST = 14.53/67990.78 = 0.0002；F = (SSR/1)/(SSE/994) = 0.2124，p = 0.6450；
SE(β₁)=√(68.39/9273.64)=0.0859；t=β₁/SE=-0.4609；p=0.6450；95%CI=[-0.2081,0.1289]。
③ 计算结果：
y = 168.3178 + -0.0396·x；R²=0.0002；β₁ t=-0.4609 p=0.6450；95%CI=[-0.2081,0.1289]
④ 结论：在 α=0.05 水平下，p=0.6450 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=-0.0396），应无影响。
案例 67. 手机尾号预测消费
简单线性回归应无影响
① 原始数据：
样本 n=305，逐条 (x, y)：[(5.39, 732.55), (7.9, 678.25), (0.74, 600.05), (5.82, 668.88), (4.07, 250.38), (11.24, 619.7), (6.33, 448.62), (8.03, 590.72), (3.29, 872.83), (5.08, 527.88), (4.28, 762.45), (4.05, 591.16), (-0.6, 326.34), (0.45, 437.69), (6.04, 249.8), (6.33, 253.05), (6.29, 707.05), (3.26, 704.19), (-2.04, 695.22), (5.97, 214.5), (1.4, 680.07), (11.07, 490.0), (4.26, 688.52), (5.66, 801.93), (3.23, 761.3), (4.27, 322.73), (5.7, 419.28), (2.43, 520.64), (4.63, 456.85), (1.69, 654.28), (4.85, 370.56), (1.87, 188.77), (8.34, 300.54), (1.36, 391.07), (2.54, 325.95), (4.19, 677.34), (6.52, 513.7), (7.24, 232.05), (4.01, 790.5), (7.79, 625.76), (4.88, 407.73), (6.89, 583.73), (6.03, 670.94), (0.89, 630.45), (4.31, 894.88), (6.94, 688.78), (0.66, 371.24), (5.14, 766.54), (7.19, 427.7), (1.91, 738.79), (4.04, 593.4), (3.26, 431.89), (7.77, 889.16), (4.67, 607.44), (4.73, 642.17), (-0.44, 163.21), (0.85, 444.42), (3.65, 710.99), (1.73, 444.61), (5.54, 499.68), (-1.52, 695.77), (1.24, 1003.01), (7.28, 630.77), (1.55, 200.36), (5.63, 292.45), (5.92, 687.36), (3.07, 735.56), (1.15, 802.08), (0.73, 252.99), (3.3, 682.51), (8.87, 600.26), (4.57, 607.73), (7.28, 635.07), (2.25, 397.07), (2.26, 729.99), (4.46, 757.24), (8.63, 711.13), (5.9, 377.7), (5.35, 485.1), (7.2, 236.61), (-0.31, 494.72), (2.52, 481.13), (2.6, 442.0), (7.64, 525.51), (2.29, 512.71), (-0.27, 419.61), (2.01, 524.88), (6.81, 604.37), (5.77, 216.8), (5.59, 476.82), (6.27, 382.7), (6.72, 663.21), (1.79, 60.43), (8.2, 364.16), (6.0, 653.22), (9.09, 437.19), (4.5, 603.27), (5.17, 293.6), (7.31, 656.39), (6.83, 700.42), (9.17, 376.82), (6.25, 557.43), (2.23, -57.18), (7.11, 219.12), (4.1, 670.53), (5.29, 678.75), (11.1, 428.89), (5.74, 600.54), (5.3, 206.55), (0.35, 402.81), (7.84, 289.47), (5.03, 359.57), (5.44, 320.68), (4.57, 658.7), (7.62, 387.72), (2.58, 738.8), (0.86, 565.33), (3.71, 425.73), (10.04, 220.35), (10.21, 316.01), (2.95, 677.34), (5.12, 622.29), (6.25, 251.68), (2.47, 948.22), (4.49, 190.02), (4.58, 828.58), (5.68, 639.06), (1.56, 514.88), (6.58, 503.17), (1.74, 456.58), (2.88, 587.24), (4.18, 356.33), (-0.27, 701.14), (6.15, 330.92), (6.01, 545.05), (7.03, 582.67), (5.73, 1099.89), (5.67, 414.6), (4.16, 351.45), (3.68, 821.17), (2.4, 924.81), (3.88, 452.29), (8.81, 669.68), (7.01, 613.1), (5.22, 312.68), (5.09, 330.79), (4.44, 484.56), (4.72, 251.25), (8.49, -60.33), (9.03, 341.09), (3.51, 513.37), (7.39, 455.6), (2.04, 280.5), (11.51, 404.59), (6.92, 445.21), (3.66, 577.77), (4.91, 503.26), (2.03, 458.45), (7.48, 531.39), (5.39, 227.4), (5.79, 587.44), (11.65, 619.03), (-2.32, 698.34), (0.08, 232.27), (10.02, 705.25), (7.14, 538.62), (6.56, 589.46), (3.92, 582.87), (2.81, 431.64), (4.09, 314.53), (5.5, 855.73), (9.76, 438.67), (5.02, 352.15), (6.05, 465.59), (8.29, 481.79), (3.9, 525.72), (7.34, 583.34), (-0.23, 225.76), (7.64, 589.8), (9.15, 534.96), (1.63, 320.81), (7.08, 305.93), (8.3, 545.07), (4.69, 265.85), (-1.66, 737.17), (3.54, 516.56), (5.92, 585.0), (2.91, 367.48), (5.63, 441.51), (5.71, 598.86), (3.42, 321.41), (2.11, 666.76), (6.73, 348.14), (4.9, 746.79), (0.89, 196.7), (4.21, 519.03), (9.49, 343.65), (4.73, 592.74), (2.1, 633.4), (6.76, 249.85), (4.78, 503.15), (3.23, 774.68), (9.53, 534.21), (8.86, 638.48), (8.97, 358.5), (7.01, 832.88), (9.93, 751.03), (7.95, 851.35), (1.5, 299.38), (5.75, 498.17), (5.73, 470.9), (4.22, 371.23), (7.89, 417.45), (5.18, 606.59), (-0.03, 932.13), (3.98, 381.64), (4.23, 679.01), (8.37, 533.09), (7.87, 458.83), (7.3, 1113.63), (2.93, 410.58), (8.27, 277.83), (2.01, 778.76), (8.36, 850.83), (4.98, 990.18), (6.39, 536.96), (7.66, 367.28), (3.2, 428.28), (5.35, 202.64), (8.87, 439.02), (8.27, 558.85), (9.61, 540.56), (4.7, 535.67), (1.41, 613.98), (3.37, 603.0), (7.09, 244.84), (4.75, 597.07), (0.43, 581.53), (2.36, 433.82), (0.45, 493.23), (5.86, 681.05), (5.53, 333.72), (5.89, 311.56), (9.37, 522.98), (5.06, 913.66), (3.65, 315.17), (8.47, 511.97), (5.76, 697.88), (6.59, 273.66), (6.42, 701.87), (5.01, 387.45), (4.03, 700.46), (8.01, 559.98), (1.03, 529.23), (4.21, 773.36), (2.42, 404.45), (6.36, 546.5), (4.2, 344.88), (10.06, 377.02), (7.46, 446.66), (4.51, 475.27), (0.43, 299.75), (4.92, 504.48), (3.77, 761.27), (7.0, 748.67), (12.43, 523.65), (7.21, 393.92), (7.81, 216.93), (1.33, 450.41), (-2.42, 238.85), (3.83, 87.99), (4.61, 458.59), (8.42, 400.28), (-2.89, 281.49), (4.97, 950.78), (2.5, 399.48), (0.88, 343.9), (1.99, 243.9), (7.06, 502.97), (5.11, 538.42), (4.66, 387.91), (1.96, 399.79), (8.03, 548.28), (2.12, 516.31), (3.13, 776.22), (0.75, 299.21), (-3.01, 402.58), (8.46, 447.03), (3.42, 402.22), (6.57, 424.62), (6.21, 501.03), (0.11, 303.09), (4.2, 305.51), (0.67, 644.19), (7.68, 595.79), (7.49, 285.13), (6.5, 438.91), (3.15, 374.42), (4.04, 545.4), (3.8, 307.12), (1.17, 461.41), (2.46, 344.57), (6.4, 617.12), (6.83, 619.97), (7.39, 331.33)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=4.8832，ȳ=507.7762；Sxx=Σ(x−x̄)²=2500.78，Sxy=2941.64；
β₁ = Sxy/Sxx = 1.1763；β₀ = ȳ−β₁x̄ = 502.0321；
R² = SSR/SST = 3460.22/11430872.51 = 0.0003；F = (SSR/1)/(SSE/303) = 0.0917，p = 0.7622；
SE(β₁)=√(37714.23/2500.78)=3.8834；t=β₁/SE=0.3029；p=0.7622；95%CI=[-6.4656,8.8182]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=4.8832，ȳ=507.7762；Sxx=Σ(x−x̄)²=2500.78，Sxy=2941.64；
β₁ = Sxy/Sxx = 1.1763；β₀ = ȳ−β₁x̄ = 502.0321；
R² = SSR/SST = 3460.22/11430872.51 = 0.0003；F = (SSR/1)/(SSE/303) = 0.0917，p = 0.7622；
SE(β₁)=√(37714.23/2500.78)=3.8834；t=β₁/SE=0.3029；p=0.7622；95%CI=[-6.4656,8.8182]。
③ 计算结果：
y = 502.0321 + 1.1763·x；R²=0.0003；β₁ t=0.3029 p=0.7622；95%CI=[-6.4656,8.8182]
④ 结论：在 α=0.05 水平下，p=0.7622 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=1.1763），应无影响。
案例 68. 星座预测月支出
简单线性回归应无影响
① 原始数据：
样本 n=557，逐条 (x, y)：[(19.46, 2323.0), (3.38, 4351.44), (3.86, 3063.37), (11.26, 2873.47), (5.85, 4183.24), (19.21, 4003.44), (6.03, 3439.4), (21.01, 4023.16), (8.29, 4430.42), (9.3, 4517.78), (13.32, 4170.33), (4.77, 3726.61), (21.91, 5506.37), (12.09, 3861.49), (10.51, 5563.94), (17.87, 3508.76), (19.29, 4942.13), (7.85, 5311.38), (4.84, 3276.77), (19.55, 3048.74), (21.72, 3181.08), (16.96, 5711.96), (2.78, 3037.87), (8.47, 3672.42), (14.89, 4450.06), (16.1, 2852.73), (15.58, 3763.94), (17.56, 3969.42), (12.06, 5761.94), (3.74, 2692.45), (26.08, 2759.05), (10.21, 3335.54), (7.78, 3430.17), (-4.5, 3348.76), (4.31, 3771.66), (-3.71, 4922.53), (8.05, 3150.31), (-4.52, 4827.0), (8.67, 5352.15), (14.97, 5213.06), (5.18, 3939.19), (24.06, 3547.9), (13.2, 6056.23), (19.95, 4456.13), (22.05, 4080.59), (17.23, 5006.6), (10.5, 4321.51), (17.27, 5527.64), (11.84, 3954.67), (6.93, 4023.9), (13.37, 2962.01), (13.31, 5824.67), (6.16, 2776.52), (14.99, 3644.27), (1.15, 2704.73), (19.62, 3043.48), (20.62, 4140.38), (13.52, 3167.4), (11.42, 5592.12), (14.5, 3845.98), (15.41, 2780.65), (21.54, 3929.53), (17.55, 3873.35), (14.53, 4701.06), (18.15, 4538.29), (2.67, 3292.24), (21.31, 4888.48), (10.6, 4119.42), (6.13, 4176.75), (10.05, 2456.61), (14.57, 4095.08), (3.89, 4438.38), (9.53, 2465.26), (5.3, 3562.14), (6.99, 3688.89), (20.05, 5025.3), (7.73, 2739.43), (6.6, 3441.56), (11.75, 5132.42), (5.76, 4723.29), (5.06, 6061.85), (8.62, 3462.7), (19.33, 3398.08), (10.83, 3937.72), (17.37, 3949.44), (8.46, 6480.02), (13.24, 5139.39), (17.49, 4715.15), (8.51, 4871.66), (8.36, 4244.52), (10.78, 3524.52), (20.81, 4243.03), (15.13, 5162.34), (7.55, 3525.4), (12.33, 3484.1), (7.05, 5535.89), (9.75, 3969.94), (9.22, 5155.02), (16.25, 3846.01), (1.37, 3215.72), (21.15, 4216.28), (4.73, 3901.67), (17.33, 3862.9), (20.24, 3566.21), (17.65, 3992.07), (23.43, 4786.72), (5.22, 3166.18), (3.84, 4437.06), (16.04, 4949.6), (10.65, 4221.01), (13.95, 3145.16), (6.85, 3670.87), (-3.15, 4214.39), (10.09, 3005.01), (10.42, 3211.25), (15.59, 4769.98), (17.42, 5068.6), (10.81, 4322.76), (7.34, 4518.94), (11.83, 3464.14), (5.63, 5387.26), (8.89, 5126.02), (10.23, 4591.09), (12.89, 2871.18), (2.32, 4101.97), (10.89, 2656.11), (18.1, 1653.44), (8.13, 3979.82), (12.57, 4762.19), (6.96, 3194.21), (22.73, 3251.52), (5.91, 4315.48), (15.9, 2532.93), (12.94, 3378.37), (11.0, 6325.27), (10.16, 4008.33), (18.11, 4483.29), (20.48, 4681.7), (26.81, 2702.88), (14.49, 4847.54), (16.52, 2941.6), (17.06, 4205.04), (21.36, 3344.06), (12.58, 5341.99), (8.32, 3968.22), (11.82, 3681.47), (17.63, 3372.19), (23.32, 3434.0), (17.13, 5878.21), (15.99, 3732.29), (23.52, 3315.37), (10.25, 3854.15), (17.83, 2817.36), (15.63, 5521.59), (11.68, 4208.04), (19.36, 4813.65), (3.1, 4981.97), (11.55, 5148.74), (11.02, 4049.57), (9.34, 6080.04), (10.84, 6059.66), (18.46, 3702.07), (5.74, 3262.01), (10.13, 2533.76), (13.15, 4352.81), (23.46, 2660.68), (21.38, 4581.51), (5.53, 6199.16), (9.1, 4991.83), (18.05, 3084.61), (9.18, 6102.83), (12.2, 3486.17), (20.14, 3345.01), (5.79, 4728.21), (15.8, 4873.52), (8.94, 4392.9), (12.82, 4664.48), (16.01, 4221.48), (16.96, 6840.86), (-4.73, 3628.09), (5.82, 3937.0), (13.99, 4556.0), (5.48, 3949.36), (6.47, 5413.4), (16.8, 3922.78), (18.02, 5026.18), (-3.75, 3251.79), (0.98, 4243.46), (9.4, 2987.56), (10.52, 4139.16), (1.32, 5022.38), (18.65, 3098.42), (1.93, 2864.32), (13.12, 4752.65), (5.89, 3008.94), (9.04, 4593.89), (-2.78, 3278.55), (12.53, 3922.87), (7.84, 3612.02), (15.26, 4742.41), (16.74, 3850.9), (13.42, 4710.04), (15.6, 3668.47), (10.21, 4583.7), (17.24, 5522.03), (6.38, 3988.69), (7.74, 4710.61), (18.71, 4476.91), (15.89, 4337.18), (3.79, 3883.87), (20.65, 4741.83), (7.21, 4652.96), (5.17, 4184.67), (11.56, 5188.27), (3.49, 3883.6), (12.54, 4303.61), (10.08, 3336.71), (2.44, 3028.5), (17.39, 3931.82), (12.23, 4193.53), (3.91, 2882.42), (13.09, 4138.68), (10.18, 5329.01), (5.84, 3241.03), (11.7, 3631.63), (16.16, 2741.79), (6.9, 5136.52), (13.72, 3249.78), (11.79, 4220.29), (7.66, 3502.92), (4.64, 4295.64), (19.72, 3330.72), (17.79, 3317.33), (10.88, 3513.81), (23.04, 2367.5), (6.09, 3722.54), (6.18, 3017.06), (13.15, 2722.99), (11.54, 2973.15), (3.82, 2988.55), (7.45, 6068.49), (25.28, 3771.95), (16.87, 4635.68), (3.11, 3964.47), (15.25, 3946.25), (19.85, 5277.07), (17.56, 4818.87), (15.98, 4013.92), (5.46, 3697.29), (10.69, 4000.4), (13.58, 5133.02), (23.34, 5107.4), (17.23, 4241.29), (4.08, 4928.69), (11.82, 3584.31), (17.6, 3346.87), (25.16, 5553.55), (21.48, 4557.92), (13.35, 4024.12), (8.43, 4433.05), (10.28, 2710.53), (2.07, 3342.16), (13.24, 5350.9), (11.14, 3761.48), (11.74, 3777.4), (17.83, 3528.5), (11.11, 3174.72), (15.5, 3775.4), (20.06, 3499.43), (18.02, 4765.2), (10.45, 4245.0), (12.22, 4305.42), (11.8, 4626.35), (1.42, 3189.59), (5.97, 3697.75), (13.2, 2943.6), (12.17, 3215.06), (16.31, 4117.15), (15.68, 3720.94), (3.91, 3003.36), (17.61, 1460.75), (8.24, 4697.08), (13.01, 5268.91), (1.12, 3679.83), (13.33, 3991.75), (15.9, 4269.57), (20.32, 3431.59), (10.3, 3644.6), (9.09, 4023.42), (8.69, 4982.01), (19.09, 3776.17), (5.27, 2942.76), (8.62, 3827.84), (13.99, 5027.64), (15.68, 4619.63), (16.5, 2162.7), (14.04, 3977.14), (12.86, 1821.71), (7.08, 2401.02), (12.98, 3709.67), (3.79, 4362.0), (9.88, 2366.09), (13.22, 2737.27), (5.36, 2554.53), (21.22, 4184.18), (-0.34, 4794.17), (29.3, 5264.86), (19.68, 5209.61), (8.81, 3503.74), (8.41, 3860.67), (8.14, 1826.77), (16.14, 5196.41), (15.64, 4415.28), (13.98, 3852.46), (11.75, 3769.64), (3.26, 4258.62), (10.95, 3036.09), (18.22, 7009.8), (3.01, 3646.89), (22.92, 3481.15), (2.03, 5762.52), (15.97, 6733.86), (12.91, 5120.57), (7.69, 2725.36), (6.74, 3801.16), (3.99, 3674.72), (6.08, 3831.61), (17.37, 3199.5), (11.7, 3689.27), (13.55, 3481.21), (32.47, 4632.32), (24.3, 3147.84), (8.75, 4764.72), (12.82, 2334.19), (19.88, 2328.14), (8.01, 4950.44), (17.85, 3983.99), (17.53, 2957.51), (3.31, 5451.51), (8.74, 4595.92), (3.6, 3033.97), (21.04, 3293.58), (6.22, 5150.9), (14.74, 4147.99), (10.47, 3549.39), (6.41, 6636.21), (7.63, 4182.38), (9.3, 5537.28), (13.95, 4130.13), (13.35, 3984.99), (12.09, 3738.05), (23.81, 4048.74), (10.66, 5035.87), (20.35, 4460.37), (-3.34, 5175.68), (11.47, 5854.74), (3.43, 5163.61), (11.72, 3631.97), (11.51, 3961.85), (11.72, 4079.1), (17.7, 3895.34), (17.58, 4998.58), (3.98, 4570.49), (19.96, 4320.99), (21.67, 3857.8), (17.06, 5781.92), (13.75, 4613.45), (12.6, 6216.14), (5.16, 1463.23), (-0.41, 4883.19), (6.99, 4944.6), (13.4, 3690.23), (11.91, 4813.14), (5.42, 3482.93), (10.86, 2869.31), (21.1, 2927.52), (2.81, 5420.5), (13.94, 2258.96), (11.88, 4037.86), (23.38, 4694.48), (13.22, 4970.97), (4.27, 4350.45), (30.46, 4332.51), (11.15, 2634.92), (11.25, 4445.55), (2.39, 3830.75), (11.11, 4030.27), (25.06, 1648.95), (9.1, 4407.68), (2.6, 5432.75), (15.93, 3165.85), (12.28, 3783.97), (11.03, 4881.25), (12.67, 3977.83), (1.91, 3687.71), (22.84, 5606.27), (16.81, 4562.42), (12.37, 4325.39), (8.58, 1887.83), (9.76, 5589.2), (16.1, 3141.27), (13.42, 4120.17), (3.81, 4415.33), (6.13, 5006.39), (7.81, 4676.19), (15.04, 2867.76), (14.7, 2338.98), (0.91, 3217.65), (20.62, 4351.56), (12.36, 3366.94), (13.95, 4494.06), (22.69, 3333.55), (9.66, 3522.28), (12.24, 4050.25), (23.76, 4951.36), (12.34, 5014.8), (17.74, 2733.87), (15.12, 3918.82), (9.25, 3530.68), (12.68, 1939.44), (21.12, 3695.38), (14.71, 4493.03), (18.0, 3835.58), (10.56, 4440.83), (8.22, 3269.24), (18.37, 2906.67), (16.88, 3548.93), (2.07, 4220.37), (4.97, 4222.07), (6.61, 3390.48), (20.83, 3592.16), (8.43, 3279.55), (8.32, 4308.61), (16.28, 4702.25), (20.49, 3577.64), (7.14, 2433.74), (15.52, 3923.89), (7.44, 4702.81), (11.13, 2650.48), (4.63, 3993.74), (6.2, 3911.98), (20.27, 3107.1), (9.76, 3769.48), (12.23, 3207.7), (12.7, 3584.57), (7.86, 4654.89), (2.42, 4617.97), (23.02, 4143.35), (18.95, 4123.41), (16.17, 3521.14), (10.95, 5128.4), (2.2, 2173.91), (10.46, 4301.09), (15.59, 6390.96), (14.95, 3077.55), (14.4, 5089.13), (14.84, 6035.21), (21.45, 3865.35), (5.37, 4913.27), (21.68, 3676.32), (12.03, 4149.52), (17.12, 4165.4), (16.57, 3599.77), (12.72, 3377.54), (13.49, 3181.76), (10.71, 2910.5), (6.42, 4202.46), (17.15, 5082.84), (15.82, 4296.9), (13.14, 2537.96), (22.12, 3589.62), (11.98, 3823.73), (19.72, 4727.53), (11.11, 4951.92), (9.69, 4644.57), (2.9, 4955.88), (11.23, 3742.55), (11.64, 3322.53), (13.56, 4237.01), (11.03, 4556.76), (12.82, 5819.92), (21.73, 2931.9), (12.08, 4419.26), (9.83, 2730.34), (23.8, 5077.5), (17.74, 4559.6), (7.26, 4069.4), (1.53, 3866.57), (17.03, 4683.8), (15.87, 4862.9), (17.05, 1901.9), (18.29, 5322.83), (1.71, 3983.93), (9.75, 4562.91), (20.33, 3625.13), (14.46, 4161.55), (5.01, 4241.7), (10.5, 6071.59), (1.25, 5128.55), (13.53, 6371.59), (13.04, 3554.18), (3.74, 3673.57), (9.42, 4314.63), (11.92, 3711.49), (13.28, 4205.96), (20.93, 3578.58), (-3.16, 4846.17), (17.17, 4701.37), (15.65, 5588.09), (3.33, 5943.35), (10.13, 2690.06), (22.62, 4480.27), (21.68, 4503.5), (8.41, 4086.14), (16.03, 3848.56), (7.81, 2451.14), (11.11, 3521.88), (16.4, 6549.78), (19.74, 3466.73), (17.99, 4817.73), (8.69, 2018.48), (12.37, 4980.46), (10.58, 3622.21), (12.05, 5400.87), (19.53, 4890.58), (11.88, 3671.09), (10.61, 4446.06), (22.46, 4741.46), (19.03, 3528.91), (10.81, 5276.44), (7.7, 3519.03), (18.34, 2569.92), (5.18, 3640.29), (11.3, 2495.29), (3.37, 4230.33), (6.73, 4393.97), (2.8, 5683.28), (27.33, 3270.71), (12.42, 3362.67), (16.31, 3348.05), (11.93, 4514.43), (7.31, 4702.67), (8.63, 3148.05), (19.23, 4730.5), (15.03, 2165.6), (15.02, 3718.18), (4.8, 4279.96), (14.57, 5200.66), (17.15, 4068.21), (18.71, 5750.28), (1.06, 3785.95), (12.88, 3942.11), (6.28, 5430.22), (10.67, 2165.46), (12.15, 3304.77), (14.26, 4426.36), (16.93, 2630.41)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=12.1999，ȳ=4059.5478；Sxx=Σ(x−x̄)²=21841.64，Sxy=-43025.02；
β₁ = Sxy/Sxx = -1.9699；β₀ = ȳ−β₁x̄ = 4083.5799；
R² = SSR/SST = 84753.36/516511561.83 = 0.0002；F = (SSR/1)/(SSE/555) = 0.0911，p = 0.7629；
SE(β₁)=√(930498.75/21841.64)=6.5270；t=β₁/SE=-0.3018；p=0.7629；95%CI=[-136.9169,132.9772]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=12.1999，ȳ=4059.5478；Sxx=Σ(x−x̄)²=21841.64，Sxy=-43025.02；
β₁ = Sxy/Sxx = -1.9699；β₀ = ȳ−β₁x̄ = 4083.5799；
R² = SSR/SST = 84753.36/516511561.83 = 0.0002；F = (SSR/1)/(SSE/555) = 0.0911，p = 0.7629；
SE(β₁)=√(930498.75/21841.64)=6.5270；t=β₁/SE=-0.3018；p=0.7629；95%CI=[-136.9169,132.9772]。
③ 计算结果：
y = 4083.5799 + -1.9699·x；R²=0.0002；β₁ t=-0.3018 p=0.7629；95%CI=[-136.9169,132.9772]
④ 结论：在 α=0.05 水平下，p=0.7629 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=-1.9699），应无影响。
案例 69. 幸运数字预测分数
简单线性回归应无影响
① 原始数据：
样本 n=817，逐条 (x, y)：[(4.41, 63.88), (2.19, 91.25), (2.93, 58.16), (4.46, 67.8), (1.36, 76.83), (4.97, 76.17), (4.16, 65.95), (2.21, 65.08), (3.21, 59.28), (4.52, 78.17), (4.15, 68.75), (0.43, 46.96), (4.35, 65.62), (5.2, 89.01), (1.87, 67.89), (2.6, 81.7), (1.74, 74.78), (2.86, 77.82), (1.2, 65.89), (3.84, 72.73), (0.01, 81.51), (2.51, 73.23), (1.33, 74.8), (2.94, 69.47), (2.53, 95.82), (6.51, 63.31), (4.79, 78.44), (1.02, 78.72), (-0.83, 55.54), (3.28, 66.43), (4.51, 73.16), (2.71, 62.59), (5.68, 81.64), (1.57, 64.41), (0.29, 80.26), (2.83, 65.69), (2.17, 66.97), (2.52, 61.19), (2.38, 69.72), (4.12, 71.59), (1.69, 91.11), (2.88, 49.08), (1.89, 87.78), (4.3, 78.13), (1.69, 68.06), (8.64, 66.14), (4.54, 66.84), (3.8, 98.32), (4.72, 46.79), (4.28, 80.67), (2.27, 87.9), (-0.36, 77.36), (3.47, 73.66), (2.43, 77.69), (3.43, 74.93), (4.24, 56.45), (3.08, 96.56), (1.84, 76.59), (4.25, 58.77), (-3.0, 49.77), (0.79, 89.15), (1.43, 73.77), (-1.01, 76.98), (3.93, 74.62), (4.78, 59.53), (0.34, 73.8), (0.29, 72.54), (5.49, 81.59), (2.54, 84.97), (5.58, 43.75), (3.79, 73.4), (1.79, 96.11), (3.5, 73.67), (1.83, 81.31), (3.31, 77.65), (2.78, 83.31), (4.36, 71.03), (2.32, 72.06), (4.33, 76.67), (3.81, 67.5), (3.14, 67.48), (-0.49, 70.71), (-0.44, 72.46), (3.62, 84.99), (1.88, 87.05), (4.56, 74.93), (1.81, 80.25), (2.13, 55.74), (4.36, 76.98), (5.49, 57.81), (5.24, 68.63), (4.89, 67.44), (0.42, 78.2), (0.51, 67.41), (2.36, 90.94), (3.74, 70.66), (3.44, 82.17), (1.28, 58.29), (6.27, 61.9), (5.15, 71.7), (-1.36, 94.49), (2.88, 76.32), (3.17, 71.65), (3.29, 73.27), (0.51, 85.6), (4.39, 63.54), (0.11, 69.57), (4.64, 41.41), (2.37, 77.98), (2.81, 84.13), (5.56, 69.58), (4.31, 96.49), (1.06, 85.28), (6.64, 66.74), (4.06, 52.15), (1.22, 80.45), (1.12, 54.22), (-0.79, 57.72), (5.49, 69.26), (2.34, 75.63), (3.54, 79.49), (3.02, 90.35), (4.71, 68.85), (6.64, 56.51), (4.95, 67.78), (4.47, 83.29), (4.85, 73.46), (0.71, 85.88), (0.83, 93.41), (2.48, 67.48), (2.22, 72.43), (-1.48, 90.98), (1.49, 82.46), (3.86, 78.91), (0.69, 53.19), (2.45, 60.91), (5.12, 92.79), (4.36, 103.52), (-0.17, 61.72), (1.8, 69.68), (4.22, 76.44), (4.14, 85.03), (6.05, 62.81), (3.49, 66.09), (0.94, 67.13), (2.83, 84.58), (1.7, 57.31), (7.11, 97.67), (3.46, 77.98), (2.74, 76.71), (1.23, 74.06), (2.48, 52.46), (2.87, 75.5), (2.78, 84.78), (3.95, 63.11), (5.04, 70.54), (1.82, 78.69), (1.48, 58.53), (1.85, 93.4), (1.39, 83.18), (4.46, 60.01), (3.39, 73.85), (2.12, 69.37), (0.82, 91.61), (2.9, 82.23), (-1.09, 77.89), (1.74, 78.83), (3.11, 70.1), (2.98, 88.8), (1.81, 91.58), (3.94, 66.29), (2.25, 69.1), (-0.5, 77.51), (4.4, 69.97), (3.92, 63.76), (1.69, 81.69), (8.04, 58.89), (2.51, 78.53), (2.77, 75.09), (4.78, 92.76), (5.7, 72.84), (6.39, 92.02), (2.34, 70.45), (7.57, 84.71), (0.07, 78.8), (3.01, 69.1), (2.31, 76.39), (0.95, 62.85), (4.37, 69.94), (4.91, 68.98), (4.35, 72.44), (7.79, 79.53), (4.39, 73.99), (3.12, 81.81), (1.39, 80.09), (3.69, 67.51), (3.04, 66.47), (3.3, 59.25), (3.14, 78.75), (1.4, 62.83), (0.82, 80.0), (4.04, 66.67), (3.65, 73.53), (4.54, 55.06), (3.13, 75.96), (2.36, 61.59), (5.15, 53.94), (-1.16, 65.41), (3.52, 56.95), (10.2, 64.98), (4.0, 54.85), (6.98, 99.75), (6.81, 60.83), (-0.07, 71.9), (3.09, 75.58), (1.89, 73.84), (0.68, 61.07), (2.96, 53.09), (1.11, 86.85), (1.49, 69.09), (4.21, 48.96), (6.91, 80.85), (3.86, 67.14), (3.16, 72.87), (3.84, 62.03), (6.65, 63.66), (1.83, 92.09), (3.68, 70.9), (2.68, 72.1), (5.26, 51.8), (2.38, 64.36), (0.71, 69.73), (4.01, 61.41), (4.25, 48.45), (3.15, 71.45), (7.02, 61.85), (0.85, 67.09), (1.07, 73.99), (3.36, 63.81), (3.87, 67.77), (5.31, 106.63), (1.98, 66.73), (0.51, 95.81), (8.89, 80.96), (4.85, 56.86), (2.72, 85.23), (4.25, 67.08), (7.72, 63.78), (2.85, 76.28), (5.23, 52.36), (2.85, 69.7), (3.7, 75.3), (4.51, 79.45), (2.49, 92.33), (2.34, 75.18), (0.48, 80.34), (0.14, 74.77), (7.07, 80.57), (6.0, 81.46), (1.21, 86.41), (3.4, 84.6), (4.63, 55.72), (5.41, 80.12), (2.97, 89.95), (2.84, 81.46), (1.57, 81.6), (6.03, 87.8), (3.67, 60.54), (-0.64, 67.11), (-0.48, 79.69), (5.45, 106.57), (0.12, 61.97), (1.44, 80.38), (4.74, 66.61), (5.97, 66.51), (5.93, 80.42), (2.09, 78.34), (3.26, 77.89), (4.01, 55.95), (-1.35, 76.99), (3.97, 80.44), (2.52, 73.99), (6.4, 81.4), (2.09, 70.31), (0.48, 64.29), (7.22, 73.45), (3.16, 78.98), (0.86, 94.18), (4.53, 89.04), (2.65, 67.46), (3.44, 86.95), (-0.19, 73.47), (4.43, 78.27), (3.62, 72.16), (5.9, 72.57), (1.62, 73.6), (4.71, 70.63), (4.34, 52.75), (4.89, 74.34), (2.09, 50.92), (3.04, 88.37), (0.57, 71.7), (6.1, 57.58), (4.79, 80.38), (4.01, 63.86), (3.9, 62.33), (2.47, 56.24), (4.78, 75.49), (5.56, 83.95), (-0.61, 58.38), (5.73, 62.32), (4.25, 74.36), (2.92, 74.6), (0.51, 72.73), (2.87, 53.73), (1.81, 57.53), (7.36, 70.37), (5.54, 90.27), (1.18, 67.67), (5.45, 64.38), (6.54, 84.53), (1.08, 66.17), (5.93, 70.59), (2.95, 83.88), (4.97, 66.65), (3.86, 82.3), (-0.23, 73.44), (-2.09, 54.26), (5.61, 73.31), (4.82, 48.69), (1.33, 105.31), (0.19, 59.33), (1.78, 82.97), (5.43, 68.26), (2.54, 78.12), (2.98, 78.19), (3.02, 72.85), (3.03, 60.2), (2.04, 87.79), (5.13, 76.43), (4.38, 82.15), (2.46, 85.68), (4.99, 71.85), (4.42, 59.2), (4.93, 69.37), (4.75, 68.46), (3.61, 70.19), (1.07, 64.45), (3.97, 66.21), (-0.2, 65.27), (2.9, 55.83), (-0.01, 67.21), (0.69, 77.58), (4.16, 66.41), (-2.38, 71.24), (3.29, 75.33), (-1.34, 69.72), (-0.1, 47.46), (3.98, 61.64), (4.4, 91.1), (0.14, 65.13), (4.61, 40.24), (-0.34, 87.99), (3.57, 73.49), (2.37, 72.02), (3.33, 87.24), (2.51, 65.95), (-0.79, 67.18), (2.31, 89.46), (5.55, 83.29), (1.3, 73.72), (4.55, 82.48), (4.7, 73.43), (0.67, 54.95), (3.08, 102.85), (2.27, 70.13), (0.3, 80.16), (1.6, 65.89), (5.05, 62.28), (3.07, 48.11), (1.51, 83.64), (0.12, 50.95), (3.46, 61.99), (1.05, 86.67), (1.64, 89.78), (0.25, 77.26), (5.64, 62.23), (2.77, 83.01), (3.25, 80.76), (1.85, 83.94), (1.23, 78.31), (5.32, 86.8), (4.36, 51.18), (3.82, 74.37), (0.59, 80.34), (0.53, 72.86), (2.26, 94.82), (2.2, 73.17), (3.15, 90.48), (3.72, 64.18), (3.79, 92.39), (7.09, 65.75), (-0.17, 61.77), (4.26, 82.95), (6.23, 84.21), (5.98, 97.08), (1.66, 76.74), (5.02, 77.37), (-0.76, 74.85), (4.38, 81.68), (3.94, 66.58), (2.59, 81.33), (1.33, 78.54), (3.91, 80.2), (6.91, 73.95), (4.19, 77.21), (3.44, 84.68), (1.28, 72.17), (0.72, 58.72), (3.36, 53.28), (2.3, 92.42), (3.45, 67.81), (3.95, 67.67), (2.21, 59.11), (5.93, 83.63), (2.22, 60.83), (3.5, 65.53), (0.6, 78.75), (3.07, 69.27), (4.05, 69.85), (2.11, 82.01), (2.52, 74.3), (9.43, 70.92), (1.73, 69.79), (0.63, 58.16), (2.72, 65.8), (2.11, 87.87), (3.67, 72.3), (2.57, 85.76), (2.71, 60.39), (5.8, 68.82), (3.69, 94.75), (-0.69, 95.87), (4.83, 39.82), (0.59, 75.51), (3.25, 74.19), (2.86, 67.12), (0.34, 61.27), (-1.62, 84.78), (-0.01, 69.09), (4.99, 69.61), (3.26, 76.59), (1.9, 86.77), (0.12, 56.37), (3.07, 50.05), (5.23, 76.82), (0.84, 62.14), (5.68, 62.24), (3.89, 53.66), (-0.1, 67.81), (1.27, 64.09), (2.06, 71.2), (5.21, 88.51), (1.57, 61.65), (3.02, 69.6), (3.64, 60.08), (4.14, 70.7), (-1.7, 67.33), (6.55, 75.83), (6.4, 81.4), (2.68, 64.0), (3.3, 71.37), (1.8, 84.27), (2.4, 69.99), (1.73, 70.15), (8.43, 71.95), (2.42, 59.92), (3.24, 58.5), (3.03, 79.7), (4.61, 81.6), (1.5, 78.75), (4.28, 71.88), (5.91, 78.67), (3.11, 57.27), (1.29, 67.38), (5.21, 70.97), (4.61, 69.42), (0.04, 91.84), (2.31, 73.27), (2.81, 55.82), (0.57, 93.4), (6.66, 73.97), (2.0, 77.36), (-0.53, 78.17), (4.26, 83.19), (2.72, 62.27), (3.64, 83.2), (0.44, 63.55), (0.58, 66.15), (2.9, 51.06), (5.07, 90.12), (0.54, 93.61), (4.69, 85.79), (2.05, 87.88), (1.95, 63.28), (1.71, 82.14), (-1.3, 90.5), (-0.82, 70.06), (3.62, 70.27), (2.78, 78.02), (0.79, 70.89), (3.75, 69.18), (1.5, 81.31), (2.7, 83.22), (1.53, 84.2), (3.34, 78.48), (0.92, 86.31), (6.14, 83.5), (4.14, 68.53), (1.67, 77.97), (3.85, 69.41), (0.37, 85.87), (1.27, 65.18), (6.23, 63.17), (2.48, 71.11), (2.18, 71.68), (0.56, 52.61), (-1.0, 60.06), (4.26, 99.98), (5.59, 79.92), (0.98, 67.46), (-0.12, 79.97), (4.47, 55.46), (4.5, 76.34), (6.49, 65.97), (1.29, 61.9), (3.74, 71.49), (1.43, 68.05), (4.48, 72.93), (1.46, 70.42), (2.19, 98.1), (3.68, 74.58), (5.45, 77.53), (7.87, 60.85), (1.12, 79.79), (6.51, 60.66), (4.04, 49.39), (6.97, 79.71), (3.04, 64.5), (3.67, 53.57), (4.92, 66.13), (0.46, 64.9), (2.69, 86.74), (7.06, 91.53), (3.64, 80.78), (1.34, 72.71), (0.98, 59.42), (2.46, 79.19), (4.33, 77.26), (2.44, 80.07), (1.44, 70.47), (2.47, 76.57), (3.27, 48.27), (5.77, 64.11), (0.33, 65.02), (3.01, 75.89), (5.95, 75.7), (3.16, 66.96), (2.71, 82.06), (4.19, 56.32), (4.63, 64.28), (6.07, 62.22), (4.09, 73.04), (1.23, 72.97), (5.26, 54.35), (2.01, 59.29), (3.29, 69.06), (0.47, 62.19), (1.3, 67.57), (-0.46, 91.91), (3.45, 56.82), (3.48, 84.51), (2.48, 84.36), (4.86, 59.64), (7.26, 84.57), (2.1, 65.07), (4.5, 69.96), (6.55, 45.54), (4.23, 65.02), (4.5, 79.43), (5.53, 58.54), (5.84, 69.85), (7.65, 68.57), (1.09, 55.2), (2.41, 50.57), (3.18, 75.74), (4.18, 68.79), (4.8, 54.26), (2.67, 50.91), (1.83, 65.33), (2.44, 85.88), (4.99, 67.56), (2.45, 103.74), (1.53, 44.4), (3.9, 95.5), (0.95, 84.87), (1.28, 60.24), (1.78, 78.72), (3.82, 59.93), (5.88, 87.93), (3.4, 98.01), (3.47, 77.41), (7.35, 79.41), (1.79, 94.23), (7.63, 86.02), (4.66, 83.07), (2.91, 64.97), (4.49, 48.11), (1.88, 50.34), (4.79, 54.57), (1.08, 63.14), (6.25, 87.91), (3.83, 68.61), (2.56, 92.34), (3.39, 75.42), (2.41, 74.04), (6.26, 86.42), (1.49, 82.86), (5.03, 95.49), (1.49, 87.53), (6.05, 72.53), (7.33, 57.08), (7.41, 68.09), (5.56, 72.15), (3.47, 65.4), (5.16, 77.02), (5.85, 67.57), (5.43, 77.21), (2.15, 76.95), (1.43, 79.78), (1.73, 63.32), (1.08, 66.34), (3.24, 67.9), (0.33, 81.27), (2.44, 90.82), (4.15, 42.93), (3.42, 48.15), (4.36, 51.01), (1.58, 61.58), (5.06, 68.32), (0.44, 87.08), (0.76, 67.84), (6.11, 67.82), (2.58, 83.67), (1.68, 55.45), (6.6, 63.56), (2.28, 69.4), (1.72, 57.75), (2.57, 75.56), (2.41, 77.86), (6.57, 66.16), (1.83, 75.3), (5.39, 68.03), (5.65, 95.83), (4.32, 79.99), (2.2, 76.73), (3.52, 73.19), (2.44, 77.87), (7.73, 58.15), (5.52, 56.09), (2.11, 84.76), (-0.65, 72.31), (1.57, 75.93), (1.5, 80.21), (2.37, 70.49), (3.72, 85.49), (2.71, 64.77), (3.76, 62.48), (2.24, 89.36), (2.08, 79.79), (3.39, 71.99), (7.47, 73.05), (1.73, 72.99), (3.1, 59.06), (0.55, 48.75), (3.74, 90.69), (3.0, 82.22), (5.87, 59.72), (3.17, 69.48), (0.89, 82.91), (0.73, 89.26), (1.14, 59.85), (0.14, 71.21), (4.04, 57.53), (4.19, 57.4), (4.58, 68.86), (5.25, 67.74), (2.92, 61.91), (4.1, 49.44), (3.5, 60.12), (-0.1, 86.91), (4.85, 60.82), (5.51, 65.4), (3.0, 66.23), (2.15, 77.55), (6.97, 75.14), (-0.06, 93.35), (1.31, 66.75), (2.97, 83.3), (2.33, 83.14), (5.32, 75.5), (4.07, 72.46), (1.32, 65.47), (3.39, 81.42), (4.52, 67.7), (-0.3, 65.21), (2.29, 67.85), (2.5, 97.24), (0.75, 91.84), (1.11, 68.57), (3.27, 67.9), (2.6, 75.16), (2.46, 72.6), (0.92, 59.23), (2.97, 74.41), (-0.75, 68.04), (3.04, 70.91), (4.55, 95.84), (3.22, 76.65), (1.19, 79.11), (-0.04, 76.9), (3.05, 60.19), (2.59, 59.51), (3.9, 77.34), (-1.94, 75.73), (1.91, 63.65), (3.36, 54.32), (-0.24, 70.65), (2.27, 60.72), (3.11, 74.32), (2.75, 80.03), (4.78, 72.71), (-0.74, 71.11), (-0.57, 49.95), (2.97, 83.19), (4.2, 86.87), (3.97, 67.17), (3.61, 92.27), (4.45, 80.64), (6.12, 76.22), (2.42, 72.31), (3.21, 60.82), (4.58, 69.67), (0.19, 66.5), (5.51, 72.41), (4.15, 84.66), (2.24, 76.63), (0.31, 84.38), (3.21, 83.44), (-0.5, 69.44), (2.76, 86.58), (5.32, 60.21), (4.77, 50.5), (0.72, 73.24), (6.17, 87.42), (6.3, 75.54), (4.84, 76.3), (3.44, 82.97), (1.41, 90.41), (3.07, 64.22), (3.92, 60.35), (4.68, 60.86), (3.06, 72.21), (3.51, 53.06), (3.51, 65.02), (2.48, 47.96), (4.38, 79.42), (4.47, 49.77), (3.33, 58.11), (-0.02, 67.67), (0.67, 81.07), (4.61, 79.36), (4.88, 70.63), (0.89, 58.26), (4.1, 75.19), (4.07, 69.34), (2.67, 81.39), (1.86, 79.95), (7.76, 70.9), (4.02, 88.16), (0.82, 78.53), (2.34, 79.3), (2.84, 73.21), (2.63, 77.51), (5.82, 40.82), (4.23, 84.17), (2.16, 79.17), (5.65, 61.01), (4.05, 96.77), (2.84, 63.92), (2.81, 83.53), (5.58, 87.53), (4.13, 78.01), (3.34, 86.98), (5.38, 67.23), (3.0, 78.74), (2.68, 62.5), (5.62, 85.85), (5.61, 55.05), (1.52, 82.3), (1.49, 69.24), (6.08, 68.98), (1.57, 68.7), (5.23, 82.82), (1.47, 54.55), (0.86, 87.57), (0.88, 75.15)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=3.0898，ȳ=72.5904；Sxx=Σ(x−x̄)²=3382.95，Sxy=-865.64；
β₁ = Sxy/Sxx = -0.2559；β₀ = ȳ−β₁x̄ = 73.3811；
R² = SSR/SST = 221.50/114450.55 = 0.0019；F = (SSR/1)/(SSE/815) = 1.5804，p = 0.2091；
SE(β₁)=√(140.16/3382.95)=0.2035；t=β₁/SE=-1.2571；p=0.2091；95%CI=[-5.1236,4.6119]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=3.0898，ȳ=72.5904；Sxx=Σ(x−x̄)²=3382.95，Sxy=-865.64；
β₁ = Sxy/Sxx = -0.2559；β₀ = ȳ−β₁x̄ = 73.3811；
R² = SSR/SST = 221.50/114450.55 = 0.0019；F = (SSR/1)/(SSE/815) = 1.5804，p = 0.2091；
SE(β₁)=√(140.16/3382.95)=0.2035；t=β₁/SE=-1.2571；p=0.2091；95%CI=[-5.1236,4.6119]。
③ 计算结果：
y = 73.3811 + -0.2559·x；R²=0.0019；β₁ t=-1.2571 p=0.2091；95%CI=[-5.1236,4.6119]
④ 结论：在 α=0.05 水平下，p=0.2091 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=-0.2559），应无影响。
案例 70. 头发颜色预测跑速
简单线性回归应无影响
① 原始数据：
样本 n=311，逐条 (x, y)：[(5.92, 14.25), (2.99, 14.63), (2.93, 11.31), (3.19, 10.55), (3.17, 16.59), (3.4, 14.04), (2.2, 14.79), (3.01, 15.0), (2.14, 21.2), (3.28, 14.22), (3.19, 11.52), (2.56, 16.88), (3.03, 13.19), (2.69, 13.81), (2.86, 12.32), (3.42, 14.58), (1.33, 15.86), (0.42, 14.81), (1.93, 11.5), (1.98, 16.73), (4.24, 14.76), (1.53, 15.34), (1.76, 13.02), (3.41, 22.24), (4.01, 12.57), (1.18, 12.89), (4.39, 18.42), (3.88, 16.01), (4.02, 17.11), (4.24, 15.19), (4.6, 19.51), (4.42, 13.63), (3.54, 5.21), (2.07, 15.07), (4.21, 19.77), (3.14, 17.73), (0.53, 20.77), (3.31, 11.04), (3.23, 14.64), (5.08, 16.78), (2.98, 9.22), (2.58, 18.85), (4.49, 13.66), (2.23, 13.03), (2.52, 17.29), (2.58, 15.0), (1.48, 16.14), (1.83, 8.2), (3.86, 16.66), (2.09, 18.86), (3.07, 13.75), (2.93, 15.57), (2.7, 11.77), (3.11, 16.19), (2.59, 18.24), (4.16, 10.86), (3.14, 15.84), (1.94, 14.68), (3.29, 14.23), (2.35, 15.87), (2.26, 14.03), (2.69, 10.55), (2.47, 16.57), (2.22, 13.54), (2.42, 17.15), (2.82, 9.89), (2.48, 11.01), (2.3, 16.47), (4.05, 14.92), (2.72, 19.3), (1.52, 11.63), (2.43, 17.3), (1.74, 15.23), (0.67, 9.32), (2.28, 13.98), (2.56, 15.3), (3.17, 16.78), (2.79, 15.35), (2.97, 16.02), (3.15, 10.44), (2.19, 17.57), (2.02, 14.93), (1.89, 22.17), (4.64, 13.38), (4.1, 11.67), (2.54, 20.38), (4.18, 10.9), (3.6, 16.23), (3.86, 13.11), (3.53, 16.45), (3.09, 19.05), (3.54, 9.91), (4.25, 17.14), (2.76, 20.72), (2.42, 14.96), (1.26, 14.57), (2.11, 11.83), (4.07, 16.6), (3.16, 12.66), (4.33, 19.89), (2.92, 15.01), (2.32, 12.63), (3.41, 13.06), (2.28, 16.06), (3.83, 22.58), (3.7, 11.72), (2.14, 18.6), (4.97, 13.18), (2.78, 18.14), (4.43, 11.75), (3.23, 17.27), (4.08, 18.58), (3.34, 12.8), (3.06, 17.56), (2.44, 15.91), (2.91, 15.59), (2.66, 14.07), (3.09, 12.69), (3.49, 13.01), (3.32, 13.48), (3.25, 18.32), (2.58, 14.74), (2.35, 18.58), (2.35, 14.31), (2.45, 10.66), (2.42, 16.97), (3.53, 16.81), (3.71, 19.07), (3.96, 17.64), (2.81, 15.28), (2.93, 15.78), (4.52, 16.02), (2.08, 10.57), (1.44, 17.79), (3.18, 12.03), (4.15, 16.44), (1.46, 20.62), (2.29, 13.33), (3.81, 13.53), (2.92, 13.49), (2.25, 11.04), (3.91, 18.91), (3.32, 18.85), (-0.19, 19.62), (3.24, 10.72), (4.35, 14.86), (3.11, 13.47), (0.81, 11.92), (3.24, 16.13), (2.19, 13.44), (2.49, 17.46), (2.81, 13.66), (1.58, 14.82), (1.69, 12.7), (4.34, 13.99), (1.96, 15.53), (2.55, 18.91), (3.06, 20.26), (2.36, 8.88), (3.66, 19.37), (3.28, 12.39), (2.88, 15.11), (1.44, 13.13), (2.87, 16.3), (3.35, 16.92), (2.49, 17.15), (3.68, 17.11), (2.99, 14.12), (3.49, 18.62), (2.18, 16.08), (0.88, 17.67), (3.05, 11.62), (1.91, 14.59), (2.38, 13.89), (3.62, 15.28), (3.8, 20.82), (2.88, 12.76), (2.68, 9.09), (4.61, 16.49), (3.14, 10.41), (2.17, 13.35), (1.44, 9.7), (2.54, 17.37), (2.33, 13.81), (2.97, 11.92), (1.5, 17.41), (2.57, 13.08), (3.78, 21.67), (3.6, 15.5), (2.82, 12.21), (2.86, 19.24), (3.75, 15.85), (2.54, 13.94), (1.35, 15.38), (3.78, 17.92), (1.54, 14.87), (3.39, 12.94), (2.88, 15.08), (3.74, 20.09), (3.76, 16.66), (2.58, 16.51), (2.41, 17.13), (3.99, 16.11), (3.77, 18.94), (2.34, 17.31), (2.93, 11.59), (3.01, 12.6), (0.46, 12.72), (2.35, 15.99), (1.12, 10.42), (2.2, 16.74), (2.45, 12.01), (3.14, 12.16), (1.44, 15.36), (3.38, 12.61), (3.52, 19.21), (1.92, 11.83), (2.4, 14.26), (3.79, 15.07), (5.11, 11.77), (1.56, 11.28), (3.45, 9.93), (3.83, 17.05), (3.26, 13.41), (4.34, 16.41), (4.04, 20.79), (3.72, 11.79), (2.78, 17.63), (3.32, 22.2), (2.88, 18.33), (2.75, 17.84), (3.0, 14.08), (3.99, 12.16), (1.76, 19.34), (3.26, 11.68), (3.08, 16.45), (3.41, 10.64), (2.76, 19.99), (3.75, 11.83), (3.64, 16.08), (2.94, 11.74), (4.86, 21.7), (4.42, 12.33), (3.38, 15.16), (2.85, 19.9), (1.77, 20.09), (2.7, 10.75), (1.91, 17.36), (4.52, 10.4), (3.6, 15.19), (3.67, 16.07), (2.92, 16.66), (1.55, 15.83), (4.15, 14.25), (3.61, 18.92), (2.68, 15.05), (5.53, 15.71), (2.75, 12.75), (3.39, 16.85), (2.19, 16.6), (3.55, 21.95), (2.59, 14.74), (2.65, 13.84), (0.23, 12.95), (2.83, 14.86), (2.91, 13.94), (2.65, 16.53), (3.7, 16.37), (1.66, 13.5), (3.34, 12.55), (3.42, 16.17), (2.75, 17.24), (3.68, 14.66), (3.54, 16.94), (3.0, 12.83), (4.17, 19.28), (2.7, 15.79), (3.73, 16.82), (3.43, 16.09), (0.81, 10.15), (2.09, 10.64), (2.56, 10.09), (3.66, 17.39), (4.53, 14.92), (3.93, 14.69), (3.37, 14.28), (2.81, 11.97), (5.63, 15.08), (1.91, 16.6), (2.69, 12.63), (1.76, 14.84), (1.13, 14.99), (4.06, 21.86), (4.03, 14.3), (2.96, 12.3), (5.22, 17.29), (4.87, 15.58), (2.28, 15.33), (3.48, 9.32), (3.16, 10.57), (2.58, 17.03), (1.55, 16.51), (3.31, 16.23), (4.2, 16.42), (2.31, 14.43), (2.79, 16.39), (2.39, 16.45), (2.75, 13.32), (1.32, 17.77), (2.94, 15.07), (1.45, 11.38)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=2.9427，ȳ=15.0980；Sxx=Σ(x−x̄)²=294.84，Sxy=81.00；
β₁ = Sxy/Sxx = 0.2747；β₀ = ȳ−β₁x̄ = 14.2896；
R² = SSR/SST = 22.25/2778.31 = 0.0080；F = (SSR/1)/(SSE/309) = 2.4947，p = 0.1152；
SE(β₁)=√(8.92/294.84)=0.1739；t=β₁/SE=1.5795；p=0.1152；95%CI=[-2.7826,3.3321]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=2.9427，ȳ=15.0980；Sxx=Σ(x−x̄)²=294.84，Sxy=81.00；
β₁ = Sxy/Sxx = 0.2747；β₀ = ȳ−β₁x̄ = 14.2896；
R² = SSR/SST = 22.25/2778.31 = 0.0080；F = (SSR/1)/(SSE/309) = 2.4947，p = 0.1152；
SE(β₁)=√(8.92/294.84)=0.1739；t=β₁/SE=1.5795；p=0.1152；95%CI=[-2.7826,3.3321]。
③ 计算结果：
y = 14.2896 + 0.2747·x；R²=0.0080；β₁ t=1.5795 p=0.1152；95%CI=[-2.7826,3.3321]
④ 结论：在 α=0.05 水平下，p=0.1152 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=0.2747），应无影响。
案例 71. 姓名笔画预测收入
简单线性回归应无影响
① 原始数据：
样本 n=610，逐条 (x, y)：[(6.56, 37.82), (20.04, 10.63), (14.08, 1.2), (6.65, 16.87), (17.3, 21.02), (17.28, 19.1), (9.42, 24.33), (8.09, 12.71), (28.69, 24.99), (15.57, 19.53), (4.76, 23.86), (10.88, 17.41), (16.85, 23.06), (11.72, 24.65), (5.11, 8.44), (9.0, 19.69), (18.04, 17.95), (19.98, 36.96), (14.52, 19.68), (11.06, 21.81), (-7.82, 26.74), (19.77, 17.09), (5.46, 15.45), (10.9, 22.76), (28.79, 34.63), (17.59, 22.61), (23.29, 20.98), (7.29, 8.68), (16.69, 30.54), (18.44, 14.25), (5.99, 13.35), (20.75, 17.4), (16.41, 28.12), (13.17, 29.15), (5.91, 30.5), (23.04, 17.75), (26.15, 22.48), (18.49, 13.59), (15.25, 11.66), (21.2, 17.9), (10.89, 10.93), (6.65, 16.16), (19.18, 17.1), (26.5, 11.82), (22.89, 20.52), (19.04, 17.21), (4.25, 2.36), (15.69, 10.54), (25.93, 19.49), (10.97, 21.25), (5.3, 18.51), (19.96, 14.45), (25.51, 28.69), (29.59, 29.77), (20.12, 24.43), (17.89, 14.1), (30.76, 21.05), (19.16, 20.95), (19.05, 9.17), (24.55, 34.83), (14.46, 21.35), (10.76, 10.91), (20.65, 17.22), (18.96, 22.62), (15.03, 18.7), (19.34, 17.92), (33.28, 25.82), (11.24, 28.08), (39.5, 21.85), (11.88, 3.83), (16.38, 23.1), (5.13, 26.95), (14.0, 30.48), (18.62, 34.29), (13.74, 20.61), (20.71, 3.03), (16.45, 5.61), (8.82, 25.97), (12.18, 23.62), (21.78, 25.5), (-1.35, 8.13), (12.41, 13.28), (14.15, 27.88), (19.04, 12.21), (15.48, 21.94), (14.6, 24.2), (23.15, 20.43), (26.53, 22.82), (14.85, 9.07), (16.26, 25.74), (16.57, 32.92), (10.52, 14.96), (15.95, 28.3), (14.59, 25.61), (23.07, 28.13), (23.83, 13.93), (15.28, 14.32), (25.44, 33.39), (24.11, 13.65), (14.49, -0.16), (2.47, 28.27), (8.98, 19.26), (-1.07, 30.14), (5.23, 18.41), (14.89, 23.66), (19.33, 18.62), (-6.22, 15.45), (2.81, -3.38), (17.7, 20.97), (25.98, 31.26), (19.18, 21.88), (17.51, 19.73), (15.03, 21.15), (8.81, 8.97), (18.38, 12.22), (5.65, 23.9), (14.38, 14.39), (15.23, 27.94), (13.34, 22.15), (11.99, 24.77), (20.94, 21.13), (18.89, 23.5), (19.95, 25.61), (12.05, 15.71), (10.08, 5.46), (19.65, 12.71), (33.02, 18.42), (13.11, 13.0), (8.33, 16.33), (17.65, 21.84), (2.18, 17.84), (15.41, 21.46), (-0.81, 19.2), (11.13, 25.55), (13.63, 24.8), (8.71, 34.44), (-5.01, 25.02), (19.75, 19.86), (7.89, 37.3), (25.96, 30.22), (23.61, 26.18), (23.21, 28.13), (31.21, 19.27), (16.18, 34.37), (8.43, 26.78), (19.89, 15.77), (16.4, 16.5), (18.24, 23.24), (15.25, 30.25), (9.68, 23.41), (12.26, 16.92), (7.76, 18.14), (15.75, 17.6), (24.45, 36.81), (18.76, 18.1), (22.83, 18.71), (5.29, 25.73), (-1.46, 18.88), (16.62, 20.78), (16.57, 18.13), (15.99, 21.89), (24.92, 16.12), (9.55, 12.21), (24.76, 20.68), (7.84, 35.03), (14.62, 20.55), (27.59, 17.81), (21.24, 19.48), (25.83, 21.06), (10.38, 25.29), (13.54, 26.94), (21.94, 24.13), (23.41, 27.34), (14.1, 20.65), (18.39, 26.74), (20.46, 19.88), (23.1, 30.47), (5.89, 28.7), (20.14, 21.96), (16.57, 9.75), (18.92, 11.45), (23.09, 34.4), (8.66, 26.32), (6.47, 24.04), (15.55, 20.18), (1.01, 31.48), (21.27, 17.4), (10.55, 19.92), (25.85, 13.66), (23.22, 18.23), (19.37, 25.41), (13.53, 24.19), (15.77, 21.72), (21.37, 29.74), (15.36, 22.04), (5.1, 32.35), (8.93, 21.06), (13.11, 1.21), (10.25, 24.41), (5.78, 12.32), (6.6, 19.15), (21.22, 12.37), (4.27, 10.52), (24.03, 22.9), (20.66, 22.03), (20.3, 30.69), (26.1, 18.71), (3.06, 22.45), (20.79, 24.53), (15.39, 26.03), (5.01, 29.85), (12.19, 18.54), (12.72, 25.55), (17.2, 21.37), (13.67, 6.59), (11.27, 26.06), (2.13, 23.85), (14.81, 24.86), (13.92, 19.81), (-2.46, 26.98), (15.43, 26.42), (8.87, 14.59), (11.3, 12.59), (24.34, 5.18), (2.14, 13.41), (13.94, 23.78), (14.37, 15.09), (33.65, 19.09), (25.11, 26.59), (8.48, 25.32), (27.12, 18.96), (6.89, 21.58), (2.99, 22.12), (20.76, 5.59), (15.41, 18.07), (8.96, 22.29), (22.87, 26.34), (20.84, 28.15), (3.14, 20.48), (32.59, 20.05), (-2.81, 18.99), (8.84, 12.69), (10.28, 9.13), (12.58, 6.11), (19.73, 18.0), (14.48, 12.61), (25.95, 32.25), (32.01, 18.75), (3.27, 27.13), (7.62, 26.32), (14.26, 14.7), (12.93, 15.22), (22.62, 38.85), (11.24, 26.71), (7.58, 27.25), (4.1, 14.34), (9.9, 21.42), (13.85, 25.86), (8.39, 13.56), (27.57, 4.15), (13.49, 19.14), (19.61, 13.54), (6.23, 20.65), (-1.64, 23.15), (17.54, 17.24), (19.27, 10.54), (15.91, 28.9), (1.13, 23.23), (2.82, 25.64), (16.22, 17.33), (-10.22, 31.49), (18.55, 27.07), (12.07, 19.18), (29.15, 11.4), (24.93, 22.32), (6.76, 24.59), (10.54, 21.97), (28.58, 18.65), (6.77, 25.2), (7.07, 26.98), (13.83, 31.74), (25.49, 28.15), (14.52, 25.44), (25.95, 13.95), (12.64, 28.52), (5.26, 31.71), (13.64, 14.32), (17.78, 27.64), (4.79, 5.87), (10.36, 16.93), (7.44, 5.43), (9.2, 23.05), (22.71, 20.81), (9.58, 20.95), (13.57, 24.38), (11.32, 13.61), (22.63, 21.72), (13.76, 26.3), (7.74, 36.43), (33.07, 23.69), (25.72, 26.22), (7.69, 16.82), (13.13, 13.99), (7.2, 17.71), (15.85, 28.02), (13.91, 19.47), (20.68, 6.41), (16.9, 29.48), (18.86, 35.5), (31.76, 14.46), (26.91, 28.29), (11.87, 10.19), (11.21, 38.05), (9.39, 24.47), (18.92, 19.33), (10.43, 21.99), (12.98, 8.69), (16.94, 15.43), (7.81, 19.44), (22.29, 27.08), (27.82, 5.11), (16.25, 13.7), (24.95, 17.86), (10.85, -1.06), (15.33, 24.06), (13.62, 23.01), (15.77, 35.66), (26.93, 14.31), (13.03, 11.82), (15.18, 30.97), (18.39, 18.14), (9.76, 17.66), (20.38, 26.04), (11.75, 18.08), (22.17, 25.85), (-9.38, 31.99), (11.94, 24.59), (15.51, 25.42), (20.75, 30.23), (20.86, 11.09), (21.33, 23.48), (9.18, 14.4), (14.96, 21.41), (3.18, 17.4), (17.87, 16.05), (8.42, 33.89), (28.53, 17.39), (24.35, 23.33), (15.32, 29.52), (6.84, 19.04), (21.11, 26.99), (32.42, 10.18), (19.93, 19.29), (8.29, 10.86), (27.38, 21.7), (5.0, 18.6), (27.94, 26.66), (9.78, 6.56), (26.3, 26.05), (17.94, 15.88), (22.54, 13.86), (7.68, 23.34), (24.82, 14.8), (13.54, 17.65), (24.67, 31.01), (32.15, 23.54), (3.68, 20.7), (18.18, 25.8), (18.28, 21.81), (5.03, 20.0), (15.72, 27.26), (24.15, 27.73), (5.03, 4.68), (-0.93, 14.87), (15.62, 11.04), (13.85, 19.85), (7.14, -0.98), (18.38, 13.07), (4.58, 7.96), (14.23, 15.69), (1.47, 19.48), (20.18, 25.75), (22.17, 23.65), (20.26, 13.24), (20.96, 15.81), (11.39, 25.52), (11.97, 28.65), (11.16, 15.87), (13.0, 34.78), (6.23, 7.3), (14.01, 17.77), (9.01, 14.84), (11.42, 17.2), (16.55, 25.14), (20.04, 14.07), (-2.52, 14.29), (12.15, 35.73), (2.98, 21.42), (13.73, 16.62), (19.75, 13.24), (12.14, 7.37), (10.85, 19.25), (8.59, 31.31), (13.11, 9.65), (10.66, 9.12), (5.71, 16.42), (14.53, 14.09), (5.24, 22.23), (6.42, 17.62), (4.53, 20.72), (23.31, 20.9), (-2.71, 11.66), (15.76, 24.3), (11.57, 12.47), (5.21, 8.58), (18.56, 23.34), (12.42, 19.99), (8.59, 22.61), (14.44, 17.97), (24.99, 16.28), (8.26, 31.55), (14.59, 16.21), (20.27, 23.25), (16.53, 28.26), (12.26, 16.28), (21.84, 21.52), (21.44, 18.54), (15.48, 24.32), (-0.48, 20.87), (32.8, 19.42), (22.15, 26.63), (24.89, 30.7), (4.24, 8.61), (20.81, 17.07), (19.38, 16.11), (11.64, 18.14), (3.86, 35.43), (19.95, 26.08), (12.47, 25.37), (0.96, 32.4), (26.44, 14.36), (18.96, 25.87), (20.25, 30.48), (13.4, 22.59), (10.88, 22.11), (21.43, 10.09), (15.44, 18.4), (9.52, 14.08), (15.95, 26.09), (15.46, 26.36), (7.13, 10.87), (14.81, 4.71), (19.98, 16.34), (14.04, 35.84), (12.67, 27.83), (22.19, 19.78), (13.5, 24.6), (21.94, 12.07), (26.06, 9.33), (14.29, 31.43), (22.25, 28.44), (21.74, 32.11), (18.41, 28.78), (11.64, 26.47), (21.29, 11.94), (6.28, 11.17), (22.66, 22.58), (13.27, 23.77), (21.05, 15.95), (27.23, 25.74), (12.88, 26.98), (10.8, 36.44), (25.56, 15.5), (11.42, 21.4), (4.3, 29.22), (7.65, 13.94), (3.77, 8.64), (21.25, 22.92), (21.05, 20.79), (13.69, 24.82), (15.4, 31.46), (6.38, 19.15), (1.99, 22.84), (7.5, 19.74), (23.29, 17.35), (9.79, 22.77), (15.13, 5.48), (15.55, 4.69), (5.71, 25.21), (-9.27, 15.21), (10.79, 15.95), (14.79, 11.22), (17.53, 13.92), (22.86, 21.68), (8.35, 18.7), (9.24, 18.27), (14.91, 20.78), (17.23, 29.62), (15.9, 30.26), (13.48, 11.42), (12.93, 1.42), (19.3, 17.01), (-7.37, 19.3), (5.54, 15.73), (15.09, 23.43), (-0.12, 21.83), (18.43, 13.31), (5.2, 26.46), (6.89, 38.46), (7.81, 14.08), (3.98, 25.4), (13.06, 25.77), (25.75, 34.32), (2.26, 42.48), (22.75, 24.54), (8.72, 19.58), (8.77, 19.54), (24.01, 10.38), (3.56, 24.85), (8.85, 5.33), (28.06, 24.67), (11.62, 23.81), (5.62, 17.42), (11.71, 16.84), (13.33, 7.79), (13.63, 25.74), (14.98, 22.75), (23.98, 22.98), (18.74, 7.28), (9.65, 12.29), (17.0, 21.56), (22.56, 22.09), (1.77, 16.56), (20.6, 30.41), (13.26, 29.09), (14.36, 24.2), (8.38, 38.51), (21.74, 19.65), (31.43, 31.48), (17.29, 21.1), (14.1, 29.22), (18.63, 21.04), (33.43, 19.55), (11.74, 21.35), (20.98, 28.27), (27.04, 17.58), (15.14, 15.8), (-6.2, 25.82), (9.18, 7.83), (10.19, 23.46), (11.67, 16.56), (1.36, 22.76), (20.27, 11.0), (13.89, 27.42), (26.12, 18.15), (1.51, 29.05), (15.01, 22.38), (19.88, 5.29), (5.43, 11.57), (11.11, 15.01), (21.05, 33.91), (13.86, 30.24), (-3.21, 26.48), (6.83, 11.77), (21.81, 21.11), (29.64, 18.33), (8.93, 14.1), (17.22, 2.35), (9.35, 33.07), (10.66, 32.74), (12.18, 27.96), (18.05, 11.96), (18.87, 17.33), (18.27, 34.42), (4.21, 8.28), (19.99, 13.97), (16.84, 23.04), (9.47, 34.01), (13.45, 20.53), (-1.13, 10.03), (20.92, 38.02), (27.47, 19.84), (6.98, 31.16), (19.08, 15.53), (9.46, 21.75), (17.65, 15.23), (21.92, 21.5), (8.5, 22.71), (0.64, 14.76), (14.93, 28.53), (14.95, 23.55), (13.48, 16.78), (11.04, 25.72), (9.0, 18.8), (6.54, 35.76), (24.83, 29.8), (16.23, 10.26), (10.99, 20.6), (18.73, 14.12), (17.99, 11.78), (6.64, 21.24), (17.21, 18.95), (11.52, 23.36), (6.45, 17.64), (30.29, 9.62), (6.97, 15.5), (19.17, 20.2), (18.6, 26.93), (8.57, 28.16), (13.06, 27.84)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=14.6423，ȳ=20.5552；Sxx=Σ(x−x̄)²=38294.14，Sxy=926.75；
β₁ = Sxy/Sxx = 0.0242；β₀ = ȳ−β₁x̄ = 20.2009；
R² = SSR/SST = 22.43/36123.56 = 0.0006；F = (SSR/1)/(SSE/608) = 0.3777，p = 0.5391；
SE(β₁)=√(59.38/38294.14)=0.0394；t=β₁/SE=0.6146；p=0.5391；95%CI=[-0.0531,0.1015]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=14.6423，ȳ=20.5552；Sxx=Σ(x−x̄)²=38294.14，Sxy=926.75；
β₁ = Sxy/Sxx = 0.0242；β₀ = ȳ−β₁x̄ = 20.2009；
R² = SSR/SST = 22.43/36123.56 = 0.0006；F = (SSR/1)/(SSE/608) = 0.3777，p = 0.5391；
SE(β₁)=√(59.38/38294.14)=0.0394；t=β₁/SE=0.6146；p=0.5391；95%CI=[-0.0531,0.1015]。
③ 计算结果：
y = 20.2009 + 0.0242·x；R²=0.0006；β₁ t=0.6146 p=0.5391；95%CI=[-0.0531,0.1015]
④ 结论：在 α=0.05 水平下，p=0.5391 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=0.0242），应无影响。
案例 72. 早餐与否预测智商
简单线性回归应无影响
① 原始数据：
