假设检验 100 个完整案例！
假设检验 100 个完整案例！
假设检验是一种统计推断方法，用来判断样本与样本、样本与总体的差异是由抽样误差引起还是本质差别造成的，其核心是通过小概率反证法对假设进行验证 。简单来说，就是先假定“没差异”，如果数据出现的概率极低，就推翻这个假定，认为“有差异” 。常用的方法包括 t 检验、Z 检验、卡方检验等，广泛应用于科学研究、医学及社会科学领域 。
核心原理：小概率反证法
假设检验的逻辑类似于法庭审判，先假设被告无罪（原假设），再看证据是否足以证明有罪（备择假设） 。
小概率事件原理：认为概率极低（通常小于 0.05 或 0.01）的事件在一次试验中几乎不会发生 。
反证法思想：先假定原假设成立，如果样本数据导致小概率事件发生，则拒绝原假设，否则接受 。
P 值含义：表示在原假设成立的前提下，出现当前样本结果或更极端结果的概率，P 值越小，拒绝原假设的理由越充分 。
操作步骤：七步走流程
进行假设检验通常遵循标准流程，结合国内外统计教学共识，可概括为以下关键节点 ：
提出假设：设定原假设（H₀，如“无差异”）和备择假设（H₁，如“有差异”） 。
选择统计量：根据数据类型（如均值、比例）和分布特征，选择合适的检验统计量（如 Z 值、t 值、F 值） 。
确定显著性水平：设定拒绝原假设的阈值α，常用值为 0.05 或 0.01 。
计算与决策：计算统计量的观测值和 P 值，若 P≤α，则拒绝 H₀，认为差异显著；若 P>α，则不拒绝 H₀ 。
得出结论：结合业务场景解释统计结论，注意结论的概率性质而非绝对真理 。
常用方法：选对工具很重要
不同的数据特征和研究目的需要匹配不同的检验方法，选错工具可能导致结论偏差 ：
Z 检验（U 检验）：适用于大样本（n>30）且总体方差已知的情况，用于比较均值差异 。
t 检验：适用于小样本（n<30）或总体方差未知的情况，分为单样本、配对样本和双独立样本 t 检验 。
卡方检验：属于非参数检验，主要用于分析分类变量之间的关联性或拟合优度 。
F 检验：常用于方差分析（ANOVA），检验多个总体均值是否相等或方差齐性 。
分类数据与连续数据的区别 📊
首先，我们要明确你的数据是分类的还是连续的。如果是分类数据，比如性别、血型等，通常我们会用卡方检验或者逻辑回归来分析。
连续数据的正态分布问题 📈
对于连续数据，情况就稍微复杂一些。首先，你得看看你的数据是否服从正态分布。如果服从正态分布，那你就可以用以下几种方法：
均值比较：可以用z检验、t检验、方差分析或者回归分析。
方差比较：可以考虑F检验、Barlett's检验、卡方检验。
非参数检验 🌱
如果你的数据不服从正态分布，那就需要用到非参数检验。以下是几种常见的非参数检验方法：
均值比较：符号秩检验、Wilcoxon检验、Mann-Whitney检验、Kruskal-Wallis检验、Mood's中位数检验、Friedman检验。
方差比较：Levene检验。
其实，假设检验的核心就是看你数据的类型和分布情况，然后选择合适的检验方法。记住以下几点：
分类数据：卡方检验、逻辑回归。
连续数据（正态分布）：z检验、t检验、方差分析、回归分析。
连续数据（非正态分布）：非参数检验（如符号秩检验、Wilcoxon检验等）。
希望这篇文章能帮你更好地理解假设检验，如果你有任何问题，欢迎在评论区留言哦！
t 检验（单样本 / 独立样本 / 配对）· 卡方检验（拟合优度 / 独立性）· z 检验（单样本 / 双样本）。每条含：原始数据（随机样本量 300–1000）· 分析过程（公式+代入）· 计算结果 · 结论。p 值由 Python 从 t/卡方/正态分布精确计算。
说明：每个案例的原始数据样本量 n 在 300–1000 之间随机生成（固定随机种子，结果可复现）。大样本会显著提升检验功效（power），因此 p 值比小样本更极端；统计显著 ≠ 实际意义重大，请结合效应量解读。
单样本 t 检验
案例 1. 某品牌电池续航是否达标 10h
原始数据
样本 n=449：[10.03, 9.29, 9.3, 10.12, 9.68, 9.55, 9.76, 9.86, 9.75, 9.61, 9.89, 9.34, 9.82, 9.94, 9.81, 9.87, 9.78, 10.22, 9.95, 9.86, 9.43, 9.9, 9.36, 9.68, 9.57, 9.46, 9.8, 10.05, 9.52, 9.72, 9.65, 9.97, 9.85, 10.13, 9.57, 9.59, 9.65, 9.85, 9.56, 9.86, 9.56, 9.57, 9.85, 9.23, 9.59, 9.44, 10.24, 9.94, 9.03, 9.03, 9.91, 9.76, 9.87, 9.69, 9.67, 9.06, 9.58, 9.45, 9.57, 9.78, 10.33, 10.15, 9.79, 9.78, 10.36, 9.97, 10.4, 9.65, 9.49, 9.4, 9.92, 9.17, 10.02, 9.79, 9.81, 9.84, 9.55, 9.53, 9.94, 9.37, 9.65, 10.12, 9.45, 9.8, 9.69, 9.92, 9.51, 10.13, 9.96, 9.53, 10.09, 10.24, 9.36, 9.57, 9.5, 9.86, 10.23, 9.88, 9.64, 9.64, 9.56, 9.55, 9.89, 9.87, 9.76, 9.71, 9.72, 10.19, 9.96, 9.57, 9.36, 9.66, 9.69, 9.42, 9.84, 10.17, 9.47, 9.88, 9.53, 9.91, 9.99, 9.44, 9.93, 9.53, 9.89, 10.01, 9.59, 10.12, 9.79, 9.21, 9.42, 9.83, 9.49, 10.28, 9.72, 9.8, 10.14, 10.02, 9.51, 9.71, 10.0, 9.63, 9.83, 9.78, 9.72, 9.61, 9.66, 9.97, 10.31, 9.74, 9.44, 9.54, 9.92, 9.93, 9.43, 9.73, 9.49, 9.71, 9.91, 9.26, 9.79, 10.06, 9.45, 9.6, 9.55, 9.85, 9.74, 10.06, 10.09, 9.92, 9.76, 9.3, 9.6, 9.55, 9.88, 9.47, 10.06, 9.55, 9.73, 9.23, 9.25, 9.35, 9.83, 9.71, 9.51, 9.28, 9.67, 10.07, 9.81, 9.51, 9.83, 10.06, 9.66, 10.55, 9.72, 9.71, 9.71, 9.77, 9.8, 9.56, 9.97, 9.36, 9.51, 10.0, 9.15, 9.04, 9.77, 9.8, 9.56, 9.87, 9.44, 9.55, 9.36, 9.77, 9.87, 9.76, 9.9, 9.45, 10.0, 9.53, 9.63, 10.15, 9.84, 9.67, 10.29, 9.63, 9.63, 10.1, 9.77, 9.54, 9.77, 9.57, 9.57, 9.81, 9.52, 10.04, 9.94, 9.66, 9.45, 9.33, 9.5, 9.71, 9.67, 9.92, 10.03, 9.65, 9.73, 9.41, 9.89, 9.64, 8.96, 9.94, 9.66, 9.41, 9.94, 9.64, 10.05, 9.21, 9.49, 9.51, 10.22, 9.39, 9.65, 9.86, 9.66, 10.24, 10.0, 9.32, 9.92, 9.49, 9.79, 9.33, 9.94, 9.2, 9.95, 9.64, 9.68, 9.68, 9.56, 9.69, 10.16, 9.56, 9.84, 9.52, 9.71, 9.54, 9.8, 9.26, 9.63, 9.79, 9.15, 9.77, 9.19, 9.67, 9.94, 10.07, 9.51, 9.71, 9.6, 9.56, 9.76, 9.74, 9.57, 9.86, 9.64, 9.62, 9.27, 9.92, 10.09, 9.7, 9.5, 9.81, 9.54, 10.3, 10.0, 9.58, 9.95, 9.99, 10.08, 9.46, 10.03, 9.51, 9.46, 9.69, 9.7, 10.02, 10.26, 9.83, 9.84, 9.85, 9.92, 9.81, 9.74, 9.51, 9.45, 10.14, 8.9, 9.44, 9.13, 9.5, 10.11, 9.68, 9.98, 9.87, 9.58, 9.7, 9.61, 9.58, 9.88, 9.5, 9.72, 9.67, 10.12, 9.59, 9.9, 9.49, 10.19, 9.68, 9.61, 9.75, 9.82, 9.44, 9.85, 9.27, 9.52, 9.43, 9.56, 9.51, 10.01, 9.47, 9.85, 9.85, 9.81, 9.76, 9.61, 9.24, 9.13, 9.78, 9.96, 9.81, 9.33, 9.47, 9.96, 9.69, 9.97, 9.18, 9.43, 9.56, 10.33, 9.88, 10.05, 9.55, 9.62, 9.7, 9.66, 9.9, 9.53, 9.87, 9.83, 9.9, 9.89, 9.6, 9.88, 9.69, 9.76, 9.49, 9.31, 9.33, 9.63, 10.03, 9.7, 9.7, 9.43, 10.24, 9.38, 9.72, 10.03, 9.93, 9.93, 9.77, 9.83, 9.1, 9.97, 9.29, 10.1, 10.35, 9.05, 9.88, 9.75, 10.06, 9.8, 10.15, 9.31, 9.67, 9.31, 9.75, 9.78, 9.09, 9.59, 9.03, 9.17, 9.82, 9.6, 9.54, 10.09, 9.06, 9.84, 9.45, 9.96]；假设 μ₀=10
分析过程
H₀: μ=10，H₁: μ≠10。
x̄ = 9.7102，s = 0.2787，SE = s/√n = 0.2787/√449 = 0.0132；
t = (x̄−μ₀)/SE = (9.7102−10)/0.0132 = -22.0347；df = n−1 = 448；双侧 p = 1.972e-73。
x̄ = 9.7102，s = 0.2787，SE = s/√n = 0.2787/√449 = 0.0132；
t = (x̄−μ₀)/SE = (9.7102−10)/0.0132 = -22.0347；df = n−1 = 448；双侧 p = 1.972e-73。
计算结果
t = -22.0347，df = 448，p = 1.972e-73
在 α=0.05 水平下，p=1.972e-73 < 0.05，拒绝原假设 H₀。若拒绝H0，说明实测续航与标称10h有显著差异。
案例 2. 灌装机标称 500ml 是否准确
原始数据
样本 n=346：[499.88, 494.63, 501.6, 499.41, 497.93, 500.4, 501.86, 498.25, 497.51, 504.45, 499.07, 501.4, 501.77, 495.0, 499.19, 498.22, 502.94, 499.33, 502.55, 499.63, 494.91, 498.88, 500.93, 497.97, 501.44, 506.27, 495.49, 500.85, 501.56, 502.06, 499.67, 503.41, 499.41, 499.76, 495.42, 499.42, 496.45, 502.67, 504.06, 500.25, 499.69, 500.69, 501.1, 501.3, 501.15, 503.06, 500.88, 500.6, 497.6, 504.5, 492.9, 495.39, 501.06, 501.99, 499.15, 496.32, 499.96, 499.58, 499.49, 504.38, 500.36, 499.83, 497.36, 500.93, 499.71, 502.99, 499.31, 500.06, 498.75, 502.02, 501.31, 501.98, 494.71, 501.73, 504.08, 501.91, 498.44, 497.68, 495.0, 496.33, 499.36, 498.4, 500.13, 496.79, 501.48, 500.61, 499.87, 501.59, 497.48, 502.4, 498.68, 499.57, 503.93, 498.44, 505.41, 502.87, 499.46, 496.56, 501.29, 500.33, 495.41, 500.75, 497.62, 498.95, 502.42, 504.66, 499.0, 498.09, 497.54, 494.97, 504.42, 502.45, 502.01, 501.86, 500.48, 500.23, 500.18, 495.73, 501.59, 500.94, 501.68, 503.83, 506.01, 500.48, 505.35, 502.46, 501.11, 501.66, 499.21, 499.1, 501.57, 499.62, 501.48, 500.24, 500.78, 498.55, 497.45, 499.54, 504.2, 497.55, 501.22, 500.5, 500.36, 502.25, 501.36, 500.36, 501.23, 501.41, 502.9, 500.7, 500.39, 498.75, 503.48, 500.43, 499.24, 500.59, 504.28, 500.69, 494.1, 503.21, 499.99, 501.73, 501.72, 499.95, 500.04, 502.49, 496.39, 499.6, 497.05, 499.92, 501.13, 497.01, 498.71, 501.09, 500.65, 500.0, 497.75, 499.85, 499.09, 505.22, 499.03, 498.32, 503.68, 501.3, 499.71, 502.85, 498.01, 502.18, 503.48, 497.19, 498.74, 502.68, 496.41, 498.72, 496.68, 498.45, 499.6, 499.28, 500.52, 501.62, 499.29, 498.98, 502.86, 498.29, 505.27, 499.65, 499.95, 500.76, 501.09, 499.7, 498.33, 497.75, 498.82, 502.42, 505.64, 499.31, 497.19, 500.08, 499.21, 500.95, 499.92, 496.99, 493.49, 498.84, 504.25, 499.43, 504.1, 499.35, 501.88, 499.84, 495.51, 500.81, 502.41, 503.41, 496.93, 503.72, 496.17, 496.63, 502.55, 504.1, 501.81, 500.74, 502.29, 498.98, 501.78, 497.68, 498.59, 497.1, 499.97, 497.53, 496.88, 498.93, 504.04, 497.09, 496.07, 500.31, 499.87, 498.69, 497.84, 505.06, 501.62, 499.91, 501.36, 493.28, 500.27, 495.0, 499.29, 500.07, 500.96, 500.57, 500.41, 500.54, 501.38, 502.16, 500.28, 498.49, 502.07, 497.88, 497.99, 500.93, 501.53, 498.95, 501.26, 503.01, 499.65, 504.67, 498.71, 497.56, 499.66, 501.36, 497.96, 496.99, 504.09, 499.61, 499.34, 500.31, 501.43, 501.3, 507.16, 501.89, 500.88, 499.6, 497.51, 498.41, 499.86, 495.33, 500.34, 493.93, 502.98, 501.01, 500.36, 500.86, 500.54, 496.38, 496.26, 498.8, 498.02, 496.52, 499.3, 498.57, 501.34, 501.99, 502.94, 498.54, 498.39, 497.28, 501.98, 497.53, 501.8, 500.93, 501.46, 501.33, 499.8, 499.02, 496.96, 502.96, 501.66, 501.32, 499.88, 503.05, 496.47, 501.31, 497.5, 498.99, 500.95, 501.87]；假设 μ₀=500
分析过程
H₀: μ=500，H₁: μ≠500。
x̄ = 500.0666，s = 2.4725，SE = s/√n = 2.4725/√346 = 0.1329；
t = (x̄−μ₀)/SE = (500.0666−500)/0.1329 = 0.5014；df = n−1 = 345；双侧 p = 0.6164。
x̄ = 500.0666，s = 2.4725，SE = s/√n = 2.4725/√346 = 0.1329；
t = (x̄−μ₀)/SE = (500.0666−500)/0.1329 = 0.5014；df = n−1 = 345；双侧 p = 0.6164。
计算结果
t = 0.5014，df = 345，p = 0.6164
在 α=0.05 水平下，p=0.6164 ≥ 0.05，不拒绝原假设 H₀。判断灌装量是否系统性偏离标称值。
案例 3. 学生平均身高是否 165cm
原始数据
样本 n=912：[170.86, 160.11, 163.25, 170.69, 159.42, 167.66, 163.41, 167.58, 174.41, 160.84, 155.45, 169.35, 167.34, 168.14, 165.97, 166.91, 167.0, 154.79, 173.13, 163.49, 164.0, 161.83, 163.59, 163.66, 161.29, 162.23, 173.08, 166.23, 159.83, 161.81, 160.3, 169.4, 170.76, 166.7, 167.83, 161.27, 163.75, 168.84, 167.53, 167.23, 160.05, 172.3, 163.01, 167.99, 168.07, 167.3, 163.48, 162.49, 163.62, 166.42, 166.68, 164.64, 168.55, 165.66, 162.1, 167.55, 164.94, 164.93, 165.74, 163.46, 166.89, 170.83, 164.82, 161.9, 165.81, 165.1, 169.09, 158.67, 171.76, 169.04, 170.62, 171.47, 162.08, 163.23, 159.64, 167.53, 167.64, 165.31, 161.62, 167.4, 166.18, 166.9, 160.21, 155.62, 164.9, 164.56, 161.12, 163.77, 160.58, 164.68, 168.97, 166.86, 166.09, 167.0, 167.58, 174.3, 164.1, 166.43, 160.58, 166.2, 161.79, 167.31, 161.69, 163.12, 167.73, 161.22, 165.1, 163.48, 160.49, 167.81, 176.6, 167.04, 157.07, 173.24, 161.63, 167.45, 169.03, 157.13, 163.12, 160.65, 173.51, 160.71, 171.97, 163.98, 161.81, 162.66, 170.17, 166.28, 170.1, 164.67, 172.93, 163.13, 165.11, 165.41, 166.27, 163.25, 160.24, 170.85, 171.84, 160.44, 163.68, 162.98, 167.46, 162.43, 168.75, 169.51, 164.51, 168.37, 161.01, 168.48, 157.35, 164.62, 162.36, 166.31, 161.12, 163.89, 161.46, 159.31, 175.33, 163.04, 161.81, 158.9, 171.91, 166.71, 169.66, 164.83, 165.34, 167.71, 163.01, 164.26, 166.4, 166.12, 164.46, 164.49, 162.22, 173.93, 159.89, 165.19, 159.55, 166.38, 162.25, 170.44, 162.86, 164.49, 167.93, 168.25, 171.95, 160.46, 172.93, 166.86, 163.02, 160.15, 166.9, 169.55, 159.28, 168.63, 163.37, 168.89, 156.45, 162.13, 165.97, 169.01, 170.56, 163.73, 167.88, 163.62, 164.59, 170.28, 164.98, 166.32, 164.11, 163.99, 162.04, 167.5, 168.69, 165.06, 167.26, 161.55, 164.95, 168.29, 160.81, 164.72, 162.88, 166.41, 163.05, 167.26, 160.13, 165.3, 164.68, 170.51, 167.82, 165.48, 163.19, 164.04, 164.38, 162.72, 156.26, 172.04, 160.86, 168.56, 168.72, 172.59, 168.39, 164.55, 163.87, 167.0, 166.68, 164.77, 165.1, 155.49, 166.31, 167.59, 163.5, 160.99, 167.72, 169.38, 154.54, 159.78, 169.19, 164.87, 161.73, 165.29, 168.36, 162.11, 170.76, 168.01, 166.41, 160.86, 169.4, 162.79, 164.32, 164.92, 167.71, 163.39, 155.32, 162.12, 173.47, 163.51, 159.96, 163.54, 160.45, 163.95, 164.18, 154.46, 164.91, 161.9, 159.24, 156.1, 165.26, 171.97, 158.65, 162.35, 161.84, 164.78, 171.86, 163.51, 166.51, 158.02, 162.77, 165.56, 161.59, 160.06, 165.4, 160.96, 168.07, 173.06, 172.0, 165.98, 167.53, 168.66, 162.14, 167.07, 171.37, 162.88, 163.88, 161.64, 158.57, 166.22, 159.35, 172.01, 158.92, 169.54, 169.35, 164.67, 170.06, 166.56, 165.11, 167.33, 160.71, 172.09, 162.97, 164.56, 166.79, 172.78, 168.76, 161.6, 165.92, 161.26, 162.12, 172.57, 164.02, 164.27, 164.81, 172.4, 162.17, 165.3, 161.18, 161.29, 163.36, 161.81, 168.9, 164.29, 163.94, 165.64, 175.27, 163.4, 159.97, 155.24, 162.87, 162.45, 160.34, 168.93, 157.75, 162.49, 165.8, 165.03, 156.43, 162.29, 158.8, 167.48, 161.0, 156.51, 164.22, 168.66, 170.3, 170.6, 164.74, 170.63, 160.96, 165.29, 165.91, 155.68, 169.24, 170.02, 175.77, 158.87, 170.85, 161.86, 158.29, 163.98, 168.96, 168.3, 165.85, 169.88, 169.11, 160.23, 163.94, 165.05, 167.99, 163.1, 171.14, 169.78, 166.42, 167.53, 161.1, 168.56, 167.63, 166.15, 171.16, 159.82, 162.39, 162.71, 164.88, 166.53, 166.05, 160.24, 164.87, 162.61, 160.29, 161.0, 168.65, 170.14, 164.78, 164.66, 162.79, 157.96, 165.01, 167.68, 164.02, 172.87, 160.39, 163.26, 168.85, 160.38, 161.81, 153.08, 168.69, 167.68, 169.75, 170.47, 165.4, 167.99, 161.94, 173.23, 172.53, 169.48, 155.17, 163.11, 165.3, 161.79, 164.17, 164.23, 160.96, 169.81, 152.35, 166.49, 163.84, 165.35, 165.59, 158.13, 173.13, 163.95, 163.68, 166.86, 171.23, 163.0, 162.07, 165.96, 158.82, 165.97, 162.43, 165.13, 170.06, 164.81, 162.11, 161.77, 159.22, 158.11, 159.47, 163.71, 159.51, 164.13, 167.18, 164.27, 165.34, 163.09, 162.59, 160.36, 163.5, 169.02, 171.05, 162.63, 161.38, 160.78, 173.48, 160.38, 160.63, 159.56, 161.36, 169.44, 162.34, 156.85, 168.28, 162.0, 166.6, 158.66, 169.74, 164.62, 166.46, 155.67, 167.05, 167.05, 164.72, 161.35, 163.31, 172.94, 165.2, 170.29, 165.02, 163.7, 167.15, 175.84, 164.3, 175.15, 165.01, 164.88, 168.59, 170.04, 162.04, 166.54, 161.78, 163.42, 174.07, 155.66, 165.63, 161.51, 172.63, 163.09, 170.88, 169.78, 157.86, 164.34, 162.73, 167.66, 164.11, 165.42, 162.63, 163.52, 169.11, 172.59, 171.23, 166.37, 162.99, 167.6, 171.13, 165.64, 167.51, 168.87, 159.85, 162.2, 172.24, 165.71, 169.83, 165.82, 163.71, 167.15, 154.1, 167.99, 161.93, 170.25, 159.25, 163.28, 166.12, 163.06, 168.45, 168.05, 167.81, 165.7, 165.04, 154.52, 157.2, 163.53, 164.14, 159.5, 163.16, 169.55, 163.66, 165.61, 168.38, 165.6, 164.83, 159.33, 159.69, 158.61, 164.79, 163.72, 174.33, 168.1, 167.8, 159.77, 161.78, 166.07, 166.49, 166.23, 162.61, 161.37, 167.34, 161.89, 167.42, 162.91, 167.67, 167.53, 164.3, 162.78, 167.23, 161.09, 159.23, 168.95, 165.72, 161.72, 150.62, 161.9, 157.14, 159.24, 162.12, 173.27, 166.21, 166.13, 171.16, 168.3, 163.48, 164.49, 162.1, 155.55, 165.88, 169.16, 167.3, 163.45, 156.88, 168.38, 164.19, 168.84, 171.92, 173.98, 170.5, 170.05, 167.77, 162.15, 153.86, 162.26, 161.78, 162.77, 163.54, 165.37, 173.64, 169.53, 168.62, 170.39, 168.02, 168.0, 168.06, 165.79, 155.01, 167.33, 166.95, 169.55, 160.54, 155.0, 159.75, 169.72, 165.47, 166.54, 165.11, 160.02, 162.08, 166.1, 165.08, 160.5, 163.29, 164.52, 169.62, 164.28, 167.05, 163.25, 168.15, 170.77, 163.29, 166.28, 171.16, 159.42, 166.64, 166.79, 165.4, 162.34, 170.07, 173.86, 172.6, 163.37, 160.48, 168.23, 176.67, 162.22, 164.68, 161.45, 165.98, 160.45, 161.35, 166.11, 168.81, 162.26, 168.81, 171.04, 165.9, 168.24, 174.16, 163.43, 165.18, 174.24, 171.79, 169.55, 158.5, 171.12, 161.16, 162.2, 170.47, 165.36, 167.97, 167.26, 162.96, 171.8, 166.14, 164.19, 167.26, 165.51, 167.22, 168.41, 165.66, 162.16, 163.41, 168.46, 163.41, 162.9, 164.72, 163.14, 162.86, 167.46, 166.28, 166.64, 173.31, 163.03, 163.96, 163.49, 167.27, 160.05, 156.92, 159.67, 165.37, 166.47, 169.32, 170.06, 160.61, 164.57, 165.08, 158.11, 173.91, 169.2, 167.78, 167.61, 161.26, 162.22, 153.77, 165.52, 169.64, 162.33, 163.54, 164.44, 165.74, 158.75, 164.57, 160.86, 165.75, 157.96, 167.09, 171.07, 165.26, 169.09, 162.63, 165.9, 160.68, 168.43, 165.26, 159.78, 159.38, 171.72, 161.79, 163.63, 164.54, 166.25, 162.21, 161.23, 166.66, 164.47, 166.47, 163.78, 168.9, 170.55, 167.46, 161.57, 160.37, 166.59, 168.95, 163.05, 161.57, 167.09, 165.01, 164.04, 162.51, 163.8, 164.23, 169.47, 158.02, 169.02, 165.51, 167.65, 160.85, 164.03, 164.09, 166.37, 165.2, 169.29, 162.9, 159.92, 166.15, 160.96, 171.24, 156.72, 165.2, 167.87, 169.3, 168.45, 162.35, 166.58, 167.18, 166.49, 165.2, 164.41, 173.28, 162.8, 160.42, 163.52, 164.16, 164.53, 174.73, 169.99, 164.37, 166.34, 163.52, 166.21, 162.78, 165.98, 162.17, 165.14, 164.07, 172.1, 162.91, 168.88, 153.57, 165.13, 166.1, 164.11, 159.43, 165.03, 164.35, 163.13, 160.64, 159.34, 156.48, 165.23, 171.95, 163.94, 170.87, 166.4, 161.89, 165.37, 169.4, 165.01, 157.9, 172.02, 168.73, 162.32, 163.15, 166.1, 160.85, 167.33, 163.62, 166.67, 167.97, 168.2, 158.96, 164.96, 164.9, 169.21, 158.79, 159.11, 172.1, 167.32, 165.89, 162.67, 165.88, 164.92, 157.92, 165.02]；假设 μ₀=165
分析过程
H₀: μ=165，H₁: μ≠165。
x̄ = 165.0369，s = 4.2411，SE = s/√n = 4.2411/√912 = 0.1404；
t = (x̄−μ₀)/SE = (165.0369−165)/0.1404 = 0.2625；df = n−1 = 911；双侧 p = 0.7930。
x̄ = 165.0369，s = 4.2411，SE = s/√n = 4.2411/√912 = 0.1404；
t = (x̄−μ₀)/SE = (165.0369−165)/0.1404 = 0.2625；df = n−1 = 911；双侧 p = 0.7930。
计算结果
t = 0.2625，df = 911，p = 0.7930
在 α=0.05 水平下，p=0.7930 ≥ 0.05，不拒绝原假设 H₀。检验抽样均值与参考身高的差异。
案例 4. 网页平均加载是否 2s
原始数据
样本 n=809：[2.18, 1.73, 1.94, 1.9, 1.93, 1.91, 2.17, 2.24, 2.07, 2.0, 2.43, 1.9, 2.05, 2.24, 1.85, 1.57, 1.74, 2.29, 1.88, 1.97, 2.11, 2.07, 2.19, 2.0, 2.15, 1.84, 1.98, 1.91, 2.17, 1.39, 1.99, 2.24, 1.72, 1.92, 2.09, 1.92, 1.81, 2.22, 2.52, 1.92, 1.88, 2.03, 1.72, 1.73, 1.62, 2.21, 2.0, 2.44, 1.75, 2.36, 2.1, 2.2, 1.75, 2.02, 2.09, 2.17, 2.12, 1.84, 1.87, 2.11, 2.25, 2.17, 2.12, 1.68, 2.03, 2.25, 1.98, 1.96, 2.01, 2.04, 1.93, 2.44, 2.27, 2.19, 1.82, 2.06, 1.81, 2.27, 2.06, 1.75, 2.3, 2.1, 2.5, 2.06, 1.95, 2.08, 2.07, 1.53, 2.15, 1.85, 1.75, 2.19, 1.96, 2.07, 1.78, 2.03, 2.37, 2.17, 2.22, 1.71, 1.39, 2.13, 1.96, 2.06, 1.84, 2.12, 1.9, 2.01, 1.89, 2.09, 2.13, 2.02, 2.02, 1.56, 2.14, 2.04, 1.8, 2.22, 2.24, 2.14, 1.98, 2.05, 2.15, 1.92, 2.22, 1.8, 2.0, 1.88, 1.97, 2.0, 1.92, 1.83, 2.05, 1.88, 1.87, 1.9, 2.21, 2.2, 1.72, 2.04, 1.96, 1.91, 2.1, 1.83, 2.01, 1.86, 1.91, 2.04, 1.79, 2.16, 1.97, 1.94, 2.03, 1.98, 1.91, 2.2, 1.65, 1.88, 2.08, 1.86, 2.02, 2.15, 1.66, 2.23, 2.34, 2.11, 1.65, 2.16, 2.15, 1.75, 1.92, 1.9, 1.94, 1.81, 1.96, 2.0, 2.18, 2.3, 2.36, 1.94, 2.09, 1.63, 1.87, 2.15, 2.06, 2.05, 2.21, 1.94, 2.17, 2.23, 2.34, 1.92, 1.93, 2.14, 2.18, 1.67, 2.01, 2.18, 2.07, 2.03, 1.57, 1.89, 2.19, 1.74, 1.71, 1.85, 1.9, 1.99, 1.8, 2.24, 2.26, 1.96, 1.97, 2.1, 1.76, 1.74, 2.03, 1.92, 1.82, 2.44, 2.05, 1.87, 1.76, 1.92, 1.91, 2.06, 2.06, 1.73, 2.11, 2.23, 1.86, 1.79, 2.07, 1.68, 1.98, 2.03, 1.81, 2.3, 1.71, 2.51, 2.14, 2.35, 1.91, 2.03, 2.08, 2.18, 1.98, 2.07, 2.22, 2.03, 2.03, 2.08, 1.82, 2.19, 2.11, 1.65, 1.63, 2.23, 2.2, 2.15, 2.16, 2.12, 2.04, 2.35, 2.2, 2.33, 2.0, 2.11, 2.12, 2.2, 2.09, 2.06, 1.93, 1.95, 2.08, 1.84, 2.3, 2.05, 2.16, 1.84, 1.97, 1.87, 2.08, 2.14, 1.85, 1.9, 2.18, 1.86, 2.08, 1.71, 2.07, 2.17, 2.11, 1.69, 1.63, 1.99, 1.69, 2.12, 1.68, 1.98, 1.91, 1.88, 1.88, 1.95, 2.29, 1.85, 2.19, 1.67, 1.8, 1.84, 2.27, 1.71, 1.99, 2.11, 1.93, 2.22, 1.76, 2.04, 1.82, 2.16, 2.05, 2.31, 2.22, 2.04, 2.2, 2.09, 2.22, 2.13, 1.98, 2.18, 2.24, 1.89, 1.92, 2.0, 2.08, 1.85, 1.97, 2.4, 1.93, 2.06, 2.25, 2.18, 1.6, 2.06, 2.08, 1.62, 1.86, 1.81, 1.87, 2.16, 1.93, 2.04, 2.11, 2.27, 2.32, 1.86, 2.15, 2.12, 2.14, 2.04, 1.81, 1.75, 1.81, 2.04, 1.97, 1.92, 1.99, 1.9, 1.81, 2.12, 1.95, 2.08, 2.15, 2.35, 2.06, 1.94, 1.81, 2.06, 2.23, 2.27, 1.8, 1.73, 2.35, 2.07, 2.21, 2.09, 2.04, 2.2, 1.79, 1.98, 2.13, 1.78, 1.85, 2.05, 1.94, 1.52, 2.06, 1.87, 2.17, 2.25, 1.89, 2.0, 2.16, 1.95, 2.22, 1.75, 2.12, 2.09, 2.23, 1.89, 2.11, 1.93, 2.14, 2.37, 2.14, 1.87, 1.93, 2.09, 2.0, 2.24, 1.96, 1.75, 2.01, 1.91, 2.24, 2.16, 1.78, 1.74, 1.95, 1.98, 1.68, 1.9, 2.08, 1.87, 2.16, 1.81, 1.87, 1.67, 2.17, 2.21, 1.73, 1.98, 2.1, 2.19, 2.15, 1.93, 1.98, 1.93, 1.98, 1.9, 2.11, 1.67, 2.32, 2.59, 1.92, 2.18, 2.11, 1.99, 2.12, 2.24, 2.19, 2.33, 2.01, 2.0, 2.06, 2.01, 1.88, 2.04, 2.15, 1.97, 2.05, 1.85, 2.02, 1.92, 2.04, 2.1, 2.08, 2.12, 1.82, 1.72, 2.35, 1.95, 2.0, 2.33, 1.76, 1.85, 1.86, 2.07, 2.3, 1.9, 1.83, 2.11, 1.58, 2.21, 1.66, 1.94, 1.94, 2.05, 2.28, 2.04, 2.13, 1.82, 2.06, 2.22, 2.29, 1.82, 1.88, 2.07, 2.13, 1.84, 1.78, 1.8, 1.69, 2.12, 2.08, 1.89, 1.81, 1.96, 2.0, 2.04, 1.87, 1.9, 1.95, 1.8, 2.11, 2.21, 1.53, 1.48, 1.9, 2.03, 2.22, 2.15, 2.22, 1.94, 2.14, 1.63, 1.7, 1.91, 1.98, 2.05, 1.59, 2.1, 1.84, 1.91, 1.98, 2.07, 1.89, 1.98, 1.95, 2.1, 2.22, 1.98, 1.81, 1.9, 2.3, 1.93, 1.72, 1.93, 1.96, 2.05, 1.99, 2.04, 1.9, 2.13, 2.38, 1.86, 1.95, 1.92, 2.05, 1.9, 1.95, 1.81, 2.0, 2.06, 1.95, 2.29, 2.18, 1.96, 2.21, 2.08, 1.68, 2.14, 1.76, 1.82, 2.04, 1.8, 1.73, 1.95, 1.99, 1.87, 1.63, 1.71, 1.62, 1.92, 1.94, 1.87, 1.57, 2.04, 2.15, 1.8, 2.12, 1.67, 2.1, 2.05, 2.09, 1.72, 2.02, 2.01, 1.98, 2.0, 1.8, 1.99, 1.76, 1.91, 1.99, 2.1, 1.52, 1.78, 1.82, 2.02, 2.07, 2.08, 1.99, 1.84, 2.0, 2.41, 2.22, 2.36, 1.9, 2.06, 2.26, 2.17, 1.89, 2.0, 2.18, 2.09, 1.86, 1.95, 2.13, 1.72, 1.78, 1.91, 2.14, 2.14, 1.79, 1.71, 2.0, 2.13, 1.91, 1.97, 2.14, 1.8, 2.02, 2.52, 2.08, 1.96, 1.93, 1.98, 2.48, 2.05, 1.9, 2.03, 1.78, 1.97, 1.95, 1.67, 1.95, 2.17, 1.95, 2.07, 1.86, 2.21, 1.96, 1.57, 2.02, 2.2, 1.97, 1.56, 2.08, 2.13, 2.32, 2.06, 1.97, 2.25, 2.06, 2.07, 1.83, 1.89, 2.22, 1.88, 1.86, 1.97, 1.75, 2.13, 1.96, 1.71, 2.0, 1.88, 2.09, 1.92, 2.07, 1.91, 1.89, 2.05, 2.38, 1.88, 2.06, 1.92, 2.05, 1.88, 2.27, 2.01, 1.76, 1.67, 1.86, 1.84, 1.86, 2.04, 2.43, 1.85, 2.17, 2.25, 1.91, 2.19, 1.9, 2.22, 2.1, 2.0, 2.29, 1.93, 1.91, 1.98, 2.11, 1.93, 1.7, 1.97, 2.15, 2.09, 1.91, 1.91, 2.15, 2.16, 2.1, 1.99, 2.26, 2.6, 1.87, 2.11, 2.14, 2.3, 1.68, 1.94, 1.7, 2.35, 2.44, 2.05, 1.92, 1.88, 2.25, 2.11, 2.06, 1.86, 1.73, 2.42, 1.73, 1.88, 2.11, 2.01, 2.17, 2.1, 2.11, 1.9, 2.03, 1.82, 2.16, 1.74, 1.99, 2.2, 1.87, 2.11, 2.04, 2.27, 1.99, 1.89, 2.03, 2.34, 1.99, 1.93, 2.01, 2.11, 1.97, 2.14, 2.04, 1.82, 2.24, 2.05, 1.88, 1.99, 2.04, 1.68, 2.02, 2.41, 1.77, 1.87, 1.98, 1.83, 1.91, 2.19, 1.89]；假设 μ₀=2
分析过程
H₀: μ=2，H₁: μ≠2。
x̄ = 2.0018，s = 0.1894，SE = s/√n = 0.1894/√809 = 0.0067；
t = (x̄−μ₀)/SE = (2.0018−2)/0.0067 = 0.2637；df = n−1 = 808；双侧 p = 0.7921。
x̄ = 2.0018，s = 0.1894，SE = s/√n = 0.1894/√809 = 0.0067；
t = (x̄−μ₀)/SE = (2.0018−2)/0.0067 = 0.2637；df = n−1 = 808；双侧 p = 0.7921。
计算结果
t = 0.2637，df = 808，p = 0.7921
在 α=0.05 水平下，p=0.7921 ≥ 0.05，不拒绝原假设 H₀。评估页面加载性能是否达标。
案例 5. 客服日均处理工单是否 30 件
原始数据
样本 n=772：[29.47, 30.11, 30.82, 27.53, 31.26, 27.78, 32.54, 25.93, 31.9, 32.96, 32.31, 28.66, 33.0, 30.63, 30.84, 31.11, 28.14, 29.42, 26.9, 30.02, 29.82, 28.69, 31.72, 30.35, 29.87, 29.18, 33.63, 29.91, 30.94, 29.57, 29.06, 29.76, 34.7, 27.27, 25.79, 29.61, 29.43, 27.97, 32.02, 33.19, 27.85, 27.64, 27.52, 31.03, 29.23, 29.79, 27.94, 29.81, 27.98, 29.88, 29.37, 29.01, 28.56, 29.04, 31.46, 29.04, 32.14, 31.31, 30.16, 31.47, 30.68, 28.41, 29.83, 26.05, 28.62, 28.02, 28.18, 30.6, 28.36, 30.29, 33.39, 28.66, 31.27, 29.49, 29.62, 28.39, 28.41, 32.82, 30.12, 30.7, 26.24, 31.2, 30.24, 28.09, 32.73, 29.64, 31.84, 30.83, 27.5, 30.84, 28.88, 30.36, 27.24, 28.58, 25.35, 32.81, 30.2, 29.2, 27.01, 28.25, 31.33, 30.33, 31.97, 28.63, 30.14, 32.25, 30.21, 30.82, 30.57, 33.78, 31.29, 26.74, 26.84, 28.34, 30.43, 29.73, 26.5, 28.02, 29.79, 32.84, 28.5, 30.8, 29.99, 31.15, 27.8, 30.05, 33.53, 29.4, 29.46, 30.76, 29.33, 29.2, 31.68, 29.7, 28.66, 30.56, 31.7, 29.34, 32.33, 33.99, 29.6, 29.69, 30.93, 31.74, 34.02, 29.01, 29.67, 29.65, 30.41, 32.47, 29.92, 31.77, 28.75, 30.28, 28.7, 26.45, 31.07, 30.57, 32.29, 31.6, 28.85, 33.44, 29.91, 27.59, 30.66, 29.83, 29.32, 30.75, 29.45, 28.43, 31.67, 29.42, 28.02, 28.34, 31.51, 27.98, 28.91, 33.39, 28.6, 32.09, 29.8, 28.53, 30.95, 29.8, 30.31, 30.07, 30.99, 32.84, 29.7, 28.26, 30.96, 31.43, 27.86, 31.48, 30.28, 29.5, 32.73, 31.73, 28.69, 31.36, 28.64, 30.9, 31.7, 28.23, 31.76, 31.21, 31.32, 29.42, 28.36, 30.82, 31.06, 28.32, 29.23, 32.02, 33.36, 31.17, 30.19, 28.26, 29.78, 29.33, 32.45, 30.46, 30.76, 27.72, 33.26, 29.62, 31.8, 26.86, 31.8, 28.78, 33.18, 31.57, 32.07, 31.33, 31.12, 26.04, 27.85, 29.33, 30.44, 31.26, 29.13, 26.08, 31.71, 28.44, 27.64, 27.71, 28.51, 30.69, 28.56, 26.12, 31.12, 32.56, 31.59, 28.56, 30.42, 32.11, 28.05, 28.87, 30.25, 28.26, 34.46, 30.8, 28.66, 32.52, 31.19, 29.06, 28.65, 30.58, 27.29, 32.02, 28.74, 31.97, 31.96, 30.54, 28.8, 28.16, 31.67, 29.8, 30.86, 29.56, 28.94, 33.57, 28.21, 28.69, 29.41, 31.99, 28.34, 27.93, 30.58, 31.53, 27.3, 30.91, 30.26, 32.18, 27.09, 28.94, 33.15, 29.38, 32.4, 28.07, 31.76, 31.62, 33.23, 26.76, 34.24, 29.96, 28.67, 27.48, 29.37, 33.08, 32.5, 29.42, 33.6, 30.86, 30.24, 30.21, 29.72, 27.88, 29.77, 25.12, 29.18, 30.45, 31.49, 30.69, 32.43, 29.45, 33.44, 31.54, 29.91, 28.42, 31.74, 26.69, 31.61, 31.78, 30.32, 28.13, 29.04, 28.43, 28.42, 29.49, 31.29, 30.3, 31.76, 29.14, 28.28, 30.2, 30.94, 27.36, 30.7, 30.86, 28.86, 30.6, 32.68, 30.28, 32.11, 31.22, 27.08, 29.95, 28.56, 32.35, 32.55, 30.35, 29.3, 32.12, 29.51, 30.16, 31.29, 31.87, 29.96, 30.79, 31.4, 29.79, 29.55, 32.7, 32.53, 27.0, 28.61, 31.13, 30.76, 27.88, 31.6, 29.15, 29.27, 26.98, 30.02, 30.99, 33.02, 30.41, 28.45, 33.83, 31.46, 30.07, 30.51, 31.73, 33.46, 29.48, 29.3, 30.44, 30.48, 29.36, 31.26, 31.16, 27.83, 27.56, 28.13, 30.94, 28.08, 29.46, 26.81, 27.13, 29.51, 30.9, 32.32, 30.25, 29.6, 27.31, 25.93, 27.42, 29.39, 28.51, 27.8, 30.72, 28.24, 33.56, 26.91, 29.05, 29.82, 33.64, 28.72, 30.42, 30.82, 28.7, 31.73, 26.34, 27.39, 30.72, 32.47, 28.88, 31.72, 28.39, 31.65, 29.32, 28.32, 26.42, 27.94, 27.5, 32.7, 30.08, 26.04, 30.8, 31.92, 29.25, 25.74, 33.42, 30.68, 28.82, 32.69, 29.49, 28.61, 31.28, 31.55, 26.06, 30.88, 29.59, 28.42, 28.73, 30.55, 32.78, 29.22, 27.57, 28.16, 29.75, 29.79, 29.74, 27.78, 32.46, 31.28, 29.14, 26.74, 31.64, 28.93, 29.78, 30.81, 31.82, 29.3, 29.71, 29.56, 30.07, 32.54, 27.14, 28.45, 29.32, 31.3, 28.24, 31.93, 29.59, 32.14, 30.4, 29.96, 25.01, 29.67, 29.43, 30.02, 32.32, 28.15, 25.86, 29.86, 26.42, 29.27, 31.84, 27.1, 29.55, 29.61, 29.91, 31.81, 29.56, 27.1, 31.41, 29.29, 29.09, 30.33, 28.49, 29.74, 31.0, 30.99, 30.35, 31.67, 31.85, 30.9, 30.06, 31.57, 31.44, 31.77, 30.97, 34.25, 26.2, 30.6, 32.42, 31.41, 30.79, 26.69, 32.51, 31.1, 31.21, 26.14, 30.43, 28.73, 28.59, 31.69, 34.48, 30.44, 29.87, 31.27, 31.91, 31.93, 26.78, 32.34, 30.85, 28.47, 29.38, 31.85, 29.17, 28.51, 31.62, 26.44, 29.15, 26.14, 32.83, 29.4, 31.52, 27.88, 30.83, 30.18, 32.63, 30.4, 30.33, 30.05, 30.5, 27.39, 29.77, 28.35, 25.2, 29.46, 30.91, 31.67, 28.27, 31.09, 31.35, 31.1, 27.99, 29.38, 31.13, 33.31, 31.32, 31.95, 29.66, 30.12, 27.72, 29.64, 31.19, 30.29, 30.47, 27.98, 30.85, 30.03, 27.73, 29.94, 30.52, 30.93, 29.24, 31.73, 27.55, 32.23, 25.18, 28.55, 31.97, 28.67, 31.29, 29.82, 30.78, 28.89, 30.07, 28.52, 31.45, 29.29, 31.2, 28.05, 29.02, 30.0, 35.86, 27.95, 31.4, 30.3, 27.65, 28.05, 30.33, 30.49, 30.09, 32.83, 29.65, 30.83, 27.48, 32.0, 29.6, 30.67, 31.38, 31.04, 29.9, 31.25, 28.81, 28.91, 28.34, 30.7, 33.23, 27.5, 30.02, 31.17, 29.04, 31.54, 29.08, 27.72, 30.74, 29.37, 31.57, 26.87, 30.04, 33.0, 30.27, 29.14, 28.87, 31.18, 30.17, 28.22, 29.56, 25.98, 30.43, 24.9, 29.0, 28.17, 30.56, 31.43, 31.13, 29.97, 29.98, 30.19, 29.12, 28.77, 31.36, 30.63, 27.65, 30.73, 30.42, 33.08, 30.64, 27.1, 32.0, 28.41, 32.31, 27.31, 25.94, 30.44, 32.17, 28.38, 31.99, 29.44, 29.95, 35.12, 29.72, 32.12, 31.31, 30.38, 28.4, 30.52, 28.14, 28.28, 31.57, 30.16, 30.57, 32.55, 31.59, 27.06, 31.89, 29.15, 30.6, 29.81, 31.9, 27.17, 28.18, 28.64, 30.59, 29.57, 26.68, 29.58, 31.36, 28.09, 25.89, 28.76, 31.53, 29.81, 26.91, 31.95, 33.48, 28.45, 30.31, 30.94, 31.85, 28.11, 26.98, 30.39, 33.4, 30.26, 29.94, 30.12, 30.13, 33.97, 31.15, 30.73, 29.88, 32.01, 33.56, 31.59, 29.94, 29.05, 31.73, 26.75, 27.62, 31.24, 29.82, 27.16, 28.82, 26.57, 28.96]；假设 μ₀=30
分析过程
H₀: μ=30，H₁: μ≠30。
x̄ = 29.9756，s = 1.8559，SE = s/√n = 1.8559/√772 = 0.0668；
t = (x̄−μ₀)/SE = (29.9756−30)/0.0668 = -0.3658；df = n−1 = 771；双侧 p = 0.7146。
x̄ = 29.9756，s = 1.8559，SE = s/√n = 1.8559/√772 = 0.0668；
t = (x̄−μ₀)/SE = (29.9756−30)/0.0668 = -0.3658；df = n−1 = 771；双侧 p = 0.7146。
计算结果
t = -0.3658，df = 771，p = 0.7146
在 α=0.05 水平下，p=0.7146 ≥ 0.05，不拒绝原假设 H₀。衡量产能是否达到标准。
案例 6. 肥料使产量是否 400kg
原始数据
样本 n=570：[391.02, 401.78, 391.49, 399.01, 407.15, 396.54, 404.81, 411.87, 389.85, 402.2, 412.4, 396.72, 403.07, 391.59, 391.04, 394.95, 393.95, 402.98, 407.68, 398.5, 396.79, 405.17, 404.49, 393.76, 396.48, 394.59, 402.49, 393.97, 392.74, 410.45, 398.79, 403.16, 400.68, 398.03, 405.22, 393.84, 401.31, 405.14, 399.24, 399.66, 406.49, 406.88, 395.53, 399.17, 403.13, 398.66, 396.68, 398.03, 391.79, 407.77, 392.09, 394.46, 401.77, 395.18, 400.74, 398.97, 389.0, 399.43, 391.94, 398.89, 401.36, 399.6, 410.46, 390.35, 410.08, 393.27, 402.12, 408.33, 385.33, 393.43, 409.1, 390.33, 400.03, 381.54, 406.32, 402.89, 398.89, 389.46, 403.65, 404.2, 390.05, 400.78, 405.18, 405.63, 398.34, 399.69, 395.94, 404.67, 400.35, 394.71, 391.04, 399.69, 406.65, 391.04, 411.98, 397.61, 405.64, 396.38, 391.5, 395.47, 403.71, 393.75, 403.51, 407.66, 395.0, 387.38, 398.96, 397.71, 404.26, 391.47, 399.76, 412.21, 398.22, 405.14, 396.0, 402.31, 398.76, 395.36, 397.83, 398.73, 394.76, 411.55, 416.04, 406.04, 391.01, 392.45, 400.38, 396.58, 398.07, 405.75, 405.12, 398.69, 397.84, 401.74, 408.65, 400.68, 408.45, 405.42, 399.97, 401.94, 402.45, 399.51, 399.25, 403.76, 393.54, 413.51, 404.3, 387.42, 393.04, 404.61, 412.6, 391.7, 399.71, 399.02, 402.92, 402.01, 394.93, 403.58, 407.46, 406.67, 392.06, 395.4, 407.63, 401.93, 408.36, 402.09, 383.16, 396.32, 402.21, 406.05, 414.48, 387.04, 399.23, 401.47, 391.33, 403.17, 395.81, 387.71, 413.92, 383.15, 401.33, 394.85, 403.8, 396.34, 400.73, 391.18, 401.17, 405.02, 411.99, 409.88, 390.27, 393.58, 398.76, 406.46, 407.74, 399.29, 389.83, 396.61, 401.61, 401.77, 397.28, 406.65, 391.52, 386.17, 398.77, 397.87, 400.67, 395.89, 401.2, 392.08, 392.71, 392.68, 403.29, 401.51, 400.64, 399.37, 396.54, 412.48, 389.17, 402.45, 405.27, 393.48, 386.54, 396.38, 404.37, 397.53, 401.27, 400.63, 394.95, 405.48, 401.46, 394.16, 415.82, 402.7, 410.58, 390.93, 410.09, 408.15, 396.47, 403.61, 396.79, 384.86, 405.31, 412.02, 402.24, 401.26, 409.78, 401.5, 399.16, 407.71, 404.22, 396.31, 405.1, 392.52, 386.94, 402.11, 407.16, 398.67, 387.63, 402.2, 405.86, 399.94, 394.62, 394.67, 396.06, 394.44, 406.74, 396.58, 394.59, 399.56, 394.03, 404.98, 413.6, 402.74, 399.18, 404.32, 403.44, 407.86, 404.99, 389.33, 412.78, 397.71, 395.74, 396.92, 399.81, 407.46, 398.16, 389.71, 396.42, 397.59, 402.11, 400.1, 402.04, 395.02, 406.3, 406.54, 392.33, 409.83, 402.64, 409.71, 393.96, 415.97, 396.72, 395.74, 401.01, 409.86, 392.7, 406.22, 400.19, 401.33, 397.02, 396.57, 407.65, 407.96, 400.26, 396.43, 390.05, 399.52, 401.28, 391.04, 403.48, 390.98, 400.65, 408.54, 405.22, 411.64, 403.66, 406.1, 404.24, 410.93, 404.66, 410.03, 401.02, 415.12, 399.77, 394.97, 403.09, 397.2, 398.65, 393.22, 400.17, 399.6, 395.38, 393.38, 405.05, 392.14, 389.91, 395.39, 403.15, 400.79, 401.54, 400.66, 407.53, 402.67, 415.74, 393.46, 384.88, 397.05, 400.65, 400.63, 393.76, 400.35, 400.28, 407.67, 391.54, 398.21, 388.2, 397.38, 408.39, 386.02, 401.99, 392.74, 406.27, 393.8, 389.74, 401.62, 406.42, 402.35, 403.51, 398.22, 397.47, 394.37, 387.72, 385.12, 395.46, 398.2, 389.3, 401.03, 395.86, 397.59, 402.8, 393.63, 400.12, 411.43, 404.08, 419.46, 396.66, 397.77, 404.06, 411.6, 401.15, 395.1, 404.33, 394.03, 400.17, 404.98, 401.8, 385.68, 407.38, 405.2, 396.45, 391.39, 403.12, 409.06, 398.7, 391.56, 402.96, 405.36, 391.05, 397.03, 409.28, 392.98, 401.5, 412.84, 395.96, 388.53, 409.9, 400.68, 396.74, 393.51, 403.39, 395.0, 402.98, 392.2, 402.85, 393.48, 394.81, 399.58, 404.04, 400.4, 395.07, 394.42, 388.92, 400.92, 408.23, 397.5, 396.49, 402.72, 393.88, 400.69, 400.41, 393.02, 402.69, 397.34, 406.48, 401.7, 416.43, 405.7, 399.58, 393.63, 405.26, 395.74, 406.09, 406.89, 399.1, 407.79, 399.43, 395.9, 397.95, 399.85, 397.55, 395.66, 391.92, 402.46, 400.34, 403.89, 396.92, 400.53, 385.88, 402.87, 409.32, 400.02, 393.97, 392.48, 413.88, 396.05, 391.79, 398.68, 399.82, 409.03, 399.66, 397.71, 404.6, 392.88, 400.64, 397.64, 397.85, 397.45, 399.76, 399.2, 394.26, 401.0, 393.49, 398.78, 413.45, 408.17, 404.68, 417.71, 394.51, 397.45, 406.54, 402.42, 387.68, 385.88, 405.97, 406.23, 416.78, 403.17, 396.33, 394.49, 398.56, 405.55, 400.17, 403.45, 393.97, 394.61, 390.37, 403.75, 394.18, 399.32, 396.35, 405.14, 399.07, 408.29, 400.52, 402.94, 398.18, 401.27, 395.0, 404.95, 400.4, 404.77, 389.92, 405.15, 395.56, 389.36, 401.67, 401.48, 398.08, 407.53, 400.69, 387.29, 405.45, 403.81, 403.08, 396.91, 392.33, 397.9, 398.43, 407.54, 397.16, 398.44, 399.28, 396.98, 409.42, 390.68, 388.47, 391.73, 390.65, 395.86]；假设 μ₀=400
分析过程
H₀: μ=400，H₁: μ≠400。
x̄ = 399.7777，s = 6.5982，SE = s/√n = 6.5982/√570 = 0.2764；
t = (x̄−μ₀)/SE = (399.7777−400)/0.2764 = -0.8042；df = n−1 = 569；双侧 p = 0.4216。
x̄ = 399.7777，s = 6.5982，SE = s/√n = 6.5982/√570 = 0.2764；
t = (x̄−μ₀)/SE = (399.7777−400)/0.2764 = -0.8042；df = n−1 = 569；双侧 p = 0.4216。
计算结果
t = -0.8042，df = 569，p = 0.4216
在 α=0.05 水平下，p=0.4216 ≥ 0.05，不拒绝原假设 H₀。验证产量是否达到预期。
案例 7. 灯泡寿命是否 1000h
原始数据
样本 n=470：[993.35, 973.34, 1006.25, 985.31, 1007.28, 994.01, 1003.06, 998.46, 996.51, 1023.55, 977.77, 999.03, 1013.55, 978.7, 1030.1, 984.05, 1031.33, 991.1, 1001.15, 988.83, 988.24, 998.02, 976.89, 980.1, 981.29, 967.01, 1016.14, 987.64, 1002.09, 981.98, 1026.48, 1040.54, 994.96, 1010.51, 989.51, 953.91, 1011.4, 990.77, 994.69, 978.09, 994.62, 1000.81, 1013.97, 999.53, 1011.65, 989.9, 1004.75, 1028.42, 1000.88, 1017.03, 993.54, 1012.56, 986.62, 1013.02, 986.37, 994.41, 979.21, 1026.42, 1024.92, 997.36, 989.02, 1019.86, 1002.77, 988.9, 998.91, 1001.46, 969.4, 1004.6, 1014.84, 1025.23, 1006.14, 993.35, 1014.09, 1039.79, 1003.16, 1012.74, 999.85, 987.22, 972.79, 1046.55, 976.01, 992.77, 979.66, 982.87, 975.34, 979.35, 1023.82, 1008.21, 995.51, 1007.47, 988.86, 1018.84, 1002.57, 992.53, 982.71, 998.97, 1004.15, 1009.46, 985.18, 987.61, 994.41, 989.85, 975.26, 1008.21, 1014.48, 1015.6, 969.93, 1003.87, 977.18, 995.85, 1003.94, 997.21, 991.49, 992.09, 1009.25, 1015.47, 997.64, 995.99, 996.45, 1004.56, 1015.46, 1006.48, 1019.49, 1005.05, 1008.27, 973.33, 980.7, 1026.77, 988.92, 1004.93, 982.44, 992.22, 982.15, 996.54, 1010.16, 1001.93, 990.02, 979.49, 1007.77, 988.01, 981.83, 1005.84, 1000.92, 1021.6, 983.23, 1005.2, 1015.22, 985.91, 994.61, 997.15, 999.83, 999.28, 1033.37, 1001.5, 1004.49, 999.21, 971.39, 997.55, 989.3, 999.61, 1008.14, 1000.56, 989.57, 995.95, 1000.91, 1001.85, 1008.14, 1011.36, 991.27, 1026.52, 988.55, 1008.27, 1023.91, 999.12, 987.06, 1008.23, 993.06, 1017.79, 999.63, 1023.33, 997.95, 1006.24, 1005.83, 990.3, 994.57, 1014.59, 1024.3, 994.93, 1006.08, 1008.5, 997.85, 989.49, 1014.86, 1017.21, 1009.34, 1008.67, 989.84, 1021.78, 989.98, 999.31, 1012.07, 990.76, 1000.15, 1027.08, 979.48, 999.68, 1011.06, 993.75, 1017.11, 992.71, 985.43, 1002.35, 1024.45, 1024.06, 969.71, 1014.25, 996.14, 994.42, 986.52, 1004.17, 986.81, 1021.31, 1010.0, 1000.91, 995.32, 1024.23, 1023.64, 998.6, 994.59, 981.65, 993.6, 1008.16, 1005.64, 979.06, 1003.04, 1016.42, 1024.58, 987.06, 1000.34, 1004.88, 987.99, 997.85, 1011.46, 992.07, 1028.84, 984.75, 1013.27, 995.1, 983.97, 975.06, 1005.54, 1019.59, 1021.69, 1009.23, 994.86, 999.69, 967.23, 989.78, 1005.79, 988.5, 1014.89, 1003.61, 976.27, 1020.4, 994.7, 986.04, 1007.58, 1002.52, 988.77, 1004.52, 1006.87, 1012.17, 993.95, 1019.57, 983.89, 980.47, 1005.32, 1006.38, 982.36, 1005.81, 971.61, 1005.96, 1022.86, 988.99, 980.53, 1009.34, 1021.94, 983.15, 986.28, 991.17, 1014.04, 1042.43, 996.32, 980.87, 1015.7, 1041.75, 969.25, 1010.26, 1021.76, 984.77, 1002.68, 988.95, 983.54, 1026.75, 1005.03, 998.8, 1028.06, 983.47, 1006.62, 982.66, 966.53, 993.99, 1018.36, 993.34, 1003.45, 1012.96, 989.74, 999.09, 988.48, 986.56, 1041.36, 988.5, 1013.01, 1010.2, 1003.71, 1025.09, 993.38, 992.17, 1018.52, 990.32, 980.13, 964.67, 1008.73, 988.56, 989.51, 992.76, 988.84, 992.71, 996.47, 1019.86, 1009.56, 989.55, 1015.19, 1010.5, 991.07, 995.12, 1013.59, 1019.0, 1003.91, 1023.86, 991.67, 991.29, 995.34, 1031.81, 985.64, 1010.58, 976.97, 1029.05, 1026.75, 994.41, 998.57, 990.98, 980.61, 984.21, 1023.48, 963.78, 1007.13, 1021.19, 1010.38, 1011.92, 1013.76, 1013.35, 990.68, 988.22, 996.0, 998.37, 1002.54, 1011.49, 976.56, 996.41, 988.19, 1016.07, 1000.6, 1015.11, 980.78, 1010.09, 972.43, 996.85, 988.51, 995.48, 980.32, 991.26, 995.0, 1009.99, 1009.44, 1003.09, 977.92, 995.5, 1004.95, 1009.03, 987.04, 988.24, 990.46, 1005.67, 987.83, 990.21, 1010.97, 988.04, 1023.89, 993.17, 1016.11, 1003.01, 992.5, 1026.9, 977.08, 990.32, 987.29, 1020.65, 1002.59, 1022.85, 981.92, 1016.24, 999.29, 1007.51, 1016.36, 981.84, 988.53, 978.82, 993.63, 987.88, 989.13, 1021.22, 997.73, 1019.4, 980.76, 989.25, 1032.46, 1022.27, 998.71, 991.78, 1003.66, 994.97, 974.55, 993.43, 990.41, 1007.69, 1001.35, 1017.58, 993.76, 1012.35, 1017.16, 1011.71, 1007.04, 982.63, 1005.86, 986.23, 1017.1, 964.58, 995.91, 986.35, 1000.32, 1000.63, 990.59, 980.54, 1015.84, 1026.16, 992.01, 979.27, 1007.5, 1030.08]；假设 μ₀=1000
分析过程
H₀: μ=1000，H₁: μ≠1000。
x̄ = 1000.0603，s = 15.5192，SE = s/√n = 15.5192/√470 = 0.7158；
t = (x̄−μ₀)/SE = (1000.0603−1000)/0.7158 = 0.0842；df = n−1 = 469；双侧 p = 0.9329。
x̄ = 1000.0603，s = 15.5192，SE = s/√n = 15.5192/√470 = 0.7158；
t = (x̄−μ₀)/SE = (1000.0603−1000)/0.7158 = 0.0842；df = n−1 = 469；双侧 p = 0.9329。
计算结果
t = 0.0842，df = 469，p = 0.9329
在 α=0.05 水平下，p=0.9329 ≥ 0.05，不拒绝原假设 H₀。判断寿命是否达标。
案例 8. 咖啡杯量是否 350ml
原始数据
样本 n=987：[346.82, 350.65, 349.62, 347.12, 350.84, 344.02, 347.63, 347.23, 344.22, 354.35, 349.01, 354.51, 353.62, 349.18, 349.38, 348.7, 349.69, 352.97, 347.88, 348.01, 353.19, 347.43, 354.51, 347.3, 353.65, 353.66, 347.59, 345.06, 351.02, 348.7, 350.59, 356.48, 341.77, 357.36, 352.07, 348.17, 345.4, 351.87, 346.08, 358.25, 345.39, 350.43, 348.05, 348.12, 351.41, 344.7, 351.82, 358.07, 345.79, 351.17, 350.48, 352.9, 347.79, 344.12, 350.65, 342.34, 346.59, 349.2, 353.83, 344.28, 350.88, 348.47, 347.93, 348.24, 345.95, 349.11, 347.82, 346.58, 347.22, 350.21, 350.7, 346.33, 348.75, 347.36, 343.2, 356.25, 349.39, 346.27, 349.7, 357.51, 348.68, 345.79, 348.23, 346.33, 353.79, 352.43, 352.21, 339.76, 353.3, 344.84, 347.28, 346.31, 349.75, 349.19, 351.47, 345.95, 348.78, 348.32, 350.49, 346.84, 349.3, 349.73, 347.92, 351.29, 347.52, 351.29, 352.69, 349.74, 350.42, 348.19, 350.07, 353.67, 354.63, 349.12, 351.01, 352.19, 348.88, 349.98, 347.42, 347.06, 350.74, 345.64, 350.9, 349.94, 351.04, 352.09, 349.08, 350.65, 347.25, 349.36, 353.69, 352.35, 350.86, 351.25, 348.15, 354.02, 347.37, 352.21, 349.69, 354.78, 346.41, 349.76, 346.51, 350.38, 345.84, 347.01, 349.21, 346.83, 347.93, 349.91, 350.27, 346.69, 347.03, 347.66, 352.96, 345.33, 349.93, 344.19, 345.81, 351.36, 354.71, 350.49, 346.46, 353.5, 353.52, 352.97, 347.86, 351.37, 352.49, 349.84, 348.99, 354.72, 350.93, 356.58, 352.47, 350.88, 350.54, 347.83, 353.64, 348.21, 353.3, 353.88, 345.11, 350.29, 347.9, 346.41, 353.3, 350.73, 349.09, 353.65, 350.6, 348.5, 347.6, 349.39, 352.04, 350.83, 348.67, 351.51, 348.02, 347.44, 348.97, 354.01, 350.92, 346.93, 352.32, 350.14, 349.72, 351.62, 349.23, 349.95, 357.11, 351.62, 346.44, 348.55, 349.87, 350.91, 350.72, 346.85, 353.03, 351.64, 345.73, 355.37, 346.81, 351.38, 351.46, 346.2, 354.43, 351.08, 347.06, 347.89, 345.01, 355.3, 354.52, 348.86, 352.29, 348.68, 343.42, 348.33, 356.64, 348.72, 358.72, 354.92, 350.42, 344.21, 346.63, 353.1, 347.71, 350.77, 350.81, 347.03, 349.18, 352.14, 348.88, 352.62, 352.72, 345.15, 345.86, 352.3, 352.36, 345.41, 351.5, 349.06, 351.49, 349.6, 353.33, 353.16, 348.6, 351.84, 347.2, 348.6, 353.51, 356.22, 352.49, 348.5, 343.57, 352.65, 349.72, 346.99, 352.15, 355.76, 352.38, 351.66, 351.48, 347.61, 345.27, 346.6, 354.37, 346.57, 358.4, 340.23, 350.66, 350.63, 356.12, 347.14, 350.08, 347.76, 347.68, 344.95, 354.16, 347.7, 353.02, 347.73, 344.51, 348.83, 354.52, 350.1, 350.99, 348.2, 352.18, 352.67, 343.76, 354.87, 351.31, 350.86, 347.85, 352.93, 350.88, 349.41, 352.55, 345.7, 346.74, 350.22, 353.28, 349.78, 346.06, 350.22, 345.81, 348.08, 348.19, 353.37, 349.09, 348.23, 343.64, 348.25, 348.54, 351.1, 344.1, 349.06, 351.45, 353.3, 351.37, 353.75, 344.73, 347.53, 345.71, 343.16, 348.13, 352.3, 349.66, 352.54, 351.96, 350.92, 349.29, 348.98, 351.17, 350.46, 352.06, 353.63, 350.6, 358.64, 348.08, 349.61, 347.07, 351.61, 349.68, 349.77, 351.57, 352.4, 347.44, 349.11, 347.61, 349.38, 349.75, 344.53, 352.46, 350.34, 345.18, 350.56, 352.17, 349.52, 349.69, 348.47, 349.09, 357.89, 357.24, 349.14, 351.65, 346.1, 350.81, 348.0, 351.74, 348.29, 346.76, 351.72, 348.75, 350.64, 350.99, 346.29, 345.98, 352.33, 352.94, 351.94, 351.49, 349.21, 348.14, 352.15, 344.77, 351.88, 354.12, 345.54, 355.9, 351.92, 351.17, 349.58, 342.27, 352.61, 348.66, 346.41, 348.56, 350.19, 347.22, 350.51, 347.32, 350.41, 352.97, 346.51, 352.33, 356.45, 352.63, 346.89, 352.2, 350.96, 355.33, 347.94, 350.07, 348.49, 343.89, 353.0, 354.48, 352.02, 351.58, 353.66, 350.42, 349.0, 355.94, 346.35, 353.96, 346.99, 354.35, 348.41, 350.89, 352.02, 348.51, 349.77, 350.01, 347.04, 351.08, 351.07, 346.98, 359.39, 347.52, 354.96, 345.78, 349.78, 352.73, 350.0, 354.39, 349.44, 347.94, 351.81, 345.43, 352.37, 345.0, 347.34, 348.34, 348.11, 348.73, 343.68, 351.79, 349.02, 352.72, 348.14, 348.0, 347.54, 349.87, 348.97, 351.34, 349.64, 353.98, 351.92, 348.31, 350.47, 348.45, 346.91, 353.88, 349.52, 353.65, 350.02, 352.62, 349.18, 347.38, 353.33, 354.89, 351.0, 352.89, 343.29, 350.99, 352.73, 349.51, 347.23, 348.54, 355.61, 355.07, 349.2, 355.46, 350.68, 347.5, 356.25, 349.87, 347.78, 357.27, 349.18, 353.75, 348.77, 352.85, 350.96, 346.11, 349.98, 347.29, 350.1, 353.56, 350.64, 347.69, 346.19, 349.89, 344.72, 349.23, 356.1, 348.38, 350.18, 348.41, 346.87, 351.78, 352.24, 348.77, 345.93, 347.43, 347.49, 347.21, 353.32, 348.8, 353.3, 349.94, 347.04, 346.98, 349.48, 346.24, 354.34, 353.94, 352.87, 350.84, 356.53, 349.11, 348.4, 350.36, 344.62, 348.04, 350.82, 349.57, 355.25, 350.09, 349.79, 348.53, 348.89, 347.07, 356.42, 344.35, 348.85, 352.84, 347.97, 350.24, 350.81, 349.38, 347.31, 343.76, 353.62, 347.01, 347.81, 352.46, 350.24, 347.13, 348.22, 348.9, 345.22, 353.7, 347.96, 350.45, 344.49, 347.65, 347.44, 347.52, 348.43, 344.98, 349.69, 351.01, 349.65, 347.32, 349.32, 349.32, 349.7, 345.93, 352.43, 350.39, 352.2, 345.2, 352.1, 345.15, 351.88, 344.89, 353.19, 352.27, 353.25, 347.75, 353.66, 347.1, 345.51, 350.12, 349.61, 349.1, 352.21, 349.41, 344.99, 351.44, 352.4, 351.75, 347.84, 349.64, 347.89, 351.86, 349.84, 352.4, 347.96, 350.36, 353.43, 348.22, 348.93, 352.42, 344.41, 352.61, 349.92, 351.83, 345.54, 349.96, 350.78, 346.46, 346.67, 354.82, 354.03, 350.97, 352.37, 348.01, 348.73, 348.96, 346.5, 354.37, 349.38, 352.48, 349.01, 350.37, 356.18, 349.04, 352.6, 348.38, 349.63, 354.92, 349.69, 354.24, 349.85, 351.13, 348.44, 349.09, 353.17, 354.53, 352.14, 342.89, 346.09, 352.18, 349.18, 348.45, 355.38, 353.15, 351.01, 344.48, 350.61, 345.58, 343.95, 350.53, 354.58, 358.97, 353.26, 348.7, 349.89, 348.71, 349.3, 350.73, 349.47, 344.83, 350.26, 349.89, 348.47, 350.45, 353.07, 344.19, 349.97, 352.92, 350.65, 354.38, 348.36, 345.34, 355.43, 352.77, 356.52, 351.72, 354.2, 349.47, 352.54, 352.69, 349.75, 355.35, 351.12, 349.58, 345.11, 350.33, 351.23, 347.87, 342.61, 349.56, 351.52, 348.12, 351.94, 345.42, 347.46, 347.72, 352.73, 346.83, 350.58, 342.38, 350.23, 347.78, 350.64, 349.66, 352.78, 351.23, 348.23, 351.56, 350.0, 349.74, 348.84, 350.27, 348.6, 345.92, 351.52, 348.17, 344.06, 348.63, 347.73, 348.28, 343.42, 346.54, 349.37, 351.25, 352.19, 344.89, 349.25, 349.42, 357.34, 346.95, 354.29, 351.27, 346.02, 348.75, 346.74, 345.32, 345.36, 348.73, 345.43, 350.41, 349.24, 351.79, 348.14, 345.8, 346.41, 349.73, 347.07, 349.83, 351.12, 349.61, 352.72, 346.78, 349.36, 348.66, 343.86, 349.61, 349.67, 340.8, 351.57, 349.27, 353.53, 354.03, 348.55, 354.75, 350.68, 353.42, 349.34, 341.96, 351.33, 353.65, 347.04, 355.39, 349.01, 350.56, 350.78, 349.96, 345.86, 354.85, 347.98, 345.6, 351.74, 346.69, 344.99, 347.57, 343.91, 344.69, 348.81, 354.24, 351.09, 351.33, 349.16, 355.61, 349.12, 354.82, 344.56, 354.2, 352.63, 347.99, 353.45, 348.84, 348.22, 349.3, 346.19, 352.26, 351.85, 346.91, 350.5, 353.06, 345.73, 353.37, 349.41, 348.77, 349.84, 350.25, 355.07, 343.25, 348.38, 348.4, 346.86, 348.29, 353.03, 351.64, 348.13, 356.45, 343.07, 347.17, 351.2, 350.42, 352.65, 353.53, 350.05, 353.78, 350.58, 350.81, 348.55, 353.01, 352.25, 350.5, 348.97, 345.68, 352.3, 350.13, 351.5, 353.06, 347.6, 349.61, 347.59, 351.12, 346.63, 351.53, 353.64, 347.17, 350.84, 345.34, 350.86, 349.57, 349.61, 351.96, 351.56, 354.27, 357.56, 349.33, 352.89, 353.91, 346.93, 350.61, 346.94, 345.72, 344.78, 352.6, 350.71, 355.78, 347.63, 350.8, 350.39, 352.42, 353.81, 350.86, 351.44, 349.27, 349.56, 353.41, 353.61, 354.45, 347.63, 340.35, 351.59, 348.77, 355.04, 351.21, 350.64, 348.8, 353.65, 347.99, 347.34, 351.49, 345.65, 352.99, 352.72, 349.02, 350.32, 346.08, 351.21, 351.49, 352.93, 352.67, 351.26, 348.9, 348.67, 348.72, 351.92, 347.1, 345.76, 340.23, 348.53, 345.56, 352.02, 353.76, 351.75, 347.47, 350.3, 350.64, 350.9, 351.02, 344.84, 348.85, 349.39, 351.99, 352.48, 356.51, 350.05, 344.51, 349.53, 347.74, 342.19, 353.16, 352.71, 357.77, 351.12, 351.68]；假设 μ₀=350
分析过程
H₀: μ=350，H₁: μ≠350。
x̄ = 349.8630，s = 3.1732，SE = s/√n = 3.1732/√987 = 0.1010；
t = (x̄−μ₀)/SE = (349.8630−350)/0.1010 = -1.3567；df = n−1 = 986；双侧 p = 0.1752。
x̄ = 349.8630，s = 3.1732，SE = s/√n = 3.1732/√987 = 0.1010；
t = (x̄−μ₀)/SE = (349.8630−350)/0.1010 = -1.3567；df = n−1 = 986；双侧 p = 0.1752。
计算结果
t = -1.3567，df = 986，p = 0.1752
在 α=0.05 水平下，p=0.1752 ≥ 0.05，不拒绝原假设 H₀。检查出杯量稳定性。
案例 9. 运动员静息心率是否 70
原始数据