样本 n=312，逐条 (x, y)：[(1.37, 105.57), (-0.03, 110.19), (0.39, 92.36), (1.37, 83.64), (0.89, 103.99), (0.68, 80.21), (0.89, 93.99), (1.0, 92.8), (1.57, 98.57), (1.01, 110.33), (1.27, 114.2), (0.68, 106.54), (1.66, 72.3), (1.15, 97.3), (0.92, 90.77), (0.62, 101.56), (1.33, 120.88), (0.78, 109.34), (0.75, 108.45), (0.02, 69.69), (0.73, 76.59), (0.9, 91.19), (1.58, 118.1), (1.28, 104.25), (0.41, 103.43), (0.45, 93.5), (1.17, 95.78), (0.56, 105.95), (0.12, 95.11), (1.47, 110.2), (0.32, 101.44), (1.2, 106.03), (0.53, 96.94), (0.89, 96.17), (0.55, 133.52), (0.66, 107.89), (1.46, 96.9), (1.15, 112.45), (0.88, 73.2), (1.18, 112.15), (0.25, 113.81), (0.92, 104.08), (1.38, 114.01), (1.81, 99.82), (1.02, 118.57), (0.88, 97.42), (1.84, 121.25), (1.54, 106.24), (1.16, 92.19), (1.61, 79.12), (1.03, 83.24), (0.71, 84.27), (1.11, 74.15), (1.53, 71.58), (1.38, 79.09), (1.42, 114.87), (0.86, 82.13), (1.11, 99.91), (1.76, 116.45), (0.59, 88.2), (0.35, 88.36), (1.69, 79.43), (0.77, 114.46), (0.88, 94.11), (0.85, 111.13), (0.61, 111.32), (0.93, 92.58), (1.09, 74.26), (0.85, 117.69), (0.97, 109.64), (1.08, 116.89), (1.37, 98.56), (0.99, 127.68), (1.03, 84.67), (0.61, 87.39), (1.26, 92.63), (0.19, 92.96), (0.8, 102.81), (1.11, 123.77), (1.3, 80.35), (0.78, 106.28), (0.68, 98.29), (2.54, 84.86), (1.58, 105.44), (0.27, 108.78), (0.08, 92.94), (1.28, 113.29), (1.65, 96.53), (2.16, 117.14), (0.53, 95.15), (0.19, 78.72), (0.87, 107.72), (1.51, 103.97), (1.08, 120.7), (1.04, 88.54), (1.16, 90.59), (0.06, 105.64), (0.91, 122.92), (0.86, 95.53), (1.63, 88.7), (0.32, 114.34), (0.13, 112.82), (-0.16, 92.97), (1.07, 96.82), (0.71, 93.04), (1.25, 107.16), (1.15, 114.85), (0.71, 112.31), (1.27, 93.16), (0.6, 88.92), (0.38, 90.43), (0.09, 117.63), (1.16, 124.63), (0.98, 56.72), (0.0, 97.36), (1.11, 110.8), (0.51, 104.75), (0.93, 107.26), (0.94, 101.35), (1.47, 80.46), (0.64, 105.04), (1.14, 122.67), (0.98, 78.02), (1.38, 113.27), (0.85, 99.12), (0.71, 85.92), (1.72, 84.36), (0.71, 117.02), (0.74, 117.15), (1.52, 89.81), (0.55, 98.23), (1.1, 76.01), (0.94, 95.64), (0.73, 93.34), (1.01, 107.69), (1.06, 111.49), (1.07, 103.78), (0.4, 97.78), (1.36, 116.29), (1.69, 118.72), (1.35, 84.26), (0.87, 105.16), (1.14, 84.3), (1.2, 112.76), (1.46, 100.05), (0.86, 119.88), (0.19, 91.72), (0.82, 95.6), (1.69, 114.29), (0.41, 94.23), (1.87, 114.99), (0.7, 100.62), (0.6, 112.54), (1.37, 93.95), (1.14, 80.21), (0.78, 95.24), (0.58, 108.3), (1.87, 85.44), (1.5, 78.63), (0.77, 98.71), (0.49, 111.29), (1.19, 94.45), (1.84, 90.08), (1.81, 112.14), (-0.18, 70.21), (-0.02, 116.71), (0.87, 89.2), (0.41, 93.04), (1.78, 80.12), (1.99, 101.23), (1.39, 87.6), (0.25, 110.19), (1.04, 111.68), (1.09, 76.35), (1.39, 92.55), (0.78, 103.46), (0.77, 78.14), (1.26, 104.12), (0.46, 116.17), (1.03, 87.01), (0.84, 121.04), (-0.13, 90.13), (1.32, 98.88), (1.09, 86.78), (0.98, 90.81), (1.81, 102.66), (1.33, 110.42), (0.39, 96.15), (1.63, 109.89), (1.52, 103.74), (1.0, 96.48), (1.05, 96.98), (0.23, 79.96), (1.82, 131.67), (1.84, 88.17), (2.6, 92.98), (0.98, 138.57), (1.12, 116.95), (0.38, 103.44), (0.88, 85.01), (0.89, 106.53), (0.45, 127.52), (0.93, 120.07), (0.63, 107.42), (1.26, 81.45), (1.16, 97.37), (1.32, 135.53), (0.28, 111.94), (0.3, 85.17), (0.73, 89.07), (0.93, 99.65), (0.99, 96.46), (0.81, 106.85), (1.12, 103.79), (0.06, 91.94), (1.15, 111.63), (2.04, 117.58), (0.96, 117.4), (-0.49, 77.71), (0.73, 91.21), (1.27, 105.06), (0.3, 110.19), (0.71, 101.87), (1.7, 133.38), (1.42, 106.28), (1.22, 94.46), (1.43, 92.46), (0.67, 102.7), (1.23, 103.21), (0.36, 93.21), (1.87, 109.4), (1.86, 100.83), (1.51, 95.0), (0.66, 109.94), (1.08, 110.37), (1.07, 89.35), (0.69, 114.14), (0.51, 92.31), (0.32, 111.15), (1.45, 85.83), (0.87, 83.3), (1.71, 106.36), (2.08, 97.35), (1.58, 122.61), (1.08, 129.28), (0.58, 75.08), (0.86, 91.23), (0.7, 109.5), (1.89, 120.61), (0.37, 124.07), (1.43, 105.1), (1.68, 99.6), (0.9, 85.52), (0.59, 93.55), (0.13, 101.21), (0.62, 88.42), (1.25, 103.73), (1.62, 94.06), (0.18, 112.11), (0.83, 119.99), (0.92, 83.07), (0.68, 95.88), (0.18, 108.27), (1.42, 93.02), (1.65, 103.67), (1.31, 92.53), (2.32, 136.35), (0.97, 86.54), (1.19, 110.42), (1.7, 114.51), (1.08, 67.06), (0.44, 90.82), (1.27, 111.78), (1.21, 88.46), (1.88, 105.59), (0.65, 100.97), (1.57, 81.93), (1.48, 110.66), (1.21, 79.5), (0.67, 87.19), (1.38, 93.66), (1.28, 105.42), (1.6, 97.41), (1.36, 82.47), (1.35, 111.94), (1.8, 130.0), (1.27, 125.27), (-0.12, 115.56), (0.95, 121.32), (1.6, 76.05), (0.43, 113.65), (0.39, 101.86), (0.91, 85.14), (0.76, 123.28), (0.52, 114.07), (1.18, 112.0), (0.5, 89.86), (1.92, 70.88), (0.96, 100.93), (2.29, 106.72), (0.97, 77.63), (1.55, 89.4), (1.33, 126.9), (0.32, 98.57), (1.09, 111.78), (-0.33, 77.81), (1.36, 115.8), (0.6, 120.2), (1.01, 87.04), (1.53, 109.05), (1.67, 101.35), (1.96, 71.75)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=1.0110，ȳ=100.4995；Sxx=Σ(x−x̄)²=86.07，Sxy=141.16；
β₁ = Sxy/Sxx = 1.6402；β₀ = ȳ−β₁x̄ = 98.8412；
R² = SSR/SST = 231.54/65666.80 = 0.0035；F = (SSR/1)/(SSE/310) = 1.0969，p = 0.2958；
SE(β₁)=√(211.08/86.07)=1.5661；t=β₁/SE=1.0473；p=0.2958；95%CI=[-24.6324,27.9128]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=1.0110，ȳ=100.4995；Sxx=Σ(x−x̄)²=86.07，Sxy=141.16；
β₁ = Sxy/Sxx = 1.6402；β₀ = ȳ−β₁x̄ = 98.8412；
R² = SSR/SST = 231.54/65666.80 = 0.0035；F = (SSR/1)/(SSE/310) = 1.0969，p = 0.2958；
SE(β₁)=√(211.08/86.07)=1.5661；t=β₁/SE=1.0473；p=0.2958；95%CI=[-24.6324,27.9128]。
③ 计算结果：
y = 98.8412 + 1.6402·x；R²=0.0035；β₁ t=1.0473 p=0.2958；95%CI=[-24.6324,27.9128]
④ 结论：在 α=0.05 水平下，p=0.2958 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=1.6402），应无影响。
案例 73. 天气预测股票收益
简单线性回归应无影响
① 原始数据：
样本 n=716，逐条 (x, y)：[(29.83, -1.37), (18.35, 1.03), (28.81, 1.58), (19.7, 0.68), (24.66, 0.97), (17.33, -2.11), (3.84, -1.07), (10.42, -0.39), (28.08, -1.71), (25.79, -0.25), (11.12, -1.12), (15.85, 1.91), (21.71, 4.06), (-7.05, -0.66), (15.6, 1.75), (15.58, -2.65), (11.97, 3.33), (13.63, 0.46), (22.91, -1.98), (29.02, -0.43), (32.7, -1.42), (23.37, 2.3), (5.04, 1.69), (26.68, 1.34), (22.32, 0.13), (6.76, 0.21), (35.26, 2.48), (22.48, -1.28), (17.68, -2.95), (30.19, 0.18), (28.48, -0.19), (25.89, -0.47), (23.79, -1.48), (25.92, 1.52), (15.77, -0.51), (25.52, -2.93), (21.76, -0.68), (33.08, 0.01), (21.71, -0.29), (28.59, -3.05), (21.13, -1.67), (14.62, 2.47), (28.84, -0.9), (9.37, 3.44), (13.15, 2.73), (17.86, -1.18), (22.58, 4.05), (17.3, -2.5), (11.62, -2.79), (19.01, 0.67), (24.63, 3.61), (-0.05, -1.43), (15.73, 0.71), (27.44, -0.8), (18.32, 2.17), (27.06, -1.09), (23.69, -3.17), (20.69, -1.26), (32.78, 1.19), (18.45, -0.92), (37.71, -1.41), (23.38, -0.89), (8.52, -0.96), (29.54, 0.44), (20.54, -1.42), (12.34, 4.23), (16.6, 1.78), (5.46, -2.29), (13.74, 1.08), (16.41, -0.21), (10.91, -3.38), (9.85, 0.3), (31.39, -0.91), (22.39, -4.08), (34.54, 3.18), (16.84, -0.91), (20.3, -2.71), (19.99, -0.43), (-6.93, -2.13), (11.58, -0.49), (25.04, 2.47), (35.08, -0.76), (11.44, 0.33), (7.24, 2.08), (0.71, -0.87), (23.88, 0.13), (30.91, -0.64), (31.43, -1.47), (29.54, 1.64), (40.68, -1.87), (16.44, -0.72), (17.12, 2.48), (38.63, -2.37), (28.99, 0.79), (29.4, -1.12), (41.5, -2.05), (33.63, 2.0), (19.63, 0.92), (20.05, 3.6), (28.42, -2.25), (31.89, 3.27), (-5.48, -4.27), (41.96, 1.88), (25.39, 0.42), (7.16, 0.47), (-1.26, -0.88), (10.48, 5.54), (35.26, 1.39), (36.48, -1.32), (8.75, 0.82), (28.93, -0.2), (21.87, -3.52), (12.08, -0.02), (21.78, 1.49), (17.41, -0.62), (14.59, -0.81), (10.47, -0.05), (29.75, 4.24), (22.46, -1.23), (40.39, -3.7), (19.57, 1.77), (43.58, 0.32), (13.38, -1.61), (18.81, -1.43), (22.43, -0.34), (26.33, 0.03), (14.39, -0.41), (13.5, 0.36), (33.11, -2.81), (12.87, 0.47), (21.13, 0.88), (41.51, 0.36), (13.2, -2.42), (29.05, -0.24), (14.56, -4.07), (19.52, -1.05), (16.62, -2.51), (18.79, -0.93), (19.13, -0.35), (28.24, -1.86), (13.03, -1.79), (30.68, -1.05), (19.67, -1.8), (35.21, -1.04), (20.57, -1.6), (13.06, 1.22), (16.08, -1.4), (10.18, -1.92), (21.05, -0.68), (23.2, -0.18), (16.95, -1.59), (13.75, -1.36), (39.68, -1.44), (10.49, 0.32), (14.97, 1.45), (46.13, -0.29), (16.59, -2.82), (25.08, -1.8), (36.95, -3.03), (14.87, -0.86), (19.22, -1.72), (10.97, 0.48), (24.93, 3.87), (12.34, -1.17), (24.43, -0.87), (28.25, 0.34), (17.69, 1.75), (40.59, 2.58), (12.89, -3.56), (19.33, 1.57), (11.59, 2.52), (2.35, -2.45), (19.09, -0.9), (23.39, -2.75), (26.33, -3.19), (14.75, -0.7), (19.38, 1.43), (15.97, -2.98), (16.34, 0.81), (-5.73, -3.69), (23.72, -0.88), (22.67, -2.58), (15.8, -0.02), (32.12, 0.85), (24.79, 1.84), (9.46, -1.12), (13.18, -3.09), (20.21, 2.36), (37.99, -0.92), (6.76, 2.74), (14.15, -0.09), (27.63, 2.94), (20.44, 1.58), (36.0, -2.99), (22.03, -0.41), (5.75, 0.54), (16.99, 1.18), (21.17, -0.26), (11.72, -1.01), (26.32, -0.71), (21.28, -0.3), (36.39, 0.47), (2.91, 1.92), (15.25, 0.71), (33.14, 1.21), (13.09, 1.21), (12.95, -1.03), (28.73, 3.42), (15.03, 0.04), (18.53, 1.17), (-2.31, -3.25), (26.97, 2.35), (33.39, -1.31), (19.22, -1.9), (22.84, 0.87), (37.47, 2.69), (8.5, -3.79), (18.85, -0.38), (11.7, -1.54), (37.95, -0.73), (22.08, -3.97), (9.21, -3.1), (14.66, -3.72), (17.57, -1.38), (20.45, -1.84), (12.91, 2.93), (11.06, 3.05), (16.31, 0.22), (2.71, 3.41), (17.54, -2.17), (11.69, 1.74), (31.59, -0.24), (18.16, -2.56), (24.77, -6.41), (1.75, -0.89), (35.21, -0.63), (16.71, 0.78), (8.72, 1.5), (33.35, -2.34), (33.36, -2.62), (5.28, -0.15), (24.54, -0.95), (22.54, -2.4), (35.54, 0.58), (18.21, -4.03), (30.5, -1.7), (10.4, 5.38), (32.83, -1.27), (27.3, -1.83), (17.9, -1.19), (4.53, 1.76), (22.79, 0.57), (18.32, -1.06), (37.57, 1.14), (16.14, 1.66), (11.06, 2.08), (22.49, -1.23), (3.19, 0.14), (18.4, 2.73), (17.67, 1.42), (2.83, 2.72), (6.4, 0.07), (12.2, -2.38), (24.55, 2.21), (27.81, 0.35), (18.08, -2.13), (20.83, 1.84), (27.69, -0.72), (3.93, -1.82), (32.56, 0.44), (37.07, -0.58), (21.32, -3.21), (26.49, 1.16), (14.74, -1.28), (12.39, 1.33), (28.56, -0.79), (29.51, 0.54), (39.84, -3.15), (10.89, 0.88), (19.66, -0.39), (23.87, -0.4), (32.82, -0.54), (26.1, 0.65), (7.09, 1.38), (13.36, 1.18), (18.08, -2.14), (35.9, 2.54), (18.53, 0.55), (17.06, 1.19), (17.73, 0.87), (15.45, -2.39), (25.26, -1.71), (18.72, -0.25), (3.36, -2.22), (29.12, -0.61), (19.7, -2.1), (-0.77, 0.79), (9.06, -3.97), (10.58, 1.8), (6.04, 1.98), (5.38, 0.51), (6.08, -1.04), (24.95, -2.44), (26.14, 0.79), (46.63, 2.21), (28.15, -0.05), (26.58, 2.38), (21.22, -3.02), (36.25, 2.61), (-1.46, -1.17), (8.35, 1.08), (6.19, -2.84), (25.61, -1.84), (32.99, 0.06), (10.97, -0.19), (12.42, 2.47), (35.01, 1.45), (40.17, 5.12), (19.9, -2.48), (10.96, -0.71), (12.15, 1.73), (8.37, 0.71), (25.8, -2.02), (19.41, 1.7), (20.61, -2.61), (28.68, 3.43), (16.02, -0.27), (-7.46, -1.46), (17.65, -1.45), (30.56, 3.32), (40.23, -2.17), (17.94, 0.08), (6.27, -0.07), (4.47, -0.8), (20.87, 2.87), (16.22, 1.34), (18.84, -2.06), (22.58, -0.8), (27.07, -0.74), (24.96, 0.89), (10.05, -0.44), (24.32, -0.8), (17.27, 2.79), (34.7, 2.2), (11.73, -0.13), (28.67, -0.56), (15.17, 0.8), (23.04, 1.64), (17.58, -1.31), (21.75, -0.06), (8.23, 1.68), (23.71, 0.04), (23.37, 2.72), (31.72, 2.59), (9.51, -1.37), (12.15, 0.05), (21.35, -1.47), (13.55, 0.35), (17.56, -0.19), (24.94, -2.09), (37.71, 2.29), (19.06, -0.83), (28.29, -0.13), (21.96, 2.8), (9.79, 3.89), (20.97, -1.03), (15.38, -1.03), (17.04, 2.12), (17.95, 0.22), (13.48, -4.59), (33.3, -2.41), (19.32, 1.61), (6.02, 0.88), (17.88, 1.68), (30.46, -1.8), (25.28, -2.14), (8.78, -0.52), (16.71, 2.68), (24.64, 3.57), (46.65, -3.16), (14.98, 0.17), (25.33, 0.48), (12.74, 2.97), (2.6, -2.95), (10.14, 0.5), (12.83, -0.48), (15.47, -2.12), (27.87, -1.11), (14.11, -0.4), (19.48, -0.29), (20.09, 2.49), (0.87, -0.45), (27.8, -2.53), (29.25, 0.45), (7.34, -0.15), (17.47, -3.81), (20.33, 1.09), (25.04, -0.62), (10.25, 2.18), (18.34, -0.32), (17.02, -0.82), (11.01, -1.36), (26.47, 0.94), (26.08, -0.86), (39.67, 1.9), (19.04, -1.01), (17.68, -0.59), (7.91, -1.38), (23.74, 0.15), (33.53, -1.36), (6.26, -2.33), (18.43, 1.04), (16.75, -1.84), (21.31, 2.92), (4.22, 3.0), (11.43, 2.64), (19.87, -1.53), (26.69, -0.63), (19.94, -2.46), (23.79, 3.03), (14.92, 1.15), (12.01, -0.75), (24.4, -2.2), (6.41, -0.44), (12.96, -0.06), (17.38, -1.7), (21.2, -2.78), (5.19, 3.06), (20.05, 0.47), (20.8, -2.92), (17.6, 1.83), (16.65, 0.99), (19.16, -0.43), (11.53, 0.27), (17.18, -1.17), (0.64, 0.22), (22.66, 1.64), (29.13, 1.97), (27.63, 0.99), (11.17, 3.58), (29.85, 0.37), (39.67, -1.73), (26.23, -0.29), (23.42, 1.5), (25.72, -1.1), (27.24, 0.23), (27.08, -1.36), (19.54, -2.54), (10.91, 0.07), (12.1, -0.78), (36.21, 0.25), (23.08, 1.11), (29.93, -0.41), (20.96, 1.29), (3.21, 1.19), (14.69, -0.16), (23.33, -2.69), (15.59, 3.95), (9.01, -0.25), (22.28, 1.34), (30.46, -0.47), (22.65, 1.06), (3.35, 4.22), (18.69, -0.53), (8.66, 1.59), (19.09, 0.67), (8.06, 0.33), (21.98, -0.98), (15.89, 0.56), (2.24, -0.54), (13.01, -0.22), (30.16, 1.0), (11.69, 1.32), (26.79, -1.33), (20.24, -0.1), (18.81, 1.36), (30.27, -2.59), (20.05, -2.49), (24.24, 1.14), (17.33, -0.51), (11.22, -1.02), (13.97, -0.04), (1.49, -0.7), (12.76, -0.08), (16.98, -1.29), (29.67, 2.85), (18.67, -3.1), (37.79, 1.22), (9.99, -0.73), (13.93, 5.51), (18.16, 0.27), (38.67, 0.73), (23.61, 2.95), (24.28, -0.69), (15.92, 2.93), (21.46, -0.66), (11.79, 0.93), (19.29, 1.51), (21.47, -1.59), (18.15, 1.33), (8.81, -2.09), (32.44, -3.61), (23.68, -1.16), (25.22, 2.38), (45.39, -0.55), (33.84, -2.82), (4.42, -1.01), (12.01, 1.21), (34.6, 1.06), (16.72, 1.64), (14.82, 0.23), (22.92, 3.01), (34.56, -1.19), (22.1, 5.95), (28.67, 0.11), (23.35, 2.42), (11.94, -1.68), (35.29, -1.1), (22.08, 2.65), (10.1, 0.46), (18.72, 0.13), (28.28, 0.47), (30.91, 0.02), (8.08, 1.69), (25.35, 0.14), (26.42, -0.07), (26.53, 1.85), (17.51, 0.62), (25.77, 0.96), (8.82, -4.31), (13.56, -1.11), (21.14, -2.17), (2.74, 0.66), (22.6, 2.94), (15.5, -0.3), (26.28, 1.25), (31.89, -0.91), (16.84, -1.22), (34.74, -1.14), (0.66, 2.77), (19.88, -0.81), (18.35, -1.83), (22.85, -2.74), (10.08, 4.05), (18.07, 0.69), (20.34, 1.0), (20.56, -2.86), (1.79, -1.66), (26.76, 1.0), (5.72, 2.45), (28.54, 0.83), (22.39, 0.46), (35.32, 1.37), (23.04, -1.28), (12.56, 0.67), (18.07, -0.25), (21.28, -0.95), (41.29, 2.51), (29.76, -0.4), (35.63, -0.35), (9.54, 0.26), (15.71, 1.39), (16.68, -1.7), (0.56, 2.5), (27.36, -2.71), (6.27, 0.22), (27.04, -1.61), (-5.83, 1.92), (27.73, -5.34), (25.2, -2.37), (14.12, -0.31), (7.88, 3.58), (16.63, 2.05), (12.7, -3.22), (23.2, -0.29), (1.29, -3.86), (31.55, 1.87), (24.54, 3.13), (20.27, 0.28), (4.37, 0.66), (35.27, 2.75), (26.92, 1.12), (14.66, 0.62), (18.89, 1.15), (31.42, 1.52), (21.73, -2.55), (17.89, 0.17), (41.84, 2.76), (25.67, -2.09), (13.55, -0.03), (10.78, -0.73), (21.92, -3.02), (21.47, -1.85), (21.39, -1.65), (14.4, 1.81), (15.37, 1.88), (35.38, 0.14), (23.69, 0.21), (16.7, 2.51), (26.4, 0.73), (39.85, -1.72), (23.15, -1.27), (30.27, 1.11), (24.01, 0.22), (28.64, 0.56), (24.46, 0.67), (32.84, 0.09), (21.81, -0.58), (17.18, 1.55), (17.08, -1.06), (8.7, -1.47), (43.16, -1.27), (24.35, -2.25), (13.96, -1.72), (20.57, 4.47), (14.75, -0.32), (28.9, 0.18), (25.11, 4.52), (26.84, 3.63), (2.9, -1.91), (2.88, 2.69), (13.87, 2.67), (31.07, 1.32), (5.79, -0.28), (-8.3, -0.1), (38.95, -1.08), (35.45, -1.65), (34.05, -2.09), (10.6, -1.96), (16.78, -1.59), (41.4, -0.75), (18.75, 0.89), (27.55, -0.3), (24.64, 2.59), (22.43, 1.59), (21.72, 0.12), (17.8, 0.84), (12.87, -0.01), (34.06, -0.78), (22.79, -1.06), (10.63, -0.69), (24.2, -1.92), (16.64, -2.82), (27.33, 1.9), (-2.51, 0.34), (23.95, -0.91), (19.76, -0.53), (6.15, -5.45), (13.65, -1.13), (25.89, 0.03), (17.86, 4.64), (35.08, 0.62), (22.38, -0.77), (12.17, -0.72), (11.04, -2.6), (10.44, 1.98), (16.83, 0.88), (17.6, -0.25), (44.11, 1.69), (22.57, 0.73), (27.51, 0.98), (24.03, 2.68), (10.23, -1.69), (9.32, 1.68), (21.22, -0.38), (20.86, -1.49), (25.33, -1.26), (27.45, 1.24), (33.72, 0.93), (15.71, -0.73), (13.51, 0.88), (4.63, 1.63), (35.16, 1.24), (17.61, 1.29), (12.95, 0.05), (5.81, -1.07), (23.74, 3.36), (9.05, -3.43), (8.97, 0.94), (21.96, 3.8), (10.26, -2.88), (24.58, -1.92), (23.34, 2.73), (29.42, 0.56), (21.11, 1.69), (19.38, -4.13), (3.74, 2.23), (27.19, 1.07), (18.17, 5.01), (25.2, 0.25), (22.04, -0.51), (28.73, -1.25), (6.5, 2.64), (20.57, 0.3), (23.28, 2.0), (-11.37, -1.12), (54.3, 2.16), (9.79, -0.83), (1.25, -2.05), (10.53, -0.1), (28.33, 1.19), (18.05, -0.73), (15.86, 0.8), (23.64, -0.94), (24.77, -4.75), (30.8, -1.16), (20.16, 2.56), (28.97, 0.17), (6.39, -1.14), (35.13, -1.7), (10.88, 0.96), (27.46, 1.3), (35.9, -2.74), (9.89, -1.24), (42.86, 2.15), (11.8, 1.31)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.0422，ȳ=-0.0337；Sxx=Σ(x−x̄)²=72012.27，Sxy=208.77；
β₁ = Sxy/Sxx = 0.0029；β₀ = ȳ−β₁x̄ = -0.0918；
R² = SSR/SST = 0.61/2635.65 = 0.0002；F = (SSR/1)/(SSE/714) = 0.1640，p = 0.6856；
SE(β₁)=√(3.69/72012.27)=0.0072；t=β₁/SE=0.4050；p=0.6856；95%CI=[-0.0112,0.0170]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=20.0422，ȳ=-0.0337；Sxx=Σ(x−x̄)²=72012.27，Sxy=208.77；
β₁ = Sxy/Sxx = 0.0029；β₀ = ȳ−β₁x̄ = -0.0918；
R² = SSR/SST = 0.61/2635.65 = 0.0002；F = (SSR/1)/(SSE/714) = 0.1640，p = 0.6856；
SE(β₁)=√(3.69/72012.27)=0.0072；t=β₁/SE=0.4050；p=0.6856；95%CI=[-0.0112,0.0170]。
③ 计算结果：
y = -0.0918 + 0.0029·x；R²=0.0002；β₁ t=0.4050 p=0.6856；95%CI=[-0.0112,0.0170]
④ 结论：在 α=0.05 水平下，p=0.6856 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=0.0029），应无影响。
案例 74. 血型预测体温
简单线性回归应无影响
① 原始数据：
样本 n=360，逐条 (x, y)：[(3.93, 37.15), (6.74, 36.66), (2.24, 36.22), (7.5, 37.39), (5.69, 36.81), (8.63, 37.01), (3.38, 36.67), (4.53, 36.69), (3.18, 36.77), (4.11, 36.94), (5.15, 37.07), (9.59, 36.65), (5.81, 36.51), (4.35, 37.35), (4.38, 36.59), (5.64, 36.99), (3.39, 37.09), (9.28, 36.61), (6.25, 36.68), (5.14, 35.89), (2.23, 36.93), (6.72, 36.26), (4.28, 35.84), (3.78, 36.68), (0.6, 36.48), (7.32, 36.69), (3.14, 37.66), (5.15, 37.01), (6.31, 35.61), (8.09, 36.94), (4.26, 36.63), (7.81, 36.81), (2.98, 36.83), (6.04, 35.97), (6.62, 36.92), (4.15, 36.71), (5.34, 36.62), (4.62, 36.42), (6.74, 35.96), (2.53, 36.14), (5.87, 36.99), (3.79, 36.79), (2.6, 37.19), (5.49, 36.81), (8.31, 37.04), (7.43, 36.9), (4.84, 36.89), (2.08, 37.06), (1.4, 36.74), (3.75, 36.8), (1.02, 37.3), (1.28, 37.26), (9.82, 37.24), (2.49, 36.59), (6.78, 37.18), (5.26, 37.14), (6.11, 36.72), (6.96, 36.68), (4.32, 36.61), (2.2, 37.05), (4.06, 36.58), (4.28, 36.74), (6.3, 36.75), (4.0, 36.93), (3.25, 36.76), (4.67, 37.61), (6.16, 36.44), (6.57, 36.77), (6.21, 36.76), (1.83, 37.05), (5.42, 36.19), (2.36, 36.79), (6.44, 36.61), (5.5, 36.75), (5.89, 36.78), (5.04, 36.8), (3.07, 36.53), (3.78, 35.96), (4.22, 37.54), (2.26, 36.79), (6.2, 36.67), (4.74, 37.26), (7.85, 36.83), (5.94, 37.39), (9.18, 37.08), (6.12, 36.46), (3.66, 36.34), (4.46, 37.23), (7.21, 37.6), (3.96, 37.41), (2.48, 36.39), (6.49, 36.65), (2.28, 36.92), (4.72, 36.65), (4.47, 35.98), (4.52, 36.57), (4.46, 36.28), (7.62, 36.69), (8.23, 36.51), (6.74, 37.7), (6.12, 36.83), (1.64, 36.64), (5.45, 36.91), (2.63, 37.2), (3.02, 36.65), (3.81, 36.99), (5.44, 36.85), (3.19, 36.46), (5.91, 37.08), (2.94, 36.79), (4.94, 36.79), (5.94, 36.73), (6.78, 37.04), (7.11, 36.85), (5.77, 36.29), (6.79, 36.34), (4.87, 36.39), (1.82, 36.68), (3.79, 35.72), (2.55, 36.85), (2.02, 36.6), (7.13, 36.81), (5.8, 36.8), (10.58, 36.18), (1.97, 36.29), (5.15, 36.74), (4.32, 37.14), (4.42, 36.37), (4.85, 36.86), (6.81, 36.94), (4.23, 36.41), (4.51, 36.76), (7.36, 36.09), (5.9, 36.98), (3.71, 37.18), (5.47, 36.69), (5.3, 36.44), (6.0, 36.77), (7.29, 36.99), (3.5, 36.15), (6.09, 36.73), (3.45, 36.81), (2.5, 37.16), (2.83, 36.68), (5.17, 37.01), (9.61, 36.71), (6.34, 36.97), (4.57, 36.7), (3.98, 37.19), (4.83, 35.75), (7.1, 36.59), (1.95, 36.37), (8.14, 36.85), (5.96, 36.79), (3.68, 37.13), (6.46, 36.89), (4.19, 36.66), (5.17, 36.03), (2.61, 37.2), (7.09, 37.15), (2.49, 37.02), (4.07, 36.33), (3.44, 36.33), (6.64, 37.27), (4.63, 36.49), (6.26, 36.86), (6.39, 36.77), (6.67, 36.77), (6.18, 35.78), (5.72, 37.0), (5.41, 36.26), (-0.18, 37.38), (6.23, 36.92), (6.23, 37.26), (6.2, 37.03), (7.02, 36.3), (7.08, 36.35), (5.74, 36.34), (8.48, 36.84), (5.94, 36.98), (5.24, 36.78), (5.61, 36.37), (6.82, 36.14), (5.29, 37.21), (6.15, 37.05), (7.62, 36.43), (4.33, 35.87), (3.99, 37.39), (4.68, 36.8), (2.89, 37.08), (2.1, 37.07), (6.12, 36.65), (3.34, 37.01), (5.13, 36.68), (5.17, 37.25), (8.01, 36.4), (6.42, 36.53), (5.64, 36.4), (3.74, 36.95), (6.07, 36.53), (3.59, 37.48), (3.33, 36.53), (3.69, 37.61), (3.77, 36.07), (6.92, 36.86), (4.35, 36.85), (8.88, 36.41), (7.29, 36.89), (6.43, 37.02), (6.26, 36.63), (4.87, 37.0), (6.47, 36.95), (3.54, 36.42), (4.28, 36.73), (4.7, 36.96), (7.76, 36.17), (3.69, 36.62), (2.95, 35.99), (7.2, 36.34), (6.77, 36.83), (5.43, 36.83), (-0.36, 36.68), (4.59, 36.78), (6.99, 36.99), (6.63, 36.39), (3.6, 36.65), (2.73, 36.6), (4.92, 36.49), (2.24, 36.76), (6.32, 36.77), (5.73, 36.41), (8.05, 37.32), (7.48, 36.56), (3.68, 37.29), (3.92, 37.04), (4.92, 36.91), (4.68, 37.18), (7.25, 36.67), (4.83, 36.57), (3.29, 36.62), (7.67, 36.68), (4.94, 36.55), (6.77, 36.28), (3.08, 36.87), (5.59, 37.36), (5.98, 37.34), (3.32, 36.52), (7.1, 37.18), (9.5, 36.99), (3.77, 37.09), (4.7, 37.13), (6.56, 37.52), (4.76, 36.32), (7.44, 36.9), (0.2, 36.64), (4.59, 36.41), (1.83, 37.85), (6.89, 37.02), (4.62, 37.46), (3.45, 36.89), (1.71, 36.93), (5.16, 36.81), (5.82, 37.21), (2.48, 35.96), (4.39, 36.21), (6.23, 37.03), (0.54, 36.23), (4.61, 37.36), (3.88, 37.37), (7.53, 36.48), (3.27, 37.01), (6.79, 36.31), (5.82, 36.56), (1.9, 36.34), (4.9, 36.67), (4.73, 36.79), (8.1, 36.58), (2.57, 35.75), (8.3, 37.01), (5.0, 37.47), (3.64, 36.51), (6.16, 36.75), (4.48, 36.13), (1.58, 37.19), (3.54, 37.17), (0.67, 36.42), (1.87, 36.18), (3.07, 37.06), (5.88, 36.5), (4.94, 37.51), (4.63, 36.78), (3.85, 36.85), (5.03, 36.45), (4.65, 37.25), (5.45, 36.74), (4.96, 37.33), (4.93, 36.51), (4.96, 37.18), (5.45, 36.69), (5.32, 36.8), (5.45, 36.2), (6.32, 36.78), (4.1, 36.87), (4.96, 36.32), (3.02, 36.92), (2.44, 36.94), (6.13, 36.55), (5.7, 37.01), (3.12, 37.15), (1.81, 37.6), (3.57, 36.66), (2.96, 37.45), (6.68, 36.06), (7.14, 36.75), (8.07, 37.69), (7.53, 36.69), (6.29, 36.96), (5.85, 36.52), (6.86, 36.65), (5.34, 36.47), (4.14, 37.11), (7.68, 36.31), (4.95, 36.85), (8.68, 37.14), (5.13, 36.73), (3.64, 36.98), (5.93, 36.95), (5.1, 37.16), (4.76, 36.66), (4.69, 37.0), (3.19, 37.21), (3.51, 36.81), (3.46, 37.24), (3.23, 36.74), (5.62, 36.29), (3.74, 36.74), (3.74, 36.27), (4.36, 37.1), (1.97, 36.42), (7.68, 36.91), (6.4, 36.35), (4.92, 37.23), (4.89, 36.81), (5.51, 36.6), (9.0, 36.69), (4.11, 37.4), (5.76, 37.13), (9.41, 36.91), (5.06, 36.68), (4.96, 36.64), (7.15, 36.7), (5.67, 36.95), (8.98, 36.26), (5.18, 36.97), (4.76, 37.53), (6.23, 36.63), (8.11, 36.65), (6.37, 37.29), (7.84, 36.78), (7.26, 37.0)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.0748，ȳ=36.7725；Sxx=Σ(x−x̄)²=1323.09，Sxy=-10.38；
β₁ = Sxy/Sxx = -0.0078；β₀ = ȳ−β₁x̄ = 36.8123；
R² = SSR/SST = 0.08/54.21 = 0.0015；F = (SSR/1)/(SSE/358) = 0.5389，p = 0.4634；
SE(β₁)=√(0.15/1323.09)=0.0107；t=β₁/SE=-0.7341；p=0.4634；95%CI=[-0.0289,0.0132]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=5.0748，ȳ=36.7725；Sxx=Σ(x−x̄)²=1323.09，Sxy=-10.38；
β₁ = Sxy/Sxx = -0.0078；β₀ = ȳ−β₁x̄ = 36.8123；
R² = SSR/SST = 0.08/54.21 = 0.0015；F = (SSR/1)/(SSE/358) = 0.5389，p = 0.4634；
SE(β₁)=√(0.15/1323.09)=0.0107；t=β₁/SE=-0.7341；p=0.4634；95%CI=[-0.0289,0.0132]。
③ 计算结果：
y = 36.8123 + -0.0078·x；R²=0.0015；β₁ t=-0.7341 p=0.4634；95%CI=[-0.0289,0.0132]
④ 结论：在 α=0.05 水平下，p=0.4634 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=-0.0078），应无影响。
案例 75. 鞋码预测月收入
简单线性回归应无影响
① 原始数据：
样本 n=698，逐条 (x, y)：[(37.53, 23.06), (42.01, 33.74), (35.68, 27.62), (37.94, 17.32), (36.72, 0.65), (42.97, 14.01), (38.26, 17.08), (40.84, 30.47), (45.0, 35.11), (40.23, 11.49), (36.37, 25.01), (41.51, 19.31), (44.41, 8.61), (34.64, 24.28), (41.76, 20.28), (41.58, 16.58), (36.3, 21.84), (40.84, 18.08), (36.03, 10.58), (40.3, 27.16), (35.91, 22.54), (41.75, 10.83), (44.06, 6.47), (39.4, 18.73), (40.24, 23.15), (40.54, 15.59), (38.8, 19.41), (41.16, 27.4), (35.19, 17.91), (45.67, 20.41), (41.3, 24.83), (42.53, 12.84), (49.34, 24.25), (41.8, 2.53), (44.56, 21.67), (42.69, 31.68), (36.19, 12.28), (39.7, 18.25), (40.41, 20.45), (37.88, 12.46), (39.02, 29.21), (38.0, 25.87), (36.21, 10.55), (40.7, 15.31), (39.37, 8.31), (47.23, 3.36), (41.52, 27.19), (38.04, 18.15), (42.1, 19.02), (40.91, 18.67), (39.85, 28.68), (46.4, 28.23), (37.29, 16.21), (39.82, 23.38), (39.86, 31.25), (35.76, 27.64), (38.31, 18.96), (41.72, 21.96), (41.45, -10.76), (38.03, 25.02), (35.4, 29.96), (45.23, 20.72), (36.64, 17.6), (43.94, 26.43), (39.89, 26.86), (35.67, 12.52), (41.03, 14.98), (36.68, 24.53), (42.53, 19.39), (41.29, 15.94), (38.33, 23.52), (39.3, 27.32), (40.38, 37.99), (39.08, 20.34), (37.57, 17.53), (32.69, 24.01), (41.73, 17.08), (44.11, 11.2), (42.5, 10.01), (40.61, 23.32), (42.04, 34.03), (41.0, 19.6), (39.75, 28.52), (37.25, 28.78), (36.93, 27.63), (42.14, 25.21), (39.04, 23.04), (40.1, 25.2), (39.98, 23.44), (45.08, 15.79), (42.03, 21.73), (40.51, 27.14), (39.81, 21.19), (41.47, 23.13), (40.38, 20.89), (39.92, 0.61), (42.22, 19.89), (42.04, 24.44), (38.95, 15.46), (40.7, 20.42), (40.44, 16.55), (41.88, 31.1), (36.69, 34.99), (38.4, 10.87), (46.21, 19.09), (38.27, 20.23), (41.93, 10.87), (35.91, 21.73), (36.09, 13.24), (40.38, 14.08), (39.24, 20.51), (41.41, 11.74), (33.82, 25.4), (38.37, 13.11), (38.66, 8.09), (42.77, 17.55), (41.67, 24.68), (41.5, 6.96), (39.41, 11.99), (39.75, 25.0), (49.48, 27.38), (45.1, 17.8), (41.23, 4.3), (32.68, 9.08), (37.92, 25.09), (43.91, 15.6), (34.49, 37.61), (45.04, 12.3), (38.94, 21.02), (35.92, 28.12), (38.29, 41.83), (40.85, 8.42), (38.48, -0.78), (41.37, 24.54), (41.73, 2.55), (36.16, 20.79), (33.8, 24.82), (35.83, 22.32), (41.74, 18.66), (38.52, 22.23), (44.64, 17.58), (37.76, 17.79), (41.79, 10.89), (38.56, 30.51), (43.15, 19.85), (41.15, 14.99), (42.52, 20.74), (40.95, 21.84), (38.18, 12.24), (37.86, 30.37), (45.16, 13.99), (39.85, 17.27), (43.1, 16.91), (44.87, 29.41), (37.58, 13.3), (40.77, 11.91), (44.08, 9.86), (40.34, 15.83), (38.13, 34.01), (43.26, 22.77), (40.82, 23.32), (40.79, 29.4), (38.32, 15.54), (42.57, 9.74), (38.69, 15.71), (37.66, 22.5), (43.34, -0.14), (42.57, 18.17), (39.98, 25.55), (41.26, 13.78), (38.78, 18.32), (40.39, 20.22), (40.76, 8.51), (41.23, 23.54), (41.77, 8.94), (42.77, 21.97), (41.83, 24.08), (37.42, 17.26), (39.46, 20.1), (40.83, 21.67), (37.38, 17.02), (42.12, 23.12), (40.36, 35.11), (36.24, 16.95), (40.56, 24.13), (35.94, 29.84), (34.84, 20.24), (37.7, 8.0), (38.46, 7.03), (40.87, 11.12), (38.67, 19.98), (40.79, 6.11), (38.47, 23.16), (32.79, 21.49), (39.57, 22.15), (38.01, 28.96), (40.59, 17.56), (42.94, 14.79), (37.15, 31.46), (45.26, 26.35), (48.82, 19.04), (39.49, 29.26), (37.41, 21.95), (40.51, 11.13), (37.81, 14.02), (33.75, 23.75), (40.62, 18.92), (41.35, 27.61), (42.01, 28.61), (38.91, 18.64), (37.34, 17.17), (39.45, 13.39), (42.37, 26.85), (38.53, 26.1), (35.43, 29.26), (40.95, 19.99), (43.52, 31.17), (48.05, 19.16), (42.52, 13.25), (41.33, 21.98), (35.07, 17.12), (34.02, 16.96), (38.46, 24.87), (34.08, 35.04), (41.49, 24.8), (44.05, 18.47), (40.81, 3.45), (39.75, 19.65), (44.42, 23.57), (40.57, 11.67), (32.94, 25.08), (40.95, 19.33), (40.1, 20.79), (38.76, 30.08), (40.46, 19.14), (37.49, 20.26), (43.57, 17.14), (44.56, 21.14), (40.53, 25.25), (37.44, 19.17), (40.17, 21.02), (43.43, 7.2), (46.6, 15.51), (39.66, 15.42), (37.0, 20.64), (32.76, 15.89), (43.49, 23.83), (43.79, 29.35), (40.82, 16.87), (41.52, 29.34), (41.21, 12.61), (38.96, 12.09), (39.08, 32.44), (39.85, 8.63), (38.05, 25.78), (43.41, 4.83), (40.95, 28.57), (38.48, 16.0), (40.41, 21.99), (42.84, 38.73), (39.29, 19.03), (40.8, 7.35), (39.21, 20.44), (33.25, 15.07), (41.17, 21.34), (38.83, 17.41), (36.12, 25.29), (41.73, 13.83), (36.23, 3.69), (38.51, 22.13), (34.84, 16.28), (41.83, 13.26), (42.87, 22.97), (43.33, 32.94), (36.93, 19.5), (36.02, 10.02), (35.14, 9.45), (36.92, 18.61), (39.94, 13.89), (38.64, 31.02), (44.57, 22.46), (35.78, 15.04), (40.96, 14.17), (38.28, 31.94), (37.92, 27.79), (39.77, 1.11), (42.59, 21.23), (39.02, 10.1), (39.52, 19.41), (35.53, 2.06), (42.11, 10.91), (37.16, 18.97), (38.25, 5.15), (40.57, 24.51), (39.0, 13.91), (34.48, 11.33), (37.34, 23.22), (38.77, 13.87), (40.65, 7.22), (42.45, 15.26), (39.47, 23.88), (44.15, 19.43), (34.99, 32.31), (44.95, 7.37), (38.07, 28.57), (45.47, 28.93), (38.38, 22.11), (38.84, 11.66), (38.0, 33.34), (33.7, 5.82), (45.48, 20.48), (35.8, 29.4), (34.15, 16.61), (38.49, 16.27), (38.46, 22.31), (41.72, 15.37), (40.89, 15.72), (43.61, 12.81), (46.48, 19.55), (37.25, 21.54), (46.77, 13.84), (43.57, 18.66), (36.97, 22.85), (39.42, 18.32), (41.69, 23.91), (34.99, 23.12), (42.38, 30.34), (42.74, 18.75), (40.45, 27.95), (39.3, 15.39), (38.95, 9.05), (40.85, 25.17), (39.47, 9.41), (42.49, 9.12), (43.03, 28.77), (38.21, 28.12), (41.44, 15.33), (39.81, 21.04), (40.55, 24.59), (41.26, 16.0), (34.35, 18.78), (44.11, 23.67), (39.67, 13.35), (41.79, 31.18), (45.92, 30.75), (39.12, 7.77), (39.65, 30.76), (40.92, 27.97), (42.56, 22.53), (38.98, 17.75), (42.4, 32.82), (43.21, 20.47), (37.97, 9.69), (32.34, 23.23), (43.26, 17.94), (38.7, 27.65), (38.34, 2.99), (43.85, 11.34), (38.43, 15.42), (42.65, 30.68), (39.14, 16.5), (41.71, 16.45), (39.49, 14.84), (35.99, 19.89), (46.45, 22.77), (35.3, 16.25), (32.59, 25.4), (35.69, 3.58), (40.96, 18.1), (40.18, 19.52), (37.15, 14.81), (44.96, 14.76), (40.48, 12.02), (39.73, 24.53), (39.77, 14.2), (38.67, 19.92), (36.17, 19.16), (38.63, 28.35), (42.2, 19.24), (40.36, 24.21), (43.21, 19.12), (39.83, 35.31), (40.1, 15.08), (39.17, 18.8), (38.04, 26.34), (38.79, 30.4), (40.11, 5.71), (37.9, 26.91), (41.55, 31.41), (43.33, 17.37), (41.39, 24.39), (42.18, 20.05), (39.48, 19.8), (40.37, 19.82), (39.21, 31.44), (36.69, 26.65), (35.74, 16.25), (38.04, 20.13), (40.45, 17.04), (38.57, 26.14), (32.45, 15.7), (44.53, 16.8), (38.91, 22.85), (37.02, 8.32), (36.23, 23.25), (40.9, 36.21), (41.92, 11.1), (42.77, 23.7), (38.99, 17.83), (41.95, 28.92), (35.46, 16.23), (40.48, 22.58), (37.5, 21.1), (34.75, 11.65), (41.76, 15.3), (38.92, 27.0), (42.99, 6.44), (38.15, 13.21), (48.1, 17.82), (43.97, 12.06), (35.01, 22.81), (42.5, 25.39), (41.47, 33.76), (42.25, 22.4), (43.02, 18.01), (34.96, 27.65), (39.27, 5.95), (42.32, 8.48), (44.98, 14.7), (40.87, 20.78), (39.54, 27.37), (37.68, 16.06), (37.96, 34.24), (38.01, 18.16), (40.69, 28.71), (46.65, 21.3), (41.21, 35.39), (35.69, 14.82), (41.85, 27.11), (37.46, 13.96), (36.7, 30.76), (42.38, 26.29), (42.36, 20.5), (33.54, 6.63), (39.65, 35.72), (42.5, 16.65), (43.68, 19.06), (43.62, 18.42), (41.2, 12.89), (37.37, 8.36), (38.75, 20.04), (35.34, 2.28), (39.31, 15.85), (43.06, 22.61), (33.94, 19.03), (47.32, 19.19), (37.0, 13.79), (36.67, 23.77), (46.0, 37.41), (44.39, 30.43), (36.88, 23.49), (46.43, 10.02), (38.62, 21.94), (41.16, 13.98), (41.06, 20.33), (42.34, 13.82), (43.6, 15.73), (42.78, 19.14), (42.58, 16.7), (42.08, 13.98), (40.67, 20.99), (39.95, 22.04), (41.93, 22.42), (42.66, 26.31), (46.2, 15.75), (44.79, 23.25), (35.84, 27.01), (37.11, 19.68), (39.34, 23.56), (37.28, 27.2), (44.1, 9.44), (40.43, 23.24), (46.4, 33.05), (35.86, 13.63), (38.19, 10.56), (34.79, 3.7), (38.71, 30.42), (42.13, 22.44), (40.89, 19.97), (37.21, 22.79), (40.29, 20.34), (37.59, -1.92), (36.42, 20.74), (41.09, 14.89), (43.23, 13.8), (41.98, 21.01), (35.0, 14.31), (40.44, 15.25), (38.73, 25.79), (43.1, 14.23), (40.46, 24.87), (37.09, 21.22), (39.94, 34.76), (38.26, 18.41), (42.46, 4.77), (32.3, 16.62), (39.52, 11.53), (43.48, 22.08), (40.56, 9.96), (40.02, 25.05), (41.75, 26.72), (44.66, 23.48), (41.68, 13.82), (42.64, 27.88), (40.32, 14.74), (45.42, 17.84), (34.69, 23.1), (35.56, 31.04), (42.71, 5.86), (37.5, 20.66), (40.59, 8.53), (37.62, 18.5), (41.24, 19.35), (41.7, 32.67), (40.16, 25.61), (39.04, 17.58), (36.48, 22.91), (39.76, 12.09), (39.35, 23.29), (40.62, 23.09), (43.96, 16.68), (40.55, 19.76), (42.65, 21.06), (44.8, 13.92), (44.44, 8.45), (45.82, 10.75), (45.86, 14.51), (41.07, 25.25), (37.73, 25.12), (35.22, 18.63), (36.64, 10.25), (42.03, 28.37), (40.94, 24.45), (40.25, 27.62), (43.29, 27.98), (39.05, 18.16), (42.02, 25.05), (40.53, 4.49), (37.1, 37.53), (41.06, 31.55), (37.57, 10.8), (42.67, 10.81), (41.46, 26.56), (38.33, 23.13), (37.17, 14.92), (39.11, 15.41), (43.31, 13.31), (45.2, 18.1), (44.43, 21.54), (39.95, 18.56), (38.34, 17.41), (38.67, 16.74), (36.75, 9.79), (43.92, 21.33), (41.33, 14.79), (44.83, 26.83), (45.37, 16.54), (40.62, 15.5), (33.87, 22.63), (40.93, 20.8), (36.42, 21.33), (40.59, 14.11), (36.68, 16.64), (38.92, 32.79), (42.29, 23.94), (37.95, 21.88), (38.42, 22.8), (42.14, 25.51), (36.65, 20.33), (43.58, 15.44), (35.96, 25.78), (34.69, 20.45), (38.38, 0.93), (36.86, 33.46), (41.44, 31.37), (37.53, 16.32), (43.25, 30.58), (39.08, 19.8), (40.39, 31.5), (33.29, 26.15), (42.61, 24.55), (39.25, 21.46), (40.94, 25.8), (39.43, 23.04), (37.44, 27.1), (38.55, 15.33), (41.17, 30.73), (32.38, 37.62), (35.8, 17.23), (40.0, 24.15), (34.31, 30.26), (37.39, 18.92), (33.48, 13.04), (41.78, 24.71), (40.52, 18.73), (42.47, 24.03), (39.17, 36.68), (38.83, 21.26), (40.04, 12.63), (39.81, 10.29), (43.98, 25.14), (38.46, 33.69), (41.17, 1.78), (41.51, 8.65), (39.32, 42.14), (43.97, 17.8), (45.23, 24.49), (34.39, 26.16), (43.85, 15.61), (37.72, 26.36), (37.07, 6.91), (44.36, 11.9), (38.84, 18.16), (39.7, 15.95), (36.65, 34.25), (39.23, 15.04), (41.56, 27.18), (39.15, 20.95), (36.34, 14.71), (34.6, 15.39), (35.39, 36.2), (41.73, 14.82), (41.77, 13.7), (44.83, 6.16), (36.31, 19.22), (38.59, 17.95), (36.62, 14.81), (40.43, 30.43), (41.82, 11.71), (40.94, 9.95), (37.19, 11.74), (40.47, 16.96), (37.46, 24.25), (42.54, 2.0), (38.61, 35.35), (50.24, 15.86), (45.03, 14.71), (45.93, 23.65), (45.46, 8.36), (42.71, 16.47), (40.56, 34.77), (35.08, 13.41), (38.24, 37.36), (41.29, 26.22), (36.73, 27.52), (44.19, 16.68), (43.9, 16.91), (39.64, 12.91), (34.14, 10.45), (42.16, 21.07), (38.02, 25.2), (38.69, 29.21), (40.03, 10.52), (43.07, 11.33), (39.53, 25.32), (34.81, 7.53), (44.27, 23.77), (39.38, 27.89), (45.07, 31.11), (36.84, 12.23), (34.62, 12.52), (37.61, 18.49), (41.22, 6.52), (39.81, 8.22), (39.63, 15.97), (40.01, 22.88), (41.9, 18.59), (41.31, 18.8), (40.75, 18.28), (41.88, 29.71), (33.82, 16.53), (37.82, 20.45), (33.62, 27.4), (35.52, 0.64), (36.81, 12.17), (42.31, 26.16), (39.71, 29.24), (42.82, 29.14), (42.95, 11.43), (45.69, 21.86), (42.72, 23.63), (41.94, 7.23), (37.14, 23.91), (44.35, 19.6), (41.78, 18.29), (41.54, 26.26), (40.45, 22.42), (35.5, 21.94)]
② 分析过程：
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=40.0126，ȳ=19.7141；Sxx=Σ(x−x̄)²=6892.49，Sxy=-443.98；
β₁ = Sxy/Sxx = -0.0644；β₀ = ȳ−β₁x̄ = 22.2915；
R² = SSR/SST = 28.60/42869.82 = 0.0007；F = (SSR/1)/(SSE/696) = 0.4646，p = 0.4957；
SE(β₁)=√(61.55/6892.49)=0.0945；t=β₁/SE=-0.6816；p=0.4957；95%CI=[-0.2500,0.1211]。
H₀: 斜率 β₁=0（x 对 y 无线性影响）；H₁: β₁≠0。
x̄=40.0126，ȳ=19.7141；Sxx=Σ(x−x̄)²=6892.49，Sxy=-443.98；
β₁ = Sxy/Sxx = -0.0644；β₀ = ȳ−β₁x̄ = 22.2915；
R² = SSR/SST = 28.60/42869.82 = 0.0007；F = (SSR/1)/(SSE/696) = 0.4646，p = 0.4957；
SE(β₁)=√(61.55/6892.49)=0.0945；t=β₁/SE=-0.6816；p=0.4957；95%CI=[-0.2500,0.1211]。
③ 计算结果：
y = 22.2915 + -0.0644·x；R²=0.0007；β₁ t=-0.6816 p=0.4957；95%CI=[-0.2500,0.1211]
④ 结论：在 α=0.05 水平下，p=0.4957 ≥ 0.05，不拒绝 H₀。x 对 y 的线性影响不显著（斜率 β₁=-0.0644），应无影响。
第3类：多元线性回归（案例 76–85）
案例 76. 广告费+促销力度预测销量
多元线性回归两因素共同解释销量
① 原始数据：
样本 n=866，逐条 (x1, x2, y)：[(31.59, 39.01, -73.46), (49.28, 42.45, 6.75), (35.43, 75.92, 42.88), (32.96, 51.93, -33.16), (84.21, 67.5, 151.76), (59.22, 40.57, -6.34), (61.05, 71.61, 81.21), (51.25, 63.94, 19.67), (43.52, 44.31, -38.18), (30.11, 45.89, -79.95), (44.2, 58.77, 7.5), (38.83, 34.32, -65.48), (29.79, 84.5, 9.81), (63.35, 45.98, 24.36), (79.42, 26.96, 53.76), (64.06, 38.0, 46.8), (38.07, 58.31, -9.72), (64.47, 58.3, 17.82), (55.31, 44.48, 37.75), (57.65, 53.02, 59.81), (66.21, 72.8, 91.1), (75.57, 37.98, 48.42), (43.78, 31.87, -62.76), (66.67, 59.81, 78.13), (19.61, 47.07, -99.52), (47.65, 66.17, 26.25), (93.47, 55.3, 155.73), (52.79, 44.37, -0.55), (55.39, 59.15, 23.0), (50.06, 65.15, 34.93), (51.45, 50.31, 22.98), (44.17, 35.24, -37.5), (56.81, 67.49, 64.48), (73.85, 60.14, 107.75), (56.36, 59.5, 54.24), (48.94, 71.78, 56.59), (56.53, 65.23, 71.18), (55.66, 38.9, 27.54), (45.92, 44.34, 13.21), (58.87, 27.11, -10.98), (55.23, 61.17, 52.0), (60.88, 71.35, 84.06), (54.97, 55.17, 65.4), (47.18, 54.51, 24.39), (58.81, 57.73, 21.83), (57.53, 72.32, 47.19), (48.44, 52.84, -0.88), (48.15, 62.3, 6.42), (59.81, 32.71, 3.46), (63.29, 61.3, 59.05), (58.27, 34.31, -22.03), (40.77, 22.66, -98.84), (27.51, 49.56, -25.34), (31.52, 6.13, -170.93), (38.71, 41.91, -17.65), (65.75, 74.48, 80.62), (33.01, 27.3, -113.71), (74.27, 73.24, 152.25), (58.69, 62.42, 81.73), (39.69, 57.59, -29.53), (36.99, 42.94, -91.11), (51.52, 64.74, 40.77), (52.95, 38.48, 5.94), (45.43, 74.78, 85.52), (57.25, 60.88, 65.22), (36.98, 39.3, -79.6), (56.36, 69.21, 83.32), (76.94, 71.28, 153.45), (49.29, 34.29, -20.35), (64.84, 71.77, 78.77), (40.66, 46.13, 14.1), (75.36, 35.17, 46.19), (0.8, 70.45, -88.0), (31.8, 43.94, -85.3), (35.02, 52.72, -59.41), (31.38, 41.55, -68.59), (58.39, 42.71, -7.99), (8.76, 50.15, -100.25), (56.86, 53.08, 27.01), (53.86, 66.15, 84.79), (59.83, 77.81, 130.14), (37.57, 45.84, -19.54), (62.22, 50.28, 79.16), (32.47, 59.61, -37.54), (56.68, 47.69, -4.44), (36.18, 58.89, 12.97), (88.85, 68.14, 165.79), (44.51, 45.77, -49.34), (50.04, 47.94, -42.29), (29.99, 42.88, -67.74), (42.82, 65.71, 41.6), (35.11, 37.35, -26.74), (31.46, 49.62, -42.16), (47.0, 37.08, -24.77), (41.22, 42.96, -49.18), (58.58, 66.56, 26.29), (52.99, 37.76, 9.44), (52.73, 43.85, 8.95), (69.62, 51.68, 72.55), (45.13, 49.27, 7.03), (71.96, 57.65, 83.12), (70.69, 64.86, 89.56), (36.51, 40.08, -71.07), (64.62, 49.64, 93.8), (60.63, 53.85, 42.52), (32.61, 36.62, -73.0), (61.59, 78.26, 80.91), (48.7, 55.74, 9.13), (80.77, 55.68, 89.41), (77.88, 68.76, 115.58), (74.59, 79.36, 150.72), (33.36, 16.77, -96.94), (55.92, 75.56, 65.53), (42.49, 51.58, -13.95), (69.86, 41.71, 79.03), (53.78, 64.18, 41.12), (43.7, 40.51, -70.15), (47.96, 59.47, 24.91), (72.19, 58.76, 106.64), (53.84, 38.28, 8.13), (59.0, 64.71, 75.78), (24.33, 50.2, -56.45), (26.97, 38.28, -78.28), (50.13, 39.58, -21.22), (36.65, 25.73, -82.78), (44.87, 50.03, -27.85), (60.6, 28.24, -25.77), (62.23, 45.08, 28.23), (46.86, 50.73, -3.21), (87.65, 66.89, 134.35), (49.04, 53.06, 52.0), (42.24, 60.35, -2.16), (54.53, 40.67, 37.32), (50.41, 43.69, 25.45), (37.09, 70.59, 20.93), (27.4, 27.26, -112.24), (67.67, 74.13, 139.29), (57.14, 47.32, -0.43), (45.03, 28.62, -81.84), (62.68, 41.56, 17.51), (33.72, 34.3, -68.58), (62.68, 54.33, 50.15), (40.15, 33.87, -65.14), (42.5, 43.41, -30.98), (47.97, 24.54, -11.1), (43.48, 26.8, -52.22), (36.6, 27.53, -95.09), (65.5, 44.24, 59.14), (19.47, 47.82, -122.55), (34.89, 56.21, -6.75), (39.91, 41.92, -23.22), (9.45, 46.16, -116.97), (47.15, 66.04, 44.16), (18.52, 63.08, -81.74), (72.78, 42.66, 56.78), (45.09, 70.94, 25.42), (49.36, 19.81, -72.22), (43.94, 38.93, -54.72), (42.71, 21.37, -43.47), (64.62, 64.08, 86.32), (39.98, 32.62, -52.03), (53.14, 72.7, 69.21), (48.95, 73.27, 57.52), (42.95, 53.67, -34.36), (39.77, 29.69, -116.56), (29.09, 52.85, -33.29), (56.79, 32.4, -1.98), (24.84, 47.87, -46.34), (28.83, 79.53, 18.71), (56.99, 47.45, 25.71), (44.48, 63.96, 36.04), (40.1, 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(50.76, 38.92, -22.79), (45.59, 60.23, -10.92), (48.23, 34.04, -36.21), (27.23, 51.93, -81.89), (73.59, 65.8, 83.94), (39.22, 48.7, -19.43), (62.16, 40.02, 17.25), (23.18, 45.69, -49.67), (70.34, 60.32, 105.23), (48.62, 61.54, 13.12), (62.32, 32.11, -5.82), (49.34, 63.86, 50.41), (35.71, 34.89, -103.23), (76.61, 56.97, 82.7), (63.92, 36.12, 24.34), (51.14, 40.92, 9.89), (51.64, 26.62, -43.83), (65.59, 75.69, 83.58), (39.64, 43.76, -22.57), (74.68, 52.09, 59.0), (39.91, 44.21, -46.89), (62.21, 71.23, 94.68), (68.7, 45.52, 67.32), (31.45, 60.92, -7.32), (57.9, 32.56, -29.43), (37.32, 48.54, -17.35), (78.47, 51.53, 114.52), (56.18, 44.39, 13.91), (51.5, 47.69, 11.45), (32.17, 48.06, -23.52), (47.54, 43.55, 8.44), (41.08, 35.3, -79.75), (20.19, 45.51, -79.31), (47.22, 39.17, -14.76), (61.97, 53.25, 96.47), (67.74, 52.54, 84.69), (22.19, 53.12, -87.03), (68.25, 56.58, 74.67), (42.24, 34.52, -86.28), (57.86, 65.05, 79.71), (54.2, 23.57, -35.44), (57.32, 60.99, 18.41), (62.5, 36.18, 31.5), (40.57, 89.99, 55.15), (66.01, 51.52, 100.34), (35.01, 34.71, -85.3), (36.41, 46.59, -44.9), (37.85, 49.48, -55.71), (29.46, 31.49, -35.68), (14.55, 19.33, -177.29), (45.77, 26.84, -57.96), (50.3, 73.69, 49.12), (25.51, 53.89, -74.48), (13.81, 29.42, -187.56), (40.85, 51.88, -42.62), (57.24, 54.57, 39.04), (55.05, 37.51, 4.6), (53.08, 51.06, -1.47), (53.37, 39.33, 3.83), (61.23, 43.19, 34.71), (61.75, 55.55, 59.03), (46.53, 39.55, -46.81), (43.69, 56.4, -12.09), (46.96, 21.48, -65.05), (60.84, 69.14, 98.77), (47.99, 50.51, -1.65), (45.99, 60.39, 24.07), (58.41, 65.12, 41.28), (63.96, 48.89, 82.95), (73.1, 55.36, 90.6), (65.7, 51.49, 63.36), (41.01, 73.98, 14.88), (43.61, 58.53, -7.73), (52.69, 67.28, 77.67), (63.57, 81.34, 130.91), (73.95, 57.87, 94.19), (58.28, 30.96, 21.54), (36.82, 56.36, -29.47), (31.56, 35.79, -55.79), (68.56, 42.2, 53.55), (56.14, 41.13, 30.64), (41.11, 32.08, -51.44), (57.01, 64.73, 63.08), (32.01, 45.86, -12.31), (63.4, 47.6, 51.55), (40.47, 41.42, -30.71), (43.92, 58.4, -23.91), (41.01, 38.26, -54.5), (39.37, 44.56, -47.85), (71.53, 57.91, 104.22), (57.2, 75.28, 68.45), (60.85, 53.65, 37.26)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=7.3771，β₁=3.0120，β₂=2.0904；
SST=3751438.74，SSR=3395382.01，SSE=356056.73；R²=0.9051，调整R²=0.9049；
整体 F=(SSR/2)/(SSE/863)=4114.8144，p=<1e-300；x1,x2 相关 r=0.2736；
β₁: SE=0.0468，t=64.3051，p=<1e-300；β₂: SE=0.0476，t=43.9559，p=1.91e-222。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=7.3771，β₁=3.0120，β₂=2.0904；
SST=3751438.74，SSR=3395382.01，SSE=356056.73；R²=0.9051，调整R²=0.9049；
整体 F=(SSR/2)/(SSE/863)=4114.8144，p=<1e-300；x1,x2 相关 r=0.2736；
β₁: SE=0.0468，t=64.3051，p=<1e-300；β₂: SE=0.0476，t=43.9559，p=1.91e-222。
③ 计算结果：
y=7.3771+3.0120x1+2.0904x2；R²=0.9051；β₁ p=<1e-300；β₂ p=1.91e-222
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=<1e-300）；变量 x2 对 y 的影响显著（p=1.91e-222）。两因素共同解释销量。
案例 77. 温度+湿度预测冰淇淋销量
多元线性回归温湿协同影响
① 原始数据：
样本 n=539，逐条 (x1, x2, y)：[(25.62, 46.54, -51.4), (29.7, 57.2, -22.78), (28.59, 69.33, 108.1), (23.36, 49.08, -52.09), (34.59, 65.14, 52.35), (21.26, 56.13, -48.37), (28.85, 63.3, 73.27), (25.57, 60.57, -28.83), (24.47, 50.34, -68.51), (15.48, 53.61, -90.37), (25.78, 57.97, -15.22), (27.45, 67.55, 6.69), (30.17, 42.06, -65.73), (26.68, 46.88, -51.88), (24.72, 65.95, 17.09), (31.66, 62.15, 45.42), (26.12, 52.25, -4.2), (31.83, 52.92, 35.79), (24.11, 54.52, -23.43), (29.26, 49.89, -30.33), (17.57, 69.95, -12.15), (37.63, 66.8, 124.55), (36.73, 61.8, 96.2), (25.59, 64.29, 20.17), (34.55, 54.06, 42.92), (31.83, 58.98, 26.24), (26.67, 78.32, 80.56), (26.35, 69.37, 68.03), (29.37, 69.26, 42.73), (31.75, 74.26, 129.16), (26.02, 51.27, -29.13), (23.15, 60.86, -18.7), (27.26, 57.82, 5.75), (19.31, 48.45, -69.74), (35.71, 60.12, 44.71), (22.49, 42.98, -117.38), (27.05, 46.8, -94.33), (36.66, 76.11, 99.56), (33.37, 64.31, 66.74), (23.12, 46.27, -83.38), (23.76, 46.48, -48.37), (21.07, 70.25, 21.95), (32.04, 63.49, 37.31), (27.25, 58.36, -8.48), (32.2, 58.15, 48.69), (34.6, 71.84, 121.71), (25.93, 52.47, -33.8), (37.32, 62.28, 101.28), (24.1, 61.92, -8.2), (26.37, 43.2, -36.6), (24.97, 55.12, -18.07), (24.26, 58.45, -7.19), (29.51, 54.22, -14.54), (27.69, 47.69, -123.45), (35.05, 44.87, -9.91), (28.12, 67.39, 39.34), (29.52, 57.23, -19.83), (29.84, 62.44, 74.18), (29.37, 60.07, 48.16), (36.41, 85.76, 151.75), (36.02, 61.23, 46.51), (28.04, 67.19, 37.54), (19.17, 80.49, 44.92), (29.17, 46.18, -28.08), (32.92, 60.49, -2.36), (34.28, 68.55, 97.8), (23.73, 50.41, -69.09), (34.37, 50.2, 65.17), (21.87, 60.15, -34.84), (29.22, 47.79, -51.09), (13.82, 52.65, -98.88), (24.87, 54.26, -10.59), (30.82, 68.55, 66.29), (19.41, 50.14, -67.04), (24.46, 63.79, -6.25), (32.84, 39.97, -15.32), (39.33, 75.93, 115.7), (26.36, 61.62, -25.48), (18.91, 52.49, -88.88), (25.35, 63.91, 4.06), (25.83, 56.76, -43.96), (22.44, 63.78, 27.02), (32.52, 63.42, 46.47), (28.12, 62.97, 77.07), (40.27, 67.6, 117.93), (39.0, 75.95, 156.51), (42.18, 69.06, 81.82), (29.44, 66.4, 46.48), (24.71, 45.09, -48.59), (31.76, 62.08, 41.18), (20.56, 75.94, 28.63), (30.47, 57.2, 22.91), (30.3, 64.94, 32.92), (38.87, 65.73, 60.94), (24.36, 53.71, -40.17), (27.39, 47.14, -42.5), (31.01, 73.14, 67.29), (27.7, 73.18, 60.74), (23.28, 53.54, -40.51), (22.51, 72.4, -13.05), (25.14, 54.53, -58.7), (32.37, 70.98, 99.07), (31.9, 61.07, 68.88), (33.63, 53.57, 12.16), (33.83, 58.97, 61.37), (19.5, 52.76, -37.05), (34.57, 46.77, -17.33), (29.63, 64.88, -2.44), (22.59, 56.04, -62.3), (35.18, 57.95, 41.54), (32.58, 67.97, 59.75), (30.28, 53.19, -31.09), (23.96, 48.88, -48.77), (25.94, 58.55, 1.57), (34.41, 61.91, 17.73), (25.77, 56.93, -34.78), (30.61, 68.3, 51.41), (30.06, 71.47, 69.04), (33.47, 59.85, 46.17), (13.39, 41.57, -169.71), (30.14, 68.1, 25.26), (31.85, 72.83, 113.95), (24.98, 43.66, -66.31), (32.02, 54.87, 9.08), (28.22, 48.02, -62.46), (31.75, 98.65, 203.55), (24.23, 79.45, 64.39), (18.25, 45.07, -102.98), (25.16, 52.79, -32.19), (31.82, 55.53, -34.91), (22.1, 61.88, -21.38), (27.11, 61.05, 5.21), (19.55, 52.48, -96.26), (30.27, 40.54, -68.86), (34.77, 51.79, 23.14), (25.68, 52.52, -5.81), (24.07, 59.79, 4.13), (37.86, 69.67, 131.04), (22.63, 60.89, -31.45), (19.12, 74.48, 33.45), (25.18, 65.34, -15.54), (27.99, 60.68, -2.43), (22.64, 58.93, 11.16), (19.15, 66.46, 36.47), (23.11, 48.96, -56.17), (26.27, 64.0, 3.11), (20.97, 55.14, -52.5), (28.91, 66.82, 4.65), (23.43, 43.49, -108.51), (27.5, 43.99, -64.47), (25.22, 53.29, -24.07), (30.53, 41.01, -21.72), (29.03, 55.57, 13.0), (20.99, 53.46, -35.85), (16.51, 64.44, -52.06), (24.31, 67.32, 22.77), (19.61, 52.22, -67.65), (31.22, 79.81, 91.1), (26.9, 54.25, -14.75), (34.08, 57.82, 17.02), (31.31, 77.18, 121.98), (34.86, 59.9, 62.11), (35.15, 63.72, 49.53), (22.19, 62.14, 21.71), (22.13, 64.55, -82.45), (35.11, 64.88, 36.61), (23.47, 62.29, 10.98), (24.78, 45.4, -85.16), (31.18, 59.28, 96.14), (21.55, 51.12, -105.38), (27.15, 57.28, -14.53), (25.45, 55.2, -26.95), (26.63, 72.14, 120.36), (28.05, 68.23, 14.57), (28.97, 57.35, 13.08), (26.22, 54.04, -27.02), (30.66, 61.3, 47.48), (26.89, 53.88, 3.24), (31.02, 53.8, -31.57), (32.13, 58.47, 8.51), (27.44, 36.93, -127.46), (16.39, 62.69, -51.32), (33.43, 62.92, 103.19), (16.58, 67.16, -4.6), (29.04, 57.5, 15.35), (26.77, 63.48, 31.73), (38.91, 61.46, 78.89), (27.68, 59.94, 29.76), (19.27, 64.65, 19.69), (37.16, 67.96, 79.95), (22.58, 52.34, -1.47), (30.05, 75.08, 56.4), (17.85, 44.94, -65.78), (26.81, 60.01, 9.36), (26.4, 37.97, -66.6), (40.3, 60.56, 92.92), (31.17, 56.88, 21.99), (24.68, 55.08, -60.58), (37.81, 51.96, 42.65), (22.4, 60.84, -18.97), (37.06, 51.32, 56.8), (26.12, 59.86, -5.24), (29.24, 63.43, 30.85), (26.3, 41.29, -78.0), (24.23, 54.71, -51.17), (29.86, 56.32, -5.12), (35.37, 49.43, 28.54), (30.07, 62.64, 29.01), (37.85, 57.0, 86.68), (25.59, 62.79, -6.32), (30.05, 60.67, 24.51), (15.31, 41.71, -162.77), (33.39, 59.4, 60.82), (31.8, 62.97, 74.57), (31.17, 62.28, 31.3), (26.11, 68.7, 13.97), (31.17, 64.33, 59.41), (28.59, 51.53, -48.24), (29.78, 56.44, -46.62), (29.35, 44.75, -76.1), (31.69, 71.08, 80.97), (22.91, 63.08, 30.8), (27.55, 33.35, -89.79), (20.77, 49.9, -52.22), (29.19, 65.29, 25.94), (22.27, 76.94, 9.44), (29.97, 42.54, -46.19), (24.91, 51.87, -41.64), (15.46, 51.95, -128.25), (24.97, 72.99, 25.32), (20.31, 50.12, -72.23), (24.31, 53.91, -56.33), (21.77, 83.62, 51.79), (25.74, 70.17, 42.5), (33.22, 66.23, 93.08), (33.43, 62.29, 72.34), (33.04, 55.85, 37.1), (30.73, 81.0, 109.96), (34.15, 63.69, 68.54), (28.69, 58.72, 24.65), (37.19, 63.73, 96.86), (31.04, 58.46, -20.92), (31.27, 67.17, 26.41), (20.08, 59.74, -6.68), (26.21, 48.42, -39.11), (27.02, 59.34, -12.43), (28.55, 47.39, 7.92), (31.3, 62.64, 65.7), (31.13, 41.41, -41.14), (27.64, 63.65, 42.72), (26.96, 51.92, 0.66), (24.82, 55.73, -41.92), (19.04, 64.23, -41.1), (31.6, 50.37, -36.39), (30.7, 60.63, 29.97), (17.96, 67.16, -10.0), (36.2, 58.07, 55.83), (34.96, 72.78, 100.82), (23.94, 64.06, -4.39), (32.69, 68.62, 71.96), (26.02, 50.44, -30.2), (29.07, 52.76, 9.49), (28.91, 41.0, -32.68), (26.15, 82.39, 115.5), (28.39, 63.8, 0.94), (21.26, 71.28, -25.41), (30.58, 80.96, 113.75), (26.54, 41.61, -59.09), (33.72, 52.32, 48.94), (28.49, 57.89, 21.03), (24.48, 69.33, 14.63), (24.1, 47.99, -31.45), (22.75, 43.01, -47.25), (27.0, 48.55, -31.87), (29.15, 61.53, -6.0), (21.52, 55.15, -26.48), (32.4, 61.12, 75.71), (26.61, 63.81, -3.21), (29.42, 55.41, -31.4), (32.98, 68.38, 106.58), (26.96, 43.67, -29.04), (33.83, 61.35, 41.11), (22.72, 60.14, -55.05), (38.61, 57.78, 70.83), (19.09, 52.87, -84.88), (23.56, 66.17, 6.18), (30.85, 55.45, -2.14), (18.22, 71.83, -17.17), (25.09, 59.25, -37.28), (29.06, 49.65, -37.62), (26.02, 65.96, -17.95), (28.42, 60.95, -2.85), (28.01, 62.92, -8.89), (41.19, 58.16, 89.42), (24.41, 67.64, 31.84), (28.49, 61.65, 21.12), (25.86, 67.34, -3.02), (33.06, 73.07, 107.0), (20.83, 60.76, -49.63), (25.24, 63.01, 26.18), (28.56, 55.63, 0.27), (31.75, 59.75, -2.32), (30.91, 54.01, -27.85), (28.57, 77.18, 57.63), (28.43, 64.77, 46.21), (25.35, 68.17, 21.92), (27.55, 67.27, 44.64), (19.58, 69.43, 8.26), (32.86, 65.52, 67.01), (21.84, 59.91, -57.89), (30.25, 74.09, 75.0), (37.08, 44.3, -4.74), (23.55, 49.97, -31.87), (19.05, 41.22, -125.18), (29.99, 41.04, -61.22), (30.92, 64.26, 77.75), (20.81, 57.69, -10.29), (24.35, 36.58, -98.32), (27.75, 64.04, 18.08), (24.3, 46.96, -44.93), (23.8, 60.06, -3.06), (24.84, 53.85, -29.23), (31.2, 65.81, 50.87), (30.76, 53.39, 18.88), (26.04, 68.91, 34.32), (21.06, 73.59, 4.89), (27.85, 55.13, -51.4), (21.64, 61.92, -7.29), (20.04, 51.1, -73.96), (19.85, 58.37, -44.03), (33.46, 74.54, 114.84), (24.43, 47.02, -52.83), (25.7, 64.05, 20.29), (22.86, 57.98, 22.0), (23.62, 53.9, -23.47), (29.32, 73.18, 112.3), (31.52, 65.5, 39.44), (25.12, 51.45, -43.78), (26.29, 74.35, 68.49), (23.53, 65.23, -53.56), (26.67, 43.33, -58.54), (28.47, 47.59, -68.25), (30.47, 52.65, -15.5), (32.07, 70.12, 68.01), (21.97, 64.0, 25.2), (28.7, 41.99, -63.38), (28.91, 49.24, -25.62), (36.98, 50.14, 23.42), (24.05, 40.33, -87.2), (24.3, 45.95, -87.4), (23.06, 48.41, -57.26), (24.72, 42.21, -72.83), (18.01, 71.13, -24.02), (31.97, 44.58, -73.41), (26.0, 64.11, -23.57), (37.79, 61.45, 87.91), (37.65, 57.02, 52.28), (28.43, 61.08, -2.58), (30.83, 71.18, 65.68), (30.47, 49.93, -32.86), (23.93, 61.81, -21.8), (26.59, 61.28, 43.08), (34.38, 45.41, -26.84), (32.34, 60.83, -16.89), (30.75, 72.75, 93.4), (31.07, 59.03, 21.88), (20.53, 47.52, -110.3), (34.13, 66.74, 129.77), (27.59, 50.72, -39.49), (29.77, 52.71, -33.0), (35.41, 60.45, 83.99), (21.6, 86.17, 80.5), (22.28, 51.38, -60.45), (16.89, 45.08, -95.09), (25.4, 54.09, -32.3), (22.17, 66.85, 15.62), (24.14, 62.67, -28.32), (16.76, 67.67, -32.83), (32.76, 47.59, 11.33), (23.41, 56.45, -47.35), (29.64, 48.47, -36.12), (21.81, 49.53, -43.83), (28.27, 54.52, 12.71), (18.8, 34.56, -157.75), (31.71, 64.68, 50.36), (18.34, 49.5, -105.34), (42.18, 69.05, 132.43), (22.08, 61.6, -65.24), (32.65, 58.06, 52.81), (32.19, 80.53, 122.53), (32.65, 66.32, 59.55), (31.44, 53.45, 12.82), (24.29, 73.26, 52.21), (23.42, 47.15, -73.88), (27.37, 61.6, 48.16), (28.23, 45.02, -0.57), (31.71, 50.18, -40.53), (29.06, 70.76, 73.28), (19.81, 48.26, -128.5), (21.84, 42.46, -125.14), (33.43, 71.36, 55.69), (31.98, 58.83, 33.11), (39.64, 62.69, 107.55), (28.86, 71.26, 64.15), (20.89, 39.8, -80.45), (30.08, 36.27, -120.86), (23.33, 49.52, -113.64), (29.64, 68.5, 7.11), (24.76, 61.87, 5.23), (34.85, 55.46, 37.08), (30.06, 61.6, -21.82), (18.21, 67.28, -20.86), (27.85, 79.54, 96.49), (26.49, 56.23, -8.36), (17.0, 54.92, -69.5), (14.11, 68.16, -73.21), (31.17, 42.53, -39.71), (26.2, 43.82, -68.23), (29.77, 58.43, 4.08), (22.71, 53.14, -9.85), (25.87, 50.38, -59.56), (30.8, 71.45, 85.46), (18.17, 63.03, -38.51), (23.63, 68.95, 24.67), (27.08, 48.06, -22.72), (18.28, 47.79, -65.63), (26.05, 58.99, -3.26), (33.97, 79.0, 133.73), (11.91, 59.65, -81.97), (25.42, 78.68, 94.44), (22.42, 51.13, -104.32), (24.23, 49.75, -84.65), (30.86, 83.99, 112.46), (32.83, 50.62, -7.32), (32.68, 78.77, 73.91), (25.97, 32.9, -131.94), (33.7, 43.97, -18.6), (29.02, 43.71, -99.28), (28.87, 66.22, 13.77), (22.99, 58.54, 3.77), (25.77, 60.63, -6.48), (17.92, 49.01, -121.04), (34.4, 63.92, 67.63), (18.73, 48.13, -62.45), (35.73, 67.37, 118.87), (29.54, 60.6, 29.71), (31.67, 55.9, 28.36), (22.93, 43.74, -66.91), (25.12, 69.92, 9.84), (23.79, 55.96, -43.86), (35.92, 76.08, 132.32), (24.81, 55.55, -37.57), (37.09, 63.3, 60.7), (27.97, 63.18, 19.76), (22.34, 63.44, -27.89), (31.02, 60.6, 2.68), (30.69, 74.11, 72.02), (27.2, 62.11, 24.98), (27.95, 80.95, 108.17), (31.19, 68.19, 65.21), (28.58, 71.5, 105.91), (21.66, 60.93, -70.78), (29.33, 57.95, -30.73), (28.88, 62.96, 57.04), (26.84, 58.71, -36.47), (26.91, 58.39, 28.27), (36.32, 65.01, 103.91), (23.37, 61.12, -49.92), (19.8, 57.79, 20.61), (22.43, 65.47, -19.61), (23.12, 53.54, -66.43), (26.68, 66.58, -29.9), (32.23, 60.97, 84.55), (20.71, 51.92, -93.35), (29.71, 56.32, 33.15), (30.3, 69.01, 132.1), (35.62, 64.7, 93.85), (35.98, 82.72, 149.47), (28.2, 70.48, 75.98), (20.74, 41.44, -137.34), (34.25, 55.66, 35.57), (30.57, 54.45, 12.62), (37.23, 75.33, 123.38), (29.18, 69.1, 43.97), (19.39, 65.62, -11.02), (19.61, 53.83, -18.98), (27.33, 55.72, -48.79), (37.37, 45.85, -6.09), (28.71, 64.08, 0.69), (32.05, 53.26, -17.41), (39.36, 67.81, 153.04), (38.0, 40.87, 12.11), (37.78, 66.64, 114.44), (27.08, 55.54, -51.02), (21.32, 60.22, -42.48), (24.74, 57.08, -31.23), (33.03, 51.35, -45.18), (30.18, 77.45, 96.76), (25.38, 58.85, -32.46), (24.01, 62.58, 4.16), (24.14, 58.04, -39.51), (26.5, 55.72, 15.62), (30.41, 61.28, 34.16), (29.05, 61.34, 8.46), (23.38, 61.38, -37.46), (26.39, 44.1, -62.81), (24.88, 56.24, -33.56), (16.72, 72.63, -35.01), (31.0, 65.51, 54.97), (18.58, 47.53, -127.02), (29.13, 43.51, -47.25), (24.76, 63.47, 50.27), (24.46, 61.2, -1.86), (24.4, 58.86, -2.44), (22.69, 58.05, -29.74), (32.27, 70.28, 121.16), (34.08, 57.84, 25.38), (34.3, 63.94, 79.77), (25.66, 61.95, 34.45), (31.49, 54.9, 7.15), (24.03, 68.95, 39.04), (33.75, 63.17, 32.8), (22.4, 77.39, 46.53), (30.14, 46.01, -33.99), (27.06, 57.9, -27.92), (25.54, 51.6, -106.91), (29.08, 65.95, -6.89), (35.31, 60.54, 15.24), (21.03, 44.31, -90.75), (37.68, 52.93, 62.04), (17.99, 68.76, -24.05), (32.85, 66.38, 79.06), (33.67, 56.75, 19.74), (22.5, 70.11, -1.37), (31.5, 63.97, -13.2), (22.1, 57.66, -57.53), (16.17, 51.75, -110.63), (25.54, 81.03, 111.81), (35.56, 51.76, 40.81)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=9.4728，β₁=6.1857，β₂=4.0984；
SST=2095506.32，SSR=1768961.87，SSE=326544.45；R²=0.8442，调整R²=0.8436；
整体 F=(SSR/2)/(SSE/536)=1451.8139，p=4.28e-217；x1,x2 相关 r=0.1666；
β₁: SE=0.1977，t=31.2919，p=4.82e-123；β₂: SE=0.1077，t=38.0436，p=2.09e-154。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=9.4728，β₁=6.1857，β₂=4.0984；
SST=2095506.32，SSR=1768961.87，SSE=326544.45；R²=0.8442，调整R²=0.8436；
整体 F=(SSR/2)/(SSE/536)=1451.8139，p=4.28e-217；x1,x2 相关 r=0.1666；
β₁: SE=0.1977，t=31.2919，p=4.82e-123；β₂: SE=0.1077，t=38.0436，p=2.09e-154。
③ 计算结果：
y=9.4728+6.1857x1+4.0984x2；R²=0.8442；β₁ p=4.82e-123；β₂ p=2.09e-154
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=4.82e-123）；变量 x2 对 y 的影响显著（p=2.09e-154）。温湿协同影响。
案例 78. 面积+房龄预测租金
多元线性回归面积主、房龄负
① 原始数据：
样本 n=684，逐条 (x1, x2, y)：[(79.29, 16.07, 919.03), (97.49, 22.91, 1041.57), (109.71, 9.94, 2011.17), (106.55, 19.1, 1700.98), (99.28, 18.41, 848.24), (71.16, 12.19, 646.66), (120.44, 10.3, 2296.72), (122.35, 10.09, 2141.88), (61.28, 16.26, -190.09), (56.33, 18.41, -816.17), (105.52, 12.21, 1170.59), (62.52, 8.88, 202.17), (123.53, 2.69, 2156.89), (81.09, 10.67, 767.17), (97.2, 15.49, 1481.33), (114.04, 8.23, 2725.42), (133.92, 19.56, 2611.39), (58.17, 12.31, 168.25), (82.22, 13.13, 858.13), (68.57, 14.03, 129.81), (63.31, 18.17, -101.84), (78.12, 15.18, 474.87), (166.69, 11.14, 4102.16), (78.46, 13.49, 486.12), (75.99, 11.53, 411.38), (90.84, 15.58, 1410.12), (70.75, 8.44, 101.0), (112.9, 17.52, 1494.86), (65.21, 10.49, 453.78), (95.89, 16.57, 1389.17), (118.36, 18.33, 2009.16), (95.65, 20.18, 701.75), (42.54, 17.3, -820.97), (61.69, 20.37, -21.58), (55.49, 18.99, -454.41), (112.41, 14.88, 2184.23), (113.86, 17.52, 1735.48), (91.63, 14.64, 1063.59), (88.72, 29.24, 960.57), (113.28, 16.24, 1909.74), (51.85, 11.35, -412.96), (64.09, 8.86, -263.48), (129.27, 15.88, 2276.32), (115.62, 20.87, 1526.2), (65.62, 9.83, 78.73), (82.16, 24.85, 533.3), (142.29, 16.82, 2911.73), (124.76, 16.67, 2338.98), (55.83, 8.55, -82.85), (141.17, 15.74, 2919.3), (73.23, 10.79, 414.18), (96.22, 13.76, 879.77), (117.63, 18.59, 1999.38), (84.71, 9.16, 1097.2), (84.92, 17.26, 634.19), (132.1, 11.19, 2341.55), (92.18, 16.55, 807.11), (71.12, 16.54, 143.38), (65.79, 18.54, -171.15), (101.86, 19.57, 1464.99), (93.86, 15.34, 1361.38), (75.05, 7.18, 276.67), (123.04, 10.4, 2452.26), (86.1, 9.99, 903.05), (108.02, 7.57, 2003.84), (119.29, 21.41, 1883.36), (58.04, 11.06, -424.93), (72.71, 11.04, 175.32), (96.8, 15.82, 1362.07), (95.8, 10.18, 1678.12), (100.51, 26.74, 1294.3), (92.68, 10.69, 1218.61), (100.58, 14.18, 1915.78), (104.7, 18.55, 1566.12), (78.44, 21.06, 553.73), (122.59, 10.44, 2690.71), (106.07, 13.56, 1402.5), (72.52, 26.52, 117.0), (69.16, 19.79, 380.64), (63.47, 16.87, -71.66), (100.01, 19.38, 1158.79), (75.72, 16.06, 389.23), (76.46, 20.09, -294.43), (82.13, 19.52, 313.72), (78.86, 21.96, 401.24), (127.39, 9.28, 2683.53), (45.17, 12.98, -739.13), (100.19, 23.42, 785.94), (112.07, 15.21, 1598.74), (112.6, 12.26, 1893.26), (115.83, 6.65, 2078.37), (105.38, 12.41, 1915.55), (92.58, 21.51, 504.79), (115.03, 16.18, 2361.7), (35.25, 5.51, -1076.12), (108.44, 18.11, 1495.56), (61.54, 18.57, -464.63), (77.28, 14.53, 774.5), (102.82, 17.5, 1370.7), (81.26, 19.47, 228.81), (101.84, 20.08, 1551.04), (78.4, 14.37, 739.28), (82.39, 15.4, 940.91), (93.76, 21.05, 1066.66), (93.33, 4.59, 1131.21), (111.95, 24.42, 2132.34), (66.25, 6.06, 226.06), (81.12, 25.55, 231.31), (66.25, 12.64, 219.06), (87.79, 16.38, 1459.62), (94.6, 7.01, 1117.03), (77.71, 12.96, 871.09), (90.38, 14.67, 501.4), (94.15, 11.32, 893.35), (55.16, 14.62, -208.35), (103.19, 23.61, 1085.83), (66.36, 15.98, -430.72), (114.22, 7.34, 1910.7), (35.1, 23.16, -1720.49), (16.98, 1.94, -1125.73), (80.93, 11.61, 550.45), (87.3, 14.6, 1202.22), (69.06, 5.94, 271.84), (91.94, 14.0, 903.5), (96.61, 10.21, 1383.81), (102.92, 9.54, 1710.98), (80.94, 18.55, 906.05), (110.66, 12.26, 2407.98), (89.57, 16.57, 909.02), (112.76, 25.63, 1625.38), (46.75, 14.24, -1078.5), (61.58, 17.56, -45.23), (120.0, 11.84, 2488.76), (67.94, 10.96, 366.39), (58.53, 13.52, -1.4), (75.24, 12.49, 60.28), (110.91, 19.3, 1290.45), (99.13, 18.49, 694.44), (98.94, 20.12, 966.7), (87.23, 13.8, 405.81), (74.15, 23.04, 340.48), (155.75, 17.65, 4034.29), (104.93, 25.01, 1775.11), (112.11, 19.22, 1287.91), (97.49, 20.53, 1772.44), (127.89, 16.32, 2620.74), (83.11, 2.6, 903.99), (57.93, 12.89, -251.6), (71.39, 19.04, 664.91), (75.57, 19.33, -240.88), (120.09, 8.46, 1918.84), (95.61, 17.39, 1352.01), (96.23, 17.45, 874.91), (72.68, 13.74, 30.34), (50.16, 16.0, -190.88), (67.8, 21.4, -599.59), (64.57, 8.35, 362.46), (110.49, 8.97, 2007.65), (132.31, 16.2, 2913.97), (74.53, 11.93, 73.56), (68.04, 14.77, 534.31), (33.53, 24.99, -1394.45), (115.42, 12.42, 1514.54), (119.98, 9.78, 1775.33), (124.25, 21.07, 1967.16), (76.54, 9.91, 331.2), (99.62, 18.75, 1322.82), (86.43, 11.41, 761.47), (76.88, 10.3, 978.84), (101.9, 13.37, 1794.12), (133.84, 19.04, 3196.31), (96.93, 15.15, 1719.43), (103.42, 10.7, 1116.76), (94.42, 14.9, 1459.02), (90.74, 15.71, 559.56), (98.01, 11.31, 1457.89), (139.8, 14.29, 2709.25), (97.78, 14.16, 1605.88), (76.12, 16.04, 160.91), (84.72, 16.77, 408.38), (67.2, 14.86, -259.42), (105.36, 17.24, 1520.98), (115.5, 11.21, 2046.09), (68.11, 15.76, 403.93), (121.4, 19.49, 2542.14), (98.29, 4.92, 1922.12), (112.73, 12.11, 2192.48), (115.2, 12.87, 1912.37), (106.59, 27.52, 1287.62), (51.31, 13.58, -844.75), (92.18, 19.19, 1210.38), (101.54, 11.85, 1633.51), (126.13, 19.01, 2816.96), (91.62, 11.55, 1169.26), (114.18, 15.0, 2303.16), (113.2, 16.54, 1879.38), (91.43, 12.47, 1703.18), (75.05, 10.47, 624.8), (94.52, 21.42, 1010.54), (101.49, 15.16, 1930.75), (49.07, 11.6, -85.1), (95.61, 13.91, 1219.79), (84.59, 19.24, 305.67), (123.82, 11.55, 2526.07), (111.09, 14.48, 1715.27), (91.12, 18.58, 1165.02), (76.97, 17.0, 948.99), (58.53, 13.19, -21.63), (87.57, 10.93, 934.54), (138.4, 19.5, 2729.75), (100.44, 9.68, 1796.81), (66.59, 12.35, 37.84), (62.73, 11.91, -293.07), (50.67, 11.66, -1140.53), (79.72, 16.21, 913.87), (86.55, 24.68, 361.65), (110.86, 16.04, 2396.89), (80.91, 4.07, 997.04), (55.6, 7.81, 332.41), (116.7, 14.33, 2149.7), (102.05, 13.86, 1420.72), (151.83, 13.08, 3104.53), (112.75, 10.25, 1893.37), (97.86, 22.92, 1723.11), (49.72, 4.64, -269.61), (131.02, 10.53, 2156.65), (103.41, 23.69, 1149.04), (104.76, 13.9, 1599.65), (67.63, 19.5, -400.83), (97.98, 11.98, 1384.15), (91.65, 22.12, 1442.46), (104.49, 7.95, 1186.93), (153.5, 9.04, 3763.63), (71.56, 16.59, 449.63), (112.9, 22.91, 1948.09), (101.79, 16.92, 1929.83), (84.47, 10.39, 738.06), (117.54, 10.67, 2233.87), (117.89, 21.9, 1879.5), (126.28, 20.33, 2332.02), (109.03, 10.0, 1622.84), (78.5, 10.28, 560.9), (110.57, 12.21, 1975.96), (104.22, 10.82, 2160.42), (95.8, 19.42, 906.99), (68.4, 13.91, -105.78), (79.05, 10.8, 241.9), (135.5, 20.05, 2815.69), (108.42, 25.86, 1689.53), (91.68, 21.11, 666.76), (94.84, 16.71, 1168.46), (74.77, 13.7, 226.22), (84.07, 14.64, 640.8), (81.45, 13.08, 1076.07), (91.6, 14.74, 1363.67), (66.78, 4.87, 103.67), (79.01, 16.26, 300.17), (73.8, 16.48, -60.38), (103.42, 14.37, 1987.01), (111.31, 7.97, 2077.97), (94.97, 13.69, 1270.85), (29.95, 7.79, -511.4), (118.42, 19.41, 2335.38), (126.2, 17.78, 2661.68), (122.1, 8.71, 2155.8), (121.72, 9.08, 2947.63), (115.06, 13.61, 2093.78), (51.22, 17.83, -652.44), (117.69, 21.28, 2140.92), (119.44, 13.4, 1848.83), (58.75, 10.76, -368.25), (62.57, 5.22, -79.82), (91.8, 16.62, 813.91), (169.92, 27.49, 3542.83), (92.32, 12.6, 731.53), (128.98, 20.72, 2524.05), (111.72, 20.16, 1748.48), (63.1, 11.49, -164.66), (103.0, 23.31, 1012.15), (99.5, 17.87, 1482.17), (72.84, 11.09, 442.37), (103.16, 12.66, 1923.2), (78.04, 23.16, 576.04), (110.38, 16.12, 1521.4), (72.47, 13.89, 534.94), (49.68, 6.79, -102.36), (110.19, 9.36, 2002.05), (105.32, 16.52, 1506.02), (77.82, 10.32, 1083.27), (108.91, 8.96, 1572.93), (86.95, 21.51, 475.83), (48.16, 11.1, -625.15), (105.16, 21.87, 1329.85), (100.9, 9.43, 1741.21), (91.4, 14.54, 1287.17), (71.82, 17.23, 741.42), (43.93, 7.8, -793.39), (63.86, 16.18, 17.52), (93.68, 21.86, 694.2), (119.12, 23.77, 1985.33), (35.14, 22.02, -1812.13), (97.27, 9.64, 1195.94), (119.55, 17.08, 1814.01), (115.06, 17.67, 2370.09), (83.8, 8.87, 944.62), (77.97, 22.85, 330.87), (85.88, 13.63, 771.55), (93.09, 17.12, 1217.52), (113.71, 15.86, 2050.25), (100.06, 17.65, 1449.35), (124.78, 14.6, 2470.28), (49.53, 20.6, -935.73), (63.62, 12.8, -161.06), (31.5, 8.09, -1187.56), (108.33, 13.07, 2034.53), (95.47, 19.29, 1238.45), (52.53, 15.15, -569.96), (109.87, 22.38, 1865.49), (105.14, 19.84, 1599.21), (69.69, 21.45, -111.35), (65.64, 10.46, 391.74), (96.85, 18.6, 914.69), (99.24, 9.3, 1427.43), (116.06, 1.12, 2278.53), (97.99, 18.56, 855.91), (127.81, 14.19, 1732.4), (127.53, 16.67, 2455.01), (115.19, 8.12, 2363.96), (91.05, 16.21, 1292.44), (92.92, 14.02, 1145.16), (104.61, 19.09, 1470.71), (94.7, 15.46, 965.19), (89.85, 16.22, 204.25), (73.27, 16.26, 368.49), (52.19, 7.71, -599.84), (68.03, 18.19, -61.46), (82.12, 10.58, 593.16), (79.75, 9.5, 1622.4), (95.2, 6.1, 1358.35), (96.97, 26.73, 675.18), (112.56, 6.61, 1781.04), (42.69, 8.43, -428.37), (116.64, 13.41, 1896.68), (35.75, 4.95, -1051.39), (45.0, 19.93, -1310.94), (63.71, 6.82, -271.86), (139.2, 18.8, 2668.92), (70.46, 18.99, -291.14), (111.51, 10.37, 1895.72), (96.59, 14.06, 1129.88), (85.78, 9.89, 1074.26), (46.97, 3.65, -661.93), (112.89, 18.88, 1404.68), (119.2, 15.01, 2255.55), (67.49, 8.24, 752.02), (90.03, 13.55, 785.68), (117.82, 12.21, 2174.11), (125.58, 14.52, 2872.68), (77.55, 10.96, 594.25), (71.75, 16.06, 256.01), (78.47, 11.86, -32.57), (102.41, 22.76, 1373.35), (62.59, 17.25, 205.12), (91.0, 12.28, 1401.98), (78.23, 15.01, -82.65), (69.34, 10.99, 608.42), (83.9, 17.95, 933.69), (94.33, 17.37, 1092.3), (93.97, 14.54, 1154.77), (76.25, 11.06, 567.46), (95.95, 22.82, 1375.18), (79.21, 16.15, 78.47), (107.14, 10.59, 1304.84), (78.04, 12.75, 468.31), (105.12, 18.33, 1587.37), (54.28, 13.56, -724.69), (118.51, 8.15, 2717.58), (94.86, 14.59, 1581.77), (65.89, 8.67, 287.07), (91.53, 9.56, 1037.03), (61.78, 18.77, -901.86), (122.22, 12.68, 2108.89), (84.66, 6.78, 952.57), (119.61, 22.0, 2148.16), (107.69, 20.33, 1439.66), (29.46, 14.35, -1656.2), (131.27, 20.65, 2799.09), (70.53, 16.29, 117.98), (88.87, 16.01, 466.67), (96.66, 13.0, 1653.49), (68.83, 18.88, 386.66), (97.84, 23.12, 1013.16), (44.07, 2.59, -654.64), (79.38, 21.39, 311.0), (97.5, 6.86, 1327.53), (71.95, 9.02, 463.15), (86.72, 19.45, 516.1), (72.29, 15.21, 371.4), (84.92, 20.05, 835.89), (96.55, 16.9, 1411.31), (117.62, 13.73, 2259.53), (101.38, 17.67, 1085.73), (125.79, 11.47, 2393.86), (127.32, 24.59, 2285.39), (100.07, 19.73, 1645.18), (115.59, 17.28, 2253.45), (117.32, 16.42, 1787.6), (68.87, 9.84, 337.63), (120.96, 20.73, 1972.75), (15.57, 14.41, -1319.04), (101.69, 11.57, 1394.14), (65.3, 12.24, 714.96), (64.9, 17.55, -64.07), (127.73, 18.34, 2782.68), (93.17, 11.05, 519.83), (56.32, 18.36, -370.57), (93.82, 9.59, 993.51), (96.23, 15.5, 1021.2), (72.91, 25.05, -402.28), (83.92, 17.67, 865.71), (72.26, 11.22, -53.85), (67.45, 15.29, 510.86), (72.98, 15.69, -118.35), (59.24, 19.78, -419.26), (68.52, 11.66, 28.45), (50.15, 14.27, -370.18), (41.63, 8.47, -511.82), (58.66, 10.89, 168.05), (74.91, 17.42, 747.75), (91.6, 22.64, 826.06), (75.87, 18.87, 467.37), (92.01, 6.7, 2086.51), (71.72, 21.39, 251.46), (106.21, 18.33, 1212.18), (101.31, 18.37, 1589.28), (89.53, 1.87, 1166.63), (93.76, 13.14, 1114.54), (79.25, 6.73, 545.56), (98.36, 13.87, 1546.33), (137.11, 15.15, 3056.74), (87.85, 20.24, 1076.44), (78.18, 13.87, 22.03), (104.71, 16.12, 2193.19), (103.39, 11.09, 1518.9), (108.62, 13.25, 2111.9), (88.09, 16.86, 418.49), (125.19, 16.84, 2069.23), (86.95, 14.12, 1378.36), (88.37, 16.15, 840.85), (86.16, 22.08, 636.68), (69.31, 11.49, 318.4), (108.96, 18.85, 1777.45), (80.65, 15.81, 1024.91), (85.1, 10.28, 1123.36), (88.91, 14.59, 763.97), (76.69, 19.42, 584.24), (87.46, 10.54, 679.61), (89.11, 23.18, 786.57), (45.7, 15.62, -996.41), (114.15, 12.41, 2195.14), (40.65, 13.93, -1145.0), (87.56, 35.17, 595.48), (101.75, 17.88, 1555.68), (81.18, 18.76, 940.46), (91.41, 15.99, 820.48), (100.9, 20.93, 1309.62), (93.1, 8.66, 852.46), (95.95, 15.88, 1019.8), (49.23, 12.19, -412.58), (79.63, 14.39, 555.12), (86.97, 3.19, 1158.81), (108.35, 12.79, 2205.99), (96.98, 16.71, 1359.17), (68.74, 13.98, 78.13), (100.7, 15.52, 1660.72), (107.27, 21.12, 1512.31), (67.21, 6.79, -206.74), (146.4, 14.83, 3438.4), (121.9, 15.38, 2116.62), (121.1, 15.4, 2680.69), (66.67, 8.16, 341.63), (124.15, 11.53, 2102.73), (87.65, 14.49, 620.23), (78.76, 22.22, 23.62), (133.63, 16.58, 3101.99), (59.56, 11.56, -308.17), (142.09, 14.58, 2880.9), (110.36, 17.05, 1485.27), (108.89, 24.86, 1677.01), (101.2, 19.78, 1678.69), (91.38, 16.29, 846.25), (70.61, 18.12, -104.87), (102.77, 13.25, 1168.62), (51.05, 10.33, -345.78), (88.8, 18.62, 1291.61), (111.55, 23.06, 1722.33), (89.64, 18.28, 844.01), (61.71, 15.43, 86.22), (84.03, 12.5, 824.73), (121.06, 14.63, 2122.76), (81.97, 11.87, 396.19), (100.58, 14.71, 1700.73), (76.09, 20.13, 424.91), (64.98, 20.46, -650.66), (125.25, 11.21, 2573.17), (63.62, 16.35, -156.45), (101.42, 4.97, 1157.06), (68.65, 19.65, 104.02), (99.03, 22.67, 835.57), (128.47, 15.37, 2943.43), (121.08, 20.97, 2049.71), (100.11, 11.54, 1799.22), (127.63, 16.19, 2790.44), (45.63, 11.74, -596.56), (85.94, 12.58, 1008.4), (69.9, 20.38, -173.32), (59.87, 17.92, -420.28), (101.68, 15.32, 974.23), (97.82, 10.28, 1585.26), (50.04, 4.55, -515.8), (117.87, 19.71, 1698.8), (114.49, 16.36, 2156.63), (60.76, 13.11, -24.56), (45.85, 9.24, -950.67), (121.76, 21.2, 2442.7), (92.94, 16.84, 389.76), (95.84, 15.5, 1229.0), (87.88, 19.34, 502.03), (119.89, 13.04, 2660.26), (97.45, 11.77, 1150.36), (65.38, 13.98, 160.44), (104.16, 18.55, 1455.18), (78.05, 8.58, 357.11), (117.05, 10.86, 2202.42), (97.2, 19.53, 1110.46), (95.67, 17.52, 1086.67), (83.57, 16.71, 248.26), (34.0, 21.01, -1574.21), (106.88, 17.29, 1759.85), (93.95, 17.24, 1012.33), (141.49, 17.04, 3407.44), (58.0, 20.16, -363.9), (105.4, 10.24, 1614.06), (101.88, 16.69, 1601.93), (24.2, 13.42, -1706.25), (84.8, 15.66, 711.75), (85.92, 5.93, 1334.49), (90.64, 13.12, 1084.29), (52.94, 11.4, -477.17), (104.24, 17.71, 1752.58), (77.71, 8.63, 407.63), (58.67, 17.49, 61.26), (115.48, 25.24, 1470.46), (102.45, 8.67, 1601.88), (90.67, 9.36, 838.28), (100.84, 22.74, 1211.06), (68.53, 17.73, -224.95), (94.11, 23.7, 581.52), (67.55, 22.62, -221.09), (124.37, 24.14, 2583.71), (110.91, 11.75, 1659.0), (77.88, 14.77, 940.53), (88.38, 13.07, 833.21), (61.65, 17.11, -624.99), (118.72, 20.22, 2164.98), (105.08, 10.72, 1684.11), (93.8, 12.28, 1724.81), (31.49, 12.16, -960.35), (106.24, 13.75, 1696.92), (95.81, 6.12, 1728.55), (104.27, 7.82, 1602.31), (81.62, 17.69, 606.03), (99.8, 13.36, 1674.17), (74.13, 5.13, 623.1), (67.59, 19.45, 80.88), (73.71, 19.35, 801.79), (91.86, 9.93, 1301.15), (108.66, 15.59, 2144.76), (103.76, 26.06, 1657.66), (69.3, 19.91, 551.84), (103.0, 18.68, 1162.83), (118.65, 32.52, 1442.77), (105.68, 19.06, 1801.39), (86.4, 18.21, 908.51), (93.45, 18.16, 837.42), (48.63, 10.89, -491.67), (70.22, 11.53, 99.27), (78.82, 13.75, 389.03), (48.33, 10.19, -520.12), (104.04, 9.41, 1569.22), (86.9, 16.99, 255.72), (78.06, 13.01, 678.8), (61.61, 13.8, -269.29), (88.6, 17.79, 829.87), (62.88, 13.39, -17.26), (92.31, 18.32, 1159.47), (51.02, 14.41, -998.53), (57.48, 12.2, -287.18), (85.68, 26.06, 802.35), (36.59, 11.45, -950.78), (72.74, 12.95, 900.27), (67.54, 5.13, 562.73), (119.64, 21.64, 2338.77), (131.58, 9.19, 2772.67), (83.35, 16.1, 988.85), (122.17, 15.96, 2055.18), (104.99, 12.52, 1794.24), (79.91, 22.91, -49.31), (118.95, 16.73, 1861.68), (109.63, 11.69, 1039.31), (52.43, 9.54, -329.23), (81.98, 15.4, 162.93), (80.33, 11.15, 321.15), (66.85, 17.93, 90.82), (78.94, 19.18, 566.49), (130.75, 12.17, 2410.55), (116.12, 15.46, 2314.79), (45.06, 18.71, -1056.08), (120.63, 19.81, 1876.74), (91.78, 16.84, 678.13), (81.15, 17.0, 950.52), (83.8, 13.85, 1196.38), (87.59, 14.89, 624.06), (38.0, 16.06, -1686.53), (97.31, 13.58, 1907.27), (115.85, 11.69, 2025.96), (108.01, 17.79, 1575.98), (90.91, 15.58, 1384.47), (88.98, 20.27, 1020.01), (78.06, 8.66, 818.51), (40.41, 17.46, -1259.0), (76.68, 23.46, 440.98), (112.62, 13.09, 1629.98), (56.55, 7.34, 154.38), (128.47, 16.4, 2929.14), (77.0, 4.75, 379.01), (117.95, 16.7, 2202.33), (152.67, 27.98, 3380.21), (131.09, 12.73, 3391.12), (102.98, 20.73, 1401.01), (112.0, 24.71, 1598.81), (66.87, 16.16, 172.49), (52.97, 11.46, -191.19), (38.84, 8.74, -605.69), (94.24, 12.02, 850.56), (149.55, 17.35, 2722.14), (68.17, 22.78, -333.98), (114.03, 9.08, 2360.12), (113.73, 20.75, 1650.62), (85.12, 6.43, 966.9), (64.14, 8.69, -192.51), (49.26, 16.02, -468.39), (79.11, 15.29, 364.29), (96.26, 16.12, 1269.38), (78.66, 16.05, 567.3), (128.04, 23.89, 2092.55), (108.01, 12.58, 1480.23), (95.99, 20.64, 1256.26), (101.41, 20.89, 1296.66), (109.92, 8.97, 2314.96), (110.99, 22.25, 2109.63), (61.96, 14.38, -475.14), (86.49, 11.59, 739.49), (61.28, 11.62, -880.88), (77.51, 16.27, 534.68), (82.46, 16.42, 686.47), (117.54, 29.62, 1246.27), (119.6, 18.8, 1931.51), (77.67, 20.29, 468.01), (75.0, 10.35, 244.87), (72.7, 30.05, -559.61), (124.08, 9.79, 2837.0), (94.91, 20.61, 773.1), (90.71, 7.97, 1051.27), (96.73, 20.32, 1148.22), (66.91, 16.0, 310.04), (52.9, 1.13, 329.82), (59.6, 14.41, -41.38), (72.91, 19.81, 154.96), (119.48, 15.66, 2015.23), (90.92, 14.56, 963.45), (89.32, 15.85, 795.49), (68.08, 11.21, -89.84), (77.45, 25.31, 340.79)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=988.4912，β₁=40.2272，β₂=-24.1707；
SST=716230652.73，SSR=657275254.46，SSE=58955398.27；R²=0.9177，调整R²=0.9174；
整体 F=(SSR/2)/(SSE/681)=3796.1278，p=<1e-300；x1,x2 相关 r=0.1786；
β₁: SE=0.4624，t=87.0036，p=<1e-300；β₂: SE=2.2267，t=-10.8548，p=-9.94e+01。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=988.4912，β₁=40.2272，β₂=-24.1707；
SST=716230652.73，SSR=657275254.46，SSE=58955398.27；R²=0.9177，调整R²=0.9174；
整体 F=(SSR/2)/(SSE/681)=3796.1278，p=<1e-300；x1,x2 相关 r=0.1786；
β₁: SE=0.4624，t=87.0036，p=<1e-300；β₂: SE=2.2267，t=-10.8548，p=-9.94e+01。
③ 计算结果：
y=988.4912+40.2272x1+-24.1707x2；R²=0.9177；β₁ p=<1e-300；β₂ p=-9.94e+01
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=<1e-300）；变量 x2 对 y 的影响显著（p=-9.94e+01）。面积主、房龄负。
案例 79. 工龄+学历预测月薪
多元线性回归两正向因素
① 原始数据：
样本 n=922，逐条 (x1, x2, y)：[(14.91, 3.18, 6571.66), (10.64, 3.01, 5860.04), (10.33, 4.0, 6746.55), (13.26, 4.53, 7893.46), (6.36, 2.79, 621.14), (11.27, 1.09, 4204.63), (12.4, 4.47, 5775.62), (2.12, 2.71, -1916.73), (7.57, 3.07, 82.52), (6.05, 4.01, 3758.0), (4.1, 0.66, -2573.51), (3.41, 2.02, -2931.94), (9.69, 0.79, 481.42), (0.78, 2.19, -2867.9), (15.98, 3.51, 10151.8), (11.55, 5.0, 9733.55), (12.62, 4.02, 5779.98), (9.4, 2.79, 4450.56), (10.22, 3.96, 8606.28), (7.74, 4.37, 8300.48), (14.01, 3.99, 7087.84), (6.15, 2.41, -1538.01), (6.62, 3.46, 3134.08), (13.22, 2.01, 5906.5), (4.87, 4.01, 147.01), (5.47, 3.0, 1159.27), (9.46, 2.84, 2170.51), (5.59, 2.16, 1122.93), (11.08, 4.26, 5980.43), (9.16, 3.57, 4231.61), (12.03, 2.92, 6018.07), (5.38, 3.13, 990.1), (7.45, 2.03, -1295.09), (10.48, 4.78, 4982.69), (9.78, 2.43, 4401.41), (12.15, 3.2, 5183.37), (7.44, 2.03, 947.81), (13.64, 2.59, 7406.25), (5.52, 1.6, -21.12), (5.55, 3.07, -437.74), (8.09, 1.97, 3752.95), (7.85, 3.12, 4084.75), (9.44, 5.01, 5637.58), (18.88, 3.42, 8041.28), (3.77, 2.98, 323.17), (15.48, 4.94, 9350.73), (14.15, 3.84, 9570.29), (14.54, 2.89, 7092.08), (5.53, 2.97, 3384.46), (5.82, 2.37, -901.8), (5.39, 4.03, 2604.35), (3.56, 2.52, -207.46), (12.8, 4.47, 7397.03), (5.98, 4.73, 4470.34), (6.72, 2.2, -484.11), (7.42, 3.04, 2765.85), (5.01, 2.1, -171.14), (2.36, 2.18, -4145.8), (6.59, 1.63, -2705.53), (12.09, 2.75, 3795.74), (10.37, 3.37, 3629.27), (11.42, 3.01, 1735.81), (15.81, 2.98, 8153.62), (9.72, 2.83, 4197.59), (15.26, 2.14, 6356.06), (10.49, 2.1, 3931.58), (13.43, 3.11, 5921.62), (5.72, 1.42, -69.25), (11.56, 2.33, 5453.01), (10.65, 2.3, 4570.99), (10.65, 1.74, 3762.8), (10.47, 1.94, 1704.37), (9.45, 1.68, 2131.35), (11.05, 3.73, 7063.63), (4.91, 3.26, 2791.48), (11.54, 3.76, 6592.43), (9.61, 2.78, 1551.67), (7.69, 4.08, 1939.88), (11.92, 2.63, 2531.06), (1.59, 2.34, -2200.88), (8.35, 2.02, 1768.24), (7.62, 3.52, 4330.55), (15.74, 4.83, 8585.83), (7.39, 2.51, 1344.18), (9.68, 4.23, 8766.83), (6.2, 2.19, -401.72), (6.75, 4.1, 6598.5), (5.89, 3.75, 3135.69), (13.61, 3.02, 8244.25), (5.86, 2.59, 1616.51), (12.76, 4.13, 6222.62), (5.46, 3.21, 2058.06), (12.18, 2.33, 2924.32), (3.98, 2.77, -1555.3), (11.1, 3.52, 2358.11), (12.06, 1.61, 2775.69), (7.35, 3.44, 1018.88), (9.65, 1.21, 2939.2), (5.31, 0.53, -1329.35), (13.31, 3.12, 7611.04), (4.25, 2.69, 1780.29), (11.34, 3.53, 7357.67), (11.31, 4.06, 6285.22), (8.18, 0.69, -2165.69), (6.99, 2.51, 1747.67), (7.08, 4.21, 7267.59), (4.11, 2.48, 1081.99), (10.14, 2.81, 2785.35), (13.74, 2.79, 5832.59), (6.12, 3.35, 1985.42), (6.64, 2.81, 2393.79), (10.14, 4.2, 7126.0), (5.59, 3.7, 2080.04), (4.7, 3.09, 828.82), (2.67, 4.29, 1592.25), (4.39, 2.37, 2436.49), (11.07, 2.53, 4257.16), (14.01, 1.96, 5647.49), (3.01, 2.49, -1287.63), (-0.18, 1.89, -2986.6), (8.15, 2.13, -1324.82), (0.94, 0.55, -6105.26), (-1.71, 2.66, -4617.5), (4.56, 2.08, 365.28), (7.74, 3.51, 2873.36), (9.34, 2.45, 3453.1), (15.52, 3.46, 6878.2), (6.32, 2.89, 1587.1), (12.19, 4.02, 6049.54), (-2.1, 2.82, -447.75), (6.92, 0.65, 985.21), (8.9, 2.71, 5525.98), (13.29, 4.05, 10724.0), (13.79, 4.7, 8581.83), (1.86, 4.27, 4078.43), (11.48, 3.21, 4298.48), (13.0, 4.53, 9973.35), (10.33, 2.67, 3015.32), (12.73, 3.69, 8584.14), (10.67, 4.68, 6892.73), (5.53, 4.13, 5478.18), (9.3, 2.36, 1303.74), (9.7, 3.56, 7139.88), (10.09, 3.15, 249.66), (8.9, 2.04, 2472.55), (12.19, 5.62, 9727.96), (3.33, 2.14, -2646.18), (7.79, 3.71, 5195.15), (11.55, 4.22, 7073.93), (12.23, 3.46, 2973.29), (5.15, 3.91, 4120.57), (4.88, 3.91, 3517.07), (8.31, 3.23, 2352.98), (0.38, 1.91, -2824.64), (7.83, 2.07, 846.39), (6.14, 1.33, 916.04), (13.76, 4.09, 9214.3), (8.59, 3.43, 3520.74), (8.96, 3.0, 2086.71), (9.22, 4.11, 3880.6), (12.07, 5.82, 9676.06), (9.94, 3.59, 5497.56), (9.24, 3.61, 4829.51), (7.59, 3.43, 2140.41), (14.29, 3.39, 8104.74), (7.43, 2.0, 1220.97), (5.64, 3.25, 294.25), (6.45, 1.93, 3057.7), (11.88, 5.24, 7941.47), (2.25, 4.18, -1569.35), (7.09, 1.71, 713.94), (10.06, 2.78, 3568.33), (7.82, 3.81, 4895.09), (13.97, 2.99, 5843.3), (7.37, 1.27, 1126.71), (5.58, 3.21, 1277.85), (9.66, 2.8, 1456.77), (3.41, 2.64, -454.71), (2.66, 4.0, 1728.44), (5.95, 2.53, -861.65), (12.72, 3.57, 7947.62), (2.01, 1.73, -2781.29), (7.41, 3.19, 4647.16), (4.57, 2.39, 1425.57), (9.59, 3.14, 3262.01), (3.95, 1.12, -3186.96), (3.12, 2.48, 229.46), (9.68, 2.71, 1455.12), (8.75, 2.88, 2335.82), (10.48, 2.2, 2699.11), (9.85, 1.71, 1930.05), (7.14, 0.31, -16.72), (0.42, 1.96, -3074.76), (4.64, 3.75, 577.82), (10.62, 3.07, 5190.13), (6.22, 1.3, -1609.64), (4.04, 3.7, 1315.3), (4.48, 2.67, -1249.69), (7.83, 2.79, 1781.76), (8.22, 2.3, 1757.23), (1.71, 1.69, -1226.88), (6.88, 2.11, -3764.04), (6.81, 0.95, 2299.35), (5.73, 2.38, 1783.12), (4.43, 4.07, 2910.19), (7.82, 2.77, 4498.63), (11.38, 3.09, 5413.01), (6.07, 2.22, -141.11), (5.76, 2.62, 3205.97), (13.07, 3.22, 7022.13), (9.26, 3.31, 4993.73), (6.09, 2.27, 1423.08), (8.29, 3.81, 1677.73), (6.92, 0.46, -1876.27), (3.32, 2.54, 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3408.5), (7.88, 2.86, 960.51), (11.43, 3.56, 4729.96), (12.29, 2.48, 1689.76), (13.19, 4.22, 8474.2), (9.19, 3.69, 4045.94), (5.76, 2.08, 2721.15), (18.88, 4.52, 13094.33), (10.42, 1.83, 1887.22), (10.3, 4.53, 6965.49), (4.67, 1.74, -2816.33), (4.58, 2.94, 2504.63), (6.36, 3.02, 1013.21), (17.12, 3.46, 9856.54), (7.24, 2.49, 3515.19), (13.73, 3.77, 8816.39), (12.42, 1.97, 3532.32), (15.14, 3.15, 4554.83), (6.37, 3.11, 3798.49), (9.76, 2.68, 3553.63), (3.86, 1.43, -2393.71), (3.96, 2.1, -659.15), (8.01, 1.7, -3015.67), (13.53, 4.4, 8610.26), (9.73, 3.57, 5019.09), (6.41, 3.18, 3442.96), (14.27, 3.41, 6560.53), (7.38, 3.21, 2471.68), (7.18, 3.74, 2922.56), (6.45, 2.65, 3840.39), (7.22, 3.78, 2780.66), (5.17, 2.6, 2156.01), (13.79, 4.37, 8858.47), (11.08, 5.75, 10966.96), (7.84, 4.82, 5552.62), (8.16, 3.27, 2151.58), (5.95, 2.67, -953.78), (12.7, 3.82, 7484.68), (7.45, 2.26, 1288.71), (8.06, 4.18, 4950.81), (13.03, 2.94, 5460.05), (10.56, 3.57, 4963.38), (7.14, 3.07, 5930.06), (6.05, 3.88, 674.46), (10.82, 3.68, 6495.25), (-0.86, 2.59, -2496.07), (12.37, 4.54, 6619.43), (14.49, 3.72, 9013.31), (10.12, 3.09, 3106.39), (16.02, 4.29, 11434.34), (6.75, 3.65, 4293.26), (9.42, 2.59, 3369.89), (8.7, 2.78, 3823.59), (7.9, 2.56, 1557.89), (8.09, 2.25, 2075.02), (12.94, 4.19, 7544.92), (8.08, 4.42, 5080.1), (7.02, 4.28, 5252.46), (10.08, 3.3, 3622.29), (9.84, 3.57, 6328.34), (1.09, 0.64, -2834.86), (5.9, 3.13, 3303.57), (9.84, 3.33, 3444.9), (7.03, 4.12, 5467.94), (12.56, 2.95, 3488.07), (3.72, 2.1, -4125.35), (10.3, 3.33, 5754.75), (7.88, 2.48, 2105.44), (7.4, 2.19, 618.92), (4.74, 2.36, 1058.65), (-3.92, 0.8, -7213.97), (8.36, 3.28, 4583.57), (5.31, 2.81, 1658.78), (11.03, 4.57, 5811.93), (5.75, 3.08, 2526.78), (6.97, 2.29, -1086.14), (10.34, 2.5, 5611.57), (8.35, 4.16, 5886.73), (8.34, 2.44, 616.3), (15.87, 3.46, 8097.41), (13.72, 2.12, 8576.05), (4.76, 3.36, 1469.61), (5.6, 2.79, 460.3), (8.89, 1.98, 277.41), (3.81, 2.1, -1641.38), (16.52, 4.7, 10207.26), (5.19, 0.61, -2847.9), (7.37, 4.2, 5226.69), (3.93, 3.33, -2461.73), (12.63, 3.71, 8186.04), (7.12, 2.68, 657.77), (9.59, 1.04, 426.76), (7.86, 1.81, 3371.72), (8.61, 3.99, 5469.52), (7.27, 3.15, 2525.3), (8.31, 2.79, 4921.42), (4.01, 3.71, 2807.99), (12.38, 3.45, 6597.57), (8.83, 3.81, 4914.47), (15.81, 2.33, 8990.99), (6.54, 0.09, -2039.03), (11.83, 2.43, 6856.08), (7.57, 2.4, 2257.55), (4.8, 3.47, 1562.87), (8.05, 3.02, 3249.49), (5.42, 3.39, 4351.39), (11.49, 3.86, 6888.7), (4.18, 2.22, 1240.91), (3.36, 2.23, -1967.88), (11.04, 2.19, 4700.21), (3.46, 2.0, -1727.33), (8.36, 3.31, 4215.45), (9.79, 4.25, 9165.21), (2.56, 2.45, -1892.26), (14.28, 2.98, 8312.05), (8.23, 2.8, 2479.05), (11.55, 2.94, 4796.55), (9.21, 4.45, 6051.9), (11.88, 2.83, 4559.85), (14.8, 4.3, 9207.28), (8.88, 1.12, -2064.25), (6.85, 3.34, 2713.42), (10.47, 3.75, 6541.83), (8.93, 3.35, 3056.16), (3.21, 2.91, -679.72), (10.15, 2.81, 3364.21), (11.74, 2.52, 1913.8), (9.26, 3.72, 2037.44), (14.51, 2.65, 8701.97), (7.19, 4.36, 3385.18), (-0.29, 1.98, -4519.56), (7.01, 1.18, 3152.45), (5.75, 2.05, -792.99), (4.7, 2.19, 2574.18), (11.76, 2.58, 5693.56), (7.0, 2.79, 1374.24), (7.62, 3.31, 3703.18), (7.93, 1.67, -1099.05), (8.04, 3.58, 3333.35), (17.77, 5.18, 14751.99), (4.36, 4.38, 1986.71), (16.79, 4.03, 10273.27), (7.49, 3.36, 2621.76), (13.43, 2.17, 3700.07), (9.99, 1.46, 1787.81), (-1.04, 0.71, -4608.52), (8.21, 3.16, 2574.7), (11.04, 4.58, 7615.14), (9.86, 3.5, 5042.41), (4.33, 0.82, -3184.82), (5.66, 3.55, 2568.98), (8.89, 2.57, 3089.92), (1.54, 2.95, -592.88), (10.19, 4.47, 6725.29), (9.34, 2.74, 3245.66), (9.39, 2.11, 4343.82), (6.45, 3.04, 4874.9), (6.64, 2.84, 4192.47), (6.41, 4.32, 6092.7), (6.53, 3.95, 3480.3), (5.99, 2.93, 901.43), (11.03, 4.05, 10452.79), (2.35, 0.94, -5365.86), (12.26, 4.46, 7036.93), (10.55, 5.04, 8157.84), (15.69, 4.67, 11249.92), (15.05, 3.79, 9133.81), (6.46, 2.39, 495.94), (6.94, 3.29, 3078.37), (10.95, 2.12, 2765.61), (10.15, 5.62, 9021.44), (5.03, 2.72, 868.79), (9.16, 3.89, 5580.84), (7.92, 2.63, 72.61), (7.1, 2.41, -918.78), (9.1, 3.31, 4467.0), (8.28, 3.08, 3874.05), (7.26, 2.63, 50.62), (8.95, 2.2, 317.9), (3.93, 2.57, 658.91), (1.36, 2.37, -1369.41), (10.2, 2.82, 5156.94), (7.9, 3.4, 3117.52), (9.64, 4.01, 6727.41), (3.04, 1.17, -3294.1), (13.69, 2.7, 3978.05), (12.47, 3.43, 5802.96), (9.19, 3.95, 7246.1), (9.46, 3.64, 5866.81), (0.73, 1.82, -1286.67), (13.51, 2.81, 7190.82), (6.03, 3.77, 1211.76), (13.07, 3.59, 9058.77), (10.18, 4.45, 8147.87), (2.06, 2.91, -1646.74), (7.31, 2.85, 2700.92), (6.18, 2.85, 1855.79), (13.48, 5.12, 7710.81), (9.55, 3.19, 4947.46), (5.08, 0.75, -5574.77), (8.31, 4.22, 5154.3), (4.71, 2.98, -72.72), (8.15, 1.76, -233.73), (5.3, 2.33, 1573.26), (10.53, 4.96, 8384.03), (16.9, 2.38, 4428.5), (5.22, 2.48, 212.28), (8.51, 3.79, 3503.81), (6.27, 3.3, 2277.49), (16.13, 4.8, 9173.12), (3.72, 1.66, -267.81), (6.62, 3.23, 4855.14), (11.94, 3.8, 7181.49), (16.13, 4.67, 8809.49), (13.01, 3.84, 6326.53), (15.09, 2.7, 7241.91), (12.91, 3.82, 6285.52), (3.19, 2.87, 766.08), (9.28, 2.93, 3682.39), (11.22, 3.88, 6430.4), (7.67, 3.17, 3712.07), (10.88, 3.36, 5455.12), (14.18, 2.46, 2780.62), (14.67, 2.81, 4452.55), (7.7, 3.64, 2801.47), (5.56, 1.13, 20.6), (3.14, 3.21, 2883.09), (13.64, 3.53, 6210.99), (8.08, 0.72, 1466.58), (2.89, 4.48, -483.58), (6.52, 4.14, 6735.59), (10.41, 3.68, 4787.46), (8.86, 3.02, 3953.55), (2.55, 1.95, -2196.23), (13.08, 3.49, 7631.54), (3.94, 3.89, 3515.1), (5.33, 4.38, 1025.69), (13.23, 2.35, 5948.33), (4.67, 2.84, -957.74), (4.92, 0.08, -2295.23), (2.69, 2.74, 656.84), (9.92, 3.33, 3443.94), (11.31, 2.28, 1602.83), (-0.13, 2.83, -2339.67), (6.7, 3.4, 3281.24), (4.66, 2.47, 93.51), (6.48, 1.88, 3552.74), (20.57, 3.48, 13684.23), (14.31, 4.07, 7283.41), (3.87, 3.19, 834.84), (5.37, 1.93, 76.09), (6.59, 2.02, 2257.87), (13.25, 4.8, 8964.64), (6.16, 3.09, 2639.5), (11.53, 3.31, 7059.47), (1.39, 2.19, -5809.65), (5.7, 3.42, 288.33), (15.4, 4.07, 9448.01), (6.17, 2.53, -199.77), (8.25, 1.82, 1531.02), (12.1, 4.49, 8300.87), (12.96, 4.3, 10394.1), (3.11, 1.49, -675.48), (0.85, 0.68, -6133.01), (14.99, 1.77, 6449.3), (3.66, 3.86, 2308.14), (13.04, 4.69, 8123.95), (3.33, 3.93, 203.06), (18.43, 3.91, 9997.4), (9.84, 2.41, 5878.89), (4.47, 3.8, 4927.58), (6.16, 3.31, 2819.03), (10.51, 3.17, 1738.0), (0.94, 1.89, -2110.29), (3.79, 2.41, 799.35), (3.65, 1.74, -1700.81), (9.15, 2.26, 3387.95), (12.43, 3.65, 6478.33), (16.2, 3.44, 8096.39), (5.99, 2.4, -419.42), (-1.1, 1.89, -4261.88), (8.27, 3.85, 2498.44), (5.74, 3.8, 1645.27), (5.55, 4.08, 4082.17), (8.26, 1.58, 3982.56), (9.8, 3.12, 971.72), (6.72, 3.05, 6576.81), (8.3, 3.35, 2962.25), (6.01, 2.28, 1399.21), (5.19, 1.63, -662.87), (11.41, 3.08, 7887.67), (14.14, 2.62, 5593.59), (10.9, 4.59, 7288.36), (7.41, 3.61, 2240.86), (2.95, 2.96, -534.75), (17.75, 5.14, 11556.09), (7.02, 2.11, 2631.22), (12.51, 3.59, 7325.03), (8.96, 3.48, 4085.74), (6.15, 1.8, -572.37), (7.07, 2.21, 181.4), (3.73, 2.48, -976.28), (9.58, 4.61, 6755.72), (13.21, 2.69, 5677.28), (10.21, 1.67, -1523.26), (10.34, 4.77, 8338.41), (0.98, 1.79, -2864.27), (2.79, 1.96, -1948.99), (9.38, 2.8, 5846.23), (13.73, 2.54, 5123.88), (15.65, 4.22, 7876.9), (6.51, 2.55, 2597.69), (16.94, 1.28, 7353.37), (0.78, 3.58, 491.17), (10.27, 3.7, 5432.61), (8.88, 1.59, 2347.61), (11.54, 3.61, 3861.09), (7.91, 3.06, 4171.66), (10.05, 2.39, 2226.63), (3.66, 3.24, 2481.49), (6.57, 2.84, 2086.25), (9.22, 3.23, 4996.24), (10.07, 3.32, 3132.6), (8.71, 3.56, 4961.67), (14.53, 3.16, 9568.48), (7.65, 3.68, 3199.78), (6.59, 3.29, 600.19), (11.1, 4.49, 4377.08), (5.62, 3.47, -910.48), (4.89, 2.9, 3431.73), (10.46, 3.97, 6193.1), (4.02, 4.07, 406.9), (14.73, 4.65, 8136.53), (7.99, 3.74, 3746.49), (9.57, 2.27, 5588.02), (6.86, 3.79, 4444.6), (4.17, 1.73, -1113.95), (7.41, 3.47, 2936.45), (5.05, 2.5, 2059.07), (1.89, 1.58, -2477.39), (1.12, 2.66, -3339.83), (13.26, 1.76, 5376.42), (10.81, 3.19, 5107.59), (7.61, 2.83, 2871.3), (10.01, 3.73, 6863.3), (1.72, 2.26, -962.87), (2.59, 2.47, 304.0), (7.05, 2.02, 3202.35), (6.08, 2.92, 3065.94), (11.4, 2.94, 6990.42), (6.54, 2.88, 1586.23), (10.53, 3.48, 5829.24), (7.23, 3.5, 4618.69), (4.49, 2.29, 1122.6), (3.3, 3.44, 66.83), (14.31, 3.64, 6237.09), (3.78, 1.77, 452.44), (0.38, 3.44, -1085.71), (6.82, 4.36, 3459.44), (11.57, 3.96, 7969.35), (6.26, 2.51, 1236.09), (4.37, 3.62, 222.61), (4.87, 2.56, 690.81), (10.08, 2.12, 2851.76), (14.12, 2.07, 5334.45), (4.73, 1.6, -2232.06), (4.47, 2.56, -593.67), (4.31, 2.26, 1527.66), (11.28, 3.13, 2923.48), (1.97, 1.49, -4260.97), (17.08, 5.25, 11371.11), (8.64, 4.51, 4187.46), (5.51, 2.2, 1796.14), (10.65, 2.4, 4923.11), (5.97, 1.26, 790.73), (7.55, 2.29, 3591.67), (9.35, 3.87, 5800.04), (6.48, 2.91, 2319.83), (3.54, 1.89, -614.38), (9.09, 2.5, 812.85), (10.23, 3.47, 6469.06), (9.62, 2.81, 4952.25), (7.41, 2.14, -326.17), (10.37, 3.8, 7434.3), (9.49, 3.25, 5132.34), (5.32, 1.12, -3202.17), (1.67, 2.46, -2363.22), (11.26, 2.73, 6823.03), (-0.81, 3.1, -1971.55), (10.46, 3.89, 5858.96), (14.44, 4.09, 12009.63), (-1.3, 1.54, -5085.9), (6.27, 2.48, 1989.51), (12.7, 3.94, 10385.07), (9.41, 2.67, 3922.0), (3.86, 1.3, -2002.97), (11.86, 2.64, 4032.67), (13.04, 3.09, 8372.96), (6.17, 3.3, 1021.3), (6.05, 2.7, 1640.25), (5.51, 2.28, -570.66), (-1.67, 2.98, -2624.33), (10.41, 2.71, 4079.9), (8.77, 2.39, 2340.57), (4.08, 2.43, 561.76), (12.78, 3.72, 6893.17), (3.0, 4.04, 3236.49), (17.9, 2.94, 8547.0), (13.22, 3.56, 8026.57), (9.77, 3.52, 4484.37), (13.61, 4.07, 10241.79), (6.53, 1.37, 2164.33), (9.59, 2.16, 1675.3), (3.57, 2.83, -338.14), (7.68, 4.94, 2749.31), (11.09, 3.94, 6114.0), (8.32, 1.49, 2978.96), (8.68, 2.81, 3711.76), (7.78, 2.27, -2395.84), (8.95, 2.37, 3731.7), (9.24, 3.89, 5963.64), (9.26, 2.9, 5415.61), (14.51, 4.27, 7486.32), (4.21, 2.73, -1384.98), (15.97, 4.49, 10945.32), (2.21, 3.11, -2421.39), (6.33, 4.02, 4769.65), (17.35, 4.6, 9879.37), (9.87, 4.4, 8947.83), (3.6, 3.69, 2018.18), (5.23, 3.47, 2517.42), (6.69, 2.69, 667.01), (3.76, 3.47, 1780.94), (12.7, 2.14, 5142.84), (1.69, 4.42, 2013.32), (8.19, 2.77, 3476.47), (7.88, 2.46, 6051.88), (6.98, 3.73, 4753.04), (2.04, 2.83, -1496.22), (9.71, 3.78, 3715.2), (5.31, 3.07, 2068.03), (8.82, 2.97, 966.75), (3.58, 1.41, -2563.16), (2.75, 3.76, 416.51), (2.08, 1.03, -5159.6), (10.81, 4.34, 6745.28), (6.48, 4.1, 3267.6), (9.81, 2.28, -524.3), (8.0, 3.13, 3908.48)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=2975.6458，β₁=600.5478，β₂=1541.2667；
SST=12004165884.12，SSR=9860068584.76，SSE=2144097299.36；R²=0.8214，调整R²=0.8210；
整体 F=(SSR/2)/(SSE/919)=2113.1044，p=<1e-300；x1,x2 相关 r=0.3803；
β₁: SE=13.6799，t=43.9001，p=8.17e-228；β₂: SE=55.7361，t=27.6530，p=-2.00e+55。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=2975.6458，β₁=600.5478，β₂=1541.2667；
SST=12004165884.12，SSR=9860068584.76，SSE=2144097299.36；R²=0.8214，调整R²=0.8210；
整体 F=(SSR/2)/(SSE/919)=2113.1044，p=<1e-300；x1,x2 相关 r=0.3803；
β₁: SE=13.6799，t=43.9001，p=8.17e-228；β₂: SE=55.7361，t=27.6530，p=-2.00e+55。
③ 计算结果：
y=2975.6458+600.5478x1+1541.2667x2；R²=0.8214；β₁ p=8.17e-228；β₂ p=-2.00e+55
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=8.17e-228）；变量 x2 对 y 的影响显著（p=-2.00e+55）。两正向因素。
案例 80. 学习时长+睡眠质量预测成绩
多元线性回归两正向因素
① 原始数据：