样本 n=939：[70.15, 69.44, 71.43, 67.73, 71.57, 68.61, 68.87, 67.33, 66.93, 70.04, 66.43, 71.47, 71.61, 71.52, 66.36, 67.44, 71.42, 65.04, 69.84, 71.15, 68.79, 69.29, 68.11, 69.7, 71.32, 67.6, 70.72, 70.27, 68.98, 71.7, 71.84, 72.82, 69.91, 71.93, 68.39, 71.92, 69.18, 69.86, 73.96, 70.64, 69.76, 72.15, 71.14, 71.51, 68.45, 70.21, 70.66, 72.95, 69.13, 69.18, 69.66, 69.94, 71.1, 70.55, 71.4, 71.59, 72.33, 70.83, 72.57, 69.85, 73.95, 69.15, 70.78, 70.99, 68.06, 71.35, 73.56, 67.53, 68.48, 70.49, 68.32, 71.12, 70.16, 71.81, 69.47, 70.23, 69.6, 71.14, 69.65, 72.28, 71.33, 68.04, 69.29, 72.02, 69.15, 71.3, 70.06, 70.33, 71.01, 69.51, 72.38, 70.49, 67.29, 67.91, 73.66, 72.25, 70.45, 70.02, 68.03, 71.63, 69.54, 74.27, 70.86, 69.02, 68.63, 71.56, 68.3, 70.8, 70.42, 65.37, 69.77, 68.79, 70.83, 69.12, 68.57, 72.89, 68.54, 68.69, 69.92, 69.69, 67.48, 72.21, 68.36, 68.51, 68.51, 67.25, 68.91, 69.33, 67.56, 66.35, 66.73, 70.83, 71.33, 71.15, 73.31, 71.76, 70.23, 69.68, 71.47, 69.05, 68.87, 69.18, 70.83, 71.21, 69.9, 69.54, 71.29, 70.04, 70.56, 70.32, 68.99, 68.8, 68.8, 69.47, 70.76, 69.45, 68.46, 71.03, 68.05, 67.73, 69.52, 65.4, 69.71, 69.34, 72.83, 68.97, 67.24, 71.09, 70.71, 73.49, 68.92, 68.5, 68.22, 70.67, 67.25, 68.7, 68.67, 67.93, 69.75, 67.27, 70.59, 72.07, 72.84, 69.7, 71.99, 72.26, 71.04, 64.39, 70.39, 67.29, 71.31, 69.93, 68.92, 67.25, 72.71, 70.37, 72.09, 67.23, 68.74, 72.93, 67.33, 69.41, 72.39, 70.7, 68.42, 68.76, 70.31, 67.74, 72.25, 74.28, 72.96, 69.78, 70.59, 70.48, 72.59, 67.55, 70.2, 68.09, 69.35, 70.3, 68.82, 68.3, 66.79, 69.97, 67.48, 70.86, 71.3, 71.26, 71.58, 68.9, 69.68, 72.44, 67.8, 69.15, 71.78, 71.73, 70.12, 67.59, 70.18, 69.82, 71.78, 71.13, 67.37, 68.01, 70.27, 74.36, 74.06, 70.37, 71.01, 68.51, 69.46, 69.49, 70.68, 69.98, 67.62, 70.58, 69.29, 70.53, 69.28, 73.91, 72.56, 71.71, 71.12, 71.36, 74.9, 73.9, 69.63, 69.21, 68.41, 70.19, 68.53, 71.45, 68.46, 69.74, 69.82, 71.23, 70.07, 73.07, 70.45, 72.0, 68.26, 70.43, 69.36, 71.22, 71.43, 72.56, 68.16, 68.15, 69.64, 72.28, 68.47, 69.29, 71.7, 69.37, 68.32, 70.74, 68.31, 67.71, 71.09, 71.71, 70.57, 70.96, 71.18, 71.85, 70.44, 69.38, 73.86, 69.15, 70.89, 72.05, 72.61, 74.29, 69.83, 68.34, 69.01, 68.69, 68.89, 69.58, 71.42, 67.11, 69.95, 69.29, 71.41, 70.26, 69.67, 70.76, 67.68, 69.95, 71.73, 67.58, 72.45, 64.94, 71.89, 72.85, 70.29, 68.87, 69.12, 69.84, 69.74, 68.94, 71.84, 69.63, 70.22, 71.61, 67.44, 67.49, 70.45, 69.4, 70.09, 71.62, 71.75, 70.32, 71.17, 69.94, 69.74, 72.12, 69.22, 68.29, 68.79, 72.19, 68.92, 72.57, 68.45, 72.71, 71.18, 66.01, 69.78, 69.58, 67.72, 74.74, 70.34, 70.02, 67.37, 65.37, 69.42, 70.09, 68.69, 67.43, 72.26, 70.03, 69.54, 67.29, 72.77, 64.9, 73.45, 68.96, 67.11, 69.4, 70.18, 70.65, 70.86, 67.51, 69.29, 68.12, 72.19, 70.7, 67.72, 68.75, 72.02, 69.34, 65.69, 72.05, 69.17, 67.74, 69.62, 69.53, 72.14, 75.23, 70.31, 70.13, 74.8, 67.96, 71.02, 71.15, 70.34, 67.83, 73.94, 70.06, 70.86, 70.97, 67.9, 70.25, 71.83, 69.32, 68.75, 69.06, 72.11, 67.33, 68.66, 68.67, 68.54, 68.99, 68.22, 70.22, 66.96, 68.66, 71.86, 69.87, 71.77, 69.06, 70.52, 70.82, 66.64, 70.33, 69.47, 68.77, 71.15, 70.81, 71.36, 67.9, 73.12, 68.05, 65.93, 70.7, 71.32, 68.6, 67.57, 69.01, 70.53, 71.17, 68.03, 70.17, 68.87, 71.14, 68.22, 71.08, 70.9, 71.13, 72.56, 68.12, 71.3, 67.24, 71.53, 71.97, 71.51, 70.04, 66.06, 68.59, 69.22, 72.23, 72.57, 67.82, 69.17, 70.38, 71.49, 71.09, 68.93, 68.75, 70.76, 70.16, 70.75, 65.76, 70.78, 72.42, 69.15, 72.11, 69.2, 67.78, 70.33, 64.15, 68.78, 68.89, 67.63, 73.12, 69.76, 74.08, 68.11, 67.92, 69.14, 71.17, 66.73, 70.15, 70.56, 72.04, 69.88, 68.67, 71.0, 70.51, 68.67, 73.07, 72.59, 71.25, 70.59, 69.27, 70.43, 69.58, 67.44, 67.51, 72.25, 69.8, 68.65, 66.51, 71.61, 67.2, 72.52, 68.28, 69.24, 72.54, 72.28, 70.27, 69.12, 69.33, 69.98, 70.94, 68.51, 70.72, 69.47, 71.35, 69.7, 69.48, 69.61, 69.57, 69.95, 70.33, 71.84, 70.26, 69.42, 74.51, 73.41, 69.34, 71.74, 70.36, 69.14, 68.0, 70.1, 68.98, 71.38, 72.02, 67.09, 74.17, 71.37, 70.13, 75.45, 74.72, 72.09, 69.34, 71.38, 65.77, 69.04, 71.61, 70.72, 72.58, 70.39, 70.08, 68.02, 70.19, 68.47, 69.38, 70.32, 66.49, 70.77, 66.14, 65.62, 70.45, 69.97, 67.13, 69.15, 70.38, 70.72, 69.3, 71.67, 71.55, 71.51, 69.96, 65.65, 69.64, 68.75, 67.34, 70.91, 70.22, 69.57, 66.76, 70.85, 65.98, 71.13, 69.91, 69.99, 67.49, 69.86, 68.8, 68.97, 70.72, 71.72, 63.18, 69.47, 71.54, 71.96, 71.84, 70.05, 67.63, 69.87, 71.19, 71.49, 68.22, 68.87, 68.18, 69.34, 67.7, 68.72, 71.19, 67.41, 70.39, 68.48, 66.8, 71.88, 69.73, 73.21, 67.83, 70.76, 71.63, 69.82, 66.57, 67.01, 72.35, 69.26, 68.72, 70.75, 67.19, 71.06, 70.67, 66.89, 69.19, 71.29, 71.56, 67.99, 71.25, 71.21, 68.9, 70.89, 69.71, 66.72, 69.91, 73.28, 71.5, 70.56, 68.66, 72.55, 71.78, 70.97, 69.32, 71.28, 68.24, 71.26, 68.76, 68.44, 68.09, 69.18, 69.45, 70.89, 68.49, 69.79, 65.63, 70.13, 66.2, 69.78, 69.77, 68.33, 69.03, 69.39, 71.81, 67.67, 64.59, 66.92, 68.08, 72.48, 70.84, 70.32, 70.34, 70.35, 71.2, 69.44, 70.57, 70.0, 69.74, 68.75, 68.99, 73.35, 70.23, 69.72, 70.93, 70.68, 68.67, 70.03, 69.51, 71.25, 69.24, 72.89, 69.08, 68.29, 68.51, 71.02, 70.07, 70.7, 67.14, 71.49, 69.48, 70.28, 67.94, 69.37, 66.23, 73.53, 67.61, 73.16, 70.8, 69.53, 71.23, 70.17, 71.61, 68.81, 68.88, 69.39, 73.12, 73.3, 69.05, 70.56, 69.13, 72.01, 72.24, 72.45, 70.6, 69.49, 67.79, 69.59, 71.42, 69.39, 70.11, 70.18, 70.25, 68.81, 69.11, 70.8, 68.35, 71.57, 66.86, 69.05, 68.3, 69.0, 68.58, 71.04, 69.51, 72.16, 69.37, 68.98, 71.15, 69.55, 67.13, 73.77, 73.94, 72.43, 69.41, 68.18, 69.14, 71.62, 68.82, 70.16, 71.15, 69.68, 66.06, 70.86, 69.7, 69.75, 71.06, 67.5, 70.14, 69.89, 72.18, 69.72, 72.17, 69.8, 70.61, 72.49, 67.3, 67.27, 70.86, 68.88, 67.45, 73.27, 68.59, 68.97, 70.59, 72.21, 68.61, 69.45, 72.07, 68.61, 70.25, 70.44, 71.68, 70.99, 68.66, 70.7, 72.45, 72.79, 69.2, 69.24, 68.09, 66.82, 72.77, 71.09, 71.96, 70.71, 69.16, 67.32, 72.07, 70.86, 68.23, 70.45, 65.64, 70.8, 72.42, 68.14, 68.35, 69.84, 67.3, 69.86, 71.31, 69.44, 68.34, 68.39, 73.43, 70.2, 70.34, 70.29, 69.84, 66.93, 73.03, 69.54, 71.93, 67.62, 69.5, 71.75, 70.43, 68.34, 70.37, 71.95, 71.23, 68.02, 66.73, 68.22, 70.1, 69.21, 68.13, 70.5, 68.88, 69.96, 68.99, 73.14, 67.57, 73.52, 72.01, 68.72, 70.51, 69.53, 72.97, 69.17, 67.75, 72.09, 71.15, 72.84, 69.09, 67.37, 72.54, 68.37, 68.52, 72.85, 68.46, 65.62, 69.36, 71.27, 73.71, 70.34, 71.88, 68.29, 72.22, 74.17, 71.92, 73.36, 72.74, 70.23, 69.68, 71.53, 73.72, 74.23, 71.02, 69.94, 70.7, 69.99, 65.48, 71.21, 70.41, 69.44, 68.6, 72.35, 68.55, 70.9, 74.49, 72.56, 70.69, 69.28]；假设 μ₀=70
分析过程
H₀: μ=70，H₁: μ≠70。
x̄ = 69.9793，s = 1.8846，SE = s/√n = 1.8846/√939 = 0.0615；
t = (x̄−μ₀)/SE = (69.9793−70)/0.0615 = -0.3358；df = n−1 = 938；双侧 p = 0.7371。
x̄ = 69.9793，s = 1.8846，SE = s/√n = 1.8846/√939 = 0.0615；
t = (x̄−μ₀)/SE = (69.9793−70)/0.0615 = -0.3358；df = n−1 = 938；双侧 p = 0.7371。
计算结果
t = -0.3358，df = 938，p = 0.7371
在 α=0.05 水平下，p=0.7371 ≥ 0.05，不拒绝原假设 H₀。评估心率是否在正常基准。
案例 10. 手机表面温度是否 35℃
原始数据
样本 n=739：[36.03, 32.99, 36.58, 33.64, 32.98, 33.72, 32.64, 38.55, 33.34, 33.99, 35.69, 34.16, 35.87, 35.12, 35.93, 37.27, 37.15, 34.8, 34.04, 34.33, 33.83, 33.22, 35.46, 32.85, 34.68, 37.11, 37.64, 36.17, 35.6, 34.58, 36.85, 33.64, 33.78, 35.22, 35.52, 35.02, 35.69, 36.87, 34.11, 33.02, 35.74, 36.62, 33.6, 35.58, 35.48, 35.24, 36.54, 37.05, 35.58, 34.69, 35.74, 35.92, 35.71, 34.66, 33.1, 35.11, 35.39, 33.52, 35.37, 34.29, 36.02, 35.72, 35.27, 36.09, 36.14, 36.2, 35.23, 36.25, 35.22, 34.53, 39.01, 34.93, 32.29, 35.33, 34.75, 34.59, 35.35, 35.13, 33.85, 35.58, 33.73, 34.34, 34.71, 35.47, 33.51, 35.4, 34.94, 34.75, 35.58, 35.65, 33.4, 34.53, 34.66, 35.16, 34.09, 35.84, 36.06, 33.72, 33.25, 34.49, 34.37, 33.96, 34.38, 36.72, 34.75, 35.38, 34.69, 36.03, 35.11, 34.69, 34.91, 34.86, 31.77, 33.84, 37.08, 32.75, 36.81, 35.24, 33.01, 33.17, 34.82, 35.26, 36.01, 34.02, 36.05, 33.2, 31.24, 35.33, 35.36, 34.65, 35.87, 34.28, 34.69, 33.83, 37.03, 34.04, 33.3, 35.29, 34.86, 35.94, 36.65, 34.47, 37.7, 33.71, 33.81, 34.1, 36.37, 36.62, 35.45, 34.79, 35.01, 34.17, 33.81, 34.91, 33.93, 34.97, 35.23, 34.84, 33.99, 36.59, 33.96, 34.68, 37.09, 35.39, 36.02, 36.64, 35.15, 33.4, 33.59, 32.55, 33.48, 37.55, 33.47, 35.9, 35.05, 35.75, 35.76, 35.33, 36.0, 34.41, 35.32, 34.35, 38.11, 31.85, 34.65, 35.49, 34.36, 34.69, 36.13, 36.76, 34.41, 33.93, 36.16, 35.38, 34.47, 35.84, 33.94, 36.82, 34.62, 36.02, 34.7, 37.22, 34.32, 35.07, 37.09, 34.01, 34.37, 35.97, 32.84, 34.79, 35.57, 35.33, 38.47, 33.86, 32.55, 37.9, 35.45, 34.3, 34.0, 35.67, 33.18, 34.68, 33.15, 35.03, 37.12, 35.08, 33.08, 35.11, 35.33, 37.08, 35.62, 33.57, 35.39, 39.74, 35.14, 33.77, 34.18, 35.75, 31.27, 32.29, 36.17, 34.28, 35.99, 35.27, 36.24, 35.12, 36.48, 34.68, 33.48, 34.12, 32.78, 36.69, 33.35, 35.7, 36.01, 37.34, 34.17, 32.97, 35.18, 34.27, 35.53, 34.48, 35.93, 34.13, 33.16, 33.18, 33.73, 35.61, 34.67, 35.32, 34.89, 33.95, 35.53, 37.07, 35.15, 35.07, 34.99, 34.87, 33.26, 31.06, 34.28, 34.44, 37.42, 32.62, 34.89, 38.7, 36.24, 33.94, 35.84, 35.81, 34.03, 34.96, 36.61, 34.33, 33.37, 34.47, 34.25, 34.94, 35.09, 35.97, 37.41, 34.13, 35.28, 36.23, 33.21, 35.26, 32.63, 35.04, 34.22, 38.35, 34.94, 33.67, 37.5, 35.76, 34.85, 35.84, 35.84, 35.77, 33.56, 35.53, 38.56, 35.68, 34.41, 31.24, 35.3, 33.55, 35.46, 37.24, 35.59, 35.64, 33.87, 34.38, 34.82, 37.16, 35.17, 32.07, 36.52, 36.09, 30.99, 35.98, 34.41, 34.02, 37.79, 35.76, 36.84, 35.39, 30.97, 35.59, 36.64, 32.98, 36.92, 35.93, 32.6, 36.72, 34.47, 37.76, 34.91, 34.63, 34.56, 34.5, 36.48, 33.97, 34.21, 35.27, 36.58, 33.83, 35.2, 33.63, 34.77, 35.26, 35.39, 33.75, 37.48, 34.19, 33.46, 35.78, 36.11, 34.21, 35.9, 34.18, 35.65, 35.43, 34.89, 34.76, 36.23, 33.73, 33.72, 37.22, 32.59, 36.52, 36.49, 36.66, 34.69, 36.7, 36.07, 37.91, 32.93, 33.36, 36.09, 34.35, 37.9, 35.38, 33.06, 34.15, 33.91, 32.75, 33.42, 34.92, 38.18, 33.94, 35.89, 37.5, 32.24, 34.43, 35.16, 35.77, 33.73, 35.24, 35.04, 36.85, 34.75, 34.07, 34.34, 36.15, 37.17, 37.07, 36.04, 36.37, 35.36, 34.27, 32.95, 37.15, 36.79, 33.47, 36.32, 34.04, 35.97, 33.52, 34.54, 33.76, 33.08, 33.42, 35.08, 31.69, 35.16, 36.75, 34.21, 34.27, 33.63, 36.19, 36.24, 34.32, 34.43, 35.25, 37.33, 34.89, 34.18, 34.59, 32.63, 36.11, 33.82, 32.72, 34.35, 35.4, 35.01, 36.57, 34.78, 35.85, 35.28, 34.9, 35.18, 36.36, 34.92, 34.95, 35.09, 34.03, 34.77, 35.07, 32.25, 36.87, 35.14, 31.91, 34.15, 36.75, 35.44, 35.33, 34.38, 36.49, 34.46, 36.42, 35.84, 34.22, 33.13, 33.86, 36.2, 34.77, 34.91, 33.92, 34.78, 34.25, 38.44, 35.63, 34.99, 34.17, 34.19, 33.73, 35.98, 34.84, 35.93, 34.68, 35.64, 35.55, 34.44, 35.92, 36.4, 34.81, 34.74, 33.98, 34.08, 35.63, 34.85, 33.16, 36.9, 36.26, 32.59, 35.16, 35.09, 35.7, 35.63, 31.83, 31.84, 35.42, 33.73, 34.07, 34.89, 35.64, 33.76, 34.86, 36.94, 34.3, 36.03, 35.81, 33.98, 35.38, 32.12, 33.51, 31.98, 35.09, 35.48, 33.74, 33.36, 37.26, 32.93, 35.88, 33.27, 36.72, 34.16, 36.73, 33.25, 33.86, 34.97, 37.33, 33.88, 35.9, 34.79, 35.05, 35.09, 34.58, 36.39, 35.9, 36.05, 36.35, 35.62, 37.21, 35.1, 35.65, 35.95, 35.27, 34.66, 36.78, 36.52, 33.86, 33.76, 35.56, 36.35, 37.42, 34.53, 36.61, 36.59, 32.78, 34.95, 35.13, 35.28, 33.07, 35.84, 34.22, 35.74, 33.88, 35.97, 33.88, 33.48, 38.08, 34.18, 34.15, 34.25, 35.58, 34.89, 35.94, 33.92, 34.46, 35.67, 35.03, 34.6, 32.77, 33.1, 35.1, 37.53, 34.99, 36.38, 33.7, 34.26, 35.34, 35.99, 36.12, 33.65, 33.6, 35.38, 35.09, 35.98, 33.27, 37.23, 36.16, 32.7, 34.77, 35.96, 36.61, 35.2, 35.48, 33.45, 33.07, 33.39, 35.86, 34.89, 35.96, 35.28, 32.54, 36.76, 34.41, 34.38, 34.49, 32.55, 34.6, 34.5, 35.58, 35.29, 34.93, 34.58, 34.94, 37.37, 34.23, 35.02, 32.37, 35.21, 35.11, 37.78, 35.83, 35.14, 31.23, 35.43, 35.25, 35.55, 36.41, 35.09, 35.28, 35.72, 35.24, 35.14, 37.27, 35.62, 36.67, 36.08, 34.33, 35.54, 35.3, 35.97, 35.23, 34.72, 35.0, 37.11, 35.17, 34.65, 34.94, 36.07, 37.71, 33.59, 36.48, 33.56, 35.93, 36.02, 35.09, 36.89, 34.96, 34.3, 33.57, 34.99, 32.21, 36.65, 33.83, 34.65, 35.21, 35.23, 35.22, 33.6, 36.21, 35.86, 35.42, 35.81, 35.15, 33.04, 33.9, 35.81, 33.99, 34.09, 32.29, 36.02, 33.49, 35.37, 34.29, 36.15, 35.52, 34.55, 35.1, 36.28, 34.18, 37.1, 36.8, 33.12, 32.6, 34.84]；假设 μ₀=35
分析过程
H₀: μ=35，H₁: μ≠35。
x̄ = 35.0014，s = 1.3635，SE = s/√n = 1.3635/√739 = 0.0502；
t = (x̄−μ₀)/SE = (35.0014−35)/0.0502 = 0.0278；df = n−1 = 738；双侧 p = 0.9778。
x̄ = 35.0014，s = 1.3635，SE = s/√n = 1.3635/√739 = 0.0502；
t = (x̄−μ₀)/SE = (35.0014−35)/0.0502 = 0.0278；df = n−1 = 738；双侧 p = 0.9778。
计算结果
t = 0.0278，df = 738，p = 0.9778
在 α=0.05 水平下，p=0.9778 ≥ 0.05，不拒绝原假设 H₀。验证温控是否达标。
案例 11. 培训后考试均分是否 75
原始数据
样本 n=326：[75.37, 75.63, 79.61, 76.91, 72.67, 74.44, 71.61, 79.19, 74.78, 76.52, 77.94, 77.32, 74.68, 74.12, 75.46, 67.49, 74.72, 75.22, 74.43, 74.87, 78.28, 75.37, 78.49, 77.34, 75.49, 69.99, 77.9, 73.06, 77.02, 70.44, 75.27, 75.15, 74.38, 75.73, 78.23, 72.29, 72.54, 77.65, 72.27, 75.18, 78.77, 77.65, 73.63, 72.0, 77.96, 69.31, 78.63, 77.89, 71.29, 74.25, 73.26, 74.42, 73.32, 76.91, 72.04, 74.82, 79.86, 76.31, 73.58, 75.11, 72.24, 76.05, 74.95, 76.11, 77.07, 72.75, 74.27, 73.9, 75.91, 74.45, 74.72, 79.35, 77.84, 76.12, 70.32, 75.65, 79.82, 68.01, 80.3, 76.06, 72.82, 73.83, 73.3, 74.47, 75.45, 74.3, 75.1, 76.08, 72.46, 71.89, 76.67, 78.6, 80.92, 74.18, 77.53, 74.25, 77.86, 73.22, 74.24, 76.14, 75.56, 73.34, 71.53, 72.96, 74.63, 73.36, 74.49, 76.14, 76.55, 74.16, 75.63, 74.11, 73.93, 72.26, 72.69, 71.49, 75.49, 71.73, 76.52, 72.84, 78.0, 78.76, 78.53, 73.79, 72.53, 76.22, 72.12, 72.33, 72.03, 71.95, 71.29, 81.33, 72.62, 69.24, 74.53, 75.95, 71.15, 75.28, 71.3, 73.43, 73.42, 78.42, 78.46, 72.57, 78.92, 72.47, 78.07, 72.97, 78.61, 76.16, 72.7, 75.37, 72.34, 72.77, 74.83, 76.16, 69.03, 77.59, 78.01, 76.14, 77.12, 78.45, 78.25, 75.12, 73.01, 74.62, 75.66, 73.07, 76.39, 76.53, 77.61, 77.67, 74.6, 78.34, 72.06, 77.43, 77.46, 75.87, 76.89, 66.76, 74.2, 75.83, 70.99, 73.84, 80.81, 72.15, 74.98, 78.32, 75.9, 78.66, 71.67, 73.86, 74.71, 75.46, 76.38, 77.07, 72.19, 73.64, 75.58, 74.54, 75.27, 74.1, 76.59, 74.85, 78.36, 72.08, 77.07, 76.17, 76.01, 76.43, 75.98, 70.64, 73.76, 76.13, 76.01, 75.16, 73.97, 77.11, 76.65, 76.97, 76.01, 75.04, 74.93, 76.86, 81.12, 70.95, 76.26, 76.85, 76.63, 76.0, 74.29, 73.45, 72.71, 78.17, 71.67, 74.15, 72.89, 78.51, 71.4, 72.37, 78.77, 76.59, 73.81, 78.07, 78.13, 71.92, 78.01, 76.05, 75.72, 73.5, 77.38, 73.87, 72.88, 78.01, 78.39, 75.35, 73.16, 74.97, 76.38, 70.42, 74.46, 74.94, 78.92, 72.19, 75.15, 76.65, 76.02, 75.21, 78.11, 75.78, 74.07, 71.64, 77.66, 75.78, 72.18, 73.48, 73.09, 76.44, 75.73, 74.35, 70.31, 75.0, 75.01, 72.95, 72.27, 74.78, 75.69, 77.76, 77.4, 74.51, 75.62, 71.62, 71.12, 74.66, 77.35, 77.8, 75.83, 74.97, 73.12, 76.01, 74.47, 74.84, 74.78, 72.79, 71.18, 75.68, 70.24, 74.25, 73.78, 75.69, 76.0, 79.29, 72.98, 79.8, 75.39, 76.16, 72.07, 73.31, 77.19, 74.58, 75.55, 81.4, 75.33, 73.79, 79.61, 80.55]；假设 μ₀=75
分析过程
H₀: μ=75，H₁: μ≠75。
x̄ = 75.0706，s = 2.5331，SE = s/√n = 2.5331/√326 = 0.1403；
t = (x̄−μ₀)/SE = (75.0706−75)/0.1403 = 0.5034；df = n−1 = 325；双侧 p = 0.6150。
x̄ = 75.0706，s = 2.5331，SE = s/√n = 2.5331/√326 = 0.1403；
t = (x̄−μ₀)/SE = (75.0706−75)/0.1403 = 0.5034；df = n−1 = 325；双侧 p = 0.6150。
计算结果
t = 0.5034，df = 325，p = 0.6150
在 α=0.05 水平下，p=0.6150 ≥ 0.05，不拒绝原假设 H₀。判断培训效果是否达到目标分。
案例 12. 包装净重是否 200g
原始数据
样本 n=614：[201.49, 199.16, 199.93, 200.57, 201.45, 197.41, 199.89, 202.1, 193.26, 203.13, 200.78, 199.06, 199.01, 203.5, 201.1, 198.68, 202.25, 201.31, 198.88, 200.01, 198.57, 201.88, 199.79, 198.38, 200.29, 199.03, 201.75, 203.3, 197.85, 199.19, 196.99, 199.53, 201.44, 200.89, 203.95, 200.42, 197.25, 200.81, 200.63, 201.49, 201.6, 199.74, 195.82, 196.41, 195.6, 199.55, 199.44, 200.53, 205.15, 200.84, 199.93, 197.74, 200.33, 200.13, 200.13, 198.04, 201.88, 199.46, 203.2, 198.45, 201.02, 201.61, 199.3, 199.03, 201.65, 201.53, 203.45, 200.16, 196.65, 198.7, 201.04, 198.12, 201.75, 199.2, 200.01, 202.76, 198.39, 200.77, 201.88, 199.09, 200.35, 198.61, 197.43, 199.47, 199.46, 200.73, 199.64, 197.28, 197.89, 200.91, 201.78, 200.11, 200.61, 199.84, 200.63, 202.95, 198.98, 200.69, 200.82, 200.26, 200.41, 198.37, 199.73, 199.39, 199.58, 199.47, 199.43, 198.85, 198.61, 201.74, 198.54, 201.6, 201.74, 199.48, 203.01, 202.41, 199.01, 197.78, 199.04, 196.85, 202.02, 198.04, 198.91, 198.05, 198.62, 200.05, 199.88, 198.63, 205.85, 201.73, 199.16, 200.57, 201.12, 201.67, 197.84, 199.79, 202.62, 199.44, 197.08, 202.07, 199.68, 202.06, 197.73, 200.55, 199.47, 199.05, 203.43, 200.01, 201.13, 200.07, 200.42, 199.77, 200.51, 202.5, 199.99, 196.19, 201.69, 197.18, 199.16, 203.06, 199.53, 200.63, 200.54, 198.44, 199.44, 199.66, 198.51, 201.17, 198.55, 198.68, 199.67, 200.05, 199.4, 201.2, 203.18, 199.66, 200.31, 199.86, 200.84, 196.98, 200.65, 198.22, 200.33, 199.7, 199.88, 200.64, 203.8, 198.07, 199.5, 202.41, 200.18, 200.33, 200.29, 200.78, 199.82, 200.3, 196.73, 199.82, 200.58, 198.04, 198.37, 202.55, 203.18, 197.96, 199.75, 201.66, 203.08, 199.45, 201.45, 201.92, 198.7, 198.23, 199.66, 197.08, 197.49, 199.62, 201.5, 200.61, 198.71, 196.8, 203.38, 202.14, 199.49, 200.5, 201.39, 197.61, 199.17, 198.83, 200.06, 200.73, 197.97, 198.13, 199.17, 198.74, 198.97, 200.76, 204.02, 198.95, 201.61, 198.9, 200.3, 199.84, 200.12, 196.88, 195.18, 199.25, 197.27, 200.16, 200.42, 200.82, 199.2, 197.89, 198.33, 199.17, 204.07, 200.4, 199.82, 201.91, 201.24, 198.72, 198.96, 198.3, 199.8, 201.85, 198.39, 200.45, 202.36, 195.63, 201.54, 203.49, 200.56, 201.42, 200.27, 200.91, 200.71, 198.63, 201.11, 199.13, 200.17, 202.7, 200.19, 198.46, 202.23, 198.75, 200.07, 203.75, 197.17, 202.16, 200.38, 202.75, 198.27, 200.78, 201.14, 197.52, 200.09, 199.41, 200.33, 199.99, 199.39, 198.01, 198.9, 204.14, 203.41, 197.62, 199.72, 202.61, 199.8, 199.13, 200.1, 198.99, 198.22, 199.63, 197.01, 195.4, 199.24, 201.42, 200.54, 198.96, 199.36, 199.3, 201.9, 199.91, 199.23, 197.6, 200.3, 198.76, 202.41, 200.91, 199.36, 197.25, 195.66, 197.3, 200.59, 200.64, 200.15, 199.71, 199.27, 200.88, 203.36, 200.47, 202.1, 200.18, 200.3, 201.04, 198.11, 200.13, 197.8, 198.46, 200.65, 200.33, 198.65, 203.0, 202.41, 199.6, 197.28, 203.52, 201.08, 198.38, 198.07, 198.11, 199.67, 198.72, 198.24, 201.44, 199.2, 202.11, 200.05, 200.76, 199.81, 201.75, 204.72, 199.3, 201.99, 199.52, 200.89, 199.91, 203.37, 198.82, 201.12, 194.59, 201.04, 201.07, 198.87, 198.69, 200.38, 198.8, 200.85, 202.49, 197.78, 200.15, 203.5, 202.01, 202.97, 199.54, 199.36, 200.74, 199.21, 199.52, 200.74, 200.23, 202.85, 202.4, 198.01, 200.42, 198.88, 199.39, 198.64, 203.96, 200.04, 201.37, 200.49, 201.24, 203.06, 204.48, 202.46, 202.5, 200.38, 197.7, 201.06, 200.71, 200.53, 200.49, 200.87, 200.15, 200.94, 201.72, 200.32, 202.3, 200.56, 197.49, 199.84, 198.1, 201.47, 198.63, 198.27, 196.32, 198.77, 201.46, 203.18, 199.6, 197.89, 198.73, 199.84, 196.94, 198.92, 200.25, 200.55, 200.0, 200.41, 199.16, 203.44, 199.92, 200.27, 199.67, 199.32, 202.15, 205.02, 202.64, 201.89, 199.38, 200.27, 202.63, 201.52, 202.12, 195.51, 202.48, 202.52, 199.88, 199.2, 202.68, 198.62, 201.33, 194.63, 202.64, 199.49, 201.29, 201.16, 197.59, 199.1, 200.13, 199.86, 198.58, 202.77, 201.04, 203.23, 197.9, 203.45, 199.75, 200.91, 198.36, 199.97, 200.42, 197.92, 197.81, 198.15, 202.78, 200.33, 198.46, 200.15, 202.19, 200.24, 202.21, 201.65, 199.67, 201.58, 198.74, 200.71, 201.69, 200.08, 201.38, 198.86, 201.47, 200.13, 198.74, 199.13, 198.92, 198.98, 200.42, 197.22, 202.03, 199.61, 199.85, 203.24, 201.17, 197.41, 200.15, 203.43, 199.62, 199.61, 195.5, 201.2, 196.94, 195.87, 199.99, 199.89, 198.32, 203.08, 200.15, 198.25, 198.89, 200.37, 198.71, 200.44, 196.12, 198.18, 198.53, 198.92, 199.31, 200.49, 199.26, 201.6, 201.2, 199.44, 202.38, 201.78, 198.1, 203.19, 202.24, 198.37, 200.06, 198.63, 200.46, 201.51, 198.87, 199.76, 197.73, 199.53, 197.24, 197.74, 204.24, 204.05, 197.56, 200.48, 199.54, 201.2, 198.43, 202.53, 202.08, 201.83, 198.32, 202.02, 200.34, 199.55, 200.78, 197.26, 199.47, 199.22, 201.76, 198.03, 200.23, 200.53, 199.48, 198.44, 200.16, 198.95, 202.28, 201.49, 196.98, 201.37, 202.34, 202.11, 201.39, 201.54, 201.39, 204.05, 201.41, 198.56, 202.83, 203.44, 201.35, 197.1, 200.2, 198.26, 201.09]；假设 μ₀=200
分析过程
H₀: μ=200，H₁: μ≠200。
x̄ = 200.0814，s = 1.8508，SE = s/√n = 1.8508/√614 = 0.0747；
t = (x̄−μ₀)/SE = (200.0814−200)/0.0747 = 1.0902；df = n−1 = 613；双侧 p = 0.2761。
x̄ = 200.0814，s = 1.8508，SE = s/√n = 1.8508/√614 = 0.0747；
t = (x̄−μ₀)/SE = (200.0814−200)/0.0747 = 1.0902；df = n−1 = 613；双侧 p = 0.2761。
计算结果
t = 1.0902，df = 613，p = 0.2761
在 α=0.05 水平下，p=0.2761 ≥ 0.05，不拒绝原假设 H₀。检测净含量是否合规。
案例 13. 用户满意度是否 4 分
原始数据
样本 n=734：[3.91, 4.09, 3.7, 4.22, 3.9, 3.97, 3.76, 3.58, 4.25, 4.11, 4.04, 4.24, 4.12, 3.6, 3.97, 3.89, 3.97, 3.92, 4.13, 3.91, 4.15, 3.78, 3.86, 4.37, 4.54, 4.02, 4.08, 4.21, 3.95, 3.63, 4.06, 4.05, 4.07, 4.15, 4.29, 4.06, 3.83, 4.16, 3.93, 3.94, 4.03, 3.89, 3.73, 3.77, 3.92, 3.52, 4.18, 3.88, 3.96, 3.76, 4.18, 4.31, 3.58, 3.94, 4.27, 4.35, 3.92, 3.69, 4.06, 3.68, 4.0, 3.75, 4.03, 4.19, 3.95, 3.93, 3.93, 4.11, 4.05, 4.04, 3.93, 3.85, 4.34, 3.9, 3.97, 3.94, 4.34, 4.0, 3.67, 4.04, 4.0, 3.78, 4.22, 4.09, 4.23, 4.31, 3.8, 3.8, 4.02, 3.53, 3.74, 3.77, 3.72, 4.13, 3.61, 3.75, 3.73, 4.07, 3.96, 3.78, 3.67, 4.23, 3.77, 4.07, 3.88, 4.05, 4.08, 3.76, 4.03, 3.91, 4.27, 4.03, 3.85, 4.15, 4.23, 3.86, 4.14, 4.03, 3.91, 3.99, 3.82, 4.08, 4.22, 4.16, 4.22, 3.86, 4.45, 3.84, 3.66, 4.09, 3.81, 3.9, 4.34, 3.91, 3.9, 3.87, 4.34, 4.19, 4.05, 4.19, 4.2, 4.08, 4.0, 4.09, 3.71, 4.42, 4.15, 3.83, 4.27, 3.9, 3.71, 4.0, 4.14, 4.25, 4.01, 4.09, 4.04, 4.18, 3.86, 3.92, 4.13, 3.75, 4.06, 3.96, 3.92, 4.11, 3.97, 4.21, 3.85, 4.18, 4.05, 4.04, 4.24, 3.94, 4.16, 4.26, 4.04, 4.08, 3.89, 4.43, 3.86, 4.11, 3.86, 4.07, 4.11, 4.44, 4.16, 3.96, 4.03, 4.31, 3.82, 3.82, 3.93, 4.13, 3.87, 3.84, 4.32, 3.95, 4.07, 4.11, 3.74, 3.7, 4.01, 3.98, 4.41, 3.98, 4.03, 4.2, 4.1, 3.79, 4.18, 3.86, 4.03, 3.8, 4.15, 3.87, 3.86, 4.16, 4.22, 3.88, 4.17, 4.11, 4.13, 3.79, 3.98, 3.74, 4.12, 4.18, 4.32, 4.16, 4.01, 4.17, 3.82, 3.91, 4.16, 4.03, 4.06, 3.82, 3.91, 3.96, 3.97, 4.23, 4.12, 3.87, 3.98, 3.88, 4.02, 4.19, 3.92, 3.81, 4.07, 4.19, 3.97, 3.75, 4.37, 4.18, 4.1, 3.95, 4.11, 4.19, 4.3, 3.8, 4.18, 3.66, 4.04, 3.84, 4.25, 4.22, 3.91, 3.89, 4.16, 3.9, 4.05, 4.05, 3.84, 4.14, 4.1, 3.82, 4.33, 4.02, 4.06, 4.33, 3.95, 4.17, 3.87, 3.98, 3.84, 3.91, 3.95, 3.72, 3.65, 4.1, 4.11, 3.84, 3.74, 4.46, 3.4, 3.91, 3.94, 3.93, 4.1, 3.68, 4.25, 4.1, 4.12, 3.77, 4.13, 4.13, 3.98, 3.9, 3.85, 3.91, 4.02, 3.62, 4.09, 3.84, 4.05, 4.15, 3.96, 3.84, 4.03, 3.85, 4.06, 4.04, 4.04, 3.78, 4.0, 4.01, 4.15, 4.04, 4.12, 4.15, 4.13, 4.12, 4.04, 4.09, 3.86, 3.96, 4.03, 3.86, 4.09, 3.83, 3.98, 4.02, 3.93, 4.18, 3.87, 4.23, 4.02, 4.04, 4.02, 4.22, 4.18, 4.19, 3.84, 4.14, 3.82, 4.02, 3.83, 3.92, 3.88, 4.13, 4.0, 3.95, 3.83, 4.05, 4.09, 4.03, 3.97, 4.06, 3.92, 3.92, 3.89, 4.14, 4.01, 3.93, 4.11, 3.85, 3.98, 4.02, 4.15, 4.06, 4.04, 3.71, 4.23, 3.9, 4.18, 3.94, 4.16, 4.22, 4.02, 3.9, 4.03, 4.2, 3.82, 4.41, 3.93, 3.69, 3.96, 4.08, 4.29, 3.88, 3.89, 4.24, 3.87, 4.13, 4.05, 4.16, 3.9, 3.87, 3.83, 4.53, 4.01, 4.28, 3.99, 4.02, 4.24, 3.87, 3.95, 3.86, 3.79, 4.24, 4.1, 3.63, 3.94, 4.22, 3.6, 3.75, 4.25, 3.79, 4.01, 4.11, 3.89, 4.15, 4.13, 3.99, 4.14, 4.11, 3.88, 3.92, 3.89, 3.69, 3.99, 3.92, 3.83, 4.08, 3.89, 4.08, 3.75, 4.26, 3.76, 3.96, 4.09, 4.13, 3.84, 4.19, 4.32, 4.05, 3.84, 4.02, 4.06, 3.95, 3.94, 3.87, 4.15, 4.1, 4.26, 3.97, 4.36, 3.89, 3.82, 4.27, 3.94, 3.96, 3.89, 4.08, 3.81, 4.32, 4.15, 4.24, 3.85, 4.06, 4.26, 4.12, 3.58, 3.99, 4.03, 3.61, 3.89, 4.2, 4.0, 3.74, 3.93, 4.2, 4.08, 3.81, 3.95, 3.7, 3.75, 3.93, 3.76, 4.18, 4.03, 4.01, 4.17, 4.08, 3.95, 3.93, 4.05, 4.12, 3.98, 4.18, 3.52, 4.38, 4.1, 4.22, 3.94, 4.09, 4.16, 3.79, 3.95, 4.08, 3.91, 4.22, 3.84, 4.1, 3.78, 3.83, 3.89, 4.06, 4.03, 4.15, 3.92, 4.11, 4.02, 4.01, 4.43, 3.9, 4.07, 4.18, 3.94, 4.0, 3.92, 4.14, 3.92, 4.14, 3.94, 3.98, 4.15, 3.88, 3.68, 4.05, 3.97, 3.87, 4.02, 3.91, 4.14, 3.93, 3.85, 4.17, 3.74, 3.84, 4.22, 4.01, 4.17, 4.01, 3.98, 4.02, 3.68, 3.87, 4.14, 3.84, 3.78, 4.19, 3.88, 4.12, 4.19, 4.1, 4.0, 4.36, 3.88, 4.13, 3.71, 3.69, 4.04, 4.05, 4.17, 3.78, 4.33, 4.04, 3.46, 4.26, 4.13, 4.22, 3.99, 4.2, 3.82, 3.47, 4.0, 3.94, 4.23, 3.66, 3.9, 4.07, 4.04, 3.91, 3.56, 3.65, 4.19, 3.66, 3.92, 4.13, 3.72, 3.93, 3.78, 3.59, 4.0, 4.03, 4.2, 3.71, 4.09, 4.16, 3.8, 4.28, 4.13, 4.13, 3.91, 3.87, 3.9, 4.21, 3.9, 3.98, 3.98, 4.01, 3.9, 4.0, 3.72, 3.86, 4.07, 4.27, 3.83, 4.26, 4.11, 4.03, 4.04, 3.92, 4.1, 4.14, 3.74, 4.11, 4.17, 3.89, 4.06, 4.05, 3.9, 4.28, 4.0, 4.11, 4.11, 3.96, 3.78, 3.77, 4.1, 3.53, 3.8, 4.1, 3.98, 4.31, 3.79, 3.94, 4.23, 4.01, 3.99, 3.93, 4.14, 4.19, 3.95, 3.66, 3.66, 4.03, 3.83, 4.29, 3.88, 4.67, 4.0, 4.01, 3.8, 3.68, 4.09, 4.03, 4.29, 3.9, 3.92, 3.89, 4.17, 3.81, 4.05, 4.37, 4.05, 3.96, 3.84, 4.16, 4.33, 3.78, 3.81, 4.38, 4.1, 3.79, 4.25, 4.12, 4.09, 4.0, 3.96, 3.58, 4.16, 4.07, 4.23, 3.76, 3.96, 4.07, 3.87, 4.25, 3.82, 3.9, 3.9, 4.2, 3.67, 3.98, 3.91, 3.88, 3.87, 4.04, 3.89, 3.86]；假设 μ₀=4
分析过程
H₀: μ=4，H₁: μ≠4。
x̄ = 3.9998，s = 0.1849，SE = s/√n = 0.1849/√734 = 0.0068；
t = (x̄−μ₀)/SE = (3.9998−4)/0.0068 = -0.0272；df = n−1 = 733；双侧 p = 0.9783。
x̄ = 3.9998，s = 0.1849，SE = s/√n = 0.1849/√734 = 0.0068；
t = (x̄−μ₀)/SE = (3.9998−4)/0.0068 = -0.0272；df = n−1 = 733；双侧 p = 0.9783。
计算结果
t = -0.0272，df = 733，p = 0.9783
在 α=0.05 水平下，p=0.9783 ≥ 0.05，不拒绝原假设 H₀。评估满意度是否达到预期。
案例 14. 零件尺寸是否 50mm
原始数据
样本 n=469：[50.16, 49.77, 50.22, 49.9, 49.67, 50.08, 50.28, 50.36, 49.55, 50.06, 49.44, 50.14, 49.61, 49.85, 49.73, 50.12, 49.91, 50.14, 50.01, 49.98, 50.11, 50.3, 49.84, 49.73, 49.93, 50.08, 50.01, 49.95, 49.81, 50.24, 50.0, 49.95, 49.93, 50.4, 49.81, 49.65, 49.99, 49.92, 50.05, 49.6, 50.03, 49.76, 50.2, 49.89, 50.13, 49.91, 49.62, 49.3, 49.98, 50.14, 49.86, 50.03, 50.04, 49.71, 50.32, 50.24, 49.54, 50.1, 50.01, 50.01, 50.35, 49.5, 49.98, 50.36, 50.3, 50.17, 50.13, 49.92, 49.84, 50.21, 49.78, 49.87, 49.89, 50.35, 49.55, 49.87, 50.44, 49.64, 50.2, 49.56, 50.13, 49.73, 49.39, 49.7, 49.82, 49.74, 50.27, 49.68, 49.88, 49.82, 50.23, 50.09, 49.81, 49.59, 49.65, 50.18, 50.31, 49.73, 49.93, 49.86, 50.3, 49.77, 50.45, 50.33, 50.07, 50.21, 50.12, 49.8, 49.68, 49.79, 49.85, 50.35, 50.05, 49.69, 50.03, 50.01, 49.73, 50.06, 50.27, 50.03, 49.99, 49.58, 50.12, 50.17, 50.2, 49.97, 50.05, 49.71, 49.48, 49.5, 50.3, 50.04, 50.12, 49.93, 49.59, 49.98, 49.88, 49.93, 49.82, 50.0, 50.38, 50.05, 49.99, 50.21, 50.26, 50.63, 50.21, 49.4, 50.37, 49.61, 49.65, 50.58, 49.99, 49.92, 49.96, 50.18, 49.92, 49.98, 49.4, 50.23, 49.66, 49.93, 50.19, 50.05, 49.96, 49.88, 49.85, 50.48, 49.81, 50.05, 49.9, 49.97, 50.28, 50.12, 50.12, 50.18, 50.01, 49.81, 50.21, 50.09, 49.95, 50.03, 49.68, 49.99, 50.1, 49.88, 50.65, 50.15, 50.14, 50.15, 50.22, 50.06, 50.31, 50.51, 49.61, 49.75, 50.14, 49.79, 50.05, 49.43, 50.11, 50.33, 49.68, 50.23, 49.39, 49.89, 49.84, 50.06, 49.78, 50.62, 49.93, 49.94, 49.82, 50.01, 49.75, 49.89, 50.21, 50.0, 49.89, 49.62, 49.8, 50.41, 49.92, 49.39, 50.19, 50.03, 49.92, 49.82, 50.04, 49.85, 50.18, 49.97, 50.11, 50.08, 49.84, 50.04, 50.14, 49.98, 49.98, 49.84, 49.99, 49.9, 49.68, 50.33, 49.81, 50.19, 50.08, 49.74, 50.33, 50.17, 50.01, 49.8, 50.18, 50.09, 50.32, 49.87, 50.23, 50.12, 49.94, 50.01, 50.16, 49.68, 49.64, 49.91, 50.12, 49.91, 49.82, 49.64, 49.86, 49.81, 50.36, 49.91, 49.98, 49.77, 49.65, 50.51, 49.93, 49.93, 49.85, 49.89, 49.91, 50.02, 49.45, 49.9, 49.86, 50.57, 49.83, 49.55, 49.88, 49.47, 49.7, 50.27, 49.86, 50.05, 49.98, 49.95, 49.98, 49.62, 50.33, 49.91, 49.91, 49.99, 50.22, 49.76, 50.31, 50.14, 49.53, 50.17, 50.17, 49.66, 49.82, 49.67, 50.06, 50.08, 49.89, 50.04, 50.09, 49.33, 50.04, 50.32, 50.67, 50.21, 50.1, 50.09, 50.14, 50.17, 50.56, 50.19, 50.19, 49.97, 50.11, 49.65, 50.3, 50.36, 50.22, 50.0, 49.73, 50.06, 50.16, 49.73, 50.2, 49.75, 49.9, 49.85, 50.06, 49.66, 49.78, 50.02, 49.98, 49.77, 50.22, 49.99, 49.36, 49.68, 50.26, 50.01, 50.09, 49.79, 49.94, 49.74, 50.17, 50.46, 50.05, 50.29, 49.96, 49.66, 50.42, 50.44, 49.98, 50.24, 49.93, 49.7, 50.04, 49.9, 49.91, 50.67, 49.8, 50.01, 49.96, 50.07, 49.99, 50.14, 50.49, 50.41, 50.32, 50.41, 50.15, 49.79, 49.93, 50.25, 49.89, 50.1, 50.32, 50.21, 50.38, 50.41, 50.03, 50.14, 50.13, 50.19, 49.93, 50.07, 49.65, 50.29, 49.97, 49.64, 49.87, 49.65, 49.64, 50.07, 49.95, 50.26, 49.92, 49.94, 50.23, 49.97, 49.84, 49.81, 50.34, 49.74, 49.88, 49.52, 50.11, 50.18, 49.92, 50.17, 50.17, 49.46, 49.64, 49.96, 50.2, 49.98, 50.02, 49.99, 50.12, 49.7, 50.17, 49.74, 50.23, 50.54, 49.92, 50.14, 49.19, 50.04, 49.8, 50.13, 49.64, 49.7, 50.2, 49.82, 50.19, 50.06, 50.33, 49.79, 50.18, 50.31, 50.1, 49.94, 50.03, 49.71, 49.91, 49.95, 50.4, 49.9, 49.57, 49.92, 49.8, 50.09, 49.98]；假设 μ₀=50
分析过程
H₀: μ=50，H₁: μ≠50。
x̄ = 49.9865，s = 0.2542，SE = s/√n = 0.2542/√469 = 0.0117；
t = (x̄−μ₀)/SE = (49.9865−50)/0.0117 = -1.1541；df = n−1 = 468；双侧 p = 0.2490。
x̄ = 49.9865，s = 0.2542，SE = s/√n = 0.2542/√469 = 0.0117；
t = (x̄−μ₀)/SE = (49.9865−50)/0.0117 = -1.1541；df = n−1 = 468；双侧 p = 0.2490。
计算结果
t = -1.1541，df = 468，p = 0.2490
在 α=0.05 水平下，p=0.2490 ≥ 0.05，不拒绝原假设 H₀。检验加工精度是否达标。
案例 15. 日均销售额是否 5000
原始数据
样本 n=617：[4742.4, 4814.62, 5119.99, 5018.86, 5217.89, 5035.41, 5107.59, 5002.47, 5077.0, 5039.29, 4942.46, 4982.72, 4876.92, 5023.53, 5221.68, 4961.81, 5152.95, 5060.67, 4756.09, 5005.34, 5058.57, 5183.4, 4978.66, 5075.78, 5005.28, 4998.25, 4857.87, 5055.92, 5009.95, 4908.78, 4963.9, 5011.54, 4909.27, 4949.34, 5125.38, 4821.38, 4937.32, 5162.73, 4941.86, 5070.42, 5056.8, 5038.75, 5105.32, 4909.13, 5179.29, 5032.09, 5075.16, 4998.88, 5022.24, 5001.54, 4865.09, 4971.77, 5106.89, 4979.37, 5030.04, 5250.14, 5032.84, 4860.25, 5080.83, 4934.19, 5034.59, 5117.29, 4931.47, 5005.11, 4838.89, 4950.43, 4869.51, 5014.67, 4913.52, 4843.42, 4974.7, 5029.99, 4826.8, 5018.23, 4991.91, 4880.15, 4961.66, 5021.45, 5113.1, 5047.7, 5118.51, 4989.36, 4938.99, 5120.37, 5044.04, 5122.9, 4818.49, 5133.11, 4763.03, 5106.81, 5050.94, 5222.58, 5072.55, 4822.28, 5089.13, 5028.25, 5104.75, 5308.36, 4996.68, 4971.71, 4872.09, 4956.24, 4964.5, 4928.28, 5085.4, 4824.76, 5097.89, 4856.06, 5066.46, 5197.08, 4667.63, 5047.83, 5042.39, 5014.22, 4846.29, 4990.72, 5143.63, 5192.82, 4939.99, 4995.08, 5041.2, 4885.58, 5059.79, 4857.72, 5036.36, 5181.94, 4933.69, 5266.45, 5043.02, 5035.99, 5065.01, 5077.93, 5066.1, 5133.11, 5013.52, 4878.65, 5120.15, 4986.98, 5003.55, 5126.12, 5199.03, 4967.36, 4968.4, 4984.8, 4850.97, 4948.67, 4840.89, 4964.58, 5153.94, 4895.98, 5103.26, 5018.96, 4856.79, 4999.38, 5078.15, 5053.78, 5048.19, 4918.12, 4923.19, 5024.99, 4848.27, 5013.37, 5019.29, 4957.32, 4980.1, 5142.69, 5130.52, 5008.13, 4813.09, 4975.46, 5198.57, 4983.71, 4693.81, 4983.51, 5009.47, 4904.93, 5115.57, 5013.31, 5042.23, 5149.22, 5101.37, 4928.37, 4748.37, 5126.85, 4973.99, 4679.48, 5056.79, 5049.93, 5126.42, 5052.01, 5014.94, 4933.68, 4878.72, 5144.94, 4981.38, 5080.79, 5021.08, 5127.19, 4953.74, 5124.25, 5027.37, 4955.23, 4963.21, 5029.91, 5119.77, 5080.1, 5009.6, 5163.2, 5117.4, 5034.11, 4915.14, 5190.75, 4860.69, 5004.93, 4740.49, 5092.39, 4999.65, 5046.32, 4837.18, 5087.72, 5026.05, 4857.75, 5037.81, 5153.39, 4999.36, 5011.27, 4984.79, 5003.39, 5051.26, 5084.98, 4936.71, 4949.83, 4678.88, 4985.89, 4896.25, 5105.36, 5274.63, 4899.92, 5001.73, 5324.85, 5138.16, 5095.04, 4882.81, 5090.72, 4842.61, 5040.52, 4904.21, 4835.67, 4832.67, 5034.56, 5004.47, 5056.46, 5148.54, 5206.45, 5081.22, 4993.97, 5185.83, 4972.37, 5076.35, 4904.18, 5024.36, 5001.12, 4993.66, 4994.91, 4954.95, 5193.26, 4965.79, 4845.88, 4785.18, 4880.74, 5018.58, 4859.51, 4829.0, 5159.55, 4876.89, 5229.86, 5316.4, 4721.07, 5079.77, 4944.69, 5022.98, 5171.84, 5034.12, 5099.12, 5036.53, 5004.97, 4956.27, 4981.16, 5090.11, 4847.03, 4818.33, 4836.83, 5028.78, 4907.26, 5136.01, 4934.93, 5114.83, 4871.82, 4990.93, 5056.42, 4760.18, 5246.99, 4926.44, 5020.1, 4797.04, 4967.78, 5157.52, 5085.81, 4976.23, 5143.93, 4957.04, 5136.02, 4864.88, 4865.1, 4934.93, 5118.23, 5070.6, 5247.48, 4780.86, 4959.03, 5022.25, 4968.04, 4990.58, 4919.88, 4873.34, 4762.1, 5014.03, 4812.14, 5071.74, 4998.53, 4746.31, 5043.32, 5250.33, 5174.28, 4997.58, 4743.33, 4942.51, 5196.6, 5049.95, 5039.75, 5193.73, 4865.16, 5060.96, 4908.62, 5054.68, 5008.45, 5024.86, 5039.29, 4898.61, 5059.49, 4800.88, 4881.4, 4916.4, 4982.44, 5457.21, 4833.02, 4974.29, 5114.31, 4868.55, 4633.43, 5005.89, 5016.44, 4974.4, 5084.94, 5216.84, 5101.73, 4826.46, 4750.45, 5325.21, 4967.22, 5300.78, 5057.0, 4958.2, 5076.94, 5055.61, 5066.84, 4862.39, 5103.87, 4928.58, 5139.66, 4854.14, 4879.5, 5109.63, 4635.98, 5104.0, 5093.45, 5020.5, 5010.57, 4717.99, 5066.99, 4883.34, 5303.69, 4874.39, 4915.2, 5039.28, 4888.73, 5209.65, 4864.08, 5111.55, 4984.3, 5033.75, 4820.39, 5032.03, 4944.97, 5167.21, 4987.58, 5115.56, 5007.98, 4858.63, 5001.79, 4922.67, 5104.04, 4936.77, 5187.58, 4945.04, 4858.63, 5123.7, 5012.45, 4957.72, 4768.53, 5170.01, 4973.23, 4810.9, 5260.69, 4912.19, 4826.09, 4990.88, 5050.07, 5052.19, 5000.13, 5159.62, 5045.83, 5151.85, 4953.39, 4755.91, 4789.82, 5111.36, 4920.47, 5131.78, 5121.75, 5006.15, 4960.22, 4929.52, 4763.84, 4748.48, 5129.38, 4791.09, 4931.51, 4985.91, 4910.18, 4957.48, 5140.23, 5000.39, 4940.56, 4917.92, 5172.03, 5067.51, 4896.05, 4972.58, 5023.7, 4917.08, 4956.96, 5166.23, 5087.17, 5197.6, 4941.89, 5060.41, 5162.86, 4925.29, 5039.32, 5077.06, 4952.2, 5048.59, 4819.78, 5112.21, 5251.31, 4981.24, 4764.84, 4945.01, 5032.8, 5068.47, 5113.54, 4893.72, 5024.39, 5035.27, 4903.24, 4885.83, 5017.02, 4965.06, 4911.93, 5179.68, 4865.33, 4899.7, 5084.49, 4978.13, 5085.45, 5074.35, 4997.01, 4928.29, 5232.24, 4875.23, 5158.16, 4972.9, 5079.65, 4844.64, 5000.51, 4926.77, 5067.93, 4839.76, 5040.54, 4839.43, 4841.7, 4995.53, 5009.74, 5226.16, 4962.09, 5046.58, 4869.01, 5088.15, 5161.9, 5065.57, 4857.91, 4996.87, 5111.1, 4921.23, 4932.24, 5107.12, 5092.06, 4927.19, 4918.95, 4783.87, 5200.78, 5051.49, 4930.96, 4901.19, 4953.87, 4973.14, 4870.05, 4922.39, 5082.63, 5044.29, 4880.78, 4942.15, 5127.37, 4888.09, 5171.81, 4912.99, 5118.63, 4999.16, 4985.47, 4989.16, 5040.26, 4986.52, 4838.67, 4994.4, 4960.13, 5010.81, 4739.83, 5185.51, 4933.87, 4965.24, 5004.73, 4983.43, 4918.28, 5132.16, 5137.91, 5091.94, 4928.36, 5208.26, 5155.81, 4868.22, 4887.25, 4821.37, 4913.61, 5066.87, 5001.21, 4990.23, 5110.45, 4948.61, 4952.56, 5079.67, 5086.21, 5097.2, 4986.83, 4827.37, 5126.96, 5064.67, 5003.38, 5144.52, 5061.25, 4875.26, 5004.29, 4854.55, 4700.82, 5150.21, 4998.89, 4874.0, 5028.04, 5003.96, 5022.9, 5030.79, 5044.29, 4744.51, 4939.74, 5096.42, 4934.35, 4735.82, 4879.11, 4779.25, 4776.81, 4981.48, 5113.17, 5180.58, 4706.22, 4959.09, 5128.73, 4951.38]；假设 μ₀=5000
分析过程
H₀: μ=5000，H₁: μ≠5000。
x̄ = 4999.1693，s = 123.7330，SE = s/√n = 123.7330/√617 = 4.9813；
t = (x̄−μ₀)/SE = (4999.1693−5000)/4.9813 = -0.1668；df = n−1 = 616；双侧 p = 0.8676。
x̄ = 4999.1693，s = 123.7330，SE = s/√n = 123.7330/√617 = 4.9813；
t = (x̄−μ₀)/SE = (4999.1693−5000)/4.9813 = -0.1668；df = n−1 = 616；双侧 p = 0.8676。
计算结果
t = -0.1668，df = 616，p = 0.8676
在 α=0.05 水平下，p=0.8676 ≥ 0.05，不拒绝原假设 H₀。判断营业额是否稳定在目标。
独立样本 t 检验
案例 16. 男女身高差异
原始数据
组1 n=314：[171.43, 169.87, 170.33, 174.57, 171.36, 170.96, 177.83, 172.84, 172.63, 172.23, 176.03, 172.24, 174.0, 173.97, 171.39, 174.69, 172.06, 170.65, 173.85, 170.74, 171.85, 170.34, 175.07, 171.43, 167.46, 173.73, 174.71, 174.65, 174.03, 168.19, 175.08, 168.36, 169.7, 171.38, 175.27, 172.15, 169.02, 173.08, 169.17, 168.54, 170.28, 170.7, 170.42, 168.47, 172.52, 169.92, 172.32, 177.04, 174.42, 175.14, 172.49, 168.1, 171.44, 171.51, 171.41, 174.26, 170.96, 175.13, 173.84, 169.64, 164.95, 173.94, 173.55, 170.03, 173.26, 170.24, 174.26, 167.62, 172.99, 171.66, 173.22, 172.53, 174.29, 176.57, 173.7, 169.29, 168.87, 172.23, 171.48, 171.53, 173.8, 176.44, 174.57, 172.22, 173.92, 168.6, 168.13, 172.55, 172.15, 176.53, 171.66, 172.33, 173.54, 174.81, 170.72, 173.62, 174.95, 174.5, 172.96, 178.55, 167.47, 173.31, 175.08, 169.86, 168.79, 176.55, 174.1, 169.41, 167.71, 170.54, 172.37, 177.42, 172.22, 174.02, 171.75, 170.92, 174.32, 171.0, 172.51, 173.04, 172.29, 173.0, 170.46, 172.25, 170.89, 171.47, 173.65, 175.94, 177.55, 172.89, 170.97, 168.4, 174.81, 174.62, 172.6, 172.14, 170.93, 174.03, 169.3, 167.85, 171.73, 171.5, 172.73, 168.5, 168.28, 170.6, 169.07, 173.1, 170.12, 167.05, 171.88, 167.92, 169.69, 178.01, 174.31, 176.96, 172.57, 170.91, 172.51, 170.07, 169.42, 172.59, 169.88, 171.73, 168.68, 173.71, 169.44, 173.12, 175.24, 173.61, 172.15, 171.01, 170.6, 171.28, 173.22, 171.59, 169.41, 170.38, 176.36, 170.07, 174.63, 172.84, 172.09, 171.93, 175.26, 172.07, 177.67, 171.6, 178.68, 172.92, 170.74, 175.14, 173.42, 168.92, 175.78, 172.74, 172.0, 173.43, 176.58, 176.08, 171.72, 171.94, 171.46, 169.99, 172.76, 170.66, 169.11, 173.73, 174.56, 171.9, 175.68, 170.0, 169.4, 169.25, 169.12, 177.36, 174.73, 169.4, 170.55, 174.04, 173.99, 167.96, 167.47, 174.55, 172.2, 173.61, 173.83, 172.1, 174.15, 168.42, 174.05, 173.22, 170.11, 171.31, 169.3, 173.64, 173.61, 166.37, 172.54, 171.81, 174.12, 165.0, 170.67, 174.95, 174.31, 174.38, 172.61, 173.97, 165.5, 174.82, 178.37, 174.56, 171.21, 171.67, 172.87, 170.06, 175.33, 176.29, 172.42, 172.01, 168.51, 175.73, 165.64, 174.11, 173.44, 175.37, 169.02, 166.57, 174.83, 171.86, 170.3, 169.29, 168.57, 171.27, 169.58, 170.27, 171.67, 167.57, 171.63, 173.51, 177.61, 169.87, 169.76, 174.23, 169.36, 168.4, 172.37, 169.43, 168.0, 170.1, 173.35, 177.4, 170.02, 174.88, 169.47, 173.47, 173.67, 172.3, 172.14, 167.03, 163.58, 169.19, 171.4, 171.89, 169.17, 172.51, 168.65, 175.09, 174.5, 169.93, 177.84, 176.69, 166.25, 172.6]；
组2 n=661：[161.08, 161.86, 163.09, 161.22, 161.21, 161.05, 162.65, 164.22, 161.13, 158.52, 162.2, 162.38, 161.04, 162.3, 156.79, 166.9, 160.93, 160.63, 164.12, 157.47, 161.74, 162.36, 160.57, 160.87, 157.42, 163.24, 159.73, 159.26, 162.45, 162.25, 163.4, 162.98, 163.35, 162.76, 158.85, 162.08, 158.4, 163.63, 161.33, 163.18, 159.25, 164.83, 161.49, 163.15, 165.04, 157.98, 164.47, 160.93, 161.72, 162.97, 161.81, 161.46, 162.3, 162.85, 162.82, 161.21, 157.23, 161.26, 161.56, 160.41, 161.43, 160.46, 161.56, 162.52, 161.76, 161.44, 157.05, 161.16, 159.87, 161.73, 163.25, 163.11, 159.22, 162.88, 155.89, 162.09, 166.31, 159.57, 160.95, 159.9, 161.22, 158.23, 160.91, 158.92, 160.5, 159.58, 161.79, 161.92, 158.72, 161.14, 160.94, 164.6, 159.56, 160.54, 162.65, 158.48, 161.88, 164.49, 163.47, 155.15, 164.04, 160.58, 158.49, 159.88, 164.47, 159.15, 163.06, 160.58, 162.54, 164.32, 157.31, 161.35, 163.32, 161.12, 160.54, 162.33, 165.39, 160.65, 163.19, 163.37, 162.35, 161.14, 161.61, 158.45, 159.76, 162.4, 161.74, 159.51, 163.03, 159.74, 159.53, 157.67, 160.57, 158.07, 161.52, 161.64, 160.95, 161.01, 161.79, 161.96, 159.64, 161.61, 161.71, 159.72, 163.56, 163.32, 164.83, 160.71, 162.36, 161.42, 157.53, 157.97, 161.61, 158.25, 157.5, 163.86, 160.3, 160.86, 161.05, 163.27, 162.71, 163.17, 160.07, 162.96, 159.66, 160.81, 165.61, 164.04, 158.42, 158.5, 161.17, 157.55, 164.37, 159.42, 162.08, 156.69, 163.99, 157.82, 159.61, 161.83, 160.34, 163.88, 162.68, 164.3, 162.19, 158.29, 160.69, 158.84, 160.61, 164.85, 161.23, 160.42, 163.41, 161.06, 164.95, 160.78, 162.62, 159.6, 162.31, 159.0, 158.47, 162.61, 158.78, 159.49, 161.63, 160.79, 161.3, 157.45, 162.72, 164.83, 164.89, 164.53, 158.29, 160.61, 163.61, 159.95, 158.49, 164.03, 160.84, 162.75, 164.71, 159.63, 162.11, 161.75, 157.16, 162.08, 158.18, 160.69, 161.35, 160.37, 162.67, 159.52, 165.31, 159.54, 162.45, 158.9, 164.25, 159.37, 163.17, 164.41, 163.08, 159.82, 160.13, 159.43, 162.08, 159.82, 161.51, 161.24, 165.3, 162.61, 159.94, 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161.73, 165.22, 164.0, 160.75, 162.66, 161.32, 158.28, 158.66, 160.9, 161.66, 164.7, 162.5, 161.74, 164.13, 156.18, 163.89, 159.23, 158.06, 159.22, 162.76, 160.37, 163.93, 163.04, 157.67, 157.29, 159.86, 163.17, 163.49, 162.15, 161.57, 158.76, 157.53, 158.03, 160.61, 162.9, 163.24, 161.09, 163.23, 162.07, 159.83, 162.75, 160.74, 163.29, 159.51, 162.48, 161.09, 164.8, 161.26, 160.01, 157.91, 159.14, 162.35, 158.96, 163.01, 158.12, 160.25, 160.85, 159.74, 163.76, 165.95, 161.18, 163.86, 162.89, 158.92, 160.02, 164.88, 161.64, 163.5, 160.27, 160.64, 159.31, 163.27, 161.23, 159.39, 158.88, 160.9, 161.61, 161.05, 160.17, 165.09, 163.24, 159.81, 160.18, 157.25, 161.21, 161.45, 158.94, 158.67, 161.29, 159.74, 160.34, 160.04, 162.08, 163.17, 159.69, 166.84, 162.18, 158.48, 161.05, 160.27, 161.34, 160.62, 154.97, 163.99, 162.51, 162.14, 164.09, 162.52, 158.65, 161.74, 163.31, 158.81, 162.61, 163.42, 159.35, 161.16, 162.24, 165.12, 161.57, 162.78, 161.02, 161.42, 158.73, 163.36, 161.18, 159.71, 164.82, 159.61, 164.26, 165.84, 159.84, 164.4, 158.22, 160.38, 161.77, 160.91, 160.71, 163.39, 161.07, 158.19, 159.77, 161.81, 165.79, 160.85, 156.88, 161.61, 158.93, 164.46, 160.58, 162.33, 165.64, 160.39, 164.09, 161.05, 161.71, 166.0, 160.41, 157.72, 161.36, 154.67, 165.28, 160.78, 158.13, 162.24, 160.28, 163.8, 162.88, 163.06, 161.73, 160.23, 161.26, 159.13, 160.22, 162.2, 161.33, 160.5, 159.56, 159.27, 160.58, 161.5, 162.3, 159.89, 160.06, 162.77, 159.1, 161.23, 159.38, 161.42, 162.28, 161.13, 160.13, 162.72, 159.45, 159.0, 161.62, 162.52, 159.41, 161.38, 163.65, 161.73, 165.31, 157.4, 165.37, 161.36, 162.71, 163.01, 162.23, 160.09, 159.28, 159.87, 162.36, 161.7, 158.05, 159.54, 160.19, 159.26, 162.85, 162.92, 161.2, 162.14, 159.01, 162.23, 163.25, 162.19, 161.71, 163.81, 155.91, 159.92, 163.99, 165.24, 162.56, 161.13, 161.09, 161.78, 160.74, 160.79, 162.33, 161.16, 162.36, 158.52, 160.73, 161.01, 162.0, 161.33, 159.18, 162.21, 159.9, 163.06, 163.91, 164.17, 157.65, 162.47, 160.85, 166.78, 162.84, 160.31, 164.33, 160.38, 162.67, 164.45, 158.45, 160.41, 159.59, 159.32, 157.87, 159.28, 158.44, 161.61, 165.17, 166.28, 162.46, 157.63, 160.83, 160.51, 160.33, 159.67, 163.01, 162.16, 161.08, 162.84, 161.32, 157.97, 163.45, 161.39]
组2 n=661：[161.08, 161.86, 163.09, 161.22, 161.21, 161.05, 162.65, 164.22, 161.13, 158.52, 162.2, 162.38, 161.04, 162.3, 156.79, 166.9, 160.93, 160.63, 164.12, 157.47, 161.74, 162.36, 160.57, 160.87, 157.42, 163.24, 159.73, 159.26, 162.45, 162.25, 163.4, 162.98, 163.35, 162.76, 158.85, 162.08, 158.4, 163.63, 161.33, 163.18, 159.25, 164.83, 161.49, 163.15, 165.04, 157.98, 164.47, 160.93, 161.72, 162.97, 161.81, 161.46, 162.3, 162.85, 162.82, 161.21, 157.23, 161.26, 161.56, 160.41, 161.43, 160.46, 161.56, 162.52, 161.76, 161.44, 157.05, 161.16, 159.87, 161.73, 163.25, 163.11, 159.22, 162.88, 155.89, 162.09, 166.31, 159.57, 160.95, 159.9, 161.22, 158.23, 160.91, 158.92, 160.5, 159.58, 161.79, 161.92, 158.72, 161.14, 160.94, 164.6, 159.56, 160.54, 162.65, 158.48, 161.88, 164.49, 163.47, 155.15, 164.04, 160.58, 158.49, 159.88, 164.47, 159.15, 163.06, 160.58, 162.54, 164.32, 157.31, 161.35, 163.32, 161.12, 160.54, 162.33, 165.39, 160.65, 163.19, 163.37, 162.35, 161.14, 161.61, 158.45, 159.76, 162.4, 161.74, 159.51, 163.03, 159.74, 159.53, 157.67, 160.57, 158.07, 161.52, 161.64, 160.95, 161.01, 161.79, 161.96, 159.64, 161.61, 161.71, 159.72, 163.56, 163.32, 164.83, 160.71, 162.36, 161.42, 157.53, 157.97, 161.61, 158.25, 157.5, 163.86, 160.3, 160.86, 161.05, 163.27, 162.71, 163.17, 160.07, 162.96, 159.66, 160.81, 165.61, 164.04, 158.42, 158.5, 161.17, 157.55, 164.37, 159.42, 162.08, 156.69, 163.99, 157.82, 159.61, 161.83, 160.34, 163.88, 162.68, 164.3, 162.19, 158.29, 160.69, 158.84, 160.61, 164.85, 161.23, 160.42, 163.41, 161.06, 164.95, 160.78, 162.62, 159.6, 162.31, 159.0, 158.47, 162.61, 158.78, 159.49, 161.63, 160.79, 161.3, 157.45, 162.72, 164.83, 164.89, 164.53, 158.29, 160.61, 163.61, 159.95, 158.49, 164.03, 160.84, 162.75, 164.71, 159.63, 162.11, 161.75, 157.16, 162.08, 158.18, 160.69, 161.35, 160.37, 162.67, 159.52, 165.31, 159.54, 162.45, 158.9, 164.25, 159.37, 163.17, 164.41, 163.08, 159.82, 160.13, 159.43, 162.08, 159.82, 161.51, 161.24, 165.3, 162.61, 159.94, 161.56, 162.63, 160.29, 163.03, 162.63, 160.99, 160.74, 160.25, 160.6, 163.9, 162.37, 165.91, 158.09, 160.77, 161.68, 162.09, 162.18, 164.15, 160.26, 161.99, 164.09, 164.12, 159.57, 159.82, 161.45, 159.72, 163.58, 162.5, 166.26, 160.93, 164.8, 159.46, 162.1, 160.95, 159.88, 160.07, 162.52, 156.98, 157.76, 162.42, 162.15, 161.68, 162.22, 164.66, 158.1, 161.24, 157.6, 159.43, 158.03, 163.22, 158.36, 165.67, 159.37, 159.48, 163.49, 161.93, 160.12, 161.25, 160.29, 162.44, 161.01, 158.91, 161.17, 161.05, 163.26, 163.0, 161.02, 160.37, 160.92, 163.42, 163.34, 162.1, 161.51, 157.7, 158.13, 161.68, 157.14, 157.77, 168.1, 161.21, 160.08, 162.35, 156.76, 163.94, 164.05, 165.75, 163.4, 159.79, 162.08, 160.18, 159.82, 165.38, 162.11, 161.48, 160.79, 159.92, 161.03, 161.09, 156.41, 165.79, 161.59, 164.23, 160.87, 164.45, 161.53, 158.12, 161.75, 155.81, 160.97, 159.88, 161.63, 163.82, 160.6, 158.31, 161.31, 167.37, 163.79, 159.84, 161.78, 159.3, 169.41, 161.53, 157.58, 166.14, 158.31, 163.19, 161.73, 165.22, 164.0, 160.75, 162.66, 161.32, 158.28, 158.66, 160.9, 161.66, 164.7, 162.5, 161.74, 164.13, 156.18, 163.89, 159.23, 158.06, 159.22, 162.76, 160.37, 163.93, 163.04, 157.67, 157.29, 159.86, 163.17, 163.49, 162.15, 161.57, 158.76, 157.53, 158.03, 160.61, 162.9, 163.24, 161.09, 163.23, 162.07, 159.83, 162.75, 160.74, 163.29, 159.51, 162.48, 161.09, 164.8, 161.26, 160.01, 157.91, 159.14, 162.35, 158.96, 163.01, 158.12, 160.25, 160.85, 159.74, 163.76, 165.95, 161.18, 163.86, 162.89, 158.92, 160.02, 164.88, 161.64, 163.5, 160.27, 160.64, 159.31, 163.27, 161.23, 159.39, 158.88, 160.9, 161.61, 161.05, 160.17, 165.09, 163.24, 159.81, 160.18, 157.25, 161.21, 161.45, 158.94, 158.67, 161.29, 159.74, 160.34, 160.04, 162.08, 163.17, 159.69, 166.84, 162.18, 158.48, 161.05, 160.27, 161.34, 160.62, 154.97, 163.99, 162.51, 162.14, 164.09, 162.52, 158.65, 161.74, 163.31, 158.81, 162.61, 163.42, 159.35, 161.16, 162.24, 165.12, 161.57, 162.78, 161.02, 161.42, 158.73, 163.36, 161.18, 159.71, 164.82, 159.61, 164.26, 165.84, 159.84, 164.4, 158.22, 160.38, 161.77, 160.91, 160.71, 163.39, 161.07, 158.19, 159.77, 161.81, 165.79, 160.85, 156.88, 161.61, 158.93, 164.46, 160.58, 162.33, 165.64, 160.39, 164.09, 161.05, 161.71, 166.0, 160.41, 157.72, 161.36, 154.67, 165.28, 160.78, 158.13, 162.24, 160.28, 163.8, 162.88, 163.06, 161.73, 160.23, 161.26, 159.13, 160.22, 162.2, 161.33, 160.5, 159.56, 159.27, 160.58, 161.5, 162.3, 159.89, 160.06, 162.77, 159.1, 161.23, 159.38, 161.42, 162.28, 161.13, 160.13, 162.72, 159.45, 159.0, 161.62, 162.52, 159.41, 161.38, 163.65, 161.73, 165.31, 157.4, 165.37, 161.36, 162.71, 163.01, 162.23, 160.09, 159.28, 159.87, 162.36, 161.7, 158.05, 159.54, 160.19, 159.26, 162.85, 162.92, 161.2, 162.14, 159.01, 162.23, 163.25, 162.19, 161.71, 163.81, 155.91, 159.92, 163.99, 165.24, 162.56, 161.13, 161.09, 161.78, 160.74, 160.79, 162.33, 161.16, 162.36, 158.52, 160.73, 161.01, 162.0, 161.33, 159.18, 162.21, 159.9, 163.06, 163.91, 164.17, 157.65, 162.47, 160.85, 166.78, 162.84, 160.31, 164.33, 160.38, 162.67, 164.45, 158.45, 160.41, 159.59, 159.32, 157.87, 159.28, 158.44, 161.61, 165.17, 166.28, 162.46, 157.63, 160.83, 160.51, 160.33, 159.67, 163.01, 162.16, 161.08, 162.84, 161.32, 157.97, 163.45, 161.39]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=172.0738, s₁=2.7185；x̄₂=161.3273, s₂=2.2029；
SE=√(s₁²/n₁+s₂²/n₂)=0.1757；t=(x̄₁−x̄₂)/SE=61.1579；
df(Welch)=514.96；双侧 p = 2.659e-238。
x̄₁=172.0738, s₁=2.7185；x̄₂=161.3273, s₂=2.2029；
SE=√(s₁²/n₁+s₂²/n₂)=0.1757；t=(x̄₁−x̄₂)/SE=61.1579；
df(Welch)=514.96；双侧 p = 2.659e-238。
计算结果
t = 61.1579，df = 514.96，p = 2.659e-238
在 α=0.05 水平下，p=2.659e-238 < 0.05，拒绝原假设 H₀。比较男女平均身高是否有显著差异。
案例 17. A/B 两版页面停留时长
原始数据