样本 n=695，逐条 (x1, x2, y)：[(13.55, 7.68, 36.85), (10.52, 6.51, 23.69), (10.38, 6.72, 17.59), (9.34, 8.16, 41.97), (11.52, 7.8, 30.24), (2.96, 6.71, 7.51), (6.88, 7.12, 18.58), (13.07, 7.3, 34.19), (5.45, 5.92, 13.24), (11.07, 7.48, 39.42), (17.31, 5.57, 59.87), (9.81, 7.14, 30.57), (5.88, 5.47, 17.73), (16.14, 7.01, 44.48), (6.32, 5.3, 25.11), (6.35, 6.02, 12.23), (9.79, 6.42, 22.85), (10.14, 8.49, 36.85), (11.27, 5.76, 26.83), (8.95, 5.93, 27.18), (10.3, 7.04, 37.95), (7.76, 7.39, 29.22), (7.61, 6.21, 17.23), (12.03, 9.34, 48.43), (10.45, 7.45, 31.09), (11.8, 7.42, 45.52), (4.46, 7.03, 0.48), (10.35, 5.59, 27.7), (9.86, 7.36, 37.27), (12.14, 6.6, 36.11), (11.4, 6.45, 33.06), (6.9, 7.29, 16.98), (10.54, 6.47, 26.94), (8.42, 6.09, 15.61), (11.47, 5.32, 35.16), (10.76, 7.32, 25.46), (12.1, 7.53, 39.79), (10.04, 8.53, 34.9), (8.65, 6.76, 24.89), (12.74, 6.99, 40.87), (11.87, 7.77, 34.79), (5.69, 5.98, 14.06), (10.44, 7.39, 44.08), (9.3, 7.02, 23.26), (10.95, 7.37, 33.93), (9.79, 6.11, 24.96), (13.47, 7.74, 38.6), (4.25, 4.46, 13.15), (8.43, 6.04, 17.08), (14.87, 7.21, 49.09), (9.25, 7.43, 37.54), (9.44, 6.28, 26.89), (13.66, 6.61, 62.41), (7.08, 7.97, 21.72), (9.1, 6.87, 25.19), (10.54, 7.01, 33.39), (8.48, 7.38, 11.72), (9.38, 7.22, 38.63), (6.75, 6.29, 21.52), (8.41, 7.73, 20.88), (10.41, 6.76, 27.38), (13.12, 6.06, 40.02), (9.19, 7.39, 23.09), (11.44, 6.46, 28.64), (5.48, 5.08, 3.19), (9.09, 7.89, 30.48), (9.66, 5.99, 29.0), (7.06, 6.26, 20.91), (11.87, 6.27, 38.24), (6.87, 7.63, 22.74), (11.72, 6.55, 28.3), (7.38, 6.3, 22.48), (14.66, 6.83, 42.98), (6.53, 6.38, 20.43), (9.7, 5.7, 24.53), (12.35, 7.2, 39.43), (10.15, 6.58, 34.25), (7.37, 8.01, 27.82), (14.12, 6.54, 41.87), (11.48, 5.6, 37.34), (10.89, 7.24, 24.97), (4.92, 7.71, 22.15), (10.37, 6.86, 22.03), (12.83, 6.03, 26.79), (6.64, 6.97, 18.46), (8.99, 6.97, 28.24), (11.52, 7.37, 32.95), (10.13, 4.72, 24.28), (10.94, 8.66, 42.34), (8.78, 7.92, 20.56), (11.03, 6.68, 36.71), (7.25, 6.53, 28.09), (11.26, 7.3, 32.75), (7.19, 6.76, 25.36), (17.08, 7.25, 50.02), (12.18, 6.68, 33.76), (8.2, 5.77, 9.68), (9.25, 8.95, 33.18), (8.29, 7.51, 19.62), (8.74, 5.92, 15.03), (11.41, 6.28, 31.0), (12.27, 6.91, 36.1), (11.24, 6.46, 39.9), (6.93, 5.95, 29.8), (9.87, 8.51, 35.12), (6.8, 7.38, 19.99), (12.18, 7.59, 39.62), (13.75, 6.78, 50.57), (15.69, 9.17, 48.06), (10.08, 5.71, 36.78), (8.9, 6.24, 24.94), (5.0, 5.76, 11.83), (13.81, 7.53, 49.28), (4.4, 5.57, 12.25), (12.34, 8.97, 36.99), (14.7, 7.23, 42.18), (7.55, 6.49, 19.59), (9.66, 7.13, 30.76), (10.11, 7.05, 34.72), (12.31, 8.99, 41.59), (7.51, 8.09, 29.23), (6.5, 6.29, 5.03), (6.44, 5.79, 16.99), (6.45, 8.63, 27.17), (10.36, 6.09, 24.63), (5.0, 7.0, 14.45), (13.39, 6.16, 41.43), (12.3, 6.78, 28.39), (8.88, 7.04, 29.14), (12.24, 7.3, 39.5), (7.81, 7.11, 23.73), (13.68, 7.66, 41.83), (6.34, 5.99, 6.13), (8.77, 5.87, 33.38), (6.6, 6.8, 30.26), (7.61, 8.16, 13.03), (10.98, 7.77, 36.39), (3.42, 5.04, -4.93), (7.5, 6.13, 24.38), (12.34, 8.34, 37.45), (12.95, 6.54, 35.92), (13.52, 6.43, 29.76), (10.64, 5.46, 24.68), (7.88, 6.34, 21.91), (13.02, 7.22, 43.08), (9.68, 7.86, 28.31), (8.86, 5.71, 26.16), (10.58, 7.94, 38.06), (6.3, 7.32, 13.75), (9.71, 7.05, 29.29), (10.74, 6.42, 36.25), (12.03, 5.95, 29.11), (11.2, 7.76, 25.98), (6.01, 5.9, 19.91), (6.79, 6.5, 17.42), (11.21, 7.52, 34.11), (12.92, 8.42, 43.23), (13.34, 6.01, 33.73), (10.82, 8.13, 29.16), (10.01, 8.07, 41.23), (10.0, 7.06, 35.56), (10.45, 7.03, 23.47), (9.52, 7.97, 31.18), (6.02, 6.23, 28.59), (6.48, 6.06, 11.72), (11.36, 6.88, 45.93), (6.23, 7.64, 18.98), (9.49, 7.59, 36.44), (11.51, 7.88, 26.52), (13.8, 7.67, 37.98), (4.19, 6.32, 19.01), (7.96, 7.03, 23.25), (4.06, 6.56, 9.88), (12.09, 6.81, 29.32), (10.39, 6.68, 35.88), (10.33, 7.1, 26.24), (8.39, 6.82, 22.61), (8.06, 7.38, 24.04), (5.57, 6.93, 16.73), (13.97, 8.9, 43.41), (13.07, 8.49, 33.06), (8.39, 5.57, 12.0), (8.27, 6.24, 18.78), (11.99, 8.19, 29.15), (9.63, 7.98, 29.36), (8.4, 7.32, 28.29), (11.76, 8.86, 34.11), (3.83, 7.79, 27.28), (13.55, 9.23, 40.85), (10.11, 7.79, 35.84), (6.99, 6.08, 15.24), (9.38, 7.71, 38.7), (12.02, 7.34, 30.68), (13.14, 7.98, 28.47), (8.64, 6.68, 26.24), (9.38, 8.3, 28.64), (7.43, 7.1, 31.23), (8.7, 6.02, 23.47), (8.96, 7.33, 33.6), (7.59, 6.07, 25.65), (10.01, 6.06, 26.58), (10.13, 7.02, 29.27), (11.24, 6.94, 31.52), (7.68, 8.28, 29.43), (4.93, 6.78, 27.23), (11.88, 7.32, 47.33), (10.21, 8.12, 40.09), (10.61, 5.37, 19.93), (11.71, 7.07, 39.81), (12.5, 7.22, 43.13), (9.85, 6.2, 25.46), (9.95, 7.57, 26.41), (7.45, 7.46, 31.4), (9.83, 7.28, 24.77), (8.22, 7.03, 26.69), (11.5, 6.18, 50.77), (4.5, 6.12, 19.06), (9.9, 7.68, 33.53), (4.73, 4.3, 2.51), (12.02, 7.85, 33.92), (13.2, 8.32, 37.29), (9.35, 6.87, 27.64), (13.86, 7.79, 34.35), (14.46, 6.85, 42.31), (1.64, 5.48, 5.07), (11.38, 8.11, 33.57), (7.53, 5.94, 19.68), (13.66, 8.6, 33.52), (9.83, 6.6, 22.14), (10.08, 7.02, 35.31), (12.31, 7.52, 34.27), (8.92, 5.77, 28.85), (11.97, 7.41, 46.14), (9.67, 7.0, 31.85), (8.7, 6.04, 30.87), (4.35, 7.0, 15.13), (10.39, 6.87, 30.87), (3.69, 5.88, 11.52), (11.46, 8.22, 40.93), (11.97, 8.06, 33.48), (5.08, 6.2, 5.09), (9.41, 6.74, 28.0), (16.57, 7.88, 61.58), (13.57, 6.92, 41.43), (14.59, 5.38, 38.85), (5.42, 7.91, 17.17), (5.25, 6.32, 27.81), (13.0, 6.94, 36.87), (13.34, 7.94, 40.83), (10.92, 6.88, 34.54), (8.81, 7.6, 18.82), (11.18, 6.88, 30.72), (12.94, 8.5, 49.27), (9.65, 6.88, 31.98), (10.55, 7.04, 32.52), (11.02, 6.24, 31.42), (14.45, 7.18, 48.55), (10.66, 7.54, 30.66), (4.51, 6.95, 15.47), (14.97, 8.74, 59.05), (7.45, 4.68, 15.58), (14.45, 6.76, 35.21), (12.51, 6.25, 45.35), (10.15, 6.57, 18.33), (10.58, 5.95, 27.7), (10.02, 6.06, 32.06), (6.11, 5.66, 18.17), (8.03, 7.34, 30.94), (7.17, 6.73, 15.62), (13.65, 7.57, 57.69), (9.79, 6.22, 27.8), (12.73, 8.12, 42.16), (8.26, 6.67, 28.7), (7.23, 5.14, 18.34), (7.04, 6.78, 24.03), (5.66, 5.47, 17.06), (4.55, 6.4, 7.67), (11.12, 7.66, 33.63), (7.41, 5.93, 21.23), (6.59, 7.64, 16.67), (10.37, 8.09, 27.14), (8.58, 5.91, 30.37), (6.14, 6.27, 16.66), (9.92, 5.77, 30.51), (10.32, 6.89, 27.5), (9.54, 6.45, 31.28), (11.34, 5.75, 32.87), (12.61, 8.26, 39.82), (8.32, 5.78, 25.06), (13.28, 8.69, 40.36), (10.18, 6.91, 23.64), (16.23, 8.29, 48.75), (7.04, 5.65, 16.43), (6.59, 5.9, 17.87), (15.24, 8.09, 45.46), (11.07, 6.83, 41.71), (12.37, 8.18, 30.86), (9.15, 6.54, 25.86), (6.12, 5.52, 15.98), (10.69, 7.49, 32.66), (8.01, 6.2, 26.76), (11.49, 6.08, 36.14), (9.63, 7.16, 25.44), (15.18, 5.73, 36.09), (9.35, 5.72, 24.31), (12.22, 6.5, 29.8), (8.07, 8.01, 22.2), (8.97, 6.0, 23.73), (12.58, 7.21, 39.78), (12.08, 6.97, 29.39), (9.8, 8.0, 32.87), (10.08, 5.02, 28.68), (10.8, 7.88, 38.22), (8.17, 7.26, 20.96), (5.75, 5.16, 17.66), (13.51, 9.24, 42.16), (9.86, 5.41, 19.93), (7.83, 5.1, 28.19), (5.61, 7.49, 14.17), (11.38, 8.3, 46.56), (8.05, 6.2, 26.39), (9.96, 5.45, 33.13), (14.18, 8.51, 44.89), (7.23, 7.56, 18.79), (11.22, 7.88, 36.99), (9.15, 8.07, 27.59), (12.82, 6.79, 30.69), (4.69, 5.57, 15.63), (6.36, 7.19, 13.07), (12.7, 7.16, 27.96), (11.87, 7.38, 19.18), (12.37, 7.86, 42.85), (11.41, 7.98, 40.85), (12.15, 7.16, 47.39), (8.22, 6.08, 31.81), (10.38, 6.32, 37.62), (10.69, 8.35, 42.55), (12.88, 6.76, 41.18), (11.44, 6.63, 34.02), (13.41, 8.72, 54.87), (8.6, 6.56, 25.48), (4.51, 6.62, 17.04), (4.74, 6.83, 10.12), (4.6, 5.47, 17.25), (12.65, 6.43, 42.28), (8.48, 6.11, 25.06), (13.76, 8.84, 41.76), (10.03, 6.4, 47.27), (10.7, 6.5, 28.31), (12.71, 6.95, 40.0), (11.66, 7.54, 35.24), (14.51, 8.8, 48.95), (12.1, 6.22, 28.33), (11.4, 7.15, 37.5), (9.67, 7.78, 28.5), (10.15, 6.37, 34.09), (9.89, 5.7, 34.4), (13.0, 8.34, 36.39), (11.81, 7.52, 28.42), (11.54, 7.45, 34.48), (10.49, 7.65, 30.36), (8.88, 6.72, 21.97), (5.59, 6.7, 31.38), (7.44, 7.43, 15.92), (6.35, 6.35, 29.13), (7.44, 7.07, 23.39), (11.62, 7.98, 32.68), (6.75, 6.53, 32.44), (10.34, 8.66, 39.26), (11.51, 8.18, 37.52), (10.35, 7.81, 30.95), (9.53, 7.29, 30.66), (14.1, 5.5, 38.07), (11.46, 6.64, 29.53), (7.66, 4.86, 8.94), (9.36, 7.1, 26.54), (11.35, 8.82, 39.94), (8.47, 6.77, 24.5), (11.05, 7.25, 30.97), (11.15, 8.0, 33.29), (9.58, 8.21, 34.41), (11.72, 7.1, 37.88), (10.95, 6.43, 35.88), (7.47, 4.82, 21.74), (12.42, 6.8, 25.82), (12.54, 6.56, 48.16), (16.36, 9.63, 46.46), (8.6, 8.66, 28.39), (18.01, 8.83, 62.0), (11.06, 7.1, 42.39), (12.3, 6.62, 28.39), (13.89, 7.08, 51.39), (4.75, 5.87, 16.61), (11.63, 7.74, 29.76), (8.88, 8.44, 22.57), (14.01, 6.92, 44.53), (8.2, 5.52, 22.32), (8.07, 6.71, 15.63), (6.54, 7.92, 22.44), (9.94, 7.45, 38.28), (12.66, 7.41, 41.92), (12.46, 6.64, 32.81), (13.39, 7.21, 42.65), (8.13, 5.17, 24.0), (10.46, 7.42, 30.56), (11.05, 7.68, 34.85), (7.74, 6.78, 30.91), (7.31, 5.72, 27.5), (7.6, 7.32, 16.69), (2.95, 5.97, 7.03), (4.45, 6.16, 5.63), (10.25, 7.19, 38.88), (6.46, 8.06, 11.36), (11.14, 7.94, 32.34), (10.14, 7.09, 35.08), (10.7, 5.63, 27.51), (13.53, 5.96, 39.05), (9.2, 7.83, 27.51), (11.28, 6.79, 25.99), (6.92, 6.38, 28.29), (10.46, 6.76, 43.29), (10.87, 6.36, 24.11), (9.48, 6.75, 32.9), (9.07, 8.2, 25.66), (11.67, 7.15, 20.82), (14.07, 9.35, 47.44), (8.58, 6.79, 28.52), (8.0, 8.71, 30.26), (10.24, 7.19, 32.61), (12.62, 8.08, 35.78), (11.47, 8.04, 36.86), (10.97, 7.21, 40.55), (10.78, 6.65, 35.04), (10.41, 7.63, 40.23), (9.0, 7.08, 23.28), (6.51, 8.58, 2.7), (12.64, 7.38, 40.64), (10.52, 7.59, 26.48), (10.99, 7.0, 37.06), (12.58, 7.57, 40.02), (10.38, 7.81, 35.47), (15.58, 6.78, 46.04), (6.6, 5.32, 23.49), (7.65, 6.28, 22.13), (14.95, 7.95, 40.23), (7.58, 6.75, 15.0), (11.64, 6.3, 24.58), (7.88, 7.56, 22.78), (12.4, 6.93, 40.33), (10.95, 8.85, 49.44), (9.89, 6.5, 20.46), (5.34, 7.49, 11.31), (11.32, 6.92, 40.35), (8.61, 6.54, 18.13), (8.36, 7.39, 23.39), (10.55, 6.9, 30.71), (11.38, 6.84, 33.61), (11.35, 7.6, 31.43), (10.34, 6.44, 28.83), (9.94, 6.17, 28.61), (12.81, 5.73, 48.58), (11.11, 8.08, 30.68), (7.34, 6.84, 18.76), (12.86, 9.32, 50.35), (14.63, 6.87, 43.96), (9.59, 6.16, 25.34), (14.18, 9.56, 49.1), (12.41, 6.23, 28.93), (8.96, 7.1, 31.1), (9.1, 6.94, 16.65), (7.21, 7.27, 24.27), (12.13, 7.94, 40.9), (9.36, 5.77, 27.8), (10.29, 6.17, 26.41), (9.47, 8.27, 28.95), (12.71, 6.55, 29.57), (14.56, 8.21, 36.41), (11.97, 8.06, 37.63), (10.6, 5.88, 28.39), (6.41, 5.07, 21.1), (13.36, 6.3, 43.91), (11.13, 8.41, 36.05), (10.99, 5.31, 19.63), (12.09, 6.5, 30.0), (16.4, 6.78, 56.86), (4.92, 6.27, 14.91), (5.32, 7.19, 13.03), (9.48, 6.0, 33.19), (10.57, 8.03, 39.27), (9.6, 5.6, 28.58), (9.87, 7.24, 41.93), (10.53, 8.85, 35.9), (10.2, 8.01, 33.71), (12.68, 8.51, 35.99), (9.05, 6.48, 28.55), (9.64, 8.22, 30.77), (8.03, 6.64, 25.05), (12.79, 7.59, 45.92), (8.17, 5.43, 25.66), (9.84, 8.21, 35.46), (11.15, 8.23, 34.63), (11.19, 5.27, 21.27), (11.29, 7.29, 30.48), (11.85, 6.83, 35.77), (13.01, 8.01, 43.91), (12.39, 6.29, 40.12), (5.33, 7.69, 15.55), (9.73, 6.33, 32.13), (9.52, 6.81, 25.55), (6.41, 5.5, 17.49), (10.62, 7.02, 29.32), (11.74, 7.71, 32.22), (13.38, 8.01, 44.71), (6.6, 6.37, 15.33), (10.5, 6.57, 36.34), (3.8, 5.23, 9.1), (7.53, 6.14, 14.49), (5.02, 6.53, 17.42), (4.12, 7.72, 7.38), (11.29, 8.13, 27.06), (12.47, 8.15, 41.78), (2.91, 5.72, 6.56), (10.42, 5.56, 29.4), (9.6, 7.46, 31.15), (8.63, 6.9, 25.9), (15.35, 6.61, 38.6), (8.91, 6.73, 22.95), (9.92, 5.77, 19.09), (12.51, 8.57, 43.39), (10.29, 7.09, 23.75), (5.81, 7.74, 4.67), (10.36, 7.12, 23.91), (11.13, 8.72, 40.26), (6.17, 7.25, 14.46), (8.08, 7.32, 27.39), (13.62, 7.62, 33.09), (13.83, 6.56, 34.78), (12.98, 6.49, 37.06), (9.02, 5.25, 20.88), (5.67, 7.59, 18.92), (12.57, 6.08, 39.88), (12.89, 7.24, 44.75), (9.39, 6.51, 26.92), (11.95, 7.79, 38.82), (10.97, 6.71, 30.8), (15.28, 7.29, 53.22), (6.95, 6.41, 29.57), (11.31, 8.16, 33.08), (13.34, 7.75, 39.89), (14.79, 7.48, 40.24), (5.55, 5.08, 4.92), (10.65, 7.24, 25.37), (7.68, 5.22, 19.25), (7.29, 5.44, 17.43), (14.86, 7.1, 38.6), (10.59, 8.9, 36.76), (6.95, 5.83, 20.76), (10.5, 6.18, 33.62), (16.52, 8.25, 51.98), (8.32, 7.11, 23.22), (7.95, 6.62, 28.36), (7.21, 7.15, 15.44), (5.54, 7.52, 11.46), (12.31, 7.37, 39.81), (12.9, 7.41, 38.11), (3.1, 5.87, -1.74), (7.54, 6.05, 20.61), (7.2, 6.47, 15.54), (4.56, 8.42, 10.01), (12.51, 6.78, 47.08), (11.54, 8.02, 41.0), (12.38, 7.1, 32.59), (12.18, 7.12, 37.33), (3.63, 7.46, 8.03), (11.39, 6.97, 26.45), (7.49, 5.72, 25.06), (10.69, 6.53, 23.35), (11.67, 6.52, 34.05), (7.92, 5.59, 14.82), (10.33, 7.29, 42.73), (4.72, 6.55, 17.39), (13.22, 7.27, 40.63), (10.36, 6.99, 24.73), (11.44, 6.8, 31.87), (10.42, 6.38, 42.55), (15.08, 7.25, 45.43), (8.27, 7.8, 27.77), (8.82, 6.89, 22.61), (14.67, 7.91, 43.39), (8.43, 6.4, 25.9), (6.61, 7.34, 8.06), (7.95, 7.31, 19.43), (4.86, 5.78, 13.08), (12.06, 5.47, 34.39), (9.03, 6.43, 16.91), (2.99, 5.4, 13.71), (6.87, 7.66, 29.31), (16.44, 7.05, 55.09), (11.92, 7.15, 40.37), (14.98, 7.28, 43.85), (7.9, 8.36, 30.31), (6.06, 4.52, 14.63), (8.03, 6.07, 16.92), (12.0, 7.16, 28.21), (9.65, 6.57, 24.52), (8.35, 6.08, 26.07), (3.56, 7.09, 5.66), (17.14, 7.96, 43.08), (14.66, 5.47, 32.89), (12.52, 7.61, 34.29), (12.33, 7.33, 40.57), (5.47, 6.26, 21.36), (12.67, 8.69, 35.96), (9.07, 5.34, 27.4), (13.45, 8.54, 41.01), (11.83, 6.18, 29.81), (15.78, 7.65, 53.03), (9.99, 5.81, 25.87), (6.73, 5.9, 17.17), (8.12, 7.81, 15.68), (15.31, 7.19, 45.13), (9.89, 8.18, 33.92), (15.14, 7.51, 41.23), (10.72, 5.64, 31.78), (9.03, 7.96, 26.17), (4.99, 6.55, 17.42), (12.92, 6.81, 35.72), (11.15, 7.39, 33.12), (7.62, 5.9, 23.21), (10.82, 6.16, 28.44), (12.29, 6.35, 35.66), (9.83, 5.27, 33.67), (3.95, 6.6, 10.3), (11.06, 6.61, 26.3), (10.26, 6.84, 25.26), (11.13, 6.95, 23.44), (5.41, 7.08, 27.48), (9.05, 7.57, 22.89), (7.55, 6.7, 25.79), (12.04, 7.32, 33.28), (7.09, 7.96, 20.16), (9.89, 6.1, 23.29), (11.95, 5.8, 43.15), (9.26, 6.69, 22.77), (12.49, 7.87, 36.74), (3.54, 5.58, 1.92), (7.31, 4.8, 20.4), (11.85, 7.21, 36.86), (8.89, 6.81, 23.59), (10.71, 8.11, 30.89), (11.74, 6.92, 31.47), (9.73, 8.53, 27.27), (12.4, 5.84, 28.33), (8.49, 6.41, 18.08), (3.94, 4.38, 10.45), (11.55, 8.57, 38.04), (11.51, 7.36, 25.45), (15.97, 8.61, 47.95), (8.61, 7.83, 23.1), (9.07, 6.78, 30.0), (8.77, 7.03, 28.85), (7.82, 7.49, 29.05), (7.49, 5.75, 17.97), (6.78, 7.56, 20.11), (11.43, 6.77, 30.17), (7.51, 7.13, 31.94), (12.84, 6.84, 33.76), (13.78, 5.88, 40.28), (10.66, 6.15, 34.0), (9.24, 7.96, 33.28), (6.58, 6.32, 15.57), (9.6, 6.43, 32.51), (14.47, 7.05, 38.62), (8.08, 6.7, 19.73), (11.1, 6.24, 29.73), (7.78, 7.49, 28.3), (9.85, 6.38, 28.65), (8.47, 7.08, 28.19), (6.31, 5.95, 14.65), (5.52, 6.68, 11.95), (9.94, 6.57, 38.09), (13.45, 7.71, 40.2), (8.26, 6.62, 30.15), (10.36, 5.37, 28.69), (10.8, 7.02, 26.9), (9.54, 7.91, 31.09), (8.45, 6.94, 28.54), (10.24, 6.58, 31.65), (7.97, 6.73, 28.34), (12.02, 5.93, 27.14), (4.05, 7.23, 20.19), (10.84, 7.76, 20.13), (13.63, 7.21, 43.77), (13.0, 6.06, 35.24), (6.93, 6.16, 17.63)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=29.8455，β₁=2.9374，β₂=1.6898；
SST=80275.84，SSR=57139.81，SSE=23136.04；R²=0.7118，调整R²=0.7110；
整体 F=(SSR/2)/(SSE/692)=854.5272，p=1.14e-187；x1,x2 相关 r=0.3539；
β₁: SE=0.0822，t=35.7209，p=3.32e-159；β₂: SE=0.2477，t=6.8207，p=1.44e-09。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=29.8455，β₁=2.9374，β₂=1.6898；
SST=80275.84，SSR=57139.81，SSE=23136.04；R²=0.7118，调整R²=0.7110；
整体 F=(SSR/2)/(SSE/692)=854.5272，p=1.14e-187；x1,x2 相关 r=0.3539；
β₁: SE=0.0822，t=35.7209，p=3.32e-159；β₂: SE=0.2477，t=6.8207，p=1.44e-09。
③ 计算结果：
y=29.8455+2.9374x1+1.6898x2；R²=0.7118；β₁ p=3.32e-159；β₂ p=1.44e-09
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=3.32e-159）；变量 x2 对 y 的影响显著（p=1.44e-09）。两正向因素。
案例 81. 训练量+饮食预测运动成绩
多元线性回归两正向因素
① 原始数据：
样本 n=857，逐条 (x1, x2, y)：[(2.77, 5.94, 27.48), (5.98, 5.58, 35.95), (9.41, 7.18, 63.14), (3.98, 4.14, 31.93), (6.06, 5.04, 42.41), (6.51, 7.16, 41.88), (7.38, 7.02, 59.23), (6.96, 6.61, 56.33), (6.43, 7.95, 46.89), (3.45, 3.87, 23.09), (4.7, 1.24, 17.68), (8.55, 3.48, 49.47), (5.07, 3.02, 38.63), (6.55, 7.83, 45.81), (7.23, 3.68, 37.33), (7.65, 4.0, 45.77), (6.58, 7.69, 61.75), (4.01, 5.32, 15.8), (7.35, 7.23, 49.26), (6.52, 5.41, 35.1), (8.62, 8.51, 62.54), (7.41, 7.2, 45.51), (5.37, 5.21, 50.79), (5.28, 3.66, 28.35), (4.03, 5.89, 32.41), (6.39, 3.28, 41.71), (5.69, 1.56, 29.19), (7.84, 3.43, 52.74), (6.15, 5.27, 36.24), (5.78, 2.59, 27.5), (4.46, 3.71, 26.79), (8.06, 7.53, 62.54), (5.02, 3.46, 31.39), (5.77, 1.39, 36.3), (3.58, 4.25, 17.69), (3.73, 4.4, 30.13), (6.69, 2.88, 44.08), (3.94, 4.32, 36.48), (5.37, 6.45, 35.19), (6.41, 6.88, 48.22), (7.61, 2.73, 33.75), (5.84, 4.76, 38.34), (3.95, 2.99, 14.62), (8.57, 5.44, 50.49), (6.58, 6.24, 39.95), (5.83, 7.74, 53.92), (4.52, 7.02, 32.7), (6.14, 3.22, 31.55), (2.91, 7.59, 22.95), (5.67, 6.03, 38.95), (4.37, 9.43, 51.0), (8.12, 6.67, 57.35), (6.18, 1.68, 34.98), (7.06, 5.04, 40.4), (4.08, 3.77, 25.12), (6.02, 6.31, 40.5), (5.41, 2.09, 36.58), (3.97, 6.39, 30.69), (6.47, 3.68, 29.22), (5.72, 3.59, 29.29), (5.75, 5.55, 40.02), (5.56, 4.91, 45.91), (8.02, 4.61, 50.91), (8.96, 3.36, 52.66), (5.31, 3.06, 31.17), (5.38, 3.35, 31.25), (4.28, 1.44, 23.91), (9.38, 6.54, 61.96), (2.08, 1.67, 11.06), (7.04, 3.2, 33.12), (7.62, 6.1, 60.13), (7.99, 6.14, 41.95), (8.75, 4.6, 44.78), (9.95, 4.01, 54.25), (6.26, 7.22, 56.79), (4.7, 5.31, 40.58), (8.53, 3.79, 46.83), (7.05, 5.99, 46.59), (3.62, 6.91, 38.32), (3.59, 3.73, 23.22), (6.85, 6.11, 46.07), (8.61, 6.31, 58.25), (8.5, 6.05, 51.98), (5.25, 5.37, 36.61), (8.41, 8.0, 58.82), (6.78, 4.71, 53.93), (6.16, 5.83, 47.55), (7.2, 6.25, 56.23), (7.25, 7.63, 55.25), (7.92, 6.79, 60.49), (5.53, 7.17, 42.66), (6.1, 1.97, 19.34), (5.31, 5.64, 37.71), (3.63, 0.75, 6.95), (-0.33, -0.03, -3.37), (5.62, 4.78, 40.7), (1.08, 6.52, 18.04), (5.46, 4.14, 31.04), (4.59, 2.84, 24.76), (3.52, 2.75, 7.82), (1.94, 3.38, 12.98), (7.82, 5.88, 50.92), (4.66, 6.6, 35.14), (5.07, 2.66, 21.47), (3.11, 3.74, 26.36), (8.92, 4.34, 50.27), (5.21, 2.12, 14.26), (9.81, 6.48, 68.41), (9.35, 2.89, 57.17), (5.57, 6.01, 34.66), (2.4, 2.79, 23.56), (5.52, 3.82, 48.27), (7.19, 2.09, 28.27), (6.11, 6.72, 45.96), (4.0, 4.97, 31.17), (8.3, 6.93, 63.43), (5.87, 2.74, 34.06), (10.28, 5.71, 58.04), (4.19, 2.51, 33.4), (8.29, 5.76, 61.43), (9.67, 5.62, 59.91), (4.95, 7.7, 39.47), (3.78, 4.77, 32.96), (7.51, 4.76, 47.61), (4.15, 4.48, 31.89), (8.32, 8.85, 56.3), (4.05, 1.27, 27.68), (5.05, 3.37, 30.39), (5.72, 2.71, 33.25), (4.4, 2.89, 13.15), (8.87, 5.36, 48.88), (6.69, 6.09, 44.76), (4.75, 4.75, 25.87), (5.13, 5.97, 41.69), (6.39, 4.35, 36.73), (4.2, 1.61, 25.52), (1.86, 6.05, 23.83), (6.33, 6.27, 39.6), (8.46, 5.63, 53.41), (5.67, 3.82, 28.09), (4.76, 2.47, 28.44), (8.14, 7.69, 57.05), (5.76, 9.06, 47.8), (4.28, 4.9, 20.97), (5.9, 5.7, 41.72), (5.32, 6.74, 36.35), (9.4, 7.91, 67.39), (6.99, 2.95, 42.32), (5.84, 6.53, 49.22), (7.08, 5.47, 46.94), (7.84, 3.85, 50.62), (7.04, 5.58, 49.89), (9.77, 9.02, 64.96), (7.61, 6.54, 58.28), (4.57, 5.12, 37.63), (5.86, 5.96, 40.08), (7.28, 8.75, 66.09), (7.34, 8.22, 47.3), (6.32, 6.13, 46.3), (0.82, 1.84, 10.67), (6.89, 4.83, 44.93), (5.12, 4.76, 30.94), (6.48, 2.95, 36.42), (8.45, 4.09, 47.36), (6.75, 4.27, 43.85), (5.02, 5.36, 30.92), (6.44, 4.7, 33.7), (4.89, 5.8, 36.23), (4.46, 3.16, 24.29), (6.14, 5.27, 37.48), (6.78, 4.33, 38.36), (8.03, 6.27, 47.98), (2.49, 2.82, 16.78), (3.19, 4.66, 23.51), (6.91, 6.43, 49.16), (2.58, 3.33, 27.19), (6.34, 4.91, 40.38), (4.87, 1.49, 16.36), (8.71, 9.91, 69.23), (4.42, 3.37, 32.45), (10.38, 5.02, 51.1), (7.48, 5.48, 48.42), (1.96, 3.16, 18.87), (7.5, 3.99, 50.95), (6.72, 4.84, 43.0), (6.58, 3.6, 29.23), (7.94, 3.69, 49.39), (5.51, 1.83, 25.98), (4.44, 3.31, 25.06), (4.67, 4.2, 31.86), (4.36, 7.86, 22.48), (6.25, 5.57, 39.43), (12.79, 7.61, 86.59), (5.59, 6.82, 41.28), (9.64, 9.45, 74.02), (4.5, 3.34, 26.43), (5.42, 8.47, 44.12), (4.56, 3.61, 28.31), (5.74, 5.36, 36.65), (5.75, 7.23, 47.82), (7.82, 6.07, 51.96), (1.83, 3.68, 21.83), (6.32, 5.34, 37.8), (8.16, 5.29, 48.24), (6.17, 5.4, 45.48), (9.55, 5.93, 61.29), (5.19, 5.14, 30.36), (6.64, 0.93, 30.02), (7.05, 6.76, 48.67), (6.83, 5.35, 44.28), (6.71, 7.11, 52.59), (7.07, 8.53, 51.27), (5.64, 7.88, 39.3), (4.98, 2.24, 20.28), (4.3, 5.51, 37.37), (1.84, 4.95, 14.82), (9.36, 7.42, 68.64), (7.4, 6.6, 45.24), (6.77, 5.12, 42.08), (5.47, 5.2, 35.73), (6.07, 4.87, 31.24), (3.64, 5.88, 32.28), (7.19, 3.53, 40.74), (8.69, 9.57, 73.88), (6.09, 4.2, 38.75), (7.91, 2.34, 41.48), (6.14, 6.94, 48.5), (5.55, 5.66, 38.99), (3.21, 4.98, 32.94), (2.13, 4.12, 18.53), (3.63, 6.15, 33.8), (5.81, 4.85, 38.43), (7.16, 5.75, 46.83), (5.22, 3.69, 35.94), (3.95, 5.14, 24.83), (6.06, 3.35, 28.03), (3.1, 4.28, 24.32), (3.71, 4.3, 26.62), (4.76, 4.39, 37.91), (6.68, 8.88, 53.41), (7.64, 4.91, 48.2), (5.87, 7.1, 52.4), (4.86, 3.66, 28.35), (4.71, 5.54, 30.63), (5.72, 3.7, 32.96), (8.9, 2.0, 39.04), (6.73, 2.24, 45.24), (7.58, 6.11, 41.79), (5.84, 5.87, 46.69), (4.15, -0.04, 11.02), (5.89, 5.38, 45.64), (4.83, 6.28, 35.76), (4.1, 4.31, 32.59), (7.62, 6.31, 52.68), (3.32, 6.15, 29.78), (10.15, 10.07, 84.48), (4.01, 4.34, 21.89), (6.02, 11.03, 60.53), (5.51, 4.26, 39.59), (6.51, 6.19, 49.53), (9.09, 5.33, 58.23), (8.13, 6.82, 56.54), (8.19, 4.91, 35.06), (7.68, 3.66, 43.98), (7.43, 5.7, 48.26), (11.19, 3.71, 68.82), (6.53, 3.52, 37.26), (6.44, 0.02, 31.53), (3.24, 5.09, 27.3), (6.76, 5.21, 41.68), (3.65, 3.57, 26.31), (4.7, 4.28, 31.0), (4.55, 7.19, 40.37), (6.48, 7.38, 53.9), (7.72, 7.42, 46.63), (10.32, 4.46, 69.17), (8.9, 8.18, 61.84), (7.36, 7.72, 55.18), (5.93, 5.78, 43.22), (11.61, 8.13, 86.57), (5.67, 5.57, 43.32), (8.38, 8.47, 62.57), (6.21, 5.03, 44.16), (3.86, 3.5, 27.41), (7.37, 1.05, 36.05), (8.09, 4.71, 48.29), (6.3, 5.73, 44.44), (6.2, 2.67, 32.53), (4.96, 5.51, 33.07), (6.55, 2.18, 37.64), (3.02, 9.5, 35.77), (5.95, 7.66, 54.26), (6.49, 2.61, 31.08), (7.04, 6.88, 53.35), (6.51, 4.9, 47.37), (5.35, 3.79, 35.53), (5.61, 6.9, 35.87), (8.13, 2.18, 30.04), (9.38, 5.67, 68.14), (2.83, 5.94, 23.89), (6.7, 4.15, 39.38), (7.69, 6.98, 48.38), (10.43, 7.53, 69.41), (5.16, 4.95, 35.86), (3.48, 4.35, 19.56), (7.31, 6.67, 51.05), (4.63, 10.2, 53.82), (8.36, 3.83, 45.36), (5.54, 4.4, 33.45), (10.12, 1.9, 48.41), (7.49, 4.73, 37.48), (6.42, 5.54, 39.72), (2.75, 5.49, 27.39), (2.79, 4.17, 23.41), (5.92, 6.12, 41.61), (2.23, 2.46, 15.23), (6.53, 6.44, 46.59), (4.67, 7.38, 49.95), (9.27, 7.14, 67.69), (7.05, 5.54, 43.88), (4.11, 3.61, 22.82), (3.56, 2.93, 28.75), (4.22, 6.18, 38.71), (7.78, 5.96, 49.4), (4.71, 3.37, 33.64), (6.39, 3.45, 41.94), (3.91, 6.23, 34.87), (5.66, 5.64, 29.7), (5.57, 7.86, 46.44), (5.0, 3.43, 30.41), (4.4, 7.49, 37.68), (8.4, 7.36, 54.33), (5.88, 4.23, 34.57), (7.77, 5.22, 53.72), (6.24, 5.09, 43.54), (5.7, 2.97, 34.95), (6.67, 9.41, 46.75), (3.77, 4.56, 23.08), (6.38, 2.09, 27.19), (5.76, 3.82, 36.39), (4.54, 5.46, 32.26), (4.1, 3.02, 23.98), (6.47, 4.42, 41.93), (8.62, 4.49, 55.41), (5.84, 4.23, 30.65), (6.64, 5.54, 49.82), (4.41, 1.03, 22.37), (2.53, 2.04, 10.0), (4.62, 4.33, 30.71), (4.55, 2.15, 19.22), (5.83, 5.6, 40.6), (3.01, 5.93, 37.69), (4.24, 5.42, 29.33), (5.63, 2.03, 26.64), (9.01, 5.1, 61.9), (7.87, 5.98, 57.4), (6.76, 3.93, 33.01), (2.34, 3.46, 13.32), (3.55, 0.27, 12.55), (7.54, 5.02, 49.94), (6.34, 4.9, 35.0), (9.97, 5.18, 63.35), (6.84, 4.28, 41.05), (7.6, 8.48, 63.41), (8.65, 4.55, 47.02), (7.61, 6.64, 54.58), (4.58, 5.39, 35.03), (6.55, 6.62, 56.08), (2.69, 5.89, 32.39), (6.22, 1.98, 28.36), (5.81, 6.06, 34.25), (8.24, 6.59, 51.41), (6.78, 3.46, 42.53), (4.12, 3.17, 31.24), (1.08, 4.05, 5.21), (5.12, 8.89, 55.24), (4.05, 4.8, 27.06), (6.86, 5.34, 52.91), (6.04, 5.97, 48.59), (5.63, 5.14, 32.18), (7.98, 6.57, 56.48), (6.13, 3.8, 43.33), (5.07, 3.22, 29.16), (6.75, 5.58, 51.39), (6.95, 3.72, 49.66), (6.19, 7.05, 50.37), (7.75, 6.53, 57.93), (3.95, 3.43, 27.09), (8.89, 5.23, 54.8), (6.52, 6.66, 46.3), (8.18, 6.67, 56.47), (4.52, 5.29, 34.25), (6.79, 7.5, 57.31), (8.82, 8.47, 73.45), (6.93, 2.57, 35.98), (8.76, 6.24, 60.15), (3.68, 2.1, 21.87), (7.86, 2.27, 44.59), (4.05, 3.92, 21.41), (5.11, 7.46, 43.06), (4.59, 7.19, 36.51), (4.17, 5.9, 26.55), (2.89, 1.99, 12.66), (7.25, 0.7, 35.16), (6.19, 9.16, 57.6), (8.17, 3.66, 41.31), (3.2, 4.12, 19.21), (8.13, 8.86, 57.02), (7.03, 5.43, 46.33), (4.49, 2.99, 28.26), (4.48, 2.44, 28.59), (4.1, 5.55, 29.07), (5.83, 4.78, 36.05), (6.85, 5.73, 48.6), (7.82, 6.24, 54.53), (5.06, 6.18, 37.32), (6.28, 4.12, 41.23), (4.33, 7.65, 35.76), (2.3, 3.28, 20.08), (7.48, 3.47, 45.55), (6.88, 4.74, 46.29), (6.27, 7.68, 44.33), (6.22, 4.9, 36.76), (2.62, 0.99, 11.6), (6.33, 5.96, 44.39), (4.79, 9.26, 47.49), (5.89, 5.13, 39.93), (3.57, 1.09, 12.03), (4.43, 5.79, 28.78), (6.43, 4.5, 35.02), (7.19, 5.98, 53.49), (7.75, 6.52, 48.99), (3.23, 1.81, 14.01), (0.17, 3.01, 15.34), (3.29, 6.81, 36.73), (3.08, 6.67, 31.5), (7.39, 4.26, 47.9), (4.28, 5.9, 38.79), (3.82, 4.75, 24.37), (5.85, 3.53, 32.4), (6.77, 5.8, 43.99), (7.01, 2.59, 40.73), (7.56, 6.48, 58.93), (4.23, 7.77, 29.31), (6.28, 4.93, 51.25), (2.7, 2.76, 16.52), (7.66, 0.8, 39.6), (5.63, 0.66, 23.34), (4.5, 5.27, 22.28), (6.29, 8.52, 55.2), (7.07, 5.49, 53.03), (0.07, -0.86, -10.69), (7.92, 3.96, 41.53), (5.61, 4.98, 36.63), (8.77, 5.3, 47.45), (11.12, 9.43, 77.48), (5.19, 5.34, 39.21), (4.77, 4.33, 32.82), (6.48, 4.2, 38.31), (8.1, 2.83, 48.74), (7.09, 4.53, 52.25), (3.28, 6.42, 34.73), (5.87, 6.67, 46.63), (7.84, 4.55, 48.4), (6.21, 2.89, 33.06), (3.63, 2.05, 15.84), (7.23, 4.34, 44.59), (9.24, 5.4, 60.72), (8.68, 6.95, 53.63), (7.54, 5.29, 51.76), (8.07, 7.39, 52.79), (6.22, 0.67, 22.17), (4.08, 4.62, 36.03), (3.31, 0.73, 21.61), (5.54, 4.63, 40.02), (0.51, -0.07, -0.11), (5.1, 5.11, 32.47), (4.48, 3.45, 31.19), (3.76, 4.32, 25.16), (8.39, 7.98, 75.51), (7.71, 5.94, 40.25), (6.39, 3.25, 38.45), (9.84, 6.41, 70.95), (8.73, 6.38, 61.15), (7.79, 4.49, 45.69), (6.97, 5.34, 50.73), (8.42, 4.2, 43.99), (3.6, 4.73, 26.93), (4.33, 1.97, 24.38), (3.41, 4.16, 25.0), (2.09, 3.15, 22.15), (6.42, 3.81, 44.77), (3.9, 4.53, 20.49), (7.28, 8.94, 68.83), (4.79, 6.38, 37.7), (4.51, 0.1, 21.16), (4.54, 6.74, 30.41), (7.53, 5.34, 48.15), (2.85, 5.03, 24.09), (5.83, 4.29, 30.45), (1.41, 2.32, 10.01), (4.13, 6.06, 26.95), (6.42, 3.81, 39.19), (3.13, 5.83, 28.16), (7.29, 6.59, 57.64), (5.79, 3.39, 35.4), (6.87, 3.76, 39.05), (7.13, 6.39, 52.75), (7.58, 5.48, 52.43), (6.72, 2.87, 32.82), (5.24, 4.38, 35.88), (3.44, 4.85, 24.27), (6.4, 5.06, 42.19), (6.82, 6.31, 49.96), (4.78, 0.85, 22.88), (11.4, 9.6, 77.71), (5.23, 4.57, 32.42), (3.96, 1.85, 14.63), (6.87, 1.93, 31.08), (7.9, 4.67, 48.82), (4.11, 6.0, 29.73), (4.33, 4.47, 27.77), (6.5, 4.46, 46.61), (4.86, 3.49, 23.25), (8.25, 5.33, 62.19), (3.49, 8.99, 37.64), (6.44, 6.09, 43.37), (9.34, 6.52, 56.89), (9.48, 5.69, 59.34), (7.05, 5.44, 42.68), (7.82, 5.6, 51.35), (4.85, 1.42, 18.15), (4.17, 4.0, 24.51), (7.36, 5.1, 53.43), (8.22, 3.9, 50.66), (5.17, 3.93, 30.52), (8.31, 7.96, 63.48), (10.14, 8.48, 62.08), (4.62, 6.09, 31.76), (7.39, 6.16, 53.5), (6.54, 5.26, 45.8), (5.33, 5.57, 44.42), (6.82, 4.43, 35.07), (6.53, 3.48, 34.66), (6.63, 6.57, 52.7), (2.6, 2.67, 8.43), (6.25, 7.16, 44.34), (2.92, 2.06, 9.16), (4.24, 2.72, 29.79), (10.25, 8.65, 74.01), (5.85, 7.89, 45.2), (5.62, 0.78, 27.54), (5.16, 4.58, 31.19), (4.0, 3.26, 24.23), (4.43, 6.77, 36.88), (1.83, 3.74, 17.96), (4.71, 1.45, 24.56), (5.62, 5.04, 31.91), (4.73, 4.09, 23.01), (6.59, 3.49, 41.1), (2.79, 0.34, 5.09), (7.56, 9.27, 66.21), (5.1, 7.45, 43.9), (7.72, 3.64, 46.59), (6.86, 7.27, 45.6), (7.22, 3.63, 38.65), (4.08, 2.6, 18.56), (4.35, 8.56, 43.18), (7.06, 3.78, 45.69), (6.1, 7.66, 53.34), (5.06, 3.58, 20.91), (6.37, 1.77, 21.76), (3.18, 5.16, 24.41), (4.08, 6.21, 31.94), (10.45, 8.32, 77.78), (6.39, 7.23, 48.87), (3.94, 5.49, 31.88), (6.56, 5.53, 46.43), (5.17, 5.31, 31.69), (10.8, 9.92, 76.43), (3.45, 5.7, 37.05), (5.56, 2.74, 26.07), (5.14, 2.8, 27.57), (7.59, 4.13, 45.96), (4.2, 4.25, 28.16), (3.89, 6.34, 33.13), (4.86, 6.01, 38.91), (5.0, 6.05, 44.36), (8.44, 7.91, 64.56), (6.38, 3.96, 36.79), (6.31, 7.04, 51.24), (6.05, 5.65, 43.58), (5.89, 4.34, 32.13), (6.77, 8.91, 60.61), (8.49, 7.55, 62.12), (5.8, 5.58, 48.08), (3.61, 7.15, 38.56), (5.22, 3.73, 38.61), (7.84, 5.24, 51.69), (9.21, 6.94, 68.21), (6.51, 7.86, 50.32), (3.42, 4.18, 23.73), (5.25, 4.12, 26.11), (5.96, 1.01, 30.82), (2.06, 5.51, 18.88), (7.5, 4.86, 50.01), (5.77, 4.42, 34.65), (5.33, 2.53, 31.23), (7.74, 6.01, 55.31), (8.95, 4.95, 47.46), (2.98, 7.2, 30.16), (5.6, 6.08, 37.1), (9.02, 6.09, 54.5), (4.22, 5.93, 35.3), (4.46, 2.94, 18.46), (4.66, 4.75, 28.24), (6.46, 6.01, 44.14), (7.53, 5.71, 56.24), (1.73, 0.24, -0.07), (5.09, 2.38, 30.12), (6.25, 6.48, 49.72), (8.41, 4.12, 46.42), (6.25, 5.0, 41.04), (8.65, 7.55, 56.43), (5.7, 7.05, 31.54), (3.67, 1.84, 12.68), (4.29, 4.79, 38.84), (9.05, 7.05, 64.72), (9.84, 6.16, 60.93), (8.78, 9.57, 68.12), (6.89, 4.03, 33.16), (7.58, 3.65, 44.19), (6.63, 3.44, 40.24), (9.48, 5.12, 61.1), (5.91, 5.47, 34.09), (11.62, 7.84, 72.67), (9.27, 3.57, 56.27), (3.81, 4.83, 31.19), (3.22, 3.66, 25.75), (7.92, 6.2, 57.22), (3.87, 6.65, 40.52), (4.51, 3.5, 33.6), (6.01, -1.49, 23.75), (7.2, 7.03, 48.06), (6.76, 5.53, 50.24), (6.75, 5.92, 40.19), (7.09, 5.72, 49.65), (4.03, 3.6, 24.68), (5.81, 3.23, 31.78), (1.16, 2.04, 8.31), (4.84, 5.92, 38.78), (6.92, 0.93, 34.43), (6.59, 1.3, 34.88), (1.7, -0.18, 0.23), (7.87, 4.95, 48.34), (7.79, 6.97, 54.6), (7.32, 4.42, 42.79), (7.0, 5.25, 38.39), (6.8, 6.46, 44.49), (8.07, 4.43, 47.99), (7.23, 6.72, 54.86), (8.78, 6.24, 58.69), (5.07, 6.36, 37.52), (7.09, 7.59, 55.44), (8.9, 8.84, 68.57), (6.79, 4.19, 44.54), (5.49, 7.02, 39.04), (9.06, 6.75, 61.12), (5.93, 3.93, 40.03), (4.87, 4.11, 19.81), (4.12, 4.73, 34.58), (6.47, 5.11, 41.06), (4.41, 2.34, 21.11), (4.82, 6.17, 28.47), (5.55, 6.76, 40.04), (5.04, 4.15, 27.03), (5.37, 6.77, 38.98), (7.27, 7.16, 51.83), (5.79, 8.31, 50.32), (9.6, 5.93, 59.81), (8.83, 3.61, 50.66), (4.13, 2.54, 21.35), (5.06, 3.85, 34.88), (5.78, 3.85, 29.91), (7.44, 7.77, 57.5), (4.27, 5.1, 29.42), (8.56, 7.6, 67.68), (4.93, 6.47, 39.8), (7.0, 0.96, 31.78), (6.82, 3.17, 41.45), (4.13, 6.29, 25.02), (9.74, 3.61, 61.46), (7.38, 6.74, 43.91), (5.86, 2.26, 27.98), (7.07, 5.26, 54.82), (0.61, 1.81, 4.76), (6.01, 5.1, 38.76), (9.14, 5.52, 57.03), (7.73, 3.85, 45.95), (7.98, 5.36, 46.37), (6.03, 6.2, 40.38), (2.19, 7.59, 35.19), (8.61, 5.42, 52.8), (5.72, 5.39, 46.4), (2.82, 5.04, 23.19), (7.12, 5.9, 58.96), (6.12, 3.52, 33.31), (7.37, 9.21, 63.87), (4.42, 3.57, 30.55), (3.99, 4.78, 28.53), (9.95, 4.1, 56.4), (7.19, 8.65, 43.5), (2.23, 3.96, 14.75), (6.91, 7.02, 49.94), (4.51, 5.04, 32.35), (7.54, 7.15, 62.89), (5.1, 7.44, 38.71), (4.65, 7.12, 43.25), (6.78, 6.29, 50.29), (5.32, 7.62, 46.71), (6.94, 8.78, 56.78), (5.23, 4.05, 31.17), (6.62, 2.3, 39.71), (4.6, 3.9, 36.73), (8.43, 2.7, 45.07), (6.01, 3.25, 31.29), (4.47, 3.81, 31.97), (8.75, 3.47, 50.18), (2.21, 2.77, 9.15), (4.01, -0.05, 16.7), (4.98, 4.33, 35.74), (5.9, 3.15, 32.79), (2.19, 4.25, 16.84), (6.49, 2.7, 33.91), (6.47, 5.4, 37.69), (5.8, 6.37, 54.43), (8.81, 4.87, 55.61), (5.21, 5.55, 38.31), (4.95, 4.33, 27.42), (9.08, 4.45, 61.74), (9.25, 4.22, 57.1), (5.23, 4.6, 34.88), (6.37, 3.5, 40.73), (6.63, 4.43, 32.28), (8.04, 2.93, 49.13), (4.42, 6.55, 43.94), (5.43, 4.94, 30.01), (6.13, 0.7, 26.19), (8.73, 3.91, 55.61), (7.17, 7.24, 58.22), (6.21, 7.95, 49.64), (6.59, 4.3, 40.42), (3.94, 4.58, 22.48), (5.68, 7.15, 46.73), (8.17, 2.56, 40.97), (4.7, 5.11, 33.48), (5.17, 1.77, 25.83), (8.0, 2.89, 50.21), (7.64, 4.68, 49.91), (2.64, 1.85, 9.28), (8.7, 3.46, 41.39), (6.57, 5.12, 39.69), (4.73, 2.88, 33.68), (5.25, 5.35, 38.99), (5.87, 3.45, 26.99), (5.22, 3.74, 26.7), (7.7, 4.28, 43.51), (7.79, 5.47, 46.66), (7.95, 5.85, 49.21), (7.03, 3.8, 33.49), (8.97, 4.76, 55.03), (6.24, 5.46, 47.55), (9.16, 7.28, 60.8), (4.75, 8.0, 49.45), (8.12, 5.56, 51.25), (6.5, 9.3, 54.36), (6.14, 1.87, 30.37), (7.21, 6.44, 54.64), (6.36, 3.91, 36.95), (7.75, 3.31, 48.69), (5.97, 5.46, 52.19), (4.26, 5.56, 28.75), (5.86, 2.3, 35.97), (4.87, 4.74, 28.65), (7.37, 5.2, 46.97), (7.05, 1.52, 34.16), (5.84, 7.03, 40.64), (8.19, 6.56, 60.63), (2.13, 6.31, 32.51), (4.83, 5.86, 40.34), (4.4, 6.76, 40.03), (9.31, 6.99, 62.18), (6.77, 8.01, 61.67), (5.17, 5.87, 38.21), (5.33, 4.88, 37.79), (6.31, 5.95, 46.98), (6.52, 4.69, 46.8), (7.95, 3.48, 40.7), (6.21, 5.65, 38.61), (6.31, 1.77, 25.87), (6.81, 5.58, 47.35), (7.0, 6.3, 52.04), (6.3, 5.49, 45.48), (5.07, 5.34, 31.99), (9.7, 5.66, 65.0), (6.12, 5.06, 41.79), (6.76, 5.51, 44.35), (5.48, 4.22, 39.59), (5.9, 6.98, 41.6), (5.06, 3.56, 24.9), (6.2, 6.98, 46.49), (5.13, 3.48, 33.5), (9.96, 4.73, 59.31), (6.21, 5.97, 38.66), (5.5, 5.15, 30.8), (8.88, 5.48, 55.15), (6.28, 3.47, 39.48), (5.38, 2.0, 37.36), (4.83, 2.33, 18.29), (7.92, 5.73, 51.73), (7.83, 7.82, 53.09), (7.31, 5.61, 44.97), (0.65, 5.96, 16.16), (-0.43, 2.9, 4.71), (5.83, 4.75, 43.14), (2.95, 5.43, 25.11), (6.14, 7.48, 45.72), (4.34, 4.84, 37.53), (7.43, 5.15, 44.88), (5.59, 6.21, 46.21), (8.77, 1.82, 47.08), (6.85, 0.71, 35.89), (2.96, 2.46, 16.19), (6.3, 3.57, 38.8), (7.69, 3.62, 46.89), (9.1, 4.55, 58.66), (5.14, 3.89, 36.13), (4.64, 4.11, 34.01), (5.82, 4.34, 34.51), (1.28, 2.9, 6.84), (0.11, 1.27, 1.66), (2.7, 6.86, 32.19), (6.54, 5.53, 49.06), (5.62, 4.34, 41.91), (6.2, 6.15, 44.4), (8.1, 5.68, 50.88), (6.59, 2.34, 43.41), (6.34, 4.66, 40.28), (8.63, 6.15, 53.75), (7.91, 4.15, 43.78), (4.73, 3.1, 33.67), (7.66, 7.61, 56.2), (8.27, 4.25, 49.4), (3.35, 2.11, 19.04), (3.4, 4.99, 28.06), (5.85, 3.35, 35.61), (4.43, 6.12, 29.34)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=39.9116，β₁=5.0931，β₂=3.0787；
SST=175143.48，SSR=154834.73，SSE=20308.75；R²=0.8840，调整R²=0.8838；
整体 F=(SSR/2)/(SSE/854)=3255.4648，p=<1e-300；x1,x2 相关 r=0.3200；
β₁: SE=0.0881，t=57.8028，p=2.02e-297；β₂: SE=0.0883，t=34.8472，p=3.27e-166。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=39.9116，β₁=5.0931，β₂=3.0787；
SST=175143.48，SSR=154834.73，SSE=20308.75；R²=0.8840，调整R²=0.8838；
整体 F=(SSR/2)/(SSE/854)=3255.4648，p=<1e-300；x1,x2 相关 r=0.3200；
β₁: SE=0.0881，t=57.8028，p=2.02e-297；β₂: SE=0.0883，t=34.8472，p=3.27e-166。
③ 计算结果：
y=39.9116+5.0931x1+3.0787x2；R²=0.8840；β₁ p=2.02e-297；β₂ p=3.27e-166
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=2.02e-297）；变量 x2 对 y 的影响显著（p=3.27e-166）。两正向因素。
案例 82. 价格+品牌力预测购买意愿
多元线性回归价负、品牌正
① 原始数据：
样本 n=391，逐条 (x1, x2, y)：[(91.29, 52.23, 7.14), (67.51, 47.25, 3.27), (68.13, 60.79, 16.78), (103.61, 30.11, -15.04), (124.09, 47.48, 1.92), (41.62, 62.99, 19.31), (97.19, 35.26, -9.65), (154.47, 50.77, 4.87), (94.02, 46.97, 2.09), (56.26, 49.41, 5.28), (154.59, 53.21, 7.08), (60.71, 36.46, -7.86), (100.49, 57.05, 11.78), (113.09, 55.17, 9.88), (144.25, 46.42, 0.05), (115.95, 56.72, 11.46), (85.51, 58.48, 13.57), (129.33, 60.41, 15.0), (81.09, 52.95, 8.71), (138.52, 53.64, 7.83), (133.08, 54.32, 8.97), (89.8, 57.31, 12.63), (140.89, 34.81, -10.98), (153.04, 62.48, 16.11), (66.18, 51.91, 7.67), (80.83, 42.63, -1.84), (63.01, 33.12, -11.28), (111.72, 79.38, 34.15), (111.6, 59.7, 14.12), (44.07, 54.46, 10.58), (102.28, 28.45, -16.61), (86.42, 49.93, 5.37), (64.91, 34.19, -10.46), (53.45, 29.85, -14.59), (-4.25, 40.73, -2.27), (109.51, 56.06, 10.89), (129.41, 48.17, 2.45), (168.49, 66.09, 20.17), (126.14, 49.38, 4.33), (131.19, 44.58, -1.3), (47.08, 48.29, 4.01), (168.87, 47.26, 0.58), (25.76, 27.31, -16.2), (159.92, 43.51, -2.68), (91.18, 32.55, -12.24), (205.32, 59.32, 12.24), (113.11, 29.77, -15.75), (73.83, 35.6, -9.19), (124.12, 63.21, 17.69), (89.59, 31.52, -13.14), (142.88, 42.69, -3.3), (66.03, 46.18, 1.61), (69.73, 59.92, 15.71), (127.72, 51.37, 5.35), (110.99, 46.86, 1.76), (121.06, 49.98, 4.68), (121.61, 64.28, 18.72), (127.72, 46.82, 0.65), (173.07, 72.88, 26.45), (109.85, 53.5, 8.9), (33.26, 48.69, 5.54), (127.71, 46.06, 0.41), (187.57, 59.93, 12.97), (58.43, 59.82, 15.32), (41.73, 31.78, -11.78), (67.35, 51.13, 6.7), (79.59, 58.62, 13.82), (72.51, 48.09, 3.72), (79.74, 57.03, 12.72), (111.48, 42.11, -3.67), (159.92, 59.63, 13.44), (105.57, 43.72, -1.52), (135.33, 46.91, 1.27), (75.84, 65.96, 21.19), (113.64, 42.79, -2.55), (80.81, 67.29, 22.88), (163.29, 60.71, 14.61), (121.78, 46.32, 0.92), (84.17, 49.51, 4.56), (87.0, 61.8, 17.36), (48.48, 57.03, 13.59), (53.9, 61.57, 17.62), (148.8, 61.5, 15.18), (124.33, 61.04, 15.57), (98.99, 50.83, 5.96), (102.94, 54.72, 9.4), (94.58, 46.47, 1.36), (153.57, 56.26, 10.02), (159.88, 75.79, 29.51), (114.59, 48.31, 3.3), (98.59, 49.66, 4.52), (166.84, 54.1, 8.18), (51.7, 35.68, -8.02), (68.96, 27.07, -17.44), (152.79, 63.67, 18.12), (125.71, 60.95, 15.24), (132.43, 33.57, -11.9), (116.29, 51.94, 6.58), (76.14, 56.25, 12.19), (67.52, 53.72, 9.6), (103.64, 44.15, -0.75), (155.44, 44.37, -1.82), (61.13, 51.77, 7.45), (115.12, 61.11, 15.94), (83.25, 56.79, 12.09), (108.59, 33.82, -11.3), (157.48, 64.7, 18.52), (74.46, 48.43, 4.11), (111.7, 46.12, 0.53), (144.63, 59.71, 13.83), (134.48, 56.88, 11.37), (124.66, 49.01, 3.94), (114.09, 30.37, -15.3), (5.32, 46.51, 2.81), (78.03, 46.67, 2.14), (57.9, 41.88, -2.63), (41.68, 49.26, 5.5), (106.89, 49.63, 4.51), (170.0, 63.39, 17.2), (96.9, 49.55, 4.74), (169.93, 57.2, 10.35), (46.67, 45.0, 1.29), (72.6, 37.46, -7.14), (104.63, 46.45, 1.45), (152.91, 64.64, 18.53), (122.92, 62.35, 16.87), (101.3, 49.85, 4.61), (130.32, 46.17, 0.52), (41.41, 46.69, 2.93), (78.11, 54.45, 10.15), (129.7, 53.24, 7.78), (40.51, 52.58, 8.81), (125.51, 51.65, 6.12), (143.53, 46.82, 1.36), (182.05, 49.77, 3.3), (56.01, 46.27, 2.01), (120.18, 55.64, 10.35), (88.43, 44.21, -0.23), (137.24, 62.29, 16.79), (121.63, 55.25, 10.03), (82.75, 50.38, 5.61), (84.02, 44.97, 0.22), (48.65, 59.85, 15.23), (141.44, 26.18, -19.91), (97.84, 49.07, 4.45), (95.88, 45.32, 0.6), (102.32, 46.33, 1.01), (94.75, 54.59, 9.41), (179.81, 41.6, -4.75), (110.82, 51.58, 6.32), (63.03, 54.15, 9.68), (111.45, 35.54, -9.78), (120.18, 52.47, 6.55), (76.2, 34.06, -10.33), (117.09, 62.04, 16.98), (113.73, 42.14, -3.27), (69.43, 32.72, -11.84), (56.8, 40.86, -3.04), (76.98, 59.2, 14.5), (38.78, 64.08, 20.69), (122.15, 44.41, -0.96), (181.53, 41.91, -4.65), (99.88, 46.18, 1.64), (99.35, 51.36, 6.0), (119.18, 56.62, 11.14), (217.96, 54.63, 6.96), (56.23, 39.49, -4.79), (63.77, 45.91, 2.05), (133.91, 67.28, 21.83), (156.84, 56.27, 9.87), (108.03, 49.56, 3.88), (129.86, 61.46, 16.1), (44.3, 58.46, 14.46), (114.51, 52.5, 7.36), (86.79, 41.68, -3.05), (51.97, 42.33, -1.91), (129.73, 72.85, 27.45), (162.02, 54.93, 9.31), (120.64, 46.7, 1.0), (143.2, 65.61, 20.05), (121.26, 38.41, -6.91), (17.23, 39.88, -3.43), (29.9, 53.04, 9.59), (149.0, 43.72, -2.08), (100.97, 57.32, 12.41), (80.02, 25.2, -19.13), (94.92, 46.81, 2.16), (75.21, 52.8, 7.66), (82.59, 46.03, 1.45), (60.64, 66.51, 22.11), (181.23, 54.42, 7.32), (78.91, 83.35, 38.68), (92.75, 51.92, 7.02), (99.14, 47.31, 2.2), (77.21, 38.96, -5.95), (102.21, 72.51, 27.6), (62.71, 59.14, 14.89), (78.78, 37.54, -6.52), (86.12, 42.72, -2.08), (165.83, 39.75, -6.27), (90.6, 42.32, -2.37), (90.84, 51.76, 6.29), (-2.04, 33.3, -9.21), (24.77, 54.11, 10.89), (59.82, 51.71, 7.67), (121.28, 50.36, 4.75), (79.72, 43.59, -0.62), (77.87, 55.22, 10.4), (65.99, 48.11, 3.88), (94.02, 44.36, -0.55), (70.74, 43.7, -0.28), (10.8, 43.05, -0.63), (142.57, 44.39, -1.78), (62.29, 56.58, 11.65), (88.26, 47.96, 3.3), (129.8, 37.87, -7.74), (116.56, 62.77, 17.35), (122.24, 51.29, 5.69), (63.43, 52.57, 8.14), (142.12, 39.76, -5.91), (55.04, 29.23, -15.33), (76.41, 55.89, 11.73), (108.13, 62.71, 16.96), (62.18, 55.95, 11.68), (179.26, 48.44, 2.35), (56.25, 37.9, -6.17), (122.15, 58.08, 13.01), (77.08, 53.23, 8.35), (117.9, 54.33, 8.61), (143.08, 57.21, 11.22), (167.98, 46.76, 0.34), (83.86, 57.25, 12.46), (77.97, 53.12, 8.34), (128.07, 44.94, -0.53), (82.07, 56.96, 12.79), (72.99, 25.21, -19.23), (57.98, 53.81, 9.98), (68.33, 35.85, -9.14), (132.17, 48.5, 2.77), (19.47, 47.85, 4.44), (147.34, 52.58, 6.61), (100.34, 45.08, -0.32), (85.04, 31.59, -12.75), (151.19, 59.9, 14.07), (103.78, 48.35, 3.37), (126.06, 58.53, 12.89), (103.59, 36.9, -7.9), (33.77, 37.5, -5.97), (78.27, 38.67, -5.82), (135.77, 64.95, 19.59), (77.38, 36.74, -7.7), (40.74, 61.21, 17.54), (65.46, 59.08, 14.42), (136.7, 38.29, -7.6), (123.29, 47.29, 1.44), (91.77, 66.65, 21.39), (136.64, 51.58, 5.9), (122.24, 53.1, 7.43), (99.36, 42.2, -3.34), (19.39, 30.2, -13.17), (119.16, 46.01, 0.33), (71.67, 35.17, -9.07), (80.5, 36.15, -8.49), (59.64, 59.41, 14.91), (124.65, 54.7, 9.59), (144.57, 49.62, 3.73), (90.91, 59.16, 14.03), (135.95, 75.56, 29.61), (129.19, 45.68, -0.4), (130.17, 56.73, 10.56), (43.71, 40.57, -3.51), (113.46, 74.46, 29.2), (137.66, 45.87, -0.12), (107.56, 48.91, 3.55), (71.88, 49.25, 4.6), (33.33, 52.19, 8.89), (94.92, 53.76, 8.56), (184.4, 48.63, 2.16), (76.01, 48.32, 4.19), (121.74, 55.71, 10.55), (62.41, 54.21, 10.09), (122.28, 47.52, 2.07), (65.0, 61.66, 17.73), (90.21, 61.01, 16.3), (67.32, 36.16, -7.94), (92.53, 46.5, 1.62), (101.47, 51.26, 6.07), (116.03, 63.25, 18.23), (51.2, 32.7, -11.74), (121.92, 49.84, 4.71), (80.82, 61.29, 16.52), (128.68, 37.43, -8.22), (111.25, 44.76, -1.02), (151.81, 54.67, 9.13), (102.2, 54.04, 8.84), (93.34, 68.08, 23.18), (65.66, 44.82, 0.5), (193.12, 56.26, 9.03), (28.34, 56.5, 12.65), (150.53, 75.7, 29.63), (174.83, 61.16, 14.41), (51.23, 51.87, 7.65), (83.91, 48.47, 3.38), (103.87, 44.28, -0.46), (40.97, 58.17, 13.92), (96.17, 57.19, 12.15), (68.43, 45.42, 1.04), (90.45, 51.49, 6.63), (145.22, 58.05, 12.07), (35.08, 46.4, 2.43), (13.33, 26.97, -15.49), (99.66, 42.02, -3.01), (122.55, 30.6, -15.3), (47.56, 42.83, -1.1), (155.72, 49.01, 2.67), (46.04, 38.81, -4.9), (146.5, 50.44, 4.46), (107.73, 37.79, -6.98), (107.9, 44.2, -1.11), (83.84, 48.03, 3.49), (109.28, 62.4, 17.58), (64.57, 51.66, 7.36), (120.34, 40.75, -4.44), (93.7, 36.37, -8.26), (79.99, 42.72, -2.0), (78.53, 53.61, 9.44), (149.88, 46.78, 0.53), (133.53, 58.21, 12.23), (149.8, 56.23, 10.69), (135.71, 41.31, -4.09), (85.17, 76.86, 32.36), (97.86, 55.0, 9.93), (94.26, 53.34, 8.97), (141.46, 36.56, -9.4), (111.66, 48.19, 2.18), (78.74, 53.36, 9.16), (57.0, 49.14, 4.43), (110.36, 47.59, 3.04), (47.5, 55.27, 11.94), (60.64, 27.75, -15.94), (28.17, 45.83, 2.31), (55.39, 53.32, 9.05), (73.59, 53.31, 8.45), (146.09, 53.15, 7.3), (59.34, 61.1, 17.01), (114.42, 34.14, -10.79), (86.65, 50.51, 5.41), (132.15, 53.75, 7.7), (151.0, 61.0, 15.17), (92.98, 46.22, 1.13), (53.12, 54.23, 10.56), (93.65, 44.07, -0.94), (140.76, 56.74, 10.54), (136.79, 42.92, -2.41), (161.57, 52.34, 6.15), (106.39, 83.44, 38.05), (204.02, 78.01, 30.91), (78.91, 56.45, 12.15), (108.15, 63.76, 18.45), (108.62, 55.06, 10.03), (95.99, 61.93, 17.03), (60.67, 32.05, -11.86), (128.16, 45.24, -0.08), (67.67, 70.91, 26.99), (168.28, 68.17, 22.21), (122.09, 51.21, 6.12), (130.72, 57.11, 11.64), (113.2, 50.88, 5.24), (139.9, 43.77, -1.96), (137.85, 62.88, 17.28), (110.04, 46.02, 1.05), (82.89, 56.94, 12.56), (100.42, 51.96, 6.59), (174.53, 51.52, 4.91), (135.75, 59.39, 13.56), (43.28, 44.04, 0.59), (97.72, 52.65, 7.55), (202.23, 59.43, 12.81), (46.44, 65.4, 21.36), (146.68, 51.16, 5.15), (116.12, 57.14, 11.73), (111.06, 48.94, 3.7), (45.89, 30.43, -13.31), (117.17, 53.01, 7.42), (6.65, 34.19, -9.07), (131.55, 61.86, 16.09), (96.63, 42.68, -2.46), (85.7, 71.6, 27.27), (78.11, 60.3, 15.89), (90.77, 54.55, 8.91), (67.09, 41.94, -2.89)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=4.9939，β₁=-0.0201，β₂=1.0007；
SST=40936.09，SSR=40904.65，SSE=31.45；R²=0.9992，调整R²=0.9992；
整体 F=(SSR/2)/(SSE/388)=252357.7353，p=<1e-300；x1,x2 相关 r=0.2533；
β₁: SE=0.0004，t=-53.8808，p=3.18e-182；β₂: SE=0.0014，t=698.9374，p=<1e-300。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=4.9939，β₁=-0.0201，β₂=1.0007；
SST=40936.09，SSR=40904.65，SSE=31.45；R²=0.9992，调整R²=0.9992；
整体 F=(SSR/2)/(SSE/388)=252357.7353，p=<1e-300；x1,x2 相关 r=0.2533；
β₁: SE=0.0004，t=-53.8808，p=3.18e-182；β₂: SE=0.0014，t=698.9374，p=<1e-300。
③ 计算结果：
y=4.9939+-0.0201x1+1.0007x2；R²=0.9992；β₁ p=3.18e-182；β₂ p=<1e-300
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=3.18e-182）；变量 x2 对 y 的影响显著（p=<1e-300）。价负、品牌正。
案例 83. 温度+风速预测用电量
多元线性回归两因素
① 原始数据：
样本 n=710，逐条 (x1, x2, y)：[(29.26, 16.74, 157.64), (15.59, 8.59, 233.28), (14.08, 2.16, 218.82), (28.24, 24.16, 133.01), (19.08, 17.11, 201.55), (25.07, 16.17, 174.52), (26.74, 21.21, 156.29), (15.27, 17.05, 177.02), (22.07, 16.92, 153.66), (15.36, 18.7, 240.9), (31.49, 23.28, 131.34), (12.76, 7.3, 278.12), (23.97, 15.65, 201.1), (11.91, 19.11, 195.46), (29.95, 21.5, 82.54), (19.49, 15.5, 217.02), (19.25, 13.95, 241.17), (4.66, 13.9, 306.04), (21.52, 7.25, 231.88), (22.27, 15.76, 222.42), (14.84, 9.93, 181.03), (14.12, 19.41, 221.58), (19.67, 14.53, 224.72), (19.99, 9.11, 199.35), (5.37, 10.28, 313.72), (15.95, 10.31, 186.64), (33.85, 19.28, 137.83), (26.71, 16.21, 176.53), (32.55, 19.79, 125.52), (26.66, 20.26, 149.32), (21.41, 19.85, 172.84), (17.12, 15.89, 176.79), (15.37, 21.91, 169.15), (13.05, 22.22, 200.52), (20.41, 13.37, 230.44), (20.72, 10.77, 180.93), (28.76, 10.78, 177.23), (9.22, 13.14, 251.38), (14.12, 10.9, 281.33), (30.96, 19.07, 154.57), (16.52, 23.27, 141.99), (31.05, 18.85, 142.38), (38.38, 15.17, 90.96), (19.62, 29.36, 107.67), (24.75, 14.61, 167.27), (-0.88, 10.25, 355.13), (21.01, 17.18, 168.32), (21.42, 5.17, 280.27), (7.38, 10.51, 233.08), (24.37, 2.39, 220.87), (23.73, 11.83, 205.85), (13.29, 14.3, 254.2), (34.88, 3.37, 167.97), (23.28, 17.06, 142.62), (17.64, 5.79, 231.29), (24.13, 23.88, 88.48), (36.24, 13.37, 126.88), (25.98, 14.32, 170.59), (16.19, 13.34, 234.21), (15.6, 10.6, 210.75), (8.49, 8.78, 257.17), (14.88, 10.41, 214.73), (14.95, 14.77, 212.18), (27.66, 14.78, 142.85), (21.07, 10.77, 212.8), (22.44, 11.46, 251.59), (13.33, 17.03, 200.04), (14.33, 18.09, 229.57), (22.19, 23.06, 161.59), (25.18, 18.41, 139.78), (23.0, 17.81, 196.29), (23.86, 18.03, 183.12), (24.56, 14.76, 135.4), (22.08, 16.32, 226.23), (12.27, 16.11, 207.82), (10.78, 16.98, 214.99), (12.95, 13.96, 281.46), (25.13, 16.6, 135.23), (36.3, 18.83, 136.13), (17.71, 14.39, 220.47), (23.23, 11.47, 237.31), (34.21, 22.53, 100.81), (27.86, 14.46, 188.6), (22.48, 24.89, 184.2), (30.34, 14.97, 203.6), (31.29, 17.48, 104.29), (20.26, 18.25, 234.44), (13.49, 13.41, 248.31), (-1.72, 18.34, 317.66), (35.51, 17.2, 66.99), (10.07, 9.01, 255.83), (20.17, 12.81, 227.77), (15.54, 16.16, 172.51), (24.88, 18.48, 127.34), (28.78, 13.25, 151.16), (12.78, 11.74, 218.08), (16.21, 15.63, 176.63), (19.85, 13.11, 208.86), (22.6, 14.43, 183.58), (27.51, 18.13, 160.49), (19.02, 10.55, 168.22), (10.58, 10.34, 244.77), (18.73, 9.96, 222.23), (21.21, 18.35, 210.66), (26.76, 17.84, 142.33), (24.48, 15.74, 178.08), (14.17, 10.03, 252.38), (15.66, 10.35, 288.97), (21.09, 20.94, 158.0), (20.72, 22.82, 114.0), (11.97, 17.78, 197.7), (10.94, 8.51, 256.39), (26.26, 17.4, 195.31), (7.3, 13.04, 246.22), (24.02, 12.26, 209.26), (22.02, 10.02, 176.11), (10.68, 11.99, 277.07), (23.25, 10.24, 204.42), (14.8, 10.34, 250.72), (23.47, 14.21, 196.53), (26.71, 18.77, 149.6), (33.65, 24.01, 85.82), (20.39, 16.82, 237.45), (28.97, 20.16, 162.92), (18.42, 11.18, 255.57), (30.51, 13.19, 185.06), (15.74, 12.06, 249.97), (18.67, 12.61, 227.46), (27.19, 14.22, 184.28), (19.7, 12.16, 231.16), (42.16, 10.13, 83.39), (23.77, 14.87, 124.72), (10.19, 7.01, 317.81), (12.54, 10.32, 224.33), (17.46, 15.3, 159.89), (18.89, 21.23, 173.96), (11.89, 17.21, 296.02), (21.37, 15.98, 198.89), (26.9, 21.1, 116.81), (26.05, 13.85, 180.94), (10.91, 11.11, 248.65), (18.87, 14.22, 194.13), (29.71, 17.7, 156.86), (14.67, 11.91, 245.17), (15.52, 20.2, 143.84), (24.18, 16.02, 194.02), (27.03, 15.63, 132.24), (27.04, 19.45, 131.96), (30.03, 15.8, 201.75), (11.99, 5.88, 223.4), (15.07, 6.39, 220.18), (29.36, 10.47, 188.57), (20.1, 17.81, 148.73), (20.48, 16.16, 222.87), (21.61, 16.84, 187.03), (26.4, 12.37, 172.18), (23.63, 25.66, 149.51), (12.84, 11.24, 262.11), (5.82, 9.06, 239.87), (3.53, 7.7, 294.94), (12.32, 15.6, 248.22), (20.83, 12.37, 220.02), (20.3, 17.03, 199.81), (3.99, 7.46, 331.77), (15.14, 17.46, 204.55), (36.47, 19.61, 98.98), (15.14, 16.51, 196.24), (15.82, 7.72, 265.22), (22.79, 8.46, 225.42), (18.5, 11.83, 226.25), (25.36, 14.33, 192.64), (16.83, 8.1, 227.75), (29.51, 20.28, 127.25), (26.45, 15.71, 192.77), (13.9, 15.77, 181.91), (20.92, 12.62, 226.53), (22.09, 15.34, 190.15), (14.6, 13.28, 222.47), (29.45, 17.8, 142.16), (9.48, 13.38, 292.43), (25.36, 13.71, 142.18), (20.92, 7.63, 207.79), (32.4, 22.71, 166.63), (4.38, 16.72, 245.15), (22.15, 9.81, 180.18), (15.74, 16.64, 207.34), (13.84, 12.71, 246.54), (24.26, 17.33, 162.67), (11.84, 15.49, 263.69), (17.69, 17.25, 254.5), (14.05, 18.3, 213.5), (24.57, 7.58, 199.54), (9.83, 12.69, 247.41), (17.11, 16.21, 210.48), (7.29, 15.84, 209.33), (18.3, 12.89, 301.71), (13.51, 17.28, 247.56), (24.6, 11.05, 167.14), (25.98, 11.39, 159.6), (17.41, 14.25, 201.75), (18.87, 11.16, 236.11), (35.79, 21.65, 104.46), (14.59, 7.38, 218.39), (37.43, 15.25, 140.4), (29.96, 13.56, 108.95), (24.73, 20.65, 206.73), (11.65, 15.13, 266.34), (20.63, 15.73, 179.41), (34.41, 21.54, 123.91), (33.06, 23.36, 145.41), (13.67, 22.05, 221.2), (22.67, 19.44, 181.78), (16.92, 19.23, 221.38), (15.55, 19.3, 184.16), (3.26, 4.9, 316.9), (15.27, 8.56, 257.67), (20.33, 14.21, 233.28), (26.55, 17.86, 245.91), (12.87, 10.01, 242.88), (38.43, 10.55, 115.61), (17.27, 16.61, 231.38), (24.84, 10.98, 163.01), (12.82, 15.57, 238.24), (24.69, 16.45, 144.64), (19.11, 11.86, 221.51), (8.31, 16.32, 247.72), (19.4, 6.81, 200.12), (23.67, 16.72, 168.7), (10.94, 10.96, 223.72), (11.65, 13.0, 230.53), (10.39, 23.25, 223.47), (15.77, 11.7, 224.36), (28.49, 17.67, 191.87), (18.41, 9.51, 273.45), (15.87, 10.2, 240.52), (19.26, 9.98, 225.14), (19.03, 10.55, 184.61), (20.57, 11.21, 205.97), (32.96, 8.59, 148.36), (25.97, 12.49, 168.97), (26.58, 18.88, 176.58), (23.66, 20.79, 107.99), (34.09, 25.67, 40.67), (2.61, 10.58, 270.48), (10.87, 19.91, 233.83), (17.93, 11.3, 207.14), (20.0, 11.15, 193.99), (17.58, 7.91, 194.78), (13.11, 16.94, 246.88), (9.84, 10.01, 244.67), (19.86, 14.67, 173.43), (11.42, 11.41, 247.26), (21.66, 14.53, 198.12), (18.67, 15.29, 229.67), (15.05, 13.63, 245.53), (18.58, 18.08, 171.58), (10.55, 11.48, 263.62), (29.55, 21.87, 110.18), (25.75, 18.77, 152.4), (17.1, 26.9, 132.96), (7.86, 13.69, 269.88), (34.16, 20.41, 78.32), (21.28, 12.96, 150.38), (12.66, 13.52, 199.81), (7.31, 13.42, 313.56), (17.86, -1.18, 300.01), (20.38, 15.54, 218.85), (14.16, 9.81, 247.12), (21.21, 18.16, 216.36), (25.32, 7.72, 166.99), (23.45, 12.68, 188.04), (18.54, 19.3, 180.05), (9.2, 15.98, 265.18), (4.54, 18.36, 239.77), (26.06, 10.86, 186.46), (22.89, 23.91, 154.95), (30.95, 13.68, 138.15), (25.73, 23.11, 129.9), (23.83, 12.87, 191.59), (27.19, 19.83, 116.34), (8.23, 6.5, 282.83), (27.48, 11.56, 198.3), (14.97, 10.96, 199.66), (26.69, 19.8, 170.93), (25.61, 10.5, 170.48), (15.32, 11.65, 234.54), (17.73, 9.66, 191.57), (32.15, 21.9, 137.39), (19.51, 25.25, 173.15), (16.52, 20.2, 164.46), (9.25, 9.48, 208.27), (16.76, 3.28, 231.83), (21.15, 12.15, 193.43), (23.35, 23.15, 164.18), (36.99, 19.31, 112.28), (11.12, 14.12, 217.55), (19.52, 11.45, 204.36), (22.03, 23.55, 136.27), (29.07, 14.9, 94.6), (17.88, 17.8, 198.06), (25.51, 15.14, 143.12), (27.54, 16.29, 179.55), (21.34, 15.96, 175.27), (17.13, 15.38, 222.63), (30.28, 11.74, 143.83), (6.94, 17.96, 289.06), (14.56, 15.8, 211.55), (-1.19, 9.84, 298.08), (27.21, 25.76, 144.7), (32.15, 11.55, 185.91), (10.98, 7.35, 225.39), (15.11, 9.87, 271.8), (12.84, 12.08, 269.77), (22.96, 16.95, 215.4), (21.37, 13.5, 167.57), (29.3, 19.55, 158.09), (16.98, 11.37, 253.95), (23.68, 9.39, 217.68), (7.79, 18.54, 255.85), (23.22, 12.88, 172.69), (15.37, 12.19, 226.42), (25.65, 11.49, 165.16), (21.01, 23.73, 157.06), (3.64, 8.91, 273.98), (6.47, 18.39, 251.53), (35.19, 18.69, 80.97), (24.73, 15.82, 191.84), (16.95, 17.67, 199.47), (25.44, 17.7, 190.92), (19.02, 5.49, 233.88), (5.77, 12.81, 322.28), (31.03, 24.52, 163.82), (9.78, 10.83, 214.49), (21.72, 13.69, 200.0), (17.2, 19.97, 115.94), (23.53, 17.75, 154.71), (11.53, 9.31, 300.31), (18.73, 17.96, 161.42), (34.64, 12.83, 193.73), (21.21, 22.84, 146.19), (11.68, 13.87, 233.28), (17.61, 14.63, 227.33), (4.47, 7.05, 283.83), (28.07, 11.57, 172.65), (26.38, 20.48, 135.82), (23.0, 10.52, 195.23), (23.42, 16.98, 184.27), (18.41, 25.64, 146.07), (16.45, 10.91, 217.08), (18.89, 9.19, 124.04), (12.03, 21.35, 247.87), (20.85, 14.35, 262.13), (21.35, 18.79, 159.66), (25.79, 14.72, 164.27), (18.05, 5.48, 206.77), (34.12, 16.57, 138.38), (12.41, 17.11, 237.68), (23.78, 19.57, 155.69), (20.42, 21.65, 107.12), (19.81, 16.27, 250.97), (21.61, 15.46, 178.08), (35.0, 14.65, 145.24), (13.14, 13.22, 239.63), (26.97, 16.01, 162.04), (21.84, 12.97, 197.89), (22.34, 16.74, 184.7), (21.15, 21.91, 193.08), (23.93, 12.14, 159.0), (22.8, 16.24, 205.91), (28.08, 21.64, 200.3), (18.51, 25.6, 166.02), (18.34, 18.52, 205.12), (10.0, 10.24, 247.9), (19.73, 9.42, 199.96), (22.22, 16.13, 205.68), (15.86, 12.51, 247.59), (23.19, 16.38, 139.17), (17.19, 16.84, 248.47), (13.83, 5.96, 243.07), (25.34, 9.55, 191.25), (22.65, 21.74, 215.03), (4.9, 8.06, 332.78), (15.13, 15.98, 199.58), (12.34, 21.94, 220.94), (24.01, 17.94, 154.83), (26.92, 25.8, 158.25), (12.22, 14.17, 261.18), (21.41, 20.9, 198.35), (18.92, 22.63, 139.23), (14.54, 16.51, 250.98), (10.44, 15.91, 271.27), (23.61, 16.68, 161.34), (7.82, 8.86, 267.81), (11.48, 9.33, 301.22), (18.56, 14.12, 206.76), (23.19, 13.28, 201.56), (25.16, 13.18, 194.61), (17.75, 16.88, 224.91), (25.19, 8.15, 220.39), (38.4, 27.84, 56.34), (23.39, 16.48, 190.24), (21.61, 14.64, 147.26), (27.26, 24.65, 119.08), (17.44, 10.14, 219.03), (23.89, 13.85, 167.63), (39.19, 24.79, 50.2), (19.55, 9.19, 218.16), (23.06, 11.03, 151.78), (24.24, 12.16, 170.77), (25.57, 20.45, 195.4), (24.63, 14.73, 226.58), (23.01, 18.35, 174.14), (19.35, 15.54, 172.2), (24.33, 15.14, 150.8), (18.76, 13.96, 186.87), (19.54, 11.85, 285.86), (32.41, 17.38, 84.86), (21.68, 15.18, 193.24), (18.88, 15.11, 211.78), (38.8, 20.11, 60.97), (10.96, 19.12, 241.79), (17.12, 15.05, 257.11), (11.79, 13.2, 205.91), (15.19, 22.37, 233.13), (22.45, 20.94, 168.15), (25.51, 22.0, 179.82), (13.25, 17.25, 221.75), (23.93, 17.19, 211.27), (16.62, 20.93, 142.36), (27.35, 15.82, 182.74), (35.4, 22.38, 141.8), (34.1, 10.35, 175.88), (25.53, 8.72, 165.85), (14.14, 6.0, 233.45), (16.3, 18.12, 208.39), (8.37, 4.0, 272.84), (22.89, 16.19, 139.16), (24.67, 22.18, 108.56), (14.32, 9.35, 288.63), (22.41, 12.04, 222.42), (15.54, 10.16, 243.45), (18.57, 15.35, 174.18), (12.3, 7.07, 266.24), (3.03, 8.63, 317.94), (30.67, 14.82, 120.33), (27.79, 13.09, 153.04), (16.66, 14.24, 204.56), (12.48, 14.67, 300.14), (18.02, 11.16, 214.12), (9.2, 9.68, 269.91), (33.11, 22.41, 72.67), (9.75, 6.91, 281.98), (19.67, 23.97, 160.67), (24.25, 20.43, 171.54), (15.25, 12.44, 280.86), (14.26, 11.8, 265.38), (7.68, 14.13, 288.86), (28.55, 12.17, 169.72), (14.08, 16.99, 245.32), (31.87, 16.97, 138.7), (30.25, 23.68, 108.98), (27.3, 15.16, 160.36), (12.6, 13.95, 210.78), (9.96, 20.05, 245.29), (21.24, 17.26, 195.9), (14.07, 11.36, 264.87), (28.44, 17.06, 244.11), (14.72, 21.08, 206.74), (6.34, 15.76, 257.31), (25.55, 17.24, 221.49), (30.63, 16.95, 170.97), (28.27, 13.38, 161.56), (34.31, 16.12, 104.78), (29.1, 14.44, 163.44), (20.47, 17.3, 217.82), (31.65, 21.26, 127.14), (11.52, 11.17, 203.62), (17.57, 10.35, 176.58), (22.08, 18.64, 223.14), (9.54, 16.06, 264.17), (9.03, 19.56, 256.91), (21.26, 15.6, 157.32), (18.26, 18.08, 186.67), (18.21, 11.53, 197.83), (15.23, 15.87, 244.1), (11.74, 12.7, 205.09), (14.17, 16.92, 265.77), (36.22, 12.95, 151.95), (16.79, 6.41, 207.81), (15.14, 21.32, 188.77), (24.1, 7.0, 213.24), (24.32, 19.94, 178.48), (25.51, 18.5, 177.92), (17.56, 17.61, 208.23), (-3.09, 17.48, 349.87), (24.07, 13.9, 177.22), (13.96, 12.06, 188.5), (23.29, 14.57, 151.32), (18.32, 20.75, 198.53), (34.86, 21.45, 69.46), (25.5, 18.23, 189.17), (20.18, 13.55, 210.4), (15.78, 13.94, 303.88), (38.6, 22.1, 145.51), (27.57, 21.28, 136.61), (32.29, 16.47, 145.22), (30.08, 21.99, 109.25), (22.22, 23.42, 145.19), (12.7, 8.07, 215.9), (18.68, 18.45, 206.28), (9.63, 15.89, 264.99), (20.58, 16.34, 169.14), (15.25, 17.18, 172.66), (18.83, 15.94, 166.46), (17.84, 10.47, 270.33), (19.29, 8.23, 194.14), (17.1, 8.41, 235.02), (5.64, 15.53, 352.85), (14.51, 15.72, 212.63), (18.65, 16.68, 168.95), (17.84, 9.67, 222.18), (35.29, 26.9, 40.86), (25.28, 10.92, 209.3), (28.22, 11.12, 153.83), (31.77, 8.22, 171.86), (13.65, 12.44, 257.72), (24.09, 21.06, 142.25), (15.35, 14.17, 203.28), (17.55, 15.95, 177.07), (38.22, 17.73, 117.31), (26.73, 18.76, 124.48), (16.14, 13.86, 200.82), (21.26, 8.64, 242.81), (24.44, 21.33, 198.89), (26.94, 12.3, 147.04), (21.24, 16.78, 236.12), (23.66, 22.9, 170.53), (26.13, 18.64, 136.01), (15.1, 13.09, 287.15), (25.42, 15.55, 189.64), (24.6, 12.95, 220.93), (24.34, 11.61, 214.56), (12.23, 19.31, 192.97), (19.33, 8.5, 143.7), (27.7, 13.38, 142.37), (12.61, 9.27, 244.36), (25.49, 9.61, 181.18), (26.63, 12.47, 168.92), (24.98, 9.84, 164.23), (30.1, 15.41, 237.13), (15.34, 13.23, 199.89), (12.52, 16.4, 197.59), (33.19, 18.98, 151.51), (32.04, 11.27, 138.11), (10.08, 15.27, 258.98), (7.1, 15.42, 211.92), (23.77, 7.84, 235.77), (23.03, 14.88, 185.61), (24.12, 13.22, 191.13), (13.08, 12.12, 227.67), (19.13, 15.36, 225.5), (21.25, 18.98, 195.82), (13.68, 13.53, 218.71), (25.35, 23.89, 169.01), (12.79, 13.78, 243.74), (19.0, 14.65, 203.01), (4.85, 8.18, 306.73), (11.72, 7.28, 223.31), (21.84, 14.68, 183.52), (3.28, 11.75, 320.94), (14.69, 9.52, 239.15), (18.19, 10.74, 191.53), (11.94, 15.25, 171.54), (25.55, 24.53, 130.65), (9.21, 15.89, 258.9), (19.12, 19.5, 202.9), (30.36, 18.14, 138.51), (26.22, 20.01, 110.6), (32.83, 22.39, 102.57), (19.53, 19.33, 232.87), (27.48, 20.41, 151.53), (11.63, 12.12, 292.61), (27.16, 22.7, 140.75), (25.63, 19.01, 122.35), (20.9, 22.19, 167.33), (23.24, 10.04, 146.58), (22.66, 16.62, 233.72), (13.87, 10.51, 254.08), (13.41, 15.87, 256.11), (29.02, 13.76, 162.15), (12.64, 8.7, 267.74), (14.4, 13.21, 251.7), (18.08, 14.81, 179.2), (3.21, 12.21, 271.28), (22.06, 15.47, 170.85), (26.34, 10.66, 151.81), (18.86, 19.55, 199.72), (21.27, 17.03, 205.37), (15.8, 5.14, 256.01), (12.31, 13.44, 295.46), (26.88, 16.2, 200.69), (20.16, 17.52, 131.37), (4.07, 12.72, 287.48), (25.4, 6.9, 217.64), (18.99, 15.1, 203.77), (14.89, 8.34, 223.47), (30.29, 16.66, 170.19), (23.64, 7.9, 192.34), (9.98, 11.84, 219.41), (16.65, 13.92, 223.92), (6.95, 23.61, 266.46), (30.26, 18.93, 143.69), (12.41, 12.89, 261.86), (32.24, 8.82, 154.5), (25.54, 15.84, 160.1), (12.06, 12.06, 242.16), (27.34, 16.78, 155.38), (28.17, 9.3, 218.77), (22.01, 19.88, 163.89), (11.37, 21.24, 203.74), (19.26, 15.69, 186.92), (8.34, 20.35, 294.85), (21.03, 16.61, 162.84), (27.87, 12.49, 140.83), (12.36, 13.97, 210.17), (16.66, 19.15, 201.92), (17.38, 15.9, 161.54), (16.05, 7.62, 293.4), (22.03, 18.61, 225.65), (11.91, 15.2, 229.53), (14.85, 3.28, 267.19), (15.42, 11.15, 241.62), (8.95, 16.79, 278.28), (29.95, 18.7, 207.49), (15.65, 17.78, 280.9), (30.71, 11.36, 120.66), (18.16, 23.11, 227.62), (9.47, 7.79, 351.53), (14.27, 19.41, 195.77), (11.62, 18.4, 260.78), (27.9, 13.24, 142.17), (22.62, 21.04, 196.95), (39.24, 22.98, 95.37), (13.71, 22.78, 185.32), (19.06, 18.87, 180.75), (15.75, 22.63, 225.8), (28.69, 10.29, 179.34), (9.22, 16.49, 237.05), (24.34, 15.4, 181.04), (26.84, 20.94, 129.05), (19.26, 9.26, 187.46), (28.87, 17.37, 92.42), (20.44, 18.32, 189.23), (18.29, 2.47, 262.21), (14.99, 22.28, 138.15), (20.17, 18.06, 187.5), (20.89, 15.84, 192.88), (19.4, 11.47, 263.68), (17.38, 10.76, 215.31), (21.04, 13.51, 197.85), (20.63, 24.65, 120.33), (28.15, 21.82, 93.38), (27.9, 15.28, 157.62), (23.67, 13.19, 197.09), (20.75, 10.97, 211.81), (17.22, 22.54, 187.17), (17.84, 12.57, 210.17), (19.49, 13.48, 250.64), (16.04, 17.75, 174.73), (13.48, 3.76, 207.23), (32.57, 12.76, 164.05), (11.82, 12.18, 289.48), (16.74, 16.2, 210.44), (30.87, 19.43, 114.59), (25.38, 12.12, 145.06), (28.71, 11.06, 181.03), (27.09, 13.96, 208.08), (17.18, 14.76, 220.57), (26.38, 23.65, 177.5), (17.64, 11.08, 192.29), (21.2, 13.83, 239.91), (6.15, 11.51, 270.88), (16.64, 14.88, 246.61), (19.69, 22.5, 150.57), (13.96, 13.09, 200.31), (23.3, 16.94, 193.32), (24.18, 11.54, 172.18), (28.15, 25.1, 117.84), (35.23, 18.94, 90.04), (20.3, 6.0, 265.29), (33.34, 20.36, 78.11), (27.94, 11.84, 146.58), (1.74, 9.1, 330.41), (20.48, 18.48, 164.46), (29.78, 15.31, 222.46), (21.53, 19.72, 203.8), (22.55, 20.36, 166.36), (35.57, 17.27, 112.7), (31.01, 22.27, 122.59), (8.96, 18.89, 201.83), (14.5, 10.23, 244.02), (12.67, 21.13, 212.58), (23.61, 8.62, 226.12), (20.0, 16.4, 193.7), (7.06, 19.9, 216.11), (11.91, 11.34, 270.78), (13.87, 10.85, 256.77), (27.3, 5.6, 193.85), (13.07, 14.46, 219.04), (23.9, 21.17, 152.23)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=199.2919，β₁=-4.8141，β₂=-3.2327；
SST=2003519.42，SSR=1384838.85，SSE=618680.58；R²=0.6912，调整R²=0.6903；
整体 F=(SSR/2)/(SSE/707)=791.2654，p=3.98e-181；x1,x2 相关 r=0.2934；
β₁: SE=0.1510，t=-31.8805，p=6.33e-139；β₂: SE=0.2414，t=-13.3939，p=-1.23e+10。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=199.2919，β₁=-4.8141，β₂=-3.2327；
SST=2003519.42，SSR=1384838.85，SSE=618680.58；R²=0.6912，调整R²=0.6903；
整体 F=(SSR/2)/(SSE/707)=791.2654，p=3.98e-181；x1,x2 相关 r=0.2934；
β₁: SE=0.1510，t=-31.8805，p=6.33e-139；β₂: SE=0.2414，t=-13.3939，p=-1.23e+10。
③ 计算结果：
y=199.2919+-4.8141x1+-3.2327x2；R²=0.6912；β₁ p=6.33e-139；β₂ p=-1.23e+10
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=6.33e-139）；变量 x2 对 y 的影响显著（p=-1.23e+10）。两因素。
案例 84. 经验+团队规模预测交付速度
多元线性回归两正向
① 原始数据：
样本 n=707，逐条 (x1, x2, y)：[(2.39, 6.7, 1.92), (8.9, 9.43, 7.16), (2.83, 8.7, 2.76), (14.87, 15.63, 17.48), (13.01, 12.67, 18.85), (6.63, 8.72, 6.72), (11.97, 9.59, 13.54), (11.57, 10.77, 10.13), (5.22, 11.59, 9.37), (5.11, 8.42, 3.07), (3.88, 7.67, 0.06), (4.24, 4.43, 4.6), (0.21, 6.15, 1.1), (8.9, 11.67, 12.39), (8.22, 7.85, 9.05), (7.52, 13.16, 11.32), (7.11, 12.25, 12.11), (11.53, 9.44, 13.37), (4.9, 8.33, 2.97), (2.34, 6.0, 6.21), (4.55, 7.41, 1.08), (7.13, 6.23, 5.72), (7.94, 8.68, 9.29), (9.16, 9.46, 14.45), (16.97, 9.5, 19.42), (4.55, 5.94, 9.63), (4.55, 7.8, 6.28), (7.95, 10.62, 8.74), (9.08, 8.17, 11.89), (12.62, 16.62, 18.92), (8.11, 9.11, 9.78), (1.19, 6.49, 1.06), (10.7, 11.65, 15.78), (10.89, 10.11, 14.8), (1.67, 10.42, 5.7), (9.21, 10.79, 13.23), (4.92, 12.67, 10.01), (4.3, 11.81, 9.39), (10.13, 9.43, 6.21), (6.09, 9.42, 2.04), (2.01, 7.15, 4.43), (6.93, 7.09, 6.63), (9.03, 9.19, 8.09), (-0.2, 9.05, 2.47), (10.52, 11.3, 17.11), (-0.55, 5.15, -2.42), (4.86, 8.07, 3.43), (10.81, 13.87, 18.67), (6.28, 12.74, 9.7), (5.73, 11.25, 3.03), (3.23, 8.78, 6.29), (14.94, 13.62, 16.12), (6.75, 8.88, 9.94), (9.87, 11.77, 10.11), (17.02, 13.72, 22.2), (-1.13, 2.1, -1.18), (5.46, 10.19, 10.4), (10.63, 11.49, 16.91), (5.55, 12.93, 12.71), (11.4, 11.2, 12.89), (9.37, 9.74, 8.94), (11.53, 8.23, 11.19), (7.41, 5.22, 9.33), (9.46, 12.7, 14.74), (8.43, 8.58, 13.82), (12.58, 16.66, 16.91), (10.26, 13.79, 13.84), (3.52, 4.76, 2.05), (9.06, 6.81, 16.15), (5.2, 9.74, 3.97), (5.89, 11.04, 7.05), (9.99, 8.19, 10.47), (12.65, 10.21, 16.0), (9.15, 11.38, 12.56), (6.29, 10.1, 10.64), (12.81, 10.45, 10.88), (8.78, 12.13, 13.63), (3.49, 4.42, 2.98), (9.41, 9.07, 13.51), (9.33, 11.49, 15.1), (0.3, 7.7, 3.59), (7.01, 10.79, 10.66), (6.25, 13.65, 7.31), (5.13, 13.77, 9.9), (9.32, 12.76, 14.1), (6.46, 11.19, 11.91), (9.0, 13.04, 15.18), (6.76, 6.06, -1.18), (6.47, 10.5, 9.7), (10.24, 9.7, 9.3), (3.55, 11.65, 16.58), (5.93, 3.35, 5.97), (8.56, 10.16, 13.07), (-0.11, 2.68, -1.32), (7.14, 9.14, 12.33), (11.76, 11.64, 17.99), (9.48, 10.72, 9.93), (9.95, 8.04, 15.61), (2.85, 9.42, 11.52), (8.5, 9.26, 13.77), (1.29, 10.52, -1.59), (11.15, 12.63, 13.41), (8.84, 13.72, 12.16), (11.53, 7.44, 8.71), (7.54, 12.17, 16.66), (10.23, 12.32, 13.47), (7.73, 11.34, 12.95), (4.68, 8.51, 11.92), (7.62, 7.23, 8.47), (13.19, 13.3, 20.91), (0.56, 13.63, 1.16), (6.14, 12.2, 8.47), (2.82, 4.89, -1.4), (5.0, 4.51, 3.3), (1.4, 8.42, 0.73), (11.11, 12.43, 17.2), (8.64, 10.52, 12.98), (8.72, 5.97, 10.57), (11.43, 13.07, 15.78), (2.11, 10.13, -0.65), (9.75, 9.65, 10.61), (10.66, 17.6, 22.24), (9.25, 10.36, 9.6), (8.44, 14.24, 6.93), (3.91, 7.4, 1.1), (9.58, 13.17, 19.95), (11.75, 10.54, 8.66), (11.02, 11.12, 9.65), (5.25, 10.58, 6.31), (11.08, 13.33, 12.68), (8.71, 9.5, 7.9), (13.77, 11.73, 14.17), (7.7, 10.29, 9.6), (8.58, 7.14, 10.42), (5.03, 12.6, 6.75), (10.03, 9.44, 8.63), (3.89, 7.49, 6.36), (6.35, 6.19, 2.42), (5.21, 8.19, 4.85), (3.08, 4.26, 4.6), (5.02, 8.83, 6.11), (11.53, 11.72, 15.92), (8.6, 6.39, 8.72), (8.57, 9.34, 10.39), (12.39, 10.25, 15.11), (4.57, 12.38, 8.76), (7.79, 13.38, 13.96), (6.67, 12.91, 9.18), (10.74, 8.17, 15.32), (16.03, 12.7, 19.87), (6.23, 12.08, 13.15), (6.95, 8.97, 9.35), (2.75, 7.66, 4.4), (15.0, 12.2, 14.59), (5.56, 5.38, 2.02), (16.37, 13.2, 22.91), (8.6, 13.02, 6.02), (10.49, 12.1, 13.73), (4.58, 10.33, 6.33), (6.95, 12.19, 9.16), (15.36, 14.49, 22.87), (3.63, 2.88, 3.36), (15.3, 14.32, 20.79), (4.42, 4.86, 2.37), (3.1, 10.96, 1.36), (14.19, 12.88, 18.25), (10.93, 14.68, 13.54), (12.33, 12.93, 17.79), (5.52, 10.6, 3.6), (10.31, 12.7, 12.01), (10.07, 9.53, 12.65), (5.07, 10.54, 7.68), (2.65, 3.56, 0.1), (13.37, 12.02, 17.33), (5.06, 7.54, 7.51), (9.25, 7.2, 12.67), (8.59, 16.59, 16.1), (10.96, 11.37, 13.29), (12.84, 10.9, 11.7), (4.2, 11.41, 6.1), (2.55, 3.22, -5.55), (15.33, 11.73, 20.52), (7.59, 9.0, 13.99), (11.14, 14.89, 14.1), (18.37, 10.72, 16.25), (14.2, 14.96, 20.18), (5.76, 12.09, 5.53), (8.0, 12.01, 10.82), (9.05, 12.8, 12.96), (4.4, 9.88, 7.08), (12.02, 11.38, 15.28), (3.79, 8.7, 6.94), (5.19, 4.81, 6.9), (13.89, 11.2, 13.08), (3.32, 12.94, 7.22), (6.2, 9.5, 7.85), (8.7, 9.26, 7.36), (1.32, 5.37, 1.61), (2.68, 8.14, 4.63), (9.17, 15.93, 18.14), (8.41, 9.59, 11.12), (12.12, 9.48, 17.04), (10.06, 10.81, 11.78), (4.85, 10.71, 6.83), (13.97, 13.3, 16.4), (5.68, 5.67, 7.62), (4.66, 13.69, 5.52), (5.66, 11.8, 10.91), (10.21, 12.64, 16.89), (0.85, 3.09, -1.0), (11.24, 10.52, 5.5), (11.03, 7.2, 14.06), (10.82, 9.83, 7.23), (6.29, 5.93, 11.34), (6.33, 8.6, 7.43), (14.46, 14.01, 13.72), (6.58, 11.89, 11.36), (0.84, 10.15, 11.96), (6.95, 5.88, 3.56), (3.7, 4.18, -0.75), (0.68, 5.02, -1.87), (10.51, 11.01, 12.21), (12.47, 11.52, 10.06), (10.94, 7.76, 13.39), (12.76, 13.48, 13.5), (0.91, 8.42, 2.18), (4.94, 11.46, 12.06), (9.01, 8.72, 7.38), (14.53, 10.75, 19.3), (8.59, 9.25, 7.66), (9.55, 8.39, 7.23), (13.54, 12.91, 20.13), (7.86, 12.1, 11.81), (-3.99, 5.28, -7.25), (8.32, 11.16, 13.14), (4.36, 12.36, 12.36), (6.99, 4.37, 4.69), (9.58, 4.89, 9.04), (4.12, 11.4, 7.11), (8.96, 12.03, 12.22), (5.6, 7.21, 2.94), (4.58, 13.37, 9.23), (8.61, 12.18, 14.9), (6.62, 10.32, 9.77), (5.68, 6.79, 5.52), (8.76, 8.74, 9.03), (8.21, 6.15, 9.28), (4.25, 5.4, 4.59), (10.52, 12.57, 17.87), (9.04, 9.94, 7.34), (8.6, 9.74, 12.92), (9.22, 12.59, 9.92), (7.38, 9.67, 15.73), (10.29, 9.13, 9.94), (3.92, 8.16, 2.48), (8.12, 12.2, 11.2), (10.01, 7.1, 12.19), (5.27, 9.89, 9.96), (9.45, 10.65, 4.29), (8.89, 10.51, 9.46), (2.38, 11.21, 4.16), (10.19, 9.89, 12.03), (5.16, 6.73, 3.47), (10.97, 8.63, 14.04), (6.82, 13.97, 16.45), (5.6, 10.19, 6.82), (14.09, 8.29, 13.73), (8.45, 6.51, 5.56), (16.06, 9.18, 18.91), (7.03, 13.75, 6.83), (9.32, 7.92, 11.81), (11.99, 14.26, 14.23), (10.13, 10.6, 14.99), (4.64, 7.42, 11.43), (9.65, 12.69, 15.62), (11.48, 10.08, 9.77), (10.14, 11.3, 13.53), (9.64, 9.33, -0.14), (5.7, 7.16, 5.71), (10.13, 12.16, 13.51), (12.99, 14.52, 18.08), (1.57, 8.31, -1.04), (2.73, 9.67, 5.59), (8.58, 10.74, 11.16), (4.31, 9.32, 4.76), (8.86, 9.52, 10.47), (11.07, 5.86, 11.42), (7.35, 11.33, 12.45), (9.62, 14.7, 12.21), (7.11, 8.21, 9.74), (8.19, 12.07, 12.87), (10.01, 10.76, 12.51), (9.29, 9.65, 14.44), (6.99, 12.36, 6.32), (5.43, 10.34, 2.6), (10.51, 10.29, 9.79), (9.14, 9.16, 10.44), (2.7, 4.73, 0.31), (4.49, 10.11, 6.19), (3.72, 9.31, 4.22), (11.2, 8.73, 9.86), (1.2, 8.35, 6.41), (10.38, 12.83, 10.32), (8.34, 8.59, 10.24), (2.72, 4.83, -2.03), (12.22, 9.52, 11.99), (5.85, 8.23, 5.58), (13.5, 20.31, 23.23), (12.01, 8.21, 16.4), (2.29, 10.79, 2.22), (20.59, 14.01, 22.57), (9.35, 10.25, 5.99), (4.11, 10.2, 2.94), (3.43, 5.42, 0.43), (8.36, 7.07, 13.58), (10.53, 12.31, 13.64), (4.8, 9.82, 7.15), (6.87, 9.31, 12.59), (14.83, 10.24, 14.43), (6.49, 9.27, 5.87), (13.22, 10.85, 10.17), (9.33, 15.97, 15.5), (4.86, 10.97, 8.51), (9.63, 12.17, 17.68), (11.99, 12.08, 17.07), (6.18, 9.86, 8.55), (10.7, 13.28, 11.92), (11.44, 5.69, 8.2), (5.2, 4.04, -2.43), (6.49, 6.07, 11.25), (6.84, 9.55, 5.37), (5.87, 10.53, 11.57), (6.95, 8.48, 5.33), (3.05, 6.98, 3.75), (2.54, 3.46, -1.23), (9.65, 13.37, 12.17), (5.29, 7.17, 6.85), (12.19, 13.02, 17.64), (5.66, 11.02, 3.48), (8.14, 15.28, 14.06), (14.8, 10.94, 14.17), (6.46, 16.53, 9.64), (7.63, 7.61, 5.59), (5.1, 6.55, 3.58), (11.34, 13.15, 16.42), (10.93, 15.01, 21.14), (7.56, 7.27, 10.88), (9.55, 11.36, 9.31), (8.2, 11.74, 11.78), (10.37, 13.68, 12.41), (7.71, 9.66, 11.7), (9.65, 13.39, 14.04), (6.78, 10.84, 9.74), (0.07, 4.52, 4.63), (5.22, 9.31, 7.32), (18.4, 15.19, 26.39), (10.12, 7.64, 16.97), (12.99, 14.16, 15.61), (13.18, 13.5, 13.69), (10.02, 14.27, 16.52), (13.28, 12.74, 20.46), (8.2, 15.34, 14.76), (2.38, 6.35, 5.44), (11.52, 11.56, 16.7), (5.15, 12.06, 12.53), (16.48, 10.53, 22.74), (-2.02, 7.97, 0.98), (12.4, 15.11, 16.29), (8.1, 9.34, 6.71), (3.45, 7.62, 6.96), (2.2, 6.61, -0.86), (2.89, 4.63, -1.89), (11.78, 10.26, 13.05), (7.26, 7.17, 8.32), (-0.54, 7.13, 4.42), (17.45, 13.51, 24.06), (6.85, 12.07, 9.24), (6.29, 9.13, 7.38), (1.15, 9.7, 5.82), (14.71, 10.11, 15.32), (9.66, 13.46, 9.71), (9.98, 8.43, 9.17), (7.51, 9.68, 8.72), (1.31, 2.14, 4.09), (11.44, 16.38, 16.88), (5.35, 8.95, 9.28), (6.06, 8.71, 5.4), (5.47, 6.03, 7.44), (6.06, 6.02, 5.01), (7.84, 10.96, 6.96), (-1.39, 9.26, 1.09), (7.37, 8.7, 7.37), (8.01, 13.16, 16.77), (14.1, 12.23, 19.95), (2.34, 4.43, -4.86), (8.9, 11.58, 11.97), (3.07, 12.47, 5.78), (7.59, 14.61, 12.38), (1.96, 7.72, -0.79), (14.61, 12.28, 17.2), (6.04, 10.19, 8.66), (13.23, 11.72, 15.09), (5.25, 11.24, 12.08), (10.97, 15.66, 20.25), (13.2, 9.01, 16.15), (15.79, 13.18, 15.59), (9.53, 10.79, 11.03), (4.78, 7.19, 0.22), (10.74, 12.6, 16.02), (11.66, 12.84, 11.32), (8.97, 9.32, 12.47), (11.26, 14.22, 18.38), (6.24, 8.68, 11.94), (8.62, 10.63, 14.2), (8.64, 13.38, 13.98), (11.3, 12.75, 11.87), (7.65, 11.28, 7.24), (6.09, 11.67, 10.28), (9.48, 11.68, 13.85), (6.08, 7.24, 7.66), (3.01, 13.39, 4.72), (14.47, 8.01, 14.29), (9.02, 12.76, 8.02), (11.47, 10.2, 13.67), (9.35, 14.35, 12.02), (9.06, 14.48, 16.6), (7.91, 13.34, 9.46), (1.57, 7.52, 3.67), (6.19, 9.35, 9.21), (5.51, 12.32, 5.56), (2.3, 8.15, 9.84), (2.97, 11.11, 6.13), (13.02, 12.08, 13.34), (10.54, 11.3, 20.2), (5.33, 4.95, 8.33), (3.96, 7.57, 8.32), (13.14, 9.92, 15.92), (1.9, 9.38, -0.94), (14.22, 9.75, 15.6), (12.57, 11.57, 11.17), (12.83, 15.13, 15.75), (10.14, 10.75, 12.67), (6.18, 8.72, 6.84), (7.28, 9.05, 11.27), (2.83, 5.39, 8.23), (15.14, 12.85, 19.7), (13.53, 13.19, 16.97), (2.83, 3.97, 6.13), (6.06, 12.0, 5.13), (4.4, 9.56, 5.41), (12.14, 14.71, 13.68), (6.19, 4.86, 3.19), (12.42, 13.16, 20.98), (15.7, 17.57, 21.06), (8.83, 8.41, 10.67), (12.53, 11.5, 20.22), (4.11, 8.96, 4.73), (9.01, 6.34, 13.15), (11.81, 11.94, 13.95), (3.31, 9.67, 3.22), (5.9, 8.09, 4.74), (11.01, 9.7, 13.8), (11.67, 11.08, 12.17), (14.0, 9.56, 14.54), (7.59, 12.79, 9.91), (18.03, 12.55, 21.16), (5.8, 12.71, 10.64), (11.92, 11.62, 20.22), (12.58, 10.28, 17.06), (9.55, 10.49, 10.74), (4.63, 4.38, 3.97), (6.74, 7.62, 10.86), (2.23, 8.72, 1.66), (6.29, 6.28, 6.9), (8.33, 10.75, 12.73), (14.36, 12.76, 14.18), (3.98, 8.82, 4.82), (9.59, 11.82, 10.27), (0.3, 4.87, -2.17), (9.03, 5.98, 13.85), (9.66, 5.55, 14.22), (8.55, 13.83, 14.09), (4.76, 7.74, 8.2), (5.06, 12.27, 6.86), (5.82, 5.39, 1.94), (13.52, 12.89, 10.88), (4.12, 9.84, 7.15), (8.98, 4.74, 9.44), (14.34, 12.85, 18.46), (9.91, 10.26, 14.82), (5.54, 8.26, 9.23), (6.39, 7.68, 8.57), (-0.79, 10.12, 2.44), (9.32, 5.65, 12.53), (9.02, 8.87, 11.87), (11.57, 13.89, 10.9), (9.89, 9.36, 8.96), (12.4, 11.63, 13.1), (15.28, 14.33, 19.26), (6.56, 6.84, 6.85), (8.75, 9.36, 12.73), (8.38, 12.71, 10.24), (12.54, 12.46, 17.53), (6.98, 9.21, 9.92), (9.37, 11.58, 12.78), (7.01, 10.09, 4.91), (-1.99, 5.59, -5.14), (4.52, 8.78, 5.21), (8.04, 9.4, 11.66), (17.05, 8.09, 14.46), (3.44, 4.01, 1.29), (5.11, 10.83, 3.5), (7.86, 9.53, 8.99), (6.48, 5.75, 12.61), (13.53, 15.33, 19.23), (3.52, 11.34, 5.9), (8.73, 12.45, 12.88), (10.12, 13.21, 11.62), (8.12, 9.17, 11.78), (10.07, 13.01, 14.18), (5.96, 10.35, 9.1), (15.03, 13.5, 14.74), (10.84, 11.87, 16.85), (9.2, 9.64, 18.1), (7.53, 6.84, 9.88), (12.73, 13.09, 12.54), (6.52, 13.95, 11.33), (12.23, 10.14, 9.82), (8.21, 13.16, 13.13), (14.22, 8.93, 19.78), (0.63, 4.58, -1.1), (0.3, 5.17, 2.36), (6.52, 5.86, 3.87), (6.65, 13.42, 6.04), (10.7, 7.93, 3.56), (12.25, 10.12, 16.55), (9.96, 12.66, 16.42), (4.76, 5.13, 1.16), (6.31, 4.92, 0.49), (6.62, 6.93, 8.49), (10.91, 15.55, 13.36), (8.78, 9.47, 9.01), (8.76, 7.09, 11.48), (6.43, 10.67, 8.37), (9.71, 10.21, 12.58), (9.89, 12.18, 13.33), (11.24, 9.4, 14.78), (7.85, 8.44, 13.88), (12.12, 10.56, 11.96), (10.07, 9.1, 14.24), (5.82, 13.62, 6.55), (4.41, 9.98, -0.42), (4.16, 9.23, 5.9), (3.77, 8.69, 8.18), (6.27, 11.03, 7.54), (4.45, 9.92, 6.59), (6.64, 13.8, 11.63), (9.24, 3.02, 6.86), (7.47, 7.87, 8.13), (6.4, 10.17, 9.99), (12.04, 11.25, 14.35), (4.87, 6.72, 6.03), (12.66, 13.15, 14.74), (8.32, 8.0, 10.74), (13.24, 9.9, 16.4), (-2.09, 9.21, -6.76), (3.84, 8.21, 7.29), (1.94, 6.73, 5.57), (10.26, 8.57, 14.23), (5.66, 8.63, 4.26), (7.25, 9.82, 8.2), (1.07, 7.04, 2.89), (9.65, 8.48, 10.45), (6.84, 7.53, 5.87), (11.23, 13.62, 15.35), (13.44, 11.29, 17.99), (7.67, 11.75, 4.46), (7.74, 10.75, 12.67), (16.21, 13.07, 18.34), (6.49, 7.19, 3.98), (4.28, 5.0, 5.7), (13.93, 15.54, 19.4), (8.5, 9.74, 10.1), (7.96, 12.17, 10.48), (7.19, 7.8, 9.17), (8.01, 11.36, 12.55), (6.77, 7.39, 6.94), (1.56, 7.88, 3.54), (11.46, 14.31, 16.77), (5.76, 7.99, 7.68), (13.03, 17.81, 22.21), (3.99, 8.0, 11.38), (4.03, 6.82, 8.81), (9.53, 10.93, 12.91), (6.87, 10.38, 6.72), (4.55, 10.25, 4.19), (1.38, 8.46, 2.41), (5.86, 10.06, 12.77), (4.22, 4.4, 2.9), (8.55, 10.98, 10.56), (1.94, 9.77, 5.4), (5.47, 12.21, 9.19), (8.07, 14.66, 9.11), (8.04, 11.0, 13.93), (8.3, 8.3, 6.31), (4.54, 7.33, 10.83), (6.28, 8.6, 8.05), (10.21, 7.6, 12.77), (6.72, 7.7, 6.88), (13.99, 12.36, 24.54), (12.53, 13.16, 18.88), (13.29, 14.34, 17.36), (5.02, 8.22, 3.76), (5.89, 8.25, 9.28), (6.87, 6.22, 6.1), (9.53, 13.67, 11.17), (12.84, 9.97, 21.78), (5.21, 10.84, -0.97), (8.88, 9.35, 14.5), (1.4, 13.67, 1.63), (5.54, 8.6, 4.72), (7.16, 10.13, 6.95), (10.47, 15.24, 18.12), (0.18, 10.74, 5.65), (9.86, 11.71, 11.58), (17.42, 14.7, 25.62), (3.36, 11.4, 4.11), (7.61, 10.91, 15.5), (13.35, 7.27, 9.28), (9.78, 11.06, 12.13), (9.98, 10.9, 9.43), (9.95, 6.86, 9.09), (5.01, 10.74, 7.86), (10.67, 7.4, 9.32), (3.61, 10.03, 6.4), (12.08, 11.46, 12.81), (7.86, 8.71, 8.76), (6.69, 13.51, 11.42), (8.82, 10.18, 13.36), (8.8, 12.08, 11.46), (11.56, 11.9, 12.83), (5.2, 7.55, 7.08), (13.16, 12.35, 21.48), (9.08, 12.68, 5.69), (7.91, 13.73, 9.51), (4.62, 6.14, 5.67), (9.29, 11.69, 11.26), (6.8, 11.19, 6.06), (9.92, 15.05, 15.44), (0.67, 9.57, 6.63), (8.24, 9.43, 11.78), (9.43, 11.03, 9.93), (11.8, 16.29, 15.15), (7.97, 10.45, 11.35), (5.97, 6.74, 2.5), (14.51, 14.76, 22.28), (8.36, 9.98, 12.68), (5.3, 8.66, 9.72), (13.61, 12.01, 14.66), (6.27, 8.82, 10.13), (10.72, 11.42, 8.82), (2.95, 12.04, 3.7), (2.74, 7.71, -2.99), (17.73, 19.13, 25.35), (3.36, 9.0, -0.34), (8.27, 7.4, 8.83), (15.71, 14.76, 18.95), (4.29, 6.71, 1.86), (8.16, 9.69, 11.3), (9.26, 11.12, 12.69), (4.12, 10.25, 8.26), (18.71, 13.69, 18.41), (14.87, 10.8, 13.59), (11.5, 10.36, 17.48), (5.88, 9.95, 10.17), (13.2, 15.12, 16.53), (2.47, 5.0, 3.87), (5.18, 7.48, 10.04), (3.09, 8.55, 5.72), (8.04, 13.69, 15.81), (2.03, 5.34, 4.36), (8.21, 9.3, 6.7), (9.58, 14.24, 11.08), (4.53, 9.2, 10.53), (1.29, 11.24, 3.57), (8.87, 12.19, 10.0), (7.07, 11.22, 12.64), (12.04, 11.89, 14.06), (11.6, 13.01, 21.27), (10.07, 16.52, 8.58), (0.46, 7.95, 0.43), (13.07, 11.64, 12.2), (11.61, 11.4, 17.78), (7.83, 11.56, 10.66), (4.84, 7.59, 5.77), (2.14, 10.81, 7.73), (11.77, 11.75, 16.46), (8.76, 13.72, 16.77), (4.28, 4.01, 9.42), (10.75, 8.38, 10.26), (8.69, 9.67, 9.68), (10.91, 10.77, 10.83), (6.86, 10.84, 12.54), (6.75, 5.16, 8.92), (9.25, 9.12, 10.2), (12.06, 9.46, 14.41), (8.06, 9.73, 13.51)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=10.0824，β₁=0.9756，β₂=0.5238；
SST=22495.17，SSR=16258.26，SSE=6236.91；R²=0.7227，调整R²=0.7220；
整体 F=(SSR/2)/(SSE/704)=917.5870，p=7.83e-197；x1,x2 相关 r=0.5217；
β₁: SE=0.0335，t=29.1046，p=7.10e-123；β₂: SE=0.0450，t=11.6324，p=-9.22e+04。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=10.0824，β₁=0.9756，β₂=0.5238；
SST=22495.17，SSR=16258.26，SSE=6236.91；R²=0.7227，调整R²=0.7220；
整体 F=(SSR/2)/(SSE/704)=917.5870，p=7.83e-197；x1,x2 相关 r=0.5217；
β₁: SE=0.0335，t=29.1046，p=7.10e-123；β₂: SE=0.0450，t=11.6324，p=-9.22e+04。
③ 计算结果：
y=10.0824+0.9756x1+0.5238x2；R²=0.7227；β₁ p=7.10e-123；β₂ p=-9.22e+04
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=7.10e-123）；变量 x2 对 y 的影响显著（p=-9.22e+04）。两正向。
案例 85. 价格+广告预测利润
多元线性回归价正、广告正
① 原始数据：
样本 n=780，逐条 (x1, x2, y)：[(13.06, 46.94, 2.98), (40.44, 66.06, 17.4), (14.81, 29.5, -4.05), (15.5, 68.48, 4.95), (30.98, 90.15, 21.33), (19.13, 46.67, 5.08), (18.98, 75.79, 11.95), (12.78, 59.02, 4.55), (26.66, 56.23, 12.93), (36.17, 61.3, 16.49), (23.65, 40.89, -2.38), (24.07, 56.67, 5.48), (16.21, 52.45, 4.14), (0.45, 19.14, -13.19), (12.57, 56.96, 2.43), (15.95, 53.65, 3.98), (17.92, 84.09, 13.34), (21.38, 51.5, 9.26), (22.0, 42.22, 7.97), (19.87, 51.25, 9.7), (27.26, 78.6, 21.07), (18.37, 61.22, 5.71), (12.53, 31.0, -6.61), (20.5, 49.08, 5.78), (-2.01, 21.64, -16.98), (18.18, 52.72, 2.99), (25.72, 63.82, 13.81), (7.61, 59.39, -3.36), (20.6, 58.95, 11.55), (13.31, 2.45, -7.47), (24.01, 68.1, 13.13), (30.11, 95.32, 24.03), (17.49, 62.85, 9.56), (24.4, 49.48, 2.28), (17.53, 43.8, 1.4), (18.9, 37.09, -1.57), (23.58, 39.18, 4.53), (29.99, 83.43, 17.86), (24.9, 39.02, 2.87), (28.93, 41.34, 13.44), (14.12, 30.82, -10.66), (16.95, 66.95, 11.01), (22.55, 39.26, -0.64), (23.57, 41.65, 8.01), (29.16, 68.32, 10.66), (25.15, 50.29, 5.86), (17.84, 52.11, 8.07), (32.12, 41.62, 11.45), (20.06, 48.52, 1.6), (20.22, 35.58, 3.63), (20.45, 48.05, 9.76), (37.43, 57.38, 19.7), (21.01, 64.34, 8.08), (16.97, 38.22, 1.35), (3.24, 10.02, -13.92), (27.74, 61.78, 9.02), (18.83, 43.9, 5.11), (26.74, 58.53, 8.2), (20.73, 54.08, 2.29), (16.65, 42.27, -3.02), (20.46, 26.93, -2.1), (21.33, 58.16, 8.16), (34.75, 47.93, 12.45), (21.34, 49.14, 4.43), (30.06, 61.53, 17.46), (27.42, 62.06, 9.73), (11.23, 17.96, -8.75), (10.77, 39.19, -5.95), (21.43, 50.73, 10.65), (14.11, 70.64, 7.6), (22.34, 73.7, 15.08), (41.86, 43.54, 12.84), (43.46, 99.71, 37.35), (21.52, 43.21, 4.99), (33.75, 66.57, 15.18), (26.13, 68.27, 13.36), (23.29, 65.8, 13.08), (30.24, 80.01, 15.55), (12.01, 35.19, -3.15), (22.07, 58.21, 3.58), (27.32, 58.1, 6.68), (14.66, 41.39, -4.53), (32.14, 54.69, 11.92), (23.54, 50.7, 4.53), (14.44, 64.14, 0.65), (27.01, 26.48, 0.59), (24.57, 52.97, 7.81), (12.5, 42.85, -2.11), (28.28, 42.41, 7.03), (31.29, 50.21, 12.58), (32.41, 49.97, 13.09), (19.64, 53.54, 3.74), (24.26, 46.12, 9.41), (14.02, 66.25, 5.23), (29.45, 61.24, 9.42), (19.42, 37.69, -0.13), (11.86, 52.21, 1.98), (13.09, 57.84, 6.04), (20.19, 53.84, 8.52), (32.25, 61.41, 18.28), (6.19, 59.06, 4.44), (29.74, 62.27, 12.87), (19.91, 53.97, 9.09), (8.94, 36.28, -3.92), (7.42, 35.54, -7.79), (29.18, 33.96, 2.85), (24.05, 64.1, 9.58), (7.32, 33.47, -6.76), (16.29, 46.4, 6.68), (30.34, 58.84, 12.72), (17.52, 34.14, 2.0), (20.6, 55.34, 8.45), (37.45, 52.03, 16.73), (23.16, 61.98, 11.15), (28.27, 61.79, 9.38), (26.62, 35.01, 6.84), (22.99, 46.68, 5.17), (17.4, 38.33, 7.22), (19.29, 36.15, 2.21), (26.1, 64.1, 14.31), (19.24, 19.21, -7.33), (24.08, 58.57, 4.61), (31.4, 56.23, 14.85), (24.87, 31.41, 4.02), (21.28, 78.6, 12.09), (29.27, 62.68, 13.07), (24.64, 41.67, 2.6), (17.53, 48.94, 5.52), (22.03, 57.86, 13.29), (25.73, 46.41, 10.1), (13.44, 28.48, 0.19), (9.83, 46.87, -2.73), (19.13, 45.64, 5.85), (36.21, 40.11, 8.7), (11.57, 28.77, -5.8), (15.49, 33.77, -0.13), (12.87, 40.06, -2.58), (23.12, 35.73, 4.35), (13.93, 45.89, -1.75), (18.45, 67.6, 7.65), (24.66, 47.89, 7.19), (39.65, 75.34, 23.62), (9.25, 43.36, -4.69), (17.42, 57.22, 9.59), (18.93, 61.05, 5.85), (22.65, 62.63, 8.47), (34.32, 69.12, 20.77), (22.17, 38.63, 5.39), (27.35, 18.97, 5.69), (11.76, 15.87, -9.27), (15.83, 25.67, -2.84), (24.41, 64.95, 10.82), (28.74, 62.85, 11.95), (19.64, 67.0, 11.51), (7.97, 48.74, -1.6), (33.24, 72.99, 12.33), (10.43, 30.52, -8.84), (20.22, 49.2, 5.46), (26.04, 48.94, 10.44), (25.73, 22.18, -3.15), (17.02, 34.97, -0.92), (3.9, 29.55, -13.45), (16.2, 50.64, 1.46), (20.0, 68.22, 7.94), (14.6, 39.97, -3.05), (19.79, 59.71, 5.48), (16.82, 35.72, -3.88), (5.15, 37.36, -9.72), (20.54, 37.45, -0.18), (31.65, 40.82, 9.28), (17.53, 21.65, -15.94), (18.67, 50.38, 4.85), (20.93, 46.32, 4.71), (9.71, 45.84, -5.2), (17.13, 61.99, 9.33), (0.94, 41.9, -4.94), (24.23, 56.32, 8.78), (15.01, 60.12, 6.77), (19.26, 67.59, 10.5), (35.72, 42.64, 4.24), (6.89, 28.19, -8.81), (25.77, 49.79, 5.56), (23.21, 50.86, 5.33), (24.08, 44.87, 16.88), (20.77, 47.12, 5.71), (19.11, 63.73, 13.2), (9.39, 36.03, -5.86), (13.9, 51.71, 3.03), (31.41, 46.63, 8.43), (27.84, 36.25, 3.15), (14.99, 48.05, 0.56), (11.57, 37.99, -1.52), (41.96, 38.24, 12.3), (24.8, 53.53, 2.66), (29.44, 55.75, 10.27), (17.9, 47.4, 7.41), (33.53, 81.9, 26.24), (33.91, 71.07, 14.1), (16.15, 49.62, 3.02), (26.34, 55.32, 9.38), (18.22, 34.27, -2.28), (34.49, 81.02, 21.95), (13.31, 71.98, 12.12), (7.7, 32.57, -9.2), (9.52, 49.1, -1.77), (17.61, 12.27, -8.91), (8.88, 30.02, -8.44), (20.74, 34.29, 0.1), (21.75, 40.15, 0.34), (0.17, 50.09, -4.73), (20.08, 51.97, 5.81), (11.85, 51.69, 7.56), (19.15, 51.35, 2.63), (11.03, 37.2, -5.58), (13.31, 69.01, 1.14), (13.46, 33.86, -4.52), (36.97, 55.13, 13.97), (22.63, 62.75, 7.77), (16.46, 34.09, -4.26), (32.24, 59.55, 11.31), (13.78, 47.99, -0.25), (38.5, 68.62, 20.42), (15.75, 22.0, -8.45), (20.62, 53.97, 2.32), (18.87, 54.14, 1.65), (21.81, 62.49, 12.82), (14.9, 68.82, 6.27), (8.6, 33.86, -4.27), (16.67, 54.26, 3.12), (34.89, 42.61, 9.71), (21.31, 50.53, 2.83), (20.82, 65.94, 15.96), (32.61, 94.47, 21.31), (11.42, 50.95, -1.26), (19.76, 35.6, 0.37), (16.28, 47.45, 1.63), (19.24, 53.04, 0.33), (21.62, 61.55, 8.2), (20.56, 54.66, 6.79), (21.9, 55.09, 6.81), (18.08, 43.49, 4.58), (29.41, 67.0, 12.72), (27.89, 61.3, 10.02), (20.63, 50.51, 0.62), (12.16, 62.53, 3.18), (23.75, 53.65, 0.96), (18.68, 58.64, 5.01), (7.07, 41.04, -3.1), (6.95, 51.75, 3.87), (16.93, 26.61, -4.42), (8.78, 17.59, -12.84), (13.63, 41.52, -5.78), (29.53, 58.55, 11.01), (19.19, 38.36, 0.7), (6.4, 39.41, -5.77), (21.29, 66.47, 12.41), (23.18, 36.41, 4.2), (23.89, 52.14, 10.94), (17.17, 51.31, 2.2), (22.25, 31.22, 0.98), (22.0, 80.3, 12.57), (16.62, 44.37, -0.23), (16.89, 30.99, -3.24), (21.89, 40.29, 3.86), (30.62, 63.86, 16.04), (13.74, 35.4, -3.65), (19.25, 45.66, 5.86), (15.14, 67.73, 8.94), (30.21, 65.04, 13.92), (7.7, 40.37, -4.74), (27.19, 39.57, 3.1), (17.91, 54.22, 5.0), (18.18, 51.04, 6.73), (1.7, 34.75, -11.88), (12.37, 39.99, -2.7), (21.07, 43.44, 5.44), (15.67, 60.61, 5.81), (5.51, 43.86, 3.45), (25.43, 39.15, 3.82), (25.25, 61.52, 6.88), (26.71, 52.72, 6.67), (19.06, 63.79, 10.15), (10.21, 47.23, -1.15), (19.4, 58.6, 10.33), (19.09, 48.88, 3.02), (28.22, 53.46, 6.0), (24.22, 69.4, 14.24), (28.48, 58.86, 8.99), (12.17, 48.14, 0.98), (8.75, 30.5, -7.06), (38.57, 94.04, 27.58), (13.83, 47.39, -1.53), (20.1, 50.97, 2.52), (19.79, 31.24, -4.01), (14.85, 27.65, -5.34), (14.77, 36.83, -0.49), (6.63, 31.79, -7.41), (28.99, 54.25, 11.38), (12.32, 41.29, 0.11), (31.52, 79.53, 18.2), (25.01, 41.99, 8.18), (22.87, 59.24, 5.83), (4.59, 40.42, -3.72), (20.43, 33.94, -1.95), (9.35, 12.21, -13.38), (26.21, 66.96, 13.41), (9.0, 51.14, -2.41), (22.23, 53.97, 7.69), (19.06, 37.12, -4.33), (20.83, 43.09, -1.14), (28.48, 46.52, 7.76), (32.42, 71.49, 16.17), (20.77, 58.85, 6.89), (11.84, 33.81, 0.8), (19.74, 27.33, -3.57), (11.78, 41.97, 0.89), (19.96, 56.09, 9.49), (30.43, 52.12, 9.88), (4.36, 33.12, -9.93), (17.77, 37.59, 0.39), (17.26, 70.08, 10.82), (24.83, 57.56, 5.5), (14.69, 53.25, 6.45), (14.47, 84.88, 10.77), (14.88, 20.66, -6.33), (8.96, 51.11, 0.33), (15.95, 56.36, 7.3), (18.73, 60.33, 5.23), (27.57, 66.71, 12.21), (27.85, 65.41, 11.69), (22.64, 33.89, 0.79), (26.45, 56.55, 9.26), (9.1, 43.91, -5.86), (11.54, 57.07, 3.21), (19.12, 50.47, 7.55), (4.3, 33.56, -5.89), (15.09, 28.76, -1.93), (7.68, 52.84, 0.37), (27.44, 62.56, 8.04), (21.88, 53.83, 6.47), (13.33, 33.33, 0.86), (25.76, 56.65, 8.91), (18.54, 42.72, 3.0), (31.82, 47.81, 9.37), (8.19, 34.17, -8.76), (20.24, 33.06, -1.07), (16.19, 55.11, 0.89), (14.12, 33.72, -7.59), (10.75, 32.04, -4.34), (17.61, 25.85, -0.93), (28.42, 39.02, 4.62), (15.7, 66.87, 10.76), (26.01, 55.66, 12.75), (17.88, 34.64, -2.38), (16.23, 36.41, -3.93), (32.95, 58.27, 11.98), (35.82, 62.64, 14.34), (15.56, 9.94, -10.59), (19.29, 31.65, -4.53), (17.45, 45.82, 4.24), (2.31, 56.27, 2.95), (20.43, 41.84, 2.38), (35.17, 63.05, 16.6), (19.22, 57.6, 7.0), (23.71, 24.92, -1.17), (28.36, 59.22, 9.54), (22.36, 70.57, 8.37), (37.18, 51.22, 19.08), (42.65, 56.9, 22.39), (23.17, 52.02, 8.48), (21.0, 42.32, 4.33), (16.73, 17.95, -5.28), (14.2, 30.01, -4.42), (17.19, 26.51, -1.94), (10.04, 51.13, -5.37), (24.53, 74.31, 17.32), (19.32, 40.03, 1.77), (24.45, 59.51, 11.39), (8.69, 40.82, -3.54), (28.69, 47.74, 7.88), (24.66, 45.11, 3.12), (17.75, 51.68, 8.98), (14.61, 70.54, 4.12), (22.21, 35.41, 0.16), (19.75, 47.43, 1.79), (23.15, 45.63, 3.9), (9.52, 58.49, -3.94), (8.31, 36.66, -1.01), (9.21, 38.09, -5.54), (20.28, 52.37, 4.82), (24.85, 50.55, 5.6), (31.71, 49.5, 9.19), (5.51, 25.97, -12.52), (19.94, 41.41, 3.79), (31.3, 59.22, 13.38), (25.51, 68.82, 15.58), (25.0, 58.88, 10.69), (15.48, 52.23, 9.24), (4.2, 30.81, -13.27), (27.65, 63.95, 15.65), (20.54, 49.3, 11.7), (11.3, 50.38, -3.84), (32.84, 90.22, 21.4), (33.28, 75.46, 16.62), (20.09, 59.41, 8.68), (20.23, 50.54, 0.32), (13.52, 53.25, 1.26), (13.8, 53.07, 4.5), (-4.36, 8.99, -19.76), (5.48, 44.52, -5.48), (14.92, 42.81, -0.91), (11.01, 40.64, -2.49), (23.66, 66.14, 3.87), (20.15, 15.78, -8.22), (28.13, 68.21, 19.81), (17.46, 43.6, -1.19), (28.35, 64.44, 14.03), (20.81, 44.6, 2.45), (8.13, 50.08, -3.23), (24.14, 46.94, 8.75), (20.64, 57.23, 6.07), (19.74, 23.79, -4.11), (23.71, 59.65, 6.71), (16.62, 60.26, 8.46), (21.16, 59.84, 4.84), (12.48, 38.06, -6.91), (18.96, 56.52, 5.1), (21.08, 79.94, 21.89), (20.07, 51.26, 1.92), (25.03, 56.86, 10.03), (22.14, 54.32, 8.73), (5.12, 48.62, 0.24), (16.63, 33.77, -1.47), (26.5, 44.45, 6.46), (18.79, 32.92, 4.05), (25.13, 21.94, -2.29), (30.35, 65.55, 10.92), (28.71, 73.51, 13.96), (26.88, 54.07, 7.9), (28.58, 55.55, 14.84), (28.05, 38.47, -2.38), (25.24, 39.78, 11.47), (12.61, 57.18, 6.48), (27.33, 73.66, 17.58), (18.11, 17.08, -3.41), (26.63, 79.09, 17.06), (46.15, 61.91, 21.97), (25.65, 47.6, 5.14), (25.38, 79.43, 16.42), (13.8, 42.97, -6.23), (23.78, 62.66, 14.39), (23.26, 58.76, 2.78), (2.43, 44.19, -7.95), (21.45, 43.04, 4.32), (23.75, 67.27, 13.65), (18.44, 50.06, 9.59), (4.0, 40.58, -7.59), (25.42, 37.34, 1.25), (25.92, 49.26, 7.7), (27.47, 46.46, 5.9), (25.16, 53.56, 5.86), (-10.9, 41.91, -16.31), (7.88, 39.02, -0.18), (11.98, 45.89, -2.87), (25.7, 63.06, 17.06), (17.52, 62.02, 9.36), (23.13, 61.32, 12.62), (13.24, 22.75, -1.16), (22.08, 52.43, 6.36), (14.59, 64.16, 7.48), (15.5, 52.6, -2.09), (18.48, 60.57, 4.43), (28.62, 48.35, 7.01), (25.75, 70.7, 9.76), (22.71, 65.44, 14.12), (29.16, 68.87, 14.57), (31.75, 62.95, 9.47), (14.47, 41.45, -4.06), (17.56, 64.17, 8.21), (11.89, 32.83, -7.05), (20.12, 55.87, 6.84), (24.15, 49.48, 5.25), (16.51, 39.87, 0.63), (11.43, 48.15, 1.38), (32.94, 61.83, 14.27), (30.12, 37.51, 3.59), (21.99, 41.57, 7.63), (26.34, 51.59, 3.46), (16.1, 83.13, 13.75), (19.28, 43.99, 4.66), (8.09, 76.56, -1.53), (21.85, 57.17, 11.92), (17.83, 48.12, 6.98), (26.3, 38.64, 3.33), (14.92, 48.88, 3.72), (24.03, 62.29, 7.99), (31.54, 38.8, 6.9), (21.95, 52.18, 5.84), (36.98, 67.56, 17.38), (18.77, 55.04, 5.5), (24.29, 58.36, 9.41), (24.98, 38.37, 6.36), (10.37, 48.27, 1.02), (25.63, 51.75, 6.07), (11.38, 52.9, 2.21), (3.89, 24.44, -16.47), (17.48, 46.04, 5.37), (16.15, 72.68, 12.62), (26.48, 47.24, 2.77), (12.88, 55.08, -0.3), (18.15, 37.96, -2.92), (20.16, 60.9, 8.85), (11.81, 42.84, 3.47), (20.05, 50.77, 5.15), (26.19, 85.59, 19.53), (24.04, 87.28, 17.54), (30.66, 69.42, 10.54), (23.14, 49.92, 10.16), (27.15, 55.14, 6.44), (17.08, 43.77, 4.23), (12.51, 61.81, 5.48), (13.67, 61.74, 4.58), (18.77, 52.81, 9.39), (15.44, 35.02, -6.66), (23.32, 34.37, 2.83), (19.42, 44.56, -1.19), (13.58, 34.09, -1.25), (22.79, 38.12, -0.87), (26.02, 78.47, 14.07), (23.52, 57.63, 5.61), (6.05, 51.29, -5.85), (10.53, 29.05, -4.91), (9.92, 46.04, 2.02), (15.45, 47.92, 4.11), (19.74, 29.76, -4.2), (18.6, 23.49, -5.61), (14.06, 47.83, 1.12), (18.49, 56.1, 6.86), (16.4, 60.73, 5.26), (24.56, 58.17, 8.06), (16.37, 47.57, 4.81), (20.96, 59.64, 7.78), (28.22, 71.69, 10.36), (21.28, 49.63, 8.08), (23.38, 42.17, 5.49), (12.1, 40.42, -1.14), (10.75, 37.22, -3.16), (15.66, 57.17, -2.56), (17.49, 47.8, 1.1), (10.96, 29.85, 0.44), (26.36, 51.73, 6.44), (18.8, 42.39, 3.19), (26.0, 58.95, 10.38), (15.48, 36.88, -11.25), (31.25, 68.8, 19.27), (38.11, 51.34, 17.75), (24.33, 77.04, 7.99), (24.2, 55.79, 6.17), (19.21, 77.56, 7.5), (30.93, 60.77, 20.41), (27.44, 49.85, 12.67), (14.99, 39.47, -3.31), (25.98, 65.52, 11.99), (26.15, 57.85, 4.77), (13.27, 37.07, -3.31), (33.58, 62.7, 19.63), (15.76, 36.92, -2.65), (10.29, 42.01, -4.49), (13.49, 42.87, 0.45), (22.18, 72.0, 12.64), (32.68, 61.4, 16.78), (17.94, 77.34, 21.7), (14.23, 46.92, -1.24), (8.45, 56.98, 0.93), (9.79, 35.7, -5.77), (25.09, 31.83, -1.12), (16.51, 24.42, -1.57), (12.6, 14.61, -10.21), (15.89, 31.12, 2.16), (24.9, 63.22, 8.34), (14.42, 37.86, -0.18), (12.18, 26.79, -3.75), (16.59, 37.03, -2.65), (23.6, 43.15, 7.57), (19.82, 67.93, 13.44), (11.39, 26.74, -10.91), (10.61, 43.75, -1.74), (14.66, 43.49, 1.39), (11.9, 30.68, -1.67), (17.86, 70.26, 14.09), (17.75, 47.87, 5.05), (23.26, 53.73, 2.19), (22.32, 52.31, 3.56), (21.04, 46.29, 12.15), (26.6, 56.49, 5.83), (6.04, 41.59, -4.63), (16.73, 74.39, 13.29), (28.19, 54.87, 10.19), (14.21, 46.73, 4.46), (15.78, 53.48, 4.65), (11.6, 40.43, -0.72), (21.25, 94.84, 17.96), (22.9, 61.24, 7.65), (14.56, 55.09, 3.47), (23.52, 70.39, 14.1), (27.81, 59.29, 9.65), (19.29, 37.08, -0.98), (32.48, 49.06, 12.75), (25.4, 45.27, 7.77), (19.39, 62.05, 10.39), (20.38, 22.66, -4.99), (7.25, 9.5, -13.91), (23.55, 63.69, 13.37), (14.87, 69.51, 12.04), (14.43, 44.02, -0.77), (19.83, 64.78, 9.21), (29.13, 49.18, 11.47), (18.41, 62.54, 8.24), (22.84, 53.79, 12.62), (21.83, 51.81, 5.19), (12.95, 59.68, 6.1), (12.85, 8.19, -12.57), (10.68, 40.82, -2.42), (21.32, 54.42, 2.08), (38.85, 50.49, 18.78), (14.28, 43.23, 1.93), (22.4, 64.56, 8.7), (35.09, 58.87, 18.55), (21.17, 46.56, 1.62), (1.85, 36.47, -8.25), (15.14, 50.6, 5.73), (15.38, 75.23, 11.38), (20.51, 29.42, -0.76), (19.23, 35.44, 1.74), (25.55, 59.19, 10.03), (31.93, 67.71, 19.55), (14.3, 44.95, 2.49), (29.19, 63.56, 15.78), (22.28, 58.51, 9.34), (28.47, 54.16, 11.61), (22.05, 68.86, 11.92), (17.83, 35.38, 1.57), (14.46, 39.73, 3.11), (21.1, 52.8, 8.98), (22.19, 46.77, 5.31), (20.79, 74.98, 13.39), (9.06, 64.29, 7.84), (23.47, 59.63, 5.07), (24.13, 33.36, -0.32), (12.45, 42.25, 1.33), (24.02, 27.64, -2.87), (22.46, 54.67, 7.75), (4.01, 43.58, -7.02), (11.94, 70.49, 8.55), (22.04, 63.17, 15.03), (19.59, 56.14, 10.43), (6.89, 38.74, -0.76), (13.33, 41.96, 2.13), (34.8, 52.57, 10.09), (14.26, 48.74, 0.48), (28.81, 52.78, 11.68), (32.19, 53.83, 9.78), (9.61, 77.62, 15.96), (19.48, 53.49, 4.16), (15.06, 42.4, 1.55), (19.5, 38.45, -1.79), (14.19, 38.24, -0.02), (19.82, 41.11, 5.07), (14.36, 49.95, 5.21), (29.72, 62.0, 14.04), (10.85, 29.66, -5.09), (12.1, 53.73, 2.46), (7.12, 22.35, -7.54), (25.21, 43.41, -0.42), (21.21, 57.43, 13.93), (4.65, 30.29, -7.09), (29.47, 53.55, 5.16), (33.69, 82.58, 23.61), (26.84, 32.78, 3.43), (22.03, 42.79, 6.23), (19.89, 59.95, 6.76), (19.65, 35.16, 3.33), (13.16, 28.18, -4.06), (4.05, 17.67, -12.4), (14.72, 73.67, 10.53), (19.46, 68.45, 11.0), (30.61, 54.02, 18.21), (18.59, 53.55, 4.82), (32.24, 58.57, 15.14), (31.37, 67.4, 14.09), (27.98, 52.27, 9.97), (26.56, 83.3, 16.51), (14.51, 51.86, 5.2), (28.72, 42.02, 8.16), (33.07, 58.27, 13.31), (9.13, 45.33, -4.27), (13.55, 37.75, 1.49), (22.43, 56.06, 7.63), (24.64, 43.39, 4.4), (26.16, 68.02, 13.33), (20.42, 51.88, -0.7), (22.35, 26.13, 1.05), (14.93, 56.11, 0.42), (33.06, 68.62, 13.95), (23.01, 44.21, 3.05), (9.23, 21.22, -8.89), (29.56, 71.95, 17.94), (20.69, 38.3, -2.26), (39.83, 92.5, 28.1), (25.96, 67.02, 14.81), (17.03, 42.34, 1.63), (-0.18, 40.96, -3.99), (28.94, 52.84, 9.87), (16.71, 42.47, -4.22), (16.35, 16.83, -5.98), (5.2, 52.36, -7.06), (22.24, 57.72, 4.55), (8.62, 26.21, -7.31), (25.87, 64.29, 10.55), (22.43, 58.51, 8.99), (29.47, 54.78, 10.52), (19.69, 37.19, -1.36), (26.02, 66.93, 14.39), (20.52, 61.13, 5.37), (16.35, 75.17, 4.67), (12.59, 22.28, -3.82), (21.87, 41.95, -1.08), (34.2, 59.53, 9.28), (4.99, 40.77, -3.8), (18.18, 46.77, 4.15), (13.65, 25.54, -11.09), (29.83, 88.93, 22.22), (6.13, 11.86, -10.21), (6.85, 40.75, -6.75), (26.42, 55.58, 14.76), (21.63, 49.54, 5.48), (21.12, 41.09, 5.08), (32.23, 55.78, 9.58), (27.63, 52.56, 5.16), (26.28, 67.22, 18.05), (12.46, 16.06, -6.57), (13.97, 36.2, 2.0), (21.76, 39.52, 3.93), (29.29, 64.33, 13.19), (25.93, 69.55, 11.58), (27.7, 46.46, 6.73), (21.35, 15.61, -9.13), (8.12, 32.54, -1.25), (16.5, 38.09, -2.67), (16.13, 32.28, -1.06), (22.99, 34.23, 2.92), (10.82, 57.79, -1.5), (9.68, 38.05, -11.02), (16.09, 51.51, 5.88), (24.76, 32.45, -0.37), (31.95, 89.88, 19.16), (25.34, 33.48, 8.46), (24.95, 43.03, 2.68), (16.44, 48.51, 1.45), (16.43, 67.87, 8.78), (28.92, 80.82, 22.28), (14.95, 21.97, -5.59), (38.62, 75.21, 20.57), (27.75, 57.52, 12.42), (25.43, 51.06, 11.56), (23.89, 51.48, 4.41), (24.94, 30.53, 1.61), (24.43, 62.2, 8.02), (27.82, 69.92, 18.46), (19.2, 51.42, 4.87), (32.07, 69.85, 17.98), (8.13, 35.19, -6.0), (17.62, 52.53, 0.33), (35.42, 50.84, 12.18), (16.45, 41.2, 3.25), (31.1, 54.62, 11.88), (17.6, 19.14, -6.42), (33.25, 46.55, 11.47), (10.0, 35.52, -2.68), (9.33, 21.52, -12.16)]
② 分析过程：
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=4.7667，β₁=0.4983，β₂=0.3020；
SST=49375.00，SSR=42457.83，SSE=6917.17；R²=0.8599，调整R²=0.8595；
整体 F=(SSR/2)/(SSE/777)=2384.6264，p=<1e-300；x1,x2 相关 r=0.4492；
β₁: SE=0.0149，t=33.5191，p=4.53e-153；β₂: SE=0.0078，t=38.8902，p=1.66e-184。
模型：y = β₀ + β₁x₁ + β₂x₂ + ε。H₀: β₁=0 且 β₂=0（整体无解释力）。
正规方程解：β₀=4.7667，β₁=0.4983，β₂=0.3020；
SST=49375.00，SSR=42457.83，SSE=6917.17；R²=0.8599，调整R²=0.8595；
整体 F=(SSR/2)/(SSE/777)=2384.6264，p=<1e-300；x1,x2 相关 r=0.4492；
β₁: SE=0.0149，t=33.5191，p=4.53e-153；β₂: SE=0.0078，t=38.8902，p=1.66e-184。
③ 计算结果：
y=4.7667+0.4983x1+0.3020x2；R²=0.8599；β₁ p=4.53e-153；β₂ p=1.66e-184
④ 结论：在 α=0.05 水平下：变量 x1 对 y 的影响显著（p=4.53e-153）；变量 x2 对 y 的影响显著（p=1.66e-184）。价正、广告正。
第4类：逻辑回归（案例 86–100）
案例 86. 广告投入预测是否购买
逻辑回归投入高更易买
① 原始数据：
样本 n=830，逐条 (x, y)：[(24.11, 0), (42.97, 0), (47.78, 0), (54.36, 1), (51.66, 0), (56.71, 1), (42.39, 0), (62.62, 1), (63.66, 0), (53.39, 0), (43.92, 1), (31.93, 1), (34.44, 0), (75.82, 0), (62.23, 1), (34.09, 0), (50.43, 0), (10.67, 0), (56.58, 1), (46.28, 0), (31.38, 0), (48.09, 1), (63.39, 1), (28.84, 0), (46.95, 0), (66.31, 0), (21.76, 0), (72.25, 0), (65.54, 1), (64.64, 1), (51.17, 0), (42.94, 0), (52.35, 0), (79.26, 1), (44.99, 0), (33.03, 0), (51.0, 0), (42.24, 0), (70.59, 0), (29.89, 0), (63.05, 1), (44.72, 0), (39.78, 0), (60.55, 0), (39.0, 0), (59.87, 0), (68.81, 0), (74.84, 1), (40.05, 0), (56.73, 0), (51.52, 1), (44.32, 1), (66.08, 1), (37.73, 0), (53.29, 0), (59.54, 1), (33.94, 1), (49.53, 0), (56.88, 0), (26.05, 0), (24.84, 0), (43.81, 0), (42.4, 0), (37.96, 1), (71.65, 1), (64.83, 1), (70.7, 1), (41.84, 0), (38.24, 1), (36.94, 0), (63.97, 0), (47.01, 1), (61.86, 1), (52.46, 0), (57.4, 1), (59.02, 0), (45.64, 0), (52.96, 0), (72.31, 0), (46.1, 0), (50.35, 0), (41.62, 0), (76.13, 1), (62.63, 1), (63.01, 0), (36.48, 0), (52.73, 0), (50.07, 0), (47.78, 0), (49.12, 0), (37.53, 0), (65.99, 1), (71.18, 0), (43.42, 0), (35.83, 0), (11.8, 0), (43.98, 0), (69.4, 1), (62.51, 1), (39.32, 0), (64.89, 0), (53.33, 0), (57.0, 0), (69.68, 0), (73.19, 1), (13.42, 0), (43.68, 0), (35.72, 0), (88.97, 1), (75.92, 0), (49.16, 0), (54.7, 0), (49.91, 1), (31.8, 0), (26.82, 0), (38.57, 0), (39.29, 0), (51.53, 0), (28.53, 0), (21.61, 0), (60.14, 0), (53.17, 0), (27.38, 0), (48.86, 1), (49.52, 0), (55.2, 1), (63.97, 1), (59.72, 1), (61.25, 0), (51.29, 1), (53.27, 0), (64.53, 0), (48.06, 0), (57.85, 0), (62.67, 1), (54.43, 0), (74.72, 0), (41.11, 0), (34.02, 0), (71.03, 0), (48.11, 1), (14.86, 0), (40.1, 1), (25.92, 0), (48.99, 0), (70.9, 1), (41.4, 0), (55.31, 0), (80.01, 1), (6.76, 0), (54.78, 0), (50.73, 0), (45.02, 0), (20.9, 0), (43.13, 0), (55.14, 1), (60.83, 1), (41.16, 0), (16.91, 0), (67.17, 0), (44.9, 0), (37.77, 0), (58.13, 0), (41.23, 0), (19.35, 0), (53.29, 0), (33.67, 0), (32.91, 0), (48.02, 1), (43.04, 1), (51.32, 0), (31.15, 1), (57.03, 1), (45.83, 0), (61.86, 1), (43.08, 0), (51.77, 0), (47.41, 0), (63.87, 1), (50.94, 0), (29.2, 0), (15.89, 0), (21.96, 0), (40.14, 0), (57.68, 0), (38.54, 0), (63.02, 1), (38.96, 0), (54.27, 1), (39.48, 1), (60.24, 0), (54.16, 0), (53.02, 0), (75.21, 0), (36.82, 0), (21.44, 0), (63.39, 0), (37.95, 0), (60.71, 1), (47.38, 1), (45.91, 0), (60.84, 0), (57.31, 0), (55.5, 0), (30.9, 0), (20.99, 0), (28.7, 0), (55.28, 1), (43.47, 0), (50.11, 1), (58.63, 0), (49.77, 0), (48.69, 1), (27.35, 1), (56.13, 0), (67.66, 1), (50.27, 0), (37.4, 0), (28.6, 0), (59.13, 1), (59.91, 1), (33.37, 0), (40.19, 0), (63.68, 1), (41.07, 0), (37.55, 0), (39.96, 0), (25.69, 0), (51.98, 1), (59.9, 0), (61.19, 1), (75.79, 0), (57.79, 1), (37.15, 0), (81.96, 1), (67.35, 0), (43.37, 0), (49.66, 1), (71.04, 1), (32.4, 0), (50.87, 1), (56.44, 0), (64.29, 1), (68.31, 1), (20.76, 0), (44.06, 0), (51.86, 0), (46.26, 0), (45.19, 0), (52.03, 0), (58.29, 0), (20.66, 0), (34.08, 1), (67.12, 0), (45.35, 0), (49.08, 1), (48.08, 0), (23.27, 0), (76.18, 1), (57.37, 0), (58.34, 0), (32.23, 0), (53.55, 0), (68.14, 1), (42.95, 0), (60.43, 0), (45.82, 0), (78.07, 0), (64.02, 0), (47.58, 0), (69.87, 1), (37.99, 0), (55.91, 0), (76.7, 1), (61.42, 0), (43.74, 1), (31.54, 0), (67.95, 1), (52.48, 0), (64.71, 1), (45.96, 0), (36.04, 0), (43.94, 0), (41.29, 0), (25.41, 0), (69.41, 1), (45.34, 0), (77.79, 0), (63.24, 0), (22.44, 0), (58.21, 1), (70.17, 0), (48.2, 0), (52.25, 0), (72.18, 1), (51.55, 0), (50.55, 0), (44.96, 0), (32.43, 0), (77.62, 1), (44.95, 0), (18.71, 0), (68.15, 1), (77.77, 0), (60.87, 0), (78.64, 1), (34.83, 0), (71.63, 0), (52.66, 1), (38.1, 0), (62.81, 0), (65.79, 0), (72.34, 0), (53.05, 0), (49.65, 0), (61.72, 0), (62.75, 0), (26.19, 0), (75.08, 1), (46.22, 0), (67.48, 0), (23.32, 0), (30.63, 0), (60.75, 1), (53.93, 0), (78.3, 0), (39.08, 0), (28.74, 1), (45.45, 0), (57.67, 0), (30.76, 0), (69.95, 1), (84.06, 0), (56.64, 0), (47.56, 0), (55.53, 0), (55.82, 0), (48.67, 1), (75.75, 0), (56.28, 1), (38.99, 0), (70.96, 1), (38.14, 0), (41.9, 0), (52.84, 0), (69.66, 0), (41.35, 0), (54.81, 0), (42.94, 0), (39.84, 0), (41.35, 0), (54.2, 0), (67.11, 0), (38.32, 0), (84.94, 0), (39.51, 0), (40.87, 0), (52.85, 1), (57.71, 1), (52.37, 0), (29.39, 0), (35.68, 0), (67.02, 0), (37.21, 1), (49.46, 0), (43.54, 0), (40.86, 0), (37.12, 1), (59.87, 0), (35.55, 0), (50.53, 0), (54.61, 0), (39.32, 0), (61.39, 0), (72.39, 1), (80.15, 1), (58.77, 0), (27.93, 0), (78.59, 1), (43.65, 0), (49.6, 0), (65.27, 1), (39.45, 0), (27.46, 0), (47.27, 0), (45.1, 1), (54.43, 1), (53.9, 0), (38.88, 0), (53.66, 0), (73.55, 1), (49.75, 0), (56.7, 0), (16.06, 0), (47.82, 1), (39.99, 0), (57.1, 0), (48.52, 1), (57.45, 1), (34.81, 0), (37.24, 0), (28.25, 0), (60.58, 1), (45.73, 0), (61.47, 0), (48.01, 1), (64.22, 0), (52.07, 0), (40.35, 0), (57.03, 0), (64.95, 0), (36.01, 0), (65.86, 1), (48.98, 0), (45.09, 0), (19.73, 0), (30.69, 0), (52.88, 1), (44.5, 0), (40.41, 0), (23.28, 0), (47.74, 0), (51.37, 1), (54.83, 0), (29.87, 0), (49.2, 1), (40.23, 0), (22.63, 0), (53.48, 1), (33.56, 0), (73.46, 0), (63.34, 1), (64.37, 1), (28.38, 1), (55.75, 1), (49.8, 0), (34.12, 0), (58.09, 0), (43.86, 0), (47.04, 0), (71.45, 0), (58.09, 0), (46.39, 0), (39.93, 0), (57.72, 1), (50.64, 0), (39.45, 0), (41.8, 0), (68.03, 0), (41.36, 0), (58.65, 0), (55.98, 1), (73.76, 1), (45.97, 0), (37.86, 0), (69.75, 1), (43.18, 0), (27.84, 0), (52.84, 0), (42.15, 0), (66.42, 0), (54.86, 1), (67.79, 1), (32.91, 0), (38.67, 0), (81.76, 1), (56.92, 0), (55.04, 1), (32.22, 0), (69.1, 1), (47.23, 0), (34.85, 0), (39.7, 0), (53.88, 1), (26.55, 0), (59.06, 0), (47.63, 0), (75.0, 1), (60.39, 1), (34.39, 0), (53.69, 1), (50.72, 0), (74.41, 1), (56.8, 0), (51.72, 1), (45.98, 0), (36.22, 0), (52.47, 0), (27.05, 0), (72.15, 1), (63.37, 0), (55.92, 0), (72.53, 1), (42.78, 0), (55.09, 0), (47.61, 0), (63.32, 0), (74.95, 1), (40.97, 0), (55.01, 0), (51.66, 1), (48.32, 0), (56.17, 0), (28.18, 0), (37.1, 0), (37.51, 0), (68.25, 1), (40.03, 0), (57.97, 1), (43.55, 0), (52.03, 0), (42.27, 0), (59.39, 0), (64.53, 1), (65.94, 1), (44.05, 0), (30.12, 0), (52.31, 1), (69.35, 1), (31.72, 0), (50.2, 0), (35.97, 0), (36.02, 1), (29.23, 0), (30.71, 0), (58.91, 1), (67.54, 0), (30.86, 0), (24.04, 0), (36.67, 0), (58.2, 0), (33.8, 0), (49.53, 0), (85.38, 0), (46.07, 1), (51.26, 0), (50.05, 0), (52.78, 0), (69.5, 0), (27.35, 0), (52.22, 1), (48.84, 0), (40.42, 0), (49.93, 1), (56.68, 0), (41.95, 0), (37.92, 0), (35.7, 0), (32.55, 0), (51.08, 0), (66.24, 0), (37.95, 1), (54.83, 0), (59.07, 1), (50.38, 1), (42.0, 0), (35.46, 0), (45.0, 0), (44.04, 0), (42.52, 0), (56.84, 0), (29.95, 0), (75.03, 1), (34.45, 0), (25.15, 0), (63.76, 0), (57.26, 0), (64.01, 1), (50.62, 0), (47.94, 0), (53.71, 0), (33.68, 0), (38.64, 0), (61.88, 1), (55.74, 1), (43.26, 0), (68.74, 0), (39.71, 0), (35.46, 0), (46.26, 0), (36.77, 0), (50.19, 1), (67.83, 1), (46.57, 1), (30.03, 0), (56.07, 1), (49.12, 0), (61.14, 0), (56.75, 0), (50.78, 0), (22.15, 0), (50.62, 0), (42.6, 0), (42.04, 0), (43.58, 0), (62.01, 1), (52.42, 0), (56.88, 0), (44.2, 0), (40.23, 0), (52.9, 1), (52.03, 0), (31.85, 1), (71.67, 1), (41.31, 0), (53.56, 0), (39.12, 0), (46.91, 0), (87.75, 1), (33.86, 0), (60.73, 0), (42.95, 0), (68.42, 0), (41.23, 0), (57.52, 0), (40.2, 1), (38.36, 1), (38.32, 0), (55.44, 0), (58.55, 0), (53.95, 0), (35.33, 0), (46.33, 0), (66.56, 1), (54.23, 1), (65.5, 0), (48.88, 1), (53.73, 0), (33.76, 1), (12.22, 0), (63.11, 0), (41.19, 1), (39.84, 0), (51.79, 0), (69.04, 1), (68.85, 1), (41.61, 0), (36.68, 0), (64.02, 0), (82.54, 0), (50.32, 0), (34.66, 0), (60.15, 1), (38.6, 0), (47.34, 0), (64.31, 0), (54.9, 1), (41.41, 0), (31.03, 0), (45.15, 0), (55.76, 0), (58.99, 1), (49.64, 0), (48.4, 1), (64.34, 0), (59.21, 1), (47.63, 0), (45.06, 0), (34.24, 0), (68.24, 0), (70.92, 0), (51.46, 0), (31.65, 1), (64.73, 0), (38.42, 0), (65.19, 0), (34.71, 1), (63.54, 1), (56.38, 0), (58.45, 0), (37.64, 0), (36.57, 0), (48.82, 1), (52.82, 0), (82.14, 1), (44.8, 0), (43.24, 0), (64.97, 0), (69.54, 0), (63.56, 0), (80.85, 1), (81.65, 0), (36.8, 0), (50.18, 0), (41.33, 0), (45.42, 0), (54.95, 0), (64.94, 0), (56.09, 1), (81.95, 1), (61.28, 0), (21.68, 0), (70.53, 1), (62.54, 0), (47.2, 0), (51.25, 1), (65.81, 0), (64.36, 1), (65.15, 0), (72.34, 0), (42.98, 0), (52.44, 1), (64.0, 1), (64.97, 0), (49.63, 1), (61.83, 0), (61.93, 0), (62.81, 1), (67.24, 1), (75.16, 0), (51.26, 0), (33.78, 0), (77.09, 0), (44.86, 1), (18.13, 0), (56.17, 1), (41.55, 0), (34.26, 0), (49.49, 0), (36.81, 0), (61.78, 0), (34.63, 0), (46.74, 0), (61.43, 1), (44.25, 0), (49.59, 0), (36.15, 0), (67.81, 0), (50.79, 0), (42.13, 0), (86.89, 1), (31.92, 0), (67.71, 1), (37.16, 1), (57.48, 1), (4.89, 0), (22.55, 0), (39.02, 0), (35.07, 0), (66.15, 0), (77.48, 1), (64.34, 0), (40.96, 0), (16.61, 0), (31.01, 0), (54.79, 0), (44.04, 0), (54.51, 0), (53.31, 1), (41.06, 0), (45.74, 0), (67.13, 1), (83.08, 0), (36.51, 0), (44.61, 0), (54.53, 0), (64.28, 0), (34.97, 0), (52.23, 0), (46.38, 0), (11.55, 0), (72.42, 1), (44.91, 0), (27.17, 0), (44.42, 0), (40.8, 0), (64.32, 0), (39.23, 0), (60.06, 1), (60.91, 1), (54.12, 1), (1.51, 0), (38.19, 0), (46.1, 0), (48.21, 0), (52.16, 0), (44.05, 0), (53.24, 0), (45.06, 0), (52.85, 0), (45.68, 1), (50.72, 0), (48.99, 0), (27.65, 0), (26.31, 0), (45.46, 0), (58.66, 1), (58.45, 0), (17.08, 0), (44.85, 0), (50.43, 0), (81.35, 1), (70.24, 0), (61.17, 0), (48.36, 0), (48.99, 0), (58.5, 0), (63.97, 1), (60.79, 1), (55.32, 0), (55.31, 0), (58.73, 1), (71.42, 1), (47.41, 0), (67.45, 1), (52.54, 1), (41.21, 1), (35.95, 0), (50.15, 0), (85.12, 1), (33.26, 0), (36.11, 0), (60.38, 1), (52.31, 0), (57.65, 0), (64.67, 1), (64.1, 0), (13.72, 0), (50.33, 0), (88.4, 1), (36.92, 0), (63.85, 1), (36.55, 0), (41.43, 0), (36.42, 0), (48.01, 0), (40.72, 1), (48.38, 0), (44.12, 1), (30.99, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1517, OR=0.3161, z=-12.7113, p=5.11e-37；x: β=0.0659, OR=1.0681, z=9.9615, p=2.25e-23；