组1 n=742：[30.5, 24.47, 31.26, 35.0, 27.77, 32.46, 33.67, 34.79, 32.56, 33.89, 38.41, 37.77, 38.86, 35.59, 34.85, 37.49, 37.93, 36.68, 37.72, 37.33, 28.41, 32.09, 28.11, 36.22, 35.73, 28.82, 30.98, 38.78, 33.93, 36.7, 31.65, 31.81, 28.38, 30.17, 26.07, 34.1, 35.42, 36.25, 28.59, 30.53, 26.5, 34.28, 34.48, 36.08, 33.88, 27.11, 36.89, 30.8, 23.99, 28.64, 37.88, 32.25, 28.57, 30.15, 30.87, 38.21, 34.35, 39.38, 33.63, 33.55, 37.26, 39.29, 22.05, 33.21, 34.77, 31.18, 32.39, 35.2, 36.81, 31.44, 28.06, 34.98, 35.16, 32.91, 26.09, 39.25, 35.89, 33.11, 32.56, 30.73, 29.42, 29.11, 32.43, 32.56, 36.38, 28.8, 33.45, 30.6, 27.75, 29.45, 33.34, 36.67, 28.08, 32.92, 31.55, 25.82, 32.45, 32.8, 34.18, 30.13, 41.74, 26.84, 32.23, 35.15, 29.06, 38.16, 34.71, 28.19, 32.16, 33.78, 33.53, 33.3, 30.38, 33.41, 27.28, 31.95, 30.61, 26.96, 24.03, 36.97, 36.6, 32.56, 28.68, 35.66, 35.41, 28.24, 31.55, 32.57, 36.73, 39.73, 33.89, 34.52, 32.04, 35.27, 25.85, 33.2, 29.59, 33.19, 34.03, 29.28, 38.27, 33.02, 34.9, 33.35, 33.76, 34.29, 35.38, 35.72, 36.06, 31.85, 30.66, 35.9, 38.31, 36.55, 34.54, 32.05, 37.46, 30.8, 36.71, 35.82, 32.89, 34.1, 33.89, 34.01, 29.67, 33.18, 34.58, 27.85, 34.08, 29.6, 32.14, 32.32, 29.69, 32.63, 24.59, 35.65, 33.37, 31.6, 31.13, 35.33, 33.95, 33.79, 34.94, 30.43, 35.69, 34.67, 39.6, 34.53, 36.0, 28.74, 33.44, 36.22, 32.26, 31.63, 31.46, 30.4, 33.14, 32.87, 35.27, 32.69, 32.56, 29.95, 33.87, 37.79, 28.6, 28.63, 31.97, 36.08, 35.3, 35.83, 33.37, 34.57, 32.86, 33.78, 29.1, 35.55, 34.07, 30.53, 36.71, 35.92, 33.36, 32.9, 29.04, 30.63, 35.45, 31.99, 33.51, 27.9, 35.09, 35.98, 33.62, 27.91, 32.11, 34.64, 33.59, 34.99, 29.93, 36.21, 35.9, 31.86, 38.5, 28.72, 28.76, 35.07, 36.55, 33.85, 19.59, 31.2, 32.63, 33.48, 31.25, 30.96, 29.69, 31.82, 32.58, 34.09, 38.0, 36.78, 29.34, 40.92, 30.44, 39.32, 36.08, 30.39, 36.26, 30.91, 29.17, 32.6, 33.1, 32.21, 24.95, 28.73, 29.04, 26.51, 28.41, 35.11, 27.14, 24.03, 34.71, 35.01, 26.19, 32.91, 34.02, 34.28, 34.45, 29.83, 32.98, 30.02, 34.39, 29.74, 23.96, 34.74, 27.42, 36.16, 31.25, 31.79, 24.76, 34.61, 24.76, 38.53, 31.03, 33.79, 24.5, 36.15, 30.73, 30.83, 28.89, 32.87, 36.8, 23.77, 28.7, 35.28, 33.75, 26.95, 34.68, 34.88, 32.53, 39.61, 36.32, 29.71, 37.51, 33.3, 28.16, 33.17, 35.53, 29.66, 30.33, 33.85, 34.6, 31.11, 36.8, 33.21, 33.11, 32.83, 31.22, 37.99, 38.32, 31.42, 36.65, 34.11, 33.38, 30.44, 29.83, 29.92, 31.47, 32.75, 36.09, 30.81, 37.57, 36.25, 27.01, 32.67, 29.61, 26.34, 38.96, 29.49, 29.94, 38.34, 42.27, 38.89, 29.87, 35.61, 34.31, 36.81, 33.95, 27.23, 31.47, 35.86, 33.96, 29.69, 29.59, 36.18, 38.04, 32.26, 36.71, 31.05, 32.33, 34.09, 29.48, 35.67, 32.94, 28.45, 30.57, 34.24, 34.71, 28.71, 36.88, 30.7, 37.71, 30.96, 29.54, 29.53, 27.86, 31.51, 37.35, 36.21, 32.59, 36.73, 30.88, 34.23, 33.67, 30.03, 40.61, 34.09, 34.27, 31.11, 35.14, 34.24, 34.75, 30.51, 38.37, 30.99, 28.81, 29.1, 31.53, 29.11, 35.41, 28.89, 37.09, 34.79, 32.69, 34.1, 37.82, 35.23, 35.83, 34.02, 33.79, 29.29, 36.96, 34.05, 32.98, 31.62, 35.15, 32.88, 27.7, 33.89, 36.26, 34.09, 35.45, 36.44, 31.49, 29.87, 34.46, 33.1, 31.8, 36.5, 27.66, 31.38, 40.96, 25.11, 29.21, 34.62, 25.66, 37.21, 42.26, 28.88, 31.46, 36.98, 29.03, 39.3, 36.69, 29.78, 32.54, 27.76, 33.32, 35.17, 34.49, 35.82, 38.11, 33.39, 36.08, 31.81, 33.01, 38.51, 34.0, 36.42, 29.09, 30.68, 33.25, 33.42, 29.33, 32.89, 28.24, 40.01, 28.26, 37.12, 34.39, 23.71, 36.24, 32.79, 36.36, 35.14, 32.12, 35.61, 30.5, 35.16, 39.8, 36.44, 30.59, 31.95, 31.47, 29.05, 31.67, 40.64, 43.33, 25.51, 28.83, 31.26, 25.8, 35.4, 33.51, 25.7, 35.71, 35.56, 33.55, 28.66, 30.88, 40.45, 31.55, 31.55, 34.9, 31.11, 33.42, 38.16, 28.76, 35.57, 33.93, 27.09, 28.18, 36.67, 32.83, 33.05, 36.09, 30.99, 27.47, 32.7, 28.16, 35.9, 34.96, 36.36, 33.99, 35.67, 27.09, 35.59, 32.56, 33.29, 37.29, 30.97, 40.81, 34.95, 25.58, 30.91, 29.9, 31.15, 31.95, 31.01, 25.57, 38.47, 33.82, 34.95, 27.51, 32.65, 28.51, 32.89, 38.0, 34.63, 33.12, 22.13, 28.4, 24.7, 31.71, 23.56, 38.27, 32.99, 33.06, 36.49, 33.72, 28.79, 34.98, 30.24, 35.4, 35.92, 31.85, 28.54, 35.1, 34.0, 31.36, 35.0, 31.19, 36.87, 28.93, 35.16, 31.25, 32.64, 36.66, 27.61, 34.21, 30.65, 37.19, 27.07, 35.23, 33.2, 34.22, 33.0, 35.91, 37.75, 29.34, 33.64, 37.23, 30.44, 27.8, 32.76, 27.16, 33.71, 30.24, 30.62, 34.99, 30.75, 30.75, 32.78, 39.17, 29.55, 36.2, 30.06, 29.41, 39.01, 25.37, 37.62, 31.74, 38.7, 29.2, 29.26, 29.07, 31.86, 27.47, 30.69, 35.11, 32.35, 36.81, 36.29, 26.55, 31.43, 31.95, 28.88, 37.22, 35.98, 29.49, 23.42, 37.64, 33.59, 34.91, 29.05, 35.52, 33.28, 33.13, 28.55, 29.35, 31.29, 26.56, 37.59, 23.32, 33.65, 30.9, 33.5, 30.06, 29.6, 32.05, 30.54, 36.81, 31.69, 33.94, 33.04, 29.66, 35.47, 39.16, 30.26, 30.31, 34.06, 35.26, 32.38, 36.37, 31.56, 37.4, 27.49, 37.28, 36.6, 29.96, 31.5, 31.57, 30.86, 27.79, 34.32, 33.31, 31.4, 30.05, 32.4, 32.36, 41.05, 31.77, 31.66, 35.34, 34.91, 32.88, 33.44, 34.78, 31.82, 29.94, 36.74, 34.15, 35.67, 38.69, 28.8, 28.66, 36.34, 34.08, 36.23, 32.62, 36.29, 32.38, 36.65, 30.84, 34.63, 22.81, 33.15, 30.03, 27.61, 30.39, 27.78, 39.82, 40.89, 32.15, 35.5, 31.22, 37.5, 30.28, 34.28, 28.22, 34.99, 30.78, 32.08, 33.02, 32.81]；
组2 n=456：[25.67, 28.36, 23.19, 23.18, 22.82, 27.42, 28.47, 26.38, 25.3, 22.8, 27.69, 29.52, 25.86, 25.81, 27.29, 27.04, 27.75, 28.15, 28.12, 25.56, 25.19, 26.53, 29.47, 21.85, 22.81, 26.69, 24.21, 24.35, 21.27, 25.46, 25.56, 29.25, 28.09, 25.74, 25.98, 27.91, 26.07, 22.4, 24.49, 23.97, 27.77, 26.43, 25.18, 27.19, 26.84, 26.67, 28.21, 23.31, 20.86, 26.07, 29.5, 21.41, 25.86, 26.12, 26.28, 22.58, 27.74, 22.58, 24.57, 24.83, 25.7, 26.53, 22.37, 23.54, 28.44, 23.82, 20.98, 25.93, 24.37, 23.36, 27.4, 28.34, 22.41, 29.01, 27.09, 19.55, 32.21, 26.87, 28.05, 23.57, 26.94, 25.91, 29.36, 31.88, 25.22, 27.09, 21.12, 25.18, 23.72, 29.92, 27.8, 29.94, 27.61, 26.83, 28.9, 27.28, 24.39, 24.23, 21.61, 22.55, 25.38, 28.08, 19.51, 24.95, 26.82, 25.92, 24.68, 30.55, 25.95, 22.94, 28.57, 27.5, 23.25, 22.62, 26.65, 27.56, 28.33, 23.75, 31.05, 25.98, 27.3, 23.77, 24.66, 28.11, 27.99, 25.03, 28.58, 24.07, 28.14, 25.75, 28.92, 27.07, 28.06, 26.79, 22.66, 24.95, 29.33, 24.97, 22.31, 26.4, 23.08, 28.48, 25.04, 32.3, 26.75, 29.03, 27.46, 25.03, 28.42, 24.11, 28.36, 24.74, 24.01, 26.15, 26.78, 27.07, 27.08, 27.53, 23.54, 22.44, 25.24, 22.9, 24.43, 23.73, 28.3, 24.95, 32.19, 28.89, 26.16, 20.54, 25.18, 29.79, 25.17, 26.95, 29.78, 26.44, 24.96, 22.06, 26.88, 27.79, 22.98, 27.62, 24.34, 28.61, 25.27, 27.74, 30.52, 25.96, 29.7, 28.4, 25.74, 26.28, 29.15, 25.3, 25.43, 18.45, 26.31, 26.4, 28.3, 23.17, 26.88, 26.59, 27.52, 23.21, 26.82, 25.3, 25.81, 28.49, 26.29, 26.25, 27.64, 22.63, 22.57, 20.72, 29.53, 25.77, 27.52, 24.04, 29.81, 27.8, 29.15, 26.17, 24.9, 26.57, 25.34, 22.65, 27.7, 26.9, 25.73, 25.7, 30.91, 26.98, 27.47, 24.49, 22.94, 27.41, 27.44, 27.36, 25.13, 27.86, 27.53, 25.61, 22.11, 25.6, 24.19, 26.1, 27.8, 27.66, 21.99, 28.63, 24.06, 26.64, 25.62, 23.52, 27.2, 22.07, 27.63, 27.92, 26.09, 26.62, 23.09, 27.56, 26.99, 26.52, 23.39, 27.17, 26.93, 28.11, 24.49, 27.65, 27.79, 27.71, 28.18, 22.91, 29.12, 30.97, 26.74, 26.11, 22.53, 26.26, 25.72, 28.04, 25.85, 21.93, 24.66, 25.65, 27.93, 24.55, 27.41, 26.03, 26.62, 26.07, 29.59, 23.8, 27.01, 30.55, 30.5, 25.34, 23.17, 27.52, 27.62, 30.05, 25.08, 25.86, 29.45, 29.15, 24.0, 24.71, 27.17, 28.27, 24.06, 26.0, 26.49, 19.11, 25.82, 30.06, 26.95, 25.14, 24.34, 25.94, 28.36, 25.5, 24.85, 23.97, 24.03, 26.46, 25.91, 30.63, 30.9, 29.8, 25.3, 21.97, 30.02, 25.55, 26.9, 27.16, 26.38, 24.77, 27.12, 25.48, 25.44, 25.82, 24.87, 28.32, 24.23, 22.98, 25.55, 26.39, 30.02, 25.47, 29.47, 25.8, 32.61, 25.35, 23.46, 26.96, 22.53, 23.93, 25.09, 24.07, 25.65, 23.81, 25.33, 23.32, 23.85, 29.65, 26.37, 25.12, 28.18, 25.36, 26.68, 28.65, 25.46, 28.35, 22.61, 31.2, 21.46, 28.99, 29.45, 23.79, 29.24, 23.09, 26.62, 25.13, 29.03, 22.77, 28.08, 27.4, 26.79, 22.75, 23.16, 21.37, 22.51, 24.0, 24.63, 25.82, 27.34, 27.4, 30.31, 23.27, 25.23, 27.38, 22.27, 30.54, 28.61, 23.19, 21.81, 28.27, 27.66, 21.36, 26.0, 31.01, 25.36, 28.72, 25.2, 26.3, 24.81, 27.3, 25.4, 24.85, 27.37, 24.31, 26.02, 26.98, 33.75, 27.46, 26.26, 25.97, 27.77, 27.99, 24.51, 26.53, 26.11, 29.59, 21.35, 25.25, 25.48, 26.06, 25.25, 23.54, 27.84, 28.75, 30.47, 26.96, 23.78, 26.29, 25.32, 26.58, 24.3, 24.11, 27.23, 25.16, 26.95, 28.06, 25.7, 25.25]
组2 n=456：[25.67, 28.36, 23.19, 23.18, 22.82, 27.42, 28.47, 26.38, 25.3, 22.8, 27.69, 29.52, 25.86, 25.81, 27.29, 27.04, 27.75, 28.15, 28.12, 25.56, 25.19, 26.53, 29.47, 21.85, 22.81, 26.69, 24.21, 24.35, 21.27, 25.46, 25.56, 29.25, 28.09, 25.74, 25.98, 27.91, 26.07, 22.4, 24.49, 23.97, 27.77, 26.43, 25.18, 27.19, 26.84, 26.67, 28.21, 23.31, 20.86, 26.07, 29.5, 21.41, 25.86, 26.12, 26.28, 22.58, 27.74, 22.58, 24.57, 24.83, 25.7, 26.53, 22.37, 23.54, 28.44, 23.82, 20.98, 25.93, 24.37, 23.36, 27.4, 28.34, 22.41, 29.01, 27.09, 19.55, 32.21, 26.87, 28.05, 23.57, 26.94, 25.91, 29.36, 31.88, 25.22, 27.09, 21.12, 25.18, 23.72, 29.92, 27.8, 29.94, 27.61, 26.83, 28.9, 27.28, 24.39, 24.23, 21.61, 22.55, 25.38, 28.08, 19.51, 24.95, 26.82, 25.92, 24.68, 30.55, 25.95, 22.94, 28.57, 27.5, 23.25, 22.62, 26.65, 27.56, 28.33, 23.75, 31.05, 25.98, 27.3, 23.77, 24.66, 28.11, 27.99, 25.03, 28.58, 24.07, 28.14, 25.75, 28.92, 27.07, 28.06, 26.79, 22.66, 24.95, 29.33, 24.97, 22.31, 26.4, 23.08, 28.48, 25.04, 32.3, 26.75, 29.03, 27.46, 25.03, 28.42, 24.11, 28.36, 24.74, 24.01, 26.15, 26.78, 27.07, 27.08, 27.53, 23.54, 22.44, 25.24, 22.9, 24.43, 23.73, 28.3, 24.95, 32.19, 28.89, 26.16, 20.54, 25.18, 29.79, 25.17, 26.95, 29.78, 26.44, 24.96, 22.06, 26.88, 27.79, 22.98, 27.62, 24.34, 28.61, 25.27, 27.74, 30.52, 25.96, 29.7, 28.4, 25.74, 26.28, 29.15, 25.3, 25.43, 18.45, 26.31, 26.4, 28.3, 23.17, 26.88, 26.59, 27.52, 23.21, 26.82, 25.3, 25.81, 28.49, 26.29, 26.25, 27.64, 22.63, 22.57, 20.72, 29.53, 25.77, 27.52, 24.04, 29.81, 27.8, 29.15, 26.17, 24.9, 26.57, 25.34, 22.65, 27.7, 26.9, 25.73, 25.7, 30.91, 26.98, 27.47, 24.49, 22.94, 27.41, 27.44, 27.36, 25.13, 27.86, 27.53, 25.61, 22.11, 25.6, 24.19, 26.1, 27.8, 27.66, 21.99, 28.63, 24.06, 26.64, 25.62, 23.52, 27.2, 22.07, 27.63, 27.92, 26.09, 26.62, 23.09, 27.56, 26.99, 26.52, 23.39, 27.17, 26.93, 28.11, 24.49, 27.65, 27.79, 27.71, 28.18, 22.91, 29.12, 30.97, 26.74, 26.11, 22.53, 26.26, 25.72, 28.04, 25.85, 21.93, 24.66, 25.65, 27.93, 24.55, 27.41, 26.03, 26.62, 26.07, 29.59, 23.8, 27.01, 30.55, 30.5, 25.34, 23.17, 27.52, 27.62, 30.05, 25.08, 25.86, 29.45, 29.15, 24.0, 24.71, 27.17, 28.27, 24.06, 26.0, 26.49, 19.11, 25.82, 30.06, 26.95, 25.14, 24.34, 25.94, 28.36, 25.5, 24.85, 23.97, 24.03, 26.46, 25.91, 30.63, 30.9, 29.8, 25.3, 21.97, 30.02, 25.55, 26.9, 27.16, 26.38, 24.77, 27.12, 25.48, 25.44, 25.82, 24.87, 28.32, 24.23, 22.98, 25.55, 26.39, 30.02, 25.47, 29.47, 25.8, 32.61, 25.35, 23.46, 26.96, 22.53, 23.93, 25.09, 24.07, 25.65, 23.81, 25.33, 23.32, 23.85, 29.65, 26.37, 25.12, 28.18, 25.36, 26.68, 28.65, 25.46, 28.35, 22.61, 31.2, 21.46, 28.99, 29.45, 23.79, 29.24, 23.09, 26.62, 25.13, 29.03, 22.77, 28.08, 27.4, 26.79, 22.75, 23.16, 21.37, 22.51, 24.0, 24.63, 25.82, 27.34, 27.4, 30.31, 23.27, 25.23, 27.38, 22.27, 30.54, 28.61, 23.19, 21.81, 28.27, 27.66, 21.36, 26.0, 31.01, 25.36, 28.72, 25.2, 26.3, 24.81, 27.3, 25.4, 24.85, 27.37, 24.31, 26.02, 26.98, 33.75, 27.46, 26.26, 25.97, 27.77, 27.99, 24.51, 26.53, 26.11, 29.59, 21.35, 25.25, 25.48, 26.06, 25.25, 23.54, 27.84, 28.75, 30.47, 26.96, 23.78, 26.29, 25.32, 26.58, 24.3, 24.11, 27.23, 25.16, 26.95, 28.06, 25.7, 25.25]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=32.7808, s₁=3.6530；x̄₂=26.1054, s₂=2.4327；
SE=√(s₁²/n₁+s₂²/n₂)=0.1760；t=(x̄₁−x̄₂)/SE=37.9363；
df(Welch)=1188.44；双侧 p = 5.460e-207。
x̄₁=32.7808, s₁=3.6530；x̄₂=26.1054, s₂=2.4327；
SE=√(s₁²/n₁+s₂²/n₂)=0.1760；t=(x̄₁−x̄₂)/SE=37.9363；
df(Welch)=1188.44；双侧 p = 5.460e-207。
计算结果
t = 37.9363，df = 1188.44，p = 5.460e-207
在 α=0.05 水平下，p=5.460e-207 < 0.05，拒绝原假设 H₀。判断新版本是否延长停留。
案例 18. 两种降压药效果
原始数据
组1 n=671：[136.09, 141.39, 141.31, 138.25, 140.07, 143.31, 141.49, 139.54, 138.82, 136.86, 145.91, 137.31, 143.47, 140.59, 136.3, 136.88, 132.67, 141.54, 135.93, 137.76, 136.32, 140.73, 137.3, 140.93, 135.44, 138.6, 139.31, 137.79, 141.35, 137.15, 135.7, 139.97, 136.06, 142.39, 142.0, 138.36, 137.85, 139.11, 140.4, 137.41, 139.58, 139.82, 142.94, 138.22, 137.91, 140.82, 138.19, 141.22, 141.22, 139.31, 138.22, 138.82, 134.84, 140.56, 136.84, 132.08, 142.45, 136.58, 136.1, 138.55, 136.26, 141.79, 134.51, 136.22, 144.44, 140.48, 137.19, 138.1, 135.47, 140.0, 142.08, 137.19, 139.03, 137.38, 141.16, 139.05, 132.88, 136.47, 142.24, 136.17, 138.73, 136.33, 139.66, 143.25, 139.78, 140.84, 135.95, 133.12, 140.46, 137.41, 138.28, 140.76, 140.34, 140.03, 138.72, 141.71, 137.27, 141.41, 135.3, 137.22, 134.76, 142.55, 144.6, 141.12, 141.51, 137.19, 137.36, 136.41, 134.76, 137.64, 138.98, 140.98, 143.21, 135.96, 143.97, 137.32, 142.62, 137.03, 136.32, 138.84, 138.3, 136.4, 141.43, 138.14, 141.88, 141.17, 143.53, 142.56, 140.19, 137.18, 133.16, 139.08, 140.3, 142.58, 133.16, 139.61, 137.61, 141.41, 136.73, 138.71, 139.13, 139.01, 138.0, 138.76, 137.1, 136.82, 137.74, 139.95, 134.33, 138.82, 133.15, 140.18, 139.27, 139.8, 137.95, 140.54, 142.2, 136.7, 139.91, 137.93, 143.06, 137.44, 137.82, 137.81, 143.94, 136.36, 139.18, 136.61, 140.8, 140.28, 136.54, 136.47, 139.78, 136.66, 141.03, 142.55, 136.28, 139.07, 140.98, 138.5, 137.48, 134.36, 141.89, 136.87, 138.82, 141.54, 137.07, 135.9, 137.15, 137.14, 137.26, 143.24, 139.92, 137.89, 141.65, 142.61, 138.35, 138.25, 142.02, 142.6, 136.18, 138.2, 142.44, 139.29, 136.43, 143.95, 136.13, 146.23, 138.1, 139.59, 139.61, 141.74, 132.53, 139.83, 136.71, 141.47, 133.12, 142.08, 139.91, 140.25, 141.62, 139.77, 135.97, 137.96, 134.39, 137.93, 141.04, 140.65, 136.21, 145.21, 140.78, 139.85, 140.25, 137.85, 136.28, 138.39, 140.71, 140.9, 141.82, 139.85, 140.21, 140.6, 140.53, 139.3, 140.44, 141.06, 145.09, 141.8, 139.61, 141.75, 137.21, 135.61, 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131.59, 131.43, 129.28, 124.41, 130.18, 128.69, 130.36, 132.03, 129.08, 130.99, 131.58, 129.15, 131.41, 129.35, 132.57, 130.35, 127.48, 131.1, 127.18, 130.06, 131.43, 129.05, 130.53, 131.9, 129.77, 131.74, 129.88, 130.19, 129.4, 127.5, 130.18, 131.99, 128.65, 128.99, 129.3, 128.86, 130.56, 130.17, 128.07, 131.4, 130.33, 130.39, 133.04, 131.74, 130.28, 128.81, 126.27, 128.23, 127.46, 126.9, 127.24, 132.66, 132.01, 129.78, 133.29, 126.79, 128.56, 127.47, 129.69, 130.87, 130.14, 130.4, 129.96, 130.56, 129.54, 131.04, 128.98, 126.98, 131.7, 130.37, 127.72, 131.06, 128.95, 129.24, 131.51, 129.21, 128.6, 127.13, 128.3, 127.8, 129.85, 127.86, 130.0, 132.47, 130.38, 127.78, 130.89, 131.82, 132.38, 130.64, 126.89, 129.99, 130.96, 130.96, 129.25, 131.58, 131.91, 129.46, 132.6, 130.63, 131.09, 131.86, 127.16, 130.98, 130.59, 129.62, 129.56, 128.14, 132.22, 130.65, 127.17, 129.29, 126.89, 130.57, 128.75, 133.21, 129.67, 128.01, 130.79, 129.8, 128.43, 130.05, 130.35, 128.92, 127.99, 129.35, 130.52, 131.13, 125.31, 128.47, 129.15, 126.91, 126.49, 128.11, 129.34, 130.58, 128.95, 129.75, 130.88, 129.79, 130.2, 129.87, 130.81, 131.26, 128.52, 130.23, 136.61, 126.97, 126.85, 127.08, 128.14, 130.29, 131.41, 128.31, 131.39, 129.92, 130.43, 132.2, 131.87, 132.12, 127.36, 131.64, 133.76, 131.0, 128.1, 125.89, 130.35, 129.22, 129.67, 130.32, 129.89, 129.87, 131.67, 130.98, 127.26, 125.49, 129.49, 131.68, 126.96, 127.88, 128.74, 128.36, 127.89, 131.67, 129.05, 131.2, 129.83, 130.16, 133.32, 130.04, 128.92, 130.64, 134.51, 127.44, 130.71, 133.41, 134.79, 128.31, 134.43, 131.63, 128.74, 129.23, 133.28, 130.11, 129.98, 128.89, 128.34, 128.7, 131.61, 129.19, 130.44, 131.74, 131.21, 128.31, 129.57, 128.43, 131.63, 128.55, 128.36, 131.08, 128.2, 133.41, 134.09, 132.98, 126.38, 128.22, 129.35, 129.43, 131.0, 135.09, 129.85, 131.38, 128.85, 130.42, 130.06, 128.12, 130.77, 130.72, 127.51, 129.83, 128.54, 131.51, 129.22, 127.75, 127.93, 128.55, 129.6, 130.55, 130.52, 130.36, 130.82, 131.09, 128.29, 128.58, 128.8, 129.48, 129.8, 127.63, 131.44, 132.53, 130.92, 129.24, 130.37, 127.27, 131.51, 131.25, 127.37, 131.05, 128.05, 130.8, 132.01, 130.36, 130.24, 128.84, 127.92, 128.0, 128.95, 130.36, 130.84, 130.42, 132.54, 127.49]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=139.1401, s₁=2.5938；x̄₂=129.9230, s₂=1.8041；
SE=√(s₁²/n₁+s₂²/n₂)=0.1166；t=(x̄₁−x̄₂)/SE=79.0611；
df(Welch)=1126.53；双侧 p = 0.000e+00。
x̄₁=139.1401, s₁=2.5938；x̄₂=129.9230, s₂=1.8041；
SE=√(s₁²/n₁+s₂²/n₂)=0.1166；t=(x̄₁−x̄₂)/SE=79.0611；
df(Welch)=1126.53；双侧 p = 0.000e+00。
计算结果
t = 79.0611，df = 1126.53，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。比较两种药降压幅度。
案例 19. 新旧工艺缺陷尺寸
原始数据
组1 n=345：[2.3, 2.1, 2.19, 2.24, 2.2, 2.29, 2.1, 2.44, 2.3, 2.12, 2.33, 2.43, 2.16, 2.16, 2.28, 2.02, 2.2, 1.87, 2.14, 2.31, 2.2, 2.4, 2.32, 2.02, 2.53, 2.15, 2.23, 2.4, 2.18, 2.08, 2.53, 2.24, 2.08, 2.36, 2.04, 1.95, 2.46, 2.09, 2.21, 2.23, 2.16, 2.38, 2.39, 2.31, 2.18, 2.27, 2.2, 2.25, 2.13, 1.83, 2.13, 2.44, 2.1, 2.04, 2.36, 2.17, 1.87, 2.22, 2.01, 2.3, 2.17, 2.27, 2.2, 2.22, 1.91, 2.22, 2.15, 2.2, 2.11, 2.47, 2.03, 2.13, 2.31, 2.04, 2.38, 2.31, 2.36, 2.08, 2.44, 2.12, 2.09, 2.32, 1.93, 2.63, 2.49, 2.4, 2.15, 1.91, 2.31, 2.24, 2.13, 1.68, 2.16, 2.08, 2.12, 2.19, 2.13, 2.38, 1.99, 2.32, 2.21, 2.27, 2.32, 2.33, 2.25, 2.33, 2.3, 2.02, 2.18, 2.36, 2.3, 2.09, 2.36, 2.51, 2.34, 2.04, 1.97, 2.13, 2.56, 1.89, 2.09, 2.16, 2.37, 2.43, 2.08, 2.19, 2.41, 2.42, 2.19, 2.33, 2.4, 2.24, 2.24, 2.33, 2.43, 2.27, 2.09, 2.16, 2.5, 2.21, 2.06, 2.17, 2.36, 2.3, 2.12, 2.01, 2.26, 2.25, 2.2, 2.14, 2.12, 1.99, 2.18, 2.36, 2.37, 2.26, 2.17, 2.06, 2.16, 2.47, 2.16, 2.16, 2.5, 2.14, 1.89, 1.92, 1.99, 2.09, 2.54, 2.17, 2.35, 2.52, 2.12, 2.22, 2.22, 2.55, 2.24, 2.0, 2.11, 2.34, 2.48, 2.29, 2.19, 2.03, 2.03, 2.34, 2.28, 2.51, 2.01, 2.56, 1.98, 2.17, 2.32, 2.11, 2.21, 2.01, 2.58, 2.26, 2.11, 2.38, 2.31, 2.29, 2.31, 2.12, 2.27, 2.46, 2.19, 2.38, 2.11, 2.51, 2.22, 2.29, 2.18, 2.25, 2.46, 2.34, 2.22, 2.17, 2.44, 2.31, 2.17, 1.85, 2.14, 2.39, 2.08, 2.5, 2.55, 2.01, 2.08, 2.16, 2.1, 2.52, 2.17, 2.27, 2.05, 2.37, 2.23, 2.25, 2.14, 2.05, 2.28, 2.19, 2.32, 2.23, 2.14, 2.28, 2.14, 2.24, 2.4, 2.16, 2.24, 1.96, 2.17, 2.2, 2.11, 2.13, 2.29, 2.31, 2.1, 2.17, 2.06, 2.52, 2.45, 2.28, 2.08, 1.99, 1.92, 2.23, 2.1, 1.87, 2.3, 1.93, 1.98, 1.87, 2.38, 2.33, 1.71, 2.28, 2.25, 2.18, 2.39, 2.28, 2.43, 2.07, 2.03, 1.94, 2.29, 2.12, 2.03, 2.43, 1.95, 2.21, 2.06, 1.97, 2.38, 2.02, 2.09, 2.11, 2.15, 2.41, 1.86, 2.15, 2.27, 2.18, 2.48, 2.08, 2.0, 2.34, 2.22, 2.66, 2.24, 2.27, 2.05, 2.16, 2.03, 2.09, 2.09, 2.08, 2.22, 2.2, 2.32, 2.49, 2.19, 2.2, 2.4, 2.09, 2.25, 2.32, 2.05, 2.08, 2.16, 2.29, 2.26, 1.98, 2.0, 2.06, 1.93, 1.99, 2.35, 2.17, 2.39, 2.3, 2.17, 2.34, 2.42]；
组2 n=676：[1.78, 1.93, 1.82, 1.6, 1.72, 1.88, 1.81, 1.73, 1.78, 1.82, 1.8, 1.8, 1.69, 1.75, 1.54, 1.65, 1.68, 1.66, 1.83, 1.41, 1.64, 2.09, 1.74, 1.61, 1.56, 1.82, 1.96, 1.82, 1.74, 1.68, 1.7, 1.97, 1.77, 1.64, 1.65, 1.5, 1.81, 1.62, 1.87, 1.58, 2.07, 1.6, 1.67, 1.61, 1.61, 1.71, 1.57, 1.54, 1.64, 1.82, 1.48, 1.94, 1.66, 2.13, 1.7, 1.69, 1.83, 1.63, 1.65, 1.87, 1.66, 2.03, 1.83, 1.91, 1.73, 1.59, 1.78, 1.74, 1.82, 1.55, 1.66, 1.72, 1.53, 1.62, 1.9, 1.69, 1.72, 1.61, 1.92, 1.69, 1.66, 1.7, 1.6, 1.87, 1.71, 1.87, 1.74, 1.65, 1.65, 1.68, 1.58, 1.73, 1.72, 1.78, 1.95, 1.65, 1.74, 1.66, 1.75, 1.78, 1.77, 1.77, 1.62, 1.72, 1.71, 1.87, 1.7, 1.66, 1.83, 1.68, 1.67, 1.77, 1.49, 1.85, 1.65, 1.7, 1.23, 1.82, 1.9, 1.57, 1.84, 1.82, 2.05, 1.61, 1.71, 1.65, 1.54, 1.9, 1.69, 1.55, 1.88, 1.78, 1.77, 1.89, 1.59, 1.61, 1.71, 1.48, 1.84, 1.59, 1.61, 1.82, 1.63, 1.68, 1.8, 1.83, 1.98, 1.59, 1.76, 1.52, 2.09, 1.72, 1.69, 1.66, 1.8, 1.65, 1.64, 1.88, 1.93, 1.86, 1.62, 1.73, 1.68, 1.53, 1.56, 1.9, 2.0, 1.92, 1.48, 1.79, 1.96, 1.82, 1.84, 1.97, 1.74, 1.64, 1.83, 1.87, 1.61, 1.44, 1.72, 1.43, 1.72, 1.77, 1.61, 1.74, 1.75, 1.56, 1.72, 1.75, 1.74, 1.77, 1.77, 1.49, 1.78, 1.73, 1.92, 1.55, 1.99, 1.74, 1.68, 1.75, 1.73, 1.81, 1.52, 1.52, 1.72, 2.07, 1.91, 1.72, 1.71, 1.5, 1.7, 1.5, 1.82, 1.69, 1.86, 1.89, 1.65, 1.81, 1.56, 1.62, 1.88, 1.73, 1.92, 1.75, 1.71, 1.55, 1.75, 1.84, 1.68, 1.86, 1.56, 1.83, 1.51, 1.79, 1.53, 1.8, 1.7, 1.59, 1.66, 1.59, 1.87, 1.7, 1.72, 1.65, 1.93, 1.69, 1.53, 1.62, 1.7, 1.61, 1.79, 1.43, 1.7, 1.66, 1.63, 1.74, 1.83, 1.87, 1.47, 1.76, 1.61, 1.67, 1.64, 1.57, 1.74, 2.02, 1.94, 1.46, 1.57, 1.59, 1.47, 1.64, 1.61, 1.64, 1.47, 1.83, 1.96, 1.69, 1.8, 1.51, 1.64, 1.74, 1.77, 2.0, 1.78, 2.07, 1.76, 1.72, 1.7, 1.72, 1.74, 1.77, 1.73, 1.82, 1.52, 1.58, 1.8, 1.63, 1.66, 1.75, 1.58, 1.72, 1.8, 1.54, 1.82, 1.82, 1.65, 1.74, 1.68, 1.77, 1.7, 1.73, 1.51, 1.75, 1.96, 1.86, 1.66, 1.54, 1.9, 1.67, 1.62, 1.71, 2.0, 1.86, 1.74, 1.41, 1.78, 1.83, 1.5, 1.51, 1.9, 1.87, 1.92, 1.63, 1.65, 1.66, 1.53, 1.69, 1.8, 1.68, 1.82, 1.8, 1.95, 1.78, 1.7, 1.46, 1.55, 1.62, 1.59, 1.95, 1.72, 1.75, 1.83, 1.72, 1.72, 1.98, 1.73, 1.76, 1.8, 1.8, 1.43, 1.82, 1.7, 1.85, 1.68, 1.4, 1.63, 1.72, 1.37, 1.98, 1.87, 1.89, 1.63, 1.77, 1.6, 1.95, 1.72, 1.84, 1.94, 1.59, 1.7, 1.7, 1.86, 1.71, 1.84, 1.6, 1.41, 1.84, 1.69, 1.73, 1.65, 1.72, 1.79, 1.9, 1.7, 1.67, 1.8, 1.51, 1.64, 1.57, 1.65, 1.74, 1.62, 1.82, 1.93, 1.69, 1.87, 1.68, 1.75, 1.8, 1.61, 1.9, 1.76, 1.69, 1.73, 1.8, 1.91, 1.44, 1.64, 1.57, 1.68, 1.75, 1.67, 1.62, 1.81, 1.94, 1.75, 1.76, 1.9, 1.65, 1.55, 1.6, 1.4, 1.84, 1.65, 1.71, 1.62, 1.48, 1.68, 1.53, 1.64, 1.74, 1.57, 1.96, 1.68, 1.51, 1.66, 1.62, 1.59, 1.91, 2.03, 1.58, 1.75, 1.95, 1.69, 1.92, 1.65, 1.5, 2.05, 1.72, 1.93, 1.81, 1.61, 1.67, 1.84, 1.6, 1.73, 1.89, 1.55, 1.83, 1.55, 1.93, 1.76, 1.82, 1.63, 1.76, 1.61, 1.52, 1.79, 1.62, 1.66, 1.74, 1.87, 1.77, 1.63, 1.71, 1.74, 1.67, 1.72, 1.72, 1.81, 1.75, 1.69, 1.65, 1.77, 2.13, 1.63, 1.74, 1.83, 1.68, 1.85, 1.74, 1.56, 1.78, 1.67, 1.75, 1.88, 1.75, 1.83, 1.94, 1.85, 1.44, 1.66, 1.53, 1.58, 1.33, 1.6, 1.58, 1.57, 1.53, 1.91, 1.74, 1.64, 1.72, 1.8, 1.76, 1.82, 1.72, 1.6, 1.69, 1.82, 1.93, 1.73, 1.9, 1.85, 1.82, 1.85, 1.68, 1.72, 1.81, 1.85, 2.02, 1.68, 1.6, 1.78, 1.65, 1.64, 1.75, 1.74, 1.72, 1.53, 1.72, 1.54, 1.56, 1.79, 1.61, 1.6, 1.73, 1.74, 1.72, 1.77, 1.81, 1.92, 1.54, 1.83, 1.81, 1.7, 1.68, 1.49, 1.69, 1.92, 1.56, 1.88, 1.6, 1.63, 1.76, 1.84, 1.56, 1.63, 1.9, 1.82, 1.89, 1.79, 1.86, 1.67, 1.66, 1.87, 1.87, 1.58, 1.58, 1.8, 1.86, 1.73, 1.5, 2.03, 1.82, 1.49, 1.62, 1.56, 1.97, 1.84, 1.9, 1.54, 1.38, 1.64, 1.91, 1.45, 1.82, 1.53, 1.77, 1.58, 1.63, 1.69, 1.53, 1.89, 1.61, 1.67, 1.75, 1.77, 1.7, 1.84, 1.66, 1.56, 1.61, 1.59, 1.63, 1.89, 1.88, 1.74, 2.1, 1.95, 1.55, 1.88, 1.67, 1.76, 1.61, 1.67, 1.6, 1.66, 1.47, 1.75, 1.79, 1.98, 1.64, 1.86, 1.86, 1.76, 1.6, 1.78, 1.8, 1.65, 1.62, 1.5, 1.98, 1.71, 1.66, 1.76, 1.88, 1.68, 1.6, 1.61, 1.58, 1.71, 1.66, 1.5, 1.73, 1.37, 1.85, 1.8, 1.77, 1.84, 1.8, 1.77, 1.66]
组2 n=676：[1.78, 1.93, 1.82, 1.6, 1.72, 1.88, 1.81, 1.73, 1.78, 1.82, 1.8, 1.8, 1.69, 1.75, 1.54, 1.65, 1.68, 1.66, 1.83, 1.41, 1.64, 2.09, 1.74, 1.61, 1.56, 1.82, 1.96, 1.82, 1.74, 1.68, 1.7, 1.97, 1.77, 1.64, 1.65, 1.5, 1.81, 1.62, 1.87, 1.58, 2.07, 1.6, 1.67, 1.61, 1.61, 1.71, 1.57, 1.54, 1.64, 1.82, 1.48, 1.94, 1.66, 2.13, 1.7, 1.69, 1.83, 1.63, 1.65, 1.87, 1.66, 2.03, 1.83, 1.91, 1.73, 1.59, 1.78, 1.74, 1.82, 1.55, 1.66, 1.72, 1.53, 1.62, 1.9, 1.69, 1.72, 1.61, 1.92, 1.69, 1.66, 1.7, 1.6, 1.87, 1.71, 1.87, 1.74, 1.65, 1.65, 1.68, 1.58, 1.73, 1.72, 1.78, 1.95, 1.65, 1.74, 1.66, 1.75, 1.78, 1.77, 1.77, 1.62, 1.72, 1.71, 1.87, 1.7, 1.66, 1.83, 1.68, 1.67, 1.77, 1.49, 1.85, 1.65, 1.7, 1.23, 1.82, 1.9, 1.57, 1.84, 1.82, 2.05, 1.61, 1.71, 1.65, 1.54, 1.9, 1.69, 1.55, 1.88, 1.78, 1.77, 1.89, 1.59, 1.61, 1.71, 1.48, 1.84, 1.59, 1.61, 1.82, 1.63, 1.68, 1.8, 1.83, 1.98, 1.59, 1.76, 1.52, 2.09, 1.72, 1.69, 1.66, 1.8, 1.65, 1.64, 1.88, 1.93, 1.86, 1.62, 1.73, 1.68, 1.53, 1.56, 1.9, 2.0, 1.92, 1.48, 1.79, 1.96, 1.82, 1.84, 1.97, 1.74, 1.64, 1.83, 1.87, 1.61, 1.44, 1.72, 1.43, 1.72, 1.77, 1.61, 1.74, 1.75, 1.56, 1.72, 1.75, 1.74, 1.77, 1.77, 1.49, 1.78, 1.73, 1.92, 1.55, 1.99, 1.74, 1.68, 1.75, 1.73, 1.81, 1.52, 1.52, 1.72, 2.07, 1.91, 1.72, 1.71, 1.5, 1.7, 1.5, 1.82, 1.69, 1.86, 1.89, 1.65, 1.81, 1.56, 1.62, 1.88, 1.73, 1.92, 1.75, 1.71, 1.55, 1.75, 1.84, 1.68, 1.86, 1.56, 1.83, 1.51, 1.79, 1.53, 1.8, 1.7, 1.59, 1.66, 1.59, 1.87, 1.7, 1.72, 1.65, 1.93, 1.69, 1.53, 1.62, 1.7, 1.61, 1.79, 1.43, 1.7, 1.66, 1.63, 1.74, 1.83, 1.87, 1.47, 1.76, 1.61, 1.67, 1.64, 1.57, 1.74, 2.02, 1.94, 1.46, 1.57, 1.59, 1.47, 1.64, 1.61, 1.64, 1.47, 1.83, 1.96, 1.69, 1.8, 1.51, 1.64, 1.74, 1.77, 2.0, 1.78, 2.07, 1.76, 1.72, 1.7, 1.72, 1.74, 1.77, 1.73, 1.82, 1.52, 1.58, 1.8, 1.63, 1.66, 1.75, 1.58, 1.72, 1.8, 1.54, 1.82, 1.82, 1.65, 1.74, 1.68, 1.77, 1.7, 1.73, 1.51, 1.75, 1.96, 1.86, 1.66, 1.54, 1.9, 1.67, 1.62, 1.71, 2.0, 1.86, 1.74, 1.41, 1.78, 1.83, 1.5, 1.51, 1.9, 1.87, 1.92, 1.63, 1.65, 1.66, 1.53, 1.69, 1.8, 1.68, 1.82, 1.8, 1.95, 1.78, 1.7, 1.46, 1.55, 1.62, 1.59, 1.95, 1.72, 1.75, 1.83, 1.72, 1.72, 1.98, 1.73, 1.76, 1.8, 1.8, 1.43, 1.82, 1.7, 1.85, 1.68, 1.4, 1.63, 1.72, 1.37, 1.98, 1.87, 1.89, 1.63, 1.77, 1.6, 1.95, 1.72, 1.84, 1.94, 1.59, 1.7, 1.7, 1.86, 1.71, 1.84, 1.6, 1.41, 1.84, 1.69, 1.73, 1.65, 1.72, 1.79, 1.9, 1.7, 1.67, 1.8, 1.51, 1.64, 1.57, 1.65, 1.74, 1.62, 1.82, 1.93, 1.69, 1.87, 1.68, 1.75, 1.8, 1.61, 1.9, 1.76, 1.69, 1.73, 1.8, 1.91, 1.44, 1.64, 1.57, 1.68, 1.75, 1.67, 1.62, 1.81, 1.94, 1.75, 1.76, 1.9, 1.65, 1.55, 1.6, 1.4, 1.84, 1.65, 1.71, 1.62, 1.48, 1.68, 1.53, 1.64, 1.74, 1.57, 1.96, 1.68, 1.51, 1.66, 1.62, 1.59, 1.91, 2.03, 1.58, 1.75, 1.95, 1.69, 1.92, 1.65, 1.5, 2.05, 1.72, 1.93, 1.81, 1.61, 1.67, 1.84, 1.6, 1.73, 1.89, 1.55, 1.83, 1.55, 1.93, 1.76, 1.82, 1.63, 1.76, 1.61, 1.52, 1.79, 1.62, 1.66, 1.74, 1.87, 1.77, 1.63, 1.71, 1.74, 1.67, 1.72, 1.72, 1.81, 1.75, 1.69, 1.65, 1.77, 2.13, 1.63, 1.74, 1.83, 1.68, 1.85, 1.74, 1.56, 1.78, 1.67, 1.75, 1.88, 1.75, 1.83, 1.94, 1.85, 1.44, 1.66, 1.53, 1.58, 1.33, 1.6, 1.58, 1.57, 1.53, 1.91, 1.74, 1.64, 1.72, 1.8, 1.76, 1.82, 1.72, 1.6, 1.69, 1.82, 1.93, 1.73, 1.9, 1.85, 1.82, 1.85, 1.68, 1.72, 1.81, 1.85, 2.02, 1.68, 1.6, 1.78, 1.65, 1.64, 1.75, 1.74, 1.72, 1.53, 1.72, 1.54, 1.56, 1.79, 1.61, 1.6, 1.73, 1.74, 1.72, 1.77, 1.81, 1.92, 1.54, 1.83, 1.81, 1.7, 1.68, 1.49, 1.69, 1.92, 1.56, 1.88, 1.6, 1.63, 1.76, 1.84, 1.56, 1.63, 1.9, 1.82, 1.89, 1.79, 1.86, 1.67, 1.66, 1.87, 1.87, 1.58, 1.58, 1.8, 1.86, 1.73, 1.5, 2.03, 1.82, 1.49, 1.62, 1.56, 1.97, 1.84, 1.9, 1.54, 1.38, 1.64, 1.91, 1.45, 1.82, 1.53, 1.77, 1.58, 1.63, 1.69, 1.53, 1.89, 1.61, 1.67, 1.75, 1.77, 1.7, 1.84, 1.66, 1.56, 1.61, 1.59, 1.63, 1.89, 1.88, 1.74, 2.1, 1.95, 1.55, 1.88, 1.67, 1.76, 1.61, 1.67, 1.6, 1.66, 1.47, 1.75, 1.79, 1.98, 1.64, 1.86, 1.86, 1.76, 1.6, 1.78, 1.8, 1.65, 1.62, 1.5, 1.98, 1.71, 1.66, 1.76, 1.88, 1.68, 1.6, 1.61, 1.58, 1.71, 1.66, 1.5, 1.73, 1.37, 1.85, 1.8, 1.77, 1.84, 1.8, 1.77, 1.66]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=2.2106, s₁=0.1662；x̄₂=1.7206, s₂=0.1395；
SE=√(s₁²/n₁+s₂²/n₂)=0.0104；t=(x̄₁−x̄₂)/SE=46.9812；
df(Welch)=596.54；双侧 p = 1.251e-202。
x̄₁=2.2106, s₁=0.1662；x̄₂=1.7206, s₂=0.1395；
SE=√(s₁²/n₁+s₂²/n₂)=0.0104；t=(x̄₁−x̄₂)/SE=46.9812；
df(Welch)=596.54；双侧 p = 1.251e-202。
计算结果
t = 46.9812，df = 596.54，p = 1.251e-202
在 α=0.05 水平下，p=1.251e-202 < 0.05，拒绝原假设 H₀。验证新工艺是否减小缺陷。
案例 20. 甲班乙班产量
原始数据
组1 n=759：[100.97, 102.43, 101.87, 100.18, 102.09, 104.11, 98.84, 100.6, 105.54, 103.32, 100.62, 99.18, 101.66, 101.79, 100.68, 100.66, 102.0, 96.32, 95.02, 96.87, 102.49, 99.08, 97.87, 109.03, 95.92, 103.53, 109.6, 105.4, 106.62, 103.18, 104.17, 103.62, 102.54, 106.84, 99.8, 102.84, 97.52, 101.76, 103.08, 102.63, 93.39, 99.31, 101.09, 99.34, 102.61, 103.26, 97.31, 100.63, 100.93, 98.56, 102.63, 103.12, 101.05, 106.14, 104.41, 97.66, 104.13, 99.51, 106.78, 102.49, 98.8, 103.91, 104.02, 103.13, 102.25, 98.89, 104.34, 104.58, 101.39, 104.23, 101.26, 101.2, 102.68, 97.25, 102.41, 99.92, 104.93, 102.69, 108.73, 99.49, 101.66, 105.98, 104.99, 99.81, 95.83, 103.09, 105.18, 98.36, 97.73, 100.64, 109.89, 102.15, 101.24, 99.95, 102.29, 104.15, 100.7, 103.43, 100.76, 104.09, 100.49, 107.62, 101.24, 103.88, 97.62, 103.24, 99.25, 103.85, 107.39, 104.07, 101.74, 102.01, 99.01, 101.79, 104.33, 100.04, 99.92, 103.45, 106.39, 101.55, 99.17, 104.38, 103.48, 102.71, 103.53, 102.29, 101.31, 95.55, 105.91, 96.39, 106.02, 96.39, 104.73, 104.15, 106.53, 102.33, 101.38, 99.17, 103.08, 103.92, 103.48, 101.25, 102.19, 100.16, 99.3, 104.04, 105.87, 105.01, 104.1, 104.53, 103.61, 101.93, 102.68, 101.37, 100.71, 104.45, 101.65, 102.64, 98.61, 106.2, 102.45, 101.01, 101.91, 105.56, 101.94, 101.04, 99.53, 100.78, 102.69, 101.33, 103.05, 100.63, 99.48, 103.67, 105.95, 104.29, 97.76, 105.18, 98.92, 99.26, 107.63, 104.28, 105.24, 102.26, 101.76, 100.72, 103.74, 98.07, 100.72, 102.9, 102.13, 103.01, 104.94, 105.11, 99.35, 106.33, 102.38, 107.08, 103.14, 103.53, 101.98, 103.1, 101.93, 106.27, 106.89, 104.38, 105.19, 99.96, 103.84, 104.19, 98.73, 98.53, 99.13, 104.53, 97.9, 99.96, 102.09, 103.94, 104.16, 97.15, 104.5, 100.86, 101.45, 99.56, 100.7, 102.62, 100.06, 98.76, 100.1, 105.65, 97.97, 98.06, 102.2, 106.96, 99.0, 101.5, 101.78, 104.15, 104.27, 103.35, 103.12, 107.16, 104.6, 104.77, 98.69, 108.89, 100.09, 97.73, 99.9, 101.8, 97.72, 103.01, 97.23, 99.08, 96.92, 99.77, 106.57, 104.45, 106.96, 102.5, 106.54, 98.29, 104.77, 100.43, 101.25, 97.06, 102.34, 103.86, 97.19, 103.64, 99.53, 103.95, 103.88, 107.0, 103.2, 101.4, 101.98, 104.78, 98.67, 106.41, 100.69, 106.3, 110.22, 104.52, 99.23, 99.1, 104.11, 105.37, 101.89, 99.28, 107.38, 102.4, 98.59, 105.71, 100.35, 102.56, 99.95, 94.4, 102.21, 96.24, 103.67, 102.04, 104.58, 100.08, 102.89, 105.24, 106.76, 99.16, 108.13, 99.69, 101.03, 100.98, 101.84, 102.13, 99.31, 104.58, 102.04, 101.35, 100.73, 96.48, 99.61, 100.94, 101.24, 105.77, 104.28, 103.06, 106.32, 104.73, 97.56, 102.49, 97.44, 97.82, 100.23, 100.94, 101.43, 102.06, 100.6, 99.38, 98.07, 101.29, 105.31, 103.81, 102.0, 106.5, 105.55, 101.4, 99.2, 101.79, 100.16, 100.35, 99.75, 100.33, 101.52, 96.21, 105.11, 106.38, 103.38, 101.82, 103.86, 106.24, 103.01, 100.04, 99.39, 99.06, 98.59, 100.6, 100.4, 102.31, 100.93, 100.31, 100.67, 105.17, 105.98, 101.8, 101.71, 100.97, 99.42, 101.39, 99.8, 101.85, 99.14, 101.89, 96.42, 99.78, 103.11, 98.79, 101.81, 102.77, 98.49, 96.75, 105.21, 101.51, 96.31, 104.86, 103.51, 98.11, 96.8, 106.68, 103.64, 102.82, 99.02, 104.4, 101.73, 103.22, 100.59, 106.65, 104.17, 105.22, 95.99, 102.42, 104.56, 103.42, 100.05, 108.15, 100.26, 101.19, 99.66, 104.03, 100.96, 101.65, 103.08, 105.11, 102.23, 98.68, 103.9, 104.04, 104.57, 98.81, 101.64, 95.18, 101.79, 100.06, 101.89, 102.32, 100.09, 100.78, 104.53, 102.2, 97.82, 98.88, 100.65, 101.44, 100.78, 105.78, 99.12, 102.26, 96.98, 103.95, 103.32, 105.83, 101.82, 99.36, 104.95, 101.25, 101.84, 102.13, 105.61, 102.08, 99.95, 107.94, 97.45, 99.8, 105.01, 98.44, 102.12, 103.34, 99.55, 102.42, 103.43, 101.64, 98.05, 104.17, 106.8, 106.78, 100.64, 103.61, 99.67, 104.62, 100.41, 101.06, 98.1, 100.95, 101.15, 100.5, 103.64, 102.54, 104.4, 100.02, 103.78, 101.49, 99.25, 102.04, 107.65, 104.69, 96.6, 102.08, 97.84, 102.22, 101.68, 103.36, 102.6, 105.26, 101.56, 104.15, 104.21, 99.84, 102.15, 105.22, 102.71, 101.22, 100.35, 103.21, 98.77, 97.38, 101.7, 98.13, 105.05, 104.46, 101.96, 99.69, 96.79, 97.32, 99.09, 103.83, 99.26, 101.51, 97.82, 100.21, 102.08, 102.53, 101.87, 104.47, 101.29, 101.99, 100.8, 100.2, 102.54, 100.32, 101.15, 103.06, 94.9, 102.36, 98.09, 103.44, 100.08, 100.36, 104.36, 101.03, 100.94, 98.5, 108.55, 107.71, 104.07, 99.47, 103.23, 103.68, 104.43, 101.32, 105.88, 105.32, 100.81, 99.85, 99.51, 100.01, 102.2, 99.42, 104.06, 103.62, 99.93, 101.89, 103.3, 100.38, 105.53, 102.52, 103.24, 102.32, 104.26, 104.43, 98.36, 100.11, 102.98, 103.04, 99.92, 101.9, 96.84, 101.97, 101.89, 100.49, 99.71, 100.07, 99.86, 101.26, 103.85, 104.93, 103.56, 102.92, 100.75, 106.1, 97.4, 103.47, 95.93, 103.27, 104.01, 101.26, 103.45, 101.75, 98.53, 102.76, 101.02, 101.24, 100.81, 105.29, 102.32, 105.52, 95.62, 104.94, 100.71, 100.48, 103.66, 102.09, 99.98, 104.07, 100.2, 102.69, 97.76, 100.66, 104.33, 106.52, 101.6, 99.65, 104.59, 101.41, 97.71, 101.25, 100.07, 96.33, 104.7, 100.42, 101.76, 97.88, 103.24, 103.54, 104.33, 106.13, 103.48, 103.2, 101.02, 103.13, 100.4, 103.37, 109.12, 97.32, 106.66, 101.31, 98.31, 99.28, 103.27, 105.5, 103.79, 105.95, 100.87, 103.37, 100.3, 100.94, 107.3, 103.36, 101.89, 106.44, 100.69, 103.0, 102.98, 97.75, 99.5, 103.79, 100.55, 106.21, 100.0, 100.27, 107.25, 100.92, 104.09, 105.38, 100.23, 100.49, 105.6, 104.08, 99.94, 104.86, 98.06, 104.75, 99.29, 101.93, 105.98, 105.5, 104.81, 102.87, 106.08, 105.51, 99.98, 98.73, 100.9, 98.18, 98.39, 102.69, 97.56, 99.76, 99.63, 102.06, 98.54, 100.81, 100.54, 107.06, 100.87, 104.44, 102.14, 107.62, 98.48, 105.42, 104.88, 100.7, 102.1, 102.97, 100.11, 99.81, 99.06, 101.96, 100.01, 103.58, 105.82, 95.28, 104.6, 96.73, 100.58, 96.66, 98.73, 103.96, 102.54, 102.11, 104.55, 99.85, 99.77, 104.25, 105.86, 105.97, 102.86, 101.63, 102.77, 97.09, 100.7, 100.55, 105.79, 99.79, 110.02, 102.36, 101.59, 99.79, 103.41, 104.18, 96.95]；
组2 n=350：[90.73, 90.89, 95.77, 99.91, 93.5, 93.04, 97.08, 92.67, 95.5, 94.81, 92.24, 93.47, 95.76, 88.01, 97.23, 91.31, 94.65, 90.75, 89.16, 92.23, 88.78, 93.29, 87.79, 92.91, 96.24, 93.11, 94.98, 89.21, 92.34, 94.07, 92.31, 97.54, 93.97, 95.17, 97.52, 93.38, 91.63, 96.61, 92.49, 90.6, 96.11, 90.89, 93.1, 98.11, 89.91, 92.4, 97.78, 95.75, 96.57, 91.28, 91.37, 94.48, 91.76, 96.04, 94.9, 95.87, 90.7, 97.58, 93.4, 94.23, 89.56, 98.48, 98.44, 94.54, 93.86, 93.36, 88.41, 96.02, 94.29, 96.29, 93.3, 93.95, 93.56, 91.54, 91.98, 92.37, 90.47, 97.56, 93.14, 96.58, 93.29, 94.56, 91.07, 93.66, 95.17, 92.59, 92.43, 92.44, 99.15, 88.06, 96.72, 96.36, 94.82, 92.35, 92.76, 95.24, 91.35, 93.76, 92.97, 97.21, 91.41, 89.4, 93.48, 91.59, 95.57, 97.27, 95.62, 97.08, 97.97, 86.15, 94.18, 92.17, 100.5, 95.08, 95.91, 98.68, 92.51, 96.48, 95.85, 97.32, 97.01, 98.52, 93.7, 95.7, 93.66, 93.43, 97.11, 92.7, 92.64, 93.67, 91.69, 95.3, 93.77, 96.43, 97.58, 91.71, 96.35, 95.39, 93.08, 94.26, 95.29, 92.22, 89.36, 89.29, 97.17, 93.63, 93.97, 95.85, 92.47, 95.39, 91.31, 95.08, 94.83, 95.48, 95.45, 92.92, 95.17, 92.06, 99.73, 94.81, 95.32, 90.69, 95.26, 94.8, 97.59, 98.05, 98.38, 91.54, 94.62, 92.81, 94.39, 91.67, 98.33, 94.29, 99.53, 92.85, 94.68, 98.49, 94.63, 93.5, 97.95, 89.64, 93.49, 93.17, 94.86, 95.82, 93.88, 95.2, 90.17, 89.24, 95.92, 98.82, 90.0, 98.41, 100.23, 91.7, 98.99, 90.76, 91.34, 93.25, 94.81, 95.36, 94.64, 93.32, 95.56, 96.81, 90.76, 93.74, 93.91, 95.04, 90.04, 99.8, 92.4, 92.19, 93.56, 99.5, 94.54, 95.82, 100.77, 97.2, 99.84, 96.0, 94.9, 95.59, 89.49, 97.63, 96.4, 91.47, 96.46, 95.38, 93.56, 93.53, 94.3, 95.56, 97.65, 100.06, 88.35, 92.13, 92.9, 96.04, 91.26, 94.18, 94.35, 93.43, 91.25, 92.95, 96.73, 89.41, 91.07, 99.05, 95.32, 90.99, 99.25, 91.42, 97.24, 94.74, 90.26, 92.67, 96.2, 92.62, 95.08, 93.98, 95.32, 98.26, 97.6, 95.66, 91.42, 95.02, 98.04, 97.81, 90.95, 97.58, 91.43, 92.4, 90.69, 96.64, 93.89, 96.46, 93.2, 95.33, 88.97, 94.19, 89.8, 95.93, 94.77, 96.62, 89.1, 97.55, 96.25, 96.6, 95.31, 94.09, 91.51, 91.92, 94.46, 92.24, 96.3, 97.16, 93.49, 95.39, 97.08, 90.57, 92.79, 92.4, 92.9, 94.52, 96.25, 94.61, 95.35, 94.72, 96.74, 93.79, 95.78, 93.92, 97.75, 93.83, 96.45, 94.72, 90.24, 93.03, 92.03, 89.0, 90.98, 93.66, 97.68, 92.06, 92.48, 98.88, 96.88, 90.33, 97.36, 95.11, 99.71, 93.34, 94.39, 93.81, 91.92, 95.05, 91.65, 92.45, 96.56, 97.46, 97.26, 89.61, 93.13, 94.31, 92.02, 97.71, 94.63, 92.19]
组2 n=350：[90.73, 90.89, 95.77, 99.91, 93.5, 93.04, 97.08, 92.67, 95.5, 94.81, 92.24, 93.47, 95.76, 88.01, 97.23, 91.31, 94.65, 90.75, 89.16, 92.23, 88.78, 93.29, 87.79, 92.91, 96.24, 93.11, 94.98, 89.21, 92.34, 94.07, 92.31, 97.54, 93.97, 95.17, 97.52, 93.38, 91.63, 96.61, 92.49, 90.6, 96.11, 90.89, 93.1, 98.11, 89.91, 92.4, 97.78, 95.75, 96.57, 91.28, 91.37, 94.48, 91.76, 96.04, 94.9, 95.87, 90.7, 97.58, 93.4, 94.23, 89.56, 98.48, 98.44, 94.54, 93.86, 93.36, 88.41, 96.02, 94.29, 96.29, 93.3, 93.95, 93.56, 91.54, 91.98, 92.37, 90.47, 97.56, 93.14, 96.58, 93.29, 94.56, 91.07, 93.66, 95.17, 92.59, 92.43, 92.44, 99.15, 88.06, 96.72, 96.36, 94.82, 92.35, 92.76, 95.24, 91.35, 93.76, 92.97, 97.21, 91.41, 89.4, 93.48, 91.59, 95.57, 97.27, 95.62, 97.08, 97.97, 86.15, 94.18, 92.17, 100.5, 95.08, 95.91, 98.68, 92.51, 96.48, 95.85, 97.32, 97.01, 98.52, 93.7, 95.7, 93.66, 93.43, 97.11, 92.7, 92.64, 93.67, 91.69, 95.3, 93.77, 96.43, 97.58, 91.71, 96.35, 95.39, 93.08, 94.26, 95.29, 92.22, 89.36, 89.29, 97.17, 93.63, 93.97, 95.85, 92.47, 95.39, 91.31, 95.08, 94.83, 95.48, 95.45, 92.92, 95.17, 92.06, 99.73, 94.81, 95.32, 90.69, 95.26, 94.8, 97.59, 98.05, 98.38, 91.54, 94.62, 92.81, 94.39, 91.67, 98.33, 94.29, 99.53, 92.85, 94.68, 98.49, 94.63, 93.5, 97.95, 89.64, 93.49, 93.17, 94.86, 95.82, 93.88, 95.2, 90.17, 89.24, 95.92, 98.82, 90.0, 98.41, 100.23, 91.7, 98.99, 90.76, 91.34, 93.25, 94.81, 95.36, 94.64, 93.32, 95.56, 96.81, 90.76, 93.74, 93.91, 95.04, 90.04, 99.8, 92.4, 92.19, 93.56, 99.5, 94.54, 95.82, 100.77, 97.2, 99.84, 96.0, 94.9, 95.59, 89.49, 97.63, 96.4, 91.47, 96.46, 95.38, 93.56, 93.53, 94.3, 95.56, 97.65, 100.06, 88.35, 92.13, 92.9, 96.04, 91.26, 94.18, 94.35, 93.43, 91.25, 92.95, 96.73, 89.41, 91.07, 99.05, 95.32, 90.99, 99.25, 91.42, 97.24, 94.74, 90.26, 92.67, 96.2, 92.62, 95.08, 93.98, 95.32, 98.26, 97.6, 95.66, 91.42, 95.02, 98.04, 97.81, 90.95, 97.58, 91.43, 92.4, 90.69, 96.64, 93.89, 96.46, 93.2, 95.33, 88.97, 94.19, 89.8, 95.93, 94.77, 96.62, 89.1, 97.55, 96.25, 96.6, 95.31, 94.09, 91.51, 91.92, 94.46, 92.24, 96.3, 97.16, 93.49, 95.39, 97.08, 90.57, 92.79, 92.4, 92.9, 94.52, 96.25, 94.61, 95.35, 94.72, 96.74, 93.79, 95.78, 93.92, 97.75, 93.83, 96.45, 94.72, 90.24, 93.03, 92.03, 89.0, 90.98, 93.66, 97.68, 92.06, 92.48, 98.88, 96.88, 90.33, 97.36, 95.11, 99.71, 93.34, 94.39, 93.81, 91.92, 95.05, 91.65, 92.45, 96.56, 97.46, 97.26, 89.61, 93.13, 94.31, 92.02, 97.71, 94.63, 92.19]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=101.9607, s₁=2.8236；x̄₂=94.2374, s₂=2.7361；
SE=√(s₁²/n₁+s₂²/n₂)=0.1786；t=(x̄₁−x̄₂)/SE=43.2465；
df(Welch)=698.39；双侧 p = 1.099e-199。
x̄₁=101.9607, s₁=2.8236；x̄₂=94.2374, s₂=2.7361；
SE=√(s₁²/n₁+s₂²/n₂)=0.1786；t=(x̄₁−x̄₂)/SE=43.2465；
df(Welch)=698.39；双侧 p = 1.099e-199。
计算结果
t = 43.2465，df = 698.39，p = 1.099e-199
在 α=0.05 水平下，p=1.099e-199 < 0.05，拒绝原假设 H₀。比较两班生产效率。
案例 21. 城市与农村收入
原始数据
组1 n=600：[4939.78, 5122.49, 5242.31, 5112.54, 4748.23, 5201.02, 4956.78, 5093.09, 4904.54, 5055.17, 5125.45, 4914.88, 4914.1, 4940.47, 4940.9, 5165.66, 5250.35, 4779.0, 4871.75, 4985.31, 4924.71, 4916.82, 5150.71, 5034.29, 5136.65, 4996.58, 4888.59, 5133.94, 4969.92, 4884.04, 4961.18, 5076.38, 5229.01, 4983.73, 5263.78, 4701.96, 5034.37, 5084.69, 5019.13, 4995.34, 4531.96, 4647.49, 5057.11, 5352.1, 4820.66, 5104.43, 5310.92, 5296.3, 4921.75, 4657.57, 5098.14, 5039.15, 4958.69, 5022.03, 5191.12, 5082.81, 4700.95, 5073.28, 4973.99, 5356.83, 4708.31, 5063.79, 4961.37, 5115.15, 5100.14, 5448.7, 5130.1, 4857.94, 5028.94, 5210.32, 4780.73, 4839.6, 4754.15, 5096.51, 5184.79, 5089.93, 4817.91, 5043.56, 4902.3, 5191.11, 4977.75, 5211.19, 4723.78, 4834.14, 4908.97, 5418.88, 4983.5, 4594.55, 5186.58, 5215.38, 4963.17, 4697.99, 4926.43, 4931.68, 4859.02, 5186.17, 5035.33, 5129.16, 5339.69, 4527.34, 4878.92, 4623.31, 5058.44, 4490.11, 5204.29, 5199.97, 4962.57, 4593.3, 5067.05, 4708.05, 4979.17, 5008.82, 4776.75, 4902.92, 4914.29, 4924.97, 4918.15, 5054.85, 4895.15, 4802.76, 5170.57, 5136.01, 4859.8, 4958.33, 5243.31, 4951.93, 4567.58, 4934.53, 4790.61, 4920.45, 4997.95, 5055.27, 4982.86, 5211.17, 4974.85, 4930.53, 5148.99, 4892.82, 4946.59, 5065.1, 5001.41, 5082.28, 4952.09, 4947.79, 4623.85, 4830.3, 5157.4, 4702.28, 5295.6, 4772.01, 4899.26, 5063.99, 5306.97, 4891.79, 4796.62, 5071.25, 4941.15, 4849.5, 4930.18, 5077.91, 4976.98, 5186.77, 4826.55, 5239.78, 5184.69, 4855.28, 5133.93, 5012.93, 5279.97, 5022.7, 4920.26, 5176.73, 5012.46, 5197.05, 5175.36, 5082.99, 4814.73, 4961.64, 4993.36, 5168.82, 4717.3, 4818.78, 5303.17, 5138.51, 4932.63, 4775.75, 4844.68, 4830.81, 5066.0, 5373.67, 4966.43, 4886.38, 5231.97, 4814.78, 4715.72, 4912.94, 5011.38, 5611.75, 4808.83, 4784.39, 5131.2, 4925.93, 5113.44, 4897.44, 4960.19, 5313.66, 5052.65, 4870.86, 4790.88, 5076.72, 4945.36, 5125.06, 4638.81, 4795.27, 4739.71, 4688.81, 4862.39, 4934.23, 5111.07, 5020.47, 5265.45, 5224.61, 4746.1, 5036.82, 5102.08, 5384.46, 5054.24, 4607.11, 4727.9, 5012.22, 4915.95, 4805.28, 4888.59, 5437.05, 4915.8, 5152.32, 5124.66, 4866.78, 4899.07, 5035.44, 4921.65, 5279.18, 4725.94, 5174.77, 4884.53, 5263.91, 5128.1, 4721.56, 4866.84, 4669.93, 5151.97, 4847.41, 4843.53, 4879.25, 5042.17, 4861.01, 4841.67, 4942.62, 4856.14, 4901.53, 4941.52, 4988.33, 5194.53, 5254.51, 5281.87, 4829.23, 4810.33, 5074.91, 5363.79, 4951.22, 4846.5, 4920.3, 4923.21, 4904.05, 5013.12, 5073.63, 4881.1, 4816.43, 4957.55, 4935.23, 4821.26, 5202.64, 5121.24, 4787.68, 4885.68, 4961.53, 4882.76, 4733.7, 5084.37, 4903.39, 5127.63, 4957.25, 4720.49, 4890.63, 4908.07, 4927.16, 4841.12, 5117.46, 4949.69, 4862.95, 4683.09, 4934.37, 4982.43, 5010.37, 5274.11, 4984.22, 4729.3, 5073.7, 5111.37, 5103.73, 5281.03, 5061.02, 5222.68, 4972.48, 4980.43, 4977.52, 5163.76, 4711.93, 5133.78, 5294.76, 4897.84, 5136.5, 5294.29, 4810.65, 5047.48, 5307.12, 4867.22, 5134.74, 5153.84, 5026.92, 4919.26, 5022.36, 4709.62, 4962.94, 5244.21, 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组2 n=780：[3183.37, 2997.58, 2684.06, 2978.75, 3035.82, 2813.04, 3108.34, 2981.15, 2842.61, 2925.86, 2798.78, 3043.43, 3006.01, 3360.23, 3044.32, 2866.19, 2987.62, 3001.78, 2753.95, 2860.68, 3105.65, 3136.76, 2809.07, 3028.03, 2898.52, 3053.99, 3245.35, 2747.8, 3238.82, 3145.38, 3231.38, 2879.34, 3035.59, 3067.14, 3164.53, 3130.61, 3134.52, 2820.85, 2796.5, 2732.82, 2898.44, 3178.3, 3323.38, 2877.15, 2998.14, 2861.02, 2987.09, 3200.84, 3108.18, 2935.12, 3166.61, 2986.86, 3062.46, 3292.41, 3169.83, 2530.84, 3108.45, 2762.36, 3141.99, 2975.92, 2907.5, 2901.04, 3036.14, 3160.14, 3081.24, 3016.8, 2815.12, 3060.12, 3147.42, 3008.18, 2996.27, 3092.24, 2840.93, 3245.3, 2894.46, 2982.98, 2793.4, 3162.59, 3152.55, 3365.17, 3359.13, 3194.88, 2864.84, 3021.81, 2986.74, 2884.24, 2883.69, 3073.14, 2934.73, 3101.28, 3148.33, 2924.26, 3043.96, 2965.1, 2999.85, 3114.17, 2831.51, 3030.65, 2695.73, 3362.01, 2928.74, 2827.5, 3116.77, 3115.4, 2650.17, 2857.95, 3105.96, 3260.8, 2899.05, 2964.72, 2806.09, 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组2 n=780：[3183.37, 2997.58, 2684.06, 2978.75, 3035.82, 2813.04, 3108.34, 2981.15, 2842.61, 2925.86, 2798.78, 3043.43, 3006.01, 3360.23, 3044.32, 2866.19, 2987.62, 3001.78, 2753.95, 2860.68, 3105.65, 3136.76, 2809.07, 3028.03, 2898.52, 3053.99, 3245.35, 2747.8, 3238.82, 3145.38, 3231.38, 2879.34, 3035.59, 3067.14, 3164.53, 3130.61, 3134.52, 2820.85, 2796.5, 2732.82, 2898.44, 3178.3, 3323.38, 2877.15, 2998.14, 2861.02, 2987.09, 3200.84, 3108.18, 2935.12, 3166.61, 2986.86, 3062.46, 3292.41, 3169.83, 2530.84, 3108.45, 2762.36, 3141.99, 2975.92, 2907.5, 2901.04, 3036.14, 3160.14, 3081.24, 3016.8, 2815.12, 3060.12, 3147.42, 3008.18, 2996.27, 3092.24, 2840.93, 3245.3, 2894.46, 2982.98, 2793.4, 3162.59, 3152.55, 3365.17, 3359.13, 3194.88, 2864.84, 3021.81, 2986.74, 2884.24, 2883.69, 3073.14, 2934.73, 3101.28, 3148.33, 2924.26, 3043.96, 2965.1, 2999.85, 3114.17, 2831.51, 3030.65, 2695.73, 3362.01, 2928.74, 2827.5, 3116.77, 3115.4, 2650.17, 2857.95, 3105.96, 3260.8, 2899.05, 2964.72, 2806.09, 2977.46, 2955.68, 2735.09, 2686.24, 2909.27, 3080.75, 2615.41, 2881.78, 3041.75, 2919.84, 2862.38, 3249.44, 3176.85, 2714.59, 2893.28, 3402.05, 3005.56, 3006.6, 3163.23, 3108.55, 3169.11, 3481.06, 3209.07, 3060.98, 2754.41, 2791.43, 3009.79, 3050.26, 2834.39, 3205.82, 3014.37, 3171.23, 3054.1, 3378.85, 3026.83, 2848.83, 3027.98, 2772.89, 2841.94, 2717.58, 3031.69, 3058.14, 3095.22, 3147.76, 2924.29, 2679.22, 3228.05, 2949.75, 3272.99, 3117.88, 2813.88, 2734.7, 2918.96, 2938.32, 2848.51, 2721.75, 3060.95, 3451.92, 3094.06, 2888.5, 3165.83, 2788.43, 3179.83, 3459.95, 2907.18, 3010.6, 3139.43, 3129.96, 2849.17, 2977.57, 3080.44, 2939.17, 3348.71, 3280.66, 3271.11, 2672.07, 3249.9, 2805.46, 2830.08, 2906.15, 3153.31, 3050.72, 2887.66, 2871.05, 3239.97, 2930.57, 2800.86, 2740.56, 2763.36, 3077.49, 2894.12, 3125.82, 2937.38, 2955.46, 2890.26, 2967.63, 3041.62, 2989.88, 3161.16, 2746.28, 2906.36, 2963.74, 3200.85, 2965.57, 2823.13, 2856.8, 3264.71, 2847.25, 3312.73, 2999.32, 3126.01, 3570.01, 3088.22, 3372.07, 3192.92, 3094.74, 3165.95, 3237.92, 2976.13, 2967.5, 2969.98, 2990.38, 2837.51, 2686.78, 2974.32, 3086.37, 2995.46, 2949.98, 2736.75, 2951.07, 3128.66, 3140.18, 3191.85, 2688.88, 3228.34, 3051.28, 3148.33, 2837.99, 2953.35, 2979.29, 3189.57, 3140.31, 3507.42, 2948.53, 3213.9, 2754.49, 2951.35, 3024.32, 2719.73, 2871.34, 2899.77, 3105.88, 2799.12, 3093.64, 3450.47, 3218.29, 2735.13, 2866.87, 3200.4, 2968.73, 3286.08, 3005.38, 3124.21, 2716.97, 2839.04, 3023.46, 2952.85, 2991.16, 2854.72, 2939.73, 2923.36, 3087.52, 2970.29, 2973.83, 2929.68, 2937.88, 2981.22, 3065.21, 2936.6, 3100.18, 3021.02, 2969.77, 3089.56, 3153.11, 3273.12, 3104.71, 3160.35, 2972.05, 3044.1, 3271.33, 2934.69, 3084.15, 3016.8, 3132.12, 3027.17, 3156.94, 2701.46, 3269.29, 3066.7, 2937.18, 3166.24, 3081.52, 3048.75, 2891.85, 3143.69, 3446.55, 3029.26, 2918.31, 2971.78, 2864.89, 2755.2, 3036.24, 3052.7, 2911.02, 2751.49, 3003.75, 2967.0, 2884.56, 3088.41, 3147.7, 2876.79, 3128.76, 2907.52, 2846.96, 3389.51, 2932.19, 3060.42, 3066.06, 3135.13, 2924.84, 2825.81, 3198.65, 3155.88, 2915.99, 2858.74, 3208.82, 3319.24, 3065.12, 3225.05, 3318.24, 3253.03, 2994.1, 2720.34, 3027.11, 3075.38, 3088.73, 2841.8, 2654.65, 2820.75, 2844.33, 2895.98, 3224.51, 3247.52, 2839.88, 3009.27, 2925.49, 2980.09, 3218.7, 3088.01, 2481.49, 3187.9, 2773.38, 2739.55, 3219.92, 3031.26, 3106.55, 3052.91, 2914.49, 3011.32, 3486.36, 2748.97, 2841.06, 2709.09, 3291.5, 3032.63, 3202.37, 3193.43, 3158.68, 3183.05, 2835.7, 3266.66, 2994.55, 3126.68, 3002.13, 2840.2, 3260.94, 2947.54, 3297.88, 2839.61, 3090.0, 3244.16, 2769.09, 3126.81, 2828.92, 2981.3, 3003.43, 3158.73, 2725.95, 3267.64, 3050.82, 3306.75, 2844.78, 2962.54, 3216.95, 2817.82, 2868.89, 2922.8, 3064.24, 3227.61, 2882.17, 2722.84, 2914.5, 3322.48, 2998.89, 2997.56, 3071.96, 2641.18, 2962.49, 2802.21, 3118.48, 3329.92, 3030.35, 3076.21, 3379.47, 2815.7, 3128.08, 2881.09, 2827.69, 2927.92, 2867.32, 3052.5, 3311.93, 3315.91, 2633.13, 2959.94, 2538.46, 3173.26, 3114.99, 2836.79, 3304.59, 3271.57, 2947.73, 3488.64, 2964.25, 3513.8, 3176.01, 3223.42, 2802.58, 2569.81, 2882.8, 3061.66, 2976.85, 2970.15, 2930.7, 3295.57, 3004.16, 3213.94, 3321.02, 3370.89, 3074.35, 2857.13, 2699.01, 3137.9, 2898.05, 2852.52, 3182.85, 2905.79, 2764.19, 2788.2, 2890.83, 3003.5, 2560.72, 2882.49, 3039.62, 2967.07, 2940.61, 2778.46, 3203.2, 3101.17, 2947.48, 3047.62, 2953.13, 2844.95, 3143.51, 3027.09, 3049.69, 2980.87, 3255.76, 2744.5, 2882.96, 3058.45, 3168.61, 2966.63, 2973.57, 3234.56, 2779.23, 3144.05, 2980.08, 3215.74, 2670.46, 3081.99, 2892.69, 3135.02, 2752.07, 2976.67, 3052.54, 2962.24, 3103.53, 3065.53, 3134.77, 3155.56, 3314.35, 3018.39, 3262.21, 2853.72, 3038.95, 2827.68, 3090.11, 3274.68, 2762.32, 2839.41, 2778.83, 3285.32, 2947.62, 2953.72, 2668.01, 2987.96, 3216.42, 3341.29, 2998.01, 3013.43, 3044.56, 2966.04, 3038.61, 2841.06, 3107.69, 3203.6, 2990.19, 2968.15, 3363.35, 3063.62, 2891.52, 3049.17, 2583.27, 3262.13, 2785.11, 2752.18, 3067.79, 3119.47, 3151.63, 3040.56, 2950.49, 2911.37, 3240.67, 2857.69, 3150.87, 3526.31, 3337.47, 3246.06, 3150.98, 3435.11, 2740.69, 3039.9, 3390.77, 3155.72, 3085.68, 2743.87, 3166.04, 3212.94, 3095.05, 2735.69, 3054.66, 3084.14, 2570.92, 2941.0, 3077.05, 3116.05, 2907.04, 3219.73, 2513.98, 2818.82, 2698.67, 2763.72, 3287.03, 3177.24, 3218.73, 2748.4, 3029.95, 3132.07, 3236.61, 3290.61, 2956.2, 2860.44, 3213.87, 3143.56, 3041.66, 2896.13, 3529.21, 3053.61, 2647.55, 3164.7, 3217.49, 3307.23, 2953.92, 3251.04, 2961.14, 3126.26, 2810.69, 3109.06, 2940.07, 3084.29, 2987.97, 2902.54, 2977.76, 3049.49, 2997.14, 2698.64, 3200.86, 3142.63, 3234.73, 2901.84, 2966.29, 3408.15, 3059.85, 3006.23, 2891.31, 3179.52, 3204.93, 2918.88, 3277.98, 3233.48, 2852.73, 3234.38, 2579.87, 2944.64, 2876.61, 3031.63, 3411.11, 2844.97, 3271.15, 3185.06, 3232.92, 2968.23, 2939.74, 3032.49, 2989.12, 3153.94, 2895.52, 3202.25, 3125.63, 2604.47, 3066.22, 3196.64, 2861.77, 3308.62, 3058.78, 3125.62, 3025.19, 2780.43, 3379.88, 2718.86, 3260.72, 3116.06, 2888.74, 3098.25, 3065.1, 3016.23, 2990.31, 3003.87, 3115.88, 2799.36, 2511.99, 2878.94, 3039.19, 3128.96, 3376.17, 3124.39, 3057.76, 3038.63, 2839.23, 2860.44, 2927.83, 2954.19, 3018.41, 3319.2, 3111.15, 3089.99, 2998.94, 3061.69, 3016.73, 3059.14, 3417.64, 3037.21, 3417.26, 2697.99, 3263.67, 2817.01, 2826.24, 3076.17, 3252.58, 2872.94, 2890.87, 2905.19, 3142.78, 2943.63, 3011.33, 3103.16, 2966.1, 3383.36, 3152.73, 2910.82, 2771.38, 2697.8, 3189.62, 2855.45, 2814.59, 2973.96, 3047.75, 3059.51, 3218.05, 3024.56, 3109.83, 3144.83, 2988.83, 2659.83, 3116.46, 3311.04, 2919.52, 2651.51, 3067.76, 3023.36, 3083.66, 3085.25, 3059.13, 3051.29, 3278.45, 3090.59, 3055.61, 3317.91, 3186.39, 3155.13, 2769.27, 2663.86, 3131.63, 3141.12, 3092.56, 3364.66, 3087.01, 2880.22, 3159.8, 3203.64, 2874.96, 2793.13, 3144.07, 3107.5, 2938.33, 3130.92, 2853.63, 3244.07, 3173.9, 3327.31, 3051.72, 2793.66, 3004.22, 3227.19, 2797.1, 3218.12, 3340.95, 2891.3]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=4983.3098, s₁=184.9401；x̄₂=3022.8967, s₂=187.7730；