McFadden 伪 R² = 1−LL/LL₀ = 0.1266；
示例预测（x=50）：P(y=1)=1/(1+e^(−-1.1517))=0.2402。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1517, OR=0.3161, z=-12.7113, p=5.11e-37；x: β=0.0659, OR=1.0681, z=9.9615, p=2.25e-23；
McFadden 伪 R² = 1−LL/LL₀ = 0.1266；
示例预测（x=50）：P(y=1)=1/(1+e^(−-1.1517))=0.2402。
③ 计算结果：
截距: β=-1.1517 OR=0.316 p=5.11e-37；x: β=0.0659 OR=1.068 p=2.25e-23
④ 结论：在 α=0.05 水平下：x显著(p=2.25e-23)。投入高更易买。（伪 R²=0.1266，模型解释力有限）
案例 87. 信用分预测是否违约
逻辑回归分低易违约
① 原始数据：
样本 n=873，逐条 (x, y)：[(685.01, 0), (690.75, 0), (683.01, 0), (699.11, 0), (647.94, 0), (696.13, 0), (697.96, 1), (638.08, 0), (819.78, 0), (634.62, 1), (576.76, 0), (835.77, 0), (692.87, 0), (773.67, 0), (901.52, 0), (709.47, 0), (768.5, 0), (703.63, 1), (722.32, 0), (722.3, 0), (751.68, 0), (736.28, 1), (751.44, 0), (675.26, 0), (682.29, 1), (639.2, 1), (627.09, 0), (888.16, 0), (664.86, 1), (855.62, 0), (544.04, 0), (620.38, 0), (713.98, 1), (694.39, 0), (769.55, 0), (833.51, 0), (798.77, 0), (674.02, 1), (725.24, 1), (718.25, 0), (683.44, 0), (678.3, 0), (754.9, 0), (771.72, 1), (677.22, 0), (771.52, 0), (794.27, 0), (701.82, 0), (642.34, 0), (707.81, 0), (680.96, 0), (763.71, 0), (732.84, 0), (744.51, 0), (760.89, 1), (709.59, 0), (581.26, 1), (699.66, 0), (700.99, 0), (638.92, 0), (797.19, 0), (828.25, 0), (786.9, 0), (674.32, 1), (720.81, 0), (692.34, 0), (699.37, 1), (765.59, 0), (736.82, 0), (720.78, 0), (631.1, 0), (690.03, 0), (805.57, 0), (729.27, 0), (708.87, 1), (707.38, 0), (749.13, 0), (560.1, 0), (627.52, 0), (691.42, 1), (697.61, 0), (722.92, 1), (648.43, 1), (549.98, 0), (753.58, 0), (578.71, 1), (671.07, 1), (655.84, 1), (691.11, 1), (594.74, 1), (675.11, 1), (655.74, 1), (719.39, 0), (668.21, 1), (496.3, 0), (647.72, 1), (748.11, 1), (652.9, 1), (711.84, 0), (760.54, 0), (772.88, 0), (694.0, 1), (777.16, 0), (864.7, 0), (853.15, 0), (669.1, 1), (692.32, 0), (833.62, 0), (794.62, 0), (699.39, 1), (677.44, 1), (590.95, 0), (934.08, 0), (775.97, 0), (647.56, 0), (664.64, 1), (543.2, 1), (686.19, 1), (597.66, 0), (672.94, 0), (677.62, 0), (682.68, 1), (703.58, 0), (698.26, 1), (704.41, 1), (603.34, 1), (706.0, 0), (670.55, 1), (690.38, 0), (668.29, 0), (743.57, 0), (867.93, 0), (818.65, 0), (815.32, 0), (628.89, 0), (752.55, 0), (728.84, 0), (610.07, 1), (652.88, 0), (632.56, 0), (720.71, 0), (708.05, 0), (794.87, 1), (682.96, 0), (706.15, 0), (803.37, 0), (686.75, 0), (656.53, 1), (751.13, 0), (527.57, 1), (502.21, 0), (712.41, 0), (720.59, 0), (776.11, 0), (780.37, 0), (546.89, 1), (791.53, 0), (740.58, 0), (574.85, 1), (581.59, 1), (698.39, 1), (713.27, 1), (705.89, 0), (608.99, 1), (726.69, 0), (519.7, 1), (780.06, 0), (672.74, 0), (600.77, 1), (696.85, 0), (710.83, 0), (684.91, 0), (627.89, 1), (623.36, 1), (654.3, 1), (838.05, 0), (747.86, 0), (695.99, 0), (666.42, 0), (653.4, 0), (573.11, 1), (635.35, 0), (581.94, 0), (839.05, 0), (794.26, 0), (622.89, 0), (702.62, 0), (786.61, 0), (784.94, 0), (640.78, 0), (652.79, 0), (774.43, 0), (627.34, 0), (717.53, 0), (730.08, 0), (603.88, 0), (627.12, 1), (672.53, 1), (747.25, 0), (676.65, 0), (686.94, 0), (744.47, 1), (699.24, 0), (731.58, 1), (612.25, 1), (713.3, 0), (723.94, 0), (678.69, 0), (764.32, 0), (666.72, 1), (605.85, 0), (796.3, 0), (639.43, 0), (751.65, 0), (711.7, 0), (725.82, 0), (728.3, 0), (732.27, 0), (641.4, 1), (687.61, 0), (733.42, 0), (526.52, 1), (703.32, 0), (707.94, 0), (635.45, 0), (855.43, 0), (622.9, 0), (810.66, 0), (758.64, 0), (616.19, 0), (635.71, 0), (558.55, 1), (698.06, 0), (645.39, 1), (709.39, 0), (693.49, 0), (615.09, 0), (746.33, 0), (728.58, 0), (771.09, 0), (675.78, 0), (574.87, 1), (705.34, 0), (764.29, 0), (742.13, 0), (776.16, 0), (750.25, 0), (647.59, 1), (674.64, 0), (562.99, 0), (635.78, 0), (564.05, 1), (626.26, 1), (560.27, 1), (625.46, 0), (631.46, 0), (754.45, 0), (670.74, 1), (735.39, 0), (727.08, 0), (886.18, 0), (558.22, 0), (683.36, 0), (590.07, 1), (611.01, 1), (708.84, 1), (639.62, 1), (711.98, 1), (761.83, 0), (851.0, 0), (863.26, 0), (714.86, 1), (761.56, 0), (613.84, 1), (749.33, 0), (643.33, 0), (593.5, 1), (729.02, 0), (807.45, 0), (792.72, 0), (782.03, 0), (683.08, 0), (668.31, 0), (733.39, 0), (705.58, 0), (665.18, 0), (628.53, 0), (712.45, 1), (696.83, 1), (690.53, 0), (839.63, 0), (853.83, 0), (596.51, 1), (552.48, 1), (673.65, 0), (687.25, 0), (591.05, 0), (856.31, 0), (721.27, 0), (652.93, 0), (816.16, 0), (709.84, 0), (822.84, 0), (741.16, 0), (557.43, 1), (559.33, 1), (716.35, 0), (775.19, 0), (564.84, 1), (564.83, 1), (591.41, 0), (565.53, 1), (599.86, 1), (775.73, 0), (721.5, 0), (753.37, 0), (683.42, 0), (607.9, 0), (679.18, 0), (620.86, 1), (648.67, 0), (682.64, 0), (707.82, 0), (754.13, 0), (565.22, 1), (682.62, 0), (830.27, 0), (864.22, 0), (628.45, 0), (643.47, 0), (720.93, 0), (855.26, 1), (721.03, 0), (752.81, 0), (605.48, 1), (688.93, 1), (800.67, 0), (695.15, 0), (572.95, 1), (721.74, 1), (740.41, 0), (868.14, 0), (692.58, 1), (811.55, 0), (613.06, 0), (771.91, 0), (766.1, 0), (654.96, 1), (666.52, 0), (635.41, 0), (648.0, 0), (672.47, 0), (732.12, 0), (728.29, 0), (723.37, 0), (638.27, 1), (769.39, 0), (849.55, 0), (956.98, 0), (688.5, 0), (659.73, 1), (799.6, 1), (663.9, 0), (741.48, 0), (786.22, 0), (664.61, 0), (678.91, 0), (690.63, 0), (742.52, 0), (744.56, 0), (723.19, 0), (775.29, 0), (881.89, 0), (818.7, 0), (709.0, 0), (822.08, 0), (709.12, 0), (722.4, 1), (713.77, 0), (618.97, 0), (641.68, 0), (797.96, 1), (744.37, 0), (543.0, 1), (725.85, 0), (711.5, 1), (676.23, 0), (520.91, 0), (676.46, 0), (822.84, 0), (696.59, 1), (650.98, 1), (644.65, 0), (696.97, 0), (704.35, 0), (666.05, 0), (692.8, 1), (873.24, 0), (684.74, 0), (609.35, 0), (756.71, 0), (668.28, 0), (725.74, 1), (587.79, 0), (722.43, 0), (648.41, 0), (672.83, 0), (596.41, 0), (737.66, 1), (723.61, 0), (626.85, 0), (765.58, 0), (778.08, 0), (810.49, 0), (540.0, 1), (678.23, 0), (768.84, 1), (689.66, 0), (767.68, 0), (719.72, 0), (625.17, 1), (836.42, 0), (783.52, 0), (748.71, 0), (750.72, 0), (903.4, 0), (749.48, 0), (713.5, 0), (726.19, 0), (605.1, 0), (678.04, 0), (563.92, 1), (613.47, 1), (630.56, 0), (701.88, 1), (748.16, 0), (603.58, 1), (728.75, 0), (632.66, 0), (626.54, 1), (668.6, 0), (689.61, 0), (616.41, 0), (621.12, 1), (521.69, 1), (714.39, 0), (736.45, 1), (567.26, 0), (645.88, 0), (722.3, 0), (637.93, 1), (683.36, 1), (717.73, 0), (760.61, 0), (794.08, 0), (737.79, 0), (665.83, 1), (745.92, 0), (774.37, 0), (677.12, 1), (688.03, 1), (791.66, 0), (631.42, 0), (572.67, 1), (765.43, 0), (822.25, 0), (703.6, 1), (608.28, 1), (774.14, 0), (719.94, 0), (636.02, 1), (573.69, 1), (639.39, 0), (681.57, 0), (597.86, 1), (809.2, 0), (648.67, 0), (781.96, 0), (646.26, 0), (695.66, 0), (692.25, 1), (712.92, 0), (680.89, 0), (803.74, 0), (704.44, 0), (746.73, 0), (637.77, 1), (653.99, 1), (683.54, 1), (582.37, 0), (586.48, 1), (718.58, 0), (663.48, 0), (684.48, 1), (765.79, 0), (615.96, 1), (611.76, 0), (632.29, 1), (809.27, 0), (632.93, 0), (722.44, 0), (642.41, 1), (812.09, 0), (778.32, 0), (772.96, 1), (558.88, 1), (729.84, 0), (559.02, 0), (643.45, 1), (757.86, 0), (599.79, 1), (698.62, 0), (596.77, 1), (739.87, 0), (837.15, 0), (793.24, 0), (767.31, 1), (746.03, 0), (659.02, 0), (739.18, 0), (700.89, 0), (687.22, 1), (659.72, 0), (470.98, 1), (802.84, 0), (814.51, 1), (741.24, 0), (776.11, 0), (677.94, 1), (782.6, 0), (626.42, 1), (735.73, 0), (776.53, 0), (839.55, 0), (640.1, 1), (718.59, 0), (741.85, 1), (711.78, 0), (725.39, 0), (681.98, 1), (705.23, 0), (747.44, 0), (662.52, 0), (773.98, 1), (729.13, 0), (701.62, 0), (661.3, 0), (677.47, 0), (642.19, 1), (796.8, 0), (759.46, 1), (714.92, 1), (667.68, 1), (604.89, 1), (706.25, 0), (743.29, 0), (595.8, 0), (579.36, 1), (720.83, 1), (732.61, 0), (558.57, 1), (762.69, 0), (745.81, 0), (706.47, 0), (672.99, 0), (693.13, 0), (692.99, 1), (770.15, 0), (618.43, 0), (731.71, 0), (761.51, 1), (600.7, 1), (700.87, 0), (713.38, 1), (670.75, 1), (698.36, 0), (662.87, 1), (664.89, 0), (647.96, 1), (656.2, 0), (681.22, 1), (648.22, 0), (595.84, 0), (780.43, 0), (497.61, 0), (554.61, 1), (646.64, 1), (584.5, 1), (818.78, 0), (762.33, 0), (701.71, 1), (752.4, 0), (619.39, 1), (671.91, 0), (691.68, 0), (664.67, 1), (854.6, 1), (618.45, 1), (616.1, 0), (789.36, 0), (728.67, 0), (857.15, 0), (649.45, 1), (623.65, 0), (819.56, 0), (821.86, 0), (767.82, 1), (816.45, 0), (740.34, 0), (615.02, 0), (686.76, 0), (729.45, 0), (863.06, 0), (710.43, 0), (712.11, 1), (753.91, 0), (635.51, 1), (834.18, 0), (804.78, 0), (674.14, 1), (683.04, 1), (553.51, 1), (741.74, 0), (715.75, 0), (630.04, 1), (546.81, 1), (803.33, 0), (653.35, 0), (699.74, 0), (595.02, 1), (800.45, 0), (746.05, 0), (638.45, 1), (522.9, 1), (741.24, 0), (817.47, 0), (561.2, 1), (623.12, 0), (759.98, 1), (577.75, 1), (786.3, 0), (557.38, 1), (694.66, 0), (861.73, 0), (700.3, 0), (708.38, 0), (655.46, 0), (678.69, 1), (576.28, 1), (835.54, 0), (693.52, 1), (768.75, 0), (805.4, 0), (812.52, 0), (712.66, 0), (645.61, 0), (763.05, 0), (613.18, 1), (777.23, 0), (690.17, 0), (692.14, 0), (809.05, 0), (624.66, 1), (629.92, 0), (795.27, 0), (728.34, 1), (783.43, 0), (572.87, 1), (793.04, 0), (731.8, 1), (706.68, 0), (700.97, 1), (697.93, 0), (670.16, 1), (815.12, 0), (817.32, 0), (769.48, 0), (652.2, 1), (699.65, 0), (725.64, 0), (779.95, 0), (765.19, 0), (625.63, 1), (567.38, 0), (634.35, 1), (614.72, 0), (710.87, 0), (640.94, 0), (568.98, 1), (592.25, 0), (591.16, 0), (730.49, 0), (692.25, 1), (684.42, 0), (686.8, 0), (684.92, 0), (687.22, 0), (655.23, 1), (765.44, 0), (753.39, 0), (749.32, 0), (678.44, 1), (740.19, 0), (830.25, 0), (796.31, 0), (510.32, 1), (707.65, 1), (670.68, 0), (704.64, 0), (806.67, 0), (686.4, 0), (773.3, 0), (715.32, 0), (649.73, 0), (674.03, 0), (501.9, 1), (801.53, 0), (726.76, 0), (668.89, 0), (540.15, 1), (662.53, 0), (775.14, 1), (722.64, 0), (743.06, 0), (703.83, 0), (698.67, 0), (764.35, 0), (680.03, 0), (645.53, 0), (682.86, 1), (732.24, 0), (823.89, 0), (615.14, 1), (617.44, 0), (732.9, 0), (817.84, 0), (601.27, 1), (618.03, 1), (719.63, 1), (704.78, 0), (733.36, 0), (604.09, 1), (688.93, 0), (712.85, 0), (650.66, 1), (699.61, 1), (742.12, 0), (783.04, 0), (787.15, 0), (729.2, 1), (665.75, 1), (669.95, 0), (629.35, 1), (936.89, 0), (635.1, 0), (532.11, 1), (755.49, 0), (710.01, 1), (609.96, 1), (588.58, 1), (670.1, 1), (733.7, 0), (654.94, 0), (799.79, 1), (745.46, 0), (703.94, 0), (885.47, 0), (680.8, 0), (606.94, 1), (717.75, 1), (786.86, 0), (709.38, 1), (738.72, 0), (646.19, 1), (714.72, 0), (783.41, 0), (717.11, 1), (720.48, 1), (651.5, 0), (689.83, 0), (716.65, 0), (742.98, 0), (565.22, 0), (655.79, 0), (619.73, 0), (629.9, 1), (613.02, 0), (635.64, 0), (706.95, 1), (934.96, 0), (673.64, 1), (736.4, 1), (581.01, 0), (700.81, 0), (645.08, 0), (601.83, 0), (735.35, 1), (694.05, 1), (546.07, 1), (724.77, 1), (803.31, 1), (848.47, 0), (632.69, 0), (732.62, 0), (805.34, 0), (667.11, 0), (716.16, 0), (776.93, 0), (902.58, 0), (631.21, 1), (680.78, 0), (721.41, 0), (749.01, 1), (690.87, 0), (606.01, 0), (821.0, 0), (782.18, 0), (637.65, 1), (831.35, 0), (812.08, 0), (703.92, 0), (724.62, 0), (767.75, 0), (497.53, 1), (718.59, 0), (743.53, 0), (564.03, 1), (706.22, 0), (751.75, 0), (638.92, 0), (684.71, 1), (591.21, 0), (724.7, 1), (856.89, 0), (642.08, 0), (752.43, 0), (681.85, 1), (669.55, 1), (915.28, 0), (738.24, 0), (656.09, 1), (741.06, 0), (694.85, 0), (646.86, 1), (641.54, 0), (767.96, 0), (710.29, 0), (725.11, 0), (620.57, 1), (647.87, 0), (543.0, 0), (761.31, 0), (713.33, 0), (652.9, 0), (746.94, 0), (623.61, 0), (574.01, 0), (735.8, 0), (662.12, 1), (709.9, 0), (729.83, 1), (755.88, 0), (762.79, 0), (720.51, 0), (616.7, 0), (727.28, 1), (689.29, 0), (696.95, 1), (631.79, 0), (763.56, 1), (682.1, 1), (759.61, 0), (796.33, 1)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-0.9533, OR=0.3855, z=-11.3687, p=5.99e-30；x: β=-0.0126, OR=0.9875, z=-10.5139, p=7.45e-26；
McFadden 伪 R² = 1−LL/LL₀ = 0.1297；
示例预测（x=700）：P(y=1)=1/(1+e^(−-0.9533))=0.2782。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-0.9533, OR=0.3855, z=-11.3687, p=5.99e-30；x: β=-0.0126, OR=0.9875, z=-10.5139, p=7.45e-26；
McFadden 伪 R² = 1−LL/LL₀ = 0.1297；
示例预测（x=700）：P(y=1)=1/(1+e^(−-0.9533))=0.2782。
③ 计算结果：
截距: β=-0.9533 OR=0.385 p=5.99e-30；x: β=-0.0126 OR=0.988 p=7.45e-26
④ 结论：在 α=0.05 水平下：x显著(p=7.45e-26)。分低易违约。（伪 R²=0.1297，模型解释力有限）
案例 88. 学习时长预测考试及格
逻辑回归学得久易过
① 原始数据：
样本 n=725，逐条 (x, y)：[(5.98, 0), (11.78, 0), (5.87, 0), (10.55, 0), (10.42, 0), (7.52, 1), (13.8, 0), (8.07, 0), (10.05, 0), (7.16, 0), (11.79, 1), (11.68, 1), (10.5, 0), (8.35, 0), (4.56, 0), (14.4, 1), (6.07, 0), (12.49, 1), (9.64, 0), (6.13, 0), (8.68, 0), (5.38, 0), (5.89, 0), (12.43, 1), (6.66, 0), (6.57, 0), (13.44, 1), (11.45, 0), (12.81, 1), (14.52, 1), (2.68, 0), (5.2, 0), (11.0, 0), (12.78, 1), (11.47, 0), (3.61, 0), (6.64, 0), (7.17, 0), (9.31, 0), (11.72, 0), (11.03, 0), (8.34, 0), (11.83, 1), (7.35, 0), (7.54, 0), (10.35, 1), (6.88, 0), (7.45, 1), (5.22, 0), (7.65, 0), (11.66, 1), (2.48, 0), (4.66, 0), (13.3, 0), (9.53, 0), (15.1, 1), (9.98, 0), (6.29, 1), (8.58, 0), (16.15, 0), (9.93, 0), (6.98, 0), (10.1, 0), (6.44, 0), (8.79, 0), (10.5, 0), (6.45, 0), (6.53, 0), (6.66, 0), (6.64, 0), (9.92, 0), (11.88, 0), (10.88, 0), (13.16, 1), (10.27, 0), (12.15, 0), (17.6, 1), (13.74, 1), (16.57, 1), (10.93, 1), (14.39, 0), (7.78, 0), (13.96, 0), (8.32, 0), (12.27, 1), (6.57, 0), (9.46, 1), (9.1, 0), (9.29, 0), (12.26, 1), (13.71, 0), (9.65, 0), (1.92, 0), (8.56, 0), (5.3, 0), (15.67, 1), (4.63, 0), (7.11, 0), (10.89, 0), (8.57, 0), (9.75, 0), (10.82, 1), (6.7, 0), (12.11, 0), (13.96, 0), (12.9, 0), (8.04, 0), (9.18, 0), (11.07, 1), (10.11, 0), (12.16, 0), (4.18, 0), (10.16, 1), (5.52, 0), (8.64, 1), (8.55, 0), (8.49, 0), (16.06, 0), (12.07, 0), (13.51, 0), (11.43, 1), (9.65, 0), (12.17, 0), (7.63, 0), (6.08, 0), (4.97, 0), (8.03, 0), (7.53, 0), (9.97, 0), (5.91, 0), (10.19, 0), (7.74, 0), (6.97, 0), (11.02, 1), (11.95, 0), (12.65, 0), (3.94, 0), (11.72, 1), (12.02, 0), (10.35, 0), (9.53, 0), (8.63, 0), (11.53, 1), (6.13, 0), (13.79, 1), (9.28, 0), (7.72, 0), (14.3, 1), (6.07, 0), (12.19, 0), (11.3, 0), (10.42, 0), (10.13, 0), (10.26, 1), (10.75, 0), (11.56, 0), (9.95, 0), (11.09, 1), (11.24, 1), (11.55, 0), (11.38, 0), (9.44, 0), (13.47, 0), (13.57, 0), (6.97, 0), (8.64, 0), (10.96, 1), (5.95, 0), (9.49, 0), (12.16, 0), (4.23, 0), (8.68, 0), (7.14, 0), (7.78, 0), (7.24, 0), (5.68, 0), (14.33, 1), (7.81, 0), (8.31, 0), (9.18, 0), (4.83, 0), (11.6, 0), (7.11, 0), (7.09, 0), (12.5, 1), (9.7, 1), (9.87, 1), (11.73, 0), (10.12, 1), (17.42, 1), (3.93, 0), (9.18, 1), (8.79, 0), (8.66, 0), (10.87, 1), (6.81, 1), (8.62, 0), (9.67, 0), (12.03, 1), (11.63, 1), (11.83, 0), (15.43, 0), (13.46, 1), (4.99, 0), (10.8, 0), (10.47, 0), (14.44, 0), (10.32, 0), (13.56, 1), (9.62, 0), (11.42, 0), (9.31, 0), (12.45, 0), (11.12, 1), (10.36, 0), (6.95, 0), (7.21, 0), (11.56, 1), (11.77, 0), (6.8, 0), (13.36, 0), (17.49, 1), (7.79, 0), (8.73, 0), (12.9, 1), (8.07, 0), (15.48, 1), (10.9, 0), (10.35, 0), (8.71, 0), (4.92, 0), (10.53, 0), (12.55, 0), (11.2, 0), (7.53, 0), (11.88, 1), (8.25, 0), (12.28, 0), (9.54, 0), (11.62, 1), (11.66, 0), (8.1, 0), (12.74, 1), (4.09, 0), (11.13, 0), (12.28, 0), (14.34, 0), (7.42, 0), (8.29, 1), (11.94, 1), (12.25, 1), (6.25, 0), (9.36, 0), (10.45, 0), (10.4, 1), (9.41, 1), (8.98, 0), (7.15, 0), (14.15, 0), (7.26, 0), (11.92, 1), (13.45, 1), (8.55, 0), (8.53, 0), (10.87, 0), (7.82, 0), (14.09, 1), (12.12, 0), (7.81, 0), (10.43, 1), (15.23, 1), (10.15, 0), (6.21, 1), (9.36, 0), (9.87, 0), (9.77, 0), (9.48, 1), (8.32, 0), (6.81, 0), (12.62, 0), (13.12, 0), (3.01, 0), (6.44, 0), (12.03, 0), (10.71, 1), (9.03, 0), (6.76, 0), (10.53, 1), (4.99, 0), (15.73, 0), (12.13, 0), (10.14, 0), (5.48, 0), (10.36, 1), (7.12, 0), (6.8, 0), (12.28, 0), (7.73, 0), (8.76, 1), (10.3, 1), (11.33, 0), (9.55, 0), (13.09, 1), (7.07, 0), (4.45, 0), (9.07, 0), (6.77, 0), (6.37, 0), (8.77, 0), (5.92, 1), (8.62, 0), (8.63, 0), (9.69, 0), (12.14, 0), (7.35, 0), (7.54, 0), (12.32, 0), (10.23, 0), (12.79, 1), (7.08, 0), (12.21, 0), (9.01, 0), (9.91, 0), (10.41, 1), (10.67, 0), (12.4, 1), (13.4, 0), (6.88, 0), (15.07, 1), (11.98, 0), (13.33, 0), (6.3, 0), (5.27, 0), (14.27, 0), (10.53, 0), (13.34, 1), (8.24, 0), (10.79, 0), (9.67, 0), (10.15, 1), (6.83, 0), (13.45, 1), (8.52, 0), (6.75, 0), (11.85, 0), (11.72, 0), (13.12, 1), (12.51, 0), (13.6, 0), (9.65, 0), (8.8, 0), (8.89, 0), (11.91, 1), (9.85, 0), (8.16, 0), (13.45, 1), (12.91, 1), (6.0, 1), (13.24, 0), (12.2, 1), (12.64, 1), (10.11, 0), (13.05, 0), (11.26, 0), (8.72, 0), (7.58, 0), (9.79, 0), (11.24, 0), (12.01, 0), (7.25, 0), (13.17, 0), (9.24, 0), (5.45, 0), (9.65, 0), (8.41, 1), (8.11, 0), (7.84, 0), (8.38, 0), (9.02, 0), (7.73, 0), (8.29, 0), (13.99, 0), (11.78, 0), (14.46, 1), (5.87, 0), (8.2, 0), (11.96, 1), (8.29, 0), (4.41, 0), (14.75, 1), (11.02, 1), (10.96, 1), (10.62, 0), (11.32, 1), (9.71, 1), (5.14, 0), (7.06, 0), (7.64, 0), (11.5, 0), (3.71, 0), (7.2, 1), (7.07, 0), (9.24, 0), (4.05, 0), (6.25, 0), (14.55, 1), (11.12, 0), (5.44, 0), (12.93, 0), (3.33, 0), (12.13, 0), (8.19, 1), (8.74, 1), (11.6, 0), (14.89, 1), (9.17, 0), (5.71, 0), (5.01, 0), (8.04, 0), (9.43, 0), (7.09, 0), (10.01, 0), (6.51, 0), (13.0, 1), (14.12, 0), (10.13, 0), (12.72, 0), (11.43, 1), (11.08, 0), (12.55, 1), (5.92, 0), (7.36, 0), (7.35, 0), (8.03, 1), (8.33, 0), (15.85, 1), (12.02, 0), (8.83, 0), (11.5, 1), (1.79, 0), (13.3, 0), (8.89, 0), (8.89, 0), (16.46, 0), (8.78, 0), (11.25, 0), (7.58, 0), (11.33, 0), (7.61, 0), (12.75, 1), (17.06, 1), (12.34, 0), (9.18, 0), (9.6, 0), (15.17, 1), (11.09, 0), (14.32, 1), (9.6, 0), (11.91, 0), (5.29, 0), (9.74, 0), (9.13, 1), (10.67, 1), (7.12, 0), (13.03, 0), (10.26, 1), (5.56, 0), (7.11, 0), (11.9, 1), (11.63, 0), (9.61, 0), (14.1, 1), (4.96, 0), (14.07, 0), (8.68, 0), (9.31, 0), (10.35, 0), (8.72, 0), (12.21, 0), (12.34, 1), (8.07, 0), (10.32, 1), (6.03, 0), (10.79, 0), (9.05, 0), (12.65, 1), (6.89, 1), (10.98, 1), (13.31, 1), (11.84, 0), (13.89, 0), (10.42, 0), (14.05, 1), (8.06, 1), (8.95, 0), (11.09, 0), (6.94, 0), (10.81, 0), (9.96, 1), (8.98, 0), (4.56, 0), (11.92, 0), (9.36, 1), (5.56, 0), (9.79, 0), (6.6, 0), (7.69, 1), (11.34, 0), (8.36, 0), (11.05, 0), (5.78, 0), (14.29, 1), (10.59, 0), (12.3, 0), (11.41, 0), (9.17, 0), (8.07, 0), (15.91, 1), (14.53, 0), (14.42, 0), (10.04, 0), (6.34, 0), (13.9, 0), (11.0, 1), (15.29, 0), (16.1, 0), (8.8, 0), (12.8, 1), (13.04, 0), (12.15, 1), (11.54, 0), (10.8, 0), (12.43, 0), (14.38, 0), (9.11, 0), (8.44, 1), (12.11, 1), (8.49, 0), (13.76, 0), (13.72, 0), (13.72, 0), (7.89, 1), (16.01, 0), (12.51, 1), (10.16, 0), (12.7, 0), (11.18, 0), (9.35, 1), (15.0, 0), (4.82, 0), (13.62, 1), (5.37, 0), (1.84, 0), (12.93, 1), (3.95, 0), (9.9, 0), (6.03, 0), (6.75, 0), (7.94, 0), (17.31, 1), (9.17, 1), (8.95, 0), (5.59, 0), (11.22, 0), (12.45, 0), (9.48, 0), (11.11, 0), (14.64, 1), (8.58, 0), (2.56, 0), (16.97, 1), (7.13, 1), (8.4, 0), (13.79, 0), (8.47, 1), (9.62, 1), (12.97, 0), (12.28, 0), (9.79, 0), (6.88, 0), (10.81, 0), (15.22, 1), (7.14, 0), (10.19, 0), (11.64, 0), (15.77, 0), (10.99, 1), (10.5, 1), (6.52, 0), (6.99, 0), (6.48, 0), (6.76, 0), (16.18, 1), (6.81, 0), (6.95, 1), (8.64, 0), (9.72, 0), (17.61, 1), (13.8, 0), (6.66, 0), (11.3, 1), (10.73, 0), (11.75, 0), (5.25, 0), (9.94, 0), (3.21, 0), (8.58, 0), (6.82, 0), (7.13, 0), (16.63, 0), (6.03, 0), (10.83, 0), (9.12, 0), (7.03, 1), (10.98, 1), (11.64, 0), (15.18, 0), (8.87, 0), (12.97, 0), (10.55, 1), (7.33, 0), (8.68, 0), (6.74, 0), (14.64, 0), (16.69, 1), (13.95, 1), (10.63, 0), (8.26, 0), (4.7, 0), (6.65, 0), (10.6, 0), (10.54, 0), (9.79, 1), (8.71, 1), (11.55, 0), (9.24, 0), (11.48, 0), (9.1, 0), (8.91, 0), (12.9, 1), (11.02, 1), (15.44, 0), (6.05, 1), (14.1, 1), (12.97, 0), (10.01, 1), (7.96, 0), (8.77, 1), (2.95, 0), (3.68, 0), (8.59, 1), (12.0, 0), (11.7, 0), (9.36, 0), (9.99, 1), (6.64, 0), (9.43, 0), (9.9, 1), (12.95, 0), (13.63, 0), (11.59, 0), (13.59, 0), (9.57, 0), (5.51, 1), (5.54, 0), (10.01, 0), (8.75, 0), (9.43, 0), (10.45, 0), (5.22, 0), (11.23, 0), (7.14, 0), (7.44, 0), (9.08, 0), (11.85, 0), (9.68, 0), (9.26, 0), (6.45, 0), (8.37, 0), (7.12, 0), (6.37, 0), (12.8, 1), (6.69, 1), (3.88, 0), (8.46, 0), (12.64, 0), (12.21, 0), (5.91, 0), (10.25, 1), (11.55, 1), (10.56, 1), (14.53, 1), (10.21, 1), (8.66, 1), (12.93, 0), (4.41, 0), (12.36, 0), (10.34, 1), (5.9, 0), (11.94, 0), (7.9, 0), (13.29, 1), (13.94, 0), (10.21, 0), (9.01, 1), (6.92, 0), (6.0, 0), (6.68, 0), (7.37, 1), (11.86, 1), (7.2, 1), (11.58, 1), (8.29, 1), (12.76, 1), (7.54, 0), (9.98, 0), (8.25, 0), (11.73, 0), (5.25, 0), (6.27, 0), (11.26, 0), (9.59, 1), (9.34, 0), (15.99, 1), (3.67, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1198, OR=0.3263, z=-11.9416, p=7.18e-33；x: β=0.2862, OR=1.3314, z=8.5616, p=1.11e-17；
McFadden 伪 R² = 1−LL/LL₀ = 0.1040；
示例预测（x=10）：P(y=1)=1/(1+e^(−-1.1198))=0.2460。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1198, OR=0.3263, z=-11.9416, p=7.18e-33；x: β=0.2862, OR=1.3314, z=8.5616, p=1.11e-17；
McFadden 伪 R² = 1−LL/LL₀ = 0.1040；
示例预测（x=10）：P(y=1)=1/(1+e^(−-1.1198))=0.2460。
③ 计算结果：
截距: β=-1.1198 OR=0.326 p=7.18e-33；x: β=0.2862 OR=1.331 p=1.11e-17
④ 结论：在 α=0.05 水平下：x显著(p=1.11e-17)。学得久易过。（伪 R²=0.1040，模型解释力有限）
案例 89. 运动量预测是否患慢病
逻辑回归运动多风险低
① 原始数据：
样本 n=374，逐条 (x, y)：[(4.31, 0), (4.52, 0), (9.02, 0), (5.46, 1), (1.13, 1), (3.32, 1), (1.47, 1), (5.81, 1), (7.11, 0), (2.51, 0), (3.52, 0), (3.4, 1), (5.56, 1), (3.29, 0), (3.75, 0), (7.46, 0), (9.83, 0), (10.5, 0), (2.36, 0), (5.77, 0), (5.39, 0), (4.79, 1), (7.17, 0), (3.02, 1), (6.59, 1), (6.16, 0), (1.02, 0), (5.96, 0), (3.88, 0), (2.3, 1), (6.33, 0), (5.37, 0), (2.92, 1), (6.56, 0), (5.3, 0), (8.51, 1), (2.89, 0), (4.44, 0), (4.75, 0), (6.0, 0), (8.33, 0), (2.52, 0), (6.73, 0), (3.79, 0), (6.11, 1), (6.6, 0), (2.9, 1), (4.52, 1), (1.47, 1), (1.74, 0), (7.59, 0), (5.86, 0), (5.47, 0), (5.97, 0), (4.64, 0), (5.23, 0), (4.47, 0), (6.77, 0), (5.55, 0), (5.38, 0), (4.71, 0), (6.85, 1), (6.72, 0), (1.21, 0), (6.72, 0), (4.75, 0), (4.86, 0), (5.44, 0), (1.52, 1), (9.97, 0), (2.69, 0), (4.94, 0), (6.99, 0), (4.97, 0), (6.2, 1), (2.17, 0), (7.07, 0), (4.32, 1), (5.4, 0), (8.25, 0), (4.39, 0), (3.76, 0), (2.56, 1), (3.95, 1), (5.19, 0), (1.03, 0), (5.09, 0), (5.36, 0), (4.61, 0), (3.94, 0), (5.86, 0), (3.25, 1), (5.67, 0), (4.64, 0), (6.8, 0), (3.82, 0), (3.86, 1), (5.53, 0), (1.62, 1), (9.15, 0), (3.21, 0), (4.58, 1), (7.47, 0), (4.61, 1), (4.76, 0), (5.5, 0), (4.64, 0), (4.87, 0), (6.57, 0), (7.34, 0), (5.4, 0), (4.34, 0), (5.39, 0), (5.99, 1), (3.31, 1), (4.85, 0), (7.09, 0), (4.05, 0), (9.79, 0), (5.38, 0), (3.7, 0), (5.32, 0), (6.18, 0), (3.89, 0), (6.27, 1), (5.33, 0), (8.62, 0), (7.4, 0), (6.93, 1), (4.58, 0), (5.93, 1), (4.75, 1), (1.82, 1), (8.35, 0), (7.25, 0), (2.59, 0), (5.95, 1), (4.45, 0), (8.61, 0), (0.79, 0), (4.91, 0), (5.73, 0), (2.82, 0), (7.11, 0), (4.86, 0), (9.79, 0), (5.19, 0), (7.77, 0), (6.58, 1), (6.33, 0), (3.22, 0), (3.38, 0), (3.21, 0), (3.64, 0), (5.59, 0), (5.58, 0), (4.07, 0), (6.17, 0), (2.31, 1), (5.08, 0), (5.1, 0), (4.99, 0), (4.37, 1), (4.89, 0), (5.0, 0), (2.52, 1), (6.12, 0), (4.71, 0), (2.73, 1), (5.32, 1), (7.29, 0), (4.65, 0), (3.12, 0), (4.49, 0), (8.1, 0), (8.25, 0), (5.11, 0), (9.37, 0), (3.15, 1), (7.65, 0), (5.45, 0), (4.56, 0), (5.87, 0), (6.02, 0), (5.28, 0), (5.61, 0), (6.58, 0), (5.83, 0), (5.09, 0), (2.16, 0), (6.89, 0), (6.99, 0), (4.24, 1), (4.15, 0), (2.94, 1), (3.92, 1), (2.35, 1), (5.92, 0), (4.47, 1), (4.63, 0), (5.9, 0), (5.61, 0), (3.77, 1), (3.55, 1), (6.88, 1), (3.27, 1), (6.1, 0), (4.23, 0), (-0.07, 0), (7.47, 0), (6.7, 1), (1.61, 1), (5.43, 0), (4.41, 0), (1.71, 0), (5.68, 1), (6.39, 0), (4.54, 1), (6.52, 0), (4.92, 0), (3.3, 0), (7.06, 0), (9.37, 0), (2.68, 0), (5.86, 0), (4.17, 0), (6.93, 0), (3.36, 1), (3.76, 0), (4.87, 0), (2.99, 1), (6.93, 0), (6.1, 1), (8.17, 0), (7.21, 1), (5.41, 0), (5.25, 0), (2.09, 0), (1.8, 0), (2.48, 0), (4.99, 1), (5.24, 0), (6.7, 0), (0.22, 0), (8.7, 0), (5.38, 0), (6.72, 0), (4.1, 1), (7.76, 0), (5.97, 0), (6.52, 0), (1.13, 1), (6.5, 0), (4.9, 0), (4.1, 1), (6.01, 0), (2.02, 0), (4.75, 0), (5.83, 0), (6.5, 0), (7.01, 0), (4.57, 0), (6.0, 0), (5.12, 0), (2.16, 0), (9.16, 0), (4.71, 0), (6.9, 0), (3.5, 1), (3.81, 0), (5.1, 1), (4.69, 1), (6.31, 0), (3.97, 0), (5.68, 0), (5.19, 1), (1.91, 0), (7.47, 0), (5.42, 0), (4.3, 0), (6.53, 0), (5.64, 1), (-2.51, 1), (8.36, 0), (4.14, 0), (3.51, 1), (7.14, 0), (8.41, 0), (3.66, 0), (1.71, 1), (6.39, 0), (8.51, 0), (6.14, 0), (6.75, 0), (7.03, 1), (4.81, 0), (6.26, 0), (6.5, 0), (3.96, 0), (6.11, 0), (4.49, 0), (6.7, 0), (7.44, 0), (2.48, 0), (2.53, 1), (6.42, 0), (5.62, 0), (0.23, 1), (7.49, 0), (3.82, 0), (5.43, 0), (4.81, 0), (5.49, 0), (4.24, 1), (5.68, 0), (6.01, 0), (8.03, 0), (4.24, 1), (4.14, 0), (7.04, 0), (6.35, 1), (5.71, 0), (5.98, 0), (5.45, 0), (4.84, 0), (7.84, 0), (6.56, 0), (3.28, 1), (4.77, 0), (8.17, 0), (3.62, 0), (2.54, 1), (6.46, 0), (6.74, 0), (2.84, 1), (3.07, 1), (7.49, 0), (0.31, 1), (8.13, 0), (6.49, 1), (2.48, 0), (0.49, 1), (7.32, 0), (5.9, 1), (2.85, 1), (5.65, 0), (2.59, 1), (2.83, 1), (4.08, 0), (8.4, 0), (4.86, 0), (4.23, 0), (3.63, 1), (5.42, 0), (6.11, 0), (0.59, 0), (7.5, 0), (3.19, 1), (2.51, 1), (6.48, 0), (6.35, 0), (2.17, 0), (7.62, 0), (3.9, 0), (4.05, 0), (4.81, 0), (6.74, 0), (5.26, 0), (6.23, 0), (4.16, 0), (5.95, 1), (1.91, 0), (5.77, 0), (4.65, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.2200, OR=0.2952, z=-9.1842, p=4.15e-20；x: β=-0.4105, OR=0.6633, z=-5.8822, p=4.05e-09；
McFadden 伪 R² = 1−LL/LL₀ = 0.0968；
示例预测（x=5）：P(y=1)=1/(1+e^(−-1.2200))=0.2279。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.2200, OR=0.2952, z=-9.1842, p=4.15e-20；x: β=-0.4105, OR=0.6633, z=-5.8822, p=4.05e-09；
McFadden 伪 R² = 1−LL/LL₀ = 0.0968；
示例预测（x=5）：P(y=1)=1/(1+e^(−-1.2200))=0.2279。
③ 计算结果：
截距: β=-1.2200 OR=0.295 p=4.15e-20；x: β=-0.4105 OR=0.663 p=4.05e-09
④ 结论：在 α=0.05 水平下：x显著(p=4.05e-09)。运动多风险低。（伪 R²=0.0968，模型解释力有限）
案例 90. 收入预测是否买豪车
逻辑回归收入高易买
① 原始数据：
样本 n=840，逐条 (x, y)：[(4.98, 0), (45.39, 0), (46.13, 0), (30.33, 1), (32.0, 0), (43.43, 0), (28.24, 1), (62.11, 1), (61.72, 1), (37.87, 0), (30.36, 0), (32.43, 0), (72.86, 1), (11.58, 0), (42.23, 1), (38.16, 0), (32.4, 0), (21.29, 0), (29.44, 0), (38.78, 1), (39.51, 1), (46.06, 1), (34.08, 0), (20.79, 0), (23.43, 0), (38.96, 0), (29.13, 0), (24.73, 0), (41.04, 1), (10.59, 0), (27.7, 1), (22.75, 1), (52.92, 1), (31.83, 1), (37.24, 1), (34.76, 0), (32.97, 0), (59.48, 1), (14.85, 0), (45.93, 1), (36.12, 0), (36.86, 0), (33.65, 0), (25.5, 0), (28.62, 0), (37.27, 0), (45.2, 0), (25.08, 1), (37.24, 1), (25.27, 0), (30.16, 0), (39.73, 0), (25.58, 0), (22.51, 0), (20.55, 0), (44.97, 0), (-5.82, 0), (17.4, 0), (51.42, 1), (30.35, 0), (45.26, 1), (11.83, 0), (20.65, 0), (17.11, 0), (44.56, 1), (28.24, 0), (27.42, 0), (48.13, 0), (44.99, 0), (44.48, 1), (21.39, 0), (37.61, 0), (14.68, 1), (44.96, 0), (55.75, 1), (24.68, 0), (34.83, 0), (29.32, 0), (27.28, 0), (55.26, 1), (36.67, 0), (-1.35, 0), (34.54, 1), (10.17, 0), (35.81, 0), (25.82, 0), (24.69, 0), (31.59, 0), (11.19, 0), (24.16, 0), (34.88, 0), (15.08, 0), (28.45, 1), (36.9, 0), (12.21, 0), (24.98, 1), (38.68, 0), (36.87, 0), (31.37, 0), (28.34, 0), (16.53, 0), (31.72, 1), (27.66, 1), (22.38, 1), (20.56, 0), (19.48, 1), (35.66, 0), (33.58, 0), (49.27, 1), (17.24, 0), (40.55, 0), (37.74, 1), (29.44, 0), (18.36, 1), (11.79, 0), (31.04, 0), (34.09, 0), (11.11, 0), (33.29, 0), (28.18, 0), (55.2, 1), (44.07, 0), (11.72, 0), (12.69, 0), (19.31, 0), (28.41, 0), (27.98, 0), (11.51, 0), (22.28, 0), (42.81, 1), (19.21, 0), (62.28, 1), (34.82, 0), (13.07, 0), (30.88, 0), (33.27, 0), (30.35, 0), (-1.61, 0), (44.15, 0), (34.8, 0), (37.59, 1), (29.74, 0), (42.8, 1), (38.92, 1), (26.41, 0), (16.4, 0), (7.39, 0), (20.37, 1), (18.92, 0), (32.77, 0), (26.06, 0), (26.91, 0), (42.9, 0), (26.89, 0), (30.79, 1), (46.67, 0), (14.74, 0), (-1.85, 0), (37.9, 0), (46.67, 0), (33.5, 0), (27.22, 0), (35.0, 1), (18.1, 0), (21.16, 0), (24.11, 1), (26.66, 0), (20.41, 0), (18.33, 0), (31.72, 1), (10.58, 0), (26.53, 0), (37.83, 1), (16.45, 0), (43.48, 0), (8.12, 0), (15.14, 0), (20.44, 1), (46.75, 1), (-14.72, 0), (51.83, 0), (3.14, 0), (16.75, 0), (27.06, 0), (36.56, 0), (29.85, 0), (33.04, 1), (35.38, 0), (37.74, 0), (30.38, 0), (27.18, 0), (40.36, 1), (10.12, 0), (33.0, 1), (33.32, 0), (23.72, 0), (27.33, 1), (19.85, 0), (20.95, 0), (19.1, 0), (41.17, 1), (21.44, 0), (40.95, 0), (27.56, 1), (20.48, 0), (27.32, 0), (31.21, 1), (29.03, 0), (40.81, 0), (42.42, 1), (14.09, 0), (13.56, 0), (23.49, 0), (43.03, 1), (29.53, 0), (31.63, 1), (15.19, 0), (32.8, 1), (29.12, 0), (28.16, 0), (23.86, 0), (33.66, 1), (39.42, 0), (30.99, 0), (23.08, 0), (54.64, 1), (-0.96, 0), (9.21, 0), (43.58, 1), (29.96, 0), (26.47, 0), (40.77, 0), (25.09, 0), (25.97, 0), (23.17, 0), (47.91, 1), (12.7, 0), (36.72, 0), (40.56, 0), (24.61, 0), (34.67, 0), (33.35, 1), (38.18, 0), (54.14, 1), (38.86, 0), (27.09, 0), (39.8, 0), (21.88, 0), (15.7, 0), (24.68, 1), (38.67, 0), (34.29, 0), (17.14, 0), (37.27, 0), (40.09, 1), (24.33, 0), (35.83, 0), (27.59, 0), (33.39, 0), (29.97, 0), (34.71, 0), (17.31, 0), (56.26, 0), (25.95, 0), (36.61, 0), (26.45, 0), (18.53, 0), (41.64, 0), (29.05, 0), (28.43, 0), (22.78, 0), (29.51, 0), (45.03, 1), (36.27, 0), (50.44, 1), (4.18, 0), (46.25, 1), (37.75, 0), (39.95, 0), (5.35, 0), (43.73, 0), (18.51, 0), (31.65, 0), (20.2, 0), (25.8, 1), (24.42, 1), (52.59, 0), (24.36, 0), (7.23, 0), (-7.83, 0), (8.43, 0), (55.79, 1), (28.48, 0), (14.65, 0), (21.96, 1), (18.37, 0), (23.46, 0), (34.82, 0), (27.29, 0), (53.14, 0), (30.95, 0), (28.34, 0), (42.45, 0), (37.02, 1), (16.97, 0), (26.49, 0), (10.68, 0), (46.23, 1), (42.31, 0), (30.03, 0), (57.14, 1), (24.18, 0), (24.53, 0), (47.32, 1), (23.61, 0), (29.11, 1), (41.2, 0), (9.32, 0), (35.37, 0), (33.75, 0), (26.34, 0), (36.77, 1), (28.07, 0), (38.95, 0), (25.85, 0), (24.0, 0), (22.64, 0), (18.67, 0), (12.81, 0), (40.14, 1), (31.4, 0), (23.71, 0), (36.08, 1), (35.06, 0), (33.53, 0), (19.82, 0), (39.06, 0), (46.78, 0), (32.72, 1), (40.46, 1), (45.53, 1), (44.86, 0), (22.57, 1), (24.75, 1), (35.84, 1), (28.93, 0), (26.55, 0), (31.44, 0), (57.77, 0), (42.46, 0), (31.58, 0), (64.94, 1), (19.52, 0), (39.89, 1), (34.3, 1), (24.52, 1), (25.19, 0), (19.3, 0), (32.89, 0), (31.39, 1), (40.5, 0), (18.9, 0), (25.74, 0), (28.66, 0), (22.26, 0), (26.28, 0), (26.95, 1), (58.16, 1), (16.48, 0), (43.5, 0), (60.79, 0), (56.16, 1), (36.13, 0), (35.86, 1), (23.81, 0), (35.06, 0), (15.48, 0), (9.33, 0), (24.63, 0), (28.81, 0), (27.37, 0), (39.87, 1), (31.13, 0), (26.96, 0), (37.41, 1), (39.39, 0), (14.07, 0), (19.92, 0), (20.31, 0), (14.1, 0), (30.73, 0), (7.25, 0), (36.88, 1), (46.7, 1), (26.4, 0), (35.24, 0), (33.6, 0), (34.33, 1), (15.03, 0), (12.39, 1), (20.5, 1), (14.2, 0), (12.4, 0), (17.54, 0), (37.02, 1), (34.53, 0), (32.12, 1), (14.27, 0), (21.66, 0), (27.14, 1), (11.84, 0), (30.23, 0), (36.05, 1), (31.38, 1), (35.39, 0), (25.62, 0), (29.67, 0), (26.37, 0), (25.56, 1), (37.77, 0), (14.26, 1), (33.56, 0), (33.32, 1), (22.2, 0), (-4.39, 0), (35.39, 1), (5.26, 0), (43.96, 1), (29.53, 0), (55.51, 1), (39.64, 0), (32.25, 0), (28.46, 0), (37.23, 1), (18.8, 0), (42.1, 0), (27.45, 0), (24.6, 1), (39.66, 0), (7.2, 1), (33.3, 1), (20.56, 0), (8.41, 0), (37.13, 0), (41.03, 1), (20.92, 1), (27.67, 0), (3.39, 0), (30.87, 0), (18.12, 0), (15.6, 0), (24.07, 0), (19.03, 0), (48.12, 0), (35.56, 0), (34.66, 0), (11.47, 0), (33.89, 1), (33.57, 1), (64.19, 1), (24.12, 0), (36.97, 1), (19.49, 0), (25.44, 0), (30.8, 0), (48.9, 1), (45.98, 0), (30.41, 1), (5.32, 0), (21.46, 0), (22.27, 1), (48.02, 0), (35.74, 0), (46.92, 0), (27.66, 0), (18.42, 0), (17.09, 1), (40.57, 0), (33.55, 1), (40.87, 1), (23.86, 0), (37.92, 0), (41.03, 0), (46.44, 1), (20.5, 0), (38.44, 1), (22.72, 0), (26.59, 0), (31.08, 0), (26.36, 0), (24.83, 1), (17.38, 0), (15.19, 0), (20.48, 0), (42.16, 1), (41.37, 0), (23.44, 0), (18.42, 0), (14.9, 0), (39.84, 0), (15.21, 0), (37.18, 0), (34.66, 1), (38.86, 0), (42.09, 0), (26.54, 0), (35.8, 0), (9.13, 0), (32.09, 0), (21.57, 0), (45.16, 0), (39.7, 1), (10.83, 0), (15.13, 0), (16.2, 0), (42.47, 1), (45.48, 1), (20.17, 0), (27.14, 0), (23.85, 0), (30.56, 0), (38.85, 0), (13.12, 0), (24.29, 0), (37.68, 0), (20.32, 0), (24.94, 0), (12.64, 0), (58.79, 0), (26.99, 0), (30.93, 0), (21.86, 0), (34.2, 0), (37.35, 1), (29.25, 0), (17.78, 0), (27.72, 0), (25.61, 1), (17.44, 0), (23.85, 0), (53.44, 1), (53.65, 0), (30.65, 1), (34.81, 1), (35.16, 0), (21.64, 0), (33.3, 0), (43.0, 0), (36.7, 0), (34.23, 0), (20.46, 0), (21.26, 0), (24.14, 0), (23.44, 0), (10.0, 0), (24.84, 0), (24.13, 0), (31.96, 1), (51.49, 1), (40.75, 0), (22.09, 0), (25.95, 0), (13.13, 0), (19.54, 0), (32.55, 0), (15.78, 0), (21.02, 0), (30.76, 0), (45.34, 1), (34.82, 1), (31.89, 0), (41.6, 0), (20.29, 0), (29.39, 0), (21.33, 0), (30.4, 1), (60.15, 1), (29.26, 0), (48.19, 1), (58.42, 0), (24.73, 1), (39.54, 1), (26.38, 0), (2.1, 0), (17.04, 0), (30.86, 0), (40.18, 0), (33.96, 1), (21.34, 0), (20.58, 0), (23.21, 0), (23.15, 0), (32.7, 1), (27.94, 0), (35.15, 0), (24.8, 0), (33.8, 1), (31.22, 0), (28.76, 0), (43.47, 1), (33.83, 1), (43.1, 0), (6.7, 0), (45.71, 1), (2.04, 0), (29.16, 0), (27.5, 0), (23.93, 0), (37.75, 1), (27.83, 0), (30.7, 0), (32.94, 0), (49.76, 1), (-3.07, 0), (15.67, 0), (21.14, 0), (43.64, 0), (35.88, 0), (41.42, 1), (35.37, 0), (30.76, 0), (28.3, 0), (31.36, 0), (21.92, 0), (42.86, 0), (19.86, 0), (17.55, 0), (27.85, 0), (34.0, 1), (19.17, 1), (39.0, 1), (8.94, 0), (20.26, 0), (10.45, 1), (13.14, 0), (1.34, 0), (47.28, 1), (65.77, 0), (48.86, 1), (33.55, 0), (21.27, 0), (39.98, 0), (20.33, 0), (29.72, 0), (41.27, 0), (14.7, 0), (23.09, 0), (25.71, 1), (37.89, 0), (25.33, 1), (33.5, 0), (33.14, 0), (25.71, 0), (14.58, 0), (40.56, 0), (51.94, 1), (37.13, 0), (15.14, 0), (34.36, 1), (37.24, 0), (48.21, 1), (14.43, 0), (16.38, 0), (39.6, 1), (30.07, 0), (25.73, 1), (24.93, 1), (14.48, 0), (11.13, 0), (44.21, 1), (11.36, 0), (5.96, 0), (35.2, 0), (20.23, 0), (9.47, 0), (46.41, 1), (36.16, 1), (32.57, 0), (43.73, 1), (25.81, 0), (35.92, 0), (23.85, 0), (23.58, 0), (7.05, 0), (30.06, 0), (38.09, 0), (43.68, 1), (29.49, 0), (27.04, 1), (24.65, 0), (21.54, 0), (5.76, 1), (33.86, 0), (31.04, 0), (6.14, 0), (42.86, 0), (27.87, 0), (25.72, 1), (31.19, 1), (33.01, 0), (48.26, 1), (26.99, 1), (27.41, 0), (37.14, 1), (22.59, 0), (40.95, 1), (37.27, 0), (53.03, 0), (33.3, 1), (33.14, 1), (41.47, 1), (41.64, 0), (15.97, 0), (34.02, 1), (43.47, 0), (25.45, 1), (30.19, 0), (30.06, 0), (18.49, 0), (2.97, 0), (27.4, 0), (2.07, 0), (16.72, 0), (12.18, 0), (47.99, 0), (30.62, 1), (30.98, 0), (22.08, 0), (17.46, 0), (21.11, 0), (31.76, 0), (12.3, 0), (25.39, 1), (40.56, 0), (28.44, 1), (16.12, 0), (35.39, 0), (48.72, 1), (21.95, 0), (22.1, 0), (46.45, 0), (51.87, 1), (22.55, 0), (23.99, 0), (38.4, 0), (18.06, 0), (28.24, 0), (32.98, 0), (17.84, 0), (29.92, 0), (35.53, 0), (47.65, 1), (28.15, 1), (32.59, 1), (25.8, 0), (11.92, 0), (9.64, 0), (15.03, 0), (43.39, 1), (32.25, 0), (29.95, 0), (25.52, 0), (47.67, 1), (37.31, 0), (34.0, 0), (52.24, 1), (25.64, 0), (43.22, 1), (33.36, 0), (36.4, 1), (0.6, 0), (40.85, 0), (37.53, 0), (37.14, 1), (30.63, 0), (29.08, 0), (25.48, 0), (31.64, 1), (30.7, 0), (15.07, 0), (29.06, 0), (28.3, 0), (48.75, 1), (35.31, 1), (15.84, 0), (15.3, 0), (30.08, 0), (26.03, 0), (23.51, 0), (41.82, 0), (29.79, 1), (25.58, 0), (54.96, 1), (34.59, 0), (47.36, 0), (21.79, 0), (30.99, 1), (3.12, 0), (25.85, 0), (14.42, 0), (26.45, 1), (40.63, 1), (41.12, 0), (16.43, 0), (18.74, 0), (31.91, 0), (31.85, 1), (26.82, 1), (13.9, 0), (26.08, 0), (21.1, 0), (6.33, 0), (36.9, 0), (25.52, 0), (12.25, 0), (23.33, 0), (9.74, 0), (23.42, 1), (40.98, 0), (40.32, 1), (45.2, 0), (30.78, 0), (31.14, 0), (32.63, 0), (32.31, 1), (21.34, 0), (19.94, 0), (57.01, 1), (30.6, 0), (26.47, 0), (32.85, 0), (49.37, 1), (28.47, 0), (14.08, 0), (24.24, 0), (35.14, 0), (51.03, 1), (34.36, 1), (25.2, 1), (33.97, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1469, OR=0.3176, z=-12.8067, p=1.50e-37；x: β=0.0825, OR=1.0860, z=10.1141, p=4.78e-24；
McFadden 伪 R² = 1−LL/LL₀ = 0.1343；
示例预测（x=30）：P(y=1)=1/(1+e^(−-1.1469))=0.2411。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1469, OR=0.3176, z=-12.8067, p=1.50e-37；x: β=0.0825, OR=1.0860, z=10.1141, p=4.78e-24；
McFadden 伪 R² = 1−LL/LL₀ = 0.1343；
示例预测（x=30）：P(y=1)=1/(1+e^(−-1.1469))=0.2411。
③ 计算结果：
截距: β=-1.1469 OR=0.318 p=1.50e-37；x: β=0.0825 OR=1.086 p=4.78e-24
④ 结论：在 α=0.05 水平下：x显著(p=4.78e-24)。收入高易买。（伪 R²=0.1343，模型解释力有限）
案例 91. 用户活跃度预测是否续费
逻辑回归活跃易续费
① 原始数据：
样本 n=435，逐条 (x, y)：[(22.01, 1), (41.2, 1), (51.58, 1), (25.49, 0), (45.21, 0), (41.45, 0), (18.04, 1), (44.2, 0), (38.76, 0), (22.89, 0), (23.55, 0), (28.77, 1), (25.59, 1), (22.04, 1), (38.14, 1), (23.28, 1), (32.39, 1), (15.65, 0), (53.92, 0), (50.44, 1), (27.26, 0), (49.34, 1), (23.66, 1), (38.5, 0), (54.01, 0), (34.18, 0), (32.53, 0), (22.02, 0), (31.43, 0), (25.34, 0), (20.14, 0), (22.35, 0), (12.95, 0), (36.32, 0), (16.42, 0), (23.52, 1), (34.65, 0), (34.02, 1), (39.61, 0), (31.66, 1), (13.13, 0), (29.53, 0), (11.75, 0), (27.25, 0), (18.64, 0), (48.77, 0), (10.12, 0), (41.81, 1), (42.52, 1), (40.63, 0), (37.61, 0), (4.01, 0), (31.78, 1), (33.4, 1), (34.74, 0), (46.94, 1), (27.19, 0), (16.59, 0), (16.67, 0), (31.08, 0), (25.4, 0), (45.32, 0), (25.88, 0), (30.18, 0), (27.2, 0), (36.77, 0), (40.51, 1), (23.7, 0), (19.84, 0), (37.28, 1), (26.7, 0), (37.12, 0), (26.24, 0), (28.6, 0), (19.16, 0), (20.0, 0), (18.16, 0), (15.94, 0), (38.84, 1), (25.36, 0), (10.46, 0), (32.62, 0), (40.31, 1), (41.5, 1), (36.25, 0), (60.43, 1), (23.82, 0), (31.36, 0), (17.23, 0), (31.4, 0), (26.34, 0), (28.2, 0), (33.44, 0), (28.61, 0), (40.4, 1), (37.84, 0), (40.02, 1), (22.85, 0), (44.97, 0), (25.61, 0), (25.22, 0), (37.36, 0), (28.11, 1), (29.39, 0), (38.78, 0), (12.66, 0), (36.07, 0), (13.42, 0), (36.11, 0), (45.53, 0), (26.92, 0), (34.12, 0), (25.25, 0), (32.64, 1), (18.66, 0), (28.62, 0), (43.11, 0), (16.29, 0), (50.0, 0), (24.13, 0), (32.01, 0), (43.96, 0), (34.17, 0), (44.92, 0), (22.14, 0), (34.93, 0), (45.32, 1), (27.93, 0), (42.58, 0), (41.28, 1), (41.06, 0), (29.77, 0), (31.73, 0), (24.62, 0), (58.41, 1), (29.69, 0), (33.09, 1), (35.39, 1), (15.56, 0), (44.43, 0), (35.02, 0), (35.48, 1), (31.19, 0), (33.03, 0), (9.25, 0), (43.93, 1), (25.46, 0), (33.96, 0), (27.11, 0), (23.34, 0), (19.74, 0), (34.65, 1), (23.49, 0), (32.79, 0), (40.07, 0), (14.86, 0), (24.91, 1), (10.57, 0), (24.77, 1), (39.42, 1), (34.3, 0), (27.9, 1), (51.25, 1), (12.27, 0), (19.19, 0), (26.84, 0), (33.43, 0), (40.07, 1), (23.15, 1), (21.49, 0), (21.73, 1), (43.08, 0), (23.77, 0), (29.54, 1), (27.08, 0), (35.25, 1), (44.28, 0), (29.18, 0), (39.41, 1), (50.2, 1), (28.14, 1), (25.22, 0), (29.72, 0), (35.85, 0), (32.53, 0), (25.67, 0), (46.79, 0), (25.19, 0), (28.09, 0), (25.04, 0), (13.04, 0), (11.45, 0), (18.26, 1), (58.34, 1), (53.75, 1), (34.5, 0), (38.71, 0), (40.09, 0), (24.28, 0), (39.36, 0), (34.89, 1), (31.67, 0), (40.57, 1), (34.16, 1), (35.82, 1), (39.88, 0), (26.51, 0), (24.95, 0), (34.13, 0), (28.86, 0), (25.87, 1), (32.65, 0), (23.62, 0), (33.52, 1), (32.88, 0), (24.85, 0), (33.21, 0), (38.25, 0), (15.72, 0), (32.03, 0), (42.48, 1), (28.64, 0), (33.33, 1), (36.16, 0), (10.67, 0), (17.89, 0), (33.7, 0), (17.95, 0), (37.51, 0), (18.32, 0), (49.51, 1), (34.86, 1), (17.21, 0), (27.06, 0), (34.15, 0), (30.41, 0), (21.01, 0), (32.27, 0), (30.17, 0), (41.62, 1), (37.45, 0), (41.31, 1), (33.56, 0), (23.29, 0), (33.76, 0), (28.67, 0), (33.44, 0), (7.45, 0), (19.96, 0), (45.61, 1), (34.61, 0), (27.82, 0), (31.74, 0), (22.24, 0), (29.03, 0), (50.48, 0), (31.3, 0), (27.43, 0), (42.19, 1), (17.51, 0), (12.84, 0), (36.42, 1), (15.49, 0), (25.68, 0), (18.57, 0), (39.48, 0), (24.81, 0), (23.21, 0), (31.1, 0), (32.01, 0), (31.62, 0), (24.07, 0), (26.29, 0), (20.84, 0), (36.33, 1), (28.59, 0), (34.67, 0), (35.2, 0), (31.35, 0), (27.71, 0), (37.83, 0), (19.42, 0), (25.26, 0), (39.17, 0), (45.66, 0), (34.97, 0), (29.22, 0), (24.67, 0), (24.39, 0), (42.92, 1), (17.62, 0), (26.03, 1), (18.12, 0), (33.78, 0), (33.03, 0), (13.42, 0), (18.45, 0), (25.23, 0), (47.4, 1), (33.4, 0), (38.98, 0), (26.83, 0), (28.94, 0), (42.88, 0), (27.08, 1), (26.09, 0), (31.58, 0), (33.01, 1), (30.34, 1), (18.23, 0), (20.74, 0), (27.04, 0), (23.75, 1), (22.2, 0), (20.05, 0), (33.29, 0), (25.89, 0), (18.75, 0), (25.71, 0), (21.6, 1), (28.61, 0), (42.71, 1), (36.18, 1), (12.97, 0), (22.78, 0), (41.69, 1), (32.84, 1), (24.58, 0), (44.98, 0), (52.73, 1), (43.91, 1), (35.2, 1), (33.89, 0), (36.16, 0), (27.12, 0), (28.48, 0), (16.54, 0), (47.31, 0), (18.62, 0), (30.41, 0), (19.83, 0), (25.46, 0), (41.8, 0), (25.15, 0), (27.65, 0), (15.18, 0), (26.65, 0), (26.31, 0), (52.64, 1), (27.68, 0), (34.06, 0), (7.33, 0), (19.16, 0), (21.68, 0), (41.92, 0), (39.5, 0), (23.86, 0), (40.37, 0), (35.64, 1), (10.8, 0), (7.37, 0), (25.69, 0), (52.02, 1), (22.8, 0), (36.82, 0), (53.44, 0), (45.86, 0), (27.32, 1), (16.44, 0), (20.29, 0), (53.05, 1), (18.25, 0), (46.46, 1), (32.39, 0), (37.66, 0), (23.34, 0), (19.32, 0), (27.98, 1), (29.83, 0), (31.33, 0), (22.77, 1), (39.17, 1), (8.83, 0), (38.06, 0), (14.3, 0), (30.28, 0), (24.0, 0), (39.02, 0), (8.28, 0), (26.48, 0), (19.5, 0), (25.07, 0), (20.39, 1), (38.06, 0), (38.36, 0), (19.79, 0), (37.09, 1), (42.16, 1), (23.69, 0), (53.39, 0), (35.85, 1), (26.95, 0), (28.55, 0), (35.7, 0), (38.53, 0), (49.28, 0), (32.96, 1), (44.9, 0), (35.52, 0), (42.48, 0), (35.47, 0), (47.96, 1), (25.35, 0), (27.89, 0), (47.06, 1), (44.9, 1), (19.86, 0), (44.52, 1), (41.17, 0), (35.57, 0), (30.93, 1), (12.24, 0), (26.03, 0), (43.97, 1), (35.6, 0), (28.44, 0), (44.53, 1), (26.16, 0), (13.85, 0), (19.44, 0), (32.36, 0), (35.8, 0), (31.17, 0), (24.51, 0), (34.19, 1)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.3657, OR=0.2552, z=-10.2366, p=1.36e-24；x: β=0.0920, OR=1.0964, z=7.0447, p=1.86e-12；
McFadden 伪 R² = 1−LL/LL₀ = 0.1245；
示例预测（x=30）：P(y=1)=1/(1+e^(−-1.3657))=0.2033。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.3657, OR=0.2552, z=-10.2366, p=1.36e-24；x: β=0.0920, OR=1.0964, z=7.0447, p=1.86e-12；
McFadden 伪 R² = 1−LL/LL₀ = 0.1245；
示例预测（x=30）：P(y=1)=1/(1+e^(−-1.3657))=0.2033。
③ 计算结果：
截距: β=-1.3657 OR=0.255 p=1.36e-24；x: β=0.0920 OR=1.096 p=1.86e-12
④ 结论：在 α=0.05 水平下：x显著(p=1.86e-12)。活跃易续费。（伪 R²=0.1245，模型解释力有限）
案例 92. 页面加载慢预测是否流失
逻辑回归慢易流失
① 原始数据：