SE=√(s₁²/n₁+s₂²/n₂)=10.1098；t=(x̄₁−x̄₂)/SE=193.9121；
df(Welch)=1298.03；双侧 p = 0.000e+00。
x̄₁=4983.3098, s₁=184.9401；x̄₂=3022.8967, s₂=187.7730；
SE=√(s₁²/n₁+s₂²/n₂)=10.1098；t=(x̄₁−x̄₂)/SE=193.9121；
df(Welch)=1298.03；双侧 p = 0.000e+00。
计算结果
t = 193.9121，df = 1298.03，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。量化城乡收入差距。
案例 22. 实验组对照组体重
原始数据
组1 n=646：[69.1, 70.68, 68.51, 69.54, 67.82, 68.61, 68.33, 71.42, 66.79, 75.76, 67.05, 70.28, 68.38, 71.67, 71.5, 71.4, 65.8, 68.54, 73.91, 72.41, 67.89, 70.8, 73.22, 70.77, 67.5, 71.02, 69.42, 69.63, 70.06, 70.12, 70.92, 70.05, 67.58, 69.03, 71.63, 70.38, 71.74, 65.66, 71.96, 66.41, 70.2, 68.78, 72.22, 72.76, 71.85, 68.99, 70.14, 66.76, 67.85, 67.76, 70.29, 69.17, 70.78, 72.38, 70.59, 68.81, 69.35, 68.16, 71.77, 70.85, 70.88, 68.64, 70.04, 68.39, 71.09, 70.92, 71.75, 69.63, 70.62, 67.72, 71.52, 71.93, 73.22, 71.77, 70.98, 69.58, 70.3, 69.51, 68.0, 64.4, 70.45, 70.82, 72.21, 66.6, 68.05, 69.8, 68.81, 66.37, 72.55, 71.48, 66.18, 68.28, 66.53, 69.97, 66.68, 69.49, 73.42, 72.38, 66.93, 69.15, 69.39, 67.76, 71.62, 68.87, 72.09, 67.19, 65.96, 70.96, 67.19, 69.75, 72.47, 67.94, 72.1, 69.82, 68.9, 71.94, 69.28, 68.94, 67.89, 69.47, 63.74, 71.08, 69.81, 71.8, 69.76, 72.75, 70.48, 71.87, 67.15, 73.45, 70.02, 68.11, 71.63, 73.75, 69.16, 72.59, 71.57, 71.9, 69.75, 66.56, 68.59, 71.62, 68.93, 72.67, 69.33, 71.49, 73.05, 67.55, 67.39, 72.12, 72.85, 72.35, 69.56, 67.7, 65.66, 68.51, 68.64, 69.45, 69.0, 69.19, 71.35, 71.97, 70.61, 68.8, 70.34, 70.05, 70.19, 69.14, 70.55, 73.72, 71.4, 73.16, 70.98, 69.12, 67.51, 74.12, 70.63, 72.3, 67.98, 71.07, 71.04, 70.04, 72.39, 68.83, 69.74, 69.1, 70.37, 69.21, 70.59, 70.5, 71.82, 71.1, 71.89, 72.36, 73.25, 71.95, 70.74, 71.59, 69.49, 67.56, 72.48, 72.33, 68.21, 67.74, 72.28, 69.8, 68.07, 70.54, 68.49, 70.67, 70.37, 70.24, 68.87, 67.86, 69.26, 70.71, 70.8, 69.32, 69.52, 71.32, 71.88, 70.76, 66.98, 70.37, 66.96, 72.24, 70.6, 68.45, 70.49, 71.0, 69.91, 71.72, 70.58, 69.54, 69.81, 67.66, 69.96, 68.57, 70.63, 74.08, 69.11, 67.61, 65.56, 69.43, 73.37, 71.28, 70.32, 69.7, 70.44, 66.4, 71.84, 70.62, 74.63, 69.53, 68.56, 72.15, 70.58, 68.28, 71.97, 70.92, 70.73, 71.5, 72.59, 70.68, 70.25, 68.54, 70.36, 69.88, 66.29, 71.95, 64.54, 70.3, 68.1, 67.6, 69.49, 65.42, 72.5, 71.64, 73.73, 71.48, 69.08, 68.59, 70.95, 69.7, 69.74, 72.36, 71.61, 65.57, 69.75, 73.42, 70.47, 69.58, 70.96, 75.2, 70.28, 71.54, 69.26, 68.97, 68.53, 69.0, 70.2, 70.62, 70.58, 68.46, 71.9, 68.21, 65.76, 69.87, 71.22, 70.61, 70.3, 70.63, 70.17, 69.47, 67.27, 71.41, 66.05, 72.61, 70.49, 67.15, 68.96, 66.3, 69.99, 70.77, 70.13, 68.74, 70.17, 70.58, 69.25, 75.96, 68.18, 71.66, 67.06, 69.88, 71.74, 71.41, 68.57, 67.41, 68.82, 69.11, 67.01, 71.87, 71.03, 66.38, 68.21, 73.14, 69.6, 68.86, 67.78, 70.04, 66.56, 67.07, 70.44, 69.23, 71.25, 69.93, 72.86, 70.03, 69.81, 69.99, 68.82, 71.35, 67.84, 66.92, 68.79, 67.75, 70.88, 70.61, 69.97, 66.68, 68.32, 68.98, 67.05, 68.67, 69.24, 69.99, 68.61, 68.01, 70.89, 72.81, 71.44, 64.56, 74.25, 70.5, 70.66, 68.91, 69.03, 68.27, 69.53, 67.37, 68.43, 69.51, 68.65, 70.82, 71.58, 69.32, 68.12, 70.77, 70.4, 71.53, 67.75, 71.58, 69.42, 72.63, 76.13, 68.84, 68.27, 71.1, 71.57, 67.96, 69.35, 69.92, 70.98, 66.64, 76.63, 71.48, 72.27, 66.37, 71.35, 64.75, 68.65, 70.82, 70.62, 69.55, 64.7, 66.15, 74.35, 69.76, 71.35, 72.91, 71.46, 69.79, 72.7, 72.0, 67.8, 69.06, 69.54, 70.52, 69.87, 71.85, 71.09, 71.92, 69.12, 67.75, 67.19, 71.76, 70.51, 72.62, 66.72, 70.5, 72.94, 70.74, 67.54, 70.28, 71.5, 72.44, 68.41, 66.76, 72.26, 72.63, 70.95, 72.63, 72.08, 72.07, 70.47, 68.72, 68.47, 68.84, 67.68, 64.51, 69.69, 67.15, 70.15, 68.68, 67.29, 74.08, 71.21, 69.47, 72.01, 68.67, 68.74, 71.61, 71.39, 68.6, 70.44, 65.5, 69.45, 68.53, 69.6, 71.8, 68.22, 64.72, 67.36, 69.48, 66.77, 70.83, 71.07, 69.34, 68.85, 69.0, 69.47, 68.06, 68.23, 70.23, 67.63, 71.38, 72.98, 69.39, 67.96, 71.16, 70.12, 72.09, 69.07, 69.4, 68.33, 71.9, 69.96, 70.56, 73.69, 68.6, 70.73, 68.97, 69.85, 72.12, 69.61, 69.99, 71.88, 70.29, 72.75, 68.48, 69.47, 69.65, 72.16, 72.13, 68.98, 69.67, 70.91, 72.62, 66.86, 67.15, 69.74, 68.65, 69.97, 69.65, 71.81, 69.09, 69.39, 70.05, 70.28, 69.34, 70.01, 72.48, 72.24, 74.47, 70.73, 73.91, 68.39, 68.08, 71.5, 68.21, 72.26, 71.7, 71.3, 69.44, 72.08, 71.95, 66.5, 71.81, 71.95, 68.49, 64.5, 73.16, 70.11, 69.13, 72.32, 66.78, 71.14, 67.72, 67.66, 70.78, 69.2, 68.99, 71.2, 71.47, 65.28, 71.99, 70.54, 69.65, 70.0, 68.4, 72.24, 71.72, 69.75, 71.23, 68.0, 70.11, 69.15, 70.72, 67.85, 69.32, 69.92, 69.9, 70.06, 69.91, 67.79, 67.79, 70.91, 71.09, 71.48, 69.98, 71.15, 71.79, 67.83, 71.85, 69.97, 69.49, 67.31, 68.81, 70.25, 69.38, 71.09, 68.16, 68.77, 74.07, 71.53, 69.58, 68.67, 70.02, 70.78, 71.64, 70.05, 69.82, 70.17, 66.73, 69.17, 72.09, 73.59, 68.74, 69.49, 70.22, 68.86, 69.44, 70.75, 70.15, 68.64, 67.81]；
组2 n=427：[74.11, 74.58, 73.99, 73.8, 74.49, 76.29, 75.08, 74.47, 74.28, 73.32, 72.71, 75.41, 73.27, 74.88, 74.46, 71.93, 75.87, 77.35, 72.39, 74.53, 75.0, 75.08, 74.66, 75.63, 75.14, 74.25, 73.92, 74.52, 77.17, 74.25, 75.52, 75.36, 74.93, 77.49, 76.9, 74.88, 75.16, 76.0, 75.71, 75.82, 75.85, 75.45, 74.68, 75.09, 76.38, 74.63, 74.97, 76.2, 74.47, 73.28, 73.75, 74.16, 73.31, 76.45, 74.84, 76.21, 76.98, 75.36, 75.49, 75.74, 77.31, 74.95, 74.83, 75.14, 77.71, 73.81, 76.1, 75.02, 76.29, 72.55, 73.58, 75.65, 76.21, 74.47, 73.28, 73.31, 73.24, 75.59, 74.29, 74.81, 73.64, 73.98, 74.63, 73.15, 76.43, 76.15, 73.93, 75.36, 73.63, 72.86, 75.17, 73.28, 74.35, 74.76, 74.67, 72.91, 75.19, 75.65, 76.81, 76.21, 73.11, 73.48, 75.47, 75.45, 74.14, 75.17, 73.7, 73.05, 75.38, 75.48, 74.15, 75.47, 73.18, 76.45, 73.46, 77.48, 75.65, 74.02, 72.48, 73.14, 74.82, 75.11, 73.89, 76.12, 76.27, 75.65, 73.95, 74.95, 73.24, 73.18, 74.6, 75.63, 73.75, 73.72, 75.32, 72.22, 73.47, 72.78, 76.02, 71.51, 70.7, 74.72, 73.06, 74.77, 75.31, 78.15, 74.36, 73.14, 74.07, 73.81, 74.99, 74.08, 76.45, 73.72, 73.02, 76.07, 72.83, 76.25, 73.42, 74.11, 75.13, 73.24, 74.96, 75.1, 76.21, 74.67, 73.02, 74.43, 73.49, 77.98, 74.06, 74.33, 75.01, 74.22, 76.51, 74.39, 74.17, 74.52, 74.03, 74.56, 75.12, 74.25, 74.8, 74.88, 75.6, 76.56, 73.99, 73.23, 75.4, 76.46, 73.35, 75.02, 74.62, 75.45, 77.93, 74.18, 74.25, 73.53, 73.59, 73.86, 72.98, 74.51, 74.9, 73.46, 76.82, 75.59, 75.42, 76.65, 74.89, 74.77, 73.25, 76.67, 74.12, 73.89, 73.22, 73.39, 71.61, 74.28, 72.49, 75.2, 75.27, 73.99, 75.49, 72.52, 77.25, 75.81, 72.89, 71.77, 76.29, 75.1, 73.62, 76.68, 74.3, 76.18, 76.13, 74.21, 77.37, 76.15, 75.78, 76.37, 73.03, 75.62, 75.35, 74.21, 74.12, 75.79, 75.94, 74.11, 75.23, 74.42, 75.03, 74.69, 74.43, 75.1, 75.15, 74.61, 77.26, 73.96, 74.99, 72.77, 76.38, 72.95, 75.31, 77.21, 74.8, 73.18, 74.88, 75.16, 73.77, 74.5, 76.54, 73.29, 74.63, 74.26, 75.03, 73.18, 72.86, 73.46, 73.75, 76.68, 74.37, 76.18, 73.31, 74.56, 73.99, 73.68, 74.79, 74.7, 73.85, 75.8, 74.25, 74.53, 74.71, 76.32, 75.24, 75.54, 74.24, 73.64, 75.9, 76.27, 73.39, 75.69, 73.57, 72.36, 74.77, 73.22, 74.38, 73.41, 74.46, 74.39, 74.5, 74.45, 73.29, 76.23, 75.04, 74.83, 76.79, 73.39, 75.03, 71.7, 75.19, 74.82, 76.16, 74.05, 74.23, 71.86, 75.39, 75.53, 75.79, 74.44, 72.05, 76.42, 73.56, 73.38, 73.31, 76.23, 76.66, 75.64, 74.03, 75.08, 77.3, 75.73, 72.08, 74.86, 75.7, 75.05, 73.34, 72.01, 72.16, 73.46, 76.93, 73.79, 72.9, 74.65, 75.13, 75.33, 78.44, 74.89, 76.15, 73.67, 74.03, 78.26, 74.16, 74.72, 77.07, 74.93, 76.21, 76.81, 76.43, 73.69, 75.11, 74.16, 77.02, 74.86, 76.91, 73.95, 74.64, 75.87, 75.54, 75.58, 74.95, 76.46, 73.4, 73.2, 75.08, 75.15, 75.97, 75.17, 76.57, 74.28, 75.16, 75.12, 77.32, 74.03, 77.38, 73.82, 73.85, 74.75, 75.76, 74.51, 74.93, 75.32, 74.07, 74.63, 75.71, 77.8, 73.29, 75.53, 74.0, 75.98, 71.62, 73.98, 72.17, 75.92, 74.58, 73.98, 73.68, 74.48, 75.5, 74.85, 75.24, 74.64, 74.12, 74.35, 74.65, 75.04, 74.94]
组2 n=427：[74.11, 74.58, 73.99, 73.8, 74.49, 76.29, 75.08, 74.47, 74.28, 73.32, 72.71, 75.41, 73.27, 74.88, 74.46, 71.93, 75.87, 77.35, 72.39, 74.53, 75.0, 75.08, 74.66, 75.63, 75.14, 74.25, 73.92, 74.52, 77.17, 74.25, 75.52, 75.36, 74.93, 77.49, 76.9, 74.88, 75.16, 76.0, 75.71, 75.82, 75.85, 75.45, 74.68, 75.09, 76.38, 74.63, 74.97, 76.2, 74.47, 73.28, 73.75, 74.16, 73.31, 76.45, 74.84, 76.21, 76.98, 75.36, 75.49, 75.74, 77.31, 74.95, 74.83, 75.14, 77.71, 73.81, 76.1, 75.02, 76.29, 72.55, 73.58, 75.65, 76.21, 74.47, 73.28, 73.31, 73.24, 75.59, 74.29, 74.81, 73.64, 73.98, 74.63, 73.15, 76.43, 76.15, 73.93, 75.36, 73.63, 72.86, 75.17, 73.28, 74.35, 74.76, 74.67, 72.91, 75.19, 75.65, 76.81, 76.21, 73.11, 73.48, 75.47, 75.45, 74.14, 75.17, 73.7, 73.05, 75.38, 75.48, 74.15, 75.47, 73.18, 76.45, 73.46, 77.48, 75.65, 74.02, 72.48, 73.14, 74.82, 75.11, 73.89, 76.12, 76.27, 75.65, 73.95, 74.95, 73.24, 73.18, 74.6, 75.63, 73.75, 73.72, 75.32, 72.22, 73.47, 72.78, 76.02, 71.51, 70.7, 74.72, 73.06, 74.77, 75.31, 78.15, 74.36, 73.14, 74.07, 73.81, 74.99, 74.08, 76.45, 73.72, 73.02, 76.07, 72.83, 76.25, 73.42, 74.11, 75.13, 73.24, 74.96, 75.1, 76.21, 74.67, 73.02, 74.43, 73.49, 77.98, 74.06, 74.33, 75.01, 74.22, 76.51, 74.39, 74.17, 74.52, 74.03, 74.56, 75.12, 74.25, 74.8, 74.88, 75.6, 76.56, 73.99, 73.23, 75.4, 76.46, 73.35, 75.02, 74.62, 75.45, 77.93, 74.18, 74.25, 73.53, 73.59, 73.86, 72.98, 74.51, 74.9, 73.46, 76.82, 75.59, 75.42, 76.65, 74.89, 74.77, 73.25, 76.67, 74.12, 73.89, 73.22, 73.39, 71.61, 74.28, 72.49, 75.2, 75.27, 73.99, 75.49, 72.52, 77.25, 75.81, 72.89, 71.77, 76.29, 75.1, 73.62, 76.68, 74.3, 76.18, 76.13, 74.21, 77.37, 76.15, 75.78, 76.37, 73.03, 75.62, 75.35, 74.21, 74.12, 75.79, 75.94, 74.11, 75.23, 74.42, 75.03, 74.69, 74.43, 75.1, 75.15, 74.61, 77.26, 73.96, 74.99, 72.77, 76.38, 72.95, 75.31, 77.21, 74.8, 73.18, 74.88, 75.16, 73.77, 74.5, 76.54, 73.29, 74.63, 74.26, 75.03, 73.18, 72.86, 73.46, 73.75, 76.68, 74.37, 76.18, 73.31, 74.56, 73.99, 73.68, 74.79, 74.7, 73.85, 75.8, 74.25, 74.53, 74.71, 76.32, 75.24, 75.54, 74.24, 73.64, 75.9, 76.27, 73.39, 75.69, 73.57, 72.36, 74.77, 73.22, 74.38, 73.41, 74.46, 74.39, 74.5, 74.45, 73.29, 76.23, 75.04, 74.83, 76.79, 73.39, 75.03, 71.7, 75.19, 74.82, 76.16, 74.05, 74.23, 71.86, 75.39, 75.53, 75.79, 74.44, 72.05, 76.42, 73.56, 73.38, 73.31, 76.23, 76.66, 75.64, 74.03, 75.08, 77.3, 75.73, 72.08, 74.86, 75.7, 75.05, 73.34, 72.01, 72.16, 73.46, 76.93, 73.79, 72.9, 74.65, 75.13, 75.33, 78.44, 74.89, 76.15, 73.67, 74.03, 78.26, 74.16, 74.72, 77.07, 74.93, 76.21, 76.81, 76.43, 73.69, 75.11, 74.16, 77.02, 74.86, 76.91, 73.95, 74.64, 75.87, 75.54, 75.58, 74.95, 76.46, 73.4, 73.2, 75.08, 75.15, 75.97, 75.17, 76.57, 74.28, 75.16, 75.12, 77.32, 74.03, 77.38, 73.82, 73.85, 74.75, 75.76, 74.51, 74.93, 75.32, 74.07, 74.63, 75.71, 77.8, 73.29, 75.53, 74.0, 75.98, 71.62, 73.98, 72.17, 75.92, 74.58, 73.98, 73.68, 74.48, 75.5, 74.85, 75.24, 74.64, 74.12, 74.35, 74.65, 75.04, 74.94]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=69.9326, s₁=2.0011；x̄₂=74.7445, s₂=1.3096；
SE=√(s₁²/n₁+s₂²/n₂)=0.1011；t=(x̄₁−x̄₂)/SE=-47.6097；
df(Welch)=1070.91；双侧 p = 1.341e-266。
x̄₁=69.9326, s₁=2.0011；x̄₂=74.7445, s₂=1.3096；
SE=√(s₁²/n₁+s₂²/n₂)=0.1011；t=(x̄₁−x̄₂)/SE=-47.6097；
df(Welch)=1070.91；双侧 p = 1.341e-266。
计算结果
t = -47.6097，df = 1070.91，p = 1.341e-266
在 α=0.05 水平下，p=1.341e-266 < 0.05，拒绝原假设 H₀。评估干预对体重的影响。
案例 23. 两种肥料产量
原始数据
组1 n=780：[408.43, 402.29, 402.24, 397.53, 401.4, 395.37, 400.26, 407.41, 399.84, 395.49, 400.09, 396.14, 403.56, 408.29, 402.71, 393.53, 400.16, 403.94, 406.06, 406.44, 403.11, 394.68, 397.94, 400.84, 408.46, 401.71, 405.96, 401.47, 401.22, 402.91, 398.35, 400.67, 409.02, 405.97, 401.47, 399.74, 405.84, 401.11, 407.99, 395.33, 404.07, 396.21, 401.91, 400.14, 402.93, 400.91, 401.12, 401.21, 396.44, 402.04, 404.03, 400.2, 404.96, 400.26, 410.45, 405.04, 403.96, 396.06, 402.05, 400.85, 407.48, 403.67, 401.85, 407.3, 397.94, 405.77, 392.44, 399.99, 397.9, 402.72, 410.19, 403.05, 396.52, 404.37, 397.55, 408.24, 394.61, 406.75, 407.24, 410.69, 397.52, 400.91, 399.83, 397.66, 409.77, 396.44, 403.25, 401.37, 401.36, 407.79, 399.22, 401.24, 398.92, 397.01, 402.59, 394.62, 402.44, 396.19, 407.77, 397.03, 397.98, 403.76, 398.27, 398.51, 394.6, 400.77, 400.51, 409.82, 399.26, 399.82, 399.53, 409.31, 401.31, 405.89, 406.77, 405.26, 397.86, 403.14, 411.83, 401.96, 408.26, 396.79, 402.3, 402.47, 406.89, 401.05, 402.71, 405.52, 402.18, 393.85, 398.48, 410.88, 407.42, 402.75, 407.23, 415.52, 398.2, 405.9, 397.32, 392.28, 397.22, 395.27, 387.18, 401.43, 398.56, 402.32, 399.96, 405.02, 401.31, 401.64, 397.16, 396.37, 392.0, 397.38, 404.76, 401.43, 404.01, 391.16, 398.98, 403.09, 405.2, 399.27, 401.78, 399.18, 389.13, 406.94, 406.65, 399.28, 410.76, 403.32, 401.27, 401.36, 401.3, 408.56, 393.54, 398.27, 397.36, 404.45, 402.64, 399.88, 402.42, 398.19, 402.2, 399.55, 405.83, 401.01, 405.17, 399.61, 395.5, 400.56, 401.47, 399.15, 396.12, 400.96, 401.3, 404.97, 409.1, 407.98, 405.37, 406.94, 408.56, 407.32, 393.56, 398.76, 401.43, 414.49, 405.29, 403.09, 401.11, 406.18, 398.6, 402.81, 394.74, 397.65, 399.53, 401.97, 398.54, 402.46, 396.71, 391.85, 402.0, 403.63, 403.27, 402.37, 403.02, 402.14, 392.54, 408.05, 397.9, 400.89, 405.94, 398.12, 405.33, 403.44, 404.88, 403.14, 404.82, 405.02, 400.27, 407.21, 396.89, 401.72, 401.49, 397.41, 397.26, 401.96, 403.65, 403.69, 401.43, 405.02, 407.96, 404.17, 410.11, 395.34, 398.88, 401.75, 403.63, 394.62, 404.84, 391.91, 404.86, 402.43, 401.77, 403.76, 400.13, 405.17, 395.52, 405.14, 402.32, 394.52, 394.35, 401.11, 406.57, 401.92, 396.4, 403.77, 398.51, 397.17, 404.54, 405.35, 408.4, 399.34, 398.99, 399.06, 399.95, 400.05, 396.64, 395.86, 399.87, 396.41, 388.47, 401.24, 406.19, 399.78, 403.12, 404.0, 400.6, 393.74, 400.99, 408.6, 397.58, 403.81, 400.7, 402.67, 408.24, 397.25, 399.66, 401.26, 403.59, 406.74, 394.04, 395.87, 395.52, 401.71, 393.63, 409.59, 401.32, 393.59, 401.16, 393.9, 408.33, 398.89, 396.12, 409.99, 399.98, 402.39, 408.09, 398.92, 399.29, 396.56, 393.39, 403.4, 401.0, 394.86, 398.48, 400.94, 402.68, 394.67, 400.71, 406.43, 395.0, 399.32, 406.13, 394.18, 401.75, 403.14, 387.07, 402.17, 403.14, 404.97, 396.89, 406.22, 404.08, 397.91, 402.94, 401.67, 405.96, 399.12, 396.54, 400.63, 408.2, 399.65, 405.27, 400.22, 400.89, 405.56, 402.02, 399.74, 406.32, 400.82, 400.56, 398.58, 404.04, 398.73, 393.52, 398.37, 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组2 n=740：[382.65, 386.35, 386.2, 381.3, 384.4, 384.17, 387.27, 383.18, 388.61, 382.96, 382.19, 390.23, 379.53, 383.92, 390.85, 376.41, 393.91, 396.03, 383.96, 382.83, 385.04, 386.59, 386.19, 396.42, 389.16, 385.98, 381.35, 388.61, 387.25, 382.9, 372.97, 382.47, 384.26, 383.41, 378.66, 387.92, 397.9, 381.96, 385.18, 385.67, 387.38, 385.02, 392.83, 377.21, 385.94, 376.7, 387.28, 381.37, 385.54, 387.92, 383.23, 392.18, 393.8, 386.43, 377.34, 378.44, 378.36, 392.56, 394.72, 384.02, 388.39, 380.98, 383.79, 380.52, 383.55, 384.56, 393.61, 386.86, 385.8, 385.45, 386.88, 389.46, 384.36, 384.23, 382.88, 388.81, 377.91, 382.59, 393.47, 383.7, 386.25, 385.27, 383.82, 380.26, 387.44, 379.61, 384.0, 386.5, 382.06, 381.09, 378.5, 380.57, 379.05, 374.12, 379.37, 379.67, 375.89, 381.7, 396.41, 381.12, 377.41, 391.56, 388.8, 385.25, 387.19, 379.66, 390.32, 372.02, 383.03, 380.04, 377.21, 379.76, 384.61, 386.77, 385.12, 383.27, 384.15, 384.27, 392.21, 386.58, 384.42, 382.62, 378.22, 387.64, 377.53, 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386.7, 384.41, 384.02, 380.63, 396.31, 387.51, 383.95, 378.24, 388.79, 392.72, 394.35, 388.34, 377.75, 379.5, 386.23, 384.5, 370.33, 380.62, 388.38, 388.26, 388.64, 385.74, 385.43, 385.03, 383.43, 389.28, 381.73, 385.67, 382.03, 393.35, 375.83, 391.16, 382.99, 389.21, 384.42, 380.69, 387.82, 381.4, 374.63, 378.93, 379.27, 385.31, 379.38, 380.26, 373.31, 375.32, 393.19, 380.94, 381.33, 381.54, 384.3, 373.15, 393.72, 384.3, 390.01, 376.41, 387.17, 388.75, 389.14, 383.24, 390.53, 377.16, 379.84, 386.82, 389.26, 377.72, 376.6, 377.58, 385.17, 386.72, 389.12, 385.21, 385.77, 392.69, 386.34, 384.46, 387.6, 376.05, 383.8, 376.6, 389.1, 381.95, 372.94, 385.97, 379.14, 386.82, 378.75, 381.84, 382.47, 387.72, 386.16, 384.41, 380.8, 378.6, 385.44, 384.24, 390.52, 387.6, 371.55, 386.39, 384.64, 389.49, 378.14, 376.53, 378.82, 386.58, 379.83, 398.61, 378.49, 383.93, 386.17, 387.64, 380.65, 377.5, 381.78, 390.02, 378.1, 387.16, 385.46, 385.07, 380.91, 388.8, 381.61, 382.84, 395.03, 397.83, 386.74, 374.21, 391.4, 380.93, 382.35, 381.52, 383.94, 387.31, 394.73, 388.1, 382.52, 385.11, 386.38, 393.75, 384.41, 378.42, 389.51, 390.97, 387.77, 386.55, 379.18, 379.49, 391.75, 381.91, 377.88, 380.73, 378.79, 381.99, 380.87, 384.89, 388.69, 383.02, 371.18, 381.67, 381.92, 374.8, 381.52, 390.58, 390.05, 391.64, 381.07, 387.92, 382.5, 383.71, 396.6, 387.57, 385.12, 387.07, 381.77, 382.32, 390.97, 387.93, 387.46, 382.07, 384.93, 381.52, 375.08, 391.27, 379.82, 380.76, 393.94, 385.27, 391.57, 385.31, 377.89, 378.74, 380.04, 391.69, 390.04, 380.39, 383.9, 389.97, 388.12, 386.72, 393.38, 391.67, 390.89, 384.1, 378.52, 386.88, 384.29, 383.71, 387.5, 379.58, 391.24, 378.18, 388.42, 390.09, 379.79, 387.52, 386.51, 389.72, 387.91, 384.22, 376.57, 393.29, 373.6, 380.61, 390.87, 386.88, 386.21, 379.92, 380.23, 388.05, 382.57, 382.59, 388.04, 372.31, 378.55, 384.92, 381.18, 379.98, 391.86, 383.37, 379.03, 395.8, 388.91, 375.05, 386.44, 376.71, 392.33, 393.18, 383.83, 380.41, 373.93, 382.64, 397.89, 391.73, 378.15, 398.44, 388.52, 384.75, 381.56, 386.24, 380.24, 382.41, 377.81, 385.47, 384.17, 380.16, 386.69, 388.87, 371.23, 379.27, 376.4, 382.74, 371.76, 378.99, 390.81, 385.87, 377.19, 384.18, 389.8, 389.19, 381.27, 379.15, 387.84, 381.88, 383.24, 378.66, 386.74, 382.98, 377.61, 374.13, 391.39, 388.33, 381.25, 384.02, 391.84, 384.49, 379.13, 393.41, 380.6, 388.72, 385.48, 393.25, 388.43, 386.2, 385.78, 393.89, 382.12, 382.97, 386.12, 385.85, 373.38, 389.53, 383.37, 393.17, 383.19, 388.03, 392.69, 386.82, 387.41, 381.33, 382.29, 385.52, 387.52, 373.64, 380.1, 378.72, 380.42, 374.58, 378.63, 391.1, 384.62, 379.74, 382.93, 382.67, 378.6, 383.65, 392.14, 386.14, 385.92, 386.17, 389.24, 388.27, 380.2, 378.71, 387.01, 388.28, 376.73, 387.77, 379.13, 385.64, 384.37, 391.27, 380.09, 392.14, 380.48, 384.42, 383.48, 380.67, 385.93, 395.53, 386.2, 387.72]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=401.4305, s₁=4.8136；x̄₂=384.1494, s₂=5.4784；
SE=√(s₁²/n₁+s₂²/n₂)=0.2651；t=(x̄₁−x̄₂)/SE=65.1939；
df(Welch)=1469.92；双侧 p = 0.000e+00。
x̄₁=401.4305, s₁=4.8136；x̄₂=384.1494, s₂=5.4784；
SE=√(s₁²/n₁+s₂²/n₂)=0.2651；t=(x̄₁−x̄₂)/SE=65.1939；
df(Welch)=1469.92；双侧 p = 0.000e+00。
计算结果
t = 65.1939，df = 1469.92，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。比较肥料增产效果。
案例 24. 早班晚班效率
原始数据
组1 n=365：[52.27, 48.99, 49.84, 47.41, 50.57, 50.06, 47.92, 46.92, 53.78, 48.51, 50.51, 53.11, 49.25, 51.46, 49.24, 50.85, 54.5, 52.95, 50.84, 51.97, 48.94, 52.13, 50.65, 49.31, 50.88, 50.04, 49.54, 49.95, 47.81, 53.7, 50.18, 50.48, 50.19, 47.43, 53.31, 50.33, 51.62, 53.18, 45.78, 50.33, 52.11, 48.41, 53.3, 48.79, 49.07, 48.14, 48.78, 50.23, 46.48, 47.83, 52.36, 48.95, 51.54, 48.04, 49.69, 49.68, 52.91, 43.62, 52.7, 46.46, 48.21, 53.18, 50.31, 45.06, 50.61, 48.26, 50.96, 48.46, 48.87, 54.66, 50.84, 48.9, 46.47, 51.25, 49.95, 47.8, 52.13, 50.85, 50.57, 52.7, 48.56, 49.18, 49.71, 48.83, 50.3, 49.55, 50.02, 48.85, 52.21, 45.85, 50.71, 51.4, 54.71, 51.15, 49.83, 53.58, 47.63, 49.85, 49.53, 53.21, 50.73, 49.18, 50.5, 49.65, 50.64, 51.82, 48.21, 50.75, 53.36, 47.39, 49.02, 47.56, 50.34, 50.42, 53.29, 50.28, 52.6, 48.86, 53.36, 53.0, 49.73, 48.29, 50.33, 48.07, 52.07, 49.9, 50.36, 48.53, 54.13, 50.43, 50.63, 51.4, 47.95, 48.38, 48.57, 51.15, 47.04, 48.8, 47.74, 47.09, 51.76, 50.29, 50.6, 50.99, 47.53, 49.55, 48.79, 48.78, 48.66, 52.9, 52.44, 48.81, 51.55, 47.97, 51.06, 50.74, 49.52, 49.88, 51.32, 51.05, 49.01, 51.46, 48.08, 49.77, 50.93, 48.56, 50.38, 51.55, 49.76, 51.39, 50.41, 50.53, 51.44, 51.2, 51.46, 52.0, 51.22, 51.98, 53.72, 49.44, 52.4, 51.92, 54.5, 50.38, 47.07, 51.28, 49.93, 51.58, 50.88, 50.13, 49.57, 52.01, 49.95, 48.55, 49.26, 47.68, 51.0, 49.29, 50.6, 48.99, 46.9, 50.43, 51.28, 48.04, 50.88, 46.09, 50.95, 48.66, 47.4, 51.86, 50.02, 49.51, 48.41, 45.36, 51.13, 47.28, 49.89, 49.04, 48.34, 53.53, 52.57, 49.96, 50.72, 49.44, 48.83, 48.08, 49.76, 49.52, 49.63, 46.39, 49.46, 47.33, 53.31, 49.68, 53.6, 48.2, 48.15, 48.74, 48.45, 51.2, 52.46, 50.32, 47.81, 49.13, 49.92, 50.24, 51.28, 47.4, 51.0, 50.26, 48.09, 47.2, 52.63, 52.33, 47.39, 50.51, 52.59, 50.74, 49.64, 46.84, 50.23, 51.36, 53.03, 52.45, 49.26, 49.96, 51.36, 48.59, 51.31, 50.18, 51.07, 51.12, 50.88, 49.45, 51.95, 50.29, 48.64, 51.47, 49.5, 55.77, 48.99, 48.83, 49.04, 49.26, 51.19, 50.38, 49.11, 50.57, 51.76, 51.71, 49.29, 52.03, 50.5, 49.7, 48.19, 47.49, 50.84, 51.3, 49.64, 48.56, 48.94, 51.01, 49.73, 51.42, 51.02, 49.14, 52.5, 46.12, 50.74, 47.51, 49.56, 54.43, 47.85, 48.75, 48.81, 48.98, 49.62, 51.15, 50.14, 48.53, 52.86, 51.5, 47.87, 49.99, 46.61, 50.75, 53.81, 50.93, 48.98, 49.57, 50.24, 50.59, 52.23, 50.42, 53.08, 51.38, 49.77, 50.32, 50.0, 49.82, 47.76, 49.3, 48.39, 50.56, 49.91, 47.2, 52.76, 51.85, 50.66, 51.23, 49.47, 49.48, 49.33, 51.06, 51.02, 52.67, 50.2, 52.84, 53.42, 45.78, 50.01, 46.65, 50.51, 50.04, 50.66]；
组2 n=382：[45.35, 45.89, 46.47, 44.74, 44.73, 46.42, 45.43, 44.59, 43.97, 45.84, 46.08, 45.24, 44.62, 45.56, 46.2, 47.34, 46.74, 45.77, 41.72, 46.61, 48.74, 45.18, 43.57, 44.57, 46.5, 46.82, 45.51, 47.66, 46.31, 48.95, 43.93, 43.08, 43.42, 46.73, 45.54, 44.01, 45.85, 45.98, 47.22, 44.39, 44.19, 45.83, 45.14, 48.95, 44.34, 47.6, 43.1, 45.76, 44.75, 44.96, 44.63, 47.49, 46.49, 46.31, 46.47, 44.2, 46.06, 43.77, 46.7, 44.07, 44.06, 42.18, 44.51, 46.57, 45.07, 42.3, 47.37, 46.21, 47.05, 46.5, 46.44, 46.41, 43.48, 47.21, 46.36, 48.4, 45.38, 45.96, 42.44, 45.57, 46.36, 44.52, 46.58, 45.2, 45.1, 47.28, 43.71, 42.53, 47.15, 47.59, 47.3, 46.44, 44.93, 42.67, 44.81, 46.73, 44.11, 46.05, 45.6, 48.75, 43.26, 42.92, 44.28, 46.43, 44.59, 44.82, 45.03, 40.81, 47.57, 45.63, 44.12, 46.13, 45.31, 45.89, 45.25, 43.47, 45.46, 42.3, 43.51, 45.55, 45.2, 49.87, 45.5, 47.56, 45.06, 45.22, 44.5, 44.44, 44.88, 43.93, 43.71, 44.44, 47.1, 45.6, 44.3, 45.36, 46.73, 44.59, 44.65, 46.68, 46.14, 47.55, 47.38, 46.86, 43.25, 47.3, 43.59, 45.54, 45.44, 44.79, 46.71, 45.38, 46.17, 45.61, 47.14, 45.14, 44.08, 44.68, 45.53, 42.83, 46.91, 48.17, 47.66, 44.07, 44.28, 48.57, 43.63, 45.82, 44.42, 45.98, 46.64, 45.35, 42.25, 44.53, 42.69, 44.0, 44.99, 45.08, 46.65, 45.76, 45.15, 47.03, 44.71, 46.52, 46.07, 48.54, 44.21, 46.35, 45.87, 46.09, 44.15, 49.84, 45.45, 45.51, 44.43, 46.18, 46.41, 46.99, 43.51, 44.6, 47.9, 45.61, 45.32, 43.74, 44.98, 44.84, 45.62, 44.8, 47.64, 44.8, 47.44, 47.63, 45.94, 46.33, 43.91, 46.18, 45.32, 46.33, 42.76, 49.25, 47.24, 44.52, 44.86, 46.98, 43.67, 43.93, 45.24, 44.29, 44.03, 44.44, 47.21, 44.94, 46.52, 45.29, 45.96, 46.17, 46.62, 47.23, 44.86, 44.18, 45.33, 48.07, 44.22, 44.73, 44.56, 48.64, 43.03, 45.57, 46.24, 47.92, 45.56, 44.05, 44.89, 45.19, 45.49, 45.9, 48.73, 45.69, 46.17, 46.04, 45.71, 43.37, 44.8, 41.63, 45.57, 46.41, 48.1, 44.37, 46.77, 45.45, 42.59, 45.37, 48.17, 43.98, 41.08, 46.62, 47.56, 45.21, 46.21, 46.95, 47.71, 44.94, 42.7, 41.06, 45.27, 46.49, 46.46, 43.59, 48.23, 44.76, 45.0, 42.95, 43.71, 46.43, 46.14, 47.26, 45.39, 45.59, 44.93, 44.43, 49.96, 48.62, 49.71, 46.09, 44.22, 45.03, 45.06, 45.75, 45.39, 46.25, 44.66, 46.36, 44.5, 45.45, 46.67, 44.55, 42.08, 43.68, 45.54, 47.47, 47.43, 44.95, 43.24, 46.5, 43.45, 43.75, 47.63, 43.59, 47.38, 48.02, 45.39, 43.72, 45.36, 47.66, 44.3, 45.64, 45.88, 47.57, 45.72, 43.23, 46.56, 45.67, 47.32, 44.25, 47.73, 45.2, 45.87, 46.6, 44.34, 45.84, 45.11, 45.74, 47.31, 45.11, 43.63, 46.02, 44.51, 43.9, 42.18, 44.91, 42.19, 45.87, 45.73, 44.64, 43.31, 42.97, 47.69, 44.94, 44.52, 44.8, 45.03, 46.69, 44.74, 45.98, 44.95, 46.82, 47.4, 45.05, 43.85, 46.8, 45.84, 47.09]
组2 n=382：[45.35, 45.89, 46.47, 44.74, 44.73, 46.42, 45.43, 44.59, 43.97, 45.84, 46.08, 45.24, 44.62, 45.56, 46.2, 47.34, 46.74, 45.77, 41.72, 46.61, 48.74, 45.18, 43.57, 44.57, 46.5, 46.82, 45.51, 47.66, 46.31, 48.95, 43.93, 43.08, 43.42, 46.73, 45.54, 44.01, 45.85, 45.98, 47.22, 44.39, 44.19, 45.83, 45.14, 48.95, 44.34, 47.6, 43.1, 45.76, 44.75, 44.96, 44.63, 47.49, 46.49, 46.31, 46.47, 44.2, 46.06, 43.77, 46.7, 44.07, 44.06, 42.18, 44.51, 46.57, 45.07, 42.3, 47.37, 46.21, 47.05, 46.5, 46.44, 46.41, 43.48, 47.21, 46.36, 48.4, 45.38, 45.96, 42.44, 45.57, 46.36, 44.52, 46.58, 45.2, 45.1, 47.28, 43.71, 42.53, 47.15, 47.59, 47.3, 46.44, 44.93, 42.67, 44.81, 46.73, 44.11, 46.05, 45.6, 48.75, 43.26, 42.92, 44.28, 46.43, 44.59, 44.82, 45.03, 40.81, 47.57, 45.63, 44.12, 46.13, 45.31, 45.89, 45.25, 43.47, 45.46, 42.3, 43.51, 45.55, 45.2, 49.87, 45.5, 47.56, 45.06, 45.22, 44.5, 44.44, 44.88, 43.93, 43.71, 44.44, 47.1, 45.6, 44.3, 45.36, 46.73, 44.59, 44.65, 46.68, 46.14, 47.55, 47.38, 46.86, 43.25, 47.3, 43.59, 45.54, 45.44, 44.79, 46.71, 45.38, 46.17, 45.61, 47.14, 45.14, 44.08, 44.68, 45.53, 42.83, 46.91, 48.17, 47.66, 44.07, 44.28, 48.57, 43.63, 45.82, 44.42, 45.98, 46.64, 45.35, 42.25, 44.53, 42.69, 44.0, 44.99, 45.08, 46.65, 45.76, 45.15, 47.03, 44.71, 46.52, 46.07, 48.54, 44.21, 46.35, 45.87, 46.09, 44.15, 49.84, 45.45, 45.51, 44.43, 46.18, 46.41, 46.99, 43.51, 44.6, 47.9, 45.61, 45.32, 43.74, 44.98, 44.84, 45.62, 44.8, 47.64, 44.8, 47.44, 47.63, 45.94, 46.33, 43.91, 46.18, 45.32, 46.33, 42.76, 49.25, 47.24, 44.52, 44.86, 46.98, 43.67, 43.93, 45.24, 44.29, 44.03, 44.44, 47.21, 44.94, 46.52, 45.29, 45.96, 46.17, 46.62, 47.23, 44.86, 44.18, 45.33, 48.07, 44.22, 44.73, 44.56, 48.64, 43.03, 45.57, 46.24, 47.92, 45.56, 44.05, 44.89, 45.19, 45.49, 45.9, 48.73, 45.69, 46.17, 46.04, 45.71, 43.37, 44.8, 41.63, 45.57, 46.41, 48.1, 44.37, 46.77, 45.45, 42.59, 45.37, 48.17, 43.98, 41.08, 46.62, 47.56, 45.21, 46.21, 46.95, 47.71, 44.94, 42.7, 41.06, 45.27, 46.49, 46.46, 43.59, 48.23, 44.76, 45.0, 42.95, 43.71, 46.43, 46.14, 47.26, 45.39, 45.59, 44.93, 44.43, 49.96, 48.62, 49.71, 46.09, 44.22, 45.03, 45.06, 45.75, 45.39, 46.25, 44.66, 46.36, 44.5, 45.45, 46.67, 44.55, 42.08, 43.68, 45.54, 47.47, 47.43, 44.95, 43.24, 46.5, 43.45, 43.75, 47.63, 43.59, 47.38, 48.02, 45.39, 43.72, 45.36, 47.66, 44.3, 45.64, 45.88, 47.57, 45.72, 43.23, 46.56, 45.67, 47.32, 44.25, 47.73, 45.2, 45.87, 46.6, 44.34, 45.84, 45.11, 45.74, 47.31, 45.11, 43.63, 46.02, 44.51, 43.9, 42.18, 44.91, 42.19, 45.87, 45.73, 44.64, 43.31, 42.97, 47.69, 44.94, 44.52, 44.8, 45.03, 46.69, 44.74, 45.98, 44.95, 46.82, 47.4, 45.05, 43.85, 46.8, 45.84, 47.09]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=50.1070, s₁=1.8927；x̄₂=45.4882, s₂=1.5846；
SE=√(s₁²/n₁+s₂²/n₂)=0.1280；t=(x̄₁−x̄₂)/SE=36.0802；
df(Welch)=710.41；双侧 p = 9.135e-163。
x̄₁=50.1070, s₁=1.8927；x̄₂=45.4882, s₂=1.5846；
SE=√(s₁²/n₁+s₂²/n₂)=0.1280；t=(x̄₁−x̄₂)/SE=36.0802；
df(Welch)=710.41；双侧 p = 9.135e-163。
计算结果
t = 36.0802，df = 710.41，p = 9.135e-163
在 α=0.05 水平下，p=9.135e-163 < 0.05，拒绝原假设 H₀。判断班次对效率影响。
案例 25. 线上线下客单价
原始数据
组1 n=312：[147.32, 161.99, 156.43, 149.46, 138.28, 146.31, 150.38, 151.75, 151.39, 166.31, 153.3, 147.58, 151.73, 146.91, 152.87, 148.54, 144.55, 140.85, 143.14, 169.88, 148.25, 150.88, 149.95, 154.98, 155.11, 152.65, 155.88, 157.66, 150.73, 155.26, 154.47, 158.85, 155.53, 145.26, 156.65, 155.35, 155.53, 156.73, 137.0, 135.5, 158.8, 151.51, 153.11, 147.71, 147.93, 151.08, 148.08, 156.48, 154.94, 169.25, 165.15, 140.9, 150.99, 162.56, 152.16, 147.51, 149.1, 156.62, 136.87, 154.01, 146.03, 146.92, 153.56, 158.75, 153.86, 152.38, 166.2, 139.3, 155.24, 157.42, 153.47, 159.86, 155.95, 153.43, 151.31, 163.05, 152.46, 146.92, 162.32, 143.06, 159.71, 143.52, 160.28, 149.03, 151.78, 159.11, 150.22, 156.96, 164.37, 154.05, 159.07, 154.3, 158.37, 158.32, 143.27, 143.18, 142.87, 162.74, 145.25, 159.13, 158.58, 142.08, 153.69, 157.25, 152.08, 159.48, 150.27, 154.86, 168.05, 153.01, 143.02, 154.36, 147.48, 152.73, 139.17, 154.71, 158.53, 160.55, 159.33, 145.54, 142.66, 141.78, 143.58, 154.31, 155.89, 158.47, 152.95, 163.63, 158.96, 155.67, 160.18, 147.17, 145.59, 147.01, 137.96, 143.96, 158.68, 143.36, 150.42, 147.81, 147.77, 150.32, 147.63, 151.89, 153.58, 129.78, 164.26, 156.03, 153.69, 154.86, 145.27, 167.4, 141.71, 149.02, 151.12, 147.63, 161.47, 156.53, 147.01, 149.98, 158.26, 166.87, 156.92, 150.94, 167.37, 155.04, 161.84, 155.96, 143.11, 154.25, 145.97, 146.21, 145.32, 153.85, 153.18, 157.64, 152.39, 152.96, 150.65, 151.4, 164.95, 144.53, 150.63, 156.93, 151.61, 142.6, 155.85, 159.71, 153.34, 151.9, 160.93, 140.09, 150.49, 152.26, 150.39, 139.67, 130.68, 158.63, 152.84, 149.8, 151.13, 140.49, 143.84, 161.07, 155.35, 142.83, 145.63, 154.95, 144.15, 146.2, 141.25, 157.21, 144.15, 154.57, 150.18, 148.47, 137.78, 158.59, 143.09, 159.04, 149.97, 160.54, 154.99, 142.78, 149.19, 144.95, 147.28, 155.23, 155.39, 162.39, 158.97, 152.8, 149.52, 148.7, 139.34, 149.34, 149.46, 155.67, 160.78, 156.73, 150.18, 156.96, 147.93, 148.45, 152.83, 160.4, 151.82, 150.09, 156.32, 153.32, 143.98, 145.44, 153.46, 159.19, 161.03, 158.81, 155.38, 158.93, 157.5, 157.55, 162.02, 155.46, 156.01, 149.92, 152.25, 146.16, 159.17, 146.88, 147.26, 144.92, 153.7, 132.31, 158.37, 150.21, 141.08, 153.24, 137.69, 152.72, 150.0, 155.62, 157.82, 151.13, 167.52, 169.74, 148.75, 147.7, 143.43, 157.35, 160.46, 146.17, 145.82, 146.33, 152.88, 156.2, 149.18, 145.65, 136.01, 162.29, 153.79, 146.73, 146.16, 151.18, 142.52, 156.91, 155.3, 158.9, 160.94, 150.93, 152.44, 162.04, 144.34, 153.21]；
组2 n=542：[125.31, 121.24, 124.75, 122.82, 120.37, 126.63, 128.35, 129.79, 118.96, 118.11, 125.34, 123.19, 119.82, 124.22, 124.73, 128.52, 118.2, 123.77, 123.6, 122.3, 116.87, 113.73, 115.55, 128.87, 124.93, 121.19, 129.52, 123.83, 114.3, 123.32, 123.0, 126.83, 123.04, 118.73, 126.01, 121.64, 124.29, 121.67, 120.94, 127.14, 124.8, 122.5, 117.16, 120.23, 121.51, 121.56, 127.2, 116.12, 123.47, 124.18, 124.3, 126.75, 122.16, 122.84, 120.51, 129.21, 123.34, 131.02, 129.3, 124.63, 123.82, 125.63, 112.48, 122.83, 131.47, 128.41, 126.57, 119.25, 117.9, 118.06, 124.5, 127.14, 124.62, 125.37, 119.91, 127.92, 125.64, 127.72, 127.61, 129.61, 125.17, 121.26, 125.63, 123.03, 126.0, 127.57, 125.77, 123.51, 114.32, 121.84, 114.17, 124.84, 119.36, 126.32, 124.18, 130.53, 123.9, 130.63, 124.79, 127.8, 122.41, 121.98, 119.97, 118.24, 124.48, 121.08, 112.53, 122.16, 122.67, 125.59, 128.52, 126.04, 125.81, 123.63, 120.98, 126.0, 124.64, 121.88, 122.58, 121.61, 128.23, 119.71, 122.13, 128.21, 121.76, 119.05, 128.85, 125.64, 116.77, 118.45, 127.16, 126.27, 121.7, 128.25, 122.58, 124.19, 133.74, 120.22, 123.54, 126.92, 118.31, 118.09, 121.44, 121.24, 125.94, 123.43, 125.85, 116.57, 122.81, 119.93, 120.51, 131.1, 123.7, 126.63, 127.86, 127.83, 118.44, 126.79, 117.66, 130.45, 122.27, 125.07, 127.83, 109.27, 116.6, 122.77, 125.39, 126.87, 126.02, 126.58, 122.48, 128.1, 119.39, 128.12, 131.61, 118.61, 125.42, 122.26, 131.01, 130.31, 125.45, 123.25, 126.11, 122.85, 123.85, 121.64, 127.04, 127.45, 128.86, 126.58, 128.02, 122.17, 123.97, 124.93, 130.29, 130.47, 123.82, 122.25, 121.38, 120.12, 123.27, 121.47, 124.04, 121.44, 111.59, 120.97, 117.45, 123.76, 121.33, 128.98, 122.45, 118.5, 121.89, 117.37, 125.29, 128.83, 119.64, 118.45, 129.71, 119.95, 126.09, 127.69, 125.22, 127.03, 123.72, 121.14, 131.46, 126.26, 122.91, 125.89, 119.08, 120.82, 125.55, 122.6, 118.77, 123.93, 121.51, 122.13, 123.91, 125.93, 125.32, 131.48, 124.03, 122.57, 129.83, 119.92, 125.72, 126.24, 115.22, 126.91, 127.62, 116.4, 120.84, 127.36, 121.19, 120.12, 127.04, 127.73, 125.15, 126.72, 130.74, 121.89, 129.42, 129.71, 123.67, 116.84, 127.69, 117.35, 123.06, 125.87, 120.69, 125.74, 126.45, 121.69, 121.75, 123.54, 120.79, 121.54, 121.85, 123.72, 119.36, 121.3, 123.03, 131.42, 121.64, 125.71, 122.08, 118.73, 122.9, 128.9, 120.39, 127.06, 119.44, 129.63, 126.27, 118.81, 130.43, 121.49, 121.92, 125.67, 121.33, 130.74, 126.24, 119.95, 125.17, 125.47, 118.84, 122.06, 123.29, 123.62, 122.26, 122.94, 118.51, 120.67, 125.56, 125.93, 129.75, 120.92, 126.4, 122.29, 128.67, 124.24, 130.0, 114.26, 125.69, 126.61, 122.94, 124.45, 121.9, 118.54, 126.26, 119.57, 126.0, 120.1, 124.37, 119.75, 124.82, 126.5, 115.94, 126.11, 125.0, 126.2, 123.13, 113.32, 125.19, 123.79, 115.76, 126.27, 129.91, 131.46, 124.88, 129.26, 126.45, 117.08, 124.82, 120.32, 125.18, 129.42, 125.04, 120.02, 128.8, 123.0, 126.41, 123.61, 120.88, 120.65, 123.38, 126.38, 121.29, 118.04, 119.96, 121.59, 121.24, 116.63, 117.53, 131.81, 125.8, 124.09, 117.05, 122.99, 129.4, 122.37, 116.11, 119.62, 121.03, 131.22, 118.71, 117.51, 126.43, 125.33, 119.94, 121.92, 122.17, 130.17, 111.56, 122.24, 120.0, 123.83, 118.79, 120.38, 115.4, 116.26, 120.67, 123.4, 123.48, 121.04, 124.27, 120.48, 125.55, 128.43, 123.45, 123.97, 118.17, 122.65, 122.58, 127.46, 126.32, 129.74, 124.49, 132.3, 119.5, 123.62, 125.18, 124.12, 122.81, 126.77, 114.6, 118.79, 123.83, 126.52, 120.66, 118.24, 120.41, 128.63, 120.1, 124.11, 127.87, 126.37, 127.42, 111.97, 122.41, 121.63, 121.35, 121.94, 123.8, 122.68, 121.99, 122.48, 113.67, 120.77, 121.49, 119.87, 120.98, 117.75, 121.37, 122.87, 125.75, 118.29, 122.86, 118.89, 118.05, 120.96, 122.71, 113.03, 119.09, 131.13, 126.49, 126.95, 130.52, 124.69, 128.44, 117.11, 125.06, 126.63, 118.53, 120.55, 119.48, 121.33, 121.42, 125.97, 119.02, 126.64, 125.29, 120.69, 122.72, 121.52, 124.35, 124.27, 121.36, 129.3, 115.34, 121.44, 125.6, 129.75, 123.0, 114.58, 121.08, 132.49, 122.02, 127.37, 119.69, 123.45, 116.15, 124.95, 121.37, 119.14, 123.5, 126.52, 126.82, 125.76, 116.46, 122.44, 124.96, 123.72, 128.47, 127.69, 121.52, 116.85, 123.95, 119.83, 117.4, 120.98, 125.11, 122.86, 130.03, 127.04, 120.96, 124.48, 128.18, 122.13, 122.51, 126.56, 124.96, 129.72, 123.45, 123.63, 121.84, 125.46, 123.25, 131.87, 125.61, 116.38]
组2 n=542：[125.31, 121.24, 124.75, 122.82, 120.37, 126.63, 128.35, 129.79, 118.96, 118.11, 125.34, 123.19, 119.82, 124.22, 124.73, 128.52, 118.2, 123.77, 123.6, 122.3, 116.87, 113.73, 115.55, 128.87, 124.93, 121.19, 129.52, 123.83, 114.3, 123.32, 123.0, 126.83, 123.04, 118.73, 126.01, 121.64, 124.29, 121.67, 120.94, 127.14, 124.8, 122.5, 117.16, 120.23, 121.51, 121.56, 127.2, 116.12, 123.47, 124.18, 124.3, 126.75, 122.16, 122.84, 120.51, 129.21, 123.34, 131.02, 129.3, 124.63, 123.82, 125.63, 112.48, 122.83, 131.47, 128.41, 126.57, 119.25, 117.9, 118.06, 124.5, 127.14, 124.62, 125.37, 119.91, 127.92, 125.64, 127.72, 127.61, 129.61, 125.17, 121.26, 125.63, 123.03, 126.0, 127.57, 125.77, 123.51, 114.32, 121.84, 114.17, 124.84, 119.36, 126.32, 124.18, 130.53, 123.9, 130.63, 124.79, 127.8, 122.41, 121.98, 119.97, 118.24, 124.48, 121.08, 112.53, 122.16, 122.67, 125.59, 128.52, 126.04, 125.81, 123.63, 120.98, 126.0, 124.64, 121.88, 122.58, 121.61, 128.23, 119.71, 122.13, 128.21, 121.76, 119.05, 128.85, 125.64, 116.77, 118.45, 127.16, 126.27, 121.7, 128.25, 122.58, 124.19, 133.74, 120.22, 123.54, 126.92, 118.31, 118.09, 121.44, 121.24, 125.94, 123.43, 125.85, 116.57, 122.81, 119.93, 120.51, 131.1, 123.7, 126.63, 127.86, 127.83, 118.44, 126.79, 117.66, 130.45, 122.27, 125.07, 127.83, 109.27, 116.6, 122.77, 125.39, 126.87, 126.02, 126.58, 122.48, 128.1, 119.39, 128.12, 131.61, 118.61, 125.42, 122.26, 131.01, 130.31, 125.45, 123.25, 126.11, 122.85, 123.85, 121.64, 127.04, 127.45, 128.86, 126.58, 128.02, 122.17, 123.97, 124.93, 130.29, 130.47, 123.82, 122.25, 121.38, 120.12, 123.27, 121.47, 124.04, 121.44, 111.59, 120.97, 117.45, 123.76, 121.33, 128.98, 122.45, 118.5, 121.89, 117.37, 125.29, 128.83, 119.64, 118.45, 129.71, 119.95, 126.09, 127.69, 125.22, 127.03, 123.72, 121.14, 131.46, 126.26, 122.91, 125.89, 119.08, 120.82, 125.55, 122.6, 118.77, 123.93, 121.51, 122.13, 123.91, 125.93, 125.32, 131.48, 124.03, 122.57, 129.83, 119.92, 125.72, 126.24, 115.22, 126.91, 127.62, 116.4, 120.84, 127.36, 121.19, 120.12, 127.04, 127.73, 125.15, 126.72, 130.74, 121.89, 129.42, 129.71, 123.67, 116.84, 127.69, 117.35, 123.06, 125.87, 120.69, 125.74, 126.45, 121.69, 121.75, 123.54, 120.79, 121.54, 121.85, 123.72, 119.36, 121.3, 123.03, 131.42, 121.64, 125.71, 122.08, 118.73, 122.9, 128.9, 120.39, 127.06, 119.44, 129.63, 126.27, 118.81, 130.43, 121.49, 121.92, 125.67, 121.33, 130.74, 126.24, 119.95, 125.17, 125.47, 118.84, 122.06, 123.29, 123.62, 122.26, 122.94, 118.51, 120.67, 125.56, 125.93, 129.75, 120.92, 126.4, 122.29, 128.67, 124.24, 130.0, 114.26, 125.69, 126.61, 122.94, 124.45, 121.9, 118.54, 126.26, 119.57, 126.0, 120.1, 124.37, 119.75, 124.82, 126.5, 115.94, 126.11, 125.0, 126.2, 123.13, 113.32, 125.19, 123.79, 115.76, 126.27, 129.91, 131.46, 124.88, 129.26, 126.45, 117.08, 124.82, 120.32, 125.18, 129.42, 125.04, 120.02, 128.8, 123.0, 126.41, 123.61, 120.88, 120.65, 123.38, 126.38, 121.29, 118.04, 119.96, 121.59, 121.24, 116.63, 117.53, 131.81, 125.8, 124.09, 117.05, 122.99, 129.4, 122.37, 116.11, 119.62, 121.03, 131.22, 118.71, 117.51, 126.43, 125.33, 119.94, 121.92, 122.17, 130.17, 111.56, 122.24, 120.0, 123.83, 118.79, 120.38, 115.4, 116.26, 120.67, 123.4, 123.48, 121.04, 124.27, 120.48, 125.55, 128.43, 123.45, 123.97, 118.17, 122.65, 122.58, 127.46, 126.32, 129.74, 124.49, 132.3, 119.5, 123.62, 125.18, 124.12, 122.81, 126.77, 114.6, 118.79, 123.83, 126.52, 120.66, 118.24, 120.41, 128.63, 120.1, 124.11, 127.87, 126.37, 127.42, 111.97, 122.41, 121.63, 121.35, 121.94, 123.8, 122.68, 121.99, 122.48, 113.67, 120.77, 121.49, 119.87, 120.98, 117.75, 121.37, 122.87, 125.75, 118.29, 122.86, 118.89, 118.05, 120.96, 122.71, 113.03, 119.09, 131.13, 126.49, 126.95, 130.52, 124.69, 128.44, 117.11, 125.06, 126.63, 118.53, 120.55, 119.48, 121.33, 121.42, 125.97, 119.02, 126.64, 125.29, 120.69, 122.72, 121.52, 124.35, 124.27, 121.36, 129.3, 115.34, 121.44, 125.6, 129.75, 123.0, 114.58, 121.08, 132.49, 122.02, 127.37, 119.69, 123.45, 116.15, 124.95, 121.37, 119.14, 123.5, 126.52, 126.82, 125.76, 116.46, 122.44, 124.96, 123.72, 128.47, 127.69, 121.52, 116.85, 123.95, 119.83, 117.4, 120.98, 125.11, 122.86, 130.03, 127.04, 120.96, 124.48, 128.18, 122.13, 122.51, 126.56, 124.96, 129.72, 123.45, 123.63, 121.84, 125.46, 123.25, 131.87, 125.61, 116.38]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=152.0864, s₁=7.1505；x̄₂=123.3713, s₂=4.1204；
SE=√(s₁²/n₁+s₂²/n₂)=0.4418；t=(x̄₁−x̄₂)/SE=64.9936；
df(Welch)=432.18；双侧 p = 3.297e-225。
x̄₁=152.0864, s₁=7.1505；x̄₂=123.3713, s₂=4.1204；
SE=√(s₁²/n₁+s₂²/n₂)=0.4418；t=(x̄₁−x̄₂)/SE=64.9936；
df(Welch)=432.18；双侧 p = 3.297e-225。
计算结果
t = 64.9936，df = 432.18，p = 3.297e-225
在 α=0.05 水平下，p=3.297e-225 < 0.05，拒绝原假设 H₀。对比渠道客单价差异。
案例 26. 两种教学法成绩
原始数据
组1 n=818：[84.1, 76.39, 81.96, 79.36, 75.74, 75.6, 80.42, 79.94, 71.81, 81.09, 76.49, 83.67, 79.32, 81.03, 80.41, 78.35, 79.67, 84.72, 78.95, 77.89, 81.65, 77.05, 81.71, 75.9, 76.94, 84.43, 75.38, 76.8, 83.81, 76.47, 78.52, 74.68, 80.55, 79.96, 78.11, 77.25, 74.71, 74.85, 80.16, 79.11, 80.85, 83.05, 76.7, 77.54, 77.1, 78.85, 86.47, 81.28, 77.31, 74.37, 79.26, 75.15, 74.72, 73.83, 77.5, 82.08, 77.73, 75.83, 77.17, 80.99, 83.34, 79.79, 82.94, 80.62, 77.33, 80.04, 76.52, 77.82, 75.3, 77.06, 82.61, 80.59, 78.01, 75.67, 79.84, 80.54, 70.75, 76.74, 75.26, 78.58, 74.98, 73.96, 78.26, 78.08, 79.6, 83.78, 77.12, 71.4, 77.43, 83.29, 75.88, 79.17, 84.21, 80.66, 77.18, 77.13, 79.95, 75.63, 80.19, 76.31, 83.17, 82.37, 75.84, 76.16, 80.51, 79.5, 76.39, 80.3, 82.28, 76.91, 81.37, 79.5, 79.9, 75.85, 77.29, 80.37, 80.01, 75.44, 80.09, 85.06, 80.04, 84.25, 79.06, 79.61, 77.5, 75.54, 80.8, 80.27, 79.25, 74.49, 77.6, 79.98, 76.34, 80.54, 79.24, 72.9, 84.67, 78.97, 80.55, 75.07, 80.89, 75.57, 75.2, 77.02, 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68.85, 71.07, 69.55, 72.11, 68.88, 73.38, 73.38, 73.56, 70.64, 73.59, 69.85, 66.66, 73.37, 71.35, 73.95, 78.47, 70.19, 67.98, 71.58, 72.97, 72.5, 71.53, 74.62, 70.1, 70.32, 70.21, 74.01, 71.62, 71.65, 71.19, 70.5, 67.22, 72.38, 72.9, 71.31, 68.25, 73.37, 71.79, 69.49, 67.69, 68.41, 70.26, 67.15, 73.89, 71.66, 71.75, 67.32, 68.62, 71.36, 73.39, 71.59, 70.47, 68.69, 72.87, 70.6, 68.61, 70.08, 74.2, 73.41, 72.11, 72.26, 69.53, 70.23, 71.96, 70.94, 72.31, 67.84, 69.65, 72.24, 71.01, 74.1, 70.8, 68.81, 72.72, 68.83, 69.55, 71.81, 73.46, 71.66, 70.81, 72.21, 67.55, 70.63, 71.98, 72.39, 74.35, 72.91, 72.19, 71.64, 72.67, 71.01, 67.56, 71.5, 73.53, 70.51, 73.73, 67.73, 73.0, 72.08, 74.2, 75.94, 65.98, 70.72, 74.24, 71.1, 71.83, 69.0, 71.85, 73.54, 73.3, 73.99, 73.5, 71.32, 69.17, 73.01, 74.27, 70.73, 68.53, 70.9, 69.83, 70.09, 72.37, 69.42, 69.99, 76.45, 71.41, 71.28, 72.56, 69.68, 68.15, 74.75, 70.18, 71.85, 73.55, 73.92, 68.67, 71.75, 71.53, 75.01, 68.74, 72.74, 71.72, 68.73, 69.85, 71.55, 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71.66, 69.43, 71.91, 74.25, 74.81, 71.17, 71.93, 69.51, 70.73, 73.26, 72.33, 71.61, 74.13, 70.99, 71.89, 71.42, 72.53, 66.2, 69.85, 70.05, 74.92, 69.22, 72.43, 71.69, 69.98, 71.73, 70.05, 73.05, 72.49, 69.66, 69.6, 70.56, 72.71, 72.77, 72.26, 69.76, 73.14, 74.18, 72.13, 73.07, 72.79, 69.37, 70.54, 72.28, 70.84, 72.17, 69.12, 71.72, 69.8, 73.4, 69.72, 71.16, 70.36, 73.36, 70.5, 71.88, 69.95, 70.5, 66.29, 72.95, 73.74, 71.68, 67.82, 69.28, 72.58, 70.23, 71.53, 67.9, 68.33, 70.53, 72.9, 74.53, 73.29, 74.12, 69.66, 69.77, 71.26, 70.34, 75.74, 71.71, 68.89, 72.26, 73.9, 71.55, 70.9, 73.54, 68.29, 71.49, 73.22, 71.66, 67.08, 74.65, 72.14, 72.08, 71.38, 69.42, 70.53, 72.83, 67.64, 74.57, 70.01, 69.3, 71.37, 68.44, 73.08, 71.3, 72.22, 68.83, 70.39, 72.53, 70.03, 73.63, 69.98, 71.19, 73.66, 69.31, 72.17, 68.33, 72.5, 71.33, 69.85, 73.14, 73.59, 71.09, 68.91, 71.75, 72.95, 70.94, 69.7, 75.44, 73.04, 75.24, 73.99, 69.84, 73.53, 73.13, 71.46, 69.44, 70.22, 69.15, 72.16, 72.67, 68.27, 70.14, 69.67, 71.32, 68.76, 74.87, 70.39, 68.02, 71.32, 71.99, 71.3, 75.05, 72.64, 71.76, 71.52, 69.23, 70.05, 72.11, 69.92, 72.52, 69.59, 73.99, 70.2, 73.77, 73.51, 72.09, 72.11, 70.3, 70.89, 71.95, 74.06, 72.35, 74.52, 71.47, 67.4, 75.68, 72.05, 68.54, 69.29, 67.47, 69.05, 73.58, 70.35, 66.36, 73.2, 72.18, 72.31, 74.66, 74.05, 72.48, 71.82, 73.0, 70.8, 70.77, 73.34, 72.44, 66.29, 74.94, 69.2, 70.07, 71.37, 68.93, 70.59, 72.15, 69.34, 68.95, 74.66, 75.06, 71.04, 71.41, 70.41, 69.86, 69.98, 71.14, 72.69, 70.03, 71.48, 70.9, 71.19, 68.11, 73.41, 75.03, 70.85, 71.37, 72.71, 68.6, 72.51, 73.27, 68.08, 71.32, 70.31, 69.13, 72.46, 67.62, 72.24, 72.55, 74.65, 70.97, 70.12, 70.35, 74.4, 70.64, 72.0, 71.96, 72.59, 72.39, 75.56, 69.96, 73.38, 71.0, 70.89, 73.04, 71.16, 74.2, 71.11, 71.52, 70.57, 72.88, 72.17, 69.31, 71.61, 71.47, 70.9, 73.73, 73.24, 72.08, 68.24, 69.47, 69.67, 68.35, 69.43, 70.76, 72.16, 70.65, 68.81, 73.26, 68.7, 72.33, 70.29, 69.69, 70.37, 73.81, 70.71, 74.19, 73.54, 72.21, 72.87, 69.28, 71.37, 70.81, 70.83, 74.02, 71.51, 69.79, 70.45, 73.46, 72.15, 72.1, 73.8, 71.62, 74.11, 70.25, 72.14, 70.27, 71.36, 72.3, 73.18, 74.36, 67.99, 72.4, 71.59, 71.87, 69.58, 73.49, 69.51, 72.25, 71.57, 71.65, 69.32, 71.1, 72.26, 69.94, 69.01, 72.16, 72.67, 75.18, 71.03, 71.07, 71.5, 69.22, 73.96, 72.33, 74.72, 70.59, 75.72, 72.44, 69.74, 76.35, 69.93, 74.44, 71.21, 69.41, 69.2, 71.8, 71.03, 72.02, 71.79, 69.57, 69.27, 75.61, 68.56, 71.03, 68.04, 73.07, 70.2, 72.86, 75.12, 71.28, 73.27, 71.02, 72.29, 70.57, 75.29, 72.51, 73.59, 73.08, 71.23, 73.63, 74.12, 72.99, 69.78]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=78.9186, s₁=2.6791；x̄₂=71.3471, s₂=1.9526；
SE=√(s₁²/n₁+s₂²/n₂)=0.1162；t=(x̄₁−x̄₂)/SE=65.1393；
df(Welch)=1494.55；双侧 p = 0.000e+00。
x̄₁=78.9186, s₁=2.6791；x̄₂=71.3471, s₂=1.9526；
SE=√(s₁²/n₁+s₂²/n₂)=0.1162；t=(x̄₁−x̄₂)/SE=65.1393；
df(Welch)=1494.55；双侧 p = 0.000e+00。
计算结果
t = 65.1393，df = 1494.55，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估教学方法优劣。
案例 27. 两品牌汽车油耗
原始数据
组1 n=545：[7.55, 7.43, 7.35, 7.08, 7.15, 7.28, 7.14, 7.18, 6.92, 6.92, 7.12, 7.44, 7.33, 7.16, 7.06, 7.21, 7.1, 7.16, 6.94, 7.16, 7.15, 7.21, 7.17, 7.26, 7.12, 7.76, 7.5, 6.92, 7.11, 7.17, 7.24, 7.43, 7.04, 7.0, 7.24, 7.56, 7.56, 7.49, 7.29, 7.42, 7.11, 7.2, 7.38, 7.7, 7.12, 7.11, 7.08, 7.86, 7.07, 7.05, 7.43, 7.32, 7.01, 7.58, 7.31, 7.15, 7.57, 7.47, 7.45, 7.1, 7.35, 7.08, 7.54, 7.32, 7.17, 7.48, 7.3, 7.34, 7.44, 7.55, 7.4, 7.02, 7.02, 7.25, 7.81, 7.0, 7.04, 7.07, 7.14, 6.95, 7.27, 7.19, 7.36, 7.29, 7.01, 7.43, 7.03, 7.19, 7.27, 7.47, 7.29, 7.04, 7.32, 7.27, 7.12, 7.12, 7.26, 7.14, 7.26, 7.51, 7.51, 7.49, 7.26, 7.08, 7.42, 7.13, 7.04, 7.41, 7.04, 7.32, 7.5, 7.05, 7.43, 7.45, 7.17, 7.25, 7.08, 7.6, 7.08, 6.96, 7.2, 7.24, 7.27, 7.05, 7.05, 7.36, 7.38, 7.64, 7.27, 6.74, 6.9, 7.29, 7.66, 6.93, 7.41, 7.26, 7.33, 7.21, 7.31, 7.09, 7.3, 6.98, 7.1, 7.46, 7.29, 7.74, 7.68, 7.24, 7.42, 7.31, 7.46, 7.26, 7.18, 7.04, 7.64, 7.34, 7.29, 7.18, 7.07, 7.32, 7.19, 7.39, 7.28, 7.63, 7.23, 7.46, 7.64, 7.29, 7.34, 7.2, 7.02, 7.33, 7.09, 7.14, 7.63, 7.19, 7.41, 7.11, 7.44, 7.44, 7.18, 7.06, 7.42, 7.4, 7.25, 7.24, 7.5, 7.45, 7.17, 7.55, 7.33, 7.52, 7.28, 7.02, 7.3, 7.6, 6.96, 7.35, 7.33, 7.41, 7.39, 7.3, 7.18, 7.39, 7.32, 7.43, 7.21, 7.14, 6.92, 7.38, 6.87, 7.18, 7.44, 7.09, 7.26, 6.91, 7.32, 7.12, 7.51, 7.27, 7.27, 7.05, 7.56, 7.45, 7.1, 7.31, 6.84, 7.15, 7.57, 7.45, 7.09, 7.38, 7.55, 7.09, 7.61, 7.31, 7.55, 7.26, 7.46, 7.36, 7.16, 7.56, 6.93, 7.36, 7.3, 7.46, 7.37, 7.22, 7.24, 7.62, 7.54, 7.31, 7.66, 7.13, 7.39, 7.21, 7.38, 7.5, 7.0, 7.24, 7.42, 7.58, 7.14, 7.5, 7.72, 7.35, 7.58, 7.23, 7.1, 7.29, 6.9, 7.27, 7.32, 7.18, 7.56, 7.14, 7.3, 7.51, 7.57, 7.32, 7.55, 6.96, 6.98, 7.04, 7.59, 6.94, 7.31, 7.42, 7.3, 7.41, 7.47, 7.08, 7.21, 6.77, 7.33, 7.59, 7.56, 7.26, 7.15, 7.43, 7.12, 7.54, 6.97, 7.06, 7.52, 7.29, 7.36, 7.41, 7.34, 7.15, 7.39, 7.47, 7.17, 7.31, 7.12, 7.12, 7.48, 7.16, 7.59, 7.43, 7.23, 7.12, 7.21, 7.37, 6.99, 7.02, 7.32, 7.69, 6.9, 7.18, 7.18, 7.57, 7.28, 7.37, 7.33, 7.27, 7.41, 7.35, 6.59, 7.35, 7.13, 7.36, 7.61, 7.45, 7.33, 7.31, 6.85, 7.65, 7.76, 7.19, 7.33, 7.19, 7.67, 6.94, 6.98, 7.33, 7.42, 7.43, 7.8, 7.36, 7.12, 7.85, 6.92, 6.71, 7.43, 6.99, 6.87, 7.43, 7.0, 7.18, 7.05, 7.27, 7.37, 7.45, 7.43, 7.18, 7.13, 7.28, 7.13, 7.08, 6.97, 7.6, 7.4, 7.46, 6.89, 7.56, 7.22, 7.33, 7.23, 7.05, 7.26, 7.29, 7.48, 7.32, 7.33, 6.9, 7.1, 7.33, 7.19, 7.46, 7.25, 7.1, 7.18, 7.02, 7.14, 7.32, 7.1, 7.23, 7.67, 7.25, 7.53, 7.42, 7.34, 7.26, 7.03, 7.52, 7.23, 7.26, 7.05, 7.07, 7.63, 7.26, 7.13, 7.54, 7.21, 7.61, 6.93, 7.34, 7.07, 7.27, 7.35, 7.35, 7.44, 7.69, 7.33, 7.19, 7.31, 7.39, 7.2, 7.16, 7.38, 7.82, 7.53, 7.66, 7.43, 7.06, 7.51, 7.43, 7.66, 7.69, 7.1, 7.13, 7.54, 7.09, 7.39, 7.3, 7.49, 7.56, 7.51, 7.53, 7.27, 7.31, 7.45, 7.11, 7.0, 7.4, 7.21, 7.55, 6.8, 7.35, 7.62, 7.03, 7.42, 7.15, 7.13, 7.04, 7.43, 7.37, 6.81, 7.47, 6.87, 7.65, 7.21, 7.32, 7.32, 7.54, 7.36, 7.22, 7.39, 7.21, 7.28, 7.33, 7.22, 7.07, 7.18, 7.11, 7.44, 7.41, 7.05, 7.27, 7.06, 7.48, 7.35, 7.15, 7.24, 7.3, 6.8, 7.06, 7.02, 7.01, 7.25, 7.78, 7.07, 7.64, 7.44, 7.28, 7.4, 7.18, 7.57, 7.53, 7.05, 7.45, 7.25, 7.48, 7.46, 7.3, 7.31, 7.42, 7.49, 7.41, 7.63, 7.23, 7.55, 6.8, 7.48, 7.37, 7.77, 7.29, 7.04, 7.57, 7.02, 7.16, 7.23, 7.52, 7.17]；
组2 n=1000：[6.73, 6.79, 6.78, 6.46, 6.76, 6.66, 6.61, 6.59, 7.06, 6.8, 6.66, 6.86, 6.49, 6.76, 6.99, 7.01, 6.66, 6.9, 6.55, 6.38, 6.39, 6.68, 6.75, 6.77, 6.65, 6.74, 6.75, 6.85, 6.65, 6.54, 6.73, 6.47, 6.81, 6.76, 6.91, 6.75, 6.89, 6.73, 6.63, 6.8, 6.62, 6.62, 6.33, 6.96, 6.61, 6.79, 6.8, 6.44, 6.41, 6.76, 6.81, 7.03, 6.8, 6.47, 6.26, 6.6, 6.67, 6.74, 6.75, 6.84, 6.32, 6.6, 6.97, 6.73, 6.45, 6.64, 6.62, 6.65, 6.93, 6.53, 6.78, 6.43, 7.01, 6.74, 6.42, 6.46, 6.42, 6.73, 7.09, 6.59, 6.48, 6.73, 6.69, 6.87, 6.54, 7.07, 6.79, 6.47, 6.62, 6.59, 6.69, 6.87, 6.5, 6.9, 6.47, 6.91, 6.62, 6.74, 6.95, 6.7, 6.78, 6.62, 6.68, 6.81, 7.04, 6.4, 6.65, 6.61, 6.62, 6.79, 6.66, 6.69, 6.94, 6.49, 7.07, 6.71, 6.57, 6.35, 6.52, 6.83, 6.64, 6.55, 6.37, 6.61, 6.66, 6.61, 6.44, 6.65, 6.33, 6.55, 6.97, 6.77, 6.92, 6.55, 6.97, 6.81, 6.73, 6.41, 7.06, 6.72, 6.28, 6.82, 6.71, 6.71, 6.62, 6.41, 6.86, 6.47, 6.3, 6.51, 6.76, 7.01, 6.5, 6.65, 6.84, 6.79, 6.62, 6.66, 6.65, 6.81, 6.37, 6.46, 6.73, 6.65, 6.45, 6.81, 6.92, 7.01, 6.58, 6.55, 6.62, 6.6, 6.97, 6.53, 6.45, 6.75, 6.68, 6.86, 6.85, 7.06, 6.47, 6.91, 6.83, 7.08, 6.6, 6.64, 6.91, 6.51, 6.66, 6.82, 6.65, 6.53, 6.74, 6.74, 6.55, 6.63, 6.43, 6.57, 6.85, 6.74, 7.06, 6.71, 6.66, 6.87, 6.87, 6.64, 6.69, 6.37, 6.72, 6.56, 6.77, 6.75, 6.46, 6.68, 6.4, 7.01, 6.78, 6.85, 6.69, 6.61, 6.54, 6.78, 6.44, 6.57, 6.7, 6.83, 6.5, 6.25, 7.04, 6.54, 6.59, 6.36, 6.38, 6.61, 5.94, 6.7, 6.85, 7.04, 6.56, 6.76, 6.89, 6.69, 6.61, 6.64, 6.49, 6.71, 6.85, 6.63, 6.64, 6.55, 6.66, 6.76, 6.31, 6.4, 6.72, 6.2, 6.69, 6.7, 6.64, 6.45, 6.86, 6.79, 6.95, 6.98, 6.81, 6.84, 6.83, 6.85, 6.78, 6.59, 6.75, 6.49, 6.66, 6.68, 6.64, 6.73, 6.87, 6.82, 6.6, 6.55, 6.6, 6.77, 7.17, 6.66, 6.28, 6.74, 7.09, 6.65, 6.71, 6.8, 6.87, 6.8, 6.56, 6.53, 6.45, 6.5, 6.53, 6.4, 6.71, 7.04, 6.79, 6.44, 6.69, 6.49, 6.9, 6.68, 6.97, 6.55, 6.95, 6.73, 6.78, 6.59, 6.76, 6.71, 6.9, 6.73, 7.01, 6.55, 6.42, 6.59, 6.5, 6.53, 6.79, 6.57, 6.69, 6.86, 6.65, 6.62, 6.78, 6.39, 6.51, 6.57, 6.77, 6.71, 7.01, 6.26, 6.54, 6.48, 6.74, 6.48, 6.37, 6.63, 6.69, 6.78, 6.8, 6.48, 6.91, 6.85, 5.97, 6.9, 6.75, 6.9, 6.56, 6.5, 6.69, 7.02, 6.67, 6.39, 6.46, 6.78, 6.68, 6.29, 6.63, 6.78, 6.47, 6.24, 6.76, 6.54, 6.69, 6.65, 6.72, 6.26, 6.49, 6.61, 6.89, 6.82, 7.05, 6.66, 6.6, 7.05, 6.76, 6.69, 6.56, 6.7, 6.7, 6.78, 6.85, 6.44, 7.04, 6.78, 6.75, 6.54, 6.57, 6.71, 6.8, 6.41, 6.45, 6.75, 6.72, 6.97, 6.87, 6.3, 6.58, 6.5, 6.23, 6.74, 6.46, 6.88, 6.8, 6.32, 6.82, 6.61, 6.83, 6.24, 6.33, 6.6, 6.87, 6.65, 6.59, 6.89, 6.5, 6.81, 6.7, 6.48, 6.71, 6.89, 6.59, 6.67, 6.62, 6.83, 6.32, 6.72, 6.26, 6.79, 7.06, 6.53, 6.81, 7.12, 6.53, 6.41, 7.34, 6.88, 6.63, 7.06, 6.87, 6.91, 6.56, 7.0, 6.77, 7.0, 6.71, 7.0, 6.78, 6.66, 6.64, 6.98, 6.71, 6.73, 6.63, 6.28, 6.72, 6.88, 6.72, 7.03, 6.78, 6.63, 7.0, 6.95, 6.73, 6.84, 6.75, 6.69, 6.59, 6.93, 6.93, 6.9, 6.55, 6.8, 6.79, 6.62, 6.44, 7.04, 6.22, 6.53, 6.81, 6.52, 6.4, 6.93, 6.76, 6.75, 6.59, 6.5, 6.72, 6.82, 7.1, 6.8, 6.71, 6.53, 6.78, 7.02, 7.02, 6.73, 6.27, 6.83, 6.66, 6.09, 6.77, 6.5, 6.47, 6.48, 6.75, 6.57, 6.92, 6.96, 6.74, 6.74, 7.01, 6.83, 6.69, 6.6, 6.62, 6.74, 6.87, 6.43, 6.58, 6.9, 6.56, 6.71, 6.76, 6.61, 6.94, 6.42, 6.59, 6.95, 6.73, 6.64, 6.8, 6.42, 6.78, 6.57, 6.7, 6.75, 6.44, 6.7, 6.49, 6.82, 6.74, 6.46, 6.6, 6.67, 7.05, 6.43, 6.49, 6.5, 6.7, 6.33, 6.73, 6.72, 6.57, 6.71, 6.36, 7.09, 6.63, 6.51, 6.61, 6.52, 6.87, 6.56, 6.72, 6.6, 6.47, 6.47, 6.84, 6.65, 6.7, 6.84, 6.86, 6.8, 6.61, 6.56, 6.85, 6.44, 6.94, 6.89, 6.79, 6.87, 6.62, 6.68, 6.67, 6.69, 6.71, 6.53, 6.82, 6.59, 6.67, 6.59, 6.73, 6.59, 6.79, 6.68, 6.89, 6.88, 6.54, 6.35, 6.53, 6.63, 6.82, 6.41, 6.54, 6.45, 6.56, 6.88, 6.93, 6.42, 6.83, 6.7, 6.81, 6.82, 7.17, 7.2, 6.65, 6.94, 6.57, 6.78, 6.82, 6.41, 6.87, 6.8, 6.55, 6.86, 6.94, 6.64, 7.06, 6.68, 6.6, 6.91, 6.73, 6.63, 6.57, 6.71, 6.42, 6.56, 6.54, 6.62, 6.79, 6.67, 6.81, 6.68, 6.5, 6.91, 6.97, 6.6, 6.63, 6.88, 6.91, 6.84, 6.65, 6.61, 6.96, 6.98, 6.85, 6.64, 6.48, 6.68, 7.07, 6.93, 6.77, 6.73, 6.53, 6.6, 6.64, 6.5, 6.77, 6.88, 6.44, 6.86, 6.71, 6.74, 7.03, 6.65, 6.89, 6.49, 6.86, 6.51, 6.72, 6.88, 6.58, 6.79, 6.53, 6.54, 6.62, 6.59, 6.87, 6.74, 6.81, 6.79, 6.67, 6.71, 6.43, 6.8, 6.44, 6.41, 6.57, 6.85, 7.13, 7.22, 6.47, 6.81, 6.71, 6.23, 6.61, 6.52, 6.57, 6.89, 7.04, 6.78, 6.68, 6.46, 6.56, 6.43, 6.81, 6.12, 6.48, 6.77, 6.66, 6.85, 6.98, 6.64, 6.89, 6.67, 6.85, 6.67, 6.65, 6.68, 6.88, 6.62, 6.73, 6.61, 6.2, 6.52, 6.82, 6.81, 6.76, 6.43, 6.78, 6.68, 6.94, 6.83, 6.66, 7.07, 6.83, 6.73, 6.89, 6.66, 6.68, 6.8, 6.94, 6.88, 6.99, 6.31, 6.79, 6.33, 6.52, 6.87, 6.86, 6.61, 6.67, 6.26, 6.83, 6.62, 6.72, 6.93, 6.57, 6.28, 6.48, 6.46, 6.64, 6.91, 6.78, 6.87, 6.87, 6.76, 6.55, 6.68, 6.27, 6.85, 6.96, 6.99, 6.6, 6.64, 6.82, 6.73, 6.86, 6.78, 6.36, 7.09, 6.61, 6.79, 7.08, 6.39, 7.07, 6.86, 6.53, 6.62, 6.78, 6.54, 6.65, 6.95, 6.75, 6.8, 6.66, 7.18, 6.76, 6.56, 6.7, 6.9, 6.56, 6.52, 6.68, 7.0, 7.05, 6.79, 6.68, 6.6, 6.64, 6.63, 6.84, 6.89, 6.64, 6.94, 6.73, 6.5, 6.67, 6.9, 6.82, 6.8, 6.52, 6.76, 6.49, 6.44, 6.44, 6.54, 6.91, 6.64, 6.8, 6.99, 6.39, 6.93, 6.84, 6.49, 6.77, 6.93, 6.53, 6.85, 6.91, 6.45, 6.74, 6.84, 6.96, 6.74, 6.39, 6.56, 6.93, 7.1, 6.67, 6.74, 6.74, 6.5, 6.76, 6.62, 6.46, 6.56, 6.47, 6.71, 6.88, 6.49, 6.68, 6.96, 6.66, 6.84, 6.7, 6.43, 6.62, 6.35, 6.74, 7.02, 6.65, 6.63, 6.75, 6.43, 6.83, 6.63, 6.61, 6.79, 6.72, 6.75, 6.71, 6.68, 6.54, 7.08, 6.69, 6.67, 6.53, 6.49, 6.82, 6.79, 6.87, 6.53, 6.62, 6.56, 6.23, 6.81, 6.23, 6.72, 6.94, 6.73, 6.44, 6.6, 6.69, 6.78, 6.59, 6.53, 6.78, 6.71, 6.79, 6.62, 6.58, 6.61, 6.49, 6.92, 6.75, 6.7, 6.9, 6.71, 6.56, 6.56, 6.6, 6.34, 6.43, 6.25, 6.45, 6.47, 6.87, 6.54, 6.52, 6.71, 6.55, 6.44, 6.93, 7.05, 6.48, 6.79, 6.59, 6.67, 6.63, 6.51, 6.71, 6.73, 6.83, 6.68, 6.68, 6.47, 6.78, 6.94, 6.52, 6.83, 6.45, 6.62, 6.48, 6.52, 6.68, 6.62, 6.54, 6.7, 6.89, 6.94, 6.54, 6.72, 6.68, 6.8, 6.55, 6.72, 6.73, 6.72, 6.72, 6.61, 6.58, 6.78, 7.0, 6.19, 6.83, 6.82, 6.71, 6.57, 6.48, 6.61, 6.68, 6.77, 6.85, 6.59]
组2 n=1000：[6.73, 6.79, 6.78, 6.46, 6.76, 6.66, 6.61, 6.59, 7.06, 6.8, 6.66, 6.86, 6.49, 6.76, 6.99, 7.01, 6.66, 6.9, 6.55, 6.38, 6.39, 6.68, 6.75, 6.77, 6.65, 6.74, 6.75, 6.85, 6.65, 6.54, 6.73, 6.47, 6.81, 6.76, 6.91, 6.75, 6.89, 6.73, 6.63, 6.8, 6.62, 6.62, 6.33, 6.96, 6.61, 6.79, 6.8, 6.44, 6.41, 6.76, 6.81, 7.03, 6.8, 6.47, 6.26, 6.6, 6.67, 6.74, 6.75, 6.84, 6.32, 6.6, 6.97, 6.73, 6.45, 6.64, 6.62, 6.65, 6.93, 6.53, 6.78, 6.43, 7.01, 6.74, 6.42, 6.46, 6.42, 6.73, 7.09, 6.59, 6.48, 6.73, 6.69, 6.87, 6.54, 7.07, 6.79, 6.47, 6.62, 6.59, 6.69, 6.87, 6.5, 6.9, 6.47, 6.91, 6.62, 6.74, 6.95, 6.7, 6.78, 6.62, 6.68, 6.81, 7.04, 6.4, 6.65, 6.61, 6.62, 6.79, 6.66, 6.69, 6.94, 6.49, 7.07, 6.71, 6.57, 6.35, 6.52, 6.83, 6.64, 6.55, 6.37, 6.61, 6.66, 6.61, 6.44, 6.65, 6.33, 6.55, 6.97, 6.77, 6.92, 6.55, 6.97, 6.81, 6.73, 6.41, 7.06, 6.72, 6.28, 6.82, 6.71, 6.71, 6.62, 6.41, 6.86, 6.47, 6.3, 6.51, 6.76, 7.01, 6.5, 6.65, 6.84, 6.79, 6.62, 6.66, 6.65, 6.81, 6.37, 6.46, 6.73, 6.65, 6.45, 6.81, 6.92, 7.01, 6.58, 6.55, 6.62, 6.6, 6.97, 6.53, 6.45, 6.75, 6.68, 6.86, 6.85, 7.06, 6.47, 6.91, 6.83, 7.08, 6.6, 6.64, 6.91, 6.51, 6.66, 6.82, 6.65, 6.53, 6.74, 6.74, 6.55, 6.63, 6.43, 6.57, 6.85, 6.74, 7.06, 6.71, 6.66, 6.87, 6.87, 6.64, 6.69, 6.37, 6.72, 6.56, 6.77, 6.75, 6.46, 6.68, 6.4, 7.01, 6.78, 6.85, 6.69, 6.61, 6.54, 6.78, 6.44, 6.57, 6.7, 6.83, 6.5, 6.25, 7.04, 6.54, 6.59, 6.36, 6.38, 6.61, 5.94, 6.7, 6.85, 7.04, 6.56, 6.76, 6.89, 6.69, 6.61, 6.64, 6.49, 6.71, 6.85, 6.63, 6.64, 6.55, 6.66, 6.76, 6.31, 6.4, 6.72, 6.2, 6.69, 6.7, 6.64, 6.45, 6.86, 6.79, 6.95, 6.98, 6.81, 6.84, 6.83, 6.85, 6.78, 6.59, 6.75, 6.49, 6.66, 6.68, 6.64, 6.73, 6.87, 6.82, 6.6, 6.55, 6.6, 6.77, 7.17, 6.66, 6.28, 6.74, 7.09, 6.65, 6.71, 6.8, 6.87, 6.8, 6.56, 6.53, 6.45, 6.5, 6.53, 6.4, 6.71, 7.04, 6.79, 6.44, 6.69, 6.49, 6.9, 6.68, 6.97, 6.55, 6.95, 6.73, 6.78, 6.59, 6.76, 6.71, 6.9, 6.73, 7.01, 6.55, 6.42, 6.59, 6.5, 6.53, 6.79, 6.57, 6.69, 6.86, 6.65, 6.62, 6.78, 6.39, 6.51, 6.57, 6.77, 6.71, 7.01, 6.26, 6.54, 6.48, 6.74, 6.48, 6.37, 6.63, 6.69, 6.78, 6.8, 6.48, 6.91, 6.85, 5.97, 6.9, 6.75, 6.9, 6.56, 6.5, 6.69, 7.02, 6.67, 6.39, 6.46, 6.78, 6.68, 6.29, 6.63, 6.78, 6.47, 6.24, 6.76, 6.54, 6.69, 6.65, 6.72, 6.26, 6.49, 6.61, 6.89, 6.82, 7.05, 6.66, 6.6, 7.05, 6.76, 6.69, 6.56, 6.7, 6.7, 6.78, 6.85, 6.44, 7.04, 6.78, 6.75, 6.54, 6.57, 6.71, 6.8, 6.41, 6.45, 6.75, 6.72, 6.97, 6.87, 6.3, 6.58, 6.5, 6.23, 6.74, 6.46, 6.88, 6.8, 6.32, 6.82, 6.61, 6.83, 6.24, 6.33, 6.6, 6.87, 6.65, 6.59, 6.89, 6.5, 6.81, 6.7, 6.48, 6.71, 6.89, 6.59, 6.67, 6.62, 6.83, 6.32, 6.72, 6.26, 6.79, 7.06, 6.53, 6.81, 7.12, 6.53, 6.41, 7.34, 6.88, 6.63, 7.06, 6.87, 6.91, 6.56, 7.0, 6.77, 7.0, 6.71, 7.0, 6.78, 6.66, 6.64, 6.98, 6.71, 6.73, 6.63, 6.28, 6.72, 6.88, 6.72, 7.03, 6.78, 6.63, 7.0, 6.95, 6.73, 6.84, 6.75, 6.69, 6.59, 6.93, 6.93, 6.9, 6.55, 6.8, 6.79, 6.62, 6.44, 7.04, 6.22, 6.53, 6.81, 6.52, 6.4, 6.93, 6.76, 6.75, 6.59, 6.5, 6.72, 6.82, 7.1, 6.8, 6.71, 6.53, 6.78, 7.02, 7.02, 6.73, 6.27, 6.83, 6.66, 6.09, 6.77, 6.5, 6.47, 6.48, 6.75, 6.57, 6.92, 6.96, 6.74, 6.74, 7.01, 6.83, 6.69, 6.6, 6.62, 6.74, 6.87, 6.43, 6.58, 6.9, 6.56, 6.71, 6.76, 6.61, 6.94, 6.42, 6.59, 6.95, 6.73, 6.64, 6.8, 6.42, 6.78, 6.57, 6.7, 6.75, 6.44, 6.7, 6.49, 6.82, 6.74, 6.46, 6.6, 6.67, 7.05, 6.43, 6.49, 6.5, 6.7, 6.33, 6.73, 6.72, 6.57, 6.71, 6.36, 7.09, 6.63, 6.51, 6.61, 6.52, 6.87, 6.56, 6.72, 6.6, 6.47, 6.47, 6.84, 6.65, 6.7, 6.84, 6.86, 6.8, 6.61, 6.56, 6.85, 6.44, 6.94, 6.89, 6.79, 6.87, 6.62, 6.68, 6.67, 6.69, 6.71, 6.53, 6.82, 6.59, 6.67, 6.59, 6.73, 6.59, 6.79, 6.68, 6.89, 6.88, 6.54, 6.35, 6.53, 6.63, 6.82, 6.41, 6.54, 6.45, 6.56, 6.88, 6.93, 6.42, 6.83, 6.7, 6.81, 6.82, 7.17, 7.2, 6.65, 6.94, 6.57, 6.78, 6.82, 6.41, 6.87, 6.8, 6.55, 6.86, 6.94, 6.64, 7.06, 6.68, 6.6, 6.91, 6.73, 6.63, 6.57, 6.71, 6.42, 6.56, 6.54, 6.62, 6.79, 6.67, 6.81, 6.68, 6.5, 6.91, 6.97, 6.6, 6.63, 6.88, 6.91, 6.84, 6.65, 6.61, 6.96, 6.98, 6.85, 6.64, 6.48, 6.68, 7.07, 6.93, 6.77, 6.73, 6.53, 6.6, 6.64, 6.5, 6.77, 6.88, 6.44, 6.86, 6.71, 6.74, 7.03, 6.65, 6.89, 6.49, 6.86, 6.51, 6.72, 6.88, 6.58, 6.79, 6.53, 6.54, 6.62, 6.59, 6.87, 6.74, 6.81, 6.79, 6.67, 6.71, 6.43, 6.8, 6.44, 6.41, 6.57, 6.85, 7.13, 7.22, 6.47, 6.81, 6.71, 6.23, 6.61, 6.52, 6.57, 6.89, 7.04, 6.78, 6.68, 6.46, 6.56, 6.43, 6.81, 6.12, 6.48, 6.77, 6.66, 6.85, 6.98, 6.64, 6.89, 6.67, 6.85, 6.67, 6.65, 6.68, 6.88, 6.62, 6.73, 6.61, 6.2, 6.52, 6.82, 6.81, 6.76, 6.43, 6.78, 6.68, 6.94, 6.83, 6.66, 7.07, 6.83, 6.73, 6.89, 6.66, 6.68, 6.8, 6.94, 6.88, 6.99, 6.31, 6.79, 6.33, 6.52, 6.87, 6.86, 6.61, 6.67, 6.26, 6.83, 6.62, 6.72, 6.93, 6.57, 6.28, 6.48, 6.46, 6.64, 6.91, 6.78, 6.87, 6.87, 6.76, 6.55, 6.68, 6.27, 6.85, 6.96, 6.99, 6.6, 6.64, 6.82, 6.73, 6.86, 6.78, 6.36, 7.09, 6.61, 6.79, 7.08, 6.39, 7.07, 6.86, 6.53, 6.62, 6.78, 6.54, 6.65, 6.95, 6.75, 6.8, 6.66, 7.18, 6.76, 6.56, 6.7, 6.9, 6.56, 6.52, 6.68, 7.0, 7.05, 6.79, 6.68, 6.6, 6.64, 6.63, 6.84, 6.89, 6.64, 6.94, 6.73, 6.5, 6.67, 6.9, 6.82, 6.8, 6.52, 6.76, 6.49, 6.44, 6.44, 6.54, 6.91, 6.64, 6.8, 6.99, 6.39, 6.93, 6.84, 6.49, 6.77, 6.93, 6.53, 6.85, 6.91, 6.45, 6.74, 6.84, 6.96, 6.74, 6.39, 6.56, 6.93, 7.1, 6.67, 6.74, 6.74, 6.5, 6.76, 6.62, 6.46, 6.56, 6.47, 6.71, 6.88, 6.49, 6.68, 6.96, 6.66, 6.84, 6.7, 6.43, 6.62, 6.35, 6.74, 7.02, 6.65, 6.63, 6.75, 6.43, 6.83, 6.63, 6.61, 6.79, 6.72, 6.75, 6.71, 6.68, 6.54, 7.08, 6.69, 6.67, 6.53, 6.49, 6.82, 6.79, 6.87, 6.53, 6.62, 6.56, 6.23, 6.81, 6.23, 6.72, 6.94, 6.73, 6.44, 6.6, 6.69, 6.78, 6.59, 6.53, 6.78, 6.71, 6.79, 6.62, 6.58, 6.61, 6.49, 6.92, 6.75, 6.7, 6.9, 6.71, 6.56, 6.56, 6.6, 6.34, 6.43, 6.25, 6.45, 6.47, 6.87, 6.54, 6.52, 6.71, 6.55, 6.44, 6.93, 7.05, 6.48, 6.79, 6.59, 6.67, 6.63, 6.51, 6.71, 6.73, 6.83, 6.68, 6.68, 6.47, 6.78, 6.94, 6.52, 6.83, 6.45, 6.62, 6.48, 6.52, 6.68, 6.62, 6.54, 6.7, 6.89, 6.94, 6.54, 6.72, 6.68, 6.8, 6.55, 6.72, 6.73, 6.72, 6.72, 6.61, 6.58, 6.78, 7.0, 6.19, 6.83, 6.82, 6.71, 6.57, 6.48, 6.61, 6.68, 6.77, 6.85, 6.59]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=7.2863, s₁=0.2126；x̄₂=6.6883, s₂=0.1939；
SE=√(s₁²/n₁+s₂²/n₂)=0.0110；t=(x̄₁−x̄₂)/SE=54.4726；
df(Welch)=1033.28；双侧 p = 5.322e-306。
x̄₁=7.2863, s₁=0.2126；x̄₂=6.6883, s₂=0.1939；
SE=√(s₁²/n₁+s₂²/n₂)=0.0110；t=(x̄₁−x̄₂)/SE=54.4726；
df(Welch)=1033.28；双侧 p = 5.322e-306。
计算结果
t = 54.4726，df = 1033.28，p = 5.322e-306
在 α=0.05 水平下，p=5.322e-306 < 0.05，拒绝原假设 H₀。比较油耗经济性。
案例 28. 吸烟与不吸烟肺活量
原始数据
组1 n=850：[3897.06, 3992.71, 4010.28, 3915.92, 4174.17, 3921.01, 4205.0, 4209.73, 4097.84, 4038.62, 4161.0, 4050.78, 4135.6, 4348.89, 4040.96, 4121.62, 4064.93, 4128.19, 4033.74, 4121.44, 4143.58, 4033.01, 3829.98, 4135.8, 4151.06, 4188.16, 4254.5, 3966.39, 4390.35, 3840.15, 4120.59, 4105.09, 3923.72, 3867.32, 3854.76, 4255.14, 4172.55, 4130.91, 4121.92, 4100.04, 4108.63, 4097.14, 4263.49, 4282.5, 4026.89, 3942.66, 4102.34, 4132.59, 4064.33, 4122.37, 4039.86, 3943.87, 4130.91, 3833.9, 4090.89, 4239.54, 4168.76, 4146.42, 4248.91, 4142.19, 3898.05, 4070.91, 3958.43, 4129.67, 3854.99, 4062.3, 3917.07, 3835.2, 4102.55, 4126.09, 4297.62, 3997.9, 4267.62, 4032.78, 3927.92, 3925.84, 4353.48, 4051.37, 4016.31, 4076.16, 3831.9, 4107.54, 3853.15, 4027.28, 4366.97, 4173.99, 3921.81, 3888.66, 4093.34, 4236.76, 3876.47, 3974.26, 4389.03, 3969.35, 4036.38, 4058.25, 4043.51, 4481.45, 4013.49, 4153.16, 4150.42, 4155.64, 4161.92, 3976.18, 4161.24, 3998.68, 4231.87, 4141.94, 4120.28, 4056.76, 4293.34, 4178.6, 4300.29, 4071.07, 3977.01, 4225.21, 4030.17, 3772.33, 3965.93, 4047.94, 3907.1, 4030.0, 4063.46, 3961.17, 4225.33, 4052.69, 4071.15, 4147.82, 4145.7, 4363.29, 4346.96, 4205.79, 4152.45, 4214.22, 4249.1, 4054.84, 3992.56, 4234.9, 4177.07, 3998.08, 4311.22, 4194.18, 3876.21, 4394.49, 3986.3, 4072.69, 4236.7, 4110.08, 4277.59, 4106.26, 3895.13, 4219.85, 4143.7, 3931.1, 4378.19, 4062.39, 3919.7, 4141.14, 4153.89, 4166.89, 4181.84, 4318.27, 4028.97, 4065.76, 4165.85, 4111.02, 4111.15, 3963.0, 4073.97, 4021.3, 4171.49, 3785.8, 4249.49, 4067.44, 4126.84, 4118.09, 3674.83, 4247.81, 4075.29, 4222.98, 4143.06, 3775.29, 4200.94, 4144.55, 4025.85, 4124.5, 3991.54, 4029.73, 3730.46, 3892.87, 3953.36, 4004.88, 3956.4, 4138.33, 4243.07, 4023.9, 4265.53, 4232.86, 4271.99, 4040.45, 4134.59, 3929.19, 3914.7, 4193.18, 4214.01, 4262.14, 4079.28, 4255.5, 3909.98, 4261.19, 4128.25, 4112.53, 4094.61, 4015.85, 3979.53, 3962.85, 4166.95, 3983.33, 4136.63, 4031.0, 3760.74, 3982.16, 4107.71, 4164.1, 4022.11, 3945.29, 3894.59, 4030.19, 3862.28, 4026.59, 3870.01, 3947.12, 4105.7, 4095.31, 4234.35, 4085.31, 4214.31, 3959.39, 3980.36, 4143.05, 4049.41, 4026.15, 4085.0, 4462.01, 3964.26, 4047.45, 4065.3, 3974.68, 4079.57, 3880.55, 4174.83, 3891.38, 3989.92, 4165.58, 3961.95, 3977.28, 3879.28, 3900.96, 3889.95, 4100.74, 4007.14, 4034.1, 4055.38, 4121.21, 4087.37, 3912.66, 4033.73, 3998.72, 4105.39, 4208.69, 4122.74, 3941.4, 4310.82, 4316.81, 3930.31, 4190.98, 4174.1, 4063.45, 4027.38, 4134.24, 4184.53, 4186.42, 3983.75, 4170.2, 3935.18, 4315.25, 4090.34, 4256.25, 4320.69, 4094.36, 3990.72, 4140.52, 3992.24, 3958.1, 4174.4, 4154.23, 4035.37, 4106.13, 4016.97, 4038.17, 4196.31, 4144.21, 4279.6, 3888.72, 3955.99, 3996.6, 4044.52, 4038.9, 3928.15, 4127.92, 4076.58, 4229.66, 4161.53, 3932.56, 3899.09, 3998.18, 3959.94, 4217.93, 4257.23, 4163.21, 3934.7, 3717.81, 4371.77, 4217.55, 4003.41, 4109.37, 4193.89, 3752.73, 3827.3, 4051.03, 4187.02, 4166.87, 3990.2, 4125.03, 3951.44, 3866.29, 3932.3, 4060.82, 4079.46, 4114.39, 4269.25, 4241.06, 4174.99, 4075.0, 4126.84, 4456.36, 4034.17, 4246.96, 4107.4, 4106.18, 4034.39, 4029.51, 3981.07, 4172.09, 3923.91, 4044.89, 3932.73, 4101.6, 4041.42, 4102.56, 4305.71, 3997.96, 4116.23, 3981.95, 4038.73, 4100.55, 4006.25, 4045.92, 4376.38, 4044.43, 4106.76, 3945.0, 4169.33, 4227.28, 3859.68, 3865.76, 4103.36, 4080.55, 3727.35, 3981.9, 4117.43, 4264.47, 3989.34, 3840.59, 4145.95, 4180.25, 3890.85, 4175.09, 4174.41, 4152.62, 4328.05, 4034.31, 4230.31, 4026.53, 3933.67, 4162.01, 4042.89, 4245.83, 3989.6, 4121.46, 4103.23, 4258.5, 3991.49, 4094.64, 4082.79, 4322.26, 4082.71, 3986.92, 4314.26, 4127.71, 3998.37, 4135.56, 4240.93, 3997.74, 4018.84, 4023.97, 4150.4, 4057.18, 4364.63, 3940.94, 4260.18, 4188.64, 4075.27, 4279.75, 4110.37, 3900.68, 4086.23, 4313.37, 4178.95, 4197.71, 4004.2, 4186.85, 4041.91, 4223.93, 4131.96, 4102.28, 4015.86, 4273.99, 4055.81, 4192.79, 3946.3, 3832.57, 4174.01, 4019.4, 4223.14, 4098.81, 4087.96, 3911.47, 4216.57, 4033.16, 4218.41, 4148.36, 4093.66, 4023.99, 4230.06, 4215.06, 4262.72, 3992.89, 4152.91, 4198.87, 3847.67, 4043.79, 4345.03, 4043.76, 3911.14, 3956.32, 4141.9, 4142.4, 4187.61, 3972.1, 4152.8, 3986.27, 4213.04, 4135.91, 4239.01, 3958.93, 4093.74, 3956.94, 4010.94, 4241.7, 4107.39, 3862.8, 3980.92, 4320.68, 4300.36, 4030.68, 4073.28, 3892.72, 4248.02, 4131.05, 4165.66, 4083.44, 4302.81, 3840.37, 3965.62, 4045.2, 4023.8, 4246.46, 4170.99, 4179.88, 4185.81, 4083.56, 4271.26, 4139.65, 4019.47, 4095.12, 4116.66, 4152.66, 4386.78, 4160.53, 4070.68, 4196.11, 4194.8, 4364.44, 3907.29, 4105.57, 4300.08, 3825.76, 4212.62, 4088.54, 4125.1, 4304.69, 4163.65, 4160.15, 4169.26, 4030.7, 3972.98, 4184.32, 4119.76, 4158.72, 4149.8, 3959.71, 4365.48, 4097.77, 4079.47, 4151.98, 3755.17, 4184.66, 4093.24, 4166.0, 4180.12, 4095.32, 4273.77, 4072.48, 3978.46, 4180.06, 4085.31, 3997.81, 4040.49, 3944.32, 3955.91, 4018.95, 4068.83, 3915.15, 3996.67, 3941.26, 4207.29, 4064.56, 4182.42, 4315.81, 4175.89, 4011.06, 4085.32, 4145.21, 4297.96, 4029.28, 4094.69, 4300.99, 4164.77, 3929.23, 4110.34, 3987.84, 4104.81, 4075.56, 3826.71, 4062.84, 4251.57, 4079.21, 4165.73, 3783.64, 4069.69, 4284.07, 4222.48, 4269.35, 4071.39, 4074.74, 4067.71, 4007.52, 4178.34, 4077.84, 3925.2, 4124.2, 4029.67, 4238.75, 4092.06, 3912.76, 4158.16, 4114.77, 4019.18, 4147.73, 4054.39, 4032.39, 4289.27, 4136.85, 3972.25, 4245.4, 4155.17, 4129.82, 3976.37, 4046.55, 4168.42, 4141.78, 3830.09, 4248.57, 4176.16, 4042.92, 4223.27, 3862.44, 4039.25, 4116.68, 4235.54, 4053.7, 4150.97, 4183.35, 3909.03, 3965.2, 4340.71, 4030.28, 4174.29, 3946.48, 4035.98, 4034.93, 4117.68, 4146.09, 4074.83, 4134.78, 3931.8, 4120.84, 4212.82, 4029.76, 4168.32, 4185.18, 3850.92, 3944.23, 4065.02, 3867.35, 4116.42, 4140.98, 4005.89, 4261.18, 3856.94, 4182.3, 4146.82, 4228.67, 4121.24, 4024.54, 3878.18, 4030.27, 3982.62, 3955.45, 4021.5, 4005.52, 3978.91, 3887.62, 4013.6, 4183.21, 3893.27, 4129.62, 4122.06, 4102.73, 3908.46, 4041.36, 4215.12, 4049.47, 4486.18, 4132.37, 4042.48, 4000.31, 4195.92, 3754.48, 3970.09, 4049.92, 3906.71, 4046.72, 4127.22, 4109.91, 4111.87, 4178.66, 4087.16, 4071.4, 4059.33, 3860.14, 3967.43, 4130.84, 4012.47, 4010.08, 4177.7, 4058.55, 4401.75, 4098.32, 4128.41, 4198.68, 4296.59, 4114.21, 4260.66, 4151.73, 4098.51, 4062.58, 4023.71, 4097.28, 4067.81, 4092.02, 3982.95, 4005.21, 4049.82, 3867.65, 4198.4, 4026.56, 3871.02, 4005.11, 4223.66, 4400.53, 4108.8, 4048.77, 4322.9, 3921.11, 4032.18, 4219.08, 4129.43, 4022.75, 4245.2, 4120.01, 4024.98, 3917.93, 3882.87, 4154.87, 4058.81, 4246.02, 4193.36, 4237.26, 4115.85, 4344.07, 3957.29, 4243.99, 4139.77, 3946.19, 4156.03, 4068.74, 4067.03, 4027.82, 3954.1, 4210.08, 4276.32, 4038.42, 4177.52, 3928.72, 4152.89, 4048.93, 4410.9, 4223.67, 4274.08, 4203.16, 4129.98, 4111.16, 4181.07, 4013.61, 3923.92, 3983.26, 3965.02, 4105.54, 4208.78, 3970.7, 4264.58, 3997.41, 4231.05, 4202.76, 3976.2, 4092.1, 4007.28, 4394.41, 4084.39, 3882.24, 4079.52, 4127.88, 4028.08, 4143.13, 3736.09, 4002.29, 4046.41, 4413.44, 4107.6, 4251.23, 3913.97, 3932.08, 4091.97, 3863.77, 4225.24, 4292.71, 4003.08, 4161.98, 4164.01, 4265.91, 3929.02, 3945.7, 3965.12, 3760.1, 3999.68, 4089.82, 4109.4, 4177.47, 4135.54, 4318.27, 3917.25, 4308.03, 4087.85, 4138.33, 3903.91, 4151.58, 4140.69, 3996.04, 3837.75, 4231.29, 4358.5, 3874.38, 3978.9, 3992.6, 4372.79, 4112.3, 4026.3, 4002.89, 4283.25, 4171.61, 4225.74, 4229.72, 4069.64, 4113.06, 4116.3, 4082.85, 3989.86, 4033.43, 4131.59, 4168.61, 4021.37, 3824.59, 4218.31, 4000.28, 3959.75, 4294.06, 4108.39, 4164.02]；
组2 n=994：[4864.61, 4496.01, 4576.77, 4620.75, 4494.67, 4471.21, 4678.37, 4635.63, 4467.29, 4498.23, 4584.82, 4677.76, 4614.34, 4376.8, 4576.45, 4501.28, 4610.41, 4727.86, 4649.68, 4579.79, 4649.88, 4521.67, 4463.93, 4464.89, 4639.53, 4504.58, 4590.1, 4514.37, 4557.32, 4486.11, 4525.49, 4598.88, 4533.72, 4519.78, 4562.25, 4508.66, 4570.68, 4596.07, 4568.83, 4557.33, 4489.28, 4558.61, 4501.59, 4514.93, 4654.45, 4653.36, 4431.24, 4745.61, 4519.77, 4581.36, 4445.22, 4631.43, 4509.7, 4590.85, 4479.94, 4687.82, 4561.39, 4644.07, 4488.41, 4565.58, 4520.65, 4620.92, 4679.8, 4456.98, 4502.51, 4348.63, 4513.09, 4758.25, 4516.84, 4641.9, 4646.23, 4661.78, 4472.73, 4666.83, 4456.0, 4419.28, 4613.18, 4444.34, 4543.28, 4662.41, 4524.53, 4449.42, 4472.1, 4581.41, 4750.26, 4480.42, 4532.32, 4529.29, 4665.71, 4720.31, 4458.61, 4655.83, 4492.49, 4497.76, 4614.64, 4645.39, 4544.77, 4313.58, 4505.59, 4570.02, 4599.76, 4532.8, 4603.39, 4497.77, 4648.96, 4736.49, 4454.44, 4541.35, 4685.08, 4239.17, 4520.13, 4714.84, 4478.07, 4793.9, 4623.06, 4654.97, 4461.67, 4594.43, 4480.55, 4744.73, 4426.25, 4541.85, 4499.22, 4488.77, 4689.43, 4541.29, 4549.85, 4692.2, 4537.09, 4508.62, 4625.28, 4657.8, 4405.01, 4615.54, 4454.45, 4653.2, 4647.2, 4702.33, 4632.02, 4686.33, 4784.99, 4686.21, 4537.32, 4560.18, 4433.07, 4542.82, 4671.85, 4598.95, 4743.09, 4520.91, 4395.04, 4715.13, 4485.16, 4603.97, 4785.06, 4574.64, 4629.02, 4708.72, 4590.06, 4576.55, 4508.69, 4756.63, 4482.24, 4707.1, 4518.59, 4513.57, 4409.99, 4550.43, 4599.28, 4590.63, 4392.69, 4647.31, 4629.54, 4488.55, 4464.68, 4567.51, 4467.94, 4374.71, 4780.22, 4499.94, 4529.13, 4507.93, 4583.28, 4468.7, 4758.05, 4668.11, 4658.81, 4713.38, 4607.53, 4591.01, 4488.04, 4385.78, 4508.02, 4615.52, 4507.42, 4688.1, 4543.38, 4647.11, 4511.53, 4818.08, 4512.72, 4563.87, 4544.54, 4549.23, 4741.6, 4534.22, 4598.94, 4553.63, 4622.79, 4521.61, 4490.97, 4387.05, 4687.88, 4658.01, 4730.42, 4730.61, 4464.46, 4559.35, 4490.24, 4702.48, 4464.55, 4529.19, 4509.69, 4539.92, 4464.65, 4606.52, 4535.86, 4634.07, 4593.58, 4546.6, 4374.01, 4548.95, 4588.36, 4726.14, 4444.1, 4400.96, 4575.46, 4694.52, 4575.22, 4463.34, 4432.8, 4711.52, 4725.32, 4565.59, 4489.76, 4554.05, 4488.04, 4509.71, 4574.82, 4630.7, 4510.61, 4624.56, 4545.7, 4631.49, 4720.62, 4502.02, 4529.0, 4590.11, 4659.73, 4347.83, 4505.95, 4514.07, 4523.59, 4576.82, 4719.7, 4638.96, 4481.66, 4669.53, 4512.15, 4554.6, 4481.96, 4509.96, 4393.87, 4597.2, 4669.75, 4536.9, 4576.13, 4466.99, 4588.45, 4658.36, 4606.33, 4600.99, 4767.61, 4503.43, 4667.12, 4593.34, 4486.21, 4406.92, 4584.8, 4386.41, 4686.41, 4552.16, 4596.01, 4619.67, 4332.87, 4510.28, 4414.17, 4600.92, 4741.86, 4580.13, 4682.1, 4648.49, 4583.15, 4583.23, 4682.53, 4673.27, 4494.4, 4661.7, 4271.34, 4548.82, 4572.94, 4552.21, 4425.71, 4772.54, 4456.95, 4572.86, 4549.72, 4545.19, 4642.54, 4483.88, 4573.7, 4669.45, 4716.12, 4614.31, 4798.18, 4584.1, 4643.35, 4480.81, 4570.37, 4627.63, 4592.13, 4430.09, 4411.45, 4434.38, 4353.31, 4761.93, 4568.47, 4610.11, 4533.26, 4428.06, 4482.8, 4597.49, 4543.91, 4496.49, 4601.49, 4513.0, 4503.87, 4618.25, 4568.91, 4596.56, 4554.5, 4473.36, 4795.5, 4531.77, 4698.42, 4689.86, 4646.76, 4565.38, 4660.72, 4369.16, 4559.32, 4867.49, 4715.71, 4591.39, 4662.15, 4617.62, 4654.95, 4396.8, 4740.23, 4556.26, 4852.1, 4669.54, 4722.64, 4753.21, 4635.95, 4585.62, 4782.04, 4490.79, 4459.72, 4581.83, 4873.0, 4499.87, 4558.3, 4677.37, 4665.57, 4605.9, 4475.37, 4708.53, 4693.28, 4560.21, 4507.37, 4539.4, 4648.38, 4538.21, 4402.2, 4557.75, 4656.56, 4633.52, 4432.45, 4632.88, 4491.49, 4653.0, 4617.32, 4670.9, 4416.74, 4490.92, 4530.2, 4629.16, 4666.52, 4726.4, 4582.66, 4628.99, 4694.75, 4552.39, 4674.23, 4586.65, 4445.45, 4456.15, 4701.9, 4554.56, 4583.56, 4542.75, 4488.62, 4596.32, 4598.77, 4577.64, 4430.6, 4459.64, 4504.15, 4653.65, 4427.13, 4461.02, 4538.69, 4378.08, 4616.56, 4580.07, 4655.82, 4468.12, 4533.37, 4801.63, 4657.95, 4681.56, 4498.69, 4582.57, 4497.03, 4589.69, 4508.72, 4599.45, 4569.38, 4657.94, 4633.57, 4548.13, 4507.02, 4657.1, 4687.71, 4597.12, 4645.96, 4606.67, 4643.12, 4601.86, 4487.78, 4687.88, 4562.25, 4661.77, 4671.64, 4650.15, 4803.82, 4728.8, 4572.75, 4598.12, 4559.47, 4755.15, 4625.93, 4560.2, 4568.06, 4656.55, 4671.84, 4615.84, 4680.92, 4583.14, 4555.73, 4445.88, 4515.02, 4562.33, 4715.82, 4728.06, 4700.49, 4462.62, 4615.62, 4659.84, 4631.07, 4673.33, 4786.85, 4530.35, 4343.64, 4799.04, 4592.6, 4575.82, 4560.92, 4590.48, 4416.17, 4592.41, 4576.28, 4630.07, 4512.13, 4463.19, 4559.74, 4600.87, 4443.45, 4652.64, 4638.27, 4604.88, 4590.57, 4529.55, 4550.91, 4469.63, 4552.56, 4711.34, 4733.66, 4673.53, 4454.55, 4566.55, 4689.28, 4616.59, 4695.9, 4507.2, 4443.48, 4731.7, 4468.11, 4606.72, 4385.56, 4800.8, 4569.08, 4559.85, 4538.64, 4535.63, 4519.68, 4741.18, 4641.36, 4664.87, 4539.96, 4551.32, 4634.4, 4433.31, 4460.6, 4592.9, 4566.85, 4540.86, 4471.05, 4682.96, 4389.21, 4649.78, 4535.84, 4419.86, 4595.2, 4706.04, 4523.79, 4746.72, 4572.86, 4716.94, 4607.03, 4490.99, 4727.76, 4493.0, 4743.47, 4685.04, 4511.12, 4562.89, 4641.0, 4677.77, 4784.51, 4648.58, 4620.75, 4329.05, 4573.47, 4805.88, 4625.49, 4684.63, 4470.4, 4516.41, 4578.36, 4650.42, 4639.09, 4540.35, 4536.87, 4485.55, 4566.21, 4633.3, 4573.02, 4499.83, 4465.66, 4589.19, 4543.57, 4457.08, 4427.09, 4615.42, 4417.89, 4706.24, 4384.82, 4619.4, 4419.98, 4549.87, 4580.88, 4760.78, 4433.64, 4581.26, 4707.06, 4406.0, 4531.71, 4479.19, 4517.3, 4636.39, 4745.02, 4709.2, 4781.05, 4559.12, 4676.3, 4442.58, 4411.84, 4747.99, 4543.04, 4606.02, 4598.62, 4607.13, 4730.28, 4799.61, 4606.91, 4586.55, 4502.0, 4510.29, 4548.51, 4731.98, 4541.31, 4636.67, 4517.42, 4615.02, 4524.65, 4461.75, 4486.68, 4669.58, 4562.58, 4504.26, 4504.75, 4588.7, 4408.22, 4632.07, 4653.85, 4474.9, 4591.23, 4819.47, 4530.65, 4654.37, 4516.25, 4432.74, 4664.2, 4728.17, 4550.82, 4554.09, 4454.24, 4700.27, 4330.51, 4662.73, 4459.49, 4466.36, 4727.87, 4570.38, 4721.66, 4576.9, 4561.7, 4671.94, 4543.42, 4576.09, 4651.05, 4658.53, 4438.07, 4339.26, 4638.71, 4696.8, 4611.97, 4543.25, 4755.0, 4574.85, 4558.99, 4432.14, 4652.53, 4649.37, 4398.32, 4525.87, 4552.77, 4734.48, 4527.95, 4614.14, 4598.17, 4469.85, 4701.73, 4723.78, 4647.11, 4670.14, 4446.24, 4481.99, 4570.41, 4542.73, 4591.75, 4369.5, 4592.42, 4363.83, 4651.65, 4605.88, 4674.29, 4678.35, 4442.54, 4461.63, 4577.35, 4584.52, 4659.39, 4846.72, 4647.89, 4431.49, 4684.7, 4481.99, 4599.21, 4703.27, 4476.28, 4492.09, 4670.32, 4590.67, 4512.82, 4548.15, 4653.79, 4448.63, 4499.38, 4619.01, 4576.3, 4420.7, 4702.22, 4505.34, 4566.77, 4752.25, 4516.76, 4626.78, 4586.46, 4631.52, 4684.54, 4616.7, 4647.33, 4635.57, 4610.21, 4476.69, 4589.18, 4510.57, 4466.68, 4465.33, 4720.93, 4559.45, 4502.03, 4515.26, 4574.92, 4360.27, 4558.31, 4657.64, 4803.43, 4456.93, 4468.14, 4599.83, 4525.7, 4714.05, 4623.93, 4695.73, 4587.68, 4579.05, 4417.93, 4629.2, 4497.05, 4529.67, 4757.1, 4463.86, 4684.5, 4486.06, 4541.3, 4686.99, 4557.75, 4626.61, 4556.11, 4638.69, 4425.23, 4478.45, 4565.94, 4524.79, 4618.41, 4626.44, 4431.43, 4800.69, 4541.57, 4528.11, 4649.04, 4673.73, 4627.01, 4789.83, 4601.85, 4393.97, 4607.49, 4706.42, 4630.12, 4570.5, 4613.86, 4507.13, 4635.5, 4620.08, 4565.06, 4477.18, 4702.14, 4465.11, 4634.31, 4623.7, 4538.39, 4675.86, 4634.8, 4475.67, 4540.14, 4730.7, 4638.66, 4575.08, 4650.4, 4694.76, 4610.2, 4712.17, 4368.53, 4619.06, 4460.39, 4640.55, 4644.25, 4782.84, 4504.21, 4414.37, 4656.78, 4570.03, 4519.61, 4494.57, 4612.02, 4415.66, 4473.85, 4644.61, 4444.42, 4607.9, 4629.62, 4405.1, 4764.79, 4464.17, 4585.5, 4653.12, 4590.67, 4795.81, 4599.44, 4448.95, 4466.68, 4454.73, 4585.13, 4653.59, 4550.54, 4552.28, 4504.24, 4394.72, 4712.52, 4725.93, 4456.86, 4667.88, 4517.91, 4556.18, 4755.36, 4763.28, 4549.27, 4631.62, 4641.09, 4591.43, 4410.58, 4713.64, 4693.83, 4526.5, 4560.4, 4697.95, 4710.93, 4536.58, 4635.35, 4418.38, 4597.47, 4485.7, 4706.08, 4502.56, 4259.8, 4595.87, 4620.54, 4573.98, 4540.33, 4409.3, 4566.17, 4522.58, 4593.02, 4547.84, 4566.03, 4628.24, 4471.06, 4435.18, 4820.39, 4514.9, 4531.99, 4605.41, 4650.99, 4731.51, 4459.61, 4686.6, 4430.62, 4590.64, 4677.3, 4361.53, 4482.91, 4607.59, 4695.17, 4555.39, 4423.77, 4532.03, 4502.15, 4538.68, 4615.08, 4714.61, 4658.95, 4721.03, 4581.74, 4661.85, 4565.88, 4480.24, 4659.18, 4521.25, 4702.42, 4613.4, 4534.29, 4621.49, 4678.63, 4272.98, 4632.08, 4718.12, 4642.54, 4617.48, 4686.65, 4531.52, 4345.78, 4487.5, 4697.66, 4593.08, 4574.02, 4612.94, 4579.51, 4503.15, 4542.96, 4707.13, 4458.35, 4643.09, 4591.37, 4368.47, 4705.28, 4543.08, 4593.19, 4425.6, 4668.48, 4666.07, 4444.45, 4605.8, 4494.0, 4554.41, 4583.67, 4809.74, 4622.39, 4568.87, 4574.74, 4676.96, 4686.21, 4602.35, 4628.41, 4428.61, 4588.34, 4464.3, 4556.77, 4567.53, 4248.18, 4641.33, 4639.46, 4691.08, 4728.4, 4678.41, 4523.96, 4330.79, 4417.48, 4579.21, 4672.2, 4635.88]
组2 n=994：[4864.61, 4496.01, 4576.77, 4620.75, 4494.67, 4471.21, 4678.37, 4635.63, 4467.29, 4498.23, 4584.82, 4677.76, 4614.34, 4376.8, 4576.45, 4501.28, 4610.41, 4727.86, 4649.68, 4579.79, 4649.88, 4521.67, 4463.93, 4464.89, 4639.53, 4504.58, 4590.1, 4514.37, 4557.32, 4486.11, 4525.49, 4598.88, 4533.72, 4519.78, 4562.25, 4508.66, 4570.68, 4596.07, 4568.83, 4557.33, 4489.28, 4558.61, 4501.59, 4514.93, 4654.45, 4653.36, 4431.24, 4745.61, 4519.77, 4581.36, 4445.22, 4631.43, 4509.7, 4590.85, 4479.94, 4687.82, 4561.39, 4644.07, 4488.41, 4565.58, 4520.65, 4620.92, 4679.8, 4456.98, 4502.51, 4348.63, 4513.09, 4758.25, 4516.84, 4641.9, 4646.23, 4661.78, 4472.73, 4666.83, 4456.0, 4419.28, 4613.18, 4444.34, 4543.28, 4662.41, 4524.53, 4449.42, 4472.1, 4581.41, 4750.26, 4480.42, 4532.32, 4529.29, 4665.71, 4720.31, 4458.61, 4655.83, 4492.49, 4497.76, 4614.64, 4645.39, 4544.77, 4313.58, 4505.59, 4570.02, 4599.76, 4532.8, 4603.39, 4497.77, 4648.96, 4736.49, 4454.44, 4541.35, 4685.08, 4239.17, 4520.13, 4714.84, 4478.07, 4793.9, 4623.06, 4654.97, 4461.67, 4594.43, 4480.55, 4744.73, 4426.25, 4541.85, 4499.22, 4488.77, 4689.43, 4541.29, 4549.85, 4692.2, 4537.09, 4508.62, 4625.28, 4657.8, 4405.01, 4615.54, 4454.45, 4653.2, 4647.2, 4702.33, 4632.02, 4686.33, 4784.99, 4686.21, 4537.32, 4560.18, 4433.07, 4542.82, 4671.85, 4598.95, 4743.09, 4520.91, 4395.04, 4715.13, 4485.16, 4603.97, 4785.06, 4574.64, 4629.02, 4708.72, 4590.06, 4576.55, 4508.69, 4756.63, 4482.24, 4707.1, 4518.59, 4513.57, 4409.99, 4550.43, 4599.28, 4590.63, 4392.69, 4647.31, 4629.54, 4488.55, 4464.68, 4567.51, 4467.94, 4374.71, 4780.22, 4499.94, 4529.13, 4507.93, 4583.28, 4468.7, 4758.05, 4668.11, 4658.81, 4713.38, 4607.53, 4591.01, 4488.04, 4385.78, 4508.02, 4615.52, 4507.42, 4688.1, 4543.38, 4647.11, 4511.53, 4818.08, 4512.72, 4563.87, 4544.54, 4549.23, 4741.6, 4534.22, 4598.94, 4553.63, 4622.79, 4521.61, 4490.97, 4387.05, 4687.88, 4658.01, 4730.42, 4730.61, 4464.46, 4559.35, 4490.24, 4702.48, 4464.55, 4529.19, 4509.69, 4539.92, 4464.65, 4606.52, 4535.86, 4634.07, 4593.58, 4546.6, 4374.01, 4548.95, 4588.36, 4726.14, 4444.1, 4400.96, 4575.46, 4694.52, 4575.22, 4463.34, 4432.8, 4711.52, 4725.32, 4565.59, 4489.76, 4554.05, 4488.04, 4509.71, 4574.82, 4630.7, 4510.61, 4624.56, 4545.7, 4631.49, 4720.62, 4502.02, 4529.0, 4590.11, 4659.73, 4347.83, 4505.95, 4514.07, 4523.59, 4576.82, 4719.7, 4638.96, 4481.66, 4669.53, 4512.15, 4554.6, 4481.96, 4509.96, 4393.87, 4597.2, 4669.75, 4536.9, 4576.13, 4466.99, 4588.45, 4658.36, 4606.33, 4600.99, 4767.61, 4503.43, 4667.12, 4593.34, 4486.21, 4406.92, 4584.8, 4386.41, 4686.41, 4552.16, 4596.01, 4619.67, 4332.87, 4510.28, 4414.17, 4600.92, 4741.86, 4580.13, 4682.1, 4648.49, 4583.15, 4583.23, 4682.53, 4673.27, 4494.4, 4661.7, 4271.34, 4548.82, 4572.94, 4552.21, 4425.71, 4772.54, 4456.95, 4572.86, 4549.72, 4545.19, 4642.54, 4483.88, 4573.7, 4669.45, 4716.12, 4614.31, 4798.18, 4584.1, 4643.35, 4480.81, 4570.37, 4627.63, 4592.13, 4430.09, 4411.45, 4434.38, 4353.31, 4761.93, 4568.47, 4610.11, 4533.26, 4428.06, 4482.8, 4597.49, 4543.91, 4496.49, 4601.49, 4513.0, 4503.87, 4618.25, 4568.91, 4596.56, 4554.5, 4473.36, 4795.5, 4531.77, 4698.42, 4689.86, 4646.76, 4565.38, 4660.72, 4369.16, 4559.32, 4867.49, 4715.71, 4591.39, 4662.15, 4617.62, 4654.95, 4396.8, 4740.23, 4556.26, 4852.1, 4669.54, 4722.64, 4753.21, 4635.95, 4585.62, 4782.04, 4490.79, 4459.72, 4581.83, 4873.0, 4499.87, 4558.3, 4677.37, 4665.57, 4605.9, 4475.37, 4708.53, 4693.28, 4560.21, 4507.37, 4539.4, 4648.38, 4538.21, 4402.2, 4557.75, 4656.56, 4633.52, 4432.45, 4632.88, 4491.49, 4653.0, 4617.32, 4670.9, 4416.74, 4490.92, 4530.2, 4629.16, 4666.52, 4726.4, 4582.66, 4628.99, 4694.75, 4552.39, 4674.23, 4586.65, 4445.45, 4456.15, 4701.9, 4554.56, 4583.56, 4542.75, 4488.62, 4596.32, 4598.77, 4577.64, 4430.6, 4459.64, 4504.15, 4653.65, 4427.13, 4461.02, 4538.69, 4378.08, 4616.56, 4580.07, 4655.82, 4468.12, 4533.37, 4801.63, 4657.95, 4681.56, 4498.69, 4582.57, 4497.03, 4589.69, 4508.72, 4599.45, 4569.38, 4657.94, 4633.57, 4548.13, 4507.02, 4657.1, 4687.71, 4597.12, 4645.96, 4606.67, 4643.12, 4601.86, 4487.78, 4687.88, 4562.25, 4661.77, 4671.64, 4650.15, 4803.82, 4728.8, 4572.75, 4598.12, 4559.47, 4755.15, 4625.93, 4560.2, 4568.06, 4656.55, 4671.84, 4615.84, 4680.92, 4583.14, 4555.73, 4445.88, 4515.02, 4562.33, 4715.82, 4728.06, 4700.49, 4462.62, 4615.62, 4659.84, 4631.07, 4673.33, 4786.85, 4530.35, 4343.64, 4799.04, 4592.6, 4575.82, 4560.92, 4590.48, 4416.17, 4592.41, 4576.28, 4630.07, 4512.13, 4463.19, 4559.74, 4600.87, 4443.45, 4652.64, 4638.27, 4604.88, 4590.57, 4529.55, 4550.91, 4469.63, 4552.56, 4711.34, 4733.66, 4673.53, 4454.55, 4566.55, 4689.28, 4616.59, 4695.9, 4507.2, 4443.48, 4731.7, 4468.11, 4606.72, 4385.56, 4800.8, 4569.08, 4559.85, 4538.64, 4535.63, 4519.68, 4741.18, 4641.36, 4664.87, 4539.96, 4551.32, 4634.4, 4433.31, 4460.6, 4592.9, 4566.85, 4540.86, 4471.05, 4682.96, 4389.21, 4649.78, 4535.84, 4419.86, 4595.2, 4706.04, 4523.79, 4746.72, 4572.86, 4716.94, 4607.03, 4490.99, 4727.76, 4493.0, 4743.47, 4685.04, 4511.12, 4562.89, 4641.0, 4677.77, 4784.51, 4648.58, 4620.75, 4329.05, 4573.47, 4805.88, 4625.49, 4684.63, 4470.4, 4516.41, 4578.36, 4650.42, 4639.09, 4540.35, 4536.87, 4485.55, 4566.21, 4633.3, 4573.02, 4499.83, 4465.66, 4589.19, 4543.57, 4457.08, 4427.09, 4615.42, 4417.89, 4706.24, 4384.82, 4619.4, 4419.98, 4549.87, 4580.88, 4760.78, 4433.64, 4581.26, 4707.06, 4406.0, 4531.71, 4479.19, 4517.3, 4636.39, 4745.02, 4709.2, 4781.05, 4559.12, 4676.3, 4442.58, 4411.84, 4747.99, 4543.04, 4606.02, 4598.62, 4607.13, 4730.28, 4799.61, 4606.91, 4586.55, 4502.0, 4510.29, 4548.51, 4731.98, 4541.31, 4636.67, 4517.42, 4615.02, 4524.65, 4461.75, 4486.68, 4669.58, 4562.58, 4504.26, 4504.75, 4588.7, 4408.22, 4632.07, 4653.85, 4474.9, 4591.23, 4819.47, 4530.65, 4654.37, 4516.25, 4432.74, 4664.2, 4728.17, 4550.82, 4554.09, 4454.24, 4700.27, 4330.51, 4662.73, 4459.49, 4466.36, 4727.87, 4570.38, 4721.66, 4576.9, 4561.7, 4671.94, 4543.42, 4576.09, 4651.05, 4658.53, 4438.07, 4339.26, 4638.71, 4696.8, 4611.97, 4543.25, 4755.0, 4574.85, 4558.99, 4432.14, 4652.53, 4649.37, 4398.32, 4525.87, 4552.77, 4734.48, 4527.95, 4614.14, 4598.17, 4469.85, 4701.73, 4723.78, 4647.11, 4670.14, 4446.24, 4481.99, 4570.41, 4542.73, 4591.75, 4369.5, 4592.42, 4363.83, 4651.65, 4605.88, 4674.29, 4678.35, 4442.54, 4461.63, 4577.35, 4584.52, 4659.39, 4846.72, 4647.89, 4431.49, 4684.7, 4481.99, 4599.21, 4703.27, 4476.28, 4492.09, 4670.32, 4590.67, 4512.82, 4548.15, 4653.79, 4448.63, 4499.38, 4619.01, 4576.3, 4420.7, 4702.22, 4505.34, 4566.77, 4752.25, 4516.76, 4626.78, 4586.46, 4631.52, 4684.54, 4616.7, 4647.33, 4635.57, 4610.21, 4476.69, 4589.18, 4510.57, 4466.68, 4465.33, 4720.93, 4559.45, 4502.03, 4515.26, 4574.92, 4360.27, 4558.31, 4657.64, 4803.43, 4456.93, 4468.14, 4599.83, 4525.7, 4714.05, 4623.93, 4695.73, 4587.68, 4579.05, 4417.93, 4629.2, 4497.05, 4529.67, 4757.1, 4463.86, 4684.5, 4486.06, 4541.3, 4686.99, 4557.75, 4626.61, 4556.11, 4638.69, 4425.23, 4478.45, 4565.94, 4524.79, 4618.41, 4626.44, 4431.43, 4800.69, 4541.57, 4528.11, 4649.04, 4673.73, 4627.01, 4789.83, 4601.85, 4393.97, 4607.49, 4706.42, 4630.12, 4570.5, 4613.86, 4507.13, 4635.5, 4620.08, 4565.06, 4477.18, 4702.14, 4465.11, 4634.31, 4623.7, 4538.39, 4675.86, 4634.8, 4475.67, 4540.14, 4730.7, 4638.66, 4575.08, 4650.4, 4694.76, 4610.2, 4712.17, 4368.53, 4619.06, 4460.39, 4640.55, 4644.25, 4782.84, 4504.21, 4414.37, 4656.78, 4570.03, 4519.61, 4494.57, 4612.02, 4415.66, 4473.85, 4644.61, 4444.42, 4607.9, 4629.62, 4405.1, 4764.79, 4464.17, 4585.5, 4653.12, 4590.67, 4795.81, 4599.44, 4448.95, 4466.68, 4454.73, 4585.13, 4653.59, 4550.54, 4552.28, 4504.24, 4394.72, 4712.52, 4725.93, 4456.86, 4667.88, 4517.91, 4556.18, 4755.36, 4763.28, 4549.27, 4631.62, 4641.09, 4591.43, 4410.58, 4713.64, 4693.83, 4526.5, 4560.4, 4697.95, 4710.93, 4536.58, 4635.35, 4418.38, 4597.47, 4485.7, 4706.08, 4502.56, 4259.8, 4595.87, 4620.54, 4573.98, 4540.33, 4409.3, 4566.17, 4522.58, 4593.02, 4547.84, 4566.03, 4628.24, 4471.06, 4435.18, 4820.39, 4514.9, 4531.99, 4605.41, 4650.99, 4731.51, 4459.61, 4686.6, 4430.62, 4590.64, 4677.3, 4361.53, 4482.91, 4607.59, 4695.17, 4555.39, 4423.77, 4532.03, 4502.15, 4538.68, 4615.08, 4714.61, 4658.95, 4721.03, 4581.74, 4661.85, 4565.88, 4480.24, 4659.18, 4521.25, 4702.42, 4613.4, 4534.29, 4621.49, 4678.63, 4272.98, 4632.08, 4718.12, 4642.54, 4617.48, 4686.65, 4531.52, 4345.78, 4487.5, 4697.66, 4593.08, 4574.02, 4612.94, 4579.51, 4503.15, 4542.96, 4707.13, 4458.35, 4643.09, 4591.37, 4368.47, 4705.28, 4543.08, 4593.19, 4425.6, 4668.48, 4666.07, 4444.45, 4605.8, 4494.0, 4554.41, 4583.67, 4809.74, 4622.39, 4568.87, 4574.74, 4676.96, 4686.21, 4602.35, 4628.41, 4428.61, 4588.34, 4464.3, 4556.77, 4567.53, 4248.18, 4641.33, 4639.46, 4691.08, 4728.4, 4678.41, 4523.96, 4330.79, 4417.48, 4579.21, 4672.2, 4635.88]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=4088.7670, s₁=133.3450；x̄₂=4578.1243, s₂=103.6038；
SE=√(s₁²/n₁+s₂²/n₂)=5.6318；t=(x̄₁−x̄₂)/SE=-86.8917；
df(Welch)=1589.61；双侧 p = 0.000e+00。
x̄₁=4088.7670, s₁=133.3450；x̄₂=4578.1243, s₂=103.6038；
SE=√(s₁²/n₁+s₂²/n₂)=5.6318；t=(x̄₁−x̄₂)/SE=-86.8917；
df(Welch)=1589.61；双侧 p = 0.000e+00。
计算结果
t = -86.8917，df = 1589.61，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。检验吸烟对肺活量的影响。
案例 29. 两地平均气温
原始数据
组1 n=591：[19.87, 20.77, 23.11, 17.8, 20.96, 19.39, 23.55, 20.68, 20.84, 20.28, 23.96, 22.85, 20.29, 19.67, 21.29, 22.17, 21.18, 20.86, 22.72, 20.07, 20.5, 21.31, 22.45, 22.35, 23.15, 20.51, 21.08, 17.58, 20.36, 19.04, 22.18, 20.6, 20.06, 18.63, 22.33, 19.8, 23.37, 22.96, 21.59, 18.91, 23.43, 20.06, 20.25, 20.61, 21.05, 20.3, 19.98, 19.67, 19.33, 20.14, 22.38, 25.11, 20.61, 18.58, 20.59, 19.97, 20.83, 22.14, 23.2, 19.4, 20.03, 23.89, 19.85, 20.0, 19.55, 19.72, 19.34, 19.15, 20.47, 20.04, 19.98, 21.96, 21.94, 21.06, 20.55, 21.5, 20.22, 22.82, 22.23, 21.1, 22.52, 21.59, 17.6, 23.29, 21.85, 23.2, 21.02, 23.04, 22.65, 20.23, 23.5, 20.46, 20.33, 22.39, 22.11, 21.15, 20.29, 22.71, 21.39, 24.52, 20.48, 21.85, 21.61, 22.72, 20.74, 21.41, 22.44, 20.51, 21.04, 20.11, 24.52, 20.72, 22.26, 17.97, 22.13, 19.35, 23.0, 23.16, 20.35, 21.74, 19.66, 22.22, 21.62, 21.22, 22.31, 21.72, 20.03, 22.33, 22.41, 21.76, 20.16, 22.26, 21.36, 20.45, 20.49, 20.48, 22.85, 22.36, 20.34, 21.28, 21.85, 20.34, 23.54, 23.76, 20.89, 19.33, 23.08, 20.88, 21.6, 24.04, 18.1, 21.76, 20.05, 21.21, 20.21, 20.95, 23.88, 21.76, 24.06, 21.3, 21.8, 19.24, 20.12, 21.62, 21.55, 21.87, 22.34, 23.59, 22.85, 22.39, 20.95, 18.57, 20.18, 20.92, 21.47, 19.55, 21.12, 23.26, 21.37, 21.53, 20.32, 20.72, 21.5, 20.07, 23.12, 18.74, 22.26, 20.57, 17.03, 20.01, 20.63, 21.98, 20.93, 21.23, 20.59, 21.44, 22.53, 17.56, 20.78, 21.03, 23.65, 23.15, 21.04, 22.27, 20.01, 22.65, 20.59, 22.86, 20.97, 20.63, 21.77, 22.36, 21.0, 20.22, 22.35, 20.21, 21.8, 23.31, 19.76, 20.23, 20.64, 24.5, 22.34, 21.46, 22.63, 21.13, 19.48, 21.17, 21.48, 23.78, 20.74, 21.6, 22.3, 23.13, 19.9, 20.93, 20.53, 21.1, 22.3, 20.6, 19.23, 20.86, 16.85, 19.88, 23.15, 21.13, 21.11, 19.88, 20.94, 20.13, 23.07, 24.38, 20.74, 20.7, 19.2, 23.65, 20.67, 20.0, 21.42, 24.02, 19.5, 19.64, 20.14, 19.83, 22.38, 20.93, 22.48, 20.1, 22.44, 19.54, 20.95, 20.19, 19.93, 20.8, 22.3, 21.82, 21.84, 21.3, 22.06, 20.02, 21.82, 22.68, 21.12, 24.68, 24.44, 21.16, 19.47, 20.6, 20.52, 23.0, 20.34, 23.07, 21.89, 19.34, 20.61, 24.6, 20.39, 18.65, 22.01, 20.76, 19.22, 22.6, 20.41, 21.35, 22.79, 19.64, 22.06, 25.13, 20.1, 22.11, 20.17, 22.37, 20.06, 22.52, 19.75, 19.36, 21.93, 22.21, 22.83, 20.51, 21.68, 21.09, 21.38, 21.61, 22.41, 20.65, 19.61, 21.26, 20.53, 22.1, 21.8, 20.25, 21.99, 22.85, 21.45, 21.99, 20.04, 23.04, 20.81, 21.86, 20.69, 21.68, 21.96, 22.79, 20.55, 19.96, 19.51, 20.21, 19.42, 20.76, 22.43, 22.34, 21.79, 22.11, 20.65, 21.36, 22.99, 18.69, 20.9, 21.66, 20.38, 20.55, 20.95, 22.89, 24.13, 20.69, 20.71, 20.74, 21.89, 17.6, 20.1, 21.39, 20.94, 20.3, 23.85, 22.07, 20.18, 23.48, 22.22, 20.4, 21.64, 19.19, 16.42, 20.63, 21.83, 23.13, 21.11, 22.82, 20.7, 19.34, 20.85, 23.47, 23.7, 22.91, 18.56, 21.28, 21.73, 23.97, 20.74, 21.69, 18.44, 22.11, 19.46, 22.3, 21.03, 23.63, 20.82, 21.19, 21.06, 19.74, 21.74, 22.75, 20.34, 21.85, 22.78, 21.13, 19.29, 20.3, 22.09, 23.74, 22.2, 25.09, 19.29, 22.76, 21.82, 19.38, 20.54, 20.71, 19.33, 20.12, 20.71, 18.87, 23.35, 21.02, 19.98, 21.43, 23.23, 21.07, 24.77, 21.08, 20.98, 22.22, 19.28, 21.82, 22.25, 21.92, 19.92, 20.01, 22.72, 22.05, 23.23, 22.09, 21.9, 20.24, 22.39, 21.0, 21.68, 22.78, 20.42, 20.5, 17.84, 20.28, 20.66, 18.01, 23.07, 20.68, 20.59, 20.48, 22.57, 19.54, 22.11, 21.53, 19.12, 19.59, 22.82, 22.44, 22.59, 20.52, 19.97, 22.43, 20.52, 21.59, 19.91, 19.6, 19.72, 22.98, 23.48, 19.88, 20.35, 20.56, 22.01, 19.71, 22.34, 20.1, 20.84, 22.67, 21.35, 20.84, 20.81, 21.11, 19.71, 21.63, 18.0, 25.31, 21.63, 23.43, 22.34, 22.39, 21.77, 23.11, 22.12, 22.38, 20.57, 22.46, 21.67, 19.39, 20.75, 22.37, 22.74, 17.76, 20.48, 18.13, 21.23, 21.97, 20.56, 20.51, 20.83, 22.99, 20.23, 21.24, 20.51, 21.91, 18.4, 21.17, 20.17, 20.7, 21.07, 19.6, 22.07, 21.73, 18.64, 20.03, 20.54, 21.32, 23.2, 20.29, 20.2, 21.04, 21.58, 20.56, 20.31, 18.89, 24.24, 22.16, 19.62, 18.45, 20.1, 20.43, 20.28, 20.62, 20.96, 21.97, 22.56, 20.49, 19.98, 21.15, 19.58, 20.58, 22.78, 22.46, 24.45, 21.96, 17.72, 21.39, 22.91, 22.56, 20.26, 20.39, 20.28, 20.84, 21.41, 18.77, 22.72, 20.94, 23.35, 20.72, 21.85, 21.79, 20.31, 19.97, 21.29]；
组2 n=366：[15.24, 13.73, 16.35, 19.22, 18.94, 16.0, 14.04, 12.41, 15.87, 15.79, 17.56, 16.75, 15.14, 16.16, 16.74, 16.04, 13.92, 15.04, 19.35, 13.46, 17.35, 17.89, 15.78, 13.85, 15.37, 17.21, 16.45, 16.43, 14.28, 15.99, 17.72, 16.84, 16.81, 15.25, 16.93, 15.19, 16.97, 15.36, 14.55, 13.28, 16.15, 17.76, 13.73, 17.38, 16.79, 15.82, 18.17, 17.51, 15.87, 16.56, 19.59, 14.25, 16.2, 17.1, 16.51, 17.17, 17.19, 15.17, 13.9, 15.43, 16.41, 18.59, 17.21, 16.88, 16.72, 16.27, 14.0, 21.06, 17.91, 18.77, 16.8, 19.79, 16.49, 16.62, 16.98, 17.41, 14.54, 14.27, 15.53, 16.06, 16.16, 14.88, 15.02, 16.01, 18.05, 15.86, 16.84, 13.61, 16.02, 16.58, 15.73, 17.37, 15.39, 15.54, 15.17, 17.25, 16.95, 17.69, 16.63, 14.2, 15.38, 14.23, 19.85, 14.78, 16.88, 16.76, 15.64, 17.93, 14.21, 15.3, 11.4, 17.89, 16.44, 16.55, 14.59, 17.09, 15.89, 14.68, 17.59, 18.23, 14.81, 18.45, 15.78, 16.41, 15.87, 20.56, 14.01, 14.43, 17.69, 15.98, 15.98, 17.16, 15.17, 13.01, 15.41, 14.75, 15.04, 17.14, 16.81, 16.52, 15.34, 14.99, 12.72, 14.89, 17.07, 17.23, 15.4, 16.4, 20.12, 15.41, 17.65, 14.58, 14.9, 16.22, 18.24, 16.65, 15.44, 15.78, 13.8, 14.61, 15.55, 15.79, 14.97, 15.64, 11.85, 18.33, 15.16, 16.12, 15.87, 14.4, 17.21, 18.44, 14.8, 16.64, 15.97, 15.67, 15.12, 18.3, 13.82, 16.1, 17.36, 15.06, 16.6, 17.46, 15.41, 15.01, 18.19, 15.24, 13.37, 13.96, 18.96, 17.2, 14.69, 16.0, 16.13, 17.37, 16.59, 16.67, 16.8, 14.07, 16.38, 17.55, 16.49, 15.0, 17.12, 17.17, 14.95, 17.75, 15.91, 16.59, 12.81, 14.77, 16.36, 17.04, 17.66, 15.62, 14.88, 14.78, 15.96, 15.63, 16.91, 17.12, 14.86, 15.71, 16.57, 15.27, 15.3, 17.73, 16.82, 19.9, 15.41, 12.64, 15.04, 13.99, 19.66, 15.87, 15.07, 17.86, 16.48, 15.08, 15.93, 14.32, 14.33, 17.48, 17.33, 17.99, 17.7, 16.18, 15.48, 15.52, 14.84, 17.62, 17.2, 18.21, 15.64, 17.68, 18.26, 15.38, 15.84, 14.94, 16.98, 14.3, 17.03, 15.8, 17.04, 17.16, 17.39, 14.34, 17.73, 16.41, 16.56, 17.41, 16.14, 18.35, 17.72, 14.36, 16.35, 17.13, 17.65, 16.13, 15.6, 17.64, 18.79, 13.69, 14.6, 15.52, 14.19, 15.15, 16.61, 17.84, 16.63, 17.25, 17.15, 14.7, 14.5, 19.38, 17.23, 16.73, 17.22, 16.57, 15.25, 15.03, 15.94, 14.6, 14.74, 13.3, 18.68, 16.98, 16.34, 17.21, 15.74, 16.12, 17.76, 17.19, 15.63, 16.06, 18.77, 13.94, 11.29, 16.75, 16.73, 16.67, 16.71, 15.33, 16.17, 15.42, 18.17, 15.2, 16.55, 15.22, 16.78, 14.28, 16.87, 12.54, 16.95, 12.76, 18.69, 15.53, 17.3, 15.28, 13.3, 16.91, 14.0, 16.76, 17.4, 16.81, 16.01, 17.87, 15.99, 17.05, 13.73, 15.86, 16.83, 16.5, 15.82, 17.81, 16.57, 18.29, 17.09, 15.33, 15.36, 15.6, 16.49, 16.24, 14.55, 17.27]