样本 n=561，逐条 (x, y)：[(3.32, 1), (0.11, 0), (3.23, 1), (3.58, 1), (3.61, 0), (1.52, 0), (2.1, 0), (2.39, 1), (0.74, 0), (3.63, 1), (3.27, 1), (1.78, 0), (4.35, 1), (2.55, 0), (1.88, 1), (2.91, 0), (3.39, 1), (0.49, 0), (2.85, 1), (0.75, 0), (1.4, 0), (2.76, 1), (1.5, 0), (1.51, 0), (3.0, 1), (1.7, 0), (1.73, 1), (2.29, 0), (1.77, 0), (4.36, 1), (0.81, 0), (2.05, 0), (1.06, 0), (1.77, 0), (2.86, 0), (4.28, 1), (3.74, 0), (3.94, 1), (2.6, 0), (2.26, 0), (2.13, 1), (2.01, 0), (2.79, 0), (0.69, 0), (1.0, 1), (0.71, 0), (2.48, 0), (3.81, 1), (1.84, 0), (0.32, 0), (3.22, 0), (1.69, 1), (1.87, 0), (0.69, 0), (1.4, 0), (1.85, 0), (1.33, 0), (1.54, 1), (1.21, 0), (2.44, 1), (2.03, 0), (2.2, 0), (2.27, 0), (1.06, 0), (1.23, 0), (3.23, 1), (1.41, 1), (2.77, 1), (3.4, 1), (2.36, 0), (2.98, 0), (1.6, 0), (1.8, 0), (1.79, 0), (1.66, 0), (3.81, 0), (2.18, 0), (1.95, 1), (2.29, 1), (3.32, 1), (1.26, 0), (1.93, 1), (0.31, 0), (3.87, 1), (1.92, 0), (3.47, 0), (1.67, 0), (2.54, 1), (3.6, 1), (2.19, 0), (2.97, 1), (1.36, 0), (3.42, 0), (3.08, 1), (3.12, 0), (1.54, 0), (1.32, 1), (1.5, 1), (-0.39, 0), (1.07, 0), (2.03, 1), (1.82, 1), (1.92, 0), (2.09, 1), (3.21, 0), (0.84, 0), (3.9, 1), (2.35, 1), (1.73, 0), (1.66, 1), (1.24, 0), (1.2, 1), (2.58, 0), (2.62, 0), (0.67, 0), (1.48, 0), (1.03, 0), (1.41, 0), (0.78, 0), (1.93, 0), (3.39, 1), (1.93, 0), (1.43, 0), (1.36, 0), (2.71, 1), (2.48, 0), (1.93, 1), (0.93, 0), (1.16, 0), (3.39, 1), (2.75, 0), (0.95, 0), (1.78, 0), (1.61, 0), (0.9, 0), (0.97, 1), (2.38, 0), (2.26, 1), (0.64, 0), (2.65, 1), (2.43, 0), (0.5, 0), (1.1, 0), (1.55, 0), (2.52, 1), (0.73, 0), (1.11, 0), (1.48, 1), (3.39, 0), (2.11, 0), (0.55, 0), (2.08, 0), (2.74, 0), (2.59, 0), (1.67, 0), (0.31, 0), (3.29, 1), (1.89, 0), (1.9, 0), (2.21, 0), (3.68, 0), (1.82, 1), (1.19, 0), (2.0, 0), (2.01, 0), (2.26, 0), (0.99, 0), (2.14, 0), (3.27, 0), (2.62, 1), (2.32, 0), (2.17, 1), (1.62, 0), (2.26, 0), (2.02, 0), (4.99, 1), (2.84, 0), (2.47, 0), (0.56, 0), (1.18, 1), (3.18, 0), (1.98, 0), (0.28, 0), (2.23, 0), (2.55, 0), (1.62, 0), (2.69, 0), (1.49, 0), (2.03, 0), (1.74, 1), (0.66, 0), (3.1, 0), (1.15, 0), (3.56, 1), (2.48, 1), (0.3, 0), (1.58, 0), (1.45, 1), (2.02, 0), (1.2, 0), (2.58, 0), (1.24, 0), (1.82, 0), (2.72, 0), (2.14, 0), (1.23, 0), (0.77, 0), (0.81, 0), (3.41, 1), (2.02, 0), (0.28, 1), (-0.09, 0), (1.86, 0), (0.91, 0), (0.23, 0), (1.74, 0), (2.51, 0), (0.91, 0), (0.71, 0), (2.69, 1), (2.1, 0), (3.62, 1), (2.06, 0), (3.59, 1), (1.39, 0), (1.76, 0), (2.19, 0), (1.7, 0), (2.67, 0), (2.63, 0), (1.86, 0), (1.75, 0), (2.6, 1), (3.52, 0), (1.75, 0), (2.85, 1), (3.78, 1), (2.14, 0), (2.6, 0), (0.65, 0), (2.25, 0), (2.92, 0), (1.14, 0), (1.19, 0), (2.49, 0), (2.68, 1), (1.25, 0), (1.1, 0), (2.11, 0), (2.47, 0), (1.05, 0), (0.44, 0), (0.89, 0), (1.41, 0), (0.7, 1), (2.18, 1), (3.34, 1), (2.82, 0), (1.35, 1), (1.99, 0), (2.91, 0), (0.97, 0), (2.5, 1), (3.64, 1), (1.86, 0), (1.86, 0), (1.54, 0), (3.27, 1), (3.27, 1), (2.38, 0), (1.2, 0), (1.11, 0), (2.18, 1), (2.97, 0), (1.52, 0), (1.62, 0), (0.69, 0), (0.54, 0), (0.12, 0), (1.07, 1), (1.9, 1), (0.64, 0), (1.22, 0), (1.62, 1), (1.14, 0), (1.62, 0), (1.99, 0), (1.28, 0), (1.38, 0), (2.21, 0), (3.5, 1), (1.46, 0), (2.9, 0), (1.44, 0), (3.33, 1), (1.67, 0), (4.31, 1), (2.35, 1), (1.91, 1), (2.09, 0), (2.7, 0), (1.21, 0), (1.91, 0), (3.04, 0), (2.93, 0), (2.24, 1), (2.08, 0), (1.88, 0), (1.32, 0), (2.39, 1), (1.62, 1), (1.06, 0), (2.79, 1), (0.19, 0), (2.97, 1), (2.71, 0), (3.12, 0), (0.05, 0), (2.26, 0), (2.95, 1), (4.4, 1), (2.35, 0), (1.24, 0), (1.82, 1), (3.45, 1), (2.06, 0), (3.37, 1), (1.53, 1), (1.96, 0), (4.11, 1), (1.26, 0), (2.81, 1), (2.58, 0), (0.66, 0), (2.49, 0), (2.33, 0), (2.32, 1), (4.36, 0), (1.9, 0), (1.87, 0), (2.19, 0), (2.29, 0), (2.61, 0), (3.2, 0), (2.41, 1), (2.04, 0), (1.04, 0), (2.31, 0), (1.51, 0), (1.79, 0), (1.17, 0), (0.43, 0), (1.64, 0), (3.13, 0), (2.87, 0), (-0.36, 0), (1.46, 1), (0.64, 1), (1.4, 0), (2.17, 0), (0.2, 0), (0.09, 0), (0.55, 0), (3.11, 0), (1.17, 0), (0.27, 0), (1.36, 0), (4.29, 1), (2.43, 0), (1.64, 0), (0.81, 0), (1.22, 0), (0.96, 0), (2.39, 0), (1.57, 0), (1.85, 0), (2.36, 1), (3.24, 1), (-0.1, 0), (3.61, 0), (1.33, 0), (0.84, 0), (3.62, 0), (3.53, 0), (-0.88, 0), (2.61, 0), (3.83, 1), (1.61, 0), (1.82, 0), (1.19, 0), (2.8, 0), (2.57, 0), (3.66, 0), (1.97, 0), (2.33, 1), (3.04, 0), (0.72, 0), (1.36, 0), (1.0, 0), (2.17, 0), (1.54, 0), (2.44, 1), (2.07, 0), (2.73, 1), (2.44, 0), (0.5, 0), (1.42, 0), (2.77, 0), (3.67, 0), (0.25, 0), (2.49, 0), (3.97, 1), (2.49, 1), (-0.04, 0), (0.6, 0), (1.98, 0), (1.13, 0), (2.69, 0), (2.85, 0), (2.89, 0), (1.37, 0), (2.68, 0), (-0.0, 0), (3.58, 1), (0.82, 0), (2.03, 1), (3.03, 0), (1.76, 0), (0.15, 0), (3.55, 1), (2.39, 0), (2.49, 0), (1.48, 0), (3.28, 1), (2.76, 0), (1.99, 1), (4.67, 1), (1.71, 0), (1.69, 0), (2.61, 0), (3.87, 1), (1.75, 1), (3.34, 0), (1.43, 0), (1.55, 0), (4.2, 1), (1.92, 0), (3.23, 0), (2.89, 1), (1.28, 0), (2.85, 0), (3.15, 1), (2.72, 0), (-0.56, 0), (3.51, 1), (2.8, 1), (2.48, 1), (3.35, 1), (2.31, 0), (1.31, 0), (1.95, 0), (1.69, 1), (1.94, 0), (1.17, 0), (3.34, 1), (2.04, 0), (0.95, 0), (2.01, 0), (0.9, 0), (2.09, 1), (3.37, 1), (1.22, 0), (2.31, 0), (1.67, 0), (2.41, 0), (1.38, 1), (2.39, 0), (3.27, 1), (3.85, 0), (0.81, 0), (-0.16, 0), (0.36, 0), (2.25, 0), (1.5, 0), (2.32, 0), (2.74, 0), (0.36, 0), (2.01, 0), (3.33, 1), (0.6, 0), (2.16, 0), (2.62, 0), (1.01, 0), (2.1, 0), (2.01, 1), (4.47, 1), (0.86, 0), (0.44, 0), (2.25, 0), (2.75, 0), (1.22, 1), (2.38, 0), (3.24, 0), (2.6, 0), (1.2, 0), (2.09, 0), (0.88, 0), (2.88, 0), (1.18, 0), (2.33, 1), (3.24, 0), (2.77, 1), (2.71, 1), (1.23, 0), (2.25, 0), (1.27, 1), (1.23, 0), (0.95, 0), (3.72, 1), (2.31, 0), (0.52, 0), (1.92, 0), (2.32, 0), (3.52, 1), (2.98, 0), (1.4, 1), (3.13, 0), (2.93, 1), (2.58, 0), (1.86, 0), (2.66, 0), (-0.05, 0), (0.63, 0), (0.05, 0), (2.41, 0), (1.39, 0), (-0.02, 0), (2.71, 0), (2.29, 1), (1.58, 0), (2.08, 1), (4.1, 0), (2.38, 1), (1.65, 0), (2.01, 1), (2.65, 1), (0.95, 0), (2.85, 0), (1.7, 0), (2.37, 1), (1.23, 0), (2.46, 1), (2.13, 0), (1.93, 1), (2.49, 0), (1.32, 0), (3.4, 1), (0.52, 0), (2.69, 0), (3.95, 1), (1.06, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1795, OR=0.3074, z=-10.4223, p=1.96e-25；x: β=1.0707, OR=2.9175, z=8.8376, p=9.78e-19；
McFadden 伪 R² = 1−LL/LL₀ = 0.1519；
示例预测（x=2）：P(y=1)=1/(1+e^(−-1.1795))=0.2351。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1795, OR=0.3074, z=-10.4223, p=1.96e-25；x: β=1.0707, OR=2.9175, z=8.8376, p=9.78e-19；
McFadden 伪 R² = 1−LL/LL₀ = 0.1519；
示例预测（x=2）：P(y=1)=1/(1+e^(−-1.1795))=0.2351。
③ 计算结果：
截距: β=-1.1795 OR=0.307 p=1.96e-25；x: β=1.0707 OR=2.918 p=9.78e-19
④ 结论：在 α=0.05 水平下：x显著(p=9.78e-19)。慢易流失。（伪 R²=0.1519，模型解释力有限）
案例 93. 会员等级与推荐(无显著关系)
逻辑回归等级不影响推荐
① 原始数据：
样本 n=804，逐条 (x, y)：[(4.6, 0), (1.46, 0), (2.96, 0), (2.46, 0), (2.09, 0), (1.72, 0), (3.63, 0), (1.73, 0), (4.13, 1), (3.02, 0), (1.91, 1), (2.58, 0), (3.16, 0), (2.95, 1), (2.5, 0), (1.89, 1), (2.3, 0), (2.98, 0), (4.7, 0), (0.97, 1), (3.55, 0), (4.37, 1), (2.74, 1), (1.28, 0), (1.58, 0), (2.78, 0), (1.81, 0), (3.27, 1), (2.76, 1), (3.23, 0), (5.03, 0), (1.45, 0), (3.98, 0), (1.82, 1), (1.63, 0), (1.98, 0), (3.4, 0), (3.59, 0), (3.16, 1), (2.33, 0), (4.01, 0), (2.68, 0), (2.96, 0), (5.57, 0), (3.22, 0), (2.12, 0), (2.93, 0), (1.74, 0), (2.37, 1), (3.22, 0), (3.52, 1), (1.95, 0), (2.85, 0), (1.79, 0), (2.06, 0), (3.86, 1), (3.79, 0), (2.04, 0), (5.1, 0), (4.72, 0), (4.08, 0), (2.69, 1), (3.9, 1), (4.28, 0), (2.55, 0), (2.96, 0), (1.92, 0), (1.41, 1), (2.28, 0), (4.06, 0), (4.27, 0), (4.15, 0), (2.79, 0), (3.85, 0), (4.18, 0), (3.69, 1), (3.18, 1), (4.46, 0), (3.78, 0), (3.36, 1), (3.17, 0), (3.32, 0), (1.95, 1), (2.77, 0), (2.17, 0), (3.58, 0), (2.83, 0), (2.48, 0), (2.97, 1), (5.12, 1), (2.49, 0), (2.2, 0), (4.99, 1), (0.82, 0), (1.0, 0), (4.15, 0), (3.9, 1), (2.31, 0), (2.52, 0), (3.65, 0), (2.22, 1), (1.59, 0), (1.65, 0), (2.69, 0), (3.13, 1), (3.03, 0), (2.38, 1), (3.58, 0), (2.63, 0), (4.99, 1), (1.35, 0), (3.69, 1), (3.02, 0), (1.52, 1), (1.24, 0), (3.73, 1), (2.98, 0), (3.03, 0), (4.47, 0), (3.17, 0), (4.96, 0), (4.12, 0), (4.26, 1), (1.61, 1), (5.32, 0), (1.92, 0), (2.52, 0), (4.21, 0), (3.62, 1), (3.11, 0), (3.92, 0), (2.24, 0), (2.65, 1), (2.74, 0), (2.84, 0), (3.67, 1), (2.61, 1), (1.94, 1), (2.63, 0), (2.49, 1), (3.05, 0), (3.74, 0), (2.19, 0), (4.26, 1), (2.2, 0), (3.15, 0), (2.74, 0), (4.49, 0), (2.76, 0), (2.8, 0), (4.74, 0), (3.46, 0), (2.11, 0), (1.72, 1), (3.97, 1), (3.0, 0), (2.58, 0), (2.72, 0), (4.08, 0), (2.6, 0), (3.53, 1), (2.77, 0), (1.64, 0), (2.98, 0), (3.87, 1), (2.45, 0), (1.89, 0), (3.21, 0), (3.35, 0), (3.28, 1), (0.78, 0), (3.21, 0), (5.12, 0), (3.79, 1), (2.76, 0), (2.19, 1), (2.47, 0), (3.51, 0), (2.38, 0), (4.26, 0), (3.12, 0), (3.62, 1), (2.92, 0), (3.8, 0), (4.37, 0), (3.04, 0), (1.46, 0), (2.24, 0), (4.06, 0), (2.1, 0), (2.11, 1), (2.35, 0), (3.29, 0), (2.8, 0), (2.08, 0), (3.29, 0), (3.66, 1), (2.45, 0), (2.85, 0), (3.47, 1), (3.04, 0), (4.14, 0), (3.95, 0), (5.58, 0), (2.84, 0), (2.13, 0), (3.81, 0), (2.33, 0), (4.02, 1), (3.06, 1), (0.96, 0), (4.65, 0), (2.9, 1), (2.83, 0), (0.52, 1), (1.32, 0), (4.03, 0), (3.25, 0), (2.53, 0), (1.02, 0), (3.58, 0), (3.87, 1), (2.24, 0), (0.43, 0), (4.43, 0), (2.42, 0), (4.39, 0), (3.95, 0), (4.26, 0), (2.54, 0), (4.4, 0), (2.61, 1), (1.66, 0), (1.24, 0), (2.87, 0), (1.59, 0), (1.84, 0), (2.19, 0), (4.23, 0), (1.77, 0), (3.94, 0), (2.72, 0), (4.15, 0), (2.82, 0), (2.55, 1), (4.05, 0), (1.69, 0), (2.01, 0), (1.6, 0), (2.46, 0), (3.06, 0), (3.98, 0), (2.06, 0), (1.81, 0), (2.58, 0), (1.69, 0), (2.11, 0), (3.05, 0), (2.91, 0), (3.58, 1), (3.72, 1), (3.63, 1), (1.58, 1), (1.28, 0), (3.34, 0), (3.37, 0), (2.38, 0), (3.56, 1), (1.79, 1), (3.77, 0), (3.03, 0), (3.1, 0), (3.71, 1), (3.76, 0), (0.66, 0), (2.17, 0), (3.38, 1), (3.68, 0), (1.87, 0), (4.3, 0), (3.76, 0), (1.84, 0), (4.36, 0), (2.27, 0), (2.52, 0), (1.44, 0), (1.7, 0), (2.54, 1), (3.75, 1), (3.91, 0), (3.15, 0), (3.02, 0), (3.66, 0), (2.86, 0), (5.29, 0), (4.82, 1), (2.15, 0), (1.81, 0), (4.13, 0), (2.99, 0), (2.55, 0), (4.58, 0), (3.1, 0), (4.62, 0), (2.36, 1), (3.71, 0), (4.24, 0), (2.82, 1), (3.26, 0), (3.27, 1), (3.62, 1), (-0.37, 1), (-0.01, 0), (5.56, 0), (3.79, 0), (3.15, 0), (3.19, 0), (1.52, 0), (3.74, 0), (2.64, 0), (2.81, 0), (2.54, 0), (3.21, 1), (3.97, 0), (4.98, 0), (3.23, 0), (2.75, 0), (4.01, 1), (2.97, 0), (2.48, 0), (2.62, 0), (3.25, 0), (2.71, 1), (2.26, 0), (2.92, 0), (1.62, 0), (3.18, 0), (2.6, 0), (4.4, 0), (1.01, 0), (3.08, 0), (1.98, 0), (3.82, 1), (4.47, 0), (3.5, 1), (5.71, 1), (3.78, 1), (4.5, 0), (3.39, 0), (0.68, 0), (2.88, 1), (4.34, 0), (1.93, 0), (3.0, 0), (2.2, 0), (3.66, 1), (4.02, 0), (2.16, 0), (3.62, 1), (3.81, 0), (2.7, 1), (2.95, 0), (3.08, 0), (2.7, 0), (3.04, 1), (3.39, 0), (2.39, 1), (3.2, 1), (2.27, 0), (5.42, 0), (2.9, 0), (2.11, 0), (2.04, 0), (2.38, 0), (1.34, 0), (3.77, 0), (3.26, 1), (3.32, 0), (2.0, 0), (1.99, 0), (3.78, 0), (2.28, 1), (2.65, 0), (3.04, 0), (2.8, 1), (3.02, 0), (2.33, 1), (2.94, 0), (1.58, 0), (4.03, 1), (3.47, 0), (2.29, 0), (3.67, 0), (3.31, 0), (2.99, 0), (1.59, 0), (2.8, 0), (2.86, 0), (2.28, 0), (3.42, 0), (2.86, 0), (3.56, 0), (4.82, 0), (2.82, 0), (2.21, 0), (3.48, 0), (3.59, 0), (3.97, 0), (1.32, 0), (3.0, 1), (3.46, 1), (2.54, 0), (2.77, 0), (2.83, 0), (3.29, 0), (2.95, 0), (3.55, 0), (3.99, 0), (3.12, 0), (3.25, 0), (4.64, 0), (3.43, 0), (2.19, 0), (3.28, 0), (4.16, 0), (4.08, 0), (3.04, 1), (3.73, 1), (3.67, 0), (3.67, 0), (2.8, 0), (1.9, 0), (4.62, 0), (2.63, 0), (3.22, 0), (2.94, 0), (2.86, 1), (2.55, 0), (3.04, 0), (3.54, 0), (2.46, 1), (2.58, 1), (5.19, 0), (2.7, 0), (3.68, 0), (3.68, 0), (0.77, 0), (2.18, 0), (3.02, 0), (3.25, 0), (2.67, 1), (2.71, 1), (3.48, 1), (1.1, 0), (2.28, 1), (2.31, 0), (4.41, 0), (4.45, 1), (4.36, 1), (3.94, 0), (3.21, 1), (2.04, 0), (0.28, 0), (4.56, 0), (1.16, 0), (2.09, 1), (3.59, 0), (2.49, 0), (3.79, 0), (3.05, 0), (3.2, 0), (4.86, 1), (3.01, 1), (4.32, 0), (3.41, 0), (3.65, 0), (2.13, 0), (2.26, 0), (3.63, 0), (3.32, 0), (2.05, 1), (1.44, 0), (2.01, 1), (1.77, 0), (2.54, 0), (1.87, 0), (2.03, 1), (2.65, 0), (3.79, 0), (2.81, 1), (0.58, 1), (3.53, 0), (3.31, 0), (2.34, 0), (4.53, 1), (3.28, 0), (2.32, 1), (3.06, 1), (3.79, 1), (4.22, 0), (3.4, 0), (1.87, 0), (3.9, 0), (1.41, 0), (3.27, 0), (2.94, 0), (1.96, 0), (4.51, 0), (1.18, 1), (2.99, 1), (3.14, 0), (4.64, 0), (3.58, 0), (2.97, 0), (1.7, 0), (3.34, 0), (3.1, 1), (3.11, 0), (2.29, 0), (0.57, 0), (2.35, 0), (2.4, 0), (3.47, 0), (1.58, 0), (1.93, 0), (3.53, 0), (3.62, 1), (2.59, 1), (1.14, 0), (1.6, 0), (5.11, 0), (3.04, 0), (2.21, 1), (4.27, 1), (2.52, 1), (3.32, 0), (3.21, 0), (2.69, 0), (4.02, 0), (2.59, 0), (2.55, 0), (4.52, 0), (4.17, 0), (2.19, 1), (2.97, 0), (3.13, 1), (4.06, 0), (2.96, 0), (2.42, 0), (2.73, 0), (3.18, 0), (1.86, 0), (2.55, 0), (4.96, 0), (3.22, 0), (2.97, 0), (3.78, 0), (2.82, 0), (4.4, 0), (2.18, 0), (3.0, 0), (2.72, 0), (2.05, 0), (3.17, 0), (2.23, 0), (2.27, 0), (3.95, 1), (3.25, 0), (2.54, 0), (2.03, 0), (4.07, 0), (3.13, 0), (3.43, 0), (3.85, 0), (3.51, 0), (2.53, 0), (4.33, 0), (2.71, 1), (4.06, 1), (3.71, 0), (3.35, 0), (3.2, 0), (2.22, 0), (2.24, 0), (1.58, 0), (1.41, 0), (2.51, 0), (3.97, 0), (3.05, 0), (2.96, 0), (3.23, 0), (4.6, 0), (3.39, 0), (3.56, 1), (3.2, 0), (4.52, 0), (1.5, 0), (3.11, 0), (3.56, 1), (2.86, 0), (2.78, 0), (3.14, 0), (3.03, 0), (2.18, 0), (4.71, 0), (1.03, 0), (3.74, 0), (3.17, 1), (1.34, 0), (3.69, 0), (4.16, 0), (3.02, 0), (2.42, 0), (2.78, 0), (-0.17, 1), (2.01, 0), (3.64, 1), (1.95, 1), (3.3, 1), (2.6, 0), (2.57, 0), (3.66, 0), (2.64, 0), (4.38, 0), (4.06, 1), (3.78, 1), (2.0, 1), (2.91, 0), (1.57, 0), (2.84, 0), (3.8, 0), (4.6, 0), (3.57, 0), (4.38, 0), (4.62, 0), (1.93, 0), (4.91, 0), (3.42, 1), (3.11, 1), (2.76, 0), (1.62, 0), (1.76, 0), (1.38, 1), (3.66, 1), (3.09, 0), (4.21, 0), (3.62, 0), (3.49, 0), (3.87, 0), (2.14, 0), (3.92, 0), (2.61, 0), (3.11, 0), (4.07, 0), (3.68, 0), (4.68, 1), (3.08, 0), (2.59, 1), (3.53, 0), (2.46, 0), (2.06, 0), (2.5, 0), (3.72, 0), (3.89, 0), (3.19, 0), (3.39, 0), (2.71, 0), (3.47, 1), (3.84, 1), (3.07, 0), (2.49, 0), (3.44, 0), (4.45, 0), (2.1, 0), (3.39, 0), (1.38, 0), (1.34, 0), (4.56, 0), (2.31, 0), (0.68, 0), (2.04, 0), (4.33, 0), (4.06, 1), (1.99, 0), (3.68, 0), (3.06, 1), (2.09, 1), (2.82, 1), (2.22, 0), (3.17, 0), (1.37, 0), (1.54, 1), (0.88, 0), (3.44, 0), (3.26, 0), (3.33, 0), (4.86, 0), (1.43, 0), (2.4, 0), (2.0, 0), (3.18, 0), (2.81, 0), (3.94, 0), (3.5, 0), (2.5, 0), (3.63, 0), (3.11, 0), (3.61, 0), (2.27, 0), (1.81, 1), (2.78, 0), (3.83, 0), (1.79, 0), (1.83, 0), (3.38, 0), (4.11, 1), (3.93, 0), (1.77, 1), (3.31, 1), (3.27, 0), (4.84, 1), (3.33, 0), (2.68, 0), (4.76, 0), (4.04, 0), (1.92, 0), (1.92, 1), (4.3, 0), (1.99, 1), (5.4, 0), (3.32, 0), (2.71, 1), (1.16, 0), (1.56, 0), (2.39, 0), (2.59, 0), (5.51, 0), (2.57, 1), (2.86, 0), (4.22, 0), (3.63, 0), (4.74, 0), (1.99, 0), (2.77, 0), (0.5, 0), (0.84, 0), (4.79, 0), (3.45, 0), (2.71, 0), (2.92, 1), (2.82, 1), (3.89, 0), (2.15, 0), (3.82, 0), (2.54, 0), (3.36, 1), (2.07, 0), (1.93, 0), (3.2, 0), (3.62, 0), (5.19, 1), (3.16, 0), (3.12, 0), (3.92, 0), (0.78, 0), (2.25, 0), (2.71, 1), (4.05, 0), (2.17, 0), (4.67, 0), (2.8, 0), (3.13, 0), (3.91, 1), (4.23, 0), (3.55, 0), (2.84, 1), (2.0, 1), (3.95, 0), (2.28, 0), (3.27, 1), (2.82, 0), (4.12, 1), (2.58, 1), (5.61, 0), (2.85, 1), (2.64, 0), (2.99, 0), (3.3, 1), (3.62, 0), (5.05, 0), (2.69, 0), (5.15, 1), (3.59, 1), (2.07, 0), (1.98, 0), (2.54, 0), (2.62, 0), (3.91, 1), (4.55, 0), (1.44, 0), (2.88, 0), (1.43, 0), (0.67, 0), (3.78, 1)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.2312, OR=0.2920, z=-14.5802, p=3.76e-48；x: β=0.0917, OR=1.0961, z=1.0959, p=0.2731；
McFadden 伪 R² = 1−LL/LL₀ = 0.0014；
示例预测（x=3）：P(y=1)=1/(1+e^(−-1.2312))=0.2260。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.2312, OR=0.2920, z=-14.5802, p=3.76e-48；x: β=0.0917, OR=1.0961, z=1.0959, p=0.2731；
McFadden 伪 R² = 1−LL/LL₀ = 0.0014；
示例预测（x=3）：P(y=1)=1/(1+e^(−-1.2312))=0.2260。
③ 计算结果：
截距: β=-1.2312 OR=0.292 p=3.76e-48；x: β=0.0917 OR=1.096 p=0.2731
④ 结论：在 α=0.05 水平下：x不显著(p=0.2731)。等级不影响推荐。（伪 R²=0.0014，模型解释力有限）
案例 94. 年龄预测是否患老花
逻辑回归年龄大风险高
① 原始数据：
样本 n=611，逐条 (x, y)：[(41.81, 1), (31.54, 0), (44.8, 0), (73.81, 1), (51.85, 0), (44.68, 0), (41.11, 0), (49.96, 0), (41.19, 0), (51.07, 0), (9.71, 0), (7.44, 0), (61.34, 1), (17.28, 0), (40.33, 1), (46.95, 1), (56.14, 1), (25.24, 0), (39.26, 0), (59.77, 0), (43.29, 0), (47.41, 0), (32.91, 0), (45.82, 0), (45.6, 0), (44.04, 0), (27.05, 0), (50.79, 0), (40.13, 1), (40.66, 1), (35.42, 0), (30.95, 0), (28.22, 0), (63.08, 0), (21.55, 0), (35.66, 0), (34.23, 0), (24.56, 0), (8.58, 0), (60.7, 0), (53.5, 1), (43.95, 1), (42.89, 1), (31.66, 0), (41.71, 0), (25.42, 0), (61.86, 0), (56.82, 0), (27.33, 0), (40.09, 0), (12.33, 0), (68.35, 0), (52.27, 1), (57.52, 1), (31.93, 0), (24.52, 0), (16.08, 0), (70.06, 1), (30.9, 0), (27.82, 0), (22.73, 0), (3.12, 0), (40.39, 0), (23.87, 0), (6.27, 0), (40.75, 0), (45.55, 0), (41.14, 1), (49.08, 1), (56.37, 1), (53.42, 0), (38.52, 0), (34.36, 0), (58.8, 1), (47.87, 0), (41.95, 0), (28.17, 0), (44.81, 0), (24.05, 0), (56.23, 0), (21.07, 0), (53.63, 1), (40.12, 0), (40.32, 0), (28.72, 0), (62.37, 0), (41.64, 0), (35.48, 0), (36.67, 0), (48.62, 0), (31.95, 0), (50.43, 0), (40.32, 0), (34.82, 0), (39.88, 0), (62.4, 1), (60.38, 0), (44.46, 0), (26.6, 0), (52.17, 1), (40.06, 0), (59.08, 1), (57.28, 1), (38.38, 0), (58.23, 0), (12.98, 0), (37.49, 0), (38.65, 0), (19.81, 1), (55.16, 0), (14.97, 0), (35.45, 0), (58.39, 0), (41.76, 0), (34.66, 1), (51.98, 1), (51.46, 1), (12.51, 0), (26.24, 1), (59.85, 0), (42.48, 0), (29.43, 0), (35.23, 0), (34.68, 0), (59.09, 0), (21.31, 0), (31.32, 1), (43.9, 0), (30.79, 1), (49.0, 0), (28.69, 0), (37.47, 0), (34.32, 0), (11.99, 0), (20.56, 0), (22.61, 0), (24.62, 0), (56.29, 0), (29.23, 0), (50.4, 1), (46.94, 1), (27.93, 0), (57.21, 1), (32.22, 0), (46.64, 0), (40.24, 0), (53.5, 0), (28.42, 1), (47.39, 0), (25.96, 1), (24.3, 0), (13.33, 0), (45.29, 0), (34.79, 0), (69.53, 1), (34.77, 1), (37.91, 0), (36.16, 0), (48.11, 0), (47.43, 0), (25.19, 1), (49.37, 1), (23.05, 0), (26.07, 0), (40.47, 0), (53.95, 0), (28.11, 0), (29.52, 0), (48.28, 1), (46.23, 1), (48.47, 1), (33.44, 0), (42.86, 1), (65.01, 1), (13.93, 0), (18.55, 0), (59.0, 1), (39.75, 0), (62.34, 0), (26.62, 0), (53.35, 0), (43.98, 0), (10.72, 0), (25.24, 0), (40.08, 0), (29.65, 0), (44.15, 1), (20.34, 0), (46.67, 0), (66.29, 1), (45.49, 1), (38.09, 0), (37.53, 0), (58.32, 1), (49.37, 0), (57.14, 0), (48.2, 0), (24.91, 0), (10.9, 0), (58.71, 0), (23.86, 0), (31.02, 0), (30.53, 0), (21.87, 0), (45.35, 1), (35.8, 0), (77.49, 0), (39.52, 0), (30.07, 0), (4.5, 0), (78.88, 1), (23.14, 0), (46.22, 0), (24.68, 0), (9.94, 1), (37.74, 0), (33.6, 0), (44.44, 0), (43.75, 0), (52.88, 0), (23.9, 0), (40.62, 1), (44.03, 1), (29.48, 0), (24.99, 0), (9.21, 0), (39.74, 0), (63.12, 1), (55.81, 1), (52.78, 0), (49.29, 0), (39.36, 0), (54.45, 1), (26.46, 0), (26.88, 0), (37.12, 1), (30.45, 0), (20.57, 0), (29.27, 0), (20.55, 0), (37.88, 0), (30.33, 0), (36.48, 1), (18.36, 0), (11.41, 0), (41.64, 0), (47.06, 0), (28.44, 0), (34.39, 0), (45.19, 0), (56.53, 0), (53.55, 0), (52.8, 1), (55.27, 0), (51.29, 0), (47.37, 1), (31.54, 0), (37.53, 1), (17.0, 0), (58.91, 1), (31.17, 0), (52.92, 0), (39.59, 1), (42.0, 1), (21.52, 0), (38.48, 0), (36.9, 0), (37.71, 0), (39.34, 0), (11.92, 0), (37.79, 0), (16.87, 0), (73.62, 1), (41.09, 1), (48.28, 0), (40.94, 0), (11.09, 1), (63.48, 1), (37.4, 1), (39.06, 0), (70.68, 0), (44.89, 1), (41.71, 0), (49.09, 1), (32.06, 0), (37.11, 0), (22.82, 1), (32.03, 0), (26.18, 0), (41.52, 0), (-0.57, 0), (53.51, 0), (58.52, 0), (29.32, 0), (53.34, 1), (24.44, 0), (32.93, 0), (53.51, 0), (1.43, 0), (28.15, 1), (67.0, 1), (38.5, 0), (36.88, 0), (40.36, 0), (34.88, 0), (24.9, 0), (45.65, 0), (24.35, 0), (58.17, 0), (11.3, 0), (36.04, 0), (64.49, 1), (18.08, 0), (39.63, 0), (22.9, 0), (37.05, 0), (19.59, 0), (30.89, 0), (46.81, 1), (30.8, 0), (28.02, 0), (32.81, 0), (41.8, 1), (55.34, 1), (54.04, 0), (43.78, 0), (29.54, 0), (45.4, 0), (46.01, 1), (41.3, 0), (27.52, 0), (35.5, 0), (55.58, 1), (47.81, 0), (22.52, 0), (32.39, 0), (26.29, 0), (25.73, 0), (68.79, 0), (45.44, 0), (33.01, 0), (28.71, 1), (44.32, 0), (28.74, 0), (42.65, 1), (69.02, 0), (53.56, 1), (54.03, 1), (34.95, 1), (47.12, 0), (49.97, 1), (63.92, 1), (3.48, 0), (31.14, 1), (55.15, 0), (60.63, 0), (51.56, 0), (17.26, 0), (21.71, 0), (28.27, 1), (32.03, 0), (21.63, 0), (50.01, 1), (22.76, 0), (5.88, 0), (32.45, 0), (51.56, 1), (12.27, 0), (36.94, 0), (52.47, 0), (22.17, 0), (43.32, 0), (35.16, 0), (66.69, 1), (13.21, 0), (36.4, 0), (61.85, 1), (41.29, 0), (26.04, 0), (49.07, 0), (56.45, 0), (52.05, 0), (29.14, 0), (54.28, 1), (43.01, 0), (29.57, 1), (46.35, 0), (38.38, 0), (29.68, 0), (47.32, 1), (29.34, 0), (30.69, 0), (26.78, 0), (30.98, 0), (45.46, 1), (27.81, 0), (26.47, 0), (34.21, 0), (36.66, 0), (69.25, 1), (36.49, 1), (36.71, 0), (18.9, 0), (58.32, 0), (54.46, 0), (41.8, 0), (30.53, 0), (5.79, 0), (46.82, 1), (49.4, 1), (36.56, 0), (73.78, 0), (43.24, 0), (31.05, 0), (43.65, 0), (41.4, 0), (54.65, 0), (33.58, 1), (22.45, 0), (32.16, 1), (9.54, 0), (16.38, 0), (45.41, 0), (30.31, 0), (28.25, 0), (64.45, 0), (58.35, 0), (25.77, 0), (37.36, 0), (22.34, 0), (50.23, 1), (39.77, 0), (34.47, 0), (47.67, 0), (43.81, 1), (50.57, 0), (40.78, 0), (32.89, 0), (42.7, 1), (50.73, 0), (25.89, 0), (26.95, 0), (47.18, 1), (21.71, 0), (41.21, 0), (24.82, 0), (25.76, 0), (36.43, 0), (17.89, 0), (45.84, 0), (39.24, 0), (42.28, 1), (50.2, 0), (27.84, 0), (22.93, 1), (28.55, 0), (21.85, 0), (45.29, 1), (39.22, 0), (36.65, 0), (58.01, 1), (17.8, 0), (70.79, 1), (36.18, 0), (42.33, 0), (25.3, 0), (12.12, 0), (32.95, 1), (34.65, 0), (45.32, 0), (21.27, 0), (40.76, 0), (35.28, 0), (61.13, 0), (33.73, 1), (13.64, 0), (29.89, 0), (39.02, 0), (37.9, 1), (41.27, 0), (46.5, 0), (19.62, 0), (32.43, 0), (9.66, 0), (19.38, 0), (28.62, 0), (43.32, 1), (30.03, 0), (47.94, 1), (39.25, 0), (39.56, 0), (33.93, 0), (23.04, 0), (59.27, 0), (16.73, 0), (37.46, 1), (61.02, 1), (25.69, 0), (54.7, 0), (55.75, 1), (52.84, 0), (59.45, 1), (52.75, 1), (56.03, 1), (48.96, 0), (42.95, 1), (47.51, 1), (49.96, 1), (22.07, 0), (34.42, 0), (31.59, 0), (30.39, 0), (35.31, 1), (46.65, 1), (41.39, 0), (57.23, 1), (32.76, 0), (39.23, 0), (16.99, 0), (47.51, 0), (43.0, 0), (34.69, 0), (26.42, 0), (32.92, 0), (42.37, 0), (19.8, 0), (52.25, 1), (36.39, 0), (41.63, 0), (26.6, 0), (20.84, 0), (63.3, 1), (38.79, 1), (32.96, 1), (51.06, 0), (50.02, 1), (16.71, 0), (58.3, 1), (70.03, 1), (55.44, 0), (46.34, 0), (22.71, 0), (23.02, 0), (42.24, 0), (57.65, 1), (39.19, 0), (29.57, 0), (36.87, 0), (58.79, 1), (32.71, 0), (23.03, 0), (31.37, 0), (61.1, 1), (52.27, 0), (29.99, 1), (41.51, 0), (43.39, 0), (37.89, 0), (52.44, 1), (50.65, 1), (52.84, 1), (35.14, 1), (43.51, 0), (42.79, 1), (57.27, 0), (30.9, 0), (43.48, 1), (49.26, 1), (28.26, 0), (10.96, 0), (58.76, 1), (42.32, 1), (48.75, 0), (35.71, 0), (42.33, 0), (7.75, 0), (11.55, 1), (34.53, 0), (28.39, 0), (52.7, 0), (31.96, 1), (53.96, 1), (48.09, 0), (39.39, 0), (51.96, 1), (26.01, 0), (54.68, 1), (49.51, 0), (16.64, 0), (20.97, 0), (42.55, 0), (34.12, 0), (29.96, 1), (8.36, 0), (40.23, 1), (24.75, 0), (25.31, 1), (44.69, 0), (54.2, 0), (43.13, 0), (49.45, 1), (22.66, 0), (27.48, 0), (37.05, 0), (17.97, 0), (48.0, 0), (49.35, 1), (8.26, 1), (45.47, 0), (39.7, 0), (8.32, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1001, OR=0.3329, z=-10.7721, p=4.66e-27；x: β=0.0633, OR=1.0653, z=8.1233, p=4.54e-16；
McFadden 伪 R² = 1−LL/LL₀ = 0.1144；
示例预测（x=40）：P(y=1)=1/(1+e^(−-1.1001))=0.2497。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1001, OR=0.3329, z=-10.7721, p=4.66e-27；x: β=0.0633, OR=1.0653, z=8.1233, p=4.54e-16；
McFadden 伪 R² = 1−LL/LL₀ = 0.1144；
示例预测（x=40）：P(y=1)=1/(1+e^(−-1.1001))=0.2497。
③ 计算结果：
截距: β=-1.1001 OR=0.333 p=4.66e-27；x: β=0.0633 OR=1.065 p=4.54e-16
④ 结论：在 α=0.05 水平下：x显著(p=4.54e-16)。年龄大风险高。（伪 R²=0.1144，模型解释力有限）
案例 95. 吸烟量预测是否患肺病
逻辑回归吸烟多风险高
① 原始数据：
样本 n=966，逐条 (x, y)：[(14.23, 0), (20.37, 0), (20.87, 1), (13.32, 0), (9.76, 0), (13.99, 1), (4.53, 0), (12.23, 0), (20.06, 1), (11.75, 0), (9.08, 0), (14.33, 0), (8.26, 0), (15.6, 1), (19.04, 0), (9.85, 1), (8.99, 0), (9.88, 0), (9.95, 0), (19.07, 0), (-5.64, 0), (1.81, 0), (18.77, 1), (3.38, 0), (6.95, 1), (1.54, 0), (8.14, 0), (7.14, 0), (3.67, 0), (5.24, 0), (9.13, 0), (4.41, 0), (1.08, 0), (6.34, 1), (9.44, 1), (8.35, 1), (10.97, 0), (7.81, 0), (6.64, 0), (6.29, 0), (15.27, 0), (13.23, 1), (7.45, 0), (5.28, 0), (19.34, 1), (14.57, 0), (21.04, 1), (18.63, 0), (17.88, 1), (6.98, 0), (13.61, 0), (13.89, 1), (5.0, 0), (11.35, 0), (0.85, 0), (20.65, 1), (8.07, 0), (9.78, 0), (-1.24, 1), (18.06, 1), (9.24, 1), (14.44, 0), (12.0, 0), (13.48, 0), (10.78, 0), (16.0, 0), (22.01, 1), (9.1, 0), (1.29, 0), (16.06, 0), (4.78, 0), (26.18, 1), (11.59, 0), (12.1, 0), (14.62, 0), (15.5, 1), (10.7, 0), (9.14, 1), (4.84, 1), (17.94, 1), (6.4, 0), (9.48, 0), (5.86, 1), (16.61, 1), (1.07, 0), (11.15, 0), (5.74, 0), (17.81, 0), (14.43, 0), (14.16, 1), (2.36, 0), (4.5, 0), (20.09, 1), (4.39, 1), (10.75, 1), (7.9, 0), (10.56, 1), (10.97, 0), (15.32, 0), (3.3, 0), (14.06, 0), (17.51, 1), (12.97, 0), (11.29, 1), (-2.33, 0), (2.96, 0), (2.35, 0), (15.81, 1), (10.18, 1), (-2.99, 0), (7.47, 1), (1.56, 0), (-1.31, 0), (14.62, 0), (1.27, 0), (15.89, 0), (6.87, 1), (13.33, 0), (16.1, 0), (11.16, 0), (6.22, 0), (0.55, 0), (-5.75, 0), (10.56, 0), (5.71, 1), (9.24, 0), (22.42, 0), (13.64, 0), (6.82, 0), (10.32, 0), (7.55, 0), (6.56, 0), (0.57, 0), (8.01, 0), (9.94, 0), (15.88, 1), (10.73, 0), (7.03, 0), (-0.63, 0), (13.0, 0), (14.16, 0), (13.22, 0), (13.76, 1), (10.92, 1), (19.03, 0), (8.84, 0), (2.2, 0), (9.34, 0), (8.78, 0), (18.13, 0), (9.37, 0), (5.06, 0), (12.0, 1), (8.21, 0), (17.58, 1), (1.81, 0), (11.93, 0), (12.17, 1), (14.98, 0), (9.72, 0), (6.89, 0), (11.08, 0), (15.68, 0), (12.63, 1), (5.53, 0), (25.11, 0), (14.68, 1), (3.02, 0), (10.42, 0), (1.39, 0), (21.1, 1), (15.42, 1), (14.47, 1), (9.33, 1), (10.68, 0), (14.85, 1), (12.14, 0), (6.07, 0), (4.11, 0), (12.97, 1), (14.9, 0), (10.49, 0), (14.51, 0), (4.82, 0), (12.01, 0), (6.52, 0), (3.01, 0), (6.52, 0), (7.48, 0), (9.56, 0), (12.56, 0), (7.83, 0), (5.64, 0), (4.48, 0), (12.18, 1), (11.1, 0), (9.53, 0), (12.58, 0), (6.37, 0), (10.11, 1), (-1.54, 0), (11.5, 0), (11.12, 0), (18.58, 1), (4.29, 0), (19.99, 1), (30.19, 1), (10.97, 0), (1.68, 0), (8.69, 0), (0.83, 0), (10.47, 0), (12.19, 0), (-4.55, 0), (9.49, 1), (-7.88, 0), (-3.2, 0), (6.82, 0), (17.39, 1), (0.83, 0), (17.65, 0), (6.18, 0), (17.12, 1), (-1.4, 0), (3.85, 1), (12.98, 0), (0.1, 0), (0.42, 0), (14.77, 0), (15.43, 0), (6.19, 0), (9.4, 0), (18.22, 0), (6.81, 1), (7.6, 0), (20.43, 0), (4.77, 0), (-2.57, 0), (17.85, 0), (12.99, 0), (11.66, 0), (1.25, 0), (10.01, 0), (-5.54, 0), (0.42, 0), (13.29, 0), (8.84, 0), (7.92, 0), (12.25, 0), (7.32, 0), (3.16, 0), (15.06, 1), (12.09, 0), (6.07, 0), (14.92, 0), (7.23, 1), (13.36, 0), (7.54, 0), (-7.42, 0), (10.48, 1), (10.83, 0), (10.97, 1), (6.17, 0), (4.44, 0), (12.98, 0), (19.57, 1), (15.85, 0), (10.61, 0), (15.58, 0), (12.4, 0), (0.55, 0), (8.46, 0), (18.24, 1), (12.62, 0), (1.59, 0), (11.77, 0), (11.77, 1), (15.33, 1), (5.21, 0), (0.07, 0), (16.43, 0), (10.69, 0), (2.01, 0), (5.32, 0), (4.02, 0), (2.58, 0), (11.64, 0), (21.74, 1), (6.99, 0), (3.62, 0), (1.72, 0), (9.25, 0), (10.34, 0), (22.03, 1), (8.16, 0), (6.9, 1), (17.5, 1), (7.16, 0), (12.76, 0), (2.27, 1), (1.85, 0), (15.27, 1), (14.12, 1), (3.15, 0), (7.54, 1), (25.11, 0), (14.45, 0), (9.66, 0), (17.4, 0), (9.6, 0), (10.39, 0), (7.29, 0), (10.13, 0), (5.55, 0), (1.61, 1), (5.65, 0), (20.58, 1), (12.26, 0), (17.92, 1), (7.46, 0), (5.99, 0), (13.56, 1), (4.88, 0), (-2.27, 0), (6.24, 1), (7.91, 0), (12.42, 0), (17.92, 1), (10.71, 0), (15.55, 1), (11.39, 0), (10.68, 0), (6.86, 1), (9.63, 0), (4.82, 0), (16.05, 1), (-2.86, 0), (6.34, 0), (16.99, 1), (-2.38, 0), (-2.87, 0), (7.06, 0), (13.83, 0), (-3.97, 0), (8.71, 1), (16.12, 1), (7.16, 0), (12.19, 1), (6.46, 1), (8.73, 0), (7.22, 1), (18.1, 1), (19.69, 0), (1.84, 0), (5.31, 0), (19.56, 1), (14.59, 1), (16.46, 0), (-4.88, 0), (16.39, 0), (28.57, 0), (9.48, 0), (23.54, 1), (17.61, 0), (1.29, 0), (6.0, 0), (9.51, 0), (0.9, 0), (10.38, 0), (12.59, 1), (19.35, 0), (15.57, 1), (10.15, 0), (20.47, 0), (6.07, 1), (16.27, 1), (13.55, 1), (14.18, 0), (15.73, 1), (11.46, 0), (12.42, 0), (-0.32, 0), (15.58, 0), (-1.18, 0), (14.64, 1), (6.88, 1), (4.82, 0), (14.43, 0), (17.23, 0), (5.76, 0), (2.22, 0), (10.24, 0), (10.28, 0), (15.28, 0), (4.98, 0), (3.87, 0), (21.97, 1), (14.56, 1), (8.43, 0), (5.83, 1), (0.39, 0), (14.09, 0), (-1.11, 0), (13.06, 0), (5.68, 0), (7.54, 0), (1.42, 0), (6.34, 0), (16.27, 1), (6.01, 0), (11.96, 0), (13.93, 1), (11.07, 0), (13.83, 0), (12.53, 0), (-1.34, 0), (7.77, 0), (15.29, 1), (15.43, 1), (10.93, 0), (6.83, 0), (9.35, 0), (8.61, 0), (9.07, 1), (-1.49, 1), (2.29, 0), (10.65, 0), (0.1, 0), (17.89, 0), (18.0, 0), (4.3, 0), (17.89, 0), (3.4, 0), (12.07, 0), (6.29, 1), (9.4, 0), (0.92, 0), (6.38, 0), (9.01, 1), (-1.4, 0), (21.92, 1), (3.18, 0), (15.3, 1), (22.91, 1), (0.87, 0), (12.57, 0), (8.76, 0), (4.06, 0), (13.98, 0), (13.75, 0), (12.46, 1), (7.23, 0), (0.12, 0), (9.01, 0), (17.61, 1), (4.08, 0), (25.3, 1), (18.93, 1), (12.21, 1), (22.96, 1), (13.35, 0), (17.04, 1), (3.58, 0), (9.5, 0), (14.41, 0), (0.58, 0), (5.33, 1), (12.67, 0), (11.84, 0), (18.79, 1), (0.41, 0), (4.04, 0), (4.95, 0), (6.09, 0), (18.14, 0), (4.38, 0), (4.13, 0), (8.54, 0), (10.51, 0), (16.24, 0), (13.3, 0), (13.55, 0), (-1.77, 0), (13.75, 0), (6.8, 0), (0.02, 0), (19.68, 1), (14.41, 0), (8.28, 0), (-1.45, 0), (12.63, 1), (12.09, 1), (3.53, 0), (11.71, 0), (12.86, 1), (24.96, 1), (13.98, 0), (14.94, 0), (5.43, 0), (11.33, 1), (16.16, 1), (6.85, 0), (0.97, 0), (3.26, 0), (-0.09, 1), (6.92, 0), (3.19, 0), (11.13, 0), (15.32, 1), (4.62, 0), (13.74, 1), (16.35, 0), (13.52, 1), (15.81, 0), (9.9, 0), (3.76, 0), (7.18, 0), (8.98, 1), (10.15, 0), (3.86, 0), (10.05, 0), (6.82, 0), (16.53, 0), (3.22, 0), (18.83, 0), (11.49, 0), (8.67, 1), (1.9, 0), (15.21, 1), (14.83, 1), (8.75, 0), (11.11, 1), (8.34, 0), (18.62, 1), (13.01, 1), (10.55, 0), (5.28, 0), (5.99, 0), (25.24, 0), (5.37, 0), (10.21, 0), (1.57, 0), (6.45, 0), (22.18, 0), (5.76, 0), (5.73, 0), (23.44, 1), (10.25, 1), (11.32, 1), (8.48, 0), (5.56, 1), (9.83, 1), (5.56, 1), (5.47, 0), (14.91, 0), (11.69, 0), (5.93, 0), (19.33, 1), (20.26, 0), (12.65, 0), (2.36, 0), (4.5, 0), (9.44, 0), (18.8, 0), (5.23, 0), (7.06, 0), (14.29, 0), (17.83, 0), (5.19, 0), (15.1, 1), (7.01, 0), (14.32, 1), (13.4, 0), (11.41, 0), (0.32, 1), (5.93, 0), (15.63, 1), (9.46, 0), (16.49, 0), (10.45, 0), (20.42, 1), (8.85, 0), (3.92, 0), (10.4, 0), (7.07, 0), (4.19, 1), (5.44, 1), (-3.28, 0), (19.44, 1), (8.14, 0), (6.96, 0), (14.76, 0), (2.39, 0), (8.09, 0), (7.08, 0), (12.21, 1), (7.81, 0), (8.17, 0), (15.23, 1), (5.02, 0), (11.6, 1), (10.46, 0), (3.82, 0), (8.84, 0), (12.67, 0), (19.19, 1), (20.16, 1), (16.07, 1), (19.57, 1), (15.56, 0), (7.11, 0), (16.34, 1), (12.86, 0), (17.6, 0), (2.95, 1), (10.64, 0), (8.97, 1), (19.28, 1), (7.15, 0), (4.03, 0), (8.64, 1), (15.71, 0), (8.47, 0), (13.26, 0), (19.36, 1), (-0.17, 0), (12.59, 0), (11.96, 0), (5.0, 0), (15.12, 0), (9.96, 1), (8.68, 0), (15.81, 1), (9.92, 0), (15.75, 0), (10.22, 0), (20.2, 1), (10.25, 0), (9.35, 0), (-0.74, 0), (14.38, 1), (7.53, 0), (10.91, 0), (8.01, 0), (10.29, 0), (19.38, 1), (9.27, 0), (14.05, 0), (14.01, 1), (22.34, 0), (10.2, 1), (15.31, 0), (12.75, 0), (4.41, 0), (1.47, 0), (8.53, 0), (18.48, 1), (11.29, 0), (16.36, 1), (13.05, 0), (12.24, 1), (21.54, 0), (11.14, 0), (5.77, 0), (12.85, 0), (1.46, 0), (13.02, 1), (12.97, 0), (23.58, 1), (13.2, 0), (10.28, 0), (15.08, 1), (9.28, 0), (6.97, 0), (7.52, 0), (3.52, 1), (22.24, 1), (10.09, 0), (8.83, 0), (22.59, 1), (13.11, 0), (16.06, 1), (4.71, 0), (12.15, 0), (15.17, 1), (15.25, 0), (7.02, 0), (16.65, 0), (3.28, 0), (0.85, 0), (7.7, 0), (9.19, 0), (9.57, 0), (15.22, 1), (7.99, 0), (13.67, 0), (17.89, 1), (12.68, 0), (9.15, 0), (3.37, 0), (6.07, 0), (11.6, 0), (3.21, 0), (15.65, 0), (3.42, 1), (9.8, 0), (9.46, 0), (6.61, 0), (5.49, 0), (5.31, 1), (5.09, 1), (11.8, 1), (21.17, 1), (13.98, 1), (7.08, 0), (13.61, 0), (8.63, 0), (5.99, 0), (-1.05, 0), (12.16, 1), (10.26, 0), (12.71, 1), (6.6, 0), (7.67, 0), (7.39, 0), (8.99, 0), (17.87, 1), (4.27, 0), (-2.16, 0), (7.83, 0), (11.5, 0), (12.01, 0), (3.96, 0), (7.05, 0), (7.53, 0), (11.14, 0), (4.91, 0), (11.21, 1), (6.82, 0), (5.44, 0), (15.15, 0), (18.67, 1), (9.95, 0), (14.97, 1), (2.23, 0), (3.03, 0), (19.16, 1), (3.88, 1), (13.13, 1), (8.01, 1), (20.8, 1), (1.99, 0), (12.36, 1), (9.31, 0), (9.46, 0), (7.59, 0), (10.18, 0), (15.33, 1), (12.92, 1), (11.49, 1), (7.75, 0), (11.85, 0), (10.73, 0), (5.91, 0), (3.81, 0), (9.27, 0), (18.74, 0), (11.9, 1), (11.92, 1), (9.66, 0), (6.64, 1), (-0.2, 0), (5.51, 0), (9.0, 0), (12.71, 0), (9.01, 0), (10.98, 0), (11.71, 0), (12.03, 0), (8.68, 0), (2.26, 0), (9.56, 0), (9.67, 0), (6.38, 0), (6.79, 0), (1.46, 0), (19.07, 1), (4.09, 0), (5.56, 0), (5.47, 0), (8.21, 0), (0.9, 0), (5.25, 0), (16.2, 1), (15.29, 0), (15.57, 0), (7.11, 0), (4.24, 0), (10.76, 0), (17.58, 1), (12.49, 0), (12.38, 0), (18.1, 0), (10.93, 0), (9.39, 1), (7.98, 0), (5.25, 0), (9.07, 0), (9.94, 1), (10.67, 0), (12.98, 0), (5.16, 1), (8.4, 0), (13.15, 0), (11.44, 0), (8.0, 1), (1.23, 1), (14.19, 1), (18.62, 1), (11.98, 0), (10.94, 0), (13.33, 1), (8.19, 0), (10.17, 1), (8.37, 0), (8.31, 1), (13.35, 0), (15.32, 1), (9.99, 0), (12.95, 0), (20.71, 0), (20.07, 1), (5.5, 0), (-4.67, 0), (14.02, 0), (2.66, 0), (7.31, 0), (15.57, 0), (1.4, 0), (13.73, 1), (17.51, 0), (9.18, 0), (9.91, 0), (10.88, 0), (8.89, 0), (7.65, 0), (20.41, 1), (3.67, 0), (3.27, 0), (14.34, 0), (24.86, 1), (12.8, 0), (4.94, 0), (5.58, 0), (17.69, 1), (7.75, 0), (18.73, 1), (8.1, 0), (9.36, 1), (11.22, 0), (-5.6, 0), (13.68, 1), (15.02, 1), (19.3, 1), (8.29, 0), (15.64, 1), (16.31, 1), (10.06, 0), (3.93, 0), (9.28, 0), (10.51, 1), (13.89, 0), (7.54, 0), (8.91, 0), (14.06, 0), (3.8, 0), (4.01, 0), (8.38, 1), (5.03, 0), (4.99, 0), (15.71, 0), (12.36, 0), (9.03, 1), (12.87, 1), (19.69, 0), (15.21, 0), (-9.28, 0), (6.85, 0), (7.2, 0), (19.89, 0), (9.21, 0), (11.86, 0), (10.63, 0), (8.39, 0), (10.78, 1), (18.55, 1), (23.68, 1), (4.76, 0), (10.62, 0), (12.12, 1), (3.02, 0), (17.12, 1), (8.71, 0), (6.57, 0), (2.07, 0), (10.76, 0), (21.77, 1), (-3.64, 0), (11.85, 0), (10.56, 0), (15.05, 0), (3.09, 0), (17.91, 0), (17.67, 1), (9.48, 1), (20.31, 0), (1.73, 0), (8.84, 0), (1.32, 0), (11.39, 1), (5.6, 0), (19.95, 0), (16.51, 1), (9.22, 0), (15.15, 1), (2.08, 0), (9.15, 1), (15.54, 0), (8.23, 1), (10.93, 0), (18.54, 0), (6.34, 0), (18.1, 1), (13.04, 0), (5.35, 0), (9.5, 1), (18.6, 0), (2.2, 0), (9.18, 0), (4.78, 0), (16.92, 0), (1.54, 0), (8.5, 0), (16.17, 1), (13.87, 0), (14.26, 0), (10.62, 1), (5.11, 0), (14.08, 0), (-1.48, 0), (20.76, 1), (3.47, 0), (12.17, 1), (9.29, 0), (6.14, 0), (8.93, 0), (17.85, 0), (8.2, 0), (17.58, 0), (7.56, 1), (9.75, 1), (3.05, 0), (19.49, 0), (18.01, 0), (3.9, 0), (2.67, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1428, OR=0.3189, z=-13.6278, p=2.74e-42；x: β=0.1633, OR=1.1774, z=10.8617, p=1.75e-27；
McFadden 伪 R² = 1−LL/LL₀ = 0.1298；
示例预测（x=10）：P(y=1)=1/(1+e^(−-1.1428))=0.2418。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.1428, OR=0.3189, z=-13.6278, p=2.74e-42；x: β=0.1633, OR=1.1774, z=10.8617, p=1.75e-27；
McFadden 伪 R² = 1−LL/LL₀ = 0.1298；
示例预测（x=10）：P(y=1)=1/(1+e^(−-1.1428))=0.2418。
③ 计算结果：
截距: β=-1.1428 OR=0.319 p=2.74e-42；x: β=0.1633 OR=1.177 p=1.75e-27
④ 结论：在 α=0.05 水平下：x显著(p=1.75e-27)。吸烟多风险高。（伪 R²=0.1298，模型解释力有限）
案例 96. 温度+湿度预测是否中暑
逻辑回归温湿协同增风险
① 原始数据：
样本 n=538，逐条 (x1, x2, y)：[(23.13, 27.65, 0), (18.54, 28.45, 0), (29.37, 17.56, 0), (25.25, 26.58, 0), (19.33, 22.86, 0), (28.64, 24.9, 0), (26.13, 25.75, 0), (27.01, 32.05, 1), (32.51, 15.07, 0), (32.6, 39.62, 1), (27.25, 20.99, 0), (30.96, 36.88, 0), (39.07, 21.57, 0), (22.65, 26.63, 0), (30.03, 25.68, 0), (31.31, 24.93, 0), (23.12, 27.92, 0), (32.07, 30.36, 0), (24.96, 27.67, 0), (42.72, 29.03, 0), (29.19, 29.6, 0), (38.38, 20.46, 0), (21.36, 30.04, 0), (26.54, 42.23, 1), (31.37, 33.45, 1), (19.35, 27.52, 0), (32.46, 29.93, 0), (13.43, 29.44, 0), (20.15, 31.88, 0), (35.53, 25.44, 0), (25.61, 18.57, 0), (27.96, 32.0, 1), (30.38, 27.66, 0), (20.98, 25.61, 0), (28.73, 34.05, 1), (23.12, 24.46, 0), (30.87, 16.8, 0), (33.9, 22.12, 0), (41.46, 23.76, 0), (33.67, 31.05, 0), (32.01, 36.6, 0), (32.12, 26.37, 1), (29.14, 24.31, 0), (25.58, 26.84, 0), (28.41, 21.58, 0), (28.06, 30.3, 0), (32.76, 24.6, 0), (28.43, 32.43, 0), (24.58, 23.24, 0), (20.47, 38.09, 1), (30.32, 33.88, 0), (34.44, 28.28, 0), (18.17, 27.47, 0), (27.1, 30.27, 0), (26.07, 27.28, 0), (24.7, 34.08, 0), (29.66, 29.88, 0), (20.37, 28.35, 0), (22.84, 29.24, 0), (30.59, 29.69, 1), (34.39, 25.4, 0), (29.7, 27.88, 1), (29.35, 25.14, 0), (17.55, 36.23, 1), (27.79, 35.75, 0), (34.74, 38.77, 1), (31.34, 31.79, 0), (31.75, 30.45, 0), (25.96, 29.29, 1), (23.69, 29.47, 1), (22.18, 20.33, 0), (26.3, 29.99, 0), (17.77, 24.33, 0), (27.91, 17.88, 0), (29.25, 20.07, 0), (23.73, 28.44, 0), (26.1, 19.85, 0), (34.59, 30.21, 0), (23.76, 34.8, 0), (23.53, 26.07, 0), (19.33, 17.12, 0), (22.03, 26.28, 0), (24.63, 23.39, 0), (35.43, 27.7, 0), (29.89, 26.91, 0), (31.05, 29.72, 0), (30.97, 37.15, 0), (28.22, 20.85, 0), (30.33, 29.02, 1), (20.62, 28.46, 0), (31.69, 15.0, 0), (29.55, 27.62, 0), (34.64, 25.55, 0), (22.62, 27.58, 0), (22.32, 36.84, 1), (43.04, 32.6, 1), (25.66, 32.56, 0), (23.33, 27.34, 0), (31.21, 30.42, 1), (16.42, 28.48, 0), (33.56, 31.01, 1), (32.59, 36.06, 1), (21.15, 23.87, 0), (24.47, 26.64, 0), (40.01, 30.76, 0), (25.83, 32.61, 1), (19.49, 33.83, 0), (25.52, 31.62, 0), (32.49, 24.01, 0), (28.63, 32.82, 0), (26.82, 19.85, 0), (29.72, 24.97, 0), (29.93, 25.52, 0), (20.47, 35.66, 1), (30.95, 28.04, 1), (27.79, 24.27, 0), (26.66, 33.46, 0), (23.49, 30.21, 0), (22.79, 25.7, 0), (36.6, 28.03, 0), (20.56, 31.29, 0), (28.41, 28.21, 0), (35.34, 22.61, 0), (24.63, 21.67, 0), (17.66, 32.61, 0), (39.11, 22.4, 0), (40.14, 24.52, 0), (37.88, 25.1, 0), (23.33, 27.81, 0), (15.87, 26.45, 0), (25.36, 18.76, 0), (28.97, 45.09, 0), (21.46, 30.08, 1), (20.96, 26.62, 0), (25.86, 34.7, 0), (37.24, 24.45, 0), (35.73, 42.97, 1), (24.76, 28.05, 0), (30.41, 30.88, 0), (27.99, 16.46, 0), (26.7, 27.9, 1), (27.99, 34.84, 0), (29.27, 23.49, 0), (21.87, 35.1, 0), (20.18, 34.19, 0), (31.05, 24.91, 0), (30.74, 21.27, 0), (21.54, 25.89, 0), (37.61, 37.99, 1), (36.26, 33.57, 1), (36.84, 31.65, 0), (31.45, 36.4, 1), (28.96, 42.8, 1), (29.27, 27.65, 0), (31.57, 29.02, 1), (28.6, 30.88, 0), (32.02, 29.27, 0), (27.67, 25.23, 0), (25.57, 23.12, 0), (23.15, 22.11, 0), (26.44, 18.98, 0), (38.93, 37.23, 1), (22.46, 26.72, 0), (34.55, 20.93, 0), (25.4, 20.81, 0), (32.83, 31.46, 0), (29.86, 14.29, 0), (32.05, 15.66, 0), (17.05, 30.72, 1), (28.87, 29.79, 0), (31.74, 32.97, 1), (23.14, 25.75, 0), (35.86, 34.03, 1), (26.25, 30.14, 1), (30.18, 37.01, 0), (34.2, 21.58, 1), (22.83, 22.3, 0), (36.73, 36.43, 1), (28.67, 27.6, 0), (17.28, 32.93, 0), (34.16, 27.71, 0), (14.89, 21.7, 0), (23.0, 32.51, 0), (29.24, 25.48, 0), (27.58, 21.42, 0), (21.49, 38.06, 1), (36.43, 27.34, 1), (43.36, 29.66, 0), (27.98, 15.09, 0), (31.55, 26.04, 0), (30.62, 37.46, 1), (20.14, 21.03, 0), (25.4, 28.56, 0), (18.37, 17.67, 0), (24.51, 32.92, 0), (25.32, 26.81, 0), (24.9, 30.81, 0), (18.88, 30.9, 0), (27.6, 18.59, 0), (21.09, 22.54, 0), (22.75, 32.51, 0), (38.86, 30.42, 1), (21.03, 33.77, 0), (28.11, 14.46, 0), (29.29, 41.45, 0), (25.37, 20.05, 0), (36.37, 29.27, 1), (23.62, 33.84, 1), (31.26, 31.61, 1), (29.16, 29.68, 0), (22.48, 32.93, 0), (17.15, 27.24, 1), (31.32, 23.41, 0), (31.25, 17.02, 0), (28.72, 28.33, 0), (33.44, 31.91, 1), (26.19, 29.22, 0), (36.84, 36.56, 1), (28.46, 25.33, 0), (25.81, 20.23, 0), (31.84, 15.79, 0), (27.33, 22.31, 0), (18.52, 28.42, 0), (23.25, 29.97, 0), (22.9, 28.53, 0), (28.64, 22.44, 0), (24.58, 40.0, 0), (15.5, 32.45, 0), (22.52, 21.59, 0), (16.55, 22.45, 0), (27.06, 28.85, 1), (27.03, 29.21, 0), (22.38, 31.17, 1), (26.26, 31.77, 0), (26.33, 24.43, 0), (22.52, 26.89, 0), (24.75, 31.42, 0), (25.31, 21.91, 0), (36.81, 32.07, 0), (30.42, 36.62, 1), (24.67, 26.76, 0), (28.81, 28.79, 0), (29.17, 37.32, 0), (26.5, 29.4, 0), (25.07, 29.61, 1), (23.31, 14.21, 0), (31.63, 25.46, 0), (26.01, 23.35, 0), (24.61, 34.64, 0), (31.18, 39.56, 0), (24.88, 27.87, 0), (32.61, 32.4, 0), (32.14, 17.2, 0), (24.09, 39.39, 1), (16.83, 30.58, 0), (24.35, 38.83, 1), (34.04, 22.17, 0), (36.35, 33.05, 0), (18.83, 23.73, 0), (22.94, 28.75, 0), (20.22, 22.66, 0), (36.72, 28.13, 1), (34.41, 34.65, 0), (29.29, 23.71, 0), (21.06, 29.43, 0), (27.05, 29.12, 0), (30.49, 34.23, 0), (24.11, 32.2, 0), (31.36, 19.55, 0), (23.37, 22.05, 0), (24.26, 30.74, 0), (12.25, 33.88, 0), (27.84, 25.6, 1), (23.07, 26.54, 0), (17.06, 29.54, 0), (19.17, 25.36, 0), (29.68, 28.97, 0), (26.32, 26.86, 0), (27.25, 34.43, 1), (21.6, 27.64, 0), (29.6, 24.45, 0), (31.22, 22.85, 0), (24.02, 33.85, 0), (30.85, 33.73, 0), (33.77, 21.34, 0), (35.29, 26.7, 0), (29.12, 27.48, 0), (33.2, 24.03, 0), (26.25, 33.45, 0), (33.97, 32.18, 0), (26.0, 28.81, 0), (21.13, 25.59, 0), (19.18, 28.46, 0), (22.48, 21.08, 0), (31.73, 27.79, 0), (26.91, 35.15, 0), (31.85, 25.82, 0), (32.46, 22.45, 0), (20.57, 31.04, 0), (20.19, 22.09, 0), (34.33, 21.15, 0), (29.0, 32.76, 1), (27.91, 30.06, 0), (30.26, 23.05, 0), (20.5, 22.4, 0), (29.27, 23.45, 0), (30.15, 35.22, 0), (27.81, 20.71, 0), (30.16, 31.66, 0), (33.1, 23.57, 0), (30.02, 26.84, 0), (13.72, 29.1, 1), (20.53, 31.59, 0), (39.56, 21.94, 0), (37.81, 26.32, 1), (30.88, 36.19, 1), (19.53, 35.84, 0), (28.8, 20.93, 0), (26.46, 35.97, 0), (30.19, 32.64, 0), (28.24, 34.94, 1), (28.78, 31.16, 1), (26.17, 25.96, 0), (28.37, 28.91, 0), (36.79, 31.75, 1), (22.05, 19.82, 0), (28.55, 29.68, 0), (33.38, 29.36, 0), (35.75, 27.37, 0), (23.76, 19.19, 0), (33.64, 27.45, 0), (28.6, 34.31, 0), (26.69, 30.02, 0), (30.72, 17.7, 0), (16.7, 20.32, 0), (36.4, 33.29, 0), (29.32, 29.03, 0), (31.56, 37.04, 0), (27.4, 23.69, 0), (23.31, 32.11, 1), (17.73, 16.97, 0), (33.59, 34.23, 0), (28.44, 27.79, 0), (35.7, 31.06, 0), (27.89, 19.02, 0), (27.21, 19.74, 0), (33.89, 29.79, 0), (36.81, 24.27, 0), (16.54, 25.61, 0), (27.62, 26.81, 1), (22.88, 31.83, 0), (39.05, 26.45, 1), (35.86, 30.43, 1), (19.65, 25.47, 0), (31.47, 32.71, 0), (35.28, 28.23, 1), (28.93, 25.01, 0), (31.13, 16.27, 0), (18.08, 40.02, 0), (31.7, 25.45, 0), (32.69, 27.23, 0), (41.35, 37.1, 1), (16.85, 28.9, 0), (22.16, 32.89, 0), (31.81, 29.95, 1), (26.98, 42.39, 0), (37.28, 18.1, 0), (28.64, 31.03, 0), (14.92, 26.77, 0), (33.36, 24.8, 1), (19.76, 27.47, 0), (25.19, 20.6, 0), (36.69, 19.86, 0), (24.96, 25.17, 0), (22.49, 21.85, 0), (17.81, 19.09, 0), (19.32, 34.23, 0), (15.74, 22.52, 0), (27.99, 22.87, 0), (37.34, 39.62, 0), (23.7, 28.56, 0), (18.68, 41.89, 1), (37.04, 21.95, 0), (17.66, 30.35, 0), (21.89, 40.25, 0), (27.57, 22.6, 0), (28.38, 25.85, 0), (31.85, 34.82, 0), (29.99, 31.83, 0), (22.22, 26.56, 0), (32.71, 21.75, 0), (26.77, 24.94, 0), (11.96, 27.89, 0), (24.43, 22.66, 0), (29.12, 26.92, 1), (17.26, 29.75, 0), (30.31, 34.8, 1), (29.85, 24.37, 0), (23.57, 28.54, 0), (29.98, 39.54, 1), (21.56, 24.73, 0), (34.65, 34.25, 0), (31.52, 20.24, 0), (34.15, 32.66, 0), (27.36, 29.04, 0), (29.35, 26.94, 1), (15.59, 21.6, 0), (25.93, 32.78, 0), (24.9, 33.33, 0), (29.54, 29.1, 0), (29.42, 31.49, 0), (31.87, 28.0, 1), (27.07, 27.18, 0), (28.95, 24.75, 1), (31.67, 40.05, 1), (37.4, 30.29, 0), (31.66, 25.57, 0), (29.72, 23.42, 0), (23.5, 29.4, 0), (23.25, 33.05, 0), (21.4, 7.02, 0), (21.88, 27.66, 0), (25.98, 26.05, 0), (30.5, 30.78, 0), (25.04, 30.11, 0), (37.09, 34.52, 1), (29.18, 26.68, 0), (24.16, 27.61, 0), (15.23, 25.63, 0), (37.09, 27.0, 0), (30.52, 23.94, 0), (30.14, 27.51, 0), (17.35, 34.25, 0), (27.98, 38.76, 1), (27.79, 28.54, 0), (19.66, 28.18, 0), (26.01, 27.37, 0), (31.26, 35.44, 0), (19.89, 23.78, 0), (30.7, 17.51, 0), (31.33, 32.47, 1), (27.26, 22.53, 0), (33.32, 26.76, 0), (14.19, 33.22, 0), (24.64, 48.35, 0), (28.56, 18.74, 0), (20.53, 24.02, 0), (28.5, 28.42, 0), (30.15, 28.56, 0), (28.58, 15.78, 0), (26.27, 31.94, 0), (40.23, 29.73, 1), (31.56, 30.46, 0), (30.27, 29.25, 0), (22.03, 32.94, 0), (31.37, 24.85, 1), (37.86, 24.53, 1), (23.03, 30.58, 1), (31.0, 22.62, 0), (26.47, 21.74, 0), (23.0, 27.65, 0), (24.54, 29.27, 0), (25.13, 29.35, 0), (30.79, 16.23, 0), (27.1, 21.79, 0), (25.41, 32.77, 1), (22.38, 27.87, 0), (39.63, 36.86, 1), (23.52, 37.83, 0), (21.27, 29.92, 0), (31.4, 30.41, 0), (39.15, 28.19, 1), (28.74, 20.54, 0), (24.12, 17.33, 0), (32.38, 23.82, 0), (21.27, 33.85, 0), (22.48, 22.84, 0), (24.35, 30.66, 0), (34.2, 33.66, 0), (31.04, 20.18, 0), (27.42, 20.14, 0), (33.73, 24.74, 0), (26.44, 28.27, 0), (18.18, 32.4, 0), (18.24, 31.55, 0), (28.62, 21.74, 0), (30.93, 29.27, 1), (32.93, 32.85, 0), (29.42, 27.67, 0), (29.75, 34.24, 1), (27.6, 41.23, 0), (35.18, 19.45, 1), (27.98, 26.29, 0), (24.28, 18.73, 0), (25.89, 19.33, 0), (17.54, 31.4, 0), (41.16, 28.01, 0), (28.04, 32.72, 0), (22.96, 31.07, 0), (30.0, 23.49, 0), (32.2, 25.73, 1), (38.17, 27.58, 0), (20.86, 23.68, 0), (28.51, 21.21, 0), (25.34, 18.22, 0), (38.77, 32.66, 0), (21.31, 22.38, 0), (32.52, 34.89, 1), (32.12, 24.04, 0), (32.64, 28.61, 0), (21.87, 30.74, 0), (15.61, 30.85, 0), (35.07, 22.85, 0), (18.3, 31.78, 0), (27.0, 31.35, 0), (31.12, 42.46, 1), (40.92, 23.31, 0), (18.0, 41.88, 1), (30.23, 26.56, 0), (24.06, 34.18, 0), (22.52, 36.18, 1), (26.22, 31.28, 0), (29.37, 26.5, 0), (29.28, 26.93, 0), (20.11, 27.35, 0), (29.78, 29.22, 0), (22.38, 22.49, 0), (32.34, 29.43, 0), (35.52, 30.74, 0), (29.78, 35.82, 0), (22.57, 22.3, 0), (30.91, 15.91, 0), (26.3, 32.85, 0), (37.71, 32.95, 1), (38.78, 27.61, 0), (28.19, 24.02, 0), (19.48, 31.87, 0), (26.25, 36.71, 1)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.8751, OR=0.1533, z=-12.6183, p=1.67e-36；x1: β=0.1012, OR=1.1065, z=4.6882, p=2.76e-06；x2: β=0.1891, OR=1.2082, z=7.6716, p=1.70e-14；
McFadden 伪 R² = 1−LL/LL₀ = 0.1903；
示例预测（x1=x2=28）：P(y=1)=1/(1+e^(−-1.8751))=0.1330。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-1.8751, OR=0.1533, z=-12.6183, p=1.67e-36；x1: β=0.1012, OR=1.1065, z=4.6882, p=2.76e-06；x2: β=0.1891, OR=1.2082, z=7.6716, p=1.70e-14；
McFadden 伪 R² = 1−LL/LL₀ = 0.1903；
示例预测（x1=x2=28）：P(y=1)=1/(1+e^(−-1.8751))=0.1330。
③ 计算结果：
截距: β=-1.8751 OR=0.153 p=1.67e-36；x1: β=0.1012 OR=1.107 p=2.76e-06；x2: β=0.1891 OR=1.208 p=1.70e-14
④ 结论：在 α=0.05 水平下：x1显著(p=2.76e-06)、x2显著(p=1.70e-14)。温湿协同增风险。（伪 R²=0.1903，模型解释力有限）
案例 97. 收入+学历预测是否晋升
逻辑回归两正向因素
① 原始数据：