组2 n=366：[15.24, 13.73, 16.35, 19.22, 18.94, 16.0, 14.04, 12.41, 15.87, 15.79, 17.56, 16.75, 15.14, 16.16, 16.74, 16.04, 13.92, 15.04, 19.35, 13.46, 17.35, 17.89, 15.78, 13.85, 15.37, 17.21, 16.45, 16.43, 14.28, 15.99, 17.72, 16.84, 16.81, 15.25, 16.93, 15.19, 16.97, 15.36, 14.55, 13.28, 16.15, 17.76, 13.73, 17.38, 16.79, 15.82, 18.17, 17.51, 15.87, 16.56, 19.59, 14.25, 16.2, 17.1, 16.51, 17.17, 17.19, 15.17, 13.9, 15.43, 16.41, 18.59, 17.21, 16.88, 16.72, 16.27, 14.0, 21.06, 17.91, 18.77, 16.8, 19.79, 16.49, 16.62, 16.98, 17.41, 14.54, 14.27, 15.53, 16.06, 16.16, 14.88, 15.02, 16.01, 18.05, 15.86, 16.84, 13.61, 16.02, 16.58, 15.73, 17.37, 15.39, 15.54, 15.17, 17.25, 16.95, 17.69, 16.63, 14.2, 15.38, 14.23, 19.85, 14.78, 16.88, 16.76, 15.64, 17.93, 14.21, 15.3, 11.4, 17.89, 16.44, 16.55, 14.59, 17.09, 15.89, 14.68, 17.59, 18.23, 14.81, 18.45, 15.78, 16.41, 15.87, 20.56, 14.01, 14.43, 17.69, 15.98, 15.98, 17.16, 15.17, 13.01, 15.41, 14.75, 15.04, 17.14, 16.81, 16.52, 15.34, 14.99, 12.72, 14.89, 17.07, 17.23, 15.4, 16.4, 20.12, 15.41, 17.65, 14.58, 14.9, 16.22, 18.24, 16.65, 15.44, 15.78, 13.8, 14.61, 15.55, 15.79, 14.97, 15.64, 11.85, 18.33, 15.16, 16.12, 15.87, 14.4, 17.21, 18.44, 14.8, 16.64, 15.97, 15.67, 15.12, 18.3, 13.82, 16.1, 17.36, 15.06, 16.6, 17.46, 15.41, 15.01, 18.19, 15.24, 13.37, 13.96, 18.96, 17.2, 14.69, 16.0, 16.13, 17.37, 16.59, 16.67, 16.8, 14.07, 16.38, 17.55, 16.49, 15.0, 17.12, 17.17, 14.95, 17.75, 15.91, 16.59, 12.81, 14.77, 16.36, 17.04, 17.66, 15.62, 14.88, 14.78, 15.96, 15.63, 16.91, 17.12, 14.86, 15.71, 16.57, 15.27, 15.3, 17.73, 16.82, 19.9, 15.41, 12.64, 15.04, 13.99, 19.66, 15.87, 15.07, 17.86, 16.48, 15.08, 15.93, 14.32, 14.33, 17.48, 17.33, 17.99, 17.7, 16.18, 15.48, 15.52, 14.84, 17.62, 17.2, 18.21, 15.64, 17.68, 18.26, 15.38, 15.84, 14.94, 16.98, 14.3, 17.03, 15.8, 17.04, 17.16, 17.39, 14.34, 17.73, 16.41, 16.56, 17.41, 16.14, 18.35, 17.72, 14.36, 16.35, 17.13, 17.65, 16.13, 15.6, 17.64, 18.79, 13.69, 14.6, 15.52, 14.19, 15.15, 16.61, 17.84, 16.63, 17.25, 17.15, 14.7, 14.5, 19.38, 17.23, 16.73, 17.22, 16.57, 15.25, 15.03, 15.94, 14.6, 14.74, 13.3, 18.68, 16.98, 16.34, 17.21, 15.74, 16.12, 17.76, 17.19, 15.63, 16.06, 18.77, 13.94, 11.29, 16.75, 16.73, 16.67, 16.71, 15.33, 16.17, 15.42, 18.17, 15.2, 16.55, 15.22, 16.78, 14.28, 16.87, 12.54, 16.95, 12.76, 18.69, 15.53, 17.3, 15.28, 13.3, 16.91, 14.0, 16.76, 17.4, 16.81, 16.01, 17.87, 15.99, 17.05, 13.73, 15.86, 16.83, 16.5, 15.82, 17.81, 16.57, 18.29, 17.09, 15.33, 15.36, 15.6, 16.49, 16.24, 14.55, 17.27]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=21.2121, s₁=1.4523；x̄₂=16.1499, s₂=1.5300；
SE=√(s₁²/n₁+s₂²/n₂)=0.0998；t=(x̄₁−x̄₂)/SE=50.7117；
df(Welch)=742.86；双侧 p = 1.868e-243。
x̄₁=21.2121, s₁=1.4523；x̄₂=16.1499, s₂=1.5300；
SE=√(s₁²/n₁+s₂²/n₂)=0.0998；t=(x̄₁−x̄₂)/SE=50.7117；
df(Welch)=742.86；双侧 p = 1.868e-243。
计算结果
t = 50.7117，df = 742.86，p = 1.868e-243
在 α=0.05 水平下，p=1.868e-243 < 0.05，拒绝原假设 H₀。比较两地温度差异。
案例 30. 两种包装破损数
原始数据
组1 n=954：[5.02, 5.86, 4.51, 3.72, 5.69, 5.29, 4.25, 5.03, 4.64, 5.32, 5.18, 5.5, 5.8, 5.45, 6.55, 6.19, 6.07, 6.52, 4.18, 4.6, 5.43, 6.17, 5.43, 3.94, 6.58, 4.92, 5.87, 4.69, 4.36, 4.74, 5.57, 4.32, 7.43, 5.16, 5.85, 8.49, 7.75, 5.71, 3.66, 5.05, 3.43, 4.57, 5.17, 6.0, 4.87, 6.53, 5.38, 3.86, 7.1, 5.47, 6.4, 3.93, 6.8, 5.99, 4.4, 5.79, 4.49, 4.57, 4.89, 6.19, 4.97, 4.82, 5.66, 6.14, 4.9, 3.9, 5.3, 6.44, 4.38, 4.73, 5.99, 5.8, 4.42, 5.04, 6.19, 4.01, 5.3, 7.08, 6.3, 4.59, 4.73, 4.68, 3.82, 4.63, 5.23, 4.05, 4.6, 6.65, 5.32, 5.52, 5.04, 3.8, 5.52, 5.44, 6.87, 5.9, 4.26, 4.91, 5.98, 5.8, 2.76, 5.72, 5.3, 5.2, 5.28, 4.73, 5.92, 7.08, 4.29, 6.59, 3.99, 5.32, 5.86, 7.15, 6.24, 5.41, 6.27, 5.48, 6.03, 3.88, 5.95, 4.29, 4.23, 4.92, 6.25, 5.57, 4.07, 5.64, 4.76, 5.16, 4.32, 7.19, 4.86, 6.2, 5.48, 4.25, 4.09, 4.57, 6.35, 4.59, 6.3, 5.11, 5.5, 5.51, 5.48, 7.1, 3.62, 5.33, 5.41, 4.56, 4.77, 5.3, 5.5, 4.32, 5.81, 4.24, 4.91, 5.92, 6.3, 5.86, 5.64, 4.46, 6.49, 5.58, 4.72, 5.65, 5.88, 4.42, 5.94, 6.18, 5.01, 5.43, 5.06, 5.53, 4.44, 5.21, 5.83, 4.32, 5.05, 6.56, 6.93, 6.67, 4.31, 5.4, 4.95, 3.3, 5.6, 5.24, 4.9, 3.9, 5.38, 5.08, 7.03, 4.75, 6.12, 3.98, 5.32, 5.92, 5.93, 6.33, 4.53, 5.31, 4.47, 5.19, 5.19, 7.86, 5.56, 3.25, 6.29, 5.41, 4.99, 4.15, 2.91, 6.24, 6.42, 6.33, 5.55, 4.5, 5.89, 5.46, 5.97, 5.06, 6.56, 6.21, 5.27, 5.52, 4.98, 6.39, 6.03, 5.43, 6.05, 5.73, 5.08, 5.36, 5.31, 4.84, 6.31, 5.5, 5.78, 7.05, 5.4, 4.24, 5.57, 6.04, 4.66, 6.63, 5.34, 5.16, 5.85, 2.85, 6.54, 4.74, 4.03, 5.07, 6.18, 5.75, 5.43, 5.25, 5.01, 7.24, 5.26, 5.77, 6.8, 4.67, 5.67, 6.28, 4.91, 5.96, 3.99, 4.42, 3.81, 3.23, 5.08, 6.25, 6.22, 6.39, 6.0, 4.4, 6.41, 4.52, 4.53, 5.5, 5.73, 5.54, 6.29, 4.66, 1.85, 6.84, 5.48, 4.69, 6.01, 5.41, 4.39, 4.44, 5.49, 5.06, 3.79, 6.49, 4.34, 6.56, 5.76, 3.74, 5.65, 6.23, 5.74, 6.45, 5.18, 5.51, 5.59, 6.37, 5.35, 4.09, 6.44, 4.55, 4.02, 4.14, 5.12, 5.24, 4.56, 5.44, 4.7, 4.37, 4.67, 4.28, 3.87, 5.25, 6.75, 5.1, 2.61, 4.52, 5.6, 6.49, 4.08, 3.72, 3.9, 4.09, 5.17, 5.68, 4.98, 5.03, 5.15, 5.88, 5.79, 6.02, 4.72, 5.68, 4.51, 5.01, 4.52, 6.04, 5.89, 4.06, 6.41, 4.58, 3.86, 5.52, 4.6, 4.61, 6.08, 7.54, 4.35, 4.41, 4.2, 2.99, 5.89, 4.74, 4.59, 5.12, 5.45, 5.26, 5.2, 5.1, 6.31, 5.71, 5.73, 3.55, 5.11, 6.1, 5.22, 4.64, 5.22, 5.07, 5.08, 5.69, 6.15, 5.85, 5.21, 5.73, 5.96, 7.56, 3.59, 5.53, 6.73, 4.82, 4.91, 6.52, 4.83, 4.32, 4.39, 5.91, 6.37, 5.76, 5.12, 4.82, 5.12, 5.85, 5.63, 4.65, 5.88, 4.09, 6.11, 5.63, 5.19, 5.14, 5.43, 5.86, 3.68, 4.57, 6.02, 5.11, 6.99, 6.03, 4.94, 5.48, 6.61, 7.41, 6.11, 4.32, 5.59, 6.31, 6.29, 6.29, 5.02, 4.5, 6.1, 5.35, 6.59, 3.51, 5.7, 5.43, 5.14, 6.41, 3.67, 5.74, 7.53, 4.86, 5.04, 5.23, 7.04, 4.57, 4.62, 7.31, 5.07, 5.08, 6.88, 5.41, 4.86, 5.39, 4.68, 5.08, 6.74, 4.81, 4.62, 6.21, 5.0, 4.51, 5.47, 5.05, 4.94, 6.09, 6.65, 3.95, 6.74, 5.04, 7.12, 5.23, 4.79, 4.2, 6.02, 5.43, 6.08, 6.64, 4.54, 5.68, 4.85, 4.65, 4.69, 4.54, 4.25, 4.44, 5.62, 4.91, 5.89, 4.66, 6.56, 6.21, 5.72, 3.99, 7.11, 4.23, 4.1, 5.83, 4.15, 5.43, 4.46, 6.87, 5.23, 4.12, 6.46, 5.17, 5.23, 7.44, 3.56, 4.54, 5.61, 4.46, 5.83, 4.18, 3.85, 7.0, 6.05, 5.83, 6.56, 3.25, 4.98, 5.52, 4.71, 4.97, 5.7, 4.7, 3.97, 5.05, 4.68, 4.72, 4.47, 7.31, 5.48, 5.45, 5.93, 5.45, 4.63, 6.76, 4.76, 5.29, 5.81, 3.94, 6.88, 4.88, 6.09, 6.72, 7.04, 5.96, 5.33, 6.57, 4.9, 5.76, 5.64, 6.76, 4.68, 6.15, 3.89, 6.91, 4.38, 4.87, 5.2, 6.27, 5.21, 5.74, 4.74, 5.89, 5.65, 5.62, 5.62, 6.74, 7.09, 3.11, 4.82, 6.97, 6.48, 6.16, 3.76, 6.32, 3.99, 4.54, 4.88, 6.9, 6.17, 6.08, 4.98, 5.93, 5.05, 4.75, 4.38, 5.57, 5.5, 6.34, 3.61, 7.02, 4.9, 5.32, 5.18, 7.09, 6.9, 7.03, 4.7, 5.37, 5.46, 6.59, 5.81, 5.5, 4.52, 5.52, 7.16, 4.39, 6.03, 4.88, 5.44, 5.64, 4.9, 4.82, 3.18, 4.34, 5.44, 4.66, 6.65, 6.36, 5.05, 4.53, 5.3, 5.86, 5.36, 5.35, 4.62, 4.4, 4.71, 6.18, 3.68, 3.83, 4.08, 5.77, 4.67, 6.7, 6.11, 2.77, 6.75, 4.63, 5.59, 4.02, 4.18, 5.62, 7.35, 6.11, 5.24, 3.65, 4.91, 5.3, 4.51, 6.2, 5.67, 4.78, 6.36, 4.96, 5.66, 6.18, 5.4, 3.27, 4.42, 6.3, 6.29, 3.83, 5.69, 4.81, 4.14, 5.87, 5.07, 6.01, 7.3, 7.47, 6.5, 7.16, 5.16, 6.0, 5.81, 5.72, 5.79, 5.52, 4.19, 6.08, 5.76, 6.08, 7.35, 6.75, 5.86, 6.32, 5.16, 3.56, 4.07, 6.44, 5.82, 6.72, 6.32, 5.21, 4.41, 6.07, 6.13, 5.79, 7.22, 4.57, 4.31, 6.04, 7.1, 5.29, 3.71, 5.02, 5.45, 6.12, 6.27, 5.87, 5.69, 4.06, 4.88, 4.48, 4.11, 4.98, 3.93, 5.74, 4.99, 6.05, 4.79, 4.93, 6.11, 6.04, 6.31, 3.82, 5.6, 4.25, 4.51, 5.36, 5.73, 5.06, 4.62, 4.66, 4.0, 6.08, 7.08, 5.16, 5.82, 3.65, 4.45, 6.52, 6.37, 3.85, 3.19, 6.48, 5.4, 5.16, 4.66, 6.09, 6.02, 6.34, 5.28, 5.54, 4.42, 6.74, 5.46, 6.15, 5.52, 4.4, 4.49, 6.05, 5.28, 5.82, 7.04, 6.26, 5.18, 6.59, 4.41, 3.52, 6.19, 5.02, 6.0, 5.0, 6.64, 5.27, 4.47, 3.92, 6.48, 6.28, 6.33, 5.21, 4.46, 4.65, 5.76, 5.63, 5.44, 4.39, 4.41, 4.42, 4.75, 4.67, 3.1, 6.5, 4.45, 4.79, 5.68, 6.58, 5.45, 3.42, 6.12, 5.17, 6.09, 6.25, 5.74, 5.7, 5.48, 4.66, 4.78, 4.24, 5.53, 4.89, 6.72, 4.39, 5.07, 6.24, 5.32, 5.14, 4.56, 5.88, 5.22, 6.66, 6.69, 5.86, 3.31, 7.31, 5.83, 4.69, 5.06, 4.53, 4.94, 3.61, 5.32, 5.41, 5.61, 5.2, 6.56, 5.54, 5.79, 5.69, 7.16, 4.92, 5.31, 6.47, 3.51, 5.17, 5.76, 4.09, 5.14, 4.44, 4.09, 4.61, 5.69, 4.17, 5.37, 4.67, 5.32, 5.54, 6.02, 2.79, 5.99, 3.73, 4.38, 4.79, 4.94, 6.09, 4.43, 5.23, 5.7, 7.02, 3.98, 5.88, 4.76, 5.11, 5.79, 5.78, 5.26, 3.58, 4.83, 3.97, 6.26, 6.56, 7.6, 5.55, 3.78, 5.16, 7.78, 6.77, 4.65, 5.88, 4.42, 4.77, 5.24, 4.44, 4.96, 5.1, 5.57, 6.57, 5.53, 5.99, 6.02, 5.35, 4.38, 5.91, 6.53, 5.27, 3.67, 4.93, 3.64, 6.47, 4.44, 4.8, 5.35, 4.51, 6.42, 4.73, 4.35, 4.38, 5.44, 5.62, 5.58, 6.28, 6.09, 5.81, 6.48, 5.28, 6.34, 5.53, 6.08, 6.77, 5.77, 4.74, 6.54, 6.32, 5.9, 5.41, 5.82, 5.42, 4.27, 5.44, 4.67, 5.85, 4.19, 5.0, 6.11, 5.69]；
组2 n=429：[1.75, 3.11, 1.95, 1.27, 3.77, 2.57, 3.0, 2.72, 2.91, 2.71, 3.69, 2.67, 2.22, 4.57, 3.16, 2.42, 3.44, 2.81, 3.59, 2.58, 4.52, 2.79, 2.54, 2.65, 2.03, 2.71, 3.21, 2.26, 3.75, 2.79, 3.68, 2.45, 3.04, 1.78, 3.5, 2.32, 3.18, 2.46, 2.26, 2.69, 2.5, 4.05, 3.39, 2.09, 3.01, 1.85, 2.47, 3.33, 2.29, 2.74, 2.33, 4.25, 2.27, 2.16, 2.38, 3.53, 1.46, 2.52, 1.57, 1.93, 3.1, 2.91, 3.01, 3.15, 3.79, 2.79, 2.78, 2.0, 1.89, 1.98, 2.78, 2.3, 3.43, 1.55, 2.2, 3.09, 1.59, 1.86, 3.89, 2.01, 3.64, 2.52, 1.71, 2.7, 3.6, 2.79, 3.31, 2.09, 2.84, 3.8, 3.4, 2.76, 3.13, 4.04, 2.28, 2.86, 3.82, 3.34, 4.19, 2.7, 2.05, 3.66, 3.03, 2.42, 3.68, 2.66, 3.14, 3.06, 2.8, 4.11, 2.58, 3.96, 2.27, 3.6, 3.12, 2.27, 3.64, 2.87, 3.16, 2.41, 2.34, 3.17, 2.84, 2.61, 1.82, 1.93, 3.39, 4.26, 2.78, 3.0, 3.03, 3.15, 2.33, 4.39, 3.32, 3.09, 4.44, 3.1, 3.44, 2.27, 3.37, 2.5, 2.55, 4.14, 3.63, 2.89, 2.29, 3.92, 2.66, 2.38, 3.27, 3.89, 3.16, 2.52, 2.81, 0.95, 3.25, 1.96, 1.23, 3.31, 3.68, 1.7, 1.85, 3.26, 3.53, 2.73, 3.66, 3.33, 3.29, 3.51, 2.53, 1.88, 2.4, 1.85, 2.16, 2.87, 2.66, 2.87, 2.09, 2.04, 2.83, 2.77, 2.84, 3.46, 2.33, 2.64, 2.84, 2.76, 1.84, 3.06, 3.21, 3.77, 2.51, 4.1, 2.83, 1.61, 3.56, 2.37, 2.37, 3.11, 3.3, 2.89, 2.63, 2.39, 2.22, 1.7, 2.8, 2.8, 2.89, 2.56, 2.93, 4.51, 3.66, 2.71, 4.37, 2.6, 2.53, 3.25, 4.17, 2.41, 3.2, 4.05, 3.22, 2.99, 2.55, 2.66, 2.93, 2.3, 3.18, 1.5, 2.01, 3.38, 3.34, 2.05, 3.24, 3.43, 2.8, 2.74, 3.57, 5.38, 2.75, 3.66, 3.15, 2.93, 1.91, 3.14, 2.97, 2.3, 1.71, 2.89, 3.41, 3.27, 1.88, 2.98, 3.13, 2.37, 1.87, 2.57, 3.58, 2.77, 3.51, 1.67, 2.71, 3.75, 2.93, 1.62, 2.86, 2.39, 2.66, 3.76, 2.23, 5.18, 2.75, 3.87, 2.08, 3.16, 3.41, 3.53, 3.99, 4.05, 3.59, 3.04, 3.66, 2.53, 2.66, 3.21, 2.21, 2.27, 3.79, 1.73, 2.89, 2.25, 3.85, 3.93, 2.32, 4.13, 2.53, 2.53, 2.81, 2.48, 2.27, 2.54, 2.71, 3.53, 3.65, 1.64, 3.79, 2.54, 2.22, 3.08, 3.11, 3.59, 2.97, 3.73, 3.59, 3.53, 3.23, 1.75, 2.83, 2.85, 2.79, 2.99, 3.3, 2.97, 2.73, 3.34, 3.33, 3.93, 3.03, 2.09, 3.23, 5.39, 1.58, 2.16, 2.94, 2.44, 2.14, 2.58, 2.64, 2.29, 2.49, 2.21, 2.95, 3.33, 1.19, 2.99, 3.48, 1.26, 2.36, 2.08, 3.29, 3.87, 3.37, 2.11, 3.81, 2.28, 2.25, 2.88, 3.11, 3.39, 2.48, 3.11, 2.14, 2.17, 2.91, 2.55, 2.73, 3.69, 2.4, 4.1, 3.84, 1.93, 3.74, 2.29, 1.12, 3.19, 3.87, 3.14, 2.38, 2.16, 3.51, 3.13, 2.9, 1.75, 2.07, 3.39, 1.84, 3.02, 3.5, 3.13, 1.93, 2.62, 2.13, 3.37, 3.42, 1.08, 3.8, 2.41, 2.35, 4.12, 2.96, 2.29, 2.46, 3.35, 3.45, 2.63, 3.63, 2.46, 3.53, 3.38, 2.92, 3.6, 3.28, 2.7, 2.57, 2.67, 2.5, 2.04, 1.83, 2.78, 3.88, 3.86, 3.81, 2.87, 3.06, 1.91, 3.35, 1.62, 2.55]
组2 n=429：[1.75, 3.11, 1.95, 1.27, 3.77, 2.57, 3.0, 2.72, 2.91, 2.71, 3.69, 2.67, 2.22, 4.57, 3.16, 2.42, 3.44, 2.81, 3.59, 2.58, 4.52, 2.79, 2.54, 2.65, 2.03, 2.71, 3.21, 2.26, 3.75, 2.79, 3.68, 2.45, 3.04, 1.78, 3.5, 2.32, 3.18, 2.46, 2.26, 2.69, 2.5, 4.05, 3.39, 2.09, 3.01, 1.85, 2.47, 3.33, 2.29, 2.74, 2.33, 4.25, 2.27, 2.16, 2.38, 3.53, 1.46, 2.52, 1.57, 1.93, 3.1, 2.91, 3.01, 3.15, 3.79, 2.79, 2.78, 2.0, 1.89, 1.98, 2.78, 2.3, 3.43, 1.55, 2.2, 3.09, 1.59, 1.86, 3.89, 2.01, 3.64, 2.52, 1.71, 2.7, 3.6, 2.79, 3.31, 2.09, 2.84, 3.8, 3.4, 2.76, 3.13, 4.04, 2.28, 2.86, 3.82, 3.34, 4.19, 2.7, 2.05, 3.66, 3.03, 2.42, 3.68, 2.66, 3.14, 3.06, 2.8, 4.11, 2.58, 3.96, 2.27, 3.6, 3.12, 2.27, 3.64, 2.87, 3.16, 2.41, 2.34, 3.17, 2.84, 2.61, 1.82, 1.93, 3.39, 4.26, 2.78, 3.0, 3.03, 3.15, 2.33, 4.39, 3.32, 3.09, 4.44, 3.1, 3.44, 2.27, 3.37, 2.5, 2.55, 4.14, 3.63, 2.89, 2.29, 3.92, 2.66, 2.38, 3.27, 3.89, 3.16, 2.52, 2.81, 0.95, 3.25, 1.96, 1.23, 3.31, 3.68, 1.7, 1.85, 3.26, 3.53, 2.73, 3.66, 3.33, 3.29, 3.51, 2.53, 1.88, 2.4, 1.85, 2.16, 2.87, 2.66, 2.87, 2.09, 2.04, 2.83, 2.77, 2.84, 3.46, 2.33, 2.64, 2.84, 2.76, 1.84, 3.06, 3.21, 3.77, 2.51, 4.1, 2.83, 1.61, 3.56, 2.37, 2.37, 3.11, 3.3, 2.89, 2.63, 2.39, 2.22, 1.7, 2.8, 2.8, 2.89, 2.56, 2.93, 4.51, 3.66, 2.71, 4.37, 2.6, 2.53, 3.25, 4.17, 2.41, 3.2, 4.05, 3.22, 2.99, 2.55, 2.66, 2.93, 2.3, 3.18, 1.5, 2.01, 3.38, 3.34, 2.05, 3.24, 3.43, 2.8, 2.74, 3.57, 5.38, 2.75, 3.66, 3.15, 2.93, 1.91, 3.14, 2.97, 2.3, 1.71, 2.89, 3.41, 3.27, 1.88, 2.98, 3.13, 2.37, 1.87, 2.57, 3.58, 2.77, 3.51, 1.67, 2.71, 3.75, 2.93, 1.62, 2.86, 2.39, 2.66, 3.76, 2.23, 5.18, 2.75, 3.87, 2.08, 3.16, 3.41, 3.53, 3.99, 4.05, 3.59, 3.04, 3.66, 2.53, 2.66, 3.21, 2.21, 2.27, 3.79, 1.73, 2.89, 2.25, 3.85, 3.93, 2.32, 4.13, 2.53, 2.53, 2.81, 2.48, 2.27, 2.54, 2.71, 3.53, 3.65, 1.64, 3.79, 2.54, 2.22, 3.08, 3.11, 3.59, 2.97, 3.73, 3.59, 3.53, 3.23, 1.75, 2.83, 2.85, 2.79, 2.99, 3.3, 2.97, 2.73, 3.34, 3.33, 3.93, 3.03, 2.09, 3.23, 5.39, 1.58, 2.16, 2.94, 2.44, 2.14, 2.58, 2.64, 2.29, 2.49, 2.21, 2.95, 3.33, 1.19, 2.99, 3.48, 1.26, 2.36, 2.08, 3.29, 3.87, 3.37, 2.11, 3.81, 2.28, 2.25, 2.88, 3.11, 3.39, 2.48, 3.11, 2.14, 2.17, 2.91, 2.55, 2.73, 3.69, 2.4, 4.1, 3.84, 1.93, 3.74, 2.29, 1.12, 3.19, 3.87, 3.14, 2.38, 2.16, 3.51, 3.13, 2.9, 1.75, 2.07, 3.39, 1.84, 3.02, 3.5, 3.13, 1.93, 2.62, 2.13, 3.37, 3.42, 1.08, 3.8, 2.41, 2.35, 4.12, 2.96, 2.29, 2.46, 3.35, 3.45, 2.63, 3.63, 2.46, 3.53, 3.38, 2.92, 3.6, 3.28, 2.7, 2.57, 2.67, 2.5, 2.04, 1.83, 2.78, 3.88, 3.86, 3.81, 2.87, 3.06, 1.91, 3.35, 1.62, 2.55]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=5.3407, s₁=0.9501；x̄₂=2.8683, s₂=0.7206；
SE=√(s₁²/n₁+s₂²/n₂)=0.0464；t=(x̄₁−x̄₂)/SE=53.2394；
df(Welch)=1066.07；双侧 p = 1.512e-302。
x̄₁=5.3407, s₁=0.9501；x̄₂=2.8683, s₂=0.7206；
SE=√(s₁²/n₁+s₂²/n₂)=0.0464；t=(x̄₁−x̄₂)/SE=53.2394；
df(Welch)=1066.07；双侧 p = 1.512e-302。
计算结果
t = 53.2394，df = 1066.07，p = 1.512e-302
在 α=0.05 水平下，p=1.512e-302 < 0.05，拒绝原假设 H₀。判断包装改进效果。
案例 31. 两组反应时
原始数据
组1 n=800：[283.89, 305.45, 328.91, 297.26, 303.49, 298.69, 294.7, 334.89, 322.39, 313.91, 326.51, 339.15, 303.88, 303.8, 310.82, 323.03, 310.65, 312.73, 325.59, 297.68, 320.49, 296.16, 325.9, 291.24, 304.95, 315.14, 282.45, 312.16, 302.45, 329.25, 305.27, 299.37, 294.5, 299.09, 313.65, 334.69, 326.15, 298.41, 312.32, 303.41, 335.19, 299.51, 298.78, 297.49, 329.65, 322.74, 315.3, 320.08, 313.16, 305.92, 304.51, 309.41, 299.31, 302.19, 297.39, 325.74, 304.61, 298.81, 305.93, 299.23, 292.33, 284.05, 311.67, 300.96, 298.36, 305.79, 338.65, 295.45, 325.02, 321.31, 345.82, 291.73, 307.27, 313.95, 310.97, 294.96, 312.46, 303.55, 300.17, 312.75, 299.42, 296.21, 320.02, 310.22, 320.97, 313.52, 351.57, 294.63, 318.13, 317.61, 312.69, 295.1, 322.38, 317.71, 308.2, 329.19, 297.4, 325.19, 320.06, 310.32, 307.91, 329.76, 319.99, 291.3, 319.59, 294.86, 308.87, 308.47, 307.25, 311.05, 284.11, 295.37, 304.07, 312.85, 296.74, 307.5, 302.07, 297.44, 301.25, 315.56, 293.93, 311.96, 305.45, 298.6, 307.66, 288.41, 307.69, 284.5, 307.87, 306.14, 286.52, 321.02, 318.61, 329.52, 313.67, 298.35, 321.9, 310.66, 322.3, 321.73, 285.72, 305.03, 320.75, 317.51, 292.65, 306.09, 313.34, 307.36, 301.68, 300.9, 328.08, 316.84, 314.71, 320.63, 290.43, 304.87, 306.07, 318.52, 316.08, 331.15, 309.86, 317.32, 307.61, 302.56, 288.05, 321.47, 303.23, 300.71, 284.24, 310.87, 289.56, 311.99, 292.59, 303.69, 323.66, 305.88, 288.81, 304.3, 318.22, 334.59, 305.6, 296.15, 314.68, 310.58, 308.72, 296.76, 308.77, 287.99, 306.4, 330.49, 296.37, 300.44, 317.1, 275.54, 304.7, 323.8, 294.49, 311.82, 294.43, 315.03, 305.12, 292.77, 318.59, 316.24, 311.44, 272.54, 316.22, 301.38, 319.42, 314.3, 323.07, 314.14, 295.37, 320.14, 298.4, 309.54, 293.27, 308.18, 314.77, 297.42, 319.27, 328.54, 294.25, 311.15, 309.01, 300.47, 311.0, 287.8, 329.05, 300.64, 316.56, 290.39, 312.75, 290.61, 307.18, 322.81, 309.4, 291.63, 295.34, 298.03, 318.06, 284.62, 322.84, 289.04, 296.76, 296.28, 302.05, 312.5, 323.14, 307.35, 320.93, 312.52, 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组2 n=541：[302.52, 270.82, 273.72, 265.64, 269.44, 286.32, 280.62, 275.77, 271.48, 262.33, 272.13, 277.6, 263.81, 274.57, 280.68, 269.65, 277.68, 272.7, 276.76, 263.91, 279.25, 286.69, 284.43, 271.05, 275.5, 282.52, 285.17, 295.05, 280.04, 277.11, 280.15, 274.07, 283.37, 265.89, 278.71, 291.55, 274.34, 273.85, 285.55, 269.6, 255.46, 282.94, 280.6, 265.65, 271.39, 286.96, 279.58, 287.76, 282.7, 291.53, 275.63, 280.47, 288.52, 284.61, 279.67, 279.75, 271.12, 271.07, 281.71, 281.38, 287.97, 277.01, 291.75, 272.79, 279.76, 280.61, 279.33, 281.35, 260.34, 275.27, 273.23, 266.28, 274.52, 294.09, 296.78, 272.62, 283.54, 284.86, 285.89, 269.11, 290.49, 278.89, 268.24, 271.72, 296.6, 284.47, 258.91, 270.67, 281.91, 273.98, 288.53, 279.68, 281.46, 267.82, 283.95, 269.93, 265.9, 290.13, 283.11, 272.1, 283.18, 257.57, 268.32, 279.56, 278.15, 271.11, 273.01, 268.29, 255.71, 293.2, 295.69, 276.31, 276.76, 277.0, 267.93, 284.79, 280.17, 293.71, 252.46, 274.5, 281.84, 296.16, 291.26, 277.87, 274.08, 295.48, 256.72, 286.4, 267.33, 279.01, 256.85, 277.94, 272.58, 283.16, 266.38, 278.2, 264.88, 278.1, 274.4, 286.41, 268.18, 286.41, 286.74, 285.45, 279.64, 277.94, 286.41, 289.93, 288.0, 287.39, 284.88, 282.93, 269.55, 276.74, 277.57, 270.14, 286.85, 285.6, 284.6, 278.65, 278.1, 280.98, 277.32, 275.58, 286.98, 261.74, 284.2, 262.22, 267.62, 272.08, 283.19, 267.93, 282.15, 280.89, 272.52, 286.51, 278.78, 282.6, 287.6, 285.93, 274.43, 277.7, 287.87, 273.42, 275.56, 286.63, 287.64, 286.61, 283.88, 276.05, 276.06, 282.87, 285.66, 255.72, 274.04, 286.12, 267.71, 281.25, 271.62, 293.97, 270.47, 282.38, 273.46, 274.22, 288.31, 285.34, 272.74, 273.87, 269.13, 293.78, 283.31, 295.01, 281.35, 294.83, 296.14, 283.57, 282.03, 285.04, 272.4, 264.33, 290.82, 298.66, 278.69, 269.18, 270.59, 272.58, 268.09, 272.95, 279.5, 276.89, 267.99, 272.57, 267.75, 279.29, 293.94, 279.19, 268.46, 296.1, 267.34, 284.76, 269.47, 278.75, 261.72, 267.4, 276.46, 271.26, 284.25, 288.33, 266.96, 296.79, 284.43, 281.16, 286.91, 267.71, 275.93, 287.78, 264.57, 280.05, 268.91, 263.81, 289.06, 283.15, 274.82, 290.04, 273.09, 281.39, 282.79, 289.67, 289.4, 280.37, 263.54, 265.59, 275.24, 289.15, 271.79, 274.67, 287.21, 268.12, 259.32, 269.58, 294.16, 292.69, 264.09, 279.5, 270.62, 282.92, 288.73, 283.77, 290.31, 297.08, 275.06, 261.35, 280.46, 280.64, 287.72, 268.68, 279.49, 260.12, 283.88, 283.96, 281.53, 277.28, 281.36, 275.83, 275.34, 294.89, 275.84, 271.38, 279.92, 276.72, 269.37, 305.0, 293.17, 281.15, 280.25, 272.86, 248.59, 279.24, 272.87, 288.44, 289.4, 273.69, 272.17, 276.95, 274.89, 291.12, 280.01, 272.72, 277.03, 286.58, 279.24, 286.71, 268.54, 283.62, 279.93, 279.44, 273.21, 285.15, 266.37, 281.95, 287.59, 281.55, 278.29, 277.09, 282.76, 277.18, 269.51, 282.76, 288.72, 281.0, 286.6, 279.53, 268.54, 296.31, 283.73, 267.37, 269.95, 285.15, 274.53, 289.45, 285.34, 292.22, 275.98, 275.62, 287.69, 283.36, 281.61, 281.66, 263.93, 294.34, 282.1, 287.47, 287.25, 274.18, 271.01, 277.14, 270.61, 278.46, 288.18, 295.65, 276.73, 282.67, 296.04, 278.41, 289.14, 274.62, 285.15, 276.64, 283.03, 277.81, 278.21, 289.21, 269.93, 285.67, 292.64, 295.05, 285.83, 282.42, 281.52, 275.86, 293.56, 289.97, 295.67, 273.65, 289.61, 275.32, 287.91, 270.79, 277.63, 283.93, 293.53, 272.63, 274.61, 264.68, 269.42, 275.08, 280.37, 280.47, 285.87, 277.41, 282.41, 283.57, 261.2, 280.28, 272.62, 286.54, 273.81, 279.71, 282.36, 281.3, 282.55, 264.12, 272.44, 279.48, 264.94, 281.45, 267.01, 260.89, 277.64, 290.86, 286.72, 283.69, 269.33, 273.38, 277.76, 286.74, 295.02, 272.64, 276.24, 283.24, 276.29, 285.24, 278.92, 280.1, 278.16, 277.93, 260.28, 271.1, 288.91, 275.33, 295.95, 281.12, 281.5, 274.51, 294.84, 269.04, 271.83, 276.96, 272.51, 277.12, 286.9, 273.68, 273.36, 265.65, 291.66, 291.85, 292.96, 285.44, 270.86, 271.62, 284.74, 276.34, 263.61, 296.28, 281.47, 262.93, 273.39, 278.21, 277.96, 280.23, 285.35, 272.26, 253.31, 303.25, 277.6, 276.42, 300.17, 286.49, 277.24, 280.29, 294.3, 278.02, 277.15, 289.46, 285.34, 281.09, 271.27, 281.02, 283.76, 281.2, 278.45, 295.86, 275.64, 274.65, 275.59, 295.28, 284.47, 275.88, 295.45, 286.12, 289.62, 274.75, 274.29, 273.87, 281.18, 276.24, 287.0, 290.16, 284.35, 286.19, 284.94, 272.96, 292.08, 290.89, 266.45, 279.06, 283.28, 286.3, 280.75, 295.06, 289.85]
组2 n=541：[302.52, 270.82, 273.72, 265.64, 269.44, 286.32, 280.62, 275.77, 271.48, 262.33, 272.13, 277.6, 263.81, 274.57, 280.68, 269.65, 277.68, 272.7, 276.76, 263.91, 279.25, 286.69, 284.43, 271.05, 275.5, 282.52, 285.17, 295.05, 280.04, 277.11, 280.15, 274.07, 283.37, 265.89, 278.71, 291.55, 274.34, 273.85, 285.55, 269.6, 255.46, 282.94, 280.6, 265.65, 271.39, 286.96, 279.58, 287.76, 282.7, 291.53, 275.63, 280.47, 288.52, 284.61, 279.67, 279.75, 271.12, 271.07, 281.71, 281.38, 287.97, 277.01, 291.75, 272.79, 279.76, 280.61, 279.33, 281.35, 260.34, 275.27, 273.23, 266.28, 274.52, 294.09, 296.78, 272.62, 283.54, 284.86, 285.89, 269.11, 290.49, 278.89, 268.24, 271.72, 296.6, 284.47, 258.91, 270.67, 281.91, 273.98, 288.53, 279.68, 281.46, 267.82, 283.95, 269.93, 265.9, 290.13, 283.11, 272.1, 283.18, 257.57, 268.32, 279.56, 278.15, 271.11, 273.01, 268.29, 255.71, 293.2, 295.69, 276.31, 276.76, 277.0, 267.93, 284.79, 280.17, 293.71, 252.46, 274.5, 281.84, 296.16, 291.26, 277.87, 274.08, 295.48, 256.72, 286.4, 267.33, 279.01, 256.85, 277.94, 272.58, 283.16, 266.38, 278.2, 264.88, 278.1, 274.4, 286.41, 268.18, 286.41, 286.74, 285.45, 279.64, 277.94, 286.41, 289.93, 288.0, 287.39, 284.88, 282.93, 269.55, 276.74, 277.57, 270.14, 286.85, 285.6, 284.6, 278.65, 278.1, 280.98, 277.32, 275.58, 286.98, 261.74, 284.2, 262.22, 267.62, 272.08, 283.19, 267.93, 282.15, 280.89, 272.52, 286.51, 278.78, 282.6, 287.6, 285.93, 274.43, 277.7, 287.87, 273.42, 275.56, 286.63, 287.64, 286.61, 283.88, 276.05, 276.06, 282.87, 285.66, 255.72, 274.04, 286.12, 267.71, 281.25, 271.62, 293.97, 270.47, 282.38, 273.46, 274.22, 288.31, 285.34, 272.74, 273.87, 269.13, 293.78, 283.31, 295.01, 281.35, 294.83, 296.14, 283.57, 282.03, 285.04, 272.4, 264.33, 290.82, 298.66, 278.69, 269.18, 270.59, 272.58, 268.09, 272.95, 279.5, 276.89, 267.99, 272.57, 267.75, 279.29, 293.94, 279.19, 268.46, 296.1, 267.34, 284.76, 269.47, 278.75, 261.72, 267.4, 276.46, 271.26, 284.25, 288.33, 266.96, 296.79, 284.43, 281.16, 286.91, 267.71, 275.93, 287.78, 264.57, 280.05, 268.91, 263.81, 289.06, 283.15, 274.82, 290.04, 273.09, 281.39, 282.79, 289.67, 289.4, 280.37, 263.54, 265.59, 275.24, 289.15, 271.79, 274.67, 287.21, 268.12, 259.32, 269.58, 294.16, 292.69, 264.09, 279.5, 270.62, 282.92, 288.73, 283.77, 290.31, 297.08, 275.06, 261.35, 280.46, 280.64, 287.72, 268.68, 279.49, 260.12, 283.88, 283.96, 281.53, 277.28, 281.36, 275.83, 275.34, 294.89, 275.84, 271.38, 279.92, 276.72, 269.37, 305.0, 293.17, 281.15, 280.25, 272.86, 248.59, 279.24, 272.87, 288.44, 289.4, 273.69, 272.17, 276.95, 274.89, 291.12, 280.01, 272.72, 277.03, 286.58, 279.24, 286.71, 268.54, 283.62, 279.93, 279.44, 273.21, 285.15, 266.37, 281.95, 287.59, 281.55, 278.29, 277.09, 282.76, 277.18, 269.51, 282.76, 288.72, 281.0, 286.6, 279.53, 268.54, 296.31, 283.73, 267.37, 269.95, 285.15, 274.53, 289.45, 285.34, 292.22, 275.98, 275.62, 287.69, 283.36, 281.61, 281.66, 263.93, 294.34, 282.1, 287.47, 287.25, 274.18, 271.01, 277.14, 270.61, 278.46, 288.18, 295.65, 276.73, 282.67, 296.04, 278.41, 289.14, 274.62, 285.15, 276.64, 283.03, 277.81, 278.21, 289.21, 269.93, 285.67, 292.64, 295.05, 285.83, 282.42, 281.52, 275.86, 293.56, 289.97, 295.67, 273.65, 289.61, 275.32, 287.91, 270.79, 277.63, 283.93, 293.53, 272.63, 274.61, 264.68, 269.42, 275.08, 280.37, 280.47, 285.87, 277.41, 282.41, 283.57, 261.2, 280.28, 272.62, 286.54, 273.81, 279.71, 282.36, 281.3, 282.55, 264.12, 272.44, 279.48, 264.94, 281.45, 267.01, 260.89, 277.64, 290.86, 286.72, 283.69, 269.33, 273.38, 277.76, 286.74, 295.02, 272.64, 276.24, 283.24, 276.29, 285.24, 278.92, 280.1, 278.16, 277.93, 260.28, 271.1, 288.91, 275.33, 295.95, 281.12, 281.5, 274.51, 294.84, 269.04, 271.83, 276.96, 272.51, 277.12, 286.9, 273.68, 273.36, 265.65, 291.66, 291.85, 292.96, 285.44, 270.86, 271.62, 284.74, 276.34, 263.61, 296.28, 281.47, 262.93, 273.39, 278.21, 277.96, 280.23, 285.35, 272.26, 253.31, 303.25, 277.6, 276.42, 300.17, 286.49, 277.24, 280.29, 294.3, 278.02, 277.15, 289.46, 285.34, 281.09, 271.27, 281.02, 283.76, 281.2, 278.45, 295.86, 275.64, 274.65, 275.59, 295.28, 284.47, 275.88, 295.45, 286.12, 289.62, 274.75, 274.29, 273.87, 281.18, 276.24, 287.0, 290.16, 284.35, 286.19, 284.94, 272.96, 292.08, 290.89, 266.45, 279.06, 283.28, 286.3, 280.75, 295.06, 289.85]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=309.4778, s₁=13.2203；x̄₂=279.2111, s₂=9.1941；
SE=√(s₁²/n₁+s₂²/n₂)=0.6121；t=(x̄₁−x̄₂)/SE=49.4437；
df(Welch)=1337.96；双侧 p = 3.076e-304。
x̄₁=309.4778, s₁=13.2203；x̄₂=279.2111, s₂=9.1941；
SE=√(s₁²/n₁+s₂²/n₂)=0.6121；t=(x̄₁−x̄₂)/SE=49.4437；
df(Welch)=1337.96；双侧 p = 3.076e-304。
计算结果
t = 49.4437，df = 1337.96，p = 3.076e-304
在 α=0.05 水平下，p=3.076e-304 < 0.05，拒绝原假设 H₀。比较处理速度差异。
案例 32. 男女生考试成绩
原始数据
组1 n=878：[82.48, 77.81, 88.45, 79.68, 85.33, 78.33, 78.58, 86.38, 80.15, 79.1, 82.93, 76.38, 81.87, 80.18, 82.44, 82.37, 81.87, 77.37, 79.81, 80.34, 80.25, 82.29, 80.92, 80.23, 86.56, 81.78, 81.52, 81.65, 80.29, 79.04, 78.39, 81.87, 82.35, 80.24, 85.86, 80.14, 82.57, 81.53, 80.8, 76.71, 81.61, 84.11, 80.4, 80.86, 74.98, 79.45, 78.83, 83.12, 80.95, 79.96, 82.05, 81.47, 79.43, 79.31, 85.38, 82.33, 82.92, 78.73, 82.11, 80.85, 82.6, 79.11, 85.71, 78.73, 81.61, 78.53, 81.91, 84.73, 79.94, 78.52, 78.24, 85.44, 83.09, 83.68, 82.33, 81.4, 80.09, 82.25, 84.47, 82.28, 81.59, 76.84, 77.11, 81.03, 80.9, 77.77, 81.44, 77.78, 81.18, 80.21, 79.17, 85.48, 80.78, 80.35, 88.51, 79.63, 81.38, 81.45, 82.26, 81.83, 79.32, 80.08, 87.34, 85.83, 78.15, 79.68, 80.94, 81.85, 87.16, 82.35, 81.37, 84.53, 81.58, 81.67, 80.39, 82.18, 85.12, 82.7, 83.0, 84.01, 85.01, 84.87, 80.35, 84.17, 84.26, 84.57, 77.22, 77.76, 77.28, 78.9, 80.86, 80.97, 79.37, 76.84, 80.73, 83.09, 81.56, 85.15, 74.59, 82.25, 78.78, 78.8, 82.67, 81.49, 83.05, 81.62, 79.53, 80.1, 79.02, 85.25, 80.84, 81.66, 79.12, 84.67, 85.23, 80.53, 79.45, 76.78, 85.52, 80.51, 84.34, 83.08, 89.05, 83.39, 81.04, 78.97, 75.65, 81.68, 86.08, 75.3, 77.46, 82.52, 83.16, 80.93, 79.08, 84.42, 79.35, 81.59, 83.48, 81.74, 78.03, 81.72, 78.23, 80.72, 83.08, 76.78, 83.29, 81.26, 83.73, 77.68, 83.42, 84.33, 81.07, 81.12, 77.99, 83.33, 80.41, 82.53, 83.84, 84.43, 84.94, 80.24, 78.98, 73.95, 83.41, 82.37, 82.49, 79.31, 82.22, 82.46, 81.05, 85.4, 80.86, 79.39, 81.5, 80.1, 77.81, 75.97, 79.83, 85.9, 78.03, 82.69, 77.09, 79.95, 85.72, 82.23, 83.51, 83.54, 77.59, 80.28, 80.42, 76.6, 83.28, 78.06, 78.47, 78.71, 80.42, 77.51, 82.04, 83.06, 78.88, 81.59, 82.48, 77.99, 81.36, 75.79, 82.09, 82.12, 81.97, 82.07, 81.02, 77.83, 77.6, 80.83, 83.15, 78.36, 81.72, 85.29, 78.31, 81.52, 80.94, 81.01, 78.99, 79.05, 82.73, 80.94, 80.32, 80.69, 79.04, 84.06, 83.33, 84.16, 83.42, 86.03, 80.95, 81.06, 77.55, 82.37, 83.64, 80.34, 77.06, 78.61, 81.19, 83.45, 80.14, 79.38, 80.41, 79.2, 80.98, 78.66, 83.03, 78.13, 81.51, 76.98, 84.37, 80.23, 81.33, 77.54, 82.24, 80.28, 81.83, 81.34, 83.11, 80.62, 85.98, 80.31, 79.75, 78.22, 83.17, 82.82, 83.83, 84.0, 77.1, 81.83, 85.19, 82.18, 77.54, 77.69, 80.99, 81.86, 81.21, 80.66, 83.66, 78.9, 81.62, 75.17, 81.97, 76.96, 84.12, 83.06, 77.76, 81.59, 80.13, 78.78, 85.51, 78.67, 77.01, 81.0, 76.98, 83.69, 79.67, 79.93, 78.47, 81.44, 81.5, 86.72, 84.65, 80.94, 84.52, 83.12, 79.6, 79.32, 83.71, 79.09, 80.89, 83.98, 77.26, 79.19, 80.15, 80.7, 84.7, 82.46, 78.77, 84.15, 83.18, 78.41, 82.35, 81.54, 75.06, 82.46, 82.1, 83.92, 80.01, 83.98, 79.09, 83.6, 85.23, 76.46, 82.12, 85.35, 81.38, 80.68, 79.07, 78.78, 86.24, 79.12, 82.9, 81.59, 83.67, 82.83, 81.1, 84.46, 85.96, 78.39, 77.12, 81.35, 78.09, 76.78, 80.26, 84.9, 79.52, 80.49, 82.69, 80.92, 77.12, 75.37, 78.84, 83.07, 78.52, 80.4, 83.35, 82.38, 81.52, 80.2, 83.75, 81.1, 80.56, 76.64, 82.06, 84.24, 81.89, 81.59, 79.15, 82.77, 79.36, 78.43, 76.22, 82.79, 80.03, 80.6, 81.13, 82.81, 78.1, 87.95, 81.34, 79.92, 81.18, 82.32, 85.84, 86.55, 80.49, 85.5, 79.77, 84.18, 82.58, 83.16, 82.91, 78.74, 78.97, 79.61, 77.43, 78.45, 84.62, 86.68, 81.46, 82.7, 79.87, 76.41, 80.31, 78.53, 81.86, 82.53, 80.61, 80.44, 77.53, 78.41, 74.39, 80.93, 81.45, 81.77, 83.54, 77.29, 77.33, 79.3, 82.06, 84.37, 77.3, 78.62, 80.85, 77.8, 81.1, 81.6, 82.09, 88.59, 85.02, 82.49, 81.87, 84.39, 81.24, 81.11, 84.48, 85.15, 78.32, 82.23, 84.6, 81.05, 82.03, 81.76, 85.59, 79.42, 81.33, 78.82, 80.2, 79.48, 81.43, 83.56, 77.54, 79.15, 79.43, 79.39, 82.23, 79.25, 80.46, 81.99, 80.21, 79.55, 78.19, 81.11, 80.69, 81.69, 87.63, 81.81, 77.09, 76.86, 81.49, 81.91, 81.39, 80.1, 82.42, 81.7, 82.08, 81.12, 80.41, 82.08, 82.26, 86.33, 81.94, 81.39, 80.28, 81.47, 77.36, 78.81, 81.49, 80.93, 81.16, 85.76, 81.96, 77.63, 81.62, 83.4, 81.55, 79.11, 77.83, 82.01, 77.27, 83.25, 80.08, 78.77, 81.5, 81.51, 78.99, 80.07, 78.78, 81.71, 83.77, 83.78, 81.44, 84.28, 78.05, 80.65, 81.78, 83.45, 83.43, 79.27, 80.44, 78.14, 81.18, 78.33, 77.63, 77.46, 83.08, 79.71, 79.19, 83.03, 83.48, 83.28, 80.49, 78.12, 81.55, 80.39, 82.47, 79.73, 82.42, 78.87, 84.02, 81.07, 79.23, 77.77, 81.76, 81.18, 83.43, 82.9, 77.09, 84.33, 78.78, 78.84, 78.47, 80.53, 78.37, 82.32, 77.84, 79.79, 83.18, 82.13, 84.16, 84.03, 82.83, 79.07, 83.87, 82.53, 83.99, 82.36, 78.88, 77.15, 78.64, 83.66, 86.23, 84.68, 86.63, 78.2, 81.57, 81.64, 81.59, 85.2, 76.69, 79.46, 80.79, 81.69, 75.82, 81.11, 80.57, 81.13, 84.13, 78.17, 78.39, 82.37, 82.28, 79.83, 82.19, 81.14, 81.63, 82.92, 80.73, 77.38, 79.7, 81.23, 79.32, 81.45, 82.43, 79.21, 84.13, 77.27, 81.42, 86.73, 85.54, 75.31, 78.98, 84.14, 78.88, 81.68, 81.42, 83.22, 80.67, 83.13, 79.49, 83.98, 80.76, 80.45, 80.39, 77.0, 86.46, 79.22, 81.73, 81.72, 88.07, 83.35, 81.23, 82.2, 83.22, 83.84, 78.52, 83.93, 85.37, 79.05, 75.88, 81.84, 80.08, 80.39, 78.97, 80.31, 83.06, 82.82, 80.42, 83.26, 83.81, 76.78, 79.63, 83.0, 85.89, 77.21, 84.75, 82.11, 84.47, 79.23, 82.03, 81.23, 76.95, 78.82, 81.13, 80.18, 80.27, 79.51, 79.98, 82.18, 80.79, 81.97, 75.47, 83.32, 78.46, 79.67, 82.77, 80.84, 80.12, 83.62, 85.85, 82.55, 79.44, 87.68, 78.46, 82.13, 79.37, 81.62, 78.93, 81.86, 79.14, 83.63, 81.56, 77.97, 84.08, 80.01, 81.38, 78.59, 81.57, 83.3, 83.62, 83.26, 82.71, 80.12, 78.4, 78.72, 79.53, 79.5, 77.4, 80.44, 83.11, 79.26, 79.53, 81.13, 78.95, 82.05, 74.84, 83.48, 84.59, 83.45, 81.01, 80.07, 79.62, 74.11, 80.11, 80.18, 80.09, 82.35, 80.0, 83.14, 80.42, 86.26, 81.57, 82.37, 77.69, 79.95, 79.42, 80.91, 82.15, 80.59, 81.77, 79.84, 77.21, 79.27, 78.96, 75.85, 82.24, 81.73, 79.43, 81.63, 83.58, 78.04, 78.61, 82.49, 80.42, 82.24, 81.23, 79.65, 88.22, 75.97, 81.22, 84.45, 81.84, 78.39, 79.12, 74.29, 82.74, 82.63, 79.35, 78.08, 81.24, 79.45, 80.08, 87.19, 81.77, 81.71, 83.56, 78.71, 79.05, 82.21, 82.53, 81.42, 82.35, 83.84, 80.15, 83.84, 81.99, 83.41, 82.05, 80.13, 77.86, 80.74, 77.19, 84.27, 83.59, 79.47, 82.73, 83.79, 82.63, 79.59, 83.49, 84.97, 86.78, 82.56, 83.56, 83.31, 83.12, 80.55, 81.06, 80.64, 81.11, 81.24, 84.35, 78.11, 83.56, 82.34, 80.07, 81.06, 81.36, 79.06, 82.38, 77.71, 79.85, 82.74]；
组2 n=977：[77.34, 77.77, 79.66, 75.32, 76.71, 77.87, 74.27, 73.21, 80.0, 74.81, 71.91, 80.84, 79.22, 80.94, 81.55, 80.59, 73.87, 76.88, 75.74, 78.59, 81.82, 78.5, 78.29, 76.56, 74.99, 80.16, 77.19, 75.32, 79.76, 81.79, 78.23, 76.56, 76.63, 77.94, 73.0, 75.93, 75.76, 74.34, 75.57, 71.91, 74.16, 78.99, 74.8, 72.88, 75.61, 81.32, 79.96, 80.57, 76.57, 77.08, 80.19, 78.83, 77.17, 74.59, 77.12, 78.05, 77.89, 75.38, 75.22, 82.74, 75.95, 73.37, 75.37, 75.84, 77.72, 78.44, 74.31, 75.9, 76.23, 69.7, 78.74, 75.17, 75.1, 81.64, 78.68, 78.23, 77.51, 80.89, 76.52, 82.15, 76.11, 77.41, 77.57, 77.26, 77.02, 78.88, 74.54, 76.43, 75.48, 71.6, 79.03, 75.82, 75.24, 76.32, 78.33, 76.68, 77.74, 80.87, 80.77, 77.48, 77.44, 78.71, 73.99, 74.37, 76.05, 81.24, 75.49, 82.79, 76.17, 75.37, 79.94, 76.87, 74.03, 77.28, 75.83, 78.77, 77.17, 75.67, 74.09, 78.37, 78.4, 79.71, 77.94, 78.16, 77.42, 82.71, 78.43, 80.65, 80.1, 77.71, 78.13, 75.24, 75.75, 73.83, 71.34, 78.91, 76.88, 75.79, 73.22, 77.1, 72.38, 78.87, 83.78, 78.91, 77.7, 74.78, 74.81, 80.8, 70.96, 81.76, 77.94, 76.46, 77.69, 74.04, 73.33, 78.94, 79.04, 78.85, 73.42, 75.96, 73.63, 75.37, 79.82, 76.26, 77.84, 77.35, 79.83, 76.73, 75.23, 76.12, 78.52, 79.17, 76.91, 78.23, 76.8, 82.65, 80.17, 76.49, 73.05, 77.19, 75.71, 70.89, 76.88, 71.97, 75.27, 77.37, 74.65, 76.96, 74.81, 79.76, 80.92, 76.14, 78.74, 79.03, 74.62, 76.92, 75.05, 77.11, 77.24, 80.1, 78.49, 78.91, 82.19, 82.34, 77.21, 76.88, 81.49, 74.06, 74.58, 80.23, 78.2, 78.3, 81.73, 77.46, 75.25, 78.4, 78.23, 76.41, 76.96, 79.16, 77.87, 78.07, 79.03, 76.39, 76.53, 78.35, 77.52, 79.62, 76.02, 78.68, 75.0, 80.29, 76.48, 75.28, 71.57, 74.64, 79.96, 77.41, 77.99, 76.21, 74.87, 75.89, 77.13, 76.07, 75.81, 78.28, 79.26, 78.75, 77.08, 78.74, 73.63, 74.83, 77.27, 79.64, 78.68, 77.17, 77.05, 81.16, 73.09, 79.47, 75.06, 76.92, 79.53, 73.63, 80.18, 73.21, 78.33, 72.41, 76.32, 73.18, 77.72, 80.82, 73.83, 78.17, 81.97, 78.81, 74.09, 74.13, 76.73, 76.34, 76.95, 73.38, 77.13, 79.84, 78.81, 73.7, 75.77, 74.54, 79.43, 78.18, 81.39, 80.25, 73.79, 74.36, 78.08, 74.4, 73.81, 76.5, 76.35, 85.05, 75.25, 77.7, 79.14, 75.09, 78.63, 73.47, 75.7, 71.86, 74.6, 81.91, 82.92, 78.31, 76.53, 68.77, 75.92, 77.53, 68.06, 76.95, 77.27, 78.44, 76.54, 74.45, 77.43, 78.49, 80.44, 71.94, 77.02, 77.58, 81.15, 81.66, 75.98, 78.69, 83.86, 74.76, 76.79, 73.94, 76.74, 79.21, 78.77, 73.58, 73.18, 74.83, 79.34, 77.43, 74.42, 75.29, 75.93, 84.34, 75.67, 73.22, 78.87, 76.83, 76.67, 77.93, 75.68, 75.59, 80.77, 75.74, 77.89, 77.73, 75.58, 78.34, 73.77, 78.04, 81.68, 77.42, 77.06, 75.05, 77.9, 71.6, 78.85, 75.21, 82.68, 73.99, 76.21, 77.5, 81.25, 78.54, 77.01, 78.81, 77.22, 80.46, 77.88, 76.27, 77.59, 73.1, 76.49, 76.26, 77.1, 71.01, 80.06, 75.04, 80.82, 76.35, 75.9, 81.49, 76.01, 76.03, 78.46, 77.31, 77.84, 77.38, 74.61, 77.5, 79.3, 74.69, 78.17, 77.67, 74.39, 73.78, 75.81, 81.01, 82.25, 78.69, 76.95, 71.31, 73.75, 82.44, 80.95, 76.6, 75.58, 76.31, 79.16, 76.87, 77.69, 78.95, 74.37, 74.57, 81.29, 81.44, 79.95, 77.23, 72.82, 77.92, 74.32, 80.06, 75.63, 77.91, 76.79, 77.66, 77.27, 82.46, 72.73, 80.05, 76.58, 77.7, 75.07, 74.3, 75.21, 74.93, 77.57, 77.3, 75.08, 73.29, 73.38, 75.32, 78.03, 73.47, 78.3, 74.94, 76.29, 75.89, 75.36, 71.42, 76.32, 75.7, 73.93, 78.75, 78.17, 74.82, 76.87, 74.7, 76.74, 73.18, 77.45, 79.32, 79.32, 78.89, 72.2, 72.73, 79.64, 73.53, 71.63, 73.76, 79.64, 84.5, 77.03, 78.17, 78.32, 83.36, 78.85, 74.83, 77.95, 79.78, 79.0, 75.73, 70.76, 71.4, 84.1, 75.58, 77.02, 75.62, 77.36, 77.86, 79.44, 79.59, 76.72, 79.43, 74.71, 75.95, 77.94, 77.1, 78.23, 78.12, 72.88, 76.91, 81.84, 76.42, 75.98, 78.61, 76.6, 73.55, 79.91, 75.81, 76.24, 78.01, 74.35, 82.31, 75.16, 74.37, 72.07, 76.28, 77.42, 76.09, 79.79, 75.58, 76.52, 77.48, 75.41, 76.85, 76.79, 80.1, 76.76, 73.75, 81.04, 74.22, 74.54, 80.1, 77.27, 74.3, 75.65, 79.46, 77.16, 74.63, 76.39, 75.16, 80.74, 76.25, 75.69, 79.78, 77.71, 77.72, 83.05, 79.5, 77.64, 80.09, 79.22, 74.92, 79.98, 84.54, 77.5, 78.27, 77.57, 76.59, 74.55, 72.99, 78.78, 77.3, 78.42, 77.68, 80.83, 77.64, 75.62, 76.29, 76.66, 77.34, 79.56, 78.98, 77.64, 77.9, 76.98, 78.62, 79.12, 72.83, 75.76, 78.89, 79.23, 74.86, 75.03, 80.16, 78.22, 75.3, 71.1, 74.65, 80.3, 78.48, 73.8, 75.36, 76.75, 83.36, 75.87, 78.33, 81.27, 77.15, 77.64, 72.81, 74.73, 73.98, 74.11, 80.96, 80.17, 79.75, 80.34, 71.69, 77.82, 72.18, 78.64, 81.39, 73.65, 76.55, 84.38, 78.1, 79.21, 80.85, 72.88, 77.77, 77.04, 76.35, 74.63, 74.63, 72.72, 78.13, 76.91, 74.78, 74.66, 75.37, 83.04, 76.04, 77.85, 77.13, 75.54, 78.16, 73.16, 75.58, 78.75, 69.12, 72.76, 75.07, 77.88, 76.87, 78.81, 76.54, 80.47, 81.35, 81.44, 73.04, 72.09, 77.85, 77.78, 79.82, 78.65, 78.78, 79.28, 80.27, 79.31, 74.71, 74.26, 77.09, 76.48, 74.51, 82.09, 79.17, 77.77, 78.93, 75.36, 72.53, 81.44, 78.56, 78.02, 74.31, 77.37, 77.73, 77.52, 77.37, 80.73, 81.75, 77.44, 81.1, 80.06, 81.36, 80.84, 78.07, 77.21, 78.91, 76.05, 70.38, 78.81, 79.8, 75.66, 77.63, 76.73, 80.17, 81.8, 78.38, 82.38, 76.7, 78.98, 77.76, 77.12, 77.81, 76.7, 80.62, 75.99, 75.24, 73.71, 80.68, 78.0, 77.53, 78.06, 81.35, 77.55, 76.47, 78.06, 74.14, 77.54, 75.43, 79.13, 81.89, 70.9, 74.29, 77.21, 77.76, 75.98, 78.49, 77.96, 74.07, 72.76, 78.48, 77.21, 75.5, 77.38, 72.72, 74.06, 78.84, 79.74, 71.09, 78.74, 78.62, 76.24, 75.98, 76.65, 74.63, 77.27, 76.51, 73.75, 76.38, 83.96, 74.56, 78.05, 74.68, 80.09, 78.37, 77.36, 79.77, 78.99, 73.49, 79.31, 75.71, 73.91, 71.38, 74.34, 72.86, 72.0, 77.09, 76.49, 74.63, 76.2, 78.45, 75.42, 75.01, 79.65, 78.93, 78.41, 75.89, 73.67, 79.79, 75.37, 76.26, 79.74, 77.59, 80.35, 73.34, 77.3, 75.06, 73.96, 76.86, 80.37, 69.91, 73.93, 79.55, 72.98, 79.29, 74.83, 76.36, 75.85, 76.76, 79.32, 73.77, 79.11, 79.86, 75.69, 77.02, 81.36, 79.88, 74.58, 75.85, 82.24, 75.52, 74.88, 72.35, 76.66, 76.37, 78.77, 79.86, 80.89, 81.96, 78.66, 77.79, 74.39, 74.46, 73.88, 78.82, 72.6, 77.78, 79.43, 74.98, 77.59, 77.95, 79.84, 76.8, 78.42, 79.25, 75.72, 78.0, 76.42, 79.11, 78.52, 73.72, 80.27, 77.18, 78.2, 78.04, 81.93, 80.32, 78.13, 79.06, 74.89, 77.2, 78.09, 76.61, 75.07, 75.91, 74.92, 78.06, 78.09, 76.32, 77.44, 75.04, 77.6, 73.19, 74.62, 81.84, 79.34, 79.04, 77.62, 75.59, 76.51, 74.47, 77.91, 75.76, 80.91, 75.44, 83.24, 79.01, 81.25, 78.4, 76.69, 80.97, 78.43, 76.02, 77.1, 74.75, 78.64, 71.81, 73.66, 78.57, 78.02, 75.15, 84.77, 78.03, 72.98, 78.2, 77.39, 78.11, 77.4, 72.7, 81.73, 73.09, 77.9, 76.56, 74.04, 78.94, 81.83, 81.61, 74.62, 78.18, 76.08, 73.25, 78.01, 79.0, 74.52, 78.11, 76.61, 71.94, 77.23, 78.71, 78.77, 80.11, 77.49, 75.75, 77.51, 77.66, 77.72, 77.93, 82.82, 78.63, 75.03, 74.13, 75.42, 77.29, 77.96, 73.38, 82.54, 76.9, 71.99, 75.89, 74.24, 80.24, 73.0, 80.39, 74.6, 78.91, 72.66, 78.23, 81.98, 75.82, 73.38, 77.3, 77.7, 74.81, 75.27, 75.2, 79.98, 78.22, 81.71, 76.41, 78.81]
组2 n=977：[77.34, 77.77, 79.66, 75.32, 76.71, 77.87, 74.27, 73.21, 80.0, 74.81, 71.91, 80.84, 79.22, 80.94, 81.55, 80.59, 73.87, 76.88, 75.74, 78.59, 81.82, 78.5, 78.29, 76.56, 74.99, 80.16, 77.19, 75.32, 79.76, 81.79, 78.23, 76.56, 76.63, 77.94, 73.0, 75.93, 75.76, 74.34, 75.57, 71.91, 74.16, 78.99, 74.8, 72.88, 75.61, 81.32, 79.96, 80.57, 76.57, 77.08, 80.19, 78.83, 77.17, 74.59, 77.12, 78.05, 77.89, 75.38, 75.22, 82.74, 75.95, 73.37, 75.37, 75.84, 77.72, 78.44, 74.31, 75.9, 76.23, 69.7, 78.74, 75.17, 75.1, 81.64, 78.68, 78.23, 77.51, 80.89, 76.52, 82.15, 76.11, 77.41, 77.57, 77.26, 77.02, 78.88, 74.54, 76.43, 75.48, 71.6, 79.03, 75.82, 75.24, 76.32, 78.33, 76.68, 77.74, 80.87, 80.77, 77.48, 77.44, 78.71, 73.99, 74.37, 76.05, 81.24, 75.49, 82.79, 76.17, 75.37, 79.94, 76.87, 74.03, 77.28, 75.83, 78.77, 77.17, 75.67, 74.09, 78.37, 78.4, 79.71, 77.94, 78.16, 77.42, 82.71, 78.43, 80.65, 80.1, 77.71, 78.13, 75.24, 75.75, 73.83, 71.34, 78.91, 76.88, 75.79, 73.22, 77.1, 72.38, 78.87, 83.78, 78.91, 77.7, 74.78, 74.81, 80.8, 70.96, 81.76, 77.94, 76.46, 77.69, 74.04, 73.33, 78.94, 79.04, 78.85, 73.42, 75.96, 73.63, 75.37, 79.82, 76.26, 77.84, 77.35, 79.83, 76.73, 75.23, 76.12, 78.52, 79.17, 76.91, 78.23, 76.8, 82.65, 80.17, 76.49, 73.05, 77.19, 75.71, 70.89, 76.88, 71.97, 75.27, 77.37, 74.65, 76.96, 74.81, 79.76, 80.92, 76.14, 78.74, 79.03, 74.62, 76.92, 75.05, 77.11, 77.24, 80.1, 78.49, 78.91, 82.19, 82.34, 77.21, 76.88, 81.49, 74.06, 74.58, 80.23, 78.2, 78.3, 81.73, 77.46, 75.25, 78.4, 78.23, 76.41, 76.96, 79.16, 77.87, 78.07, 79.03, 76.39, 76.53, 78.35, 77.52, 79.62, 76.02, 78.68, 75.0, 80.29, 76.48, 75.28, 71.57, 74.64, 79.96, 77.41, 77.99, 76.21, 74.87, 75.89, 77.13, 76.07, 75.81, 78.28, 79.26, 78.75, 77.08, 78.74, 73.63, 74.83, 77.27, 79.64, 78.68, 77.17, 77.05, 81.16, 73.09, 79.47, 75.06, 76.92, 79.53, 73.63, 80.18, 73.21, 78.33, 72.41, 76.32, 73.18, 77.72, 80.82, 73.83, 78.17, 81.97, 78.81, 74.09, 74.13, 76.73, 76.34, 76.95, 73.38, 77.13, 79.84, 78.81, 73.7, 75.77, 74.54, 79.43, 78.18, 81.39, 80.25, 73.79, 74.36, 78.08, 74.4, 73.81, 76.5, 76.35, 85.05, 75.25, 77.7, 79.14, 75.09, 78.63, 73.47, 75.7, 71.86, 74.6, 81.91, 82.92, 78.31, 76.53, 68.77, 75.92, 77.53, 68.06, 76.95, 77.27, 78.44, 76.54, 74.45, 77.43, 78.49, 80.44, 71.94, 77.02, 77.58, 81.15, 81.66, 75.98, 78.69, 83.86, 74.76, 76.79, 73.94, 76.74, 79.21, 78.77, 73.58, 73.18, 74.83, 79.34, 77.43, 74.42, 75.29, 75.93, 84.34, 75.67, 73.22, 78.87, 76.83, 76.67, 77.93, 75.68, 75.59, 80.77, 75.74, 77.89, 77.73, 75.58, 78.34, 73.77, 78.04, 81.68, 77.42, 77.06, 75.05, 77.9, 71.6, 78.85, 75.21, 82.68, 73.99, 76.21, 77.5, 81.25, 78.54, 77.01, 78.81, 77.22, 80.46, 77.88, 76.27, 77.59, 73.1, 76.49, 76.26, 77.1, 71.01, 80.06, 75.04, 80.82, 76.35, 75.9, 81.49, 76.01, 76.03, 78.46, 77.31, 77.84, 77.38, 74.61, 77.5, 79.3, 74.69, 78.17, 77.67, 74.39, 73.78, 75.81, 81.01, 82.25, 78.69, 76.95, 71.31, 73.75, 82.44, 80.95, 76.6, 75.58, 76.31, 79.16, 76.87, 77.69, 78.95, 74.37, 74.57, 81.29, 81.44, 79.95, 77.23, 72.82, 77.92, 74.32, 80.06, 75.63, 77.91, 76.79, 77.66, 77.27, 82.46, 72.73, 80.05, 76.58, 77.7, 75.07, 74.3, 75.21, 74.93, 77.57, 77.3, 75.08, 73.29, 73.38, 75.32, 78.03, 73.47, 78.3, 74.94, 76.29, 75.89, 75.36, 71.42, 76.32, 75.7, 73.93, 78.75, 78.17, 74.82, 76.87, 74.7, 76.74, 73.18, 77.45, 79.32, 79.32, 78.89, 72.2, 72.73, 79.64, 73.53, 71.63, 73.76, 79.64, 84.5, 77.03, 78.17, 78.32, 83.36, 78.85, 74.83, 77.95, 79.78, 79.0, 75.73, 70.76, 71.4, 84.1, 75.58, 77.02, 75.62, 77.36, 77.86, 79.44, 79.59, 76.72, 79.43, 74.71, 75.95, 77.94, 77.1, 78.23, 78.12, 72.88, 76.91, 81.84, 76.42, 75.98, 78.61, 76.6, 73.55, 79.91, 75.81, 76.24, 78.01, 74.35, 82.31, 75.16, 74.37, 72.07, 76.28, 77.42, 76.09, 79.79, 75.58, 76.52, 77.48, 75.41, 76.85, 76.79, 80.1, 76.76, 73.75, 81.04, 74.22, 74.54, 80.1, 77.27, 74.3, 75.65, 79.46, 77.16, 74.63, 76.39, 75.16, 80.74, 76.25, 75.69, 79.78, 77.71, 77.72, 83.05, 79.5, 77.64, 80.09, 79.22, 74.92, 79.98, 84.54, 77.5, 78.27, 77.57, 76.59, 74.55, 72.99, 78.78, 77.3, 78.42, 77.68, 80.83, 77.64, 75.62, 76.29, 76.66, 77.34, 79.56, 78.98, 77.64, 77.9, 76.98, 78.62, 79.12, 72.83, 75.76, 78.89, 79.23, 74.86, 75.03, 80.16, 78.22, 75.3, 71.1, 74.65, 80.3, 78.48, 73.8, 75.36, 76.75, 83.36, 75.87, 78.33, 81.27, 77.15, 77.64, 72.81, 74.73, 73.98, 74.11, 80.96, 80.17, 79.75, 80.34, 71.69, 77.82, 72.18, 78.64, 81.39, 73.65, 76.55, 84.38, 78.1, 79.21, 80.85, 72.88, 77.77, 77.04, 76.35, 74.63, 74.63, 72.72, 78.13, 76.91, 74.78, 74.66, 75.37, 83.04, 76.04, 77.85, 77.13, 75.54, 78.16, 73.16, 75.58, 78.75, 69.12, 72.76, 75.07, 77.88, 76.87, 78.81, 76.54, 80.47, 81.35, 81.44, 73.04, 72.09, 77.85, 77.78, 79.82, 78.65, 78.78, 79.28, 80.27, 79.31, 74.71, 74.26, 77.09, 76.48, 74.51, 82.09, 79.17, 77.77, 78.93, 75.36, 72.53, 81.44, 78.56, 78.02, 74.31, 77.37, 77.73, 77.52, 77.37, 80.73, 81.75, 77.44, 81.1, 80.06, 81.36, 80.84, 78.07, 77.21, 78.91, 76.05, 70.38, 78.81, 79.8, 75.66, 77.63, 76.73, 80.17, 81.8, 78.38, 82.38, 76.7, 78.98, 77.76, 77.12, 77.81, 76.7, 80.62, 75.99, 75.24, 73.71, 80.68, 78.0, 77.53, 78.06, 81.35, 77.55, 76.47, 78.06, 74.14, 77.54, 75.43, 79.13, 81.89, 70.9, 74.29, 77.21, 77.76, 75.98, 78.49, 77.96, 74.07, 72.76, 78.48, 77.21, 75.5, 77.38, 72.72, 74.06, 78.84, 79.74, 71.09, 78.74, 78.62, 76.24, 75.98, 76.65, 74.63, 77.27, 76.51, 73.75, 76.38, 83.96, 74.56, 78.05, 74.68, 80.09, 78.37, 77.36, 79.77, 78.99, 73.49, 79.31, 75.71, 73.91, 71.38, 74.34, 72.86, 72.0, 77.09, 76.49, 74.63, 76.2, 78.45, 75.42, 75.01, 79.65, 78.93, 78.41, 75.89, 73.67, 79.79, 75.37, 76.26, 79.74, 77.59, 80.35, 73.34, 77.3, 75.06, 73.96, 76.86, 80.37, 69.91, 73.93, 79.55, 72.98, 79.29, 74.83, 76.36, 75.85, 76.76, 79.32, 73.77, 79.11, 79.86, 75.69, 77.02, 81.36, 79.88, 74.58, 75.85, 82.24, 75.52, 74.88, 72.35, 76.66, 76.37, 78.77, 79.86, 80.89, 81.96, 78.66, 77.79, 74.39, 74.46, 73.88, 78.82, 72.6, 77.78, 79.43, 74.98, 77.59, 77.95, 79.84, 76.8, 78.42, 79.25, 75.72, 78.0, 76.42, 79.11, 78.52, 73.72, 80.27, 77.18, 78.2, 78.04, 81.93, 80.32, 78.13, 79.06, 74.89, 77.2, 78.09, 76.61, 75.07, 75.91, 74.92, 78.06, 78.09, 76.32, 77.44, 75.04, 77.6, 73.19, 74.62, 81.84, 79.34, 79.04, 77.62, 75.59, 76.51, 74.47, 77.91, 75.76, 80.91, 75.44, 83.24, 79.01, 81.25, 78.4, 76.69, 80.97, 78.43, 76.02, 77.1, 74.75, 78.64, 71.81, 73.66, 78.57, 78.02, 75.15, 84.77, 78.03, 72.98, 78.2, 77.39, 78.11, 77.4, 72.7, 81.73, 73.09, 77.9, 76.56, 74.04, 78.94, 81.83, 81.61, 74.62, 78.18, 76.08, 73.25, 78.01, 79.0, 74.52, 78.11, 76.61, 71.94, 77.23, 78.71, 78.77, 80.11, 77.49, 75.75, 77.51, 77.66, 77.72, 77.93, 82.82, 78.63, 75.03, 74.13, 75.42, 77.29, 77.96, 73.38, 82.54, 76.9, 71.99, 75.89, 74.24, 80.24, 73.0, 80.39, 74.6, 78.91, 72.66, 78.23, 81.98, 75.82, 73.38, 77.3, 77.7, 74.81, 75.27, 75.2, 79.98, 78.22, 81.71, 76.41, 78.81]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=81.1277, s₁=2.5525；x̄₂=77.1141, s₂=2.6911；
SE=√(s₁²/n₁+s₂²/n₂)=0.1218；t=(x̄₁−x̄₂)/SE=32.9549；
df(Welch)=1847.61；双侧 p = 9.718e-188。
x̄₁=81.1277, s₁=2.5525；x̄₂=77.1141, s₂=2.6911；
SE=√(s₁²/n₁+s₂²/n₂)=0.1218；t=(x̄₁−x̄₂)/SE=32.9549；
df(Welch)=1847.61；双侧 p = 9.718e-188。
计算结果
t = 32.9549，df = 1847.61，p = 9.718e-188
在 α=0.05 水平下，p=9.718e-188 < 0.05，拒绝原假设 H₀。检验性别对成绩影响。
案例 33. 两种饲料增重
原始数据