样本 n=625，逐条 (x1, x2, y)：[(32.68, 18.54, 0), (25.87, 25.32, 0), (7.92, 41.07, 0), (32.85, 25.67, 0), (15.59, 19.75, 1), (42.1, 33.3, 0), (27.98, 11.87, 0), (44.35, 23.5, 0), (30.09, 28.52, 0), (33.59, 39.26, 0), (27.08, 33.04, 1), (24.79, 27.8, 0), (31.81, 30.14, 0), (35.95, 43.9, 0), (37.78, 22.38, 0), (12.63, 34.41, 0), (36.1, 38.0, 0), (18.65, 44.75, 1), (33.29, 36.86, 0), (18.53, 40.23, 0), (10.65, 22.09, 0), (43.19, 14.8, 0), (27.03, 8.69, 0), (38.89, 34.17, 0), (40.79, 27.74, 0), (54.16, 19.28, 0), (35.19, 11.87, 0), (42.81, 37.02, 1), (32.02, 30.08, 0), (16.63, 36.76, 0), (38.55, 35.39, 0), (48.75, 42.12, 1), (32.25, 29.92, 0), (46.62, 42.4, 1), (22.83, 24.63, 0), (45.3, 31.72, 0), (34.35, 27.3, 0), (15.87, 24.2, 0), (35.19, 23.35, 0), (28.39, 40.66, 0), (54.19, 37.71, 0), (31.0, 37.68, 1), (26.98, 41.1, 0), (33.91, 12.21, 0), (39.49, 13.19, 0), (25.83, 14.14, 0), (27.13, 17.66, 0), (30.13, 47.37, 1), (36.98, 39.15, 1), (36.74, 23.66, 0), (24.65, 34.37, 0), (12.62, 27.15, 1), (43.54, 23.95, 0), (21.08, 39.45, 0), (30.91, 44.16, 1), (36.91, 19.23, 0), (29.91, 45.98, 0), (30.05, 31.63, 0), (15.87, 33.12, 0), (17.68, 47.48, 0), (18.82, 30.84, 0), (17.98, 32.61, 0), (50.76, 6.75, 0), (26.44, 18.18, 0), (27.24, 31.16, 1), (28.84, 35.35, 0), (-0.88, 33.16, 0), (32.86, 43.17, 0), (40.1, 36.71, 0), (24.43, 54.97, 0), (44.94, 17.18, 0), (55.0, 7.18, 0), (39.27, 19.65, 0), (47.98, 51.46, 1), (19.96, 47.43, 1), (11.32, 36.56, 0), (14.17, 19.6, 0), (32.77, -3.46, 0), (12.56, 7.93, 0), (18.65, 28.72, 0), (42.78, 8.58, 0), (29.55, 28.27, 0), (51.4, 34.02, 0), (25.75, 25.07, 0), (27.56, 36.99, 0), (37.88, 22.96, 0), (24.82, 44.86, 0), (32.43, 18.48, 0), (60.52, 52.58, 1), (58.52, 25.85, 0), (29.61, 17.8, 0), (24.3, 54.64, 0), (27.67, 18.19, 0), (41.78, 24.04, 0), (33.27, 5.92, 0), (36.21, 50.12, 0), (37.18, 35.21, 1), (28.39, 14.36, 0), (50.92, 40.29, 1), (40.3, 19.98, 1), (36.42, 28.42, 0), (20.88, 35.08, 0), (27.58, 34.98, 0), (46.67, 18.73, 0), (38.63, 3.08, 0), (11.14, 11.88, 0), (41.97, 10.57, 0), (34.55, 22.8, 0), (23.98, 51.11, 1), (33.92, 24.16, 0), (36.81, 45.02, 1), (36.45, 48.71, 0), (24.71, 17.26, 0), (35.69, 35.3, 0), (22.49, 8.78, 0), (35.44, 55.13, 1), (33.45, 26.38, 0), (22.52, 35.0, 0), (36.24, 19.74, 0), (5.39, 52.06, 0), (10.31, 14.97, 0), (35.57, 26.93, 0), (28.27, 26.76, 0), (46.72, 15.06, 0), (38.39, 29.61, 1), (14.28, 39.68, 0), (24.75, 25.59, 0), (20.87, 19.27, 0), (43.01, 28.43, 0), (52.73, 24.8, 0), (49.04, 42.0, 1), (19.51, 48.9, 0), (30.85, 46.71, 0), (16.73, 27.43, 0), (36.89, 11.21, 0), (33.84, 19.25, 0), (31.85, 44.07, 0), (22.98, 27.61, 0), (21.52, 31.25, 0), (12.69, 22.75, 0), (39.22, 34.49, 1), (14.96, 37.43, 0), (27.54, 55.48, 1), (34.62, 9.4, 0), (7.31, 21.73, 0), (23.22, 39.72, 0), (31.75, 31.88, 0), (45.09, 32.53, 0), (24.93, 50.47, 0), (11.66, 26.92, 0), (32.65, 8.39, 0), (36.11, 76.01, 0), (30.43, 9.07, 0), (30.22, 31.03, 0), (36.98, 30.55, 0), (17.19, 47.27, 1), (34.68, 24.6, 0), (21.22, 24.47, 0), (32.28, 31.87, 0), (32.63, 18.57, 0), (10.02, 29.14, 0), (33.44, 51.21, 0), (35.31, 13.26, 0), (20.96, 17.37, 0), (35.05, 30.67, 0), (48.23, 52.52, 1), (39.38, 38.65, 0), (16.14, 44.14, 0), (22.83, 20.65, 0), (26.23, 35.41, 0), (17.8, 31.47, 0), (32.16, 25.5, 0), (25.83, 39.89, 0), (41.3, 29.82, 1), (46.02, 38.06, 0), (26.97, 10.66, 0), (20.57, 32.66, 1), (18.1, 21.91, 0), (35.81, 30.71, 1), (39.91, 41.52, 1), (27.16, 34.04, 0), (25.51, 36.06, 1), (34.01, 34.31, 0), (22.24, 33.81, 0), (15.54, 34.8, 0), (27.8, 29.88, 0), (33.66, 44.28, 0), (25.56, 30.99, 0), (22.7, 24.46, 0), (32.21, 42.02, 0), (33.56, 43.3, 0), (23.88, 20.39, 0), (32.06, 6.61, 0), (34.69, 20.68, 0), (60.4, 14.32, 0), (36.71, 26.53, 0), (30.28, 54.24, 0), (39.32, 52.07, 1), (19.47, 11.81, 1), (51.71, 55.16, 0), (38.78, 15.94, 0), (12.91, 3.28, 0), (14.84, 37.71, 0), (8.08, 35.68, 0), (14.67, 34.07, 0), (34.46, 19.01, 0), (40.22, 28.65, 0), (14.01, 12.98, 0), (30.73, 16.31, 0), (20.75, 31.88, 0), (29.45, 44.17, 0), (26.88, 28.12, 0), (45.49, 25.29, 0), (23.81, 40.45, 1), (27.03, 18.47, 0), (29.64, 15.99, 0), (26.3, 8.21, 0), (51.89, 16.03, 0), (25.64, 30.57, 0), (58.2, 25.13, 0), (23.75, 20.89, 0), (39.96, 23.49, 0), (34.61, 44.46, 0), (21.97, 30.56, 0), (20.81, 39.18, 0), (22.04, 19.44, 0), (63.48, 28.08, 1), (31.89, 27.66, 0), (31.32, 6.8, 0), (18.46, 9.02, 0), (52.17, 43.88, 1), (22.93, 25.58, 0), (23.69, 23.85, 0), (38.98, 35.99, 0), (18.08, 31.67, 0), (25.79, 19.95, 0), (36.67, 24.2, 0), (47.33, 21.32, 0), (38.08, 33.81, 0), (49.57, 28.66, 0), (29.28, 20.22, 0), (39.52, 43.93, 1), (24.3, 13.97, 0), (22.0, 31.48, 0), (26.03, 22.92, 0), (20.11, 33.48, 0), (9.38, 44.25, 0), (22.66, 23.63, 0), (22.72, 42.25, 0), (24.74, 40.46, 0), (10.72, 26.36, 0), (40.19, 22.99, 0), (-0.37, 24.26, 0), (45.53, 43.52, 1), (24.6, 43.39, 0), (37.39, 26.23, 0), (18.58, 12.12, 0), (28.86, 20.34, 0), (29.46, 27.62, 0), (30.6, 51.05, 0), (14.85, 29.87, 0), (1.77, 29.33, 0), (30.72, 22.69, 0), (23.07, 16.53, 0), (48.08, 54.94, 1), (28.87, 46.77, 0), (15.37, 14.84, 0), (26.64, 11.77, 0), (22.02, 31.65, 0), (17.6, 37.68, 0), (25.82, 33.54, 0), (38.03, 47.24, 1), (31.47, 37.88, 0), (27.0, 34.63, 0), (46.74, 39.22, 1), (24.78, 21.13, 0), (16.81, 30.3, 0), (22.09, 42.35, 0), (32.21, 17.35, 0), (33.53, 38.86, 0), (30.08, 33.37, 0), (24.6, 15.51, 0), (34.66, 36.2, 0), (50.93, 35.67, 0), (23.58, 23.43, 0), (55.32, 28.09, 1), (32.11, 28.62, 1), (20.96, 18.3, 0), (28.86, 40.6, 0), (30.57, 18.71, 0), (9.73, -0.97, 0), (16.92, 42.18, 0), (11.47, 43.51, 0), (16.7, 42.89, 0), (12.72, 34.36, 0), (28.9, 39.37, 0), (37.98, 51.92, 1), (43.96, 38.94, 0), (25.92, 18.6, 0), (17.24, 18.52, 0), (23.08, 24.37, 1), (48.54, 40.68, 0), (40.33, 22.3, 0), (8.81, 40.14, 0), (55.21, 33.31, 1), (32.5, 26.8, 0), (48.88, 34.65, 0), (19.66, 15.49, 0), (33.35, 22.14, 0), (11.77, 30.94, 0), (38.8, 20.56, 0), (20.74, 46.89, 0), (-0.87, 23.31, 0), (55.99, 30.74, 0), (44.64, 29.71, 0), (29.94, 17.82, 0), (21.23, 28.15, 0), (65.27, 35.54, 0), (26.8, 5.37, 0), (31.95, 41.96, 0), (30.86, 18.9, 0), (17.04, 22.65, 0), (27.1, 35.99, 0), (34.6, 40.81, 0), (26.63, 1.16, 0), (21.13, 16.62, 0), (29.55, 20.67, 0), (36.11, 28.43, 0), (30.28, 15.01, 0), (37.65, 14.69, 0), (-5.51, 28.81, 0), (26.77, 32.42, 0), (25.48, 29.75, 0), (34.69, 56.59, 0), (25.55, 17.52, 0), (5.72, 39.17, 0), (30.69, 15.52, 0), (26.63, 39.53, 0), (48.09, 27.9, 1), (38.03, 34.14, 0), (26.93, 30.7, 0), (51.77, 31.92, 1), (41.61, 15.44, 0), (37.35, 9.66, 0), (51.74, 19.66, 0), (32.7, 30.42, 0), (24.08, 17.77, 0), (38.56, 41.32, 0), (20.54, 35.94, 0), (32.0, 19.24, 0), (-0.1, 29.97, 0), (61.61, 23.28, 0), (38.4, 13.67, 0), (9.05, 50.16, 0), (28.29, 11.97, 0), (22.23, 26.85, 0), (11.16, 37.48, 0), (36.29, 27.49, 0), (34.95, 8.83, 0), (18.4, 32.87, 0), (27.68, 21.91, 0), (41.66, 41.12, 0), (20.81, 33.26, 0), (24.54, 42.83, 0), (27.89, 20.63, 0), (49.49, 18.86, 0), (25.37, 29.49, 0), (26.11, 60.38, 1), (45.78, 23.63, 0), (45.43, 30.56, 0), (30.06, 11.84, 0), (15.3, 51.53, 0), (14.8, 29.92, 0), (33.72, 38.94, 0), (26.4, 31.17, 0), (29.62, 38.49, 0), (25.66, 30.34, 0), (27.57, 28.7, 0), (31.36, 39.52, 0), (26.29, 29.33, 0), (35.0, 31.37, 0), (29.67, 34.64, 0), (28.28, 26.99, 0), (16.04, 18.29, 0), (35.14, 31.28, 0), (25.44, 35.38, 0), (30.53, 41.67, 0), (21.29, 30.1, 0), (24.39, 36.26, 0), (31.0, 35.0, 1), (61.4, 9.21, 0), (36.1, 52.56, 1), (41.68, 43.72, 0), (36.07, 29.29, 0), (38.31, 33.29, 0), (16.11, -3.23, 0), (9.97, 27.95, 0), (20.79, 34.62, 0), (51.15, 31.64, 0), (20.58, 61.94, 1), (13.89, 31.05, 0), (26.67, 26.76, 0), (40.37, 22.55, 0), (17.94, 44.97, 0), (47.06, 23.51, 0), (47.99, 25.19, 0), (42.73, 40.99, 0), (16.64, 50.23, 0), (21.53, 28.97, 0), (8.31, 41.45, 0), (31.14, 48.02, 1), (18.66, 18.74, 1), (33.73, 38.29, 0), (38.95, 47.8, 0), (45.05, 49.75, 0), (9.99, 25.39, 0), (33.54, 19.37, 0), (40.2, 43.28, 0), (26.73, 29.82, 0), (40.12, 12.14, 0), (7.54, 59.85, 0), (16.37, 33.92, 0), (42.99, 33.31, 0), (41.45, 18.35, 0), (30.76, 21.17, 0), (34.54, 21.13, 0), (9.81, 41.48, 1), (24.6, 3.67, 0), (53.82, 31.14, 1), (33.55, 45.26, 0), (37.87, 26.02, 0), (29.7, 28.58, 0), (58.98, 9.85, 0), (31.07, 37.22, 0), (27.4, 40.97, 0), (23.31, 16.17, 0), (39.6, 38.97, 1), (42.16, 39.94, 1), (18.84, 1.79, 0), (1.52, 38.23, 0), (39.74, 24.46, 0), (36.34, 35.54, 0), (38.19, 23.58, 0), (30.21, 35.19, 0), (28.59, 31.97, 0), (24.6, 20.38, 0), (25.99, 43.96, 0), (25.13, 33.52, 0), (41.55, 32.82, 0), (34.78, 36.35, 0), (23.71, 44.1, 0), (36.79, 34.03, 0), (24.99, 40.03, 0), (33.37, 32.07, 1), (25.34, 26.14, 1), (11.69, 42.33, 0), (27.29, 48.95, 0), (52.64, 38.04, 0), (37.89, 20.44, 0), (17.12, 38.83, 0), (31.82, 40.53, 0), (24.44, 22.16, 0), (36.81, 26.29, 0), (36.29, 17.32, 0), (30.38, 41.63, 0), (40.23, 19.69, 0), (49.71, 32.24, 0), (24.4, 20.99, 0), (42.63, 48.08, 1), (24.35, 61.73, 0), (45.34, 16.43, 0), (27.97, 35.72, 0), (21.66, 44.5, 0), (34.42, 20.84, 0), (21.51, 27.91, 0), (30.98, 40.1, 0), (27.37, 25.32, 0), (23.38, 40.18, 0), (21.16, 19.96, 0), (32.03, 3.88, 0), (38.29, 11.77, 0), (41.34, 23.9, 0), (18.32, 31.28, 0), (35.29, 25.48, 0), (13.66, 45.17, 0), (36.28, 48.87, 0), (42.93, 23.93, 0), (35.69, 38.46, 0), (32.65, 22.58, 0), (51.08, 32.55, 0), (38.01, 19.64, 0), (18.19, 18.44, 0), (47.81, 26.12, 0), (35.25, 38.72, 0), (14.75, 51.48, 0), (47.87, 9.46, 0), (56.41, 54.38, 1), (23.25, 14.81, 0), (56.09, 20.58, 0), (36.76, 35.5, 0), (51.77, 35.5, 1), (29.12, 34.22, 0), (22.52, 24.87, 0), (37.02, 26.34, 0), (41.15, 20.99, 0), (29.4, 33.61, 0), (35.85, 17.66, 1), (27.92, 25.9, 0), (23.63, 38.74, 0), (50.13, 31.92, 1), (25.09, 29.92, 0), (29.5, 38.82, 0), (28.86, 24.66, 0), (24.61, 35.0, 0), (34.41, 26.02, 1), (36.58, 33.67, 0), (24.76, 31.63, 0), (24.87, 15.81, 0), (17.91, 31.48, 0), (37.12, 32.88, 0), (3.6, 28.22, 0), (43.04, 31.02, 0), (40.85, 28.7, 0), (34.04, 32.7, 0), (13.93, 59.33, 1), (37.68, 40.77, 0), (35.97, 44.16, 1), (30.55, 28.36, 1), (38.66, 38.52, 0), (34.3, 22.6, 0), (19.35, 21.99, 0), (5.35, 18.78, 0), (44.01, 31.54, 1), (-4.08, 8.07, 0), (16.63, 10.45, 0), (27.23, 24.73, 0), (26.41, 30.82, 0), (32.83, 26.27, 1), (4.33, 5.51, 0), (28.71, 27.3, 0), (46.69, 35.02, 1), (20.37, 24.98, 0), (0.16, 61.29, 0), (29.21, 37.83, 1), (16.98, 1.28, 0), (32.71, 34.9, 0), (24.99, 35.83, 0), (22.69, 24.75, 0), (36.49, 43.1, 1), (48.93, 29.4, 0), (33.46, 35.19, 0), (26.01, 50.46, 0), (15.53, 28.25, 0), (39.3, 17.31, 0), (35.66, 6.85, 0), (12.4, 28.65, 0), (31.27, 30.35, 0), (56.27, 7.58, 0), (19.69, 36.58, 0), (40.48, 28.78, 0), (28.28, 26.65, 0), (33.73, 29.09, 1), (32.94, 33.18, 0), (31.89, 26.55, 0), (38.71, 43.04, 1), (23.31, 24.17, 0), (42.29, 30.99, 1), (41.27, 44.64, 1), (35.64, 56.33, 0), (12.38, 5.54, 0), (29.29, 20.94, 0), (20.77, 24.55, 0), (26.66, 43.39, 1), (47.84, 22.98, 1), (34.47, 17.41, 0), (13.46, 46.13, 0), (25.08, 32.91, 0), (18.01, 26.05, 0), (32.37, 16.33, 0), (45.7, 33.55, 1), (28.35, 42.67, 0), (26.06, 52.63, 0), (31.24, 48.49, 0), (15.26, 41.04, 0), (51.61, 51.57, 0), (10.92, 43.8, 0), (36.77, 34.84, 0), (33.23, 44.46, 0), (22.99, 42.98, 0), (16.08, 45.56, 0), (28.01, 22.91, 0), (49.67, 33.58, 1), (20.89, 25.42, 0), (67.09, 27.83, 1), (29.61, 30.29, 0), (27.4, 26.62, 0), (3.15, 22.01, 0), (36.03, 45.2, 0), (36.48, 31.44, 0), (31.29, 36.46, 0), (33.84, 27.51, 0), (26.52, 16.73, 0), (37.54, 28.65, 0), (13.8, 47.86, 1), (31.34, 20.16, 0), (35.76, 23.17, 0), (19.53, 31.31, 0), (29.07, 11.52, 0), (22.61, 29.77, 0), (41.03, 44.95, 1), (59.46, 28.6, 0), (19.9, 12.43, 0), (21.14, 33.66, 0), (47.97, 35.44, 1), (9.89, 10.17, 0), (28.16, 42.2, 1), (22.87, 49.69, 0), (29.13, 20.7, 0), (31.91, 30.72, 0), (25.81, 27.32, 0), (25.89, 15.87, 0), (25.48, 5.74, 0), (31.5, 42.72, 1), (11.9, 33.37, 0), (46.19, 39.35, 1)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.3004, OR=0.1002, z=-14.0458, p=8.17e-45；x1: β=0.0678, OR=1.0702, z=6.1039, p=1.03e-09；x2: β=0.0837, OR=1.0873, z=7.0264, p=2.12e-12；
McFadden 伪 R² = 1−LL/LL₀ = 0.1857；
示例预测（x1=x2=30）：P(y=1)=1/(1+e^(−-2.3004))=0.0911。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.3004, OR=0.1002, z=-14.0458, p=8.17e-45；x1: β=0.0678, OR=1.0702, z=6.1039, p=1.03e-09；x2: β=0.0837, OR=1.0873, z=7.0264, p=2.12e-12；
McFadden 伪 R² = 1−LL/LL₀ = 0.1857；
示例预测（x1=x2=30）：P(y=1)=1/(1+e^(−-2.3004))=0.0911。
③ 计算结果：
截距: β=-2.3004 OR=0.100 p=8.17e-45；x1: β=0.0678 OR=1.070 p=1.03e-09；x2: β=0.0837 OR=1.087 p=2.12e-12
④ 结论：在 α=0.05 水平下：x1显著(p=1.03e-09)、x2显著(p=2.12e-12)。两正向因素。（伪 R²=0.1857，模型解释力有限）
案例 98. 广告+促销预测是否加购
逻辑回归两正向因素
① 原始数据：
样本 n=515，逐条 (x1, x2, y)：[(-38.77, 39.86, 0), (61.32, 55.54, 0), (93.81, -50.42, 0), (-48.38, 128.09, 0), (79.16, -3.93, 0), (43.5, 79.42, 0), (44.54, 68.24, 0), (57.07, 92.89, 0), (43.26, -25.8, 0), (103.9, 38.92, 0), (5.94, 57.56, 0), (23.25, 62.83, 0), (84.25, 36.61, 1), (100.15, -12.36, 0), (72.76, 43.0, 0), (69.56, 43.32, 0), (-64.19, 173.22, 0), (85.32, -1.31, 0), (152.15, 110.43, 1), (33.17, 54.86, 0), (125.19, -8.18, 0), (125.15, 130.76, 1), (58.67, -36.08, 0), (20.05, 49.09, 0), (42.53, 67.83, 0), (35.23, 66.03, 1), (118.19, 26.14, 0), (97.22, 20.36, 1), (63.54, 110.63, 0), (19.79, 2.01, 0), (18.67, 112.2, 0), (89.4, 86.53, 1), (164.88, -36.68, 0), (25.36, 35.78, 0), (15.0, 82.01, 1), (10.17, 208.69, 1), (61.23, 60.83, 0), (125.19, 9.67, 1), (142.5, 76.8, 1), (63.83, 9.77, 0), (38.42, 66.56, 0), (7.0, 57.87, 0), (7.27, 34.88, 0), (21.13, 6.04, 0), (40.98, 28.61, 0), (52.33, 63.61, 0), (75.98, 82.78, 1), (56.41, 25.71, 0), (107.63, 33.72, 0), (66.8, 41.72, 0), (-0.3, 33.1, 0), (103.9, 92.69, 1), (72.79, -16.77, 0), (32.2, 102.39, 0), (83.78, 4.82, 1), (50.62, 74.94, 0), (23.93, 54.26, 0), (13.13, 92.92, 0), (67.9, -18.91, 0), (100.28, 34.01, 0), (118.29, 25.5, 0), (47.73, -10.83, 0), (31.64, 28.27, 1), (20.71, 40.76, 0), (28.8, 34.54, 0), (50.7, 83.15, 0), (61.44, 59.58, 1), (39.66, 7.13, 0), (63.28, -17.77, 1), (21.81, 145.92, 0), (72.89, 14.96, 0), (21.35, -41.49, 0), (107.14, -13.79, 0), (30.42, 35.25, 0), (-48.56, 28.52, 0), (78.72, 41.85, 1), (50.88, 41.59, 0), (78.85, 26.08, 0), (79.76, -13.81, 0), (82.1, 23.28, 1), (9.01, 79.71, 0), (-25.61, 35.54, 0), (65.18, 74.04, 0), (69.05, 19.74, 0), (-8.12, 43.36, 0), (64.2, 6.61, 0), (-40.74, -21.05, 0), (95.85, 108.94, 1), (65.58, 81.15, 0), (16.6, 23.85, 0), (-38.32, 49.0, 0), (12.12, 60.05, 0), (40.53, 61.64, 0), (152.6, 21.64, 0), (147.88, -37.44, 0), (127.77, 39.1, 0), (59.52, 107.86, 0), (68.35, 20.9, 0), (-15.45, 90.21, 1), (-10.86, 63.92, 0), (32.88, 71.52, 0), (52.98, 81.43, 1), (-43.36, 15.51, 0), (29.01, -47.23, 0), (60.16, 9.09, 0), (135.99, 134.62, 1), (37.72, 3.51, 0), (14.92, 61.36, 0), (-100.99, 52.36, 0), (100.79, 182.49, 1), (-16.94, 25.62, 0), (138.84, 81.47, 0), (13.97, 67.97, 0), (29.3, 5.51, 0), (4.57, 54.6, 0), (82.97, 62.66, 1), (-64.54, 78.17, 0), (88.1, 11.47, 0), (-103.26, 60.21, 0), (8.6, -22.73, 0), (70.49, 65.7, 0), (91.2, 9.66, 0), (42.86, 62.98, 0), (-8.64, 54.8, 0), (41.62, 9.85, 0), (70.86, 85.28, 0), (95.46, 110.92, 0), (34.61, 74.93, 1), (6.75, 107.77, 1), (107.46, 40.78, 1), (161.08, 72.36, 0), (91.88, 85.39, 1), (75.73, 33.4, 0), (65.11, 33.21, 0), (62.63, 61.64, 0), (-22.42, -37.97, 0), (45.33, 122.08, 0), (-30.28, 43.84, 0), (34.04, 77.19, 1), (66.01, 23.09, 0), (90.89, 51.64, 0), (35.27, 15.18, 0), (79.59, 22.69, 0), (31.57, -64.17, 0), (140.92, 23.4, 0), (-19.91, -9.05, 0), (17.05, 27.18, 0), (-11.04, 117.5, 0), (0.33, 21.95, 1), (78.6, -22.28, 0), (-16.0, 7.2, 0), (-40.57, 39.68, 0), (182.06, 41.62, 1), (28.06, 119.0, 0), (132.19, 40.9, 0), (109.64, 32.61, 0), (14.26, 56.61, 0), (31.5, -60.81, 0), (120.34, 113.25, 0), (42.92, 126.26, 1), (118.3, 122.0, 1), (72.13, 83.06, 0), (68.96, 77.18, 0), (139.82, 118.73, 1), (79.67, 110.12, 0), (3.74, 30.88, 0), (80.01, 151.91, 1), (47.82, 134.44, 1), (-20.94, 30.72, 0), (88.69, 8.58, 0), (-68.44, 20.68, 0), (22.6, 16.3, 0), (104.56, -12.03, 0), (121.65, 59.74, 0), (63.03, 33.52, 0), (47.54, 82.36, 0), (63.62, 125.73, 0), (-28.75, 111.57, 0), (40.2, 57.39, 0), (70.1, 28.21, 0), (75.01, -52.17, 0), (9.28, 21.64, 0), (99.26, 7.4, 0), (78.54, 74.94, 0), (-55.34, 115.01, 0), (-12.38, 61.03, 0), (61.23, 8.73, 0), (0.11, 187.65, 0), (61.34, -16.6, 0), (-18.11, 71.27, 0), (24.32, 84.71, 1), (80.81, 125.16, 1), (35.12, 84.24, 0), (123.35, 98.24, 0), (-8.49, 74.14, 0), (53.01, 48.32, 0), (85.86, 66.6, 0), (78.74, 71.54, 0), (84.54, 57.63, 0), (35.58, 44.6, 0), (52.86, 33.34, 0), (99.18, -29.51, 0), (-25.02, 61.44, 0), (114.43, -2.65, 0), (64.92, 110.31, 0), (11.88, 13.69, 0), (62.32, 78.97, 0), (27.64, 57.04, 1), (43.13, 224.24, 1), (-4.23, 27.54, 0), (112.22, 48.38, 0), (169.7, 85.36, 1), (32.42, 19.15, 0), (35.11, 76.16, 0), (-50.58, 21.81, 0), (14.95, -27.5, 0), (-21.31, 105.78, 0), (9.94, 14.87, 0), (152.7, 111.99, 0), (-11.03, 97.07, 0), (50.47, 85.18, 0), (-53.91, 114.39, 0), (-44.15, -9.08, 0), (38.02, 33.46, 0), (-5.33, -19.85, 0), (72.18, 81.72, 0), (44.4, 81.99, 0), (25.83, 63.67, 0), (20.48, 127.91, 0), (159.04, -44.3, 0), (5.02, 39.25, 0), (24.67, 25.04, 0), (-29.18, 9.22, 0), (56.26, 156.03, 0), (108.29, 123.94, 1), (52.72, -13.79, 1), (161.38, 93.03, 1), (70.67, -32.72, 0), (-19.19, 9.21, 0), (85.18, 6.83, 0), (105.49, 146.17, 0), (62.89, 61.2, 0), (29.68, 55.65, 0), (69.23, 114.53, 1), (-7.52, 54.87, 0), (121.85, 53.8, 1), (92.04, 42.21, 1), (125.05, 30.1, 0), (45.49, 118.86, 0), (47.18, 62.77, 0), (88.35, 63.05, 1), (-2.48, 47.93, 0), (114.94, -1.2, 0), (17.65, 66.4, 0), (54.91, 117.39, 0), (41.14, 15.51, 0), (18.79, -48.2, 0), (8.73, 80.94, 0), (38.85, 45.45, 0), (26.91, -90.01, 0), (29.48, 80.71, 0), (52.76, 44.8, 0), (-35.21, 106.16, 0), (138.76, 45.36, 0), (71.06, 149.37, 1), (-6.93, -1.33, 0), (17.01, 104.21, 1), (80.2, 78.07, 0), (203.29, -26.81, 1), (-63.03, 61.15, 0), (26.33, 47.1, 0), (81.88, -51.07, 0), (-55.73, 139.7, 0), (45.47, 99.86, 1), (42.4, 110.14, 0), (12.12, 61.97, 1), (-34.64, 58.39, 0), (6.42, 31.1, 0), (0.43, 39.7, 0), (59.61, -6.91, 0), (48.6, 54.63, 0), (94.66, 26.9, 0), (25.93, 12.7, 0), (69.66, 1.65, 0), (58.97, 24.24, 0), (99.67, 156.26, 1), (120.13, 14.1, 0), (-31.88, 3.21, 0), (-6.51, 39.83, 0), (95.31, 19.12, 0), (60.78, 52.4, 0), (-15.07, 5.22, 0), (99.82, 108.84, 0), (0.76, 46.42, 0), (2.79, 91.18, 0), (80.58, 57.23, 0), (-2.53, 47.53, 0), (43.9, 44.83, 0), (5.85, 116.87, 1), (68.21, 150.0, 1), (159.39, 17.32, 1), (119.46, 69.04, 1), (68.23, 98.81, 0), (124.77, 12.28, 0), (51.37, 79.45, 0), (123.59, 40.98, 1), (39.21, 3.48, 0), (34.1, 47.87, 0), (101.92, -43.25, 0), (20.27, -1.4, 0), (37.03, 162.96, 0), (61.04, 122.51, 0), (22.84, 64.81, 0), (21.83, 48.0, 0), (41.96, 44.4, 0), (23.06, 76.03, 0), (26.36, -32.58, 0), (54.77, 15.89, 0), (23.08, 111.96, 0), (25.3, 14.28, 0), (14.83, 104.21, 0), (-23.66, 58.2, 0), (102.73, 35.18, 0), (109.0, -16.78, 0), (-26.94, -34.25, 0), (75.85, 68.48, 0), (121.14, 76.96, 0), (3.79, -32.97, 0), (59.22, 22.79, 0), (52.85, -57.62, 0), (18.23, 69.53, 0), (-1.6, 72.48, 0), (142.38, 80.08, 0), (117.4, 36.11, 0), (67.03, 93.76, 1), (-3.79, 52.3, 0), (77.88, 39.17, 0), (45.84, -11.76, 1), (30.84, 26.48, 0), (-12.58, 42.61, 0), (49.36, 63.23, 0), (158.73, 1.95, 0), (87.52, 74.13, 0), (6.18, 72.95, 0), (99.71, 55.69, 0), (119.75, -13.79, 1), (93.7, 16.61, 0), (-7.17, -14.5, 0), (108.85, 56.33, 1), (44.84, 53.4, 0), (103.86, 29.71, 0), (14.38, 41.83, 0), (116.72, 90.6, 0), (-34.53, 108.72, 0), (60.44, 49.34, 0), (11.49, 35.64, 0), (82.45, 80.28, 0), (13.33, 76.09, 0), (160.14, -22.94, 0), (4.18, 28.36, 1), (55.97, 76.82, 0), (74.02, 71.6, 1), (51.84, 3.28, 0), (105.65, -7.14, 0), (9.18, 42.7, 0), (-6.11, 49.17, 0), (-19.76, 76.61, 0), (153.25, 90.65, 0), (-9.94, 15.02, 0), (-46.84, 49.67, 0), (61.31, 103.12, 0), (63.15, 46.76, 1), (80.34, 39.84, 0), (47.84, 45.49, 0), (79.54, 12.7, 0), (74.68, 54.44, 1), (-0.2, 92.01, 0), (68.41, 44.51, 0), (9.39, 78.53, 0), (66.33, 154.88, 0), (46.98, -78.39, 0), (15.11, 62.18, 0), (65.86, -2.69, 0), (-61.24, 88.08, 0), (2.26, 125.41, 0), (36.99, -37.47, 0), (45.75, -64.1, 0), (-31.91, 81.67, 0), (85.86, 59.75, 0), (1.59, 10.81, 0), (26.95, 48.97, 1), (27.54, -8.47, 0), (-12.01, 149.6, 0), (-18.86, 43.21, 0), (26.96, -15.8, 0), (62.43, 37.58, 0), (75.65, 89.14, 0), (61.33, 156.69, 0), (24.51, 72.1, 0), (9.91, 105.64, 1), (8.89, 11.42, 0), (59.82, 34.61, 0), (153.11, 43.99, 0), (72.08, 17.66, 0), (70.43, -60.32, 0), (71.93, 35.22, 0), (50.53, 82.16, 0), (64.39, 127.34, 0), (90.82, 49.1, 0), (34.9, 105.26, 1), (30.15, 80.7, 0), (-61.31, -22.29, 0), (32.7, -15.92, 0), (29.34, 12.49, 0), (64.55, 125.52, 0), (56.58, -44.07, 0), (107.46, 32.95, 1), (100.74, 13.58, 0), (30.74, 51.57, 0), (28.97, 71.22, 0), (33.38, 166.83, 0), (96.46, 88.05, 0), (31.99, 6.59, 0), (55.43, 125.72, 0), (132.69, 132.31, 0), (17.99, 126.52, 0), (28.59, 118.44, 0), (47.78, 52.34, 0), (134.64, -19.57, 0), (74.85, 96.02, 0), (-22.64, 20.07, 0), (105.21, -32.53, 0), (30.91, -69.17, 0), (117.17, 58.13, 1), (76.17, 105.38, 0), (52.32, 28.53, 0), (37.95, 83.75, 0), (10.9, 38.67, 0), (21.52, 82.74, 0), (64.57, 42.53, 0), (0.17, 88.6, 0), (2.3, 2.14, 0), (58.2, 85.99, 0), (103.62, 6.69, 0), (77.67, 27.6, 1), (-20.44, 94.9, 0), (93.93, 17.01, 0), (28.49, 12.83, 0), (125.12, -12.02, 0), (33.98, 56.64, 0), (154.2, 97.74, 0), (-21.84, 59.1, 0), (21.37, -26.54, 0), (-37.58, -17.91, 0), (29.86, 25.8, 0), (38.89, 75.26, 0), (33.0, 61.86, 0), (3.01, 24.4, 0), (165.22, 63.63, 1), (54.19, 30.92, 0), (81.92, 91.08, 0), (-46.01, 76.85, 0), (118.76, 110.94, 0), (76.69, 29.69, 0), (69.15, 30.1, 0), (20.83, 49.17, 0), (5.78, 29.0, 0), (99.69, 45.61, 0), (-15.11, 125.08, 0), (135.82, -28.69, 0), (63.55, 34.09, 0), (96.89, 67.87, 0), (15.93, 109.77, 0), (42.43, 36.54, 0), (85.71, 93.66, 0), (39.9, 43.36, 0), (133.76, 16.36, 0), (183.69, -7.56, 0), (11.6, 71.41, 0), (18.93, 110.15, 0), (-15.94, 73.86, 0), (80.25, 66.61, 0), (-1.73, 118.6, 0), (72.04, 111.75, 0), (86.19, 78.6, 0), (69.35, -67.26, 0), (66.91, -1.78, 0), (22.74, 84.23, 0), (61.53, 64.03, 0), (28.98, 124.97, 0), (93.23, 59.29, 0), (77.66, 46.45, 0), (13.04, 133.78, 1), (46.33, 23.19, 0), (-12.58, 15.67, 0), (-22.71, 18.18, 0), (39.17, 29.32, 0), (24.8, 50.06, 0), (96.33, 88.1, 0), (67.07, 84.92, 0), (3.44, -67.83, 0), (104.08, 39.25, 0), (99.63, 4.17, 0), (65.98, 0.21, 0), (102.71, -41.05, 0), (52.17, 112.71, 0), (27.06, 25.79, 0), (93.71, 134.23, 0), (20.82, 68.16, 0), (-23.23, 51.19, 0), (18.71, 88.96, 0), (78.44, 79.74, 1), (33.94, 9.93, 0), (86.57, 58.84, 0), (29.04, 74.71, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.1116, OR=0.1210, z=-13.0562, p=5.86e-39；x1: β=0.0162, OR=1.0163, z=5.6483, p=1.62e-08；x2: β=0.0156, OR=1.0158, z=5.5394, p=3.04e-08；
McFadden 伪 R² = 1−LL/LL₀ = 0.1487；
示例预测（x1=x2=50）：P(y=1)=1/(1+e^(−-2.1116))=0.1080。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.1116, OR=0.1210, z=-13.0562, p=5.86e-39；x1: β=0.0162, OR=1.0163, z=5.6483, p=1.62e-08；x2: β=0.0156, OR=1.0158, z=5.5394, p=3.04e-08；
McFadden 伪 R² = 1−LL/LL₀ = 0.1487；
示例预测（x1=x2=50）：P(y=1)=1/(1+e^(−-2.1116))=0.1080。
③ 计算结果：
截距: β=-2.1116 OR=0.121 p=5.86e-39；x1: β=0.0162 OR=1.016 p=1.62e-08；x2: β=0.0156 OR=1.016 p=3.04e-08
④ 结论：在 α=0.05 水平下：x1显著(p=1.62e-08)、x2显著(p=3.04e-08)。两正向因素。（伪 R²=0.1487，模型解释力有限）
案例 99. 年龄+运动预测是否健康
逻辑回归年龄负运动正
① 原始数据：
样本 n=785，逐条 (x1, x2, y)：[(26.15, 34.11, 0), (41.36, 33.49, 1), (39.14, 20.1, 0), (42.02, 35.17, 0), (38.89, 30.07, 0), (48.46, 25.59, 0), (40.87, 51.13, 0), (40.83, 56.9, 0), (54.42, 39.72, 0), (50.55, 72.31, 0), (81.33, 29.73, 0), (37.89, 35.24, 0), (42.41, 50.74, 0), (25.27, 46.75, 1), (38.56, 30.99, 0), (50.32, 42.83, 0), (36.51, 34.21, 0), (57.8, 36.47, 0), (28.52, 12.11, 0), (56.13, 35.72, 0), (63.12, 68.3, 0), (36.65, 40.07, 0), (54.22, 31.9, 0), (26.89, 24.81, 0), (23.13, 34.85, 1), (32.2, 28.05, 0), (28.24, 57.26, 0), (35.97, 34.08, 0), (38.84, 46.07, 1), (43.65, 45.51, 0), (27.95, 19.26, 0), (17.21, 31.3, 0), (39.53, 33.54, 0), (33.41, 33.41, 0), (40.1, 19.12, 0), (39.69, 29.88, 0), (40.37, 26.49, 0), (53.87, 62.54, 1), (13.55, 21.13, 0), (45.16, 46.47, 0), (56.65, 22.04, 0), (64.15, 59.84, 0), (50.39, 18.44, 0), (48.07, 14.21, 0), (49.3, 28.86, 0), (39.27, 37.03, 1), (48.75, 54.79, 0), (60.02, 46.61, 0), (34.06, 6.13, 0), (46.79, 35.06, 0), (34.88, 28.19, 0), (35.32, 30.28, 0), (37.76, 48.62, 0), (44.95, 66.36, 1), (66.77, 26.0, 0), (20.45, 38.35, 0), (25.22, 45.62, 0), (26.23, 7.53, 0), (50.05, 44.57, 1), (40.31, 40.27, 0), (30.43, 29.48, 0), (40.98, 70.39, 0), (48.77, 39.76, 0), (26.59, 18.96, 0), (24.82, 49.18, 0), (34.71, 51.59, 1), (22.15, 10.07, 0), (31.06, 31.22, 0), (19.61, 20.24, 0), (11.05, 25.57, 1), (45.08, 39.4, 0), (40.86, 45.35, 0), (51.14, 68.68, 1), (62.5, 36.2, 0), (53.29, 44.34, 0), (20.69, 31.72, 0), (45.12, 49.92, 0), (5.51, 54.86, 0), (34.07, 20.87, 0), (25.74, 38.87, 0), (46.02, 32.69, 0), (43.67, 37.17, 0), (42.3, 62.42, 0), (69.13, 24.85, 0), (28.19, 40.48, 0), (8.55, 40.05, 0), (39.0, 37.54, 0), (50.05, 42.82, 0), (22.8, 59.8, 0), (16.6, 37.7, 0), (39.93, 41.97, 1), (50.14, 66.86, 0), (52.83, 44.17, 0), (36.17, 22.3, 0), (15.22, 17.21, 1), (55.91, 36.75, 0), (33.68, 51.2, 0), (14.0, 30.42, 1), (55.24, 43.5, 0), (61.15, 21.41, 0), (59.46, 25.53, 0), (49.66, 33.39, 0), (44.57, 20.09, 0), (63.35, 44.0, 0), (40.86, 24.83, 0), (50.81, 20.99, 0), (40.25, 47.73, 0), (30.83, 45.65, 0), (58.06, 28.05, 0), (58.21, 31.19, 0), (44.92, 38.81, 0), (18.76, 30.34, 0), (49.12, 32.89, 0), (62.86, 14.14, 0), (41.2, 56.96, 0), (32.59, 50.99, 1), (46.78, 32.44, 0), (49.47, 18.92, 0), (72.48, 51.99, 0), (26.98, 41.51, 0), (43.17, 49.2, 0), (44.86, 28.27, 0), (24.51, 48.14, 1), (51.54, 58.5, 0), (49.7, 34.15, 0), (17.23, 40.39, 0), (48.68, 41.05, 0), (47.06, 40.55, 0), (36.34, 31.97, 0), (46.03, 31.28, 0), (32.43, 30.46, 0), (56.55, 46.11, 0), (44.69, 43.98, 0), (55.85, 15.86, 0), (55.81, 50.48, 1), (39.51, 17.02, 0), (36.49, 26.81, 0), (42.92, 29.95, 0), (28.17, 57.57, 1), (49.03, 57.78, 0), (54.64, 35.1, 1), (58.83, 39.84, 0), (53.89, 34.41, 0), (25.1, 43.38, 0), (49.72, 32.15, 0), (27.77, 71.36, 1), (56.4, 34.28, 0), (22.82, 51.51, 0), (15.87, 60.82, 1), (36.02, 35.85, 1), (34.55, 49.43, 1), (14.91, 17.96, 0), (35.4, 54.35, 0), (36.1, 28.26, 0), (18.65, 23.7, 0), (31.11, 25.43, 0), (22.66, 30.05, 0), (27.22, 22.97, 0), (52.87, 51.39, 0), (26.05, 12.94, 0), (49.71, 35.93, 1), (67.63, 23.52, 0), (47.25, 30.97, 0), (48.95, 50.18, 0), (30.65, 47.59, 0), (48.8, 23.16, 0), (38.62, 22.78, 0), (28.31, 39.43, 0), (65.43, 46.22, 0), (34.1, 53.74, 1), (48.04, 39.97, 0), (55.37, 45.44, 0), (50.3, 43.08, 0), (48.94, 35.03, 0), (54.57, 42.16, 0), (20.74, 51.21, 0), (33.64, 40.6, 0), (43.55, 42.09, 0), (42.57, 36.27, 0), (34.01, 37.64, 0), (42.67, 41.32, 0), (22.51, 36.54, 0), (14.34, 46.24, 0), (47.31, 24.84, 0), (25.81, 29.91, 0), (32.04, 46.51, 1), (22.21, 43.75, 0), (15.08, 36.58, 0), (42.49, 55.84, 0), (35.1, 24.4, 0), (19.93, 57.63, 1), (29.64, 35.3, 0), (29.06, 34.62, 0), (7.75, 64.56, 1), (53.85, 21.25, 0), (38.84, 41.13, 0), (33.65, 14.29, 0), (46.87, 37.96, 1), (49.69, 18.88, 0), (35.73, 9.39, 0), (48.71, 33.01, 0), (45.39, 42.05, 0), (26.41, 39.0, 0), (56.18, 53.18, 0), (54.4, 15.23, 0), (45.2, 39.73, 0), (38.77, 17.65, 0), (52.12, 51.97, 0), (46.67, 21.22, 0), (60.7, 24.18, 0), (41.12, 33.09, 0), (54.79, 54.96, 1), (52.21, 68.58, 0), (2.29, 31.92, 1), (45.51, 32.32, 0), (67.03, 40.89, 0), (48.88, 45.57, 0), (21.72, 26.53, 0), (37.37, 42.8, 1), (48.9, 28.16, 0), (30.98, 80.15, 1), (29.38, 31.54, 0), (29.24, 32.18, 0), (58.78, 30.52, 0), (57.28, 41.25, 0), (18.44, 27.33, 1), (29.13, 28.97, 0), (48.53, 36.36, 1), (58.02, 49.65, 0), (13.75, 62.07, 1), (44.59, 49.18, 1), (24.72, 12.42, 0), (49.02, 53.32, 0), (44.78, 23.92, 0), (22.28, 42.6, 0), (30.98, 28.12, 0), (58.97, 46.55, 0), (42.36, 30.59, 0), (31.64, 40.59, 1), (52.2, 38.25, 0), (28.26, 44.22, 0), (46.62, 17.12, 0), (55.61, 44.15, 0), (59.12, 52.07, 0), (42.13, 32.92, 0), (78.17, 54.92, 0), (33.43, 31.35, 0), (41.12, 23.38, 0), (60.7, 35.0, 0), (62.28, 49.37, 0), (39.68, 62.66, 1), (35.72, 73.19, 1), (31.37, 54.92, 0), (50.93, 72.15, 0), (44.93, 56.89, 0), (59.26, 24.93, 0), (36.22, 61.55, 1), (12.21, 36.55, 1), (35.33, 55.57, 0), (56.56, 48.36, 0), (33.02, 44.39, 0), (18.63, 28.31, 0), (14.44, 51.73, 0), (23.2, 9.95, 0), (40.73, 28.9, 0), (16.62, 43.88, 1), (38.46, 62.27, 0), (34.69, 24.5, 0), (26.4, 39.93, 0), (37.56, 35.2, 0), (44.2, 63.73, 0), (28.72, 33.22, 0), (32.86, 25.48, 0), (47.2, 61.74, 0), (22.0, 26.97, 0), (36.45, 56.5, 0), (22.37, 41.34, 1), (32.26, 39.23, 1), (45.21, 40.35, 0), (37.56, 43.27, 0), (37.25, 50.86, 0), (34.92, 12.52, 0), (55.85, 36.74, 0), (34.87, 33.2, 0), (37.63, 42.47, 0), (28.72, 17.83, 1), (47.75, 62.3, 1), (31.06, 23.15, 0), (-2.56, 16.6, 0), (17.03, 30.42, 1), (33.52, 46.02, 0), (23.81, 51.63, 0), (45.18, 50.02, 0), (41.78, 51.83, 0), (43.49, 54.2, 0), (61.66, 42.66, 0), (16.98, 18.86, 0), (50.77, 14.33, 0), (30.67, 5.02, 0), (58.48, 48.25, 0), (40.66, 17.13, 0), (42.35, 56.91, 0), (19.28, 38.14, 1), (20.02, 24.02, 0), (61.16, 29.8, 0), (26.05, 33.22, 0), (29.09, 29.53, 0), (16.86, 1.19, 0), (23.88, 28.02, 0), (46.45, 48.67, 0), (26.07, 45.4, 0), (64.61, 18.94, 0), (39.29, 40.72, 1), (56.48, 60.36, 0), (28.61, 52.92, 0), (30.37, 51.79, 1), (47.18, 53.56, 0), (42.56, 40.86, 0), (47.71, 39.31, 0), (42.58, 46.75, 0), (34.61, 50.81, 0), (29.35, 31.83, 1), (52.3, 41.31, 0), (47.51, 38.38, 0), (31.02, 52.79, 0), (29.76, 56.19, 0), (47.52, 51.51, 0), (55.88, 35.65, 0), (46.0, 46.72, 0), (51.46, 40.08, 0), (50.0, 38.79, 0), (13.82, 52.47, 1), (41.05, 66.66, 0), (47.12, 21.82, 0), (41.46, 8.39, 0), (35.14, 30.7, 0), (16.89, 41.72, 0), (33.96, 29.64, 0), (51.25, 44.46, 0), (36.73, 35.61, 0), (64.08, 62.44, 0), (20.13, 77.53, 1), (49.64, 61.41, 1), (26.88, 55.28, 0), (49.37, 52.22, 0), (34.79, 42.58, 0), (47.87, 25.56, 0), (36.95, 54.92, 1), (42.72, 60.86, 0), (51.75, 39.19, 0), (44.86, 24.57, 0), (28.14, 28.16, 0), (63.56, 25.67, 0), (31.12, 35.4, 1), (21.49, 60.57, 0), (19.25, 21.14, 0), (51.3, 6.62, 0), (29.26, 41.03, 0), (25.84, -5.21, 0), (20.96, 39.22, 1), (6.81, 37.64, 1), (45.99, 41.67, 0), (60.16, 48.17, 0), (56.87, 24.61, 0), (34.13, 23.63, 0), (68.06, 43.51, 0), (74.16, 36.85, 0), (24.13, 53.93, 0), (32.69, 33.31, 0), (36.79, 38.15, 0), (63.2, 28.47, 0), (16.0, 31.66, 1), (22.38, 50.89, 1), (50.06, 36.83, 0), (19.53, 52.8, 0), (24.83, 28.77, 1), (56.91, 28.17, 0), (51.99, 35.25, 0), (37.55, 9.73, 0), (44.89, 39.99, 0), (46.35, 39.68, 0), (44.0, 16.7, 0), (35.67, 58.34, 0), (40.43, 59.94, 0), (28.74, 38.21, 0), (41.09, 21.4, 0), (44.1, 45.68, 0), (49.12, 40.47, 0), (53.57, 23.08, 0), (39.52, 37.08, 0), (35.73, 40.33, 0), (62.18, 42.58, 0), (70.85, 55.7, 0), (40.04, 38.45, 0), (44.56, 25.9, 0), (38.05, 33.58, 0), (24.14, 2.79, 0), (56.28, 35.95, 0), (37.55, 22.83, 0), (49.97, 48.53, 1), (54.68, 29.74, 0), (28.1, 21.15, 0), (21.54, 27.92, 0), (44.66, 40.65, 0), (38.78, 22.1, 0), (38.91, 60.81, 0), (42.24, 64.56, 1), (39.74, 39.68, 0), (32.68, 65.46, 0), (23.81, 28.21, 1), (37.39, 56.99, 0), (42.23, 32.78, 0), (32.83, 52.39, 0), (9.13, 41.27, 1), (17.66, 34.25, 1), (78.82, 54.64, 0), (31.51, 29.49, 0), (50.37, 59.21, 0), (39.81, 38.56, 0), (35.36, 47.85, 0), (36.59, 16.33, 0), (44.03, 29.09, 0), (53.46, 31.12, 0), (46.96, 62.68, 0), (41.68, 79.14, 1), (44.41, 26.53, 0), (29.29, 23.29, 0), (22.65, 10.89, 0), (19.44, 48.78, 0), (41.59, 45.15, 1), (46.35, 37.5, 0), (78.09, 34.8, 0), (29.63, 55.69, 1), (25.77, 27.53, 0), (59.66, 23.09, 0), (53.32, 36.88, 0), (36.79, 31.59, 0), (36.57, 35.51, 0), (38.98, 68.9, 0), (45.91, 50.11, 0), (41.46, 58.26, 0), (43.94, 71.21, 0), (51.77, 43.19, 0), (35.67, 57.16, 1), (49.87, 16.16, 0), (55.28, 20.18, 0), (28.23, 51.83, 1), (23.95, 38.06, 1), (45.11, 42.69, 0), (54.92, 31.69, 0), (34.0, 36.74, 0), (77.1, 31.31, 0), (56.44, 25.97, 0), (56.73, 59.36, 0), (47.71, 13.26, 0), (41.36, 40.18, 0), (31.18, 42.88, 0), (41.21, 31.74, 0), (25.48, 26.92, 0), (35.03, 44.67, 0), (48.88, 19.48, 0), (29.08, 24.87, 1), (26.34, 41.63, 0), (47.77, 44.1, 0), (33.55, 38.82, 0), (72.13, 54.28, 0), (26.11, 44.69, 1), (85.54, 22.97, 0), (75.14, 8.61, 0), (50.28, 29.19, 0), (84.12, 46.36, 0), (28.59, 34.13, 0), (10.98, 52.0, 0), (38.2, 52.2, 0), (43.76, 26.4, 0), (36.48, 44.85, 1), (40.82, 41.5, 1), (26.15, 58.33, 0), (6.79, 51.67, 1), (38.94, 48.88, 0), (26.04, 46.11, 0), (53.01, 36.29, 0), (9.85, 42.39, 1), (32.16, 15.74, 0), (40.65, 50.16, 0), (50.76, 39.25, 0), (15.51, 30.25, 0), (27.26, 42.68, 1), (48.25, 35.67, 0), (25.82, 34.41, 0), (25.14, 49.94, 0), (33.02, 55.95, 1), (26.25, 33.76, 0), (35.26, 60.18, 0), (52.4, 42.04, 1), (45.51, 53.83, 0), (47.35, 18.09, 0), (52.06, 44.63, 0), (67.56, 36.95, 0), (29.76, 48.01, 0), (44.38, 62.04, 0), (38.4, 49.05, 0), (56.05, 65.24, 0), (50.86, 36.97, 0), (56.6, 50.84, 0), (43.47, 36.66, 0), (35.19, 43.66, 0), (44.11, 20.17, 0), (33.56, 37.98, 0), (35.62, 32.28, 0), (37.32, 26.27, 0), (63.55, 35.49, 0), (70.45, 53.11, 0), (45.77, 51.37, 0), (32.59, 34.4, 0), (41.68, 45.58, 0), (26.61, 35.08, 0), (41.16, 38.34, 0), (44.29, 42.86, 0), (37.11, 17.39, 0), (35.21, 54.03, 0), (43.74, 16.09, 0), (40.25, 58.12, 0), (36.89, 27.11, 0), (31.7, 65.86, 0), (54.26, 2.91, 0), (20.76, 40.98, 0), (4.04, 33.69, 1), (28.42, 34.06, 0), (34.73, 31.98, 0), (47.72, 72.33, 0), (37.47, 39.0, 0), (37.49, 33.5, 0), (49.42, 48.98, 0), (44.77, 59.47, 1), (12.93, 60.88, 1), (55.42, 41.84, 0), (41.29, 58.46, 0), (23.92, 35.34, 0), (29.99, 32.35, 0), (39.87, 40.53, 0), (56.8, 32.6, 0), (45.13, 44.0, 0), (40.46, 63.14, 0), (54.06, 52.64, 0), (36.45, 29.97, 0), (37.89, 49.9, 1), (27.8, 60.27, 0), (75.19, 47.67, 0), (41.04, -3.64, 0), (37.78, 47.37, 0), (34.44, 64.06, 0), (34.59, 24.33, 0), (55.57, 37.28, 0), (35.9, 44.97, 0), (42.92, 25.66, 0), (37.92, 34.08, 0), (34.07, 20.22, 0), (37.08, 10.29, 0), (50.25, 10.07, 0), (78.04, 10.55, 0), (29.26, 56.7, 1), (46.45, 63.04, 0), (59.9, 43.39, 0), (36.8, 45.07, 0), (43.86, 43.8, 0), (63.52, 22.27, 0), (57.22, 49.32, 0), (3.67, 18.33, 0), (24.07, 36.25, 1), (23.59, 56.24, 0), (34.93, 38.4, 0), (54.39, 27.96, 0), (38.76, 45.58, 0), (53.05, 38.03, 0), (34.87, 59.89, 0), (38.42, 25.05, 0), (34.01, 42.42, 0), (33.97, 45.19, 0), (54.93, 28.42, 0), (68.88, 54.35, 0), (36.6, 35.63, 0), (77.84, 55.48, 0), (38.09, 36.97, 0), (51.04, 62.11, 1), (59.42, 12.2, 0), (44.11, 31.23, 0), (63.89, 46.49, 0), (26.25, 39.22, 0), (36.39, 47.46, 0), (32.65, 52.46, 0), (64.59, 36.37, 0), (16.96, 38.29, 0), (23.69, 50.92, 1), (45.24, 43.17, 0), (21.99, 24.62, 0), (15.63, 59.93, 0), (35.19, 46.07, 0), (37.52, 14.63, 0), (36.26, 45.83, 0), (54.05, 61.12, 0), (11.85, 29.67, 1), (55.89, 12.15, 0), (54.0, 40.73, 0), (46.27, 62.54, 0), (52.97, 43.37, 0), (31.93, 45.33, 0), (37.93, 54.1, 1), (32.59, 37.51, 0), (20.91, 44.47, 0), (47.68, 57.89, 0), (38.05, 32.82, 0), (49.01, 60.93, 0), (39.58, 31.55, 0), (34.3, 61.75, 1), (20.18, 58.84, 0), (31.86, 18.02, 0), (57.47, 37.32, 0), (36.16, 39.49, 0), (42.83, 55.15, 0), (55.21, 49.39, 0), (51.64, 42.63, 0), (37.19, 36.55, 0), (42.11, 40.45, 0), (47.32, 41.68, 0), (45.11, 31.38, 0), (39.27, 57.37, 0), (42.79, 29.58, 0), (34.73, 30.93, 0), (46.33, 23.78, 0), (37.15, 30.02, 0), (44.56, 39.8, 0), (37.98, 42.83, 0), (32.36, 54.44, 0), (56.13, 59.36, 0), (26.04, 50.88, 0), (20.29, 39.79, 0), (50.73, 56.17, 0), (31.92, 29.56, 0), (35.08, 35.6, 1), (20.29, 59.73, 1), (19.57, 51.75, 0), (42.38, 30.96, 0), (65.26, 23.62, 0), (48.7, 37.58, 0), (35.28, 20.23, 0), (8.39, 52.89, 1), (21.26, 25.86, 0), (51.17, 45.8, 0), (36.32, 28.35, 0), (35.84, 53.58, 0), (52.95, 23.5, 0), (52.91, 59.38, 1), (62.57, 43.1, 1), (52.64, 57.75, 1), (85.86, 60.55, 0), (20.99, 43.53, 1), (40.49, 29.67, 0), (43.38, 43.63, 0), (10.55, 27.89, 0), (6.31, 60.87, 0), (44.17, 23.82, 0), (46.63, 56.67, 0), (33.62, 24.68, 0), (56.61, 26.55, 0), (41.05, 47.85, 0), (59.97, 37.11, 0), (37.21, 56.74, 1), (42.29, 28.06, 0), (73.59, 55.1, 0), (40.36, 44.7, 0), (37.53, 43.59, 0), (59.84, 54.64, 1), (49.13, 62.96, 0), (61.6, 37.51, 0), (46.66, 19.43, 0), (38.68, 39.34, 0), (58.73, 53.27, 0), (49.29, 53.09, 0), (39.26, 44.04, 0), (38.66, 27.17, 0), (43.04, 26.82, 1), (45.8, 31.74, 0), (33.94, 34.37, 0), (16.33, 40.41, 0), (14.12, 25.05, 1), (46.45, 22.45, 0), (35.77, 38.19, 0), (25.6, 23.89, 0), (41.73, 59.29, 0), (33.31, 47.14, 0), (26.91, 34.39, 1), (49.91, 50.31, 0), (51.01, 54.2, 1), (30.45, 50.02, 1), (19.58, 46.93, 0), (38.54, 50.75, 0), (66.54, 30.41, 0), (35.52, 65.99, 1), (37.66, 36.71, 0), (28.85, 49.89, 0), (39.02, 47.68, 0), (43.58, 33.0, 1), (42.55, 44.17, 0), (35.18, 33.2, 0), (45.28, 32.62, 0), (54.99, 39.65, 0), (16.15, 40.46, 0), (24.98, 30.13, 1), (39.56, 25.84, 0), (39.61, 26.94, 0), (20.58, 57.99, 1), (32.92, 29.14, 0), (25.74, 22.9, 0), (39.74, 30.05, 0), (32.01, 34.84, 0), (-1.04, 43.67, 0), (59.07, 49.19, 0), (29.86, 61.29, 0), (27.98, 50.08, 0), (63.63, 27.5, 0), (30.67, 41.86, 0), (45.85, 47.95, 0), (34.13, 56.92, 0), (21.53, 37.77, 0), (15.0, 44.55, 0), (7.23, 49.46, 1), (41.04, 46.64, 0), (26.91, 9.03, 1), (22.29, 33.64, 0), (55.68, 51.31, 0), (26.29, 13.46, 0), (43.3, 55.53, 0), (28.47, 17.3, 0), (28.99, 30.45, 0), (20.55, 17.41, 0), (53.65, 25.38, 0), (19.55, 26.37, 0), (18.54, 73.14, 1), (31.5, 21.54, 0), (38.16, 35.12, 0), (15.84, 32.04, 0), (48.54, 30.0, 0), (23.48, 55.8, 0), (16.3, 19.73, 0), (33.14, 41.81, 0), (45.36, 16.5, 0), (28.48, 26.12, 0), (22.55, 38.11, 0), (17.58, 48.36, 1), (67.91, 29.49, 0), (29.22, 47.72, 0), (46.09, 34.16, 0), (34.92, 38.81, 0), (21.73, 10.57, 0), (70.88, 37.45, 0), (40.13, 58.23, 0), (59.8, 22.02, 0), (50.86, 51.95, 0), (48.68, 73.98, 0), (37.96, 26.17, 0), (25.36, 49.58, 0), (45.77, 50.82, 0), (54.51, 34.12, 0), (80.8, 46.61, 0), (33.08, 45.56, 0), (47.89, 46.51, 0), (24.98, 25.58, 0), (47.26, 55.35, 1), (25.51, 58.22, 0), (48.32, 4.04, 0), (22.45, 1.99, 0), (45.64, 45.92, 0), (10.13, 20.41, 0), (43.29, 7.48, 0), (27.58, 36.41, 1), (20.17, 36.59, 0), (36.99, 43.59, 0), (13.41, 9.13, 0), (24.42, 56.42, 1), (27.22, 25.22, 0), (19.26, 55.99, 0), (36.46, 22.68, 0), (26.03, 61.19, 0), (44.5, 64.89, 1)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.1081, OR=0.1215, z=-15.8649, p=1.11e-56；x1: β=-0.0611, OR=0.9408, z=-7.2136, p=5.45e-13；x2: β=0.0558, OR=1.0574, z=6.8501, p=7.38e-12；
McFadden 伪 R² = 1−LL/LL₀ = 0.1591；
示例预测（x1=x2=40）：P(y=1)=1/(1+e^(−-2.1081))=0.1083。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.1081, OR=0.1215, z=-15.8649, p=1.11e-56；x1: β=-0.0611, OR=0.9408, z=-7.2136, p=5.45e-13；x2: β=0.0558, OR=1.0574, z=6.8501, p=7.38e-12；
McFadden 伪 R² = 1−LL/LL₀ = 0.1591；
示例预测（x1=x2=40）：P(y=1)=1/(1+e^(−-2.1081))=0.1083。
③ 计算结果：
截距: β=-2.1081 OR=0.121 p=1.11e-56；x1: β=-0.0611 OR=0.941 p=5.45e-13；x2: β=0.0558 OR=1.057 p=7.38e-12
④ 结论：在 α=0.05 水平下：x1显著(p=5.45e-13)、x2显著(p=7.38e-12)。年龄负运动正。（伪 R²=0.1591，模型解释力有限）
案例 100. 价格+品牌预测是否购买
逻辑回归价负品牌正
① 原始数据：
样本 n=479，逐条 (x1, x2, y)：[(82.31, 80.2, 0), (115.94, 139.32, 0), (145.71, 73.68, 0), (110.67, 112.84, 0), (78.48, 47.93, 0), (89.51, 76.07, 0), (109.01, 77.54, 0), (73.4, 26.23, 0), (152.01, 185.64, 1), (149.72, 83.22, 0), (111.83, 98.21, 0), (120.71, 105.99, 0), (113.79, 71.12, 0), (110.93, 105.26, 0), (123.3, 91.31, 0), (162.29, 115.42, 0), (122.29, 134.64, 0), (78.27, 72.71, 0), (144.22, 18.32, 0), (77.78, 147.59, 0), (-2.61, 128.9, 0), (114.13, 200.66, 0), (102.75, 105.83, 0), (7.04, 98.29, 0), (110.85, 150.11, 0), (133.89, 105.82, 0), (50.75, 104.96, 1), (110.31, 96.43, 0), (88.31, 37.27, 0), (29.79, 103.26, 1), (85.53, 110.9, 0), (89.09, 83.12, 0), (100.12, 79.08, 0), (69.27, 80.49, 1), (44.61, 90.79, 0), (47.2, 148.52, 0), (72.26, 107.1, 0), (103.61, 107.55, 0), (114.62, 80.93, 0), (64.94, 99.18, 0), (144.29, 56.79, 0), (107.58, 41.11, 0), (89.33, 21.03, 0), (100.47, 137.46, 1), (151.77, 153.62, 0), (43.71, 113.87, 0), (84.67, 95.52, 0), (47.59, 76.0, 0), (84.54, 96.51, 0), (50.19, 65.62, 0), (160.2, 38.43, 0), (64.72, 38.53, 0), (64.28, 98.46, 0), (5.3, 133.79, 0), (187.67, 171.72, 0), (39.19, 148.67, 1), (130.29, 66.92, 0), (82.22, 85.21, 1), (139.65, 138.79, 0), (7.65, 238.73, 0), (114.06, 5.23, 0), (69.4, 148.13, 0), (128.54, 48.08, 0), (57.91, 45.71, 0), (102.04, 20.8, 0), (138.42, 18.14, 0), (119.49, 122.47, 0), (92.23, 32.13, 0), (58.99, 137.15, 1), (144.87, 4.65, 0), (77.14, 73.66, 0), (57.65, 128.11, 0), (137.9, 140.19, 0), (99.28, 92.07, 0), (123.04, 149.46, 0), (144.23, 57.87, 0), (112.81, 98.98, 0), (45.95, 41.88, 0), (36.76, 55.68, 0), (116.69, 155.65, 0), (38.34, 161.03, 0), (113.99, 116.67, 1), (179.36, 126.42, 0), (81.17, 138.38, 0), (63.49, 114.5, 0), (114.15, 83.7, 0), (13.68, 108.08, 0), (110.29, 93.8, 0), (101.98, 99.76, 0), (221.09, 213.16, 0), (54.35, 77.11, 0), (112.74, -16.06, 0), (141.08, 115.11, 0), (92.19, 78.16, 0), (24.33, 211.39, 1), (165.86, 79.18, 0), (28.94, 66.19, 0), (149.64, 74.51, 0), (69.59, 151.7, 1), (116.17, 132.83, 0), (180.57, 143.41, 0), (118.51, 168.68, 0), (71.1, 79.64, 0), (109.94, 86.14, 0), (12.4, 33.23, 0), (97.37, 117.38, 0), (131.53, 167.84, 0), (91.56, 12.83, 0), (60.53, 55.96, 0), (88.81, 15.77, 0), (115.87, 120.34, 0), (45.39, 57.02, 1), (217.48, 53.28, 0), (93.17, 36.83, 0), (77.54, 110.96, 0), (13.61, 233.45, 1), (82.13, 189.1, 1), (98.32, 51.41, 0), (-20.8, 58.06, 1), (124.84, 80.63, 0), (42.02, 67.81, 0), (165.65, 279.32, 0), (153.4, 122.66, 0), (57.83, 202.64, 1), (125.95, 205.1, 0), (101.13, 143.67, 0), (69.01, 74.64, 0), (149.46, 21.72, 0), (85.47, 91.32, 0), (131.74, 88.68, 0), (86.15, 105.62, 0), (86.82, 105.47, 0), (138.03, 223.45, 1), (58.77, 141.61, 1), (138.47, 36.42, 0), (111.08, 194.83, 0), (114.44, 243.32, 1), (152.07, -9.14, 0), (9.41, 87.72, 0), (70.56, 12.13, 0), (60.89, 20.44, 0), (111.43, 138.6, 0), (188.22, 68.24, 0), (147.02, 97.86, 0), (74.98, 176.3, 0), (133.82, 115.36, 0), (115.1, 190.42, 0), (185.53, 125.04, 0), (77.1, 114.63, 0), (66.06, 116.07, 0), (139.89, 77.98, 0), (26.21, 51.7, 0), (65.62, 130.67, 1), (51.8, 84.48, 0), (81.07, 136.68, 0), (61.05, 163.67, 0), (139.53, 72.24, 0), (128.36, 245.31, 1), (55.15, 110.3, 0), (70.24, 113.16, 0), (112.27, 140.18, 1), (113.64, 75.04, 0), (65.58, 5.15, 0), (56.79, 46.19, 0), (3.95, 129.7, 0), (29.74, 182.18, 1), (82.54, 171.23, 0), (126.64, 65.16, 0), (48.34, 19.67, 0), (136.72, 33.53, 0), (116.41, 48.27, 0), (101.14, 92.96, 0), (123.72, 22.19, 0), (163.68, 84.09, 0), (175.83, 62.66, 0), (95.69, 172.35, 0), (169.22, 152.39, 0), (96.74, 151.93, 0), (128.08, 154.22, 0), (40.64, 95.33, 0), (90.17, 107.87, 0), (84.49, 31.82, 0), (44.78, 146.77, 0), (99.0, 171.96, 0), (118.1, 53.12, 0), (134.86, 134.45, 0), (114.94, 30.65, 0), (121.04, 42.7, 0), (183.9, 120.19, 0), (4.5, 139.87, 1), (129.82, 117.42, 0), (119.36, 82.07, 0), (26.85, 137.29, 1), (122.82, 172.9, 0), (108.57, 38.22, 1), (115.18, 185.58, 0), (160.49, 83.12, 0), (163.94, 80.62, 0), (61.44, 135.25, 0), (82.73, 131.93, 1), (130.4, 97.22, 0), (94.33, 95.64, 0), (185.34, 168.93, 0), (92.36, 112.44, 0), (136.88, 138.69, 0), (8.98, 88.56, 0), (28.95, 170.53, 0), (32.26, 75.03, 0), (150.38, 117.4, 0), (15.09, 79.72, 0), (-10.05, 89.33, 1), (59.99, 160.96, 0), (139.05, 68.09, 0), (90.77, 119.55, 0), (189.01, 77.62, 0), (115.38, 86.14, 0), (76.66, 121.25, 1), (74.06, 112.3, 0), (111.65, 138.88, 1), (157.98, 32.72, 0), (100.36, 115.71, 0), (81.33, 100.08, 0), (94.19, 115.47, 0), (224.98, 103.57, 0), (86.38, 79.53, 0), (89.12, 91.79, 1), (72.79, 209.4, 1), (183.5, 179.08, 0), (99.24, 36.39, 0), (107.07, 113.17, 1), (53.47, 57.48, 0), (47.84, 118.65, 0), (61.18, 36.01, 1), (81.56, 75.11, 1), (148.25, 122.32, 0), (105.99, 146.21, 1), (99.76, 91.94, 0), (7.85, 102.02, 0), (91.3, 40.17, 1), (81.07, 140.02, 1), (37.77, 159.13, 0), (74.91, 168.01, 0), (96.52, 148.61, 0), (169.04, 136.83, 0), (82.62, 111.96, 0), (22.27, 128.55, 0), (105.16, 25.87, 0), (117.62, 152.51, 0), (76.88, 97.58, 0), (133.42, 91.84, 0), (61.05, 81.32, 0), (97.59, 57.13, 0), (105.38, 61.11, 0), (184.73, 128.22, 0), (34.73, 228.34, 1), (172.37, 79.63, 0), (97.09, 153.54, 0), (90.91, 143.7, 0), (126.1, 99.34, 0), (155.37, 7.38, 0), (161.84, 22.94, 0), (-52.67, 44.95, 0), (127.34, 166.61, 1), (90.72, 154.44, 1), (206.87, 105.66, 0), (46.59, 166.3, 1), (123.8, 155.32, 0), (96.19, 48.04, 0), (64.85, 75.99, 0), (131.95, 160.25, 0), (105.31, 81.29, 0), (177.74, 159.89, 0), (126.66, 121.51, 0), (111.19, 143.7, 0), (154.96, 1.79, 0), (115.14, 89.87, 0), (113.12, 156.92, 0), (12.33, 108.95, 0), (127.8, 72.93, 0), (80.88, 45.3, 0), (129.52, 93.16, 0), (167.86, 36.08, 0), (130.12, 163.58, 0), (144.8, 187.11, 0), (173.87, 166.87, 0), (128.77, 148.67, 1), (154.19, 118.94, 0), (104.68, 55.66, 0), (94.89, 61.63, 0), (120.78, 112.94, 0), (134.4, 23.74, 0), (68.62, 56.77, 0), (98.8, 125.37, 1), (112.03, 136.12, 0), (121.45, 185.49, 0), (115.93, 88.75, 0), (17.09, 126.5, 1), (116.97, 190.84, 0), (94.52, 121.21, 0), (8.6, 91.87, 1), (75.23, 129.03, 0), (130.92, 21.12, 0), (62.03, 81.8, 0), (132.98, 91.14, 0), (55.36, 98.03, 0), (74.49, 160.99, 0), (91.83, 115.73, 0), (170.84, 105.17, 0), (130.52, 58.8, 0), (148.69, 139.51, 0), (96.13, 123.82, 0), (89.41, 111.05, 0), (87.68, 123.65, 1), (130.8, 76.35, 0), (167.87, 26.0, 0), (31.79, 176.31, 1), (165.37, 108.59, 0), (174.23, 103.68, 0), (67.58, 230.25, 0), (134.81, 157.1, 0), (98.43, 40.25, 0), (176.69, 100.2, 0), (106.55, 105.09, 0), (164.98, 198.74, 0), (48.0, 76.31, 0), (87.68, 94.06, 1), (78.07, 119.28, 0), (71.33, 161.82, 1), (126.97, 58.84, 0), (161.04, 10.32, 0), (85.73, 161.55, 0), (67.21, 21.28, 1), (84.53, 120.16, 0), (132.71, 75.34, 1), (-20.59, 57.67, 0), (70.31, 129.53, 0), (39.39, 85.71, 0), (72.96, 69.95, 0), (48.08, 13.25, 0), (19.14, 143.31, 1), (23.18, 122.02, 1), (54.94, 92.66, 1), (123.1, 63.02, 0), (26.06, 177.11, 1), (67.79, 125.59, 0), (101.04, 86.64, 0), (112.95, 108.99, 0), (144.72, 86.01, 0), (73.15, 108.04, 0), (139.98, 180.17, 0), (168.58, 143.13, 0), (170.76, 103.66, 0), (121.21, 59.72, 0), (181.42, -5.47, 0), (105.87, 28.37, 0), (95.56, 44.93, 0), (86.22, 96.35, 0), (169.71, 136.3, 0), (147.69, 143.63, 0), (58.7, 125.88, 0), (91.33, 166.46, 1), (59.7, 163.04, 0), (99.58, 175.39, 1), (92.18, 64.93, 0), (96.06, -13.17, 0), (150.19, 133.2, 0), (147.76, 75.11, 0), (95.49, 133.18, 1), (91.97, 128.42, 0), (159.34, 134.5, 0), (104.39, 101.97, 0), (147.97, 87.51, 0), (34.88, 130.31, 1), (116.97, 31.27, 0), (70.54, 54.04, 0), (108.54, 187.33, 0), (152.71, 66.3, 0), (24.97, 119.46, 1), (118.39, -14.22, 0), (139.61, 102.81, 0), (85.78, 91.43, 0), (69.54, 166.05, 0), (146.43, 86.85, 0), (168.76, 27.83, 0), (56.62, 86.79, 1), (8.03, 77.68, 0), (165.89, 100.13, 0), (100.0, 167.36, 0), (152.79, 146.97, 0), (41.66, 87.14, 0), (80.34, 125.84, 0), (87.13, 101.9, 0), (132.65, 150.02, 0), (52.12, 204.32, 1), (97.39, 29.25, 0), (107.26, 70.69, 0), (74.52, 60.0, 0), (76.6, 45.24, 0), (211.55, 5.15, 0), (169.48, 111.41, 0), (149.24, 111.99, 0), (118.35, 71.28, 0), (85.06, 30.87, 0), (110.3, 22.72, 0), (11.21, 28.2, 0), (58.43, 74.49, 0), (181.79, 91.59, 0), (101.04, 129.5, 0), (112.0, 111.18, 0), (132.14, 129.1, 0), (45.49, 138.08, 1), (82.35, 94.84, 0), (98.37, 16.03, 0), (155.57, 50.54, 0), (39.43, 96.97, 0), (61.28, 119.27, 1), (200.15, 136.87, 0), (130.83, -30.36, 0), (171.24, 76.0, 0), (129.2, 122.42, 0), (49.83, 143.97, 1), (23.82, 61.49, 0), (99.85, 164.38, 0), (64.96, 104.11, 0), (104.95, 92.67, 0), (186.35, 137.94, 0), (90.5, 107.04, 0), (33.28, 43.01, 0), (89.19, 151.07, 0), (160.88, 149.67, 0), (120.53, 235.57, 0), (84.07, 82.07, 1), (85.44, 142.97, 0), (154.04, 64.15, 0), (96.11, 173.43, 0), (-9.44, 2.68, 0), (185.76, 35.95, 0), (142.95, 117.34, 1), (138.04, 145.6, 0), (101.98, 122.51, 0), (86.05, 18.59, 0), (78.97, 29.54, 0), (56.06, 111.74, 1), (33.46, 85.75, 0), (15.58, 36.74, 0), (100.19, 54.09, 0), (57.16, 61.48, 1), (57.5, 76.89, 0), (135.79, 58.49, 0), (134.52, 93.28, 0), (134.71, 113.75, 1), (158.86, 112.17, 1), (114.41, 76.8, 0), (143.98, 69.19, 0), (56.08, 113.91, 1), (75.63, 85.55, 0), (149.72, 167.63, 1), (46.63, 127.62, 0), (-13.96, 137.46, 1), (108.74, -24.76, 0), (59.61, 150.04, 1), (57.68, 131.21, 0), (49.02, 136.84, 0), (124.13, 91.02, 0), (46.02, 153.87, 1), (112.24, 48.62, 0), (14.2, 89.26, 0), (15.68, 128.02, 0), (90.53, 129.76, 1), (137.23, 74.03, 0), (209.3, 174.04, 0), (123.24, 125.36, 0), (132.84, 70.69, 0), (73.96, 80.36, 0), (135.63, 73.96, 1), (59.43, 108.51, 1), (96.51, 36.48, 0), (130.23, 177.53, 0), (110.65, 65.35, 0)]
② 分析过程：
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.0644, OR=0.1269, z=-12.2046, p=2.94e-34；x1: β=-0.0180, OR=0.9821, z=-5.8546, p=4.78e-09；x2: β=0.0164, OR=1.0166, z=5.6882, p=1.28e-08；
McFadden 伪 R² = 1−LL/LL₀ = 0.1693；
示例预测（x1=x2=100）：P(y=1)=1/(1+e^(−-2.0644))=0.1126。
二分类 logistic 回归：logit(P(y=1)) = β₀ + Σβⱼxⱼ。
用 IRLS(牛顿法) 极大似然估计得：截距: β=-2.0644, OR=0.1269, z=-12.2046, p=2.94e-34；x1: β=-0.0180, OR=0.9821, z=-5.8546, p=4.78e-09；x2: β=0.0164, OR=1.0166, z=5.6882, p=1.28e-08；
McFadden 伪 R² = 1−LL/LL₀ = 0.1693；
示例预测（x1=x2=100）：P(y=1)=1/(1+e^(−-2.0644))=0.1126。
③ 计算结果：
截距: β=-2.0644 OR=0.127 p=2.94e-34；x1: β=-0.0180 OR=0.982 p=4.78e-09；x2: β=0.0164 OR=1.017 p=1.28e-08
④ 结论：在 α=0.05 水平下：x1显著(p=4.78e-09)、x2显著(p=1.28e-08)。价负品牌正。（伪 R²=0.1693，模型解释力有限）
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