组1 n=784：[203.88, 205.97, 201.46, 205.95, 205.5, 196.42, 214.31, 206.62, 205.41, 205.59, 209.66, 212.45, 201.19, 207.19, 194.93, 205.25, 197.03, 200.4, 200.41, 202.79, 209.0, 201.43, 203.19, 206.45, 207.46, 209.68, 206.29, 212.5, 202.99, 201.64, 214.43, 205.66, 204.05, 192.31, 203.08, 192.98, 201.08, 199.28, 196.42, 200.56, 213.37, 197.36, 201.33, 202.67, 205.51, 205.7, 197.34, 209.61, 208.27, 207.96, 198.07, 200.41, 207.66, 208.96, 214.4, 203.56, 205.35, 196.61, 194.09, 197.23, 206.34, 206.36, 201.31, 208.2, 200.17, 209.18, 212.83, 203.76, 205.99, 203.31, 210.01, 198.73, 196.61, 200.6, 218.16, 211.89, 204.41, 204.43, 194.51, 209.03, 207.52, 208.33, 203.98, 199.26, 215.43, 207.93, 198.66, 198.98, 209.61, 201.91, 199.55, 203.95, 200.28, 218.61, 208.08, 208.79, 206.11, 206.21, 201.74, 209.29, 202.31, 206.26, 211.95, 202.16, 205.51, 206.09, 207.33, 200.23, 202.04, 206.02, 203.65, 201.65, 203.87, 207.49, 196.93, 200.69, 205.4, 207.23, 200.61, 198.85, 203.49, 208.62, 198.47, 201.36, 194.91, 202.01, 203.18, 213.9, 207.44, 197.84, 205.49, 201.34, 205.87, 201.43, 203.12, 200.9, 194.66, 206.77, 210.27, 195.36, 200.47, 203.78, 197.54, 204.12, 208.41, 195.51, 198.93, 193.85, 202.86, 202.48, 200.43, 209.94, 203.05, 202.75, 197.99, 210.67, 208.6, 209.09, 206.11, 202.17, 206.37, 199.48, 204.45, 206.22, 202.65, 210.1, 197.29, 203.5, 214.86, 202.62, 194.29, 195.69, 210.37, 208.28, 200.88, 206.88, 199.11, 206.14, 206.49, 202.78, 202.96, 205.61, 214.51, 199.89, 202.51, 205.77, 206.04, 205.52, 204.8, 198.25, 203.22, 211.36, 201.71, 197.01, 205.09, 204.72, 201.3, 201.69, 198.81, 197.32, 197.19, 201.55, 207.47, 198.07, 212.49, 201.2, 205.78, 212.47, 209.48, 201.67, 198.29, 209.36, 208.06, 211.81, 201.25, 201.48, 202.12, 215.37, 210.52, 212.01, 212.17, 203.3, 202.48, 203.08, 197.75, 208.45, 206.46, 205.81, 210.04, 207.48, 207.33, 208.2, 200.11, 206.6, 199.1, 195.65, 205.12, 214.2, 206.77, 203.18, 193.39, 207.38, 206.63, 207.63, 196.94, 200.59, 197.79, 196.18, 203.98, 202.49, 198.9, 208.73, 194.87, 202.47, 205.29, 207.48, 196.8, 202.22, 198.31, 201.01, 203.14, 197.8, 209.34, 204.05, 206.47, 217.07, 201.3, 196.19, 192.41, 212.02, 203.91, 198.74, 200.61, 214.39, 199.9, 215.51, 200.08, 208.13, 199.53, 196.4, 201.82, 197.19, 205.31, 199.49, 206.92, 210.49, 204.21, 202.99, 206.98, 200.27, 197.53, 197.95, 191.88, 210.36, 205.02, 195.88, 198.25, 199.4, 209.16, 195.87, 208.09, 205.73, 203.28, 203.48, 199.52, 192.36, 205.07, 210.04, 199.9, 208.81, 208.58, 198.13, 213.72, 192.53, 204.4, 198.39, 206.77, 210.31, 189.58, 201.55, 205.2, 198.99, 203.07, 209.35, 206.76, 203.93, 206.94, 202.06, 210.94, 200.28, 203.23, 208.43, 207.93, 205.88, 195.14, 207.52, 197.9, 203.51, 204.65, 205.69, 196.41, 210.25, 206.37, 198.28, 192.62, 200.63, 212.45, 210.08, 192.45, 203.85, 206.98, 204.79, 211.71, 201.75, 204.57, 203.52, 200.63, 208.38, 203.08, 203.45, 202.43, 201.75, 193.92, 204.45, 204.52, 199.49, 211.98, 197.34, 215.77, 192.69, 200.84, 197.2, 203.92, 206.88, 207.03, 204.1, 197.22, 191.36, 209.83, 208.64, 206.58, 204.21, 209.25, 206.62, 204.62, 207.64, 199.22, 207.26, 210.22, 207.17, 202.14, 210.48, 202.99, 211.69, 201.25, 199.71, 205.18, 208.86, 207.22, 206.58, 208.0, 210.3, 191.2, 203.65, 205.01, 203.75, 194.91, 199.7, 202.12, 200.97, 203.12, 204.31, 209.13, 198.27, 207.41, 192.92, 212.39, 200.98, 216.88, 199.26, 205.13, 203.53, 206.05, 209.02, 205.85, 203.68, 198.34, 211.8, 208.68, 195.97, 199.88, 210.54, 203.56, 203.65, 202.14, 215.34, 209.1, 202.25, 195.77, 203.36, 207.43, 210.26, 203.86, 200.32, 200.86, 191.96, 206.01, 199.41, 200.38, 206.91, 205.82, 205.44, 192.18, 206.25, 204.3, 202.92, 206.1, 211.28, 198.03, 206.96, 208.87, 203.97, 206.25, 203.53, 212.39, 208.04, 212.84, 203.0, 204.94, 201.38, 213.31, 197.08, 207.04, 202.65, 212.08, 200.98, 212.68, 200.75, 198.34, 201.23, 209.73, 203.28, 208.73, 198.07, 204.43, 203.24, 205.86, 202.37, 205.06, 203.45, 200.62, 211.15, 200.23, 209.28, 188.83, 194.55, 197.86, 199.78, 207.02, 205.13, 206.66, 199.86, 213.0, 207.99, 193.83, 201.69, 198.58, 204.14, 214.14, 207.3, 202.06, 207.72, 198.95, 198.38, 210.94, 217.83, 209.82, 200.93, 202.03, 198.29, 194.79, 207.64, 206.57, 203.84, 203.82, 195.21, 198.48, 211.68, 199.67, 203.74, 204.46, 204.3, 204.0, 205.76, 215.06, 208.77, 214.5, 203.72, 206.63, 202.55, 194.59, 209.92, 201.55, 198.95, 202.04, 197.33, 209.61, 203.35, 211.3, 205.11, 197.03, 197.84, 199.35, 193.06, 204.55, 210.19, 185.38, 207.88, 203.17, 200.18, 205.56, 189.09, 209.42, 204.81, 198.1, 205.28, 204.87, 201.48, 204.67, 198.81, 209.77, 206.24, 199.56, 203.13, 208.64, 209.42, 204.29, 205.35, 205.09, 202.59, 209.36, 206.13, 198.61, 199.2, 198.25, 212.03, 211.58, 215.81, 196.94, 193.1, 198.64, 202.4, 191.39, 201.34, 201.92, 206.38, 201.43, 198.96, 205.83, 200.1, 198.3, 205.07, 202.4, 200.45, 204.09, 199.42, 202.78, 207.78, 205.64, 209.94, 208.82, 201.85, 205.64, 208.89, 201.04, 207.41, 191.14, 205.29, 203.86, 200.67, 199.01, 194.09, 198.16, 199.91, 204.91, 198.57, 205.41, 210.56, 201.9, 205.47, 213.35, 196.94, 201.98, 211.32, 199.84, 196.82, 202.92, 203.9, 207.82, 204.13, 203.09, 206.82, 206.59, 201.14, 205.69, 205.21, 195.74, 203.93, 206.78, 202.58, 203.55, 206.6, 209.16, 201.02, 198.25, 198.35, 202.88, 205.44, 207.31, 207.54, 208.91, 207.13, 200.69, 207.72, 197.12, 199.9, 195.83, 199.94, 209.41, 194.97, 208.81, 206.16, 207.63, 215.39, 201.26, 204.83, 198.59, 203.23, 198.48, 194.44, 210.13, 201.53, 193.54, 198.25, 200.46, 199.35, 207.56, 197.37, 201.92, 202.79, 201.06, 210.91, 206.2, 211.5, 201.57, 205.07, 199.73, 200.23, 200.85, 202.45, 202.03, 203.05, 209.53, 197.28, 202.71, 205.6, 205.93, 208.15, 213.94, 200.93, 198.95, 197.17, 209.68, 197.48, 203.23, 212.3, 193.62, 201.83, 209.46, 204.26, 207.53, 210.76, 205.01, 210.16, 213.13, 203.53, 200.63, 204.38, 202.31, 204.8, 202.81, 211.51, 199.32, 199.34, 209.13, 199.91, 192.64, 196.14, 211.22, 203.88, 206.21, 201.28, 203.63, 203.14, 198.15, 205.35, 205.48, 211.97, 206.71, 205.86, 194.37, 204.68, 200.32, 200.16, 203.13, 198.39, 203.32, 193.58, 202.6, 208.53, 214.62, 196.59, 211.32, 196.12, 209.63, 206.07, 202.89, 203.72, 205.94, 202.06, 197.36, 195.63, 198.31, 195.84, 216.13, 207.45, 202.58, 205.82, 198.12, 191.63, 207.96, 199.08, 201.42, 209.18]；
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组2 n=503：[181.28, 174.74, 188.09, 189.19, 189.96, 186.6, 189.22, 187.81, 184.19, 183.45, 184.13, 175.94, 184.82, 185.11, 183.01, 186.53, 199.7, 192.58, 173.17, 192.88, 187.57, 176.92, 175.58, 173.21, 181.04, 187.63, 182.73, 184.6, 183.92, 174.82, 177.24, 190.04, 182.91, 186.98, 182.28, 190.25, 190.93, 180.17, 192.88, 182.61, 182.19, 178.34, 190.72, 178.55, 175.3, 184.01, 188.85, 189.88, 170.9, 185.58, 191.69, 192.51, 179.25, 188.37, 180.46, 191.26, 192.43, 188.35, 194.76, 180.98, 176.36, 185.44, 178.22, 185.58, 179.61, 184.77, 184.97, 187.57, 178.2, 185.69, 181.98, 184.17, 188.59, 186.92, 182.46, 188.43, 192.52, 194.76, 185.31, 181.03, 185.76, 185.4, 186.09, 190.35, 182.4, 187.3, 176.1, 179.71, 178.72, 180.21, 181.06, 188.06, 190.29, 185.12, 173.69, 187.41, 177.39, 181.57, 180.15, 179.1, 182.22, 190.07, 190.56, 195.36, 184.89, 190.1, 184.6, 185.03, 180.89, 184.3, 173.68, 183.97, 197.43, 184.11, 189.56, 192.11, 168.49, 187.15, 192.42, 179.43, 184.07, 180.51, 186.9, 177.67, 191.47, 176.69, 183.61, 178.84, 182.46, 181.57, 186.79, 183.49, 188.48, 179.8, 181.57, 190.04, 189.72, 182.47, 186.32, 186.09, 176.17, 182.64, 186.67, 178.21, 177.05, 186.61, 179.89, 186.79, 182.17, 188.14, 183.99, 186.12, 179.74, 181.61, 187.8, 183.16, 179.7, 183.37, 182.34, 187.07, 183.86, 183.46, 187.22, 187.39, 185.87, 190.88, 188.18, 191.28, 184.07, 177.41, 188.23, 192.69, 184.19, 180.48, 178.27, 187.7, 184.23, 181.86, 175.86, 173.23, 180.21, 184.53, 184.6, 179.43, 192.49, 181.37, 185.34, 180.78, 180.29, 175.71, 189.09, 182.39, 185.04, 175.79, 182.39, 188.67, 184.39, 174.58, 181.07, 191.1, 180.87, 191.17, 183.76, 198.09, 183.33, 185.83, 184.0, 186.58, 185.38, 177.73, 177.56, 186.38, 185.64, 183.75, 192.54, 177.66, 185.16, 189.45, 186.47, 185.95, 186.15, 183.25, 186.99, 185.39, 180.25, 188.43, 189.58, 177.26, 188.99, 179.0, 194.51, 180.48, 179.85, 188.44, 178.36, 182.3, 175.06, 185.32, 181.33, 185.98, 187.6, 183.67, 186.2, 191.1, 176.35, 191.6, 177.33, 188.35, 176.28, 182.63, 193.45, 181.44, 178.26, 180.57, 177.35, 187.17, 183.24, 186.33, 188.7, 179.93, 173.27, 191.45, 190.53, 178.85, 187.04, 181.83, 191.81, 182.4, 182.79, 179.64, 189.64, 181.43, 183.77, 188.93, 179.69, 188.36, 189.47, 184.77, 183.22, 187.47, 185.8, 187.38, 192.45, 178.56, 181.66, 188.71, 184.47, 183.92, 183.82, 186.62, 188.05, 189.72, 187.27, 183.97, 189.59, 179.72, 175.25, 182.92, 189.01, 174.64, 179.77, 179.81, 178.66, 194.06, 182.76, 192.59, 178.57, 189.46, 182.29, 184.33, 177.71, 181.74, 186.41, 189.46, 185.67, 183.94, 181.16, 177.62, 178.1, 188.16, 187.13, 198.67, 184.48, 184.24, 186.59, 189.06, 181.94, 185.2, 187.02, 188.11, 188.29, 187.2, 175.18, 184.98, 186.33, 181.2, 191.39, 179.73, 187.91, 187.94, 181.96, 187.6, 172.77, 184.49, 191.48, 181.81, 189.01, 192.51, 185.38, 183.3, 180.19, 172.37, 184.22, 182.69, 187.83, 181.7, 187.06, 177.89, 188.97, 189.93, 187.63, 187.21, 179.39, 180.27, 188.38, 176.13, 174.3, 191.25, 184.34, 186.85, 179.83, 189.56, 177.87, 188.4, 183.24, 200.17, 191.42, 185.49, 186.66, 186.26, 188.0, 187.0, 181.95, 179.45, 178.64, 190.71, 180.84, 182.93, 184.36, 188.79, 188.93, 184.43, 183.58, 181.87, 181.73, 179.56, 185.58, 178.08, 179.99, 182.45, 191.83, 189.1, 186.8, 177.67, 186.89, 174.84, 184.17, 182.37, 180.84, 181.18, 197.03, 185.08, 174.18, 176.27, 174.76, 194.35, 173.08, 195.89, 184.19, 183.2, 184.21, 195.59, 184.56, 184.97, 185.01, 186.78, 181.56, 197.79, 187.83, 187.73, 181.97, 178.24, 188.22, 179.69, 196.07, 173.98, 191.62, 185.01, 182.93, 171.03, 183.97, 184.25, 176.28, 176.24, 183.78, 184.22, 183.84, 189.66, 186.31, 185.31, 173.42, 189.28, 190.73, 184.02, 192.35, 179.55, 175.37, 174.36, 175.86, 177.46, 184.4, 178.86, 186.5, 174.37, 190.96, 184.01, 189.42, 183.77, 183.51, 179.02, 187.07, 193.22, 178.65, 183.71, 181.24, 205.46, 184.67, 176.38, 184.07, 174.62, 187.07, 171.22, 178.91, 189.73, 187.84, 184.44, 180.93, 190.18, 181.06, 183.57, 178.39, 180.88, 179.86, 180.97, 181.83, 180.82, 186.24, 181.08, 188.03, 187.85, 179.23, 190.95, 191.75]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=203.6965, s₁=5.3584；x̄₂=184.2228, s₂=5.4677；
SE=√(s₁²/n₁+s₂²/n₂)=0.3099；t=(x̄₁−x̄₂)/SE=62.8323；
df(Welch)=1054.55；双侧 p = 0.000e+00。
x̄₁=203.6965, s₁=5.3584；x̄₂=184.2228, s₂=5.4677；
SE=√(s₁²/n₁+s₂²/n₂)=0.3099；t=(x̄₁−x̄₂)/SE=62.8323；
df(Welch)=1054.55；双侧 p = 0.000e+00。
计算结果
t = 62.8323，df = 1054.55，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。比较饲料促生长效果。
案例 34. 新老员工绩效评分
原始数据
组1 n=804：[3.61, 3.67, 3.52, 3.46, 3.66, 3.61, 3.68, 3.55, 3.7, 3.41, 3.49, 3.5, 3.55, 3.62, 3.93, 3.82, 3.48, 3.52, 3.67, 3.17, 3.7, 3.7, 3.65, 3.6, 3.47, 3.54, 3.56, 3.43, 3.77, 3.36, 3.91, 3.19, 3.59, 3.64, 3.89, 3.72, 3.62, 3.5, 3.62, 3.66, 3.44, 4.01, 3.38, 3.64, 3.23, 3.77, 4.07, 3.38, 3.7, 3.33, 3.59, 3.36, 3.65, 3.82, 3.4, 3.45, 3.87, 3.62, 3.81, 3.8, 3.95, 3.84, 3.07, 3.56, 3.74, 3.45, 3.61, 3.57, 3.64, 3.22, 3.85, 3.47, 3.29, 3.24, 3.51, 3.76, 3.74, 3.15, 3.87, 3.47, 3.65, 3.57, 3.48, 3.73, 3.82, 3.58, 3.39, 3.37, 3.87, 3.81, 3.68, 3.79, 3.92, 3.7, 3.28, 3.54, 3.03, 3.54, 3.52, 3.33, 3.96, 3.56, 4.23, 3.71, 3.86, 3.15, 3.86, 4.02, 3.43, 3.55, 3.41, 3.35, 3.65, 3.5, 3.49, 3.42, 3.66, 3.7, 3.46, 3.43, 3.47, 3.63, 3.6, 3.64, 3.48, 3.58, 3.08, 3.6, 3.62, 3.54, 3.47, 3.59, 4.05, 3.64, 3.62, 3.63, 3.31, 3.73, 3.64, 4.09, 3.28, 3.49, 3.54, 3.65, 3.76, 3.29, 3.15, 3.88, 3.73, 3.81, 2.77, 3.33, 4.09, 3.74, 3.43, 3.69, 3.65, 3.49, 3.38, 3.74, 3.44, 3.82, 3.81, 3.79, 3.71, 3.36, 3.61, 3.18, 3.32, 3.43, 3.51, 3.22, 3.6, 3.72, 3.51, 3.37, 3.15, 3.6, 3.34, 3.72, 3.72, 3.9, 3.68, 3.47, 3.79, 3.34, 3.53, 3.89, 3.76, 3.7, 3.74, 3.42, 3.58, 3.79, 3.78, 3.65, 3.65, 3.65, 3.14, 3.68, 3.23, 3.53, 3.43, 3.46, 3.94, 3.71, 3.2, 4.03, 3.92, 3.93, 3.8, 3.78, 3.6, 3.64, 3.6, 3.85, 3.29, 3.82, 3.75, 3.59, 3.61, 3.81, 3.45, 3.48, 3.78, 3.27, 3.16, 3.15, 3.67, 3.42, 3.89, 3.93, 3.61, 3.52, 3.53, 3.73, 3.53, 3.48, 3.46, 3.52, 3.53, 4.08, 3.48, 3.42, 3.48, 3.44, 3.39, 3.49, 3.66, 3.52, 3.46, 3.44, 3.28, 3.94, 3.86, 3.31, 3.27, 3.67, 3.38, 3.64, 3.81, 3.26, 3.37, 3.57, 3.18, 3.5, 3.75, 3.35, 3.64, 3.34, 3.43, 3.68, 3.5, 3.57, 3.3, 3.23, 3.31, 3.6, 3.39, 3.27, 3.5, 3.85, 3.54, 3.43, 3.74, 3.33, 3.5, 3.55, 3.39, 3.53, 3.75, 3.45, 3.3, 3.48, 3.35, 3.72, 3.63, 3.68, 3.57, 3.24, 3.32, 3.66, 3.35, 3.81, 3.69, 3.64, 3.21, 3.28, 3.66, 3.3, 3.26, 3.44, 3.74, 3.22, 3.32, 3.79, 3.79, 3.72, 3.34, 3.65, 3.64, 3.76, 3.71, 3.51, 3.41, 3.11, 3.46, 3.67, 3.84, 3.48, 3.42, 3.6, 3.77, 3.44, 3.62, 3.42, 3.41, 3.34, 3.59, 3.85, 3.72, 3.67, 3.89, 4.19, 3.58, 3.35, 3.53, 3.51, 3.49, 3.11, 3.61, 3.37, 3.22, 3.4, 3.56, 3.5, 3.71, 3.6, 3.68, 3.68, 3.59, 3.48, 3.36, 3.87, 3.34, 3.56, 3.83, 3.13, 3.4, 3.28, 3.36, 3.24, 3.56, 3.39, 3.38, 3.56, 3.58, 3.88, 3.48, 3.73, 3.39, 3.36, 3.55, 3.87, 3.68, 3.73, 3.37, 3.63, 3.89, 3.84, 3.13, 3.56, 3.16, 3.3, 3.96, 3.67, 3.65, 3.42, 3.8, 3.59, 3.46, 3.79, 3.77, 4.08, 3.63, 3.79, 3.17, 3.2, 3.73, 3.51, 3.32, 3.45, 3.35, 3.55, 3.64, 3.81, 3.85, 3.74, 3.35, 3.4, 3.86, 3.42, 3.07, 3.29, 3.15, 3.46, 3.52, 3.79, 3.75, 3.23, 3.76, 3.39, 3.37, 3.95, 3.43, 3.68, 3.46, 3.8, 3.7, 3.37, 3.58, 3.47, 3.4, 3.13, 3.29, 3.4, 3.65, 3.29, 3.31, 3.34, 3.62, 3.9, 3.22, 3.77, 3.78, 3.45, 3.76, 3.6, 3.6, 3.64, 3.91, 3.82, 3.48, 3.64, 3.11, 3.46, 3.53, 3.29, 3.1, 3.36, 3.22, 3.73, 3.37, 3.16, 3.14, 3.5, 3.67, 3.94, 3.67, 3.72, 3.52, 3.42, 3.35, 3.19, 3.36, 3.26, 3.68, 4.0, 3.22, 3.45, 3.28, 3.57, 3.3, 3.62, 3.44, 3.65, 3.45, 4.13, 3.69, 3.45, 3.49, 3.59, 3.71, 3.83, 3.32, 3.35, 3.23, 3.54, 3.49, 3.19, 3.57, 3.87, 3.41, 3.4, 3.54, 3.45, 3.98, 3.64, 3.64, 3.22, 3.13, 3.88, 3.58, 3.11, 3.81, 3.67, 3.7, 3.63, 3.33, 3.34, 3.26, 3.75, 3.37, 3.46, 3.72, 3.72, 4.08, 3.14, 3.46, 3.53, 3.45, 3.38, 3.67, 3.66, 3.51, 3.65, 3.73, 3.35, 3.71, 2.96, 3.28, 3.73, 3.55, 3.01, 3.57, 3.67, 3.72, 4.11, 3.6, 3.9, 3.77, 3.33, 3.19, 3.64, 3.27, 3.19, 3.44, 3.44, 3.41, 3.78, 3.78, 3.43, 3.63, 3.34, 3.74, 3.4, 3.47, 3.65, 3.53, 3.26, 3.73, 3.66, 3.6, 3.64, 3.97, 3.56, 3.72, 3.7, 3.67, 3.57, 3.71, 3.38, 3.51, 3.27, 3.77, 3.5, 3.46, 3.87, 3.3, 3.23, 3.49, 3.45, 3.33, 3.21, 3.53, 3.48, 3.37, 3.84, 3.44, 4.0, 3.79, 3.72, 3.77, 3.77, 3.43, 3.32, 3.26, 3.42, 3.44, 3.26, 3.44, 3.81, 3.53, 3.31, 3.58, 3.72, 4.03, 3.21, 3.48, 3.63, 3.58, 3.52, 3.61, 3.55, 3.51, 3.65, 3.4, 3.33, 3.54, 3.48, 3.3, 3.68, 3.45, 3.72, 3.56, 3.35, 3.43, 3.78, 3.51, 3.89, 3.23, 3.33, 3.67, 3.84, 3.5, 3.77, 3.35, 3.52, 3.71, 3.73, 4.02, 3.59, 3.66, 3.54, 3.75, 3.44, 3.61, 3.51, 3.83, 3.55, 3.64, 3.62, 3.59, 3.45, 3.4, 3.92, 3.91, 3.75, 3.87, 3.54, 3.78, 3.34, 3.52, 3.55, 4.03, 3.57, 3.57, 2.83, 3.52, 3.39, 3.49, 2.97, 3.48, 3.46, 3.5, 3.48, 3.34, 3.54, 3.81, 3.48, 3.84, 3.67, 3.44, 3.66, 3.38, 3.33, 3.5, 3.88, 3.56, 3.51, 2.88, 3.65, 3.69, 3.07, 3.39, 3.37, 3.41, 3.66, 3.69, 3.53, 3.55, 3.69, 3.51, 3.46, 3.67, 3.18, 3.3, 3.66, 3.12, 3.69, 3.5, 3.28, 3.73, 3.73, 3.14, 3.52, 3.7, 3.56, 3.47, 3.62, 3.94, 3.57, 3.62, 3.76, 3.67, 3.4, 3.36, 3.57, 3.39, 3.66, 3.22, 3.68, 3.74, 3.75, 3.35, 3.83, 3.76, 3.79, 3.64, 3.67, 3.12, 3.74, 3.32, 3.54, 3.05, 3.74, 3.63, 3.63, 4.16, 3.33, 3.52, 3.44, 3.44, 3.68, 3.43, 3.64, 3.7, 3.62, 3.41, 3.77, 3.35, 3.14, 3.31, 3.42, 3.68, 3.54, 3.7, 3.45, 3.51, 3.59, 3.49, 3.94, 3.45, 3.23, 3.37, 3.69, 3.53, 3.46, 4.07, 3.3, 3.99, 3.57, 3.81, 3.4]；
组2 n=740：[3.08, 2.94, 2.86, 3.21, 2.84, 2.73, 2.99, 2.76, 3.09, 2.97, 2.98, 3.0, 2.88, 2.96, 3.32, 2.91, 3.15, 2.76, 3.06, 2.84, 2.88, 2.7, 2.83, 3.07, 3.06, 2.89, 3.18, 2.99, 2.91, 3.05, 3.0, 3.09, 3.15, 3.08, 3.2, 2.9, 3.47, 3.2, 2.76, 3.23, 3.11, 3.07, 3.36, 3.21, 3.08, 2.99, 2.99, 3.09, 2.8, 2.83, 3.15, 3.15, 3.22, 3.24, 3.08, 3.14, 3.04, 3.13, 3.19, 3.16, 2.79, 3.22, 3.11, 3.05, 3.3, 3.01, 2.89, 3.13, 2.86, 2.99, 2.94, 2.91, 3.04, 3.33, 3.13, 3.37, 3.07, 3.17, 3.2, 3.25, 3.1, 3.07, 2.99, 2.84, 3.06, 3.08, 2.95, 3.04, 2.89, 3.12, 2.96, 3.14, 3.06, 3.11, 2.8, 3.16, 2.99, 2.98, 3.02, 3.54, 3.09, 2.87, 3.14, 3.15, 3.18, 3.02, 3.15, 3.15, 3.12, 3.03, 2.86, 3.1, 3.04, 3.02, 3.38, 3.22, 3.09, 2.92, 3.04, 3.12, 2.99, 3.25, 2.82, 3.07, 3.2, 3.07, 2.95, 3.18, 2.6, 3.06, 2.83, 2.93, 3.18, 2.9, 2.91, 2.92, 3.31, 2.95, 3.22, 3.15, 2.69, 2.77, 3.04, 2.78, 3.01, 3.06, 2.81, 3.07, 3.19, 3.1, 3.18, 3.23, 3.16, 3.25, 3.22, 2.98, 2.96, 3.14, 2.78, 3.06, 3.17, 3.17, 2.99, 3.17, 2.56, 2.99, 3.23, 3.11, 3.13, 3.26, 3.08, 2.79, 3.48, 2.91, 3.23, 2.94, 2.98, 3.09, 2.72, 3.03, 3.28, 3.0, 3.03, 3.05, 2.94, 2.99, 3.15, 2.97, 2.92, 2.81, 2.82, 2.76, 3.41, 3.3, 3.2, 2.94, 2.93, 3.17, 2.84, 3.32, 3.09, 3.32, 3.08, 3.08, 2.94, 2.8, 3.09, 3.16, 3.01, 2.9, 3.11, 3.35, 3.11, 2.97, 2.81, 2.99, 2.96, 2.88, 2.8, 2.95, 2.9, 2.78, 3.02, 3.25, 3.12, 2.98, 3.33, 2.79, 3.04, 2.88, 2.87, 3.1, 3.13, 2.9, 2.99, 3.08, 3.17, 3.37, 3.21, 3.02, 2.93, 3.26, 2.91, 2.97, 2.91, 3.07, 3.25, 3.08, 2.79, 3.05, 2.89, 2.9, 3.18, 3.1, 2.86, 3.31, 2.83, 3.01, 2.98, 2.94, 2.95, 3.27, 3.09, 3.06, 2.87, 2.94, 3.07, 3.04, 3.0, 3.29, 3.16, 2.76, 2.95, 2.99, 3.19, 3.29, 2.94, 2.79, 2.91, 2.91, 3.26, 2.82, 3.17, 3.15, 3.14, 3.11, 2.87, 3.09, 3.32, 3.06, 3.05, 3.29, 3.13, 3.08, 2.92, 3.06, 2.84, 2.88, 2.85, 3.0, 3.05, 2.67, 2.92, 3.1, 3.0, 2.98, 3.09, 3.21, 2.66, 3.6, 3.07, 3.07, 3.1, 3.22, 3.18, 3.13, 3.32, 2.53, 3.0, 3.21, 3.19, 3.0, 3.32, 2.82, 3.12, 3.21, 3.3, 2.94, 2.76, 2.98, 2.98, 3.06, 3.11, 3.13, 3.35, 3.11, 2.7, 3.4, 3.15, 3.2, 3.1, 2.92, 3.11, 3.15, 3.06, 3.03, 3.03, 2.93, 3.21, 3.27, 3.22, 3.04, 3.16, 3.19, 2.77, 3.22, 3.06, 2.86, 2.8, 3.28, 3.05, 2.97, 2.9, 3.01, 2.99, 3.05, 3.01, 2.88, 3.06, 3.16, 3.09, 2.77, 3.25, 2.98, 2.96, 2.91, 2.98, 2.73, 2.85, 3.11, 2.9, 3.07, 3.16, 2.97, 3.03, 2.9, 2.81, 2.77, 2.91, 3.33, 3.15, 3.37, 2.6, 2.91, 3.16, 2.76, 2.96, 2.83, 3.38, 3.41, 2.87, 3.06, 3.35, 3.3, 2.82, 3.1, 2.87, 3.19, 2.5, 3.06, 2.87, 3.05, 3.22, 3.24, 2.7, 3.12, 3.35, 3.26, 3.13, 3.16, 3.35, 3.2, 3.06, 3.01, 2.85, 3.21, 2.86, 3.24, 2.76, 2.92, 3.24, 3.51, 3.07, 3.15, 2.96, 2.77, 2.87, 3.19, 3.3, 2.99, 3.3, 3.1, 2.81, 3.18, 2.91, 2.48, 2.7, 3.24, 2.96, 3.05, 3.15, 3.02, 2.9, 3.12, 3.0, 2.99, 2.75, 3.15, 3.05, 3.1, 3.22, 3.3, 2.61, 2.93, 3.19, 3.29, 2.82, 3.01, 2.88, 3.24, 3.21, 3.27, 3.08, 2.78, 3.37, 3.01, 2.93, 3.17, 3.24, 3.0, 3.31, 3.18, 3.18, 3.31, 3.04, 3.1, 2.7, 2.8, 3.1, 2.85, 2.89, 2.83, 3.07, 3.05, 2.85, 3.04, 3.25, 3.2, 2.94, 3.02, 3.11, 3.03, 2.67, 3.21, 3.13, 3.15, 3.21, 3.07, 3.08, 2.8, 2.97, 2.93, 2.87, 2.96, 3.2, 2.74, 2.68, 3.05, 2.86, 2.87, 3.0, 3.17, 3.12, 3.39, 2.79, 2.99, 3.26, 3.19, 2.75, 3.27, 3.13, 2.91, 2.81, 3.16, 3.02, 2.98, 3.01, 2.97, 2.82, 3.15, 3.17, 2.97, 2.89, 3.06, 3.39, 3.01, 2.68, 3.16, 3.08, 3.07, 3.06, 2.88, 3.03, 3.27, 3.2, 3.34, 2.99, 2.96, 3.12, 3.35, 2.76, 2.73, 2.9, 2.75, 3.07, 3.26, 3.28, 2.95, 3.32, 2.94, 3.16, 2.98, 3.12, 3.19, 3.21, 3.16, 3.2, 3.0, 2.77, 3.27, 3.06, 3.19, 2.96, 2.92, 2.97, 2.51, 3.02, 3.04, 3.07, 3.15, 2.99, 3.01, 2.89, 2.89, 3.14, 3.06, 2.72, 2.94, 2.81, 3.04, 3.01, 2.98, 2.97, 2.97, 3.25, 2.89, 2.89, 3.38, 2.85, 2.78, 3.17, 2.89, 3.09, 2.95, 2.82, 2.87, 3.16, 2.98, 3.04, 3.08, 3.16, 3.12, 2.93, 3.37, 3.07, 3.14, 3.25, 3.09, 3.12, 2.93, 2.62, 3.11, 2.8, 3.16, 3.21, 2.87, 3.04, 3.16, 2.78, 3.12, 3.16, 3.11, 3.31, 3.23, 2.93, 3.29, 2.94, 3.0, 3.38, 2.76, 3.12, 2.99, 2.92, 3.02, 3.03, 2.98, 3.14, 2.98, 3.03, 3.03, 3.04, 2.96, 3.1, 3.05, 2.86, 3.22, 2.8, 3.16, 2.92, 3.02, 3.02, 2.9, 2.86, 3.14, 3.05, 2.91, 3.09, 2.7, 2.84, 2.77, 3.18, 2.68, 2.92, 3.0, 3.02, 3.29, 2.86, 3.02, 3.0, 2.98, 3.17, 2.98, 2.97, 2.85, 3.19, 2.69, 2.79, 3.07, 3.16, 3.23, 3.18, 3.22, 2.99, 2.85, 3.29, 2.88, 3.15, 2.99, 3.31, 3.12, 2.93, 2.77, 3.08, 2.96, 3.0, 3.02, 2.95, 3.38, 2.92, 2.86, 3.12, 3.29, 3.2, 3.22, 2.92, 3.16, 2.93, 2.7, 2.72, 3.17, 2.89, 3.12, 3.06, 2.8, 2.72, 3.12, 2.85, 2.95, 3.08, 3.21]
组2 n=740：[3.08, 2.94, 2.86, 3.21, 2.84, 2.73, 2.99, 2.76, 3.09, 2.97, 2.98, 3.0, 2.88, 2.96, 3.32, 2.91, 3.15, 2.76, 3.06, 2.84, 2.88, 2.7, 2.83, 3.07, 3.06, 2.89, 3.18, 2.99, 2.91, 3.05, 3.0, 3.09, 3.15, 3.08, 3.2, 2.9, 3.47, 3.2, 2.76, 3.23, 3.11, 3.07, 3.36, 3.21, 3.08, 2.99, 2.99, 3.09, 2.8, 2.83, 3.15, 3.15, 3.22, 3.24, 3.08, 3.14, 3.04, 3.13, 3.19, 3.16, 2.79, 3.22, 3.11, 3.05, 3.3, 3.01, 2.89, 3.13, 2.86, 2.99, 2.94, 2.91, 3.04, 3.33, 3.13, 3.37, 3.07, 3.17, 3.2, 3.25, 3.1, 3.07, 2.99, 2.84, 3.06, 3.08, 2.95, 3.04, 2.89, 3.12, 2.96, 3.14, 3.06, 3.11, 2.8, 3.16, 2.99, 2.98, 3.02, 3.54, 3.09, 2.87, 3.14, 3.15, 3.18, 3.02, 3.15, 3.15, 3.12, 3.03, 2.86, 3.1, 3.04, 3.02, 3.38, 3.22, 3.09, 2.92, 3.04, 3.12, 2.99, 3.25, 2.82, 3.07, 3.2, 3.07, 2.95, 3.18, 2.6, 3.06, 2.83, 2.93, 3.18, 2.9, 2.91, 2.92, 3.31, 2.95, 3.22, 3.15, 2.69, 2.77, 3.04, 2.78, 3.01, 3.06, 2.81, 3.07, 3.19, 3.1, 3.18, 3.23, 3.16, 3.25, 3.22, 2.98, 2.96, 3.14, 2.78, 3.06, 3.17, 3.17, 2.99, 3.17, 2.56, 2.99, 3.23, 3.11, 3.13, 3.26, 3.08, 2.79, 3.48, 2.91, 3.23, 2.94, 2.98, 3.09, 2.72, 3.03, 3.28, 3.0, 3.03, 3.05, 2.94, 2.99, 3.15, 2.97, 2.92, 2.81, 2.82, 2.76, 3.41, 3.3, 3.2, 2.94, 2.93, 3.17, 2.84, 3.32, 3.09, 3.32, 3.08, 3.08, 2.94, 2.8, 3.09, 3.16, 3.01, 2.9, 3.11, 3.35, 3.11, 2.97, 2.81, 2.99, 2.96, 2.88, 2.8, 2.95, 2.9, 2.78, 3.02, 3.25, 3.12, 2.98, 3.33, 2.79, 3.04, 2.88, 2.87, 3.1, 3.13, 2.9, 2.99, 3.08, 3.17, 3.37, 3.21, 3.02, 2.93, 3.26, 2.91, 2.97, 2.91, 3.07, 3.25, 3.08, 2.79, 3.05, 2.89, 2.9, 3.18, 3.1, 2.86, 3.31, 2.83, 3.01, 2.98, 2.94, 2.95, 3.27, 3.09, 3.06, 2.87, 2.94, 3.07, 3.04, 3.0, 3.29, 3.16, 2.76, 2.95, 2.99, 3.19, 3.29, 2.94, 2.79, 2.91, 2.91, 3.26, 2.82, 3.17, 3.15, 3.14, 3.11, 2.87, 3.09, 3.32, 3.06, 3.05, 3.29, 3.13, 3.08, 2.92, 3.06, 2.84, 2.88, 2.85, 3.0, 3.05, 2.67, 2.92, 3.1, 3.0, 2.98, 3.09, 3.21, 2.66, 3.6, 3.07, 3.07, 3.1, 3.22, 3.18, 3.13, 3.32, 2.53, 3.0, 3.21, 3.19, 3.0, 3.32, 2.82, 3.12, 3.21, 3.3, 2.94, 2.76, 2.98, 2.98, 3.06, 3.11, 3.13, 3.35, 3.11, 2.7, 3.4, 3.15, 3.2, 3.1, 2.92, 3.11, 3.15, 3.06, 3.03, 3.03, 2.93, 3.21, 3.27, 3.22, 3.04, 3.16, 3.19, 2.77, 3.22, 3.06, 2.86, 2.8, 3.28, 3.05, 2.97, 2.9, 3.01, 2.99, 3.05, 3.01, 2.88, 3.06, 3.16, 3.09, 2.77, 3.25, 2.98, 2.96, 2.91, 2.98, 2.73, 2.85, 3.11, 2.9, 3.07, 3.16, 2.97, 3.03, 2.9, 2.81, 2.77, 2.91, 3.33, 3.15, 3.37, 2.6, 2.91, 3.16, 2.76, 2.96, 2.83, 3.38, 3.41, 2.87, 3.06, 3.35, 3.3, 2.82, 3.1, 2.87, 3.19, 2.5, 3.06, 2.87, 3.05, 3.22, 3.24, 2.7, 3.12, 3.35, 3.26, 3.13, 3.16, 3.35, 3.2, 3.06, 3.01, 2.85, 3.21, 2.86, 3.24, 2.76, 2.92, 3.24, 3.51, 3.07, 3.15, 2.96, 2.77, 2.87, 3.19, 3.3, 2.99, 3.3, 3.1, 2.81, 3.18, 2.91, 2.48, 2.7, 3.24, 2.96, 3.05, 3.15, 3.02, 2.9, 3.12, 3.0, 2.99, 2.75, 3.15, 3.05, 3.1, 3.22, 3.3, 2.61, 2.93, 3.19, 3.29, 2.82, 3.01, 2.88, 3.24, 3.21, 3.27, 3.08, 2.78, 3.37, 3.01, 2.93, 3.17, 3.24, 3.0, 3.31, 3.18, 3.18, 3.31, 3.04, 3.1, 2.7, 2.8, 3.1, 2.85, 2.89, 2.83, 3.07, 3.05, 2.85, 3.04, 3.25, 3.2, 2.94, 3.02, 3.11, 3.03, 2.67, 3.21, 3.13, 3.15, 3.21, 3.07, 3.08, 2.8, 2.97, 2.93, 2.87, 2.96, 3.2, 2.74, 2.68, 3.05, 2.86, 2.87, 3.0, 3.17, 3.12, 3.39, 2.79, 2.99, 3.26, 3.19, 2.75, 3.27, 3.13, 2.91, 2.81, 3.16, 3.02, 2.98, 3.01, 2.97, 2.82, 3.15, 3.17, 2.97, 2.89, 3.06, 3.39, 3.01, 2.68, 3.16, 3.08, 3.07, 3.06, 2.88, 3.03, 3.27, 3.2, 3.34, 2.99, 2.96, 3.12, 3.35, 2.76, 2.73, 2.9, 2.75, 3.07, 3.26, 3.28, 2.95, 3.32, 2.94, 3.16, 2.98, 3.12, 3.19, 3.21, 3.16, 3.2, 3.0, 2.77, 3.27, 3.06, 3.19, 2.96, 2.92, 2.97, 2.51, 3.02, 3.04, 3.07, 3.15, 2.99, 3.01, 2.89, 2.89, 3.14, 3.06, 2.72, 2.94, 2.81, 3.04, 3.01, 2.98, 2.97, 2.97, 3.25, 2.89, 2.89, 3.38, 2.85, 2.78, 3.17, 2.89, 3.09, 2.95, 2.82, 2.87, 3.16, 2.98, 3.04, 3.08, 3.16, 3.12, 2.93, 3.37, 3.07, 3.14, 3.25, 3.09, 3.12, 2.93, 2.62, 3.11, 2.8, 3.16, 3.21, 2.87, 3.04, 3.16, 2.78, 3.12, 3.16, 3.11, 3.31, 3.23, 2.93, 3.29, 2.94, 3.0, 3.38, 2.76, 3.12, 2.99, 2.92, 3.02, 3.03, 2.98, 3.14, 2.98, 3.03, 3.03, 3.04, 2.96, 3.1, 3.05, 2.86, 3.22, 2.8, 3.16, 2.92, 3.02, 3.02, 2.9, 2.86, 3.14, 3.05, 2.91, 3.09, 2.7, 2.84, 2.77, 3.18, 2.68, 2.92, 3.0, 3.02, 3.29, 2.86, 3.02, 3.0, 2.98, 3.17, 2.98, 2.97, 2.85, 3.19, 2.69, 2.79, 3.07, 3.16, 3.23, 3.18, 3.22, 2.99, 2.85, 3.29, 2.88, 3.15, 2.99, 3.31, 3.12, 2.93, 2.77, 3.08, 2.96, 3.0, 3.02, 2.95, 3.38, 2.92, 2.86, 3.12, 3.29, 3.2, 3.22, 2.92, 3.16, 2.93, 2.7, 2.72, 3.17, 2.89, 3.12, 3.06, 2.8, 2.72, 3.12, 2.85, 2.95, 3.08, 3.21]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=3.5517, s₁=0.2231；x̄₂=3.0369, s₂=0.1744；
SE=√(s₁²/n₁+s₂²/n₂)=0.0101；t=(x̄₁−x̄₂)/SE=50.7243；
df(Welch)=1503.24；双侧 p = 0.000e+00。
x̄₁=3.5517, s₁=0.2231；x̄₂=3.0369, s₂=0.1744；
SE=√(s₁²/n₁+s₂²/n₂)=0.0101；t=(x̄₁−x̄₂)/SE=50.7243；
df(Welch)=1503.24；双侧 p = 0.000e+00。
计算结果
t = 50.7243，df = 1503.24，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估经验对绩效影响。
案例 35. 两医院治愈天数
原始数据
组1 n=712：[7.8, 7.99, 8.77, 8.86, 6.82, 5.09, 8.74, 7.54, 8.77, 6.82, 8.52, 7.16, 8.59, 8.02, 7.02, 6.92, 8.51, 9.82, 7.54, 8.98, 7.02, 9.9, 4.48, 5.76, 6.81, 7.33, 9.12, 9.08, 6.55, 7.39, 8.27, 7.36, 8.57, 9.56, 6.61, 5.84, 6.47, 9.83, 7.79, 8.51, 9.44, 9.4, 8.61, 4.69, 8.91, 7.85, 8.56, 5.83, 8.6, 6.84, 7.11, 6.08, 6.98, 5.87, 7.01, 8.04, 7.39, 7.43, 7.58, 7.67, 7.74, 8.16, 6.94, 10.0, 9.65, 4.43, 6.96, 5.6, 7.7, 6.36, 7.85, 8.41, 9.92, 7.29, 6.3, 6.83, 5.36, 6.95, 7.19, 7.52, 7.8, 7.63, 7.17, 8.05, 7.08, 8.95, 6.52, 8.39, 8.11, 7.02, 7.37, 7.42, 7.3, 7.42, 7.76, 5.67, 4.89, 5.62, 7.27, 6.71, 7.15, 9.0, 6.14, 7.54, 7.55, 6.43, 6.52, 7.16, 8.18, 6.99, 9.48, 4.87, 7.09, 6.35, 8.22, 7.4, 8.27, 6.45, 9.28, 7.01, 7.45, 8.09, 6.87, 7.09, 8.25, 8.69, 6.32, 8.55, 6.32, 9.05, 5.9, 8.45, 6.81, 6.2, 7.33, 6.35, 5.96, 7.25, 6.91, 7.48, 5.17, 9.26, 7.6, 6.53, 8.07, 7.63, 8.54, 7.65, 6.49, 6.37, 5.37, 6.86, 7.37, 7.72, 9.75, 6.41, 9.16, 6.81, 9.69, 5.77, 8.71, 5.07, 8.11, 6.86, 8.87, 6.84, 6.66, 7.95, 7.25, 5.91, 8.28, 9.63, 7.74, 6.89, 8.07, 7.86, 7.68, 7.6, 7.18, 8.35, 7.71, 9.06, 6.09, 7.8, 8.25, 6.48, 7.44, 7.51, 7.98, 5.25, 7.61, 6.54, 8.71, 6.92, 6.31, 7.22, 8.67, 8.05, 8.35, 10.68, 8.02, 7.93, 6.64, 8.5, 8.31, 8.17, 6.5, 8.78, 7.41, 7.81, 7.24, 7.56, 7.73, 6.79, 7.76, 6.43, 6.88, 9.05, 5.51, 7.02, 9.57, 7.83, 6.67, 8.13, 9.32, 6.33, 7.87, 7.47, 7.14, 7.37, 7.11, 8.82, 7.38, 7.14, 6.55, 8.21, 5.67, 7.79, 6.9, 7.12, 7.62, 7.92, 6.62, 7.57, 7.91, 5.77, 9.39, 9.51, 6.49, 8.17, 8.8, 8.48, 7.67, 7.02, 6.72, 5.95, 6.95, 5.21, 7.45, 6.89, 6.17, 7.92, 8.38, 8.05, 7.6, 6.72, 6.75, 6.9, 6.8, 6.45, 7.73, 7.45, 8.46, 10.85, 6.54, 7.68, 9.74, 8.29, 5.31, 6.34, 8.52, 7.12, 7.08, 6.27, 9.29, 8.12, 8.31, 7.4, 9.06, 6.43, 7.42, 8.0, 7.11, 9.28, 8.32, 7.84, 7.54, 6.83, 8.46, 8.64, 8.62, 8.39, 5.87, 6.97, 7.03, 7.58, 7.62, 7.66, 6.39, 6.81, 7.54, 6.96, 6.95, 7.53, 7.48, 9.32, 8.97, 6.94, 7.44, 6.85, 8.17, 7.42, 7.89, 6.9, 7.57, 9.09, 7.4, 7.05, 8.13, 8.6, 8.57, 8.51, 6.53, 6.46, 7.15, 5.43, 8.28, 7.37, 7.31, 5.58, 7.19, 8.4, 7.69, 7.99, 8.14, 6.95, 8.89, 6.89, 8.29, 6.88, 7.14, 9.53, 6.39, 9.38, 6.35, 7.9, 8.71, 7.55, 6.59, 7.78, 6.04, 8.05, 9.19, 7.16, 6.82, 6.91, 8.66, 9.14, 7.45, 9.46, 7.13, 8.95, 8.63, 7.85, 6.62, 7.71, 7.35, 8.93, 6.98, 7.43, 6.46, 9.26, 8.89, 9.27, 8.36, 6.07, 8.47, 5.74, 7.29, 7.73, 7.12, 8.51, 7.19, 8.35, 6.92, 7.76, 7.96, 7.68, 9.11, 9.1, 7.54, 7.7, 8.1, 9.36, 9.03, 8.29, 6.31, 6.0, 8.2, 7.15, 9.47, 8.86, 5.79, 8.53, 8.39, 7.49, 6.91, 8.57, 5.66, 6.91, 7.51, 8.15, 7.65, 7.26, 7.33, 5.25, 6.25, 6.11, 8.35, 6.31, 8.37, 7.67, 7.43, 9.71, 8.8, 7.09, 8.95, 8.38, 5.87, 7.98, 6.77, 8.44, 7.7, 7.87, 7.23, 7.11, 7.07, 6.5, 7.58, 5.62, 8.32, 8.49, 8.78, 4.73, 6.03, 6.07, 6.64, 8.22, 7.79, 7.76, 7.58, 7.73, 6.58, 6.03, 8.32, 8.19, 6.17, 6.54, 7.94, 7.17, 6.65, 8.55, 8.82, 7.34, 8.16, 7.48, 6.67, 7.16, 6.19, 5.72, 5.16, 6.58, 4.81, 7.19, 8.38, 8.74, 7.9, 8.5, 8.84, 5.84, 7.62, 8.2, 6.75, 6.39, 7.09, 8.99, 8.71, 9.58, 7.76, 7.27, 6.96, 7.57, 8.71, 7.81, 7.97, 8.52, 7.84, 8.16, 7.1, 7.99, 6.7, 7.95, 6.42, 7.71, 8.93, 8.7, 8.67, 8.32, 7.43, 6.27, 8.24, 7.72, 8.22, 7.11, 8.33, 7.67, 9.12, 8.14, 7.94, 7.33, 8.0, 8.35, 6.55, 9.01, 7.24, 7.39, 8.05, 7.12, 5.5, 9.94, 7.43, 7.73, 8.19, 8.14, 7.51, 7.75, 8.18, 6.56, 7.84, 7.39, 9.73, 8.16, 6.14, 7.9, 6.17, 6.94, 6.42, 5.93, 7.23, 7.67, 6.25, 7.88, 7.81, 8.51, 4.74, 6.9, 7.66, 7.6, 9.41, 7.42, 7.31, 5.84, 9.09, 7.25, 5.79, 8.15, 6.75, 9.2, 7.67, 6.29, 6.91, 7.68, 5.76, 6.65, 6.8, 7.97, 8.09, 8.4, 8.49, 6.96, 8.38, 7.78, 8.03, 8.55, 7.78, 6.47, 6.36, 8.13, 8.91, 8.28, 5.2, 6.78, 8.31, 7.46, 7.42, 7.83, 7.47, 6.74, 8.13, 9.11, 8.24, 9.56, 6.95, 5.15, 8.33, 8.36, 8.75, 7.4, 7.95, 5.43, 7.37, 7.02, 7.01, 7.89, 7.73, 6.94, 8.96, 7.8, 7.78, 6.86, 9.19, 9.36, 6.93, 7.48, 8.35, 8.19, 5.39, 6.54, 6.05, 6.32, 8.83, 6.19, 4.84, 7.25, 7.53, 8.31, 5.82, 6.81, 8.27, 6.77, 8.04, 7.44, 6.64, 7.54, 8.67, 8.66, 8.22, 7.06, 6.66, 6.32, 7.49, 8.1, 6.68, 7.04, 8.39, 8.21, 7.6, 7.31, 8.97, 7.95, 6.29, 8.37, 7.68, 6.68, 6.77, 8.7, 7.13, 7.97, 7.46, 9.11, 8.84, 9.06, 6.38, 6.37, 6.71, 6.05, 6.14, 8.55, 6.76, 6.12, 7.97, 6.08, 8.32, 7.69, 6.14, 7.89, 6.58, 7.36, 8.37, 7.47, 8.94, 8.14, 7.23, 6.83, 7.53, 7.64, 6.71, 8.16, 8.33, 9.31, 7.38, 6.23]；
组2 n=855：[10.84, 8.24, 8.32, 8.12, 8.27, 8.96, 9.83, 10.21, 9.59, 9.64, 8.08, 9.67, 10.79, 8.47, 9.63, 8.81, 8.05, 11.16, 8.34, 8.49, 8.98, 10.24, 8.62, 9.7, 10.22, 8.88, 9.02, 8.4, 9.31, 9.36, 9.51, 10.53, 8.92, 7.89, 10.88, 9.36, 8.35, 10.11, 9.41, 8.99, 8.5, 9.82, 7.54, 8.8, 10.68, 9.81, 9.97, 9.11, 8.45, 8.68, 9.75, 9.76, 9.82, 12.6, 11.24, 9.39, 9.87, 11.53, 11.97, 10.33, 9.3, 8.6, 11.66, 10.77, 8.44, 9.62, 10.04, 6.4, 9.83, 9.57, 8.93, 7.89, 9.17, 7.94, 8.6, 10.55, 9.72, 9.27, 9.14, 10.17, 9.65, 9.58, 8.71, 9.23, 8.75, 8.05, 10.49, 10.03, 11.45, 10.14, 9.5, 9.35, 9.15, 11.14, 10.23, 9.91, 9.71, 9.2, 9.69, 10.25, 10.6, 11.03, 9.36, 9.71, 8.65, 11.11, 6.87, 8.51, 8.87, 9.93, 8.79, 11.45, 9.95, 8.16, 9.27, 10.87, 6.5, 10.15, 8.77, 8.21, 8.73, 8.76, 10.75, 10.21, 8.17, 9.61, 10.41, 9.64, 9.19, 9.4, 9.02, 9.19, 10.18, 11.6, 8.82, 8.69, 10.08, 9.37, 8.87, 9.81, 8.06, 9.67, 10.54, 10.12, 8.91, 8.89, 9.21, 11.55, 9.46, 8.49, 10.43, 9.25, 11.63, 9.92, 8.6, 7.17, 10.81, 11.47, 11.61, 10.63, 9.5, 8.61, 10.15, 9.71, 11.35, 9.04, 7.83, 9.38, 8.24, 7.23, 10.27, 10.88, 11.18, 10.49, 11.23, 10.74, 10.65, 8.59, 10.47, 9.38, 10.22, 11.09, 8.68, 9.27, 7.67, 9.43, 9.26, 9.91, 9.85, 9.8, 8.45, 10.45, 9.76, 9.51, 10.2, 10.83, 10.67, 8.63, 9.29, 10.46, 10.74, 10.64, 9.74, 10.58, 7.85, 8.25, 10.25, 8.63, 10.25, 11.74, 8.19, 9.03, 10.94, 8.37, 11.18, 7.73, 9.65, 10.38, 8.55, 9.98, 9.66, 9.95, 8.8, 8.1, 10.51, 9.58, 10.78, 9.4, 8.56, 8.31, 10.09, 8.86, 8.49, 9.89, 11.37, 11.02, 7.25, 7.87, 11.52, 10.13, 9.09, 7.61, 8.21, 10.38, 11.72, 9.16, 7.79, 9.36, 10.77, 10.12, 10.72, 8.65, 9.81, 10.72, 10.25, 9.65, 9.46, 9.32, 9.46, 9.75, 8.18, 8.61, 8.24, 9.75, 8.24, 11.45, 10.21, 10.36, 10.8, 13.25, 9.89, 8.64, 10.74, 7.91, 10.26, 11.58, 9.41, 10.72, 10.16, 9.35, 7.78, 8.92, 8.58, 7.88, 10.13, 9.76, 9.14, 9.69, 9.72, 8.31, 8.82, 9.9, 9.81, 9.78, 10.91, 8.44, 9.49, 9.89, 9.52, 8.33, 9.17, 12.47, 9.52, 9.3, 9.92, 8.92, 9.52, 9.17, 8.1, 9.42, 9.57, 10.86, 8.98, 6.7, 9.14, 9.28, 10.27, 9.82, 9.68, 8.18, 10.1, 10.05, 9.67, 8.55, 10.7, 9.41, 9.7, 8.31, 9.68, 8.79, 8.8, 7.71, 9.51, 11.0, 10.06, 9.7, 9.62, 8.57, 9.96, 8.18, 7.32, 10.88, 9.51, 10.9, 10.0, 8.7, 9.3, 10.05, 8.31, 10.37, 11.03, 8.29, 9.45, 8.32, 10.97, 11.24, 7.93, 7.53, 8.84, 10.56, 9.14, 10.6, 12.74, 8.52, 10.8, 9.82, 10.59, 7.69, 7.92, 9.25, 10.6, 9.83, 7.12, 8.09, 9.17, 9.52, 11.09, 12.47, 9.83, 7.76, 8.67, 8.77, 7.98, 10.57, 9.37, 9.47, 13.88, 8.97, 9.56, 9.49, 8.61, 7.6, 8.61, 8.66, 8.95, 8.65, 7.93, 9.28, 9.77, 8.99, 8.4, 10.76, 8.1, 10.89, 9.01, 10.32, 9.29, 10.79, 11.32, 9.46, 7.84, 8.15, 10.49, 10.41, 7.56, 9.96, 8.51, 9.11, 10.05, 8.1, 8.35, 9.63, 9.7, 7.51, 11.32, 8.59, 11.65, 11.04, 7.24, 10.15, 10.4, 12.93, 9.22, 8.47, 8.65, 10.15, 8.28, 11.0, 8.92, 9.07, 8.68, 10.35, 8.34, 9.2, 9.17, 9.16, 11.15, 9.38, 9.8, 12.26, 9.79, 10.52, 10.51, 8.76, 9.82, 9.53, 10.97, 11.04, 10.27, 9.62, 9.58, 9.88, 9.62, 9.18, 8.48, 9.44, 9.55, 12.77, 10.94, 9.01, 10.15, 8.01, 9.54, 8.07, 9.62, 11.11, 8.5, 11.35, 8.42, 7.04, 11.29, 9.5, 9.58, 9.33, 10.17, 7.65, 11.16, 9.45, 9.74, 10.53, 8.6, 10.98, 8.07, 8.82, 9.44, 10.34, 7.24, 8.49, 10.91, 8.65, 10.15, 9.19, 11.3, 7.51, 10.12, 8.17, 10.49, 9.95, 9.92, 9.06, 10.54, 10.89, 9.04, 8.52, 11.04, 8.76, 9.41, 11.54, 6.94, 9.95, 10.19, 7.55, 9.26, 9.2, 10.69, 10.53, 8.38, 8.75, 9.94, 8.1, 6.93, 9.59, 9.14, 8.9, 11.17, 8.73, 9.85, 10.57, 9.68, 9.19, 7.68, 9.08, 9.58, 8.81, 9.44, 8.9, 10.43, 11.51, 10.16, 9.16, 10.92, 11.05, 9.78, 9.89, 8.53, 11.8, 10.15, 9.15, 8.32, 9.55, 9.14, 11.26, 7.46, 10.2, 8.95, 10.56, 10.89, 9.18, 10.18, 9.55, 7.75, 9.82, 9.73, 10.58, 8.64, 8.64, 10.13, 8.77, 8.44, 9.69, 9.49, 8.65, 10.13, 8.95, 10.48, 7.48, 10.8, 7.68, 11.78, 10.31, 10.39, 7.52, 9.94, 10.0, 9.51, 9.6, 9.84, 8.07, 7.93, 9.05, 12.09, 9.88, 10.12, 8.23, 7.63, 10.1, 10.14, 8.9, 9.67, 9.16, 10.6, 10.07, 8.6, 8.66, 8.24, 10.24, 10.3, 10.35, 11.66, 9.09, 7.25, 11.08, 9.76, 9.21, 10.04, 8.94, 8.18, 11.27, 9.23, 10.78, 8.54, 9.81, 8.74, 9.88, 9.28, 9.9, 8.96, 10.92, 9.17, 11.14, 9.6, 8.74, 10.27, 8.72, 10.01, 7.78, 9.85, 10.14, 8.93, 7.83, 10.42, 9.47, 8.2, 9.25, 7.82, 11.86, 9.56, 9.58, 7.6, 10.24, 9.12, 8.62, 9.1, 8.98, 10.25, 11.68, 9.42, 11.49, 9.42, 8.54, 8.85, 10.41, 10.09, 6.43, 9.83, 9.57, 9.93, 8.15, 10.42, 10.85, 9.68, 11.18, 10.87, 10.09, 9.43, 8.6, 8.56, 7.89, 9.24, 10.22, 9.81, 9.11, 8.61, 8.52, 8.75, 10.05, 10.98, 11.45, 11.14, 11.48, 9.38, 10.19, 9.13, 9.01, 8.86, 9.63, 10.57, 7.96, 8.82, 7.41, 9.42, 9.92, 10.71, 9.95, 10.99, 9.05, 9.31, 8.28, 8.24, 9.24, 7.27, 6.58, 8.73, 10.63, 10.95, 9.19, 9.48, 10.68, 9.38, 9.29, 8.75, 10.26, 10.78, 10.27, 10.51, 10.01, 8.88, 9.37, 9.27, 9.09, 8.28, 11.42, 10.91, 10.52, 10.7, 8.9, 9.16, 12.34, 10.65, 9.07, 8.41, 9.23, 10.35, 10.15, 9.06, 9.21, 8.62, 11.13, 9.81, 10.32, 10.9, 8.05, 8.99, 9.03, 9.46, 9.29, 8.84, 11.65, 10.67, 10.75, 10.59, 9.6, 10.88, 8.96, 9.58, 10.73, 10.08, 10.72, 7.71, 11.3, 9.33, 8.84, 9.6, 9.35, 10.65, 9.15, 10.04, 10.29, 10.32, 9.7, 9.03, 10.97, 10.23, 9.82, 10.6, 10.0, 9.01, 7.98, 10.75, 8.46, 9.57, 8.67, 9.51, 9.31, 9.75, 9.06, 7.9, 8.33, 8.86, 8.51, 7.46, 7.67, 9.72, 8.91, 9.82, 10.09, 6.53, 9.37, 8.77, 9.62, 8.28, 8.73, 9.33, 9.96, 7.84, 11.32, 10.18, 8.3, 9.68, 12.02, 10.5, 10.32, 9.15, 10.36, 11.36, 10.2, 11.26, 10.59, 9.85, 9.96, 10.11, 9.64, 9.54, 12.36, 8.32, 8.71, 9.81, 9.51, 9.81, 9.17, 8.68, 8.92, 9.56, 9.17, 8.25]
组2 n=855：[10.84, 8.24, 8.32, 8.12, 8.27, 8.96, 9.83, 10.21, 9.59, 9.64, 8.08, 9.67, 10.79, 8.47, 9.63, 8.81, 8.05, 11.16, 8.34, 8.49, 8.98, 10.24, 8.62, 9.7, 10.22, 8.88, 9.02, 8.4, 9.31, 9.36, 9.51, 10.53, 8.92, 7.89, 10.88, 9.36, 8.35, 10.11, 9.41, 8.99, 8.5, 9.82, 7.54, 8.8, 10.68, 9.81, 9.97, 9.11, 8.45, 8.68, 9.75, 9.76, 9.82, 12.6, 11.24, 9.39, 9.87, 11.53, 11.97, 10.33, 9.3, 8.6, 11.66, 10.77, 8.44, 9.62, 10.04, 6.4, 9.83, 9.57, 8.93, 7.89, 9.17, 7.94, 8.6, 10.55, 9.72, 9.27, 9.14, 10.17, 9.65, 9.58, 8.71, 9.23, 8.75, 8.05, 10.49, 10.03, 11.45, 10.14, 9.5, 9.35, 9.15, 11.14, 10.23, 9.91, 9.71, 9.2, 9.69, 10.25, 10.6, 11.03, 9.36, 9.71, 8.65, 11.11, 6.87, 8.51, 8.87, 9.93, 8.79, 11.45, 9.95, 8.16, 9.27, 10.87, 6.5, 10.15, 8.77, 8.21, 8.73, 8.76, 10.75, 10.21, 8.17, 9.61, 10.41, 9.64, 9.19, 9.4, 9.02, 9.19, 10.18, 11.6, 8.82, 8.69, 10.08, 9.37, 8.87, 9.81, 8.06, 9.67, 10.54, 10.12, 8.91, 8.89, 9.21, 11.55, 9.46, 8.49, 10.43, 9.25, 11.63, 9.92, 8.6, 7.17, 10.81, 11.47, 11.61, 10.63, 9.5, 8.61, 10.15, 9.71, 11.35, 9.04, 7.83, 9.38, 8.24, 7.23, 10.27, 10.88, 11.18, 10.49, 11.23, 10.74, 10.65, 8.59, 10.47, 9.38, 10.22, 11.09, 8.68, 9.27, 7.67, 9.43, 9.26, 9.91, 9.85, 9.8, 8.45, 10.45, 9.76, 9.51, 10.2, 10.83, 10.67, 8.63, 9.29, 10.46, 10.74, 10.64, 9.74, 10.58, 7.85, 8.25, 10.25, 8.63, 10.25, 11.74, 8.19, 9.03, 10.94, 8.37, 11.18, 7.73, 9.65, 10.38, 8.55, 9.98, 9.66, 9.95, 8.8, 8.1, 10.51, 9.58, 10.78, 9.4, 8.56, 8.31, 10.09, 8.86, 8.49, 9.89, 11.37, 11.02, 7.25, 7.87, 11.52, 10.13, 9.09, 7.61, 8.21, 10.38, 11.72, 9.16, 7.79, 9.36, 10.77, 10.12, 10.72, 8.65, 9.81, 10.72, 10.25, 9.65, 9.46, 9.32, 9.46, 9.75, 8.18, 8.61, 8.24, 9.75, 8.24, 11.45, 10.21, 10.36, 10.8, 13.25, 9.89, 8.64, 10.74, 7.91, 10.26, 11.58, 9.41, 10.72, 10.16, 9.35, 7.78, 8.92, 8.58, 7.88, 10.13, 9.76, 9.14, 9.69, 9.72, 8.31, 8.82, 9.9, 9.81, 9.78, 10.91, 8.44, 9.49, 9.89, 9.52, 8.33, 9.17, 12.47, 9.52, 9.3, 9.92, 8.92, 9.52, 9.17, 8.1, 9.42, 9.57, 10.86, 8.98, 6.7, 9.14, 9.28, 10.27, 9.82, 9.68, 8.18, 10.1, 10.05, 9.67, 8.55, 10.7, 9.41, 9.7, 8.31, 9.68, 8.79, 8.8, 7.71, 9.51, 11.0, 10.06, 9.7, 9.62, 8.57, 9.96, 8.18, 7.32, 10.88, 9.51, 10.9, 10.0, 8.7, 9.3, 10.05, 8.31, 10.37, 11.03, 8.29, 9.45, 8.32, 10.97, 11.24, 7.93, 7.53, 8.84, 10.56, 9.14, 10.6, 12.74, 8.52, 10.8, 9.82, 10.59, 7.69, 7.92, 9.25, 10.6, 9.83, 7.12, 8.09, 9.17, 9.52, 11.09, 12.47, 9.83, 7.76, 8.67, 8.77, 7.98, 10.57, 9.37, 9.47, 13.88, 8.97, 9.56, 9.49, 8.61, 7.6, 8.61, 8.66, 8.95, 8.65, 7.93, 9.28, 9.77, 8.99, 8.4, 10.76, 8.1, 10.89, 9.01, 10.32, 9.29, 10.79, 11.32, 9.46, 7.84, 8.15, 10.49, 10.41, 7.56, 9.96, 8.51, 9.11, 10.05, 8.1, 8.35, 9.63, 9.7, 7.51, 11.32, 8.59, 11.65, 11.04, 7.24, 10.15, 10.4, 12.93, 9.22, 8.47, 8.65, 10.15, 8.28, 11.0, 8.92, 9.07, 8.68, 10.35, 8.34, 9.2, 9.17, 9.16, 11.15, 9.38, 9.8, 12.26, 9.79, 10.52, 10.51, 8.76, 9.82, 9.53, 10.97, 11.04, 10.27, 9.62, 9.58, 9.88, 9.62, 9.18, 8.48, 9.44, 9.55, 12.77, 10.94, 9.01, 10.15, 8.01, 9.54, 8.07, 9.62, 11.11, 8.5, 11.35, 8.42, 7.04, 11.29, 9.5, 9.58, 9.33, 10.17, 7.65, 11.16, 9.45, 9.74, 10.53, 8.6, 10.98, 8.07, 8.82, 9.44, 10.34, 7.24, 8.49, 10.91, 8.65, 10.15, 9.19, 11.3, 7.51, 10.12, 8.17, 10.49, 9.95, 9.92, 9.06, 10.54, 10.89, 9.04, 8.52, 11.04, 8.76, 9.41, 11.54, 6.94, 9.95, 10.19, 7.55, 9.26, 9.2, 10.69, 10.53, 8.38, 8.75, 9.94, 8.1, 6.93, 9.59, 9.14, 8.9, 11.17, 8.73, 9.85, 10.57, 9.68, 9.19, 7.68, 9.08, 9.58, 8.81, 9.44, 8.9, 10.43, 11.51, 10.16, 9.16, 10.92, 11.05, 9.78, 9.89, 8.53, 11.8, 10.15, 9.15, 8.32, 9.55, 9.14, 11.26, 7.46, 10.2, 8.95, 10.56, 10.89, 9.18, 10.18, 9.55, 7.75, 9.82, 9.73, 10.58, 8.64, 8.64, 10.13, 8.77, 8.44, 9.69, 9.49, 8.65, 10.13, 8.95, 10.48, 7.48, 10.8, 7.68, 11.78, 10.31, 10.39, 7.52, 9.94, 10.0, 9.51, 9.6, 9.84, 8.07, 7.93, 9.05, 12.09, 9.88, 10.12, 8.23, 7.63, 10.1, 10.14, 8.9, 9.67, 9.16, 10.6, 10.07, 8.6, 8.66, 8.24, 10.24, 10.3, 10.35, 11.66, 9.09, 7.25, 11.08, 9.76, 9.21, 10.04, 8.94, 8.18, 11.27, 9.23, 10.78, 8.54, 9.81, 8.74, 9.88, 9.28, 9.9, 8.96, 10.92, 9.17, 11.14, 9.6, 8.74, 10.27, 8.72, 10.01, 7.78, 9.85, 10.14, 8.93, 7.83, 10.42, 9.47, 8.2, 9.25, 7.82, 11.86, 9.56, 9.58, 7.6, 10.24, 9.12, 8.62, 9.1, 8.98, 10.25, 11.68, 9.42, 11.49, 9.42, 8.54, 8.85, 10.41, 10.09, 6.43, 9.83, 9.57, 9.93, 8.15, 10.42, 10.85, 9.68, 11.18, 10.87, 10.09, 9.43, 8.6, 8.56, 7.89, 9.24, 10.22, 9.81, 9.11, 8.61, 8.52, 8.75, 10.05, 10.98, 11.45, 11.14, 11.48, 9.38, 10.19, 9.13, 9.01, 8.86, 9.63, 10.57, 7.96, 8.82, 7.41, 9.42, 9.92, 10.71, 9.95, 10.99, 9.05, 9.31, 8.28, 8.24, 9.24, 7.27, 6.58, 8.73, 10.63, 10.95, 9.19, 9.48, 10.68, 9.38, 9.29, 8.75, 10.26, 10.78, 10.27, 10.51, 10.01, 8.88, 9.37, 9.27, 9.09, 8.28, 11.42, 10.91, 10.52, 10.7, 8.9, 9.16, 12.34, 10.65, 9.07, 8.41, 9.23, 10.35, 10.15, 9.06, 9.21, 8.62, 11.13, 9.81, 10.32, 10.9, 8.05, 8.99, 9.03, 9.46, 9.29, 8.84, 11.65, 10.67, 10.75, 10.59, 9.6, 10.88, 8.96, 9.58, 10.73, 10.08, 10.72, 7.71, 11.3, 9.33, 8.84, 9.6, 9.35, 10.65, 9.15, 10.04, 10.29, 10.32, 9.7, 9.03, 10.97, 10.23, 9.82, 10.6, 10.0, 9.01, 7.98, 10.75, 8.46, 9.57, 8.67, 9.51, 9.31, 9.75, 9.06, 7.9, 8.33, 8.86, 8.51, 7.46, 7.67, 9.72, 8.91, 9.82, 10.09, 6.53, 9.37, 8.77, 9.62, 8.28, 8.73, 9.33, 9.96, 7.84, 11.32, 10.18, 8.3, 9.68, 12.02, 10.5, 10.32, 9.15, 10.36, 11.36, 10.2, 11.26, 10.59, 9.85, 9.96, 10.11, 9.64, 9.54, 12.36, 8.32, 8.71, 9.81, 9.51, 9.81, 9.17, 8.68, 8.92, 9.56, 9.17, 8.25]
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（Welch）。
x̄₁=7.5309, s₁=1.0719；x̄₂=9.5443, s₂=1.1202；
SE=√(s₁²/n₁+s₂²/n₂)=0.0555；t=(x̄₁−x̄₂)/SE=-36.2703；
df(Welch)=1535.20；双侧 p = 1.419e-208。
x̄₁=7.5309, s₁=1.0719；x̄₂=9.5443, s₂=1.1202；
SE=√(s₁²/n₁+s₂²/n₂)=0.0555；t=(x̄₁−x̄₂)/SE=-36.2703；
df(Welch)=1535.20；双侧 p = 1.419e-208。
计算结果
t = -36.2703，df = 1535.20，p = 1.419e-208
在 α=0.05 水平下，p=1.419e-208 < 0.05，拒绝原假设 H₀。比较医院治疗效率。
配对样本 t 检验
案例 36. 训练前后体重
原始数据
前 n=551：[76.09, 78.76, 84.71, 78.49, 79.64, 81.44, 83.31, 80.26, 80.84, 80.78, 78.21, 80.15, 71.55, 82.62, 78.56, 84.65, 79.68, 77.08, 81.41, 79.02, 78.69, 78.42, 78.93, 81.97, 76.59, 79.46, 78.65, 76.63, 81.09, 78.75, 73.72, 75.24, 84.4, 81.49, 77.39, 81.28, 80.24, 79.25, 81.71, 79.81, 79.63, 74.57, 79.37, 77.14, 79.33, 80.81, 81.36, 76.82, 76.54, 75.13, 82.95, 77.66, 76.63, 76.57, 80.3, 75.28, 76.41, 72.83, 76.52, 83.56, 77.3, 74.71, 78.26, 83.28, 75.73, 83.26, 77.52, 77.56, 76.15, 78.54, 81.96, 78.78, 81.49, 76.15, 79.14, 78.97, 82.01, 80.57, 77.67, 77.3, 77.2, 81.6, 80.68, 81.34, 81.75, 79.69, 75.48, 81.33, 77.86, 80.11, 79.06, 81.53, 79.95, 81.81, 78.58, 80.93, 78.55, 80.23, 73.18, 77.52, 75.24, 75.71, 77.14, 76.05, 80.77, 80.53, 73.48, 77.54, 79.18, 78.27, 75.07, 79.63, 76.3, 77.76, 82.73, 76.74, 77.05, 77.59, 77.91, 78.63, 75.65, 81.6, 78.5, 75.76, 74.34, 77.89, 78.24, 76.86, 79.19, 81.92, 79.79, 78.85, 78.79, 79.4, 78.75, 79.9, 79.05, 80.11, 81.2, 75.78, 78.97, 75.49, 75.4, 80.25, 77.77, 80.49, 81.94, 74.83, 76.22, 80.77, 80.73, 77.05, 84.24, 79.94, 80.1, 76.06, 84.19, 76.7, 82.02, 80.92, 74.61, 85.37, 79.4, 82.31, 80.84, 77.67, 78.71, 79.56, 76.71, 80.55, 81.92, 76.64, 82.29, 86.29, 78.62, 80.51, 76.83, 79.49, 80.37, 81.11, 83.2, 81.11, 76.61, 72.25, 77.81, 80.36, 78.38, 81.12, 76.37, 80.71, 81.84, 77.7, 79.27, 74.71, 78.49, 77.18, 79.12, 79.73, 77.84, 76.26, 75.62, 78.95, 81.39, 76.31, 83.23, 81.26, 83.57, 80.03, 79.97, 77.32, 80.63, 81.88, 76.94, 77.21, 76.75, 75.57, 83.3, 75.53, 73.57, 75.22, 79.63, 79.51, 76.8, 79.1, 82.67, 77.98, 76.35, 77.89, 81.93, 79.55, 78.48, 78.15, 81.74, 77.87, 80.89, 78.5, 74.65, 78.99, 80.57, 82.39, 75.91, 79.28, 81.63, 80.97, 82.02, 76.12, 79.04, 74.54, 78.05, 77.36, 78.27, 79.81, 77.98, 78.57, 75.89, 84.71, 82.89, 78.93, 79.31, 76.61, 84.01, 77.84, 79.68, 78.59, 81.92, 82.36, 77.83, 75.35, 78.96, 80.37, 73.82, 82.0, 79.48, 73.56, 83.55, 81.19, 81.51, 77.98, 80.6, 80.56, 77.86, 73.54, 79.11, 77.49, 77.09, 79.36, 78.74, 77.96, 78.2, 79.56, 77.68, 81.55, 79.67, 77.04, 82.56, 79.66, 80.56, 78.93, 84.09, 77.79, 76.41, 83.21, 78.39, 77.4, 81.72, 81.27, 79.89, 79.95, 80.4, 71.55, 81.73, 78.52, 82.75, 82.1, 79.86, 77.49, 80.4, 79.23, 75.71, 79.6, 78.02, 75.09, 76.83, 83.45, 79.36, 78.37, 76.07, 83.22, 79.91, 75.48, 78.26, 74.39, 79.96, 75.23, 80.47, 82.07, 76.68, 74.25, 77.26, 80.18, 81.52, 75.2, 83.65, 86.98, 76.35, 74.01, 78.07, 80.02, 83.71, 81.16, 76.82, 77.63, 74.95, 81.59, 78.76, 83.43, 76.81, 76.07, 80.63, 82.63, 80.23, 81.56, 80.51, 81.05, 78.83, 77.96, 79.4, 75.15, 77.33, 74.6, 80.72, 78.33, 82.81, 81.06, 76.2, 77.68, 76.06, 79.3, 78.03, 79.56, 75.82, 75.22, 80.32, 82.83, 79.09, 72.32, 79.7, 78.5, 82.56, 77.23, 79.83, 76.18, 77.68, 80.15, 78.84, 81.58, 79.8, 81.13, 81.87, 74.21, 78.82, 82.97, 80.74, 79.05, 78.8, 77.09, 76.79, 77.87, 79.53, 73.9, 82.12, 76.81, 81.4, 79.15, 77.57, 79.37, 78.53, 79.2, 79.85, 77.09, 78.56, 81.3, 83.08, 83.35, 80.02, 82.77, 79.77, 82.69, 77.7, 77.75, 76.69, 73.25, 76.91, 81.01, 79.13, 81.36, 82.38, 77.01, 77.93, 79.19, 80.13, 78.95, 78.08, 81.89, 78.01, 73.64, 83.75, 73.81, 83.29, 75.49, 78.5, 79.31, 77.21, 78.49, 74.08, 80.31, 79.17, 79.52, 79.98, 81.64, 80.32, 74.33, 75.87, 81.63, 78.85, 80.28, 76.51, 76.07, 79.35, 79.03, 78.43, 78.63, 77.23, 81.63, 85.24, 81.04, 78.1, 80.59, 79.0, 75.49, 77.94, 79.31, 81.92, 78.42, 78.72, 85.02, 79.42, 83.36, 84.26, 77.94, 81.99, 77.42, 78.03, 77.09, 76.94, 80.22, 81.18, 76.99, 76.94, 81.24, 78.53, 76.43, 81.65, 79.18, 76.08, 78.85, 76.18, 78.39, 83.87, 78.02, 79.47, 79.35, 80.11, 80.75, 86.52, 83.24, 79.84, 79.62, 78.5, 80.98, 77.0, 81.45, 77.37, 83.26, 80.14, 80.47, 78.12, 78.91, 80.27, 81.19, 78.67, 78.98, 79.31, 79.6, 77.01, 80.94, 78.62, 81.57, 77.25, 81.72, 76.93, 76.89, 81.5, 80.38, 79.31, 80.98, 80.68, 78.46, 81.62, 81.19, 81.48]；
后 n=551：[72.94, 74.7, 81.75, 75.99, 77.32, 77.22, 80.8, 75.95, 78.45, 76.9, 75.37, 77.43, 68.12, 80.59, 74.71, 80.95, 76.04, 73.92, 79.6, 76.06, 74.94, 75.22, 76.3, 79.69, 72.66, 75.94, 75.57, 72.11, 77.11, 75.15, 71.69, 71.5, 81.29, 77.35, 73.99, 78.52, 76.08, 75.75, 78.02, 75.61, 76.21, 72.48, 75.95, 74.32, 76.91, 78.35, 77.88, 74.04, 73.31, 72.83, 78.01, 74.66, 73.28, 73.48, 76.01, 72.1, 72.64, 69.48, 73.28, 79.64, 74.25, 71.94, 74.54, 80.47, 72.43, 80.56, 75.3, 72.67, 74.17, 75.6, 78.82, 75.69, 79.6, 73.41, 76.24, 76.33, 78.35, 78.25, 74.41, 74.68, 73.13, 79.04, 75.92, 77.13, 78.07, 77.03, 72.58, 77.99, 73.15, 76.55, 76.1, 79.34, 76.99, 77.58, 74.51, 76.79, 74.99, 75.84, 70.35, 74.66, 72.57, 71.74, 73.72, 74.03, 78.67, 76.44, 69.81, 73.47, 75.47, 74.72, 72.48, 77.0, 73.26, 74.19, 78.08, 73.25, 74.01, 74.72, 74.08, 76.07, 72.41, 77.55, 76.2, 72.32, 70.64, 75.55, 75.63, 72.68, 75.86, 78.82, 76.66, 75.33, 74.89, 75.57, 75.66, 76.59, 75.85, 76.95, 77.6, 72.69, 76.31, 72.88, 72.14, 76.61, 74.56, 76.35, 78.71, 71.26, 73.35, 76.76, 77.5, 73.81, 79.85, 76.69, 76.89, 72.48, 81.26, 74.02, 79.68, 78.32, 71.51, 82.66, 75.14, 79.5, 76.87, 74.51, 75.08, 76.52, 72.89, 77.0, 79.75, 73.92, 78.75, 82.81, 75.07, 77.48, 74.2, 77.12, 75.95, 79.11, 79.44, 77.71, 72.39, 68.3, 73.92, 77.09, 75.25, 78.3, 72.21, 79.38, 77.41, 74.2, 76.77, 73.17, 75.06, 74.81, 75.19, 75.99, 75.19, 73.67, 72.27, 75.87, 78.31, 73.84, 79.06, 77.74, 80.04, 76.82, 75.95, 75.37, 76.95, 78.19, 73.31, 73.75, 72.66, 72.34, 80.05, 72.25, 68.35, 72.49, 76.85, 76.07, 73.11, 75.82, 79.9, 74.58, 73.5, 74.77, 78.01, 75.61, 75.26, 73.32, 78.6, 74.37, 78.43, 74.84, 71.64, 76.11, 77.11, 79.41, 71.89, 76.42, 78.24, 78.2, 77.54, 72.92, 75.91, 70.98, 74.48, 72.9, 74.34, 76.48, 73.73, 75.06, 73.28, 82.24, 79.34, 73.67, 75.66, 72.54, 80.53, 75.15, 76.62, 76.18, 77.96, 78.25, 74.64, 72.63, 75.57, 76.75, 70.96, 79.45, 75.62, 69.62, 80.95, 78.66, 77.17, 74.84, 77.36, 77.4, 75.32, 71.21, 75.8, 73.72, 74.62, 75.55, 75.92, 76.11, 74.69, 76.48, 75.19, 77.51, 77.04, 72.67, 79.97, 76.67, 76.73, 74.2, 79.79, 74.4, 73.04, 79.83, 75.04, 74.92, 78.02, 78.23, 76.31, 75.65, 77.63, 68.51, 79.21, 75.25, 79.94, 77.88, 76.72, 74.15, 77.65, 75.97, 72.18, 76.27, 73.53, 71.86, 73.2, 80.39, 75.63, 74.46, 74.08, 80.49, 76.06, 71.98, 75.06, 70.82, 77.52, 71.29, 76.57, 79.2, 74.77, 71.15, 73.14, 76.83, 78.54, 71.82, 79.79, 84.57, 73.23, 70.44, 74.97, 76.53, 82.04, 78.41, 73.54, 73.6, 71.29, 79.32, 74.88, 80.64, 72.73, 73.57, 77.7, 79.59, 76.58, 78.23, 77.58, 79.05, 75.42, 73.73, 76.49, 71.18, 74.79, 71.54, 76.41, 74.6, 79.38, 77.83, 72.22, 74.9, 72.61, 76.2, 74.58, 75.22, 72.03, 70.93, 77.78, 79.34, 76.78, 69.24, 76.93, 74.38, 79.12, 73.71, 76.99, 72.71, 73.59, 77.04, 75.35, 77.81, 77.62, 78.13, 77.24, 69.81, 75.64, 80.59, 78.33, 76.18, 74.41, 74.38, 73.72, 74.7, 76.18, 71.63, 78.49, 74.95, 78.87, 74.65, 73.99, 76.57, 75.82, 75.83, 76.0, 74.35, 76.15, 75.78, 80.45, 80.28, 75.86, 79.26, 76.51, 78.99, 74.79, 74.68, 71.76, 68.89, 72.38, 78.47, 75.43, 78.43, 79.59, 73.94, 74.5, 76.59, 76.65, 74.64, 75.2, 79.07, 75.31, 71.1, 80.95, 69.85, 80.0, 71.64, 75.11, 76.7, 73.8, 74.96, 70.96, 76.53, 76.8, 76.0, 76.29, 78.06, 75.28, 70.67, 73.2, 78.93, 75.12, 77.89, 72.25, 72.85, 76.52, 75.79, 75.7, 75.8, 74.24, 79.23, 82.57, 77.65, 74.77, 77.76, 76.75, 71.61, 73.98, 76.67, 79.17, 75.69, 75.65, 82.88, 76.68, 80.34, 80.44, 74.77, 78.08, 73.02, 74.25, 74.16, 73.75, 77.42, 77.3, 72.72, 73.14, 78.17, 75.16, 71.94, 78.3, 76.05, 72.85, 74.89, 73.04, 75.89, 80.93, 75.91, 75.24, 75.19, 75.47, 77.75, 82.55, 79.01, 76.44, 76.51, 75.1, 77.72, 74.6, 79.01, 74.96, 79.56, 76.77, 77.98, 75.78, 76.28, 76.01, 77.71, 75.53, 75.43, 76.6, 76.13, 72.78, 77.42, 75.09, 78.47, 74.01, 78.29, 73.03, 74.7, 78.03, 76.27, 75.84, 78.41, 79.33, 75.6, 78.33, 77.41, 77.22]；
差值(后−前) n=551：[-3.15, -4.06, -2.96, -2.5, -2.32, -4.21, -2.5, -4.31, -2.39, -3.88, -2.84, -2.72, -3.43, -2.03, -3.86, -3.7, -3.64, -3.17, -1.8, -2.96, -3.74, -3.21, -2.64, -2.28, -3.93, -3.52, -3.08, -4.52, -3.98, -3.6, -2.03, -3.74, -3.11, -4.14, -3.4, -2.76, -4.16, -3.49, -3.68, -4.2, -3.43, -2.08, -3.42, -2.82, -2.43, -2.46, -3.48, -2.78, -3.22, -2.3, -4.95, -3.0, -3.35, -3.08, -4.29, -3.18, -3.77, -3.34, -3.24, -3.91, -3.05, -2.76, -3.72, -2.82, -3.29, -2.7, -2.22, -4.9, -1.98, -2.94, -3.14, -3.09, -1.89, -2.74, -2.9, -2.64, -3.66, -2.32, -3.26, -2.61, -4.07, -2.56, -4.76, -4.22, -3.68, -2.66, -2.9, -3.34, -4.71, -3.56, -2.97, -2.19, -2.96, -4.23, -4.07, -4.15, -3.56, -4.39, -2.83, -2.87, -2.67, -3.97, -3.43, -2.02, -2.1, -4.09, -3.67, -4.07, -3.71, -3.55, -2.58, -2.63, -3.04, -3.57, -4.65, -3.49, -3.03, -2.87, -3.83, -2.56, -3.24, -4.05, -2.29, -3.44, -3.69, -2.34, -2.61, -4.18, -3.33, -3.1, -3.13, -3.52, -3.9, -3.84, -3.09, -3.31, -3.2, -3.17, -3.6, -3.09, -2.67, -2.61, -3.26, -3.64, -3.21, -4.14, -3.23, -3.57, -2.87, -4.01, -3.23, -3.24, -4.39, -3.25, -3.21, -3.59, -2.94, -2.68, -2.35, -2.6, -3.1, -2.71, -4.26, -2.81, -3.97, -3.15, -3.62, -3.05, -3.81, -3.55, -2.17, -2.72, -3.54, -3.48, -3.56, -3.04, -2.62, -2.37, -4.42, -2.0, -3.76, -3.4, -4.22, -3.95, -3.88, -3.26, -3.13, -2.82, -4.16, -1.34, -4.43, -3.5, -2.5, -1.54, -3.43, -2.37, -3.93, -3.74, -2.65, -2.59, -3.35, -3.08, -3.08, -2.47, -4.17, -3.52, -3.52, -3.21, -4.02, -1.95, -3.68, -3.7, -3.64, -3.46, -4.1, -3.23, -3.25, -3.28, -5.22, -2.73, -2.79, -3.44, -3.69, -3.28, -2.77, -3.4, -2.84, -3.13, -3.93, -3.94, -3.22, -4.84, -3.14, -3.5, -2.46, -3.67, -3.01, -2.88, -3.46, -2.98, -4.03, -2.86, -3.39, -2.77, -4.49, -3.21, -3.13, -3.56, -3.57, -4.46, -3.93, -3.33, -4.25, -3.51, -2.61, -2.47, -3.55, -5.26, -3.65, -4.07, -3.47, -2.69, -3.07, -2.42, -3.96, -4.11, -3.2, -2.73, -3.39, -3.62, -2.86, -2.56, -3.87, -3.94, -2.6, -2.52, -4.34, -3.14, -3.24, -3.17, -2.54, -2.32, -3.31, -3.78, -2.47, -3.8, -2.82, -1.85, -3.51, -3.08, -2.5, -4.04, -2.63, -4.37, -2.59, -2.99, -3.83, -4.72, -4.29, -3.39, -3.37, -3.38, -3.35, -2.47, -3.7, -3.04, -3.57, -4.3, -2.76, -3.04, -2.52, -3.27, -2.81, -4.22, -3.15, -3.34, -2.75, -3.26, -3.53, -3.33, -4.49, -3.23, -3.63, -3.06, -3.72, -3.91, -1.99, -2.73, -3.85, -3.5, -3.2, -3.57, -2.44, -3.94, -3.89, -2.87, -1.91, -3.11, -4.12, -3.35, -2.97, -3.38, -3.85, -2.41, -3.12, -3.56, -3.1, -3.49, -1.68, -2.76, -3.29, -4.03, -3.66, -2.26, -3.88, -2.79, -4.08, -2.5, -2.93, -3.04, -3.65, -3.33, -2.93, -2.0, -3.42, -4.24, -2.9, -3.97, -2.54, -3.07, -4.31, -3.73, -3.43, -3.24, -3.97, -2.78, -3.45, -3.1, -3.46, -4.35, -3.78, -4.28, -2.54, -3.49, -2.32, -3.08, -2.77, -4.12, -3.45, -3.52, -2.84, -3.47, -4.1, -3.11, -3.48, -3.77, -2.19, -3.01, -4.63, -4.4, -3.18, -2.38, -2.42, -2.86, -4.39, -2.71, -3.06, -3.17, -3.35, -2.27, -3.63, -1.86, -2.52, -4.5, -3.58, -2.79, -2.71, -3.37, -3.86, -2.73, -2.42, -5.52, -2.63, -3.07, -4.15, -3.5, -3.26, -3.7, -2.91, -3.07, -4.93, -4.36, -4.53, -2.54, -3.7, -2.93, -2.79, -3.07, -3.44, -2.59, -3.48, -4.31, -2.88, -2.83, -2.7, -2.53, -2.8, -3.97, -3.29, -3.86, -3.39, -2.61, -3.41, -3.53, -3.12, -3.78, -2.37, -3.52, -3.69, -3.58, -5.04, -3.66, -2.67, -2.7, -3.73, -2.39, -4.26, -3.23, -2.84, -3.23, -2.73, -2.83, -2.99, -2.39, -2.67, -3.39, -3.33, -2.83, -2.25, -3.88, -3.96, -2.65, -2.74, -2.73, -3.07, -2.15, -2.73, -3.02, -3.82, -3.17, -3.92, -4.4, -3.78, -2.93, -3.19, -2.8, -3.87, -4.28, -3.8, -3.07, -3.38, -4.5, -3.35, -3.13, -3.23, -3.96, -3.14, -2.5, -2.94, -2.11, -4.22, -4.16, -4.64, -3.0, -3.97, -4.23, -3.39, -3.11, -3.4, -3.26, -2.4, -2.44, -2.42, -3.7, -3.37, -2.49, -2.33, -2.63, -4.26, -3.48, -3.14, -3.56, -2.72, -3.47, -4.22, -3.52, -3.53, -3.1, -3.24, -3.43, -3.9, -2.19, -3.46, -4.11, -3.47, -2.57, -1.35, -2.85, -3.28, -3.79, -4.26]
后 n=551：[72.94, 74.7, 81.75, 75.99, 77.32, 77.22, 80.8, 75.95, 78.45, 76.9, 75.37, 77.43, 68.12, 80.59, 74.71, 80.95, 76.04, 73.92, 79.6, 76.06, 74.94, 75.22, 76.3, 79.69, 72.66, 75.94, 75.57, 72.11, 77.11, 75.15, 71.69, 71.5, 81.29, 77.35, 73.99, 78.52, 76.08, 75.75, 78.02, 75.61, 76.21, 72.48, 75.95, 74.32, 76.91, 78.35, 77.88, 74.04, 73.31, 72.83, 78.01, 74.66, 73.28, 73.48, 76.01, 72.1, 72.64, 69.48, 73.28, 79.64, 74.25, 71.94, 74.54, 80.47, 72.43, 80.56, 75.3, 72.67, 74.17, 75.6, 78.82, 75.69, 79.6, 73.41, 76.24, 76.33, 78.35, 78.25, 74.41, 74.68, 73.13, 79.04, 75.92, 77.13, 78.07, 77.03, 72.58, 77.99, 73.15, 76.55, 76.1, 79.34, 76.99, 77.58, 74.51, 76.79, 74.99, 75.84, 70.35, 74.66, 72.57, 71.74, 73.72, 74.03, 78.67, 76.44, 69.81, 73.47, 75.47, 74.72, 72.48, 77.0, 73.26, 74.19, 78.08, 73.25, 74.01, 74.72, 74.08, 76.07, 72.41, 77.55, 76.2, 72.32, 70.64, 75.55, 75.63, 72.68, 75.86, 78.82, 76.66, 75.33, 74.89, 75.57, 75.66, 76.59, 75.85, 76.95, 77.6, 72.69, 76.31, 72.88, 72.14, 76.61, 74.56, 76.35, 78.71, 71.26, 73.35, 76.76, 77.5, 73.81, 79.85, 76.69, 76.89, 72.48, 81.26, 74.02, 79.68, 78.32, 71.51, 82.66, 75.14, 79.5, 76.87, 74.51, 75.08, 76.52, 72.89, 77.0, 79.75, 73.92, 78.75, 82.81, 75.07, 77.48, 74.2, 77.12, 75.95, 79.11, 79.44, 77.71, 72.39, 68.3, 73.92, 77.09, 75.25, 78.3, 72.21, 79.38, 77.41, 74.2, 76.77, 73.17, 75.06, 74.81, 75.19, 75.99, 75.19, 73.67, 72.27, 75.87, 78.31, 73.84, 79.06, 77.74, 80.04, 76.82, 75.95, 75.37, 76.95, 78.19, 73.31, 73.75, 72.66, 72.34, 80.05, 72.25, 68.35, 72.49, 76.85, 76.07, 73.11, 75.82, 79.9, 74.58, 73.5, 74.77, 78.01, 75.61, 75.26, 73.32, 78.6, 74.37, 78.43, 74.84, 71.64, 76.11, 77.11, 79.41, 71.89, 76.42, 78.24, 78.2, 77.54, 72.92, 75.91, 70.98, 74.48, 72.9, 74.34, 76.48, 73.73, 75.06, 73.28, 82.24, 79.34, 73.67, 75.66, 72.54, 80.53, 75.15, 76.62, 76.18, 77.96, 78.25, 74.64, 72.63, 75.57, 76.75, 70.96, 79.45, 75.62, 69.62, 80.95, 78.66, 77.17, 74.84, 77.36, 77.4, 75.32, 71.21, 75.8, 73.72, 74.62, 75.55, 75.92, 76.11, 74.69, 76.48, 75.19, 77.51, 77.04, 72.67, 79.97, 76.67, 76.73, 74.2, 79.79, 74.4, 73.04, 79.83, 75.04, 74.92, 78.02, 78.23, 76.31, 75.65, 77.63, 68.51, 79.21, 75.25, 79.94, 77.88, 76.72, 74.15, 77.65, 75.97, 72.18, 76.27, 73.53, 71.86, 73.2, 80.39, 75.63, 74.46, 74.08, 80.49, 76.06, 71.98, 75.06, 70.82, 77.52, 71.29, 76.57, 79.2, 74.77, 71.15, 73.14, 76.83, 78.54, 71.82, 79.79, 84.57, 73.23, 70.44, 74.97, 76.53, 82.04, 78.41, 73.54, 73.6, 71.29, 79.32, 74.88, 80.64, 72.73, 73.57, 77.7, 79.59, 76.58, 78.23, 77.58, 79.05, 75.42, 73.73, 76.49, 71.18, 74.79, 71.54, 76.41, 74.6, 79.38, 77.83, 72.22, 74.9, 72.61, 76.2, 74.58, 75.22, 72.03, 70.93, 77.78, 79.34, 76.78, 69.24, 76.93, 74.38, 79.12, 73.71, 76.99, 72.71, 73.59, 77.04, 75.35, 77.81, 77.62, 78.13, 77.24, 69.81, 75.64, 80.59, 78.33, 76.18, 74.41, 74.38, 73.72, 74.7, 76.18, 71.63, 78.49, 74.95, 78.87, 74.65, 73.99, 76.57, 75.82, 75.83, 76.0, 74.35, 76.15, 75.78, 80.45, 80.28, 75.86, 79.26, 76.51, 78.99, 74.79, 74.68, 71.76, 68.89, 72.38, 78.47, 75.43, 78.43, 79.59, 73.94, 74.5, 76.59, 76.65, 74.64, 75.2, 79.07, 75.31, 71.1, 80.95, 69.85, 80.0, 71.64, 75.11, 76.7, 73.8, 74.96, 70.96, 76.53, 76.8, 76.0, 76.29, 78.06, 75.28, 70.67, 73.2, 78.93, 75.12, 77.89, 72.25, 72.85, 76.52, 75.79, 75.7, 75.8, 74.24, 79.23, 82.57, 77.65, 74.77, 77.76, 76.75, 71.61, 73.98, 76.67, 79.17, 75.69, 75.65, 82.88, 76.68, 80.34, 80.44, 74.77, 78.08, 73.02, 74.25, 74.16, 73.75, 77.42, 77.3, 72.72, 73.14, 78.17, 75.16, 71.94, 78.3, 76.05, 72.85, 74.89, 73.04, 75.89, 80.93, 75.91, 75.24, 75.19, 75.47, 77.75, 82.55, 79.01, 76.44, 76.51, 75.1, 77.72, 74.6, 79.01, 74.96, 79.56, 76.77, 77.98, 75.78, 76.28, 76.01, 77.71, 75.53, 75.43, 76.6, 76.13, 72.78, 77.42, 75.09, 78.47, 74.01, 78.29, 73.03, 74.7, 78.03, 76.27, 75.84, 78.41, 79.33, 75.6, 78.33, 77.41, 77.22]；
差值(后−前) n=551：[-3.15, -4.06, -2.96, -2.5, -2.32, -4.21, -2.5, -4.31, -2.39, -3.88, -2.84, -2.72, -3.43, -2.03, -3.86, -3.7, -3.64, -3.17, -1.8, -2.96, -3.74, -3.21, -2.64, -2.28, -3.93, -3.52, -3.08, -4.52, -3.98, -3.6, -2.03, -3.74, -3.11, -4.14, -3.4, -2.76, -4.16, -3.49, -3.68, -4.2, -3.43, -2.08, -3.42, -2.82, -2.43, -2.46, -3.48, -2.78, -3.22, -2.3, -4.95, -3.0, -3.35, -3.08, -4.29, -3.18, -3.77, -3.34, -3.24, -3.91, -3.05, -2.76, -3.72, -2.82, -3.29, -2.7, -2.22, -4.9, -1.98, -2.94, -3.14, -3.09, -1.89, -2.74, -2.9, -2.64, -3.66, -2.32, -3.26, -2.61, -4.07, -2.56, -4.76, -4.22, -3.68, -2.66, -2.9, -3.34, -4.71, -3.56, -2.97, -2.19, -2.96, -4.23, -4.07, -4.15, -3.56, -4.39, -2.83, -2.87, -2.67, -3.97, -3.43, -2.02, -2.1, -4.09, -3.67, -4.07, -3.71, -3.55, -2.58, -2.63, -3.04, -3.57, -4.65, -3.49, -3.03, -2.87, -3.83, -2.56, -3.24, -4.05, -2.29, -3.44, -3.69, -2.34, -2.61, -4.18, -3.33, -3.1, -3.13, -3.52, -3.9, -3.84, -3.09, -3.31, -3.2, -3.17, -3.6, -3.09, -2.67, -2.61, -3.26, -3.64, -3.21, -4.14, -3.23, -3.57, -2.87, -4.01, -3.23, -3.24, -4.39, -3.25, -3.21, -3.59, -2.94, -2.68, -2.35, -2.6, -3.1, -2.71, -4.26, -2.81, -3.97, -3.15, -3.62, -3.05, -3.81, -3.55, -2.17, -2.72, -3.54, -3.48, -3.56, -3.04, -2.62, -2.37, -4.42, -2.0, -3.76, -3.4, -4.22, -3.95, -3.88, -3.26, -3.13, -2.82, -4.16, -1.34, -4.43, -3.5, -2.5, -1.54, -3.43, -2.37, -3.93, -3.74, -2.65, -2.59, -3.35, -3.08, -3.08, -2.47, -4.17, -3.52, -3.52, -3.21, -4.02, -1.95, -3.68, -3.7, -3.64, -3.46, -4.1, -3.23, -3.25, -3.28, -5.22, -2.73, -2.79, -3.44, -3.69, -3.28, -2.77, -3.4, -2.84, -3.13, -3.93, -3.94, -3.22, -4.84, -3.14, -3.5, -2.46, -3.67, -3.01, -2.88, -3.46, -2.98, -4.03, -2.86, -3.39, -2.77, -4.49, -3.21, -3.13, -3.56, -3.57, -4.46, -3.93, -3.33, -4.25, -3.51, -2.61, -2.47, -3.55, -5.26, -3.65, -4.07, -3.47, -2.69, -3.07, -2.42, -3.96, -4.11, -3.2, -2.73, -3.39, -3.62, -2.86, -2.56, -3.87, -3.94, -2.6, -2.52, -4.34, -3.14, -3.24, -3.17, -2.54, -2.32, -3.31, -3.78, -2.47, -3.8, -2.82, -1.85, -3.51, -3.08, -2.5, -4.04, -2.63, -4.37, -2.59, -2.99, -3.83, -4.72, -4.29, -3.39, -3.37, -3.38, -3.35, -2.47, -3.7, -3.04, -3.57, -4.3, -2.76, -3.04, -2.52, -3.27, -2.81, -4.22, -3.15, -3.34, -2.75, -3.26, -3.53, -3.33, -4.49, -3.23, -3.63, -3.06, -3.72, -3.91, -1.99, -2.73, -3.85, -3.5, -3.2, -3.57, -2.44, -3.94, -3.89, -2.87, -1.91, -3.11, -4.12, -3.35, -2.97, -3.38, -3.85, -2.41, -3.12, -3.56, -3.1, -3.49, -1.68, -2.76, -3.29, -4.03, -3.66, -2.26, -3.88, -2.79, -4.08, -2.5, -2.93, -3.04, -3.65, -3.33, -2.93, -2.0, -3.42, -4.24, -2.9, -3.97, -2.54, -3.07, -4.31, -3.73, -3.43, -3.24, -3.97, -2.78, -3.45, -3.1, -3.46, -4.35, -3.78, -4.28, -2.54, -3.49, -2.32, -3.08, -2.77, -4.12, -3.45, -3.52, -2.84, -3.47, -4.1, -3.11, -3.48, -3.77, -2.19, -3.01, -4.63, -4.4, -3.18, -2.38, -2.42, -2.86, -4.39, -2.71, -3.06, -3.17, -3.35, -2.27, -3.63, -1.86, -2.52, -4.5, -3.58, -2.79, -2.71, -3.37, -3.86, -2.73, -2.42, -5.52, -2.63, -3.07, -4.15, -3.5, -3.26, -3.7, -2.91, -3.07, -4.93, -4.36, -4.53, -2.54, -3.7, -2.93, -2.79, -3.07, -3.44, -2.59, -3.48, -4.31, -2.88, -2.83, -2.7, -2.53, -2.8, -3.97, -3.29, -3.86, -3.39, -2.61, -3.41, -3.53, -3.12, -3.78, -2.37, -3.52, -3.69, -3.58, -5.04, -3.66, -2.67, -2.7, -3.73, -2.39, -4.26, -3.23, -2.84, -3.23, -2.73, -2.83, -2.99, -2.39, -2.67, -3.39, -3.33, -2.83, -2.25, -3.88, -3.96, -2.65, -2.74, -2.73, -3.07, -2.15, -2.73, -3.02, -3.82, -3.17, -3.92, -4.4, -3.78, -2.93, -3.19, -2.8, -3.87, -4.28, -3.8, -3.07, -3.38, -4.5, -3.35, -3.13, -3.23, -3.96, -3.14, -2.5, -2.94, -2.11, -4.22, -4.16, -4.64, -3.0, -3.97, -4.23, -3.39, -3.11, -3.4, -3.26, -2.4, -2.44, -2.42, -3.7, -3.37, -2.49, -2.33, -2.63, -4.26, -3.48, -3.14, -3.56, -2.72, -3.47, -4.22, -3.52, -3.53, -3.1, -3.24, -3.43, -3.9, -2.19, -3.46, -4.11, -3.47, -2.57, -1.35, -2.85, -3.28, -3.79, -4.26]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -3.2875，s_d = 0.6710，SE = s_d/√n = 0.0286；
t = d̄/SE = -115.0026；df = n−1 = 550；双侧 p = 0.000e+00。
d̄ = -3.2875，s_d = 0.6710，SE = s_d/√n = 0.0286；
t = d̄/SE = -115.0026；df = n−1 = 550；双侧 p = 0.000e+00。
计算结果
t = -115.0026，df = 550，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。判断训练是否显著减重。
案例 37. 用药前后血压
原始数据
前 n=323：[151.13, 156.86, 151.69, 154.1, 155.22, 155.07, 153.64, 154.27, 156.5, 149.94, 151.65, 149.02, 153.85, 151.02, 155.86, 151.05, 148.38, 154.4, 152.5, 152.34, 147.27, 157.09, 151.25, 155.83, 158.38, 151.37, 156.79, 152.86, 153.58, 155.43, 146.19, 154.67, 150.2, 150.75, 149.87, 150.67, 152.32, 153.04, 151.36, 151.36, 150.52, 153.84, 147.15, 154.46, 148.82, 160.04, 156.34, 149.96, 152.59, 150.12, 151.23, 151.38, 156.08, 150.84, 148.48, 149.71, 150.11, 154.83, 151.95, 148.83, 146.78, 153.95, 158.73, 154.69, 156.99, 149.54, 152.13, 155.68, 149.07, 154.13, 150.26, 152.88, 153.48, 148.14, 150.61, 155.39, 146.06, 152.02, 156.59, 143.68, 154.79, 150.04, 156.78, 151.48, 144.74, 150.51, 153.21, 152.21, 153.83, 149.36, 155.47, 151.6, 154.18, 154.98, 156.95, 154.64, 147.9, 154.06, 155.55, 148.42, 154.17, 149.13, 149.53, 156.41, 152.66, 156.82, 148.34, 154.37, 154.98, 150.12, 152.58, 158.69, 158.85, 153.85, 148.61, 151.64, 152.08, 151.49, 151.47, 158.99, 154.61, 153.7, 157.27, 153.81, 154.68, 157.99, 153.66, 152.73, 149.28, 153.69, 156.26, 156.0, 152.59, 153.83, 148.18, 152.87, 148.18, 154.22, 148.14, 148.57, 152.44, 151.54, 152.36, 156.55, 149.14, 149.86, 148.17, 154.68, 151.41, 155.77, 153.62, 151.47, 150.0, 151.43, 152.83, 154.94, 149.85, 152.84, 151.73, 152.76, 153.57, 155.01, 154.02, 153.72, 154.32, 153.17, 151.76, 154.59, 147.67, 157.23, 145.22, 151.12, 152.13, 152.98, 150.22, 155.57, 157.94, 153.07, 154.05, 150.43, 150.13, 147.06, 144.93, 151.66, 150.57, 155.37, 154.41, 155.94, 157.86, 151.32, 153.55, 157.62, 160.06, 150.8, 149.56, 154.06, 148.14, 150.63, 148.57, 154.53, 152.12, 157.75, 157.85, 148.81, 154.59, 156.88, 150.24, 152.8, 152.21, 149.79, 157.08, 146.88, 152.55, 151.63, 145.9, 153.55, 152.86, 152.25, 154.15, 151.72, 148.66, 146.18, 145.67, 155.13, 152.71, 153.11, 149.92, 151.52, 153.64, 151.11, 153.84, 149.4, 146.02, 155.76, 154.93, 143.61, 155.49, 150.32, 153.5, 153.11, 156.52, 148.27, 156.3, 152.65, 154.33, 152.76, 153.2, 152.64, 151.74, 148.96, 146.73, 151.82, 151.48, 152.06, 149.31, 154.12, 158.14, 157.08, 148.2, 152.28, 150.29, 150.63, 159.65, 150.15, 147.92, 148.33, 151.39, 153.37, 151.74, 153.33, 155.52, 155.33, 150.19, 157.19, 153.99, 154.53, 155.18, 151.81, 150.54, 154.98, 152.48, 151.08, 152.33, 156.56, 154.14, 147.5, 151.72, 159.43, 149.79, 148.8, 152.85, 155.45, 154.28, 156.29, 151.11, 151.08, 149.54, 154.66, 152.99, 152.0, 149.07, 152.62, 149.48, 148.58, 151.65, 151.41, 150.01, 150.12, 150.55, 150.45, 148.81, 152.42, 158.38, 149.56, 151.69, 149.08, 153.73, 153.86, 152.3, 147.84, 158.98, 151.52, 146.75]；
后 n=323：[142.69, 147.76, 142.82, 143.22, 147.15, 146.63, 145.61, 145.82, 148.26, 140.75, 143.71, 140.08, 144.34, 143.14, 147.94, 142.32, 138.77, 145.75, 143.8, 143.53, 137.72, 147.03, 141.98, 147.16, 147.97, 143.02, 146.0, 145.18, 145.05, 146.15, 137.81, 146.84, 141.24, 141.89, 140.99, 142.76, 144.27, 145.46, 141.91, 142.98, 140.96, 145.8, 136.52, 145.09, 140.11, 150.8, 147.33, 140.83, 143.65, 140.93, 144.35, 143.69, 145.92, 141.91, 139.86, 140.81, 140.63, 145.38, 143.62, 139.25, 138.29, 145.7, 149.44, 146.48, 147.77, 140.53, 142.75, 146.68, 139.25, 144.27, 139.63, 143.97, 144.08, 140.01, 142.08, 147.84, 135.67, 143.77, 147.52, 135.02, 147.52, 140.24, 148.21, 142.39, 136.42, 142.42, 144.65, 144.06, 143.85, 139.7, 146.98, 142.24, 145.06, 143.73, 147.44, 147.01, 136.06, 145.02, 147.17, 140.11, 145.19, 139.46, 141.95, 146.85, 143.46, 148.24, 140.17, 143.9, 146.07, 140.3, 143.22, 150.31, 150.63, 145.41, 139.24, 142.49, 144.75, 143.54, 142.25, 151.19, 146.04, 143.89, 148.97, 144.45, 146.36, 147.3, 144.4, 143.43, 140.13, 144.45, 149.49, 148.15, 143.26, 145.06, 138.99, 144.87, 138.18, 145.05, 139.99, 138.75, 143.31, 140.88, 142.74, 148.56, 140.14, 139.99, 139.84, 144.81, 142.68, 145.76, 146.95, 141.74, 143.02, 142.43, 144.06, 145.81, 142.03, 144.63, 143.34, 144.09, 146.2, 145.41, 144.63, 145.78, 144.56, 144.06, 142.66, 145.56, 138.37, 149.06, 135.12, 142.4, 143.55, 142.75, 140.07, 147.91, 148.56, 143.61, 145.24, 142.44, 139.63, 140.25, 134.82, 143.87, 140.97, 147.22, 146.87, 148.83, 148.81, 142.24, 145.41, 148.64, 151.76, 141.93, 140.78, 144.27, 139.74, 140.96, 140.79, 144.06, 144.05, 148.29, 149.04, 138.63, 146.36, 148.74, 140.41, 143.11, 143.3, 139.74, 147.8, 138.76, 144.0, 142.98, 135.76, 145.33, 144.73, 141.36, 144.99, 142.3, 140.4, 136.81, 135.26, 146.77, 143.49, 143.7, 142.21, 143.31, 144.24, 141.03, 145.12, 139.53, 136.34, 146.91, 146.26, 133.38, 146.5, 142.05, 144.84, 144.09, 147.83, 138.93, 147.2, 144.26, 145.13, 143.72, 144.62, 143.74, 142.39, 139.76, 136.58, 142.05, 141.95, 143.56, 139.73, 144.62, 148.45, 148.47, 138.92, 142.57, 141.34, 141.07, 150.49, 141.02, 139.17, 139.81, 142.36, 143.65, 142.66, 143.4, 146.53, 147.09, 141.6, 147.15, 145.74, 146.36, 145.93, 143.58, 141.77, 147.87, 143.51, 141.94, 143.4, 147.99, 145.63, 137.73, 142.31, 150.72, 141.35, 139.53, 144.39, 146.88, 144.21, 146.79, 141.17, 143.35, 140.78, 146.02, 145.77, 142.32, 139.17, 144.07, 140.55, 138.66, 142.53, 142.54, 143.18, 140.77, 141.76, 140.6, 140.45, 143.92, 147.94, 140.41, 142.5, 139.98, 144.0, 146.69, 143.95, 137.49, 151.09, 141.27, 137.59]；
差值(后−前) n=323：[-8.44, -9.1, -8.87, -10.88, -8.07, -8.44, -8.03, -8.45, -8.24, -9.19, -7.94, -8.94, -9.51, -7.88, -7.92, -8.72, -9.61, -8.65, -8.7, -8.81, -9.55, -10.06, -9.27, -8.66, -10.41, -8.36, -10.79, -7.67, -8.53, -9.28, -8.38, -7.82, -8.96, -8.86, -8.88, -7.91, -8.05, -7.58, -9.45, -8.38, -9.56, -8.05, -10.63, -9.37, -8.71, -9.24, -9.01, -9.13, -8.94, -9.19, -6.87, -7.69, -10.16, -8.93, -8.62, -8.9, -9.48, -9.45, -8.33, -9.58, -8.49, -8.25, -9.29, -8.21, -9.22, -9.01, -9.38, -8.99, -9.82, -9.86, -10.63, -8.91, -9.4, -8.12, -8.52, -7.56, -10.39, -8.25, -9.07, -8.66, -7.27, -9.8, -8.57, -9.09, -8.33, -8.09, -8.56, -8.15, -9.98, -9.67, -8.48, -9.36, -9.12, -11.25, -9.52, -7.62, -11.84, -9.05, -8.38, -8.31, -8.97, -9.67, -7.58, -9.56, -9.2, -8.58, -8.18, -10.48, -8.91, -9.82, -9.36, -8.38, -8.22, -8.44, -9.38, -9.15, -7.33, -7.95, -9.22, -7.81, -8.57, -9.81, -8.3, -9.36, -8.32, -10.68, -9.26, -9.3, -9.15, -9.24, -6.77, -7.85, -9.33, -8.77, -9.18, -8.0, -10.01, -9.17, -8.15, -9.82, -9.12, -10.67, -9.62, -7.99, -9.01, -9.87, -8.33, -9.87, -8.73, -10.0, -6.67, -9.72, -6.97, -9.0, -8.76, -9.13, -7.82, -8.21, -8.38, -8.67, -7.37, -9.59, -9.4, -7.94, -9.76, -9.11, -9.11, -9.03, -9.31, -8.17, -10.1, -8.73, -8.57, -10.23, -10.15, -7.66, -9.38, -9.46, -8.81, -7.99, -10.5, -6.81, -10.11, -7.79, -9.6, -8.15, -7.54, -7.11, -9.04, -9.08, -8.13, -8.99, -8.3, -8.86, -8.78, -9.79, -8.4, -9.67, -7.78, -10.48, -8.07, -9.46, -8.8, -10.18, -8.23, -8.14, -9.83, -9.69, -8.91, -10.05, -9.28, -8.12, -8.55, -8.65, -10.15, -8.22, -8.13, -10.89, -9.16, -9.42, -8.26, -9.38, -10.41, -8.36, -9.21, -9.42, -7.71, -8.21, -9.4, -10.07, -8.72, -9.88, -9.68, -8.85, -8.66, -10.23, -8.99, -8.27, -8.66, -9.02, -8.69, -9.34, -9.1, -8.39, -9.2, -9.04, -8.59, -8.9, -9.35, -9.2, -10.15, -9.77, -9.53, -8.5, -9.58, -9.5, -9.69, -8.61, -9.28, -9.71, -8.95, -9.56, -9.15, -9.13, -8.75, -8.52, -9.03, -9.72, -9.07, -9.94, -8.99, -8.24, -8.59, -10.04, -8.24, -8.17, -9.25, -8.23, -8.77, -7.1, -8.97, -9.14, -8.93, -8.58, -8.51, -9.77, -9.41, -8.71, -8.45, -9.27, -8.45, -8.57, -10.07, -9.5, -9.94, -7.73, -8.76, -8.65, -7.22, -9.68, -9.89, -8.55, -8.93, -9.93, -9.13, -8.87, -6.84, -9.35, -8.8, -9.85, -8.36, -8.51, -10.43, -9.15, -9.19, -9.1, -9.73, -7.18, -8.36, -10.36, -7.88, -10.25, -9.16]
后 n=323：[142.69, 147.76, 142.82, 143.22, 147.15, 146.63, 145.61, 145.82, 148.26, 140.75, 143.71, 140.08, 144.34, 143.14, 147.94, 142.32, 138.77, 145.75, 143.8, 143.53, 137.72, 147.03, 141.98, 147.16, 147.97, 143.02, 146.0, 145.18, 145.05, 146.15, 137.81, 146.84, 141.24, 141.89, 140.99, 142.76, 144.27, 145.46, 141.91, 142.98, 140.96, 145.8, 136.52, 145.09, 140.11, 150.8, 147.33, 140.83, 143.65, 140.93, 144.35, 143.69, 145.92, 141.91, 139.86, 140.81, 140.63, 145.38, 143.62, 139.25, 138.29, 145.7, 149.44, 146.48, 147.77, 140.53, 142.75, 146.68, 139.25, 144.27, 139.63, 143.97, 144.08, 140.01, 142.08, 147.84, 135.67, 143.77, 147.52, 135.02, 147.52, 140.24, 148.21, 142.39, 136.42, 142.42, 144.65, 144.06, 143.85, 139.7, 146.98, 142.24, 145.06, 143.73, 147.44, 147.01, 136.06, 145.02, 147.17, 140.11, 145.19, 139.46, 141.95, 146.85, 143.46, 148.24, 140.17, 143.9, 146.07, 140.3, 143.22, 150.31, 150.63, 145.41, 139.24, 142.49, 144.75, 143.54, 142.25, 151.19, 146.04, 143.89, 148.97, 144.45, 146.36, 147.3, 144.4, 143.43, 140.13, 144.45, 149.49, 148.15, 143.26, 145.06, 138.99, 144.87, 138.18, 145.05, 139.99, 138.75, 143.31, 140.88, 142.74, 148.56, 140.14, 139.99, 139.84, 144.81, 142.68, 145.76, 146.95, 141.74, 143.02, 142.43, 144.06, 145.81, 142.03, 144.63, 143.34, 144.09, 146.2, 145.41, 144.63, 145.78, 144.56, 144.06, 142.66, 145.56, 138.37, 149.06, 135.12, 142.4, 143.55, 142.75, 140.07, 147.91, 148.56, 143.61, 145.24, 142.44, 139.63, 140.25, 134.82, 143.87, 140.97, 147.22, 146.87, 148.83, 148.81, 142.24, 145.41, 148.64, 151.76, 141.93, 140.78, 144.27, 139.74, 140.96, 140.79, 144.06, 144.05, 148.29, 149.04, 138.63, 146.36, 148.74, 140.41, 143.11, 143.3, 139.74, 147.8, 138.76, 144.0, 142.98, 135.76, 145.33, 144.73, 141.36, 144.99, 142.3, 140.4, 136.81, 135.26, 146.77, 143.49, 143.7, 142.21, 143.31, 144.24, 141.03, 145.12, 139.53, 136.34, 146.91, 146.26, 133.38, 146.5, 142.05, 144.84, 144.09, 147.83, 138.93, 147.2, 144.26, 145.13, 143.72, 144.62, 143.74, 142.39, 139.76, 136.58, 142.05, 141.95, 143.56, 139.73, 144.62, 148.45, 148.47, 138.92, 142.57, 141.34, 141.07, 150.49, 141.02, 139.17, 139.81, 142.36, 143.65, 142.66, 143.4, 146.53, 147.09, 141.6, 147.15, 145.74, 146.36, 145.93, 143.58, 141.77, 147.87, 143.51, 141.94, 143.4, 147.99, 145.63, 137.73, 142.31, 150.72, 141.35, 139.53, 144.39, 146.88, 144.21, 146.79, 141.17, 143.35, 140.78, 146.02, 145.77, 142.32, 139.17, 144.07, 140.55, 138.66, 142.53, 142.54, 143.18, 140.77, 141.76, 140.6, 140.45, 143.92, 147.94, 140.41, 142.5, 139.98, 144.0, 146.69, 143.95, 137.49, 151.09, 141.27, 137.59]；
差值(后−前) n=323：[-8.44, -9.1, -8.87, -10.88, -8.07, -8.44, -8.03, -8.45, -8.24, -9.19, -7.94, -8.94, -9.51, -7.88, -7.92, -8.72, -9.61, -8.65, -8.7, -8.81, -9.55, -10.06, -9.27, -8.66, -10.41, -8.36, -10.79, -7.67, -8.53, -9.28, -8.38, -7.82, -8.96, -8.86, -8.88, -7.91, -8.05, -7.58, -9.45, -8.38, -9.56, -8.05, -10.63, -9.37, -8.71, -9.24, -9.01, -9.13, -8.94, -9.19, -6.87, -7.69, -10.16, -8.93, -8.62, -8.9, -9.48, -9.45, -8.33, -9.58, -8.49, -8.25, -9.29, -8.21, -9.22, -9.01, -9.38, -8.99, -9.82, -9.86, -10.63, -8.91, -9.4, -8.12, -8.52, -7.56, -10.39, -8.25, -9.07, -8.66, -7.27, -9.8, -8.57, -9.09, -8.33, -8.09, -8.56, -8.15, -9.98, -9.67, -8.48, -9.36, -9.12, -11.25, -9.52, -7.62, -11.84, -9.05, -8.38, -8.31, -8.97, -9.67, -7.58, -9.56, -9.2, -8.58, -8.18, -10.48, -8.91, -9.82, -9.36, -8.38, -8.22, -8.44, -9.38, -9.15, -7.33, -7.95, -9.22, -7.81, -8.57, -9.81, -8.3, -9.36, -8.32, -10.68, -9.26, -9.3, -9.15, -9.24, -6.77, -7.85, -9.33, -8.77, -9.18, -8.0, -10.01, -9.17, -8.15, -9.82, -9.12, -10.67, -9.62, -7.99, -9.01, -9.87, -8.33, -9.87, -8.73, -10.0, -6.67, -9.72, -6.97, -9.0, -8.76, -9.13, -7.82, -8.21, -8.38, -8.67, -7.37, -9.59, -9.4, -7.94, -9.76, -9.11, -9.11, -9.03, -9.31, -8.17, -10.1, -8.73, -8.57, -10.23, -10.15, -7.66, -9.38, -9.46, -8.81, -7.99, -10.5, -6.81, -10.11, -7.79, -9.6, -8.15, -7.54, -7.11, -9.04, -9.08, -8.13, -8.99, -8.3, -8.86, -8.78, -9.79, -8.4, -9.67, -7.78, -10.48, -8.07, -9.46, -8.8, -10.18, -8.23, -8.14, -9.83, -9.69, -8.91, -10.05, -9.28, -8.12, -8.55, -8.65, -10.15, -8.22, -8.13, -10.89, -9.16, -9.42, -8.26, -9.38, -10.41, -8.36, -9.21, -9.42, -7.71, -8.21, -9.4, -10.07, -8.72, -9.88, -9.68, -8.85, -8.66, -10.23, -8.99, -8.27, -8.66, -9.02, -8.69, -9.34, -9.1, -8.39, -9.2, -9.04, -8.59, -8.9, -9.35, -9.2, -10.15, -9.77, -9.53, -8.5, -9.58, -9.5, -9.69, -8.61, -9.28, -9.71, -8.95, -9.56, -9.15, -9.13, -8.75, -8.52, -9.03, -9.72, -9.07, -9.94, -8.99, -8.24, -8.59, -10.04, -8.24, -8.17, -9.25, -8.23, -8.77, -7.1, -8.97, -9.14, -8.93, -8.58, -8.51, -9.77, -9.41, -8.71, -8.45, -9.27, -8.45, -8.57, -10.07, -9.5, -9.94, -7.73, -8.76, -8.65, -7.22, -9.68, -9.89, -8.55, -8.93, -9.93, -9.13, -8.87, -6.84, -9.35, -8.8, -9.85, -8.36, -8.51, -10.43, -9.15, -9.19, -9.1, -9.73, -7.18, -8.36, -10.36, -7.88, -10.25, -9.16]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -8.9447，s_d = 0.8403，SE = s_d/√n = 0.0468；
t = d̄/SE = -191.3010；df = n−1 = 322；双侧 p = 0.000e+00。
d̄ = -8.9447，s_d = 0.8403，SE = s_d/√n = 0.0468；
t = d̄/SE = -191.3010；df = n−1 = 322；双侧 p = 0.000e+00。
计算结果
t = -191.3010，df = 322，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。验证药物降压效果。
案例 38. 培训前后成绩
原始数据
前 n=638：[60.73, 63.6, 68.77, 59.69, 66.15, 62.95, 60.47, 51.65, 62.34, 58.99, 64.88, 62.73, 62.7, 61.6, 62.5, 64.87, 60.15, 64.3, 61.33, 64.38, 56.93, 66.57, 65.66, 63.92, 60.5, 67.79, 65.2, 61.07, 62.21, 65.08, 60.08, 61.49, 63.64, 65.22, 60.37, 66.22, 60.95, 62.44, 62.65, 62.81, 64.13, 66.45, 64.73, 65.3, 65.48, 66.56, 62.18, 62.83, 60.1, 58.16, 62.67, 65.8, 65.84, 60.32, 67.91, 64.34, 57.28, 61.33, 62.9, 55.2, 62.67, 60.02, 63.87, 60.15, 64.7, 62.14, 68.95, 65.03, 59.04, 68.1, 65.24, 65.18, 58.8, 61.69, 59.68, 70.38, 63.08, 61.25, 71.35, 58.15, 63.68, 57.62, 60.21, 63.03, 61.94, 68.52, 62.86, 64.05, 58.66, 63.1, 63.99, 65.4, 64.12, 59.14, 60.6, 59.74, 67.14, 57.23, 62.19, 65.21, 64.13, 63.76, 63.89, 63.66, 61.26, 62.55, 61.05, 60.77, 66.39, 61.39, 54.81, 60.76, 63.06, 61.32, 65.61, 56.55, 57.91, 68.22, 65.54, 63.44, 61.0, 63.19, 57.15, 70.13, 59.91, 66.1, 60.21, 64.23, 64.36, 63.21, 63.51, 62.13, 66.07, 61.61, 61.63, 60.24, 58.4, 63.35, 63.66, 61.5, 62.51, 61.79, 63.22, 64.29, 57.93, 66.96, 63.67, 63.44, 67.25, 61.32, 66.08, 62.04, 64.29, 67.29, 68.34, 62.89, 65.71, 64.39, 60.46, 66.31, 61.48, 60.81, 59.02, 65.28, 63.09, 62.0, 65.16, 61.19, 59.91, 55.13, 61.44, 63.78, 63.37, 64.95, 61.87, 62.62, 70.16, 60.57, 63.9, 64.46, 56.71, 57.78, 64.06, 59.83, 56.23, 58.34, 56.88, 58.05, 61.05, 65.08, 62.91, 63.74, 60.03, 64.48, 63.77, 57.07, 63.92, 60.47, 54.12, 63.98, 60.75, 64.2, 60.89, 62.81, 66.46, 62.23, 63.53, 59.49, 59.38, 57.83, 63.93, 62.22, 58.14, 59.66, 64.44, 65.45, 65.37, 67.59, 64.9, 57.04, 59.28, 65.39, 63.1, 61.14, 58.12, 61.21, 63.03, 63.66, 62.75, 49.37, 64.17, 64.01, 65.59, 68.22, 63.5, 60.58, 61.37, 63.63, 64.94, 57.96, 62.37, 57.25, 62.28, 65.01, 56.71, 60.77, 60.93, 59.26, 61.09, 61.83, 59.24, 62.88, 56.6, 59.58, 63.38, 66.1, 57.43, 61.22, 63.85, 64.81, 64.79, 62.68, 59.34, 66.14, 65.49, 62.13, 61.51, 57.34, 61.71, 63.98, 59.72, 63.77, 64.97, 63.04, 61.22, 60.42, 61.66, 60.6, 56.23, 57.98, 60.27, 55.79, 61.67, 66.23, 64.84, 58.63, 65.74, 63.06, 62.75, 59.48, 56.69, 63.7, 65.84, 64.72, 57.68, 64.64, 56.75, 65.93, 62.06, 65.31, 59.78, 64.13, 61.42, 63.23, 66.94, 64.36, 59.04, 59.67, 61.67, 64.9, 65.34, 64.02, 65.01, 58.63, 65.22, 61.54, 63.98, 59.01, 58.77, 63.32, 61.7, 61.26, 57.05, 62.77, 58.77, 56.56, 59.12, 66.94, 60.08, 63.81, 64.74, 68.7, 64.37, 69.89, 69.79, 57.81, 63.52, 62.28, 54.95, 58.67, 59.29, 65.12, 63.51, 61.54, 55.04, 62.11, 60.83, 64.68, 65.45, 61.35, 61.37, 60.95, 59.92, 59.75, 63.73, 61.91, 60.41, 65.35, 60.46, 65.65, 61.91, 64.4, 60.46, 63.19, 64.08, 61.29, 68.14, 60.25, 63.46, 67.04, 56.27, 57.77, 63.76, 59.56, 56.43, 71.83, 66.43, 62.34, 64.83, 56.34, 56.7, 63.88, 62.07, 59.88, 60.17, 55.9, 62.64, 63.51, 64.02, 67.21, 63.94, 64.64, 63.99, 64.62, 59.91, 66.79, 59.74, 61.78, 63.53, 66.19, 65.34, 66.66, 65.54, 55.54, 62.18, 60.5, 70.08, 64.27, 59.64, 62.71, 62.56, 65.29, 60.48, 63.63, 59.78, 67.42, 62.88, 62.43, 66.38, 66.02, 64.59, 60.6, 66.32, 60.62, 62.52, 66.34, 61.38, 66.76, 68.68, 70.93, 65.91, 59.11, 61.21, 63.62, 61.66, 62.91, 64.31, 59.96, 65.52, 62.46, 67.27, 65.58, 63.32, 58.18, 65.92, 66.07, 62.82, 59.6, 56.44, 59.85, 62.37, 57.42, 63.37, 65.81, 64.79, 65.76, 61.33, 61.35, 62.88, 61.94, 60.71, 66.42, 59.63, 63.84, 65.35, 63.59, 66.4, 60.68, 59.63, 58.63, 65.87, 62.78, 59.71, 62.89, 58.49, 60.8, 63.01, 65.22, 60.69, 67.8, 58.96, 60.31, 63.55, 63.61, 59.4, 57.35, 60.96, 62.19, 63.59, 64.43, 65.42, 64.1, 62.54, 65.57, 64.81, 63.28, 59.84, 56.13, 64.0, 59.96, 62.16, 67.25, 63.01, 62.41, 61.57, 62.58, 64.96, 66.34, 66.24, 66.01, 60.0, 60.62, 62.07, 60.02, 55.47, 56.36, 62.95, 60.07, 63.1, 65.0, 62.51, 60.28, 65.83, 67.83, 63.59, 54.54, 65.18, 61.73, 66.96, 64.51, 65.12, 63.27, 59.55, 60.09, 64.07, 60.34, 62.59, 61.05, 59.67, 62.93, 58.99, 59.52, 61.6, 60.31, 68.11, 62.2, 62.19, 64.66, 64.16, 66.98, 55.88, 62.86, 60.81, 63.12, 59.6, 57.65, 60.7, 60.39, 65.57, 65.13, 60.97, 66.56, 65.4, 66.1, 59.35, 57.69, 61.6, 61.94, 62.04, 63.02, 67.95, 64.08, 61.07, 65.47, 66.91, 64.51, 60.58, 57.39, 64.29, 61.64, 62.6, 59.62, 64.89, 63.08, 62.48, 64.8, 58.01, 67.71, 62.93, 64.91, 62.23, 58.26, 69.41, 64.94, 67.05, 66.9, 62.42, 59.01, 65.77, 62.25, 61.63, 62.78, 64.43, 59.9, 59.84, 64.45, 65.18, 62.82, 65.13, 61.6, 60.09, 66.35, 64.96, 65.68, 60.46, 56.61, 57.88, 66.21, 67.04, 60.54, 58.91, 61.71, 62.88, 68.26, 61.71, 62.78, 67.92, 67.21, 61.61, 60.98, 61.33, 62.47, 60.78, 59.25, 61.95, 54.68, 64.65, 61.2]；
后 n=638：[71.37, 74.36, 78.6, 70.01, 74.88, 72.68, 69.33, 62.86, 71.36, 69.58, 75.64, 72.8, 74.13, 72.01, 72.47, 75.2, 69.44, 72.58, 71.33, 75.04, 67.89, 75.17, 75.53, 74.51, 69.31, 76.34, 72.85, 70.73, 72.71, 77.25, 72.25, 72.57, 73.19, 75.14, 71.27, 74.85, 67.77, 72.26, 71.9, 72.92, 74.08, 76.26, 75.69, 75.53, 75.76, 76.02, 72.92, 72.97, 70.95, 66.99, 72.25, 76.1, 73.56, 70.09, 76.41, 74.01, 67.02, 70.11, 73.47, 64.84, 71.8, 69.94, 74.69, 69.36, 72.51, 71.16, 78.71, 73.9, 70.99, 78.17, 75.42, 75.16, 69.59, 71.71, 70.47, 80.59, 72.87, 72.5, 81.25, 68.58, 74.9, 67.32, 70.33, 74.18, 72.62, 80.73, 73.07, 74.7, 67.84, 73.7, 75.16, 74.58, 74.61, 68.7, 70.33, 70.78, 77.52, 65.92, 71.47, 74.69, 74.04, 73.37, 72.7, 73.59, 72.88, 73.49, 70.67, 71.17, 76.83, 71.52, 64.88, 70.92, 72.95, 71.4, 76.86, 67.42, 67.67, 79.84, 75.73, 72.98, 71.31, 73.77, 68.72, 79.3, 68.37, 76.91, 71.83, 74.78, 73.9, 73.39, 73.46, 72.73, 75.03, 72.04, 71.36, 71.97, 68.03, 71.66, 73.92, 73.32, 72.73, 71.99, 74.06, 73.57, 67.64, 76.68, 73.26, 73.6, 77.58, 70.65, 75.59, 71.6, 75.76, 76.73, 77.61, 72.46, 75.63, 73.66, 69.1, 74.39, 70.27, 71.17, 69.29, 75.36, 73.3, 74.58, 75.16, 71.68, 70.33, 65.1, 72.18, 72.48, 73.53, 73.37, 71.85, 71.14, 80.13, 70.09, 74.15, 72.65, 66.23, 68.0, 73.52, 69.44, 66.89, 67.72, 68.49, 67.2, 70.8, 75.46, 73.09, 73.1, 69.92, 74.27, 74.45, 66.41, 73.93, 70.48, 65.21, 73.07, 70.94, 74.78, 72.04, 75.08, 77.75, 72.24, 74.52, 71.34, 68.03, 67.75, 75.0, 72.85, 66.92, 69.62, 75.57, 76.14, 74.62, 78.71, 75.86, 67.04, 68.98, 75.18, 72.8, 69.77, 67.24, 70.57, 73.68, 73.85, 71.95, 58.64, 74.3, 72.96, 75.85, 79.16, 73.55, 70.46, 70.9, 74.2, 73.76, 68.68, 72.56, 66.88, 73.23, 74.3, 67.85, 71.61, 70.67, 70.11, 69.58, 72.15, 70.3, 72.35, 68.42, 69.46, 74.66, 76.57, 68.11, 70.57, 73.67, 75.6, 73.93, 72.69, 69.81, 76.92, 75.7, 72.23, 70.37, 68.74, 71.78, 74.42, 69.56, 74.15, 75.73, 72.44, 71.42, 70.58, 71.57, 71.8, 67.13, 67.44, 70.96, 65.27, 70.79, 78.68, 74.26, 67.64, 76.64, 71.84, 73.31, 68.62, 67.04, 74.1, 75.25, 74.5, 68.69, 74.32, 68.07, 75.71, 71.85, 75.62, 69.59, 74.66, 71.61, 72.82, 76.64, 74.87, 70.22, 70.27, 70.13, 74.49, 75.57, 72.91, 75.67, 68.73, 76.02, 71.98, 74.26, 69.51, 68.02, 72.35, 72.69, 70.54, 65.34, 73.13, 67.84, 65.82, 68.7, 77.31, 70.14, 73.23, 74.92, 79.31, 74.37, 80.62, 79.19, 68.57, 74.27, 72.38, 64.98, 68.51, 68.16, 75.71, 74.02, 73.02, 64.67, 73.27, 70.52, 74.89, 73.18, 70.94, 70.25, 71.77, 69.58, 69.21, 74.69, 73.63, 70.58, 76.09, 69.3, 76.18, 72.93, 73.24, 69.53, 72.71, 73.64, 70.88, 79.17, 70.55, 73.12, 76.84, 66.57, 66.92, 72.43, 69.25, 66.85, 82.69, 76.15, 71.67, 75.19, 64.12, 65.8, 73.29, 73.71, 70.89, 70.05, 64.45, 71.91, 72.73, 73.57, 77.37, 74.08, 75.89, 73.14, 74.69, 68.93, 76.37, 68.57, 70.58, 73.25, 76.72, 74.64, 76.15, 76.59, 66.13, 72.66, 68.27, 79.94, 74.2, 70.95, 73.61, 72.32, 76.3, 69.95, 73.31, 69.3, 78.78, 72.54, 74.06, 75.78, 76.88, 73.59, 71.97, 77.13, 71.99, 71.98, 75.72, 71.75, 77.34, 80.23, 81.65, 74.85, 70.48, 72.22, 73.31, 71.02, 72.03, 73.38, 69.92, 75.8, 71.65, 77.14, 76.51, 73.0, 68.42, 75.01, 75.62, 72.15, 69.31, 68.11, 70.21, 72.69, 67.5, 73.19, 75.85, 75.1, 74.51, 71.92, 71.49, 73.55, 72.65, 70.78, 76.59, 69.8, 74.13, 76.16, 74.52, 76.76, 71.94, 69.69, 68.99, 74.47, 72.95, 71.36, 72.84, 69.56, 70.9, 74.68, 76.14, 72.38, 76.4, 67.9, 70.65, 72.41, 73.45, 68.28, 66.6, 72.46, 73.4, 73.75, 73.75, 75.42, 74.41, 72.08, 74.72, 75.5, 73.61, 69.97, 65.6, 74.85, 69.02, 72.43, 77.92, 71.41, 72.73, 71.07, 71.8, 75.62, 76.29, 75.3, 75.9, 69.8, 70.39, 74.28, 70.31, 65.57, 68.04, 71.24, 70.32, 73.3, 74.69, 72.56, 70.89, 76.0, 77.28, 73.41, 64.56, 76.1, 71.99, 75.97, 74.89, 74.79, 72.98, 70.14, 69.48, 72.27, 70.22, 72.96, 70.64, 69.38, 74.23, 67.65, 69.32, 73.5, 70.44, 77.29, 71.69, 72.04, 73.9, 72.57, 77.1, 63.51, 74.42, 70.75, 72.89, 69.85, 67.26, 70.83, 70.73, 76.22, 73.82, 71.18, 75.85, 75.61, 77.29, 70.51, 67.65, 71.87, 71.67, 72.41, 74.86, 78.72, 75.45, 71.1, 76.11, 76.49, 74.18, 70.9, 67.7, 75.01, 71.91, 73.98, 70.7, 74.56, 73.45, 73.78, 74.78, 68.29, 77.2, 70.18, 75.05, 73.5, 69.17, 78.34, 75.29, 76.85, 76.94, 72.6, 68.84, 74.94, 71.15, 71.92, 71.55, 74.74, 69.1, 71.51, 74.88, 75.53, 72.1, 74.44, 71.21, 69.78, 76.27, 74.39, 74.24, 71.12, 66.62, 68.6, 75.36, 76.76, 69.87, 67.85, 72.69, 72.95, 78.26, 70.63, 73.39, 78.38, 77.41, 72.33, 72.0, 70.52, 72.57, 70.45, 68.88, 71.83, 65.78, 75.14, 71.34]；
差值(后−前) n=638：[10.64, 10.76, 9.83, 10.32, 8.73, 9.72, 8.87, 11.2, 9.02, 10.59, 10.76, 10.07, 11.43, 10.41, 9.96, 10.33, 9.3, 8.28, 10.0, 10.66, 10.96, 8.6, 9.86, 10.6, 8.81, 8.55, 7.65, 9.66, 10.51, 12.17, 12.17, 11.08, 9.54, 9.92, 10.91, 8.63, 6.83, 9.82, 9.25, 10.11, 9.96, 9.8, 10.96, 10.22, 10.28, 9.47, 10.74, 10.14, 10.85, 8.83, 9.57, 10.3, 7.72, 9.77, 8.5, 9.67, 9.74, 8.79, 10.57, 9.64, 9.13, 9.92, 10.82, 9.21, 7.82, 9.02, 9.76, 8.87, 11.95, 10.07, 10.18, 9.97, 10.79, 10.02, 10.79, 10.22, 9.79, 11.25, 9.91, 10.43, 11.22, 9.7, 10.12, 11.15, 10.68, 12.21, 10.21, 10.65, 9.17, 10.6, 11.18, 9.18, 10.5, 9.56, 9.73, 11.04, 10.38, 8.69, 9.28, 9.48, 9.91, 9.61, 8.81, 9.93, 11.62, 10.94, 9.61, 10.4, 10.44, 10.13, 10.08, 10.15, 9.88, 10.08, 11.25, 10.87, 9.76, 11.62, 10.19, 9.54, 10.31, 10.59, 11.57, 9.17, 8.46, 10.8, 11.61, 10.55, 9.53, 10.18, 9.95, 10.6, 8.96, 10.44, 9.73, 11.73, 9.64, 8.3, 10.27, 11.82, 10.22, 10.2, 10.84, 9.28, 9.71, 9.72, 9.59, 10.17, 10.32, 9.33, 9.51, 9.57, 11.47, 9.44, 9.27, 9.57, 9.92, 9.28, 8.64, 8.08, 8.78, 10.36, 10.28, 10.08, 10.21, 12.59, 10.0, 10.49, 10.42, 9.97, 10.74, 8.69, 10.16, 8.42, 9.97, 8.52, 9.97, 9.52, 10.25, 8.19, 9.53, 10.22, 9.46, 9.6, 10.65, 9.37, 11.62, 9.15, 9.75, 10.38, 10.18, 9.36, 9.89, 9.79, 10.68, 9.34, 10.01, 10.02, 11.08, 9.09, 10.19, 10.58, 11.15, 12.27, 11.28, 10.01, 10.99, 11.85, 8.65, 9.92, 11.07, 10.63, 8.77, 9.95, 11.13, 10.69, 9.25, 11.12, 10.95, 10.0, 9.7, 9.79, 9.7, 8.63, 9.12, 9.35, 10.65, 10.19, 9.19, 9.27, 10.13, 8.94, 10.26, 10.95, 10.05, 9.88, 9.53, 10.57, 8.82, 10.72, 10.19, 9.63, 10.95, 9.29, 11.14, 10.83, 9.74, 10.84, 8.49, 10.33, 11.06, 9.47, 11.82, 9.88, 11.29, 10.47, 10.68, 9.35, 9.82, 10.78, 9.14, 10.01, 10.47, 10.78, 10.21, 10.09, 8.87, 11.4, 10.07, 10.45, 9.84, 10.38, 10.75, 9.4, 10.2, 10.16, 9.91, 11.2, 10.89, 9.46, 10.69, 9.48, 9.12, 12.46, 9.41, 9.01, 10.9, 8.79, 10.56, 9.14, 10.35, 10.4, 9.41, 9.78, 11.01, 9.69, 11.32, 9.77, 9.79, 10.32, 9.81, 10.53, 10.19, 9.59, 9.69, 10.52, 11.18, 10.61, 8.46, 9.59, 10.23, 8.89, 10.66, 10.1, 10.8, 10.44, 10.29, 10.49, 9.26, 9.04, 10.99, 9.28, 8.29, 10.36, 9.08, 9.26, 9.58, 10.38, 10.06, 9.43, 10.19, 10.62, 9.99, 10.73, 9.4, 10.76, 10.75, 10.09, 10.03, 9.84, 8.87, 10.6, 10.5, 11.48, 9.63, 11.15, 9.69, 10.2, 7.73, 9.6, 8.88, 10.82, 9.66, 9.47, 10.96, 11.71, 10.17, 10.74, 8.84, 10.53, 11.01, 8.84, 9.07, 9.52, 9.57, 9.6, 11.04, 10.3, 9.66, 9.8, 10.3, 9.14, 8.67, 9.7, 10.42, 10.86, 9.72, 9.33, 10.36, 7.78, 9.1, 9.42, 11.63, 11.01, 9.88, 8.55, 9.27, 9.22, 9.55, 10.16, 10.13, 11.25, 9.15, 10.07, 9.02, 9.58, 8.84, 8.8, 9.72, 10.52, 9.3, 9.49, 11.05, 10.6, 10.48, 7.77, 9.86, 9.94, 11.32, 10.89, 9.76, 11.0, 9.48, 9.68, 9.52, 11.36, 9.67, 11.63, 9.4, 10.85, 8.99, 11.37, 10.81, 11.37, 9.46, 9.38, 10.38, 10.58, 11.55, 10.71, 8.94, 11.37, 11.01, 9.7, 9.35, 9.12, 9.08, 9.96, 10.28, 9.19, 9.87, 10.93, 9.69, 10.23, 9.09, 9.55, 9.33, 9.7, 11.67, 10.35, 10.32, 10.08, 9.82, 10.04, 10.31, 8.75, 10.59, 10.15, 10.67, 10.71, 10.07, 10.17, 10.17, 10.29, 10.81, 10.92, 10.36, 11.25, 10.06, 10.37, 8.6, 10.16, 11.65, 9.94, 11.07, 10.1, 11.66, 10.92, 11.68, 8.6, 8.94, 10.33, 8.86, 9.85, 8.89, 9.25, 11.5, 11.21, 10.15, 9.31, 10.0, 10.31, 9.54, 9.15, 10.69, 10.32, 10.12, 9.47, 10.85, 9.06, 10.26, 10.67, 8.41, 10.33, 9.5, 9.22, 10.66, 9.96, 9.06, 9.89, 9.8, 9.76, 12.2, 10.28, 10.11, 11.68, 8.29, 10.25, 10.2, 9.68, 10.05, 10.62, 10.17, 9.45, 9.81, 10.02, 10.92, 10.26, 9.0, 10.38, 9.67, 9.71, 10.59, 9.39, 8.2, 9.88, 10.38, 9.6, 9.71, 11.29, 8.66, 9.8, 11.9, 10.13, 9.18, 9.49, 9.85, 9.24, 8.41, 10.12, 7.63, 11.56, 9.94, 9.77, 10.25, 9.61, 10.14, 10.33, 10.66, 8.7, 10.21, 9.29, 10.21, 11.19, 11.17, 9.97, 10.27, 9.73, 10.37, 11.84, 10.76, 11.37, 10.03, 10.64, 9.58, 9.67, 10.32, 10.31, 10.72, 10.27, 11.38, 11.08, 9.67, 10.37, 11.3, 9.97, 10.28, 9.48, 7.25, 10.15, 11.27, 10.92, 8.93, 10.36, 9.8, 10.04, 10.18, 9.84, 9.17, 8.9, 10.29, 8.77, 10.31, 9.2, 11.67, 10.43, 10.34, 9.28, 9.31, 9.61, 9.69, 9.92, 9.43, 8.56, 10.66, 10.01, 10.72, 9.15, 9.73, 9.33, 8.94, 10.97, 10.08, 10.0, 8.92, 10.61, 10.47, 10.21, 10.71, 11.01, 9.19, 10.1, 9.68, 9.63, 9.88, 11.1, 10.49, 10.14]
后 n=638：[71.37, 74.36, 78.6, 70.01, 74.88, 72.68, 69.33, 62.86, 71.36, 69.58, 75.64, 72.8, 74.13, 72.01, 72.47, 75.2, 69.44, 72.58, 71.33, 75.04, 67.89, 75.17, 75.53, 74.51, 69.31, 76.34, 72.85, 70.73, 72.71, 77.25, 72.25, 72.57, 73.19, 75.14, 71.27, 74.85, 67.77, 72.26, 71.9, 72.92, 74.08, 76.26, 75.69, 75.53, 75.76, 76.02, 72.92, 72.97, 70.95, 66.99, 72.25, 76.1, 73.56, 70.09, 76.41, 74.01, 67.02, 70.11, 73.47, 64.84, 71.8, 69.94, 74.69, 69.36, 72.51, 71.16, 78.71, 73.9, 70.99, 78.17, 75.42, 75.16, 69.59, 71.71, 70.47, 80.59, 72.87, 72.5, 81.25, 68.58, 74.9, 67.32, 70.33, 74.18, 72.62, 80.73, 73.07, 74.7, 67.84, 73.7, 75.16, 74.58, 74.61, 68.7, 70.33, 70.78, 77.52, 65.92, 71.47, 74.69, 74.04, 73.37, 72.7, 73.59, 72.88, 73.49, 70.67, 71.17, 76.83, 71.52, 64.88, 70.92, 72.95, 71.4, 76.86, 67.42, 67.67, 79.84, 75.73, 72.98, 71.31, 73.77, 68.72, 79.3, 68.37, 76.91, 71.83, 74.78, 73.9, 73.39, 73.46, 72.73, 75.03, 72.04, 71.36, 71.97, 68.03, 71.66, 73.92, 73.32, 72.73, 71.99, 74.06, 73.57, 67.64, 76.68, 73.26, 73.6, 77.58, 70.65, 75.59, 71.6, 75.76, 76.73, 77.61, 72.46, 75.63, 73.66, 69.1, 74.39, 70.27, 71.17, 69.29, 75.36, 73.3, 74.58, 75.16, 71.68, 70.33, 65.1, 72.18, 72.48, 73.53, 73.37, 71.85, 71.14, 80.13, 70.09, 74.15, 72.65, 66.23, 68.0, 73.52, 69.44, 66.89, 67.72, 68.49, 67.2, 70.8, 75.46, 73.09, 73.1, 69.92, 74.27, 74.45, 66.41, 73.93, 70.48, 65.21, 73.07, 70.94, 74.78, 72.04, 75.08, 77.75, 72.24, 74.52, 71.34, 68.03, 67.75, 75.0, 72.85, 66.92, 69.62, 75.57, 76.14, 74.62, 78.71, 75.86, 67.04, 68.98, 75.18, 72.8, 69.77, 67.24, 70.57, 73.68, 73.85, 71.95, 58.64, 74.3, 72.96, 75.85, 79.16, 73.55, 70.46, 70.9, 74.2, 73.76, 68.68, 72.56, 66.88, 73.23, 74.3, 67.85, 71.61, 70.67, 70.11, 69.58, 72.15, 70.3, 72.35, 68.42, 69.46, 74.66, 76.57, 68.11, 70.57, 73.67, 75.6, 73.93, 72.69, 69.81, 76.92, 75.7, 72.23, 70.37, 68.74, 71.78, 74.42, 69.56, 74.15, 75.73, 72.44, 71.42, 70.58, 71.57, 71.8, 67.13, 67.44, 70.96, 65.27, 70.79, 78.68, 74.26, 67.64, 76.64, 71.84, 73.31, 68.62, 67.04, 74.1, 75.25, 74.5, 68.69, 74.32, 68.07, 75.71, 71.85, 75.62, 69.59, 74.66, 71.61, 72.82, 76.64, 74.87, 70.22, 70.27, 70.13, 74.49, 75.57, 72.91, 75.67, 68.73, 76.02, 71.98, 74.26, 69.51, 68.02, 72.35, 72.69, 70.54, 65.34, 73.13, 67.84, 65.82, 68.7, 77.31, 70.14, 73.23, 74.92, 79.31, 74.37, 80.62, 79.19, 68.57, 74.27, 72.38, 64.98, 68.51, 68.16, 75.71, 74.02, 73.02, 64.67, 73.27, 70.52, 74.89, 73.18, 70.94, 70.25, 71.77, 69.58, 69.21, 74.69, 73.63, 70.58, 76.09, 69.3, 76.18, 72.93, 73.24, 69.53, 72.71, 73.64, 70.88, 79.17, 70.55, 73.12, 76.84, 66.57, 66.92, 72.43, 69.25, 66.85, 82.69, 76.15, 71.67, 75.19, 64.12, 65.8, 73.29, 73.71, 70.89, 70.05, 64.45, 71.91, 72.73, 73.57, 77.37, 74.08, 75.89, 73.14, 74.69, 68.93, 76.37, 68.57, 70.58, 73.25, 76.72, 74.64, 76.15, 76.59, 66.13, 72.66, 68.27, 79.94, 74.2, 70.95, 73.61, 72.32, 76.3, 69.95, 73.31, 69.3, 78.78, 72.54, 74.06, 75.78, 76.88, 73.59, 71.97, 77.13, 71.99, 71.98, 75.72, 71.75, 77.34, 80.23, 81.65, 74.85, 70.48, 72.22, 73.31, 71.02, 72.03, 73.38, 69.92, 75.8, 71.65, 77.14, 76.51, 73.0, 68.42, 75.01, 75.62, 72.15, 69.31, 68.11, 70.21, 72.69, 67.5, 73.19, 75.85, 75.1, 74.51, 71.92, 71.49, 73.55, 72.65, 70.78, 76.59, 69.8, 74.13, 76.16, 74.52, 76.76, 71.94, 69.69, 68.99, 74.47, 72.95, 71.36, 72.84, 69.56, 70.9, 74.68, 76.14, 72.38, 76.4, 67.9, 70.65, 72.41, 73.45, 68.28, 66.6, 72.46, 73.4, 73.75, 73.75, 75.42, 74.41, 72.08, 74.72, 75.5, 73.61, 69.97, 65.6, 74.85, 69.02, 72.43, 77.92, 71.41, 72.73, 71.07, 71.8, 75.62, 76.29, 75.3, 75.9, 69.8, 70.39, 74.28, 70.31, 65.57, 68.04, 71.24, 70.32, 73.3, 74.69, 72.56, 70.89, 76.0, 77.28, 73.41, 64.56, 76.1, 71.99, 75.97, 74.89, 74.79, 72.98, 70.14, 69.48, 72.27, 70.22, 72.96, 70.64, 69.38, 74.23, 67.65, 69.32, 73.5, 70.44, 77.29, 71.69, 72.04, 73.9, 72.57, 77.1, 63.51, 74.42, 70.75, 72.89, 69.85, 67.26, 70.83, 70.73, 76.22, 73.82, 71.18, 75.85, 75.61, 77.29, 70.51, 67.65, 71.87, 71.67, 72.41, 74.86, 78.72, 75.45, 71.1, 76.11, 76.49, 74.18, 70.9, 67.7, 75.01, 71.91, 73.98, 70.7, 74.56, 73.45, 73.78, 74.78, 68.29, 77.2, 70.18, 75.05, 73.5, 69.17, 78.34, 75.29, 76.85, 76.94, 72.6, 68.84, 74.94, 71.15, 71.92, 71.55, 74.74, 69.1, 71.51, 74.88, 75.53, 72.1, 74.44, 71.21, 69.78, 76.27, 74.39, 74.24, 71.12, 66.62, 68.6, 75.36, 76.76, 69.87, 67.85, 72.69, 72.95, 78.26, 70.63, 73.39, 78.38, 77.41, 72.33, 72.0, 70.52, 72.57, 70.45, 68.88, 71.83, 65.78, 75.14, 71.34]；
差值(后−前) n=638：[10.64, 10.76, 9.83, 10.32, 8.73, 9.72, 8.87, 11.2, 9.02, 10.59, 10.76, 10.07, 11.43, 10.41, 9.96, 10.33, 9.3, 8.28, 10.0, 10.66, 10.96, 8.6, 9.86, 10.6, 8.81, 8.55, 7.65, 9.66, 10.51, 12.17, 12.17, 11.08, 9.54, 9.92, 10.91, 8.63, 6.83, 9.82, 9.25, 10.11, 9.96, 9.8, 10.96, 10.22, 10.28, 9.47, 10.74, 10.14, 10.85, 8.83, 9.57, 10.3, 7.72, 9.77, 8.5, 9.67, 9.74, 8.79, 10.57, 9.64, 9.13, 9.92, 10.82, 9.21, 7.82, 9.02, 9.76, 8.87, 11.95, 10.07, 10.18, 9.97, 10.79, 10.02, 10.79, 10.22, 9.79, 11.25, 9.91, 10.43, 11.22, 9.7, 10.12, 11.15, 10.68, 12.21, 10.21, 10.65, 9.17, 10.6, 11.18, 9.18, 10.5, 9.56, 9.73, 11.04, 10.38, 8.69, 9.28, 9.48, 9.91, 9.61, 8.81, 9.93, 11.62, 10.94, 9.61, 10.4, 10.44, 10.13, 10.08, 10.15, 9.88, 10.08, 11.25, 10.87, 9.76, 11.62, 10.19, 9.54, 10.31, 10.59, 11.57, 9.17, 8.46, 10.8, 11.61, 10.55, 9.53, 10.18, 9.95, 10.6, 8.96, 10.44, 9.73, 11.73, 9.64, 8.3, 10.27, 11.82, 10.22, 10.2, 10.84, 9.28, 9.71, 9.72, 9.59, 10.17, 10.32, 9.33, 9.51, 9.57, 11.47, 9.44, 9.27, 9.57, 9.92, 9.28, 8.64, 8.08, 8.78, 10.36, 10.28, 10.08, 10.21, 12.59, 10.0, 10.49, 10.42, 9.97, 10.74, 8.69, 10.16, 8.42, 9.97, 8.52, 9.97, 9.52, 10.25, 8.19, 9.53, 10.22, 9.46, 9.6, 10.65, 9.37, 11.62, 9.15, 9.75, 10.38, 10.18, 9.36, 9.89, 9.79, 10.68, 9.34, 10.01, 10.02, 11.08, 9.09, 10.19, 10.58, 11.15, 12.27, 11.28, 10.01, 10.99, 11.85, 8.65, 9.92, 11.07, 10.63, 8.77, 9.95, 11.13, 10.69, 9.25, 11.12, 10.95, 10.0, 9.7, 9.79, 9.7, 8.63, 9.12, 9.35, 10.65, 10.19, 9.19, 9.27, 10.13, 8.94, 10.26, 10.95, 10.05, 9.88, 9.53, 10.57, 8.82, 10.72, 10.19, 9.63, 10.95, 9.29, 11.14, 10.83, 9.74, 10.84, 8.49, 10.33, 11.06, 9.47, 11.82, 9.88, 11.29, 10.47, 10.68, 9.35, 9.82, 10.78, 9.14, 10.01, 10.47, 10.78, 10.21, 10.09, 8.87, 11.4, 10.07, 10.45, 9.84, 10.38, 10.75, 9.4, 10.2, 10.16, 9.91, 11.2, 10.89, 9.46, 10.69, 9.48, 9.12, 12.46, 9.41, 9.01, 10.9, 8.79, 10.56, 9.14, 10.35, 10.4, 9.41, 9.78, 11.01, 9.69, 11.32, 9.77, 9.79, 10.32, 9.81, 10.53, 10.19, 9.59, 9.69, 10.52, 11.18, 10.61, 8.46, 9.59, 10.23, 8.89, 10.66, 10.1, 10.8, 10.44, 10.29, 10.49, 9.26, 9.04, 10.99, 9.28, 8.29, 10.36, 9.08, 9.26, 9.58, 10.38, 10.06, 9.43, 10.19, 10.62, 9.99, 10.73, 9.4, 10.76, 10.75, 10.09, 10.03, 9.84, 8.87, 10.6, 10.5, 11.48, 9.63, 11.15, 9.69, 10.2, 7.73, 9.6, 8.88, 10.82, 9.66, 9.47, 10.96, 11.71, 10.17, 10.74, 8.84, 10.53, 11.01, 8.84, 9.07, 9.52, 9.57, 9.6, 11.04, 10.3, 9.66, 9.8, 10.3, 9.14, 8.67, 9.7, 10.42, 10.86, 9.72, 9.33, 10.36, 7.78, 9.1, 9.42, 11.63, 11.01, 9.88, 8.55, 9.27, 9.22, 9.55, 10.16, 10.13, 11.25, 9.15, 10.07, 9.02, 9.58, 8.84, 8.8, 9.72, 10.52, 9.3, 9.49, 11.05, 10.6, 10.48, 7.77, 9.86, 9.94, 11.32, 10.89, 9.76, 11.0, 9.48, 9.68, 9.52, 11.36, 9.67, 11.63, 9.4, 10.85, 8.99, 11.37, 10.81, 11.37, 9.46, 9.38, 10.38, 10.58, 11.55, 10.71, 8.94, 11.37, 11.01, 9.7, 9.35, 9.12, 9.08, 9.96, 10.28, 9.19, 9.87, 10.93, 9.69, 10.23, 9.09, 9.55, 9.33, 9.7, 11.67, 10.35, 10.32, 10.08, 9.82, 10.04, 10.31, 8.75, 10.59, 10.15, 10.67, 10.71, 10.07, 10.17, 10.17, 10.29, 10.81, 10.92, 10.36, 11.25, 10.06, 10.37, 8.6, 10.16, 11.65, 9.94, 11.07, 10.1, 11.66, 10.92, 11.68, 8.6, 8.94, 10.33, 8.86, 9.85, 8.89, 9.25, 11.5, 11.21, 10.15, 9.31, 10.0, 10.31, 9.54, 9.15, 10.69, 10.32, 10.12, 9.47, 10.85, 9.06, 10.26, 10.67, 8.41, 10.33, 9.5, 9.22, 10.66, 9.96, 9.06, 9.89, 9.8, 9.76, 12.2, 10.28, 10.11, 11.68, 8.29, 10.25, 10.2, 9.68, 10.05, 10.62, 10.17, 9.45, 9.81, 10.02, 10.92, 10.26, 9.0, 10.38, 9.67, 9.71, 10.59, 9.39, 8.2, 9.88, 10.38, 9.6, 9.71, 11.29, 8.66, 9.8, 11.9, 10.13, 9.18, 9.49, 9.85, 9.24, 8.41, 10.12, 7.63, 11.56, 9.94, 9.77, 10.25, 9.61, 10.14, 10.33, 10.66, 8.7, 10.21, 9.29, 10.21, 11.19, 11.17, 9.97, 10.27, 9.73, 10.37, 11.84, 10.76, 11.37, 10.03, 10.64, 9.58, 9.67, 10.32, 10.31, 10.72, 10.27, 11.38, 11.08, 9.67, 10.37, 11.3, 9.97, 10.28, 9.48, 7.25, 10.15, 11.27, 10.92, 8.93, 10.36, 9.8, 10.04, 10.18, 9.84, 9.17, 8.9, 10.29, 8.77, 10.31, 9.2, 11.67, 10.43, 10.34, 9.28, 9.31, 9.61, 9.69, 9.92, 9.43, 8.56, 10.66, 10.01, 10.72, 9.15, 9.73, 9.33, 8.94, 10.97, 10.08, 10.0, 8.92, 10.61, 10.47, 10.21, 10.71, 11.01, 9.19, 10.1, 9.68, 9.63, 9.88, 11.1, 10.49, 10.14]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = 10.0306，s_d = 0.8613，SE = s_d/√n = 0.0341；
t = d̄/SE = 294.1629；df = n−1 = 637；双侧 p = 0.000e+00。
d̄ = 10.0306，s_d = 0.8613，SE = s_d/√n = 0.0341；
t = d̄/SE = 294.1629；df = n−1 = 637；双侧 p = 0.000e+00。
计算结果
t = 294.1629，df = 637，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估培训提升效果。
案例 39. 广告前后销量
原始数据
前 n=365：[111.8, 103.96, 97.82, 110.05, 117.55, 101.94, 108.34, 106.18, 97.36, 91.93, 107.33, 98.67, 106.24, 106.48, 99.87, 102.8, 103.89, 106.37, 103.98, 95.95, 108.61, 110.7, 100.18, 106.17, 103.82, 98.65, 102.63, 97.76, 110.26, 108.77, 100.53, 104.88, 103.57, 100.6, 100.75, 109.25, 101.61, 103.03, 94.2, 103.57, 98.84, 94.24, 98.48, 104.74, 103.28, 103.07, 99.0, 103.35, 101.08, 107.64, 116.26, 103.0, 110.47, 103.02, 106.13, 96.86, 110.69, 110.54, 108.57, 101.58, 97.68, 104.39, 102.08, 98.51, 111.64, 98.63, 98.28, 100.49, 102.89, 111.62, 96.55, 96.57, 110.89, 96.76, 99.48, 104.49, 108.74, 91.1, 107.06, 102.91, 104.65, 95.26, 96.22, 100.89, 98.46, 94.27, 94.7, 111.41, 101.88, 112.12, 113.07, 101.95, 111.21, 99.17, 97.63, 102.57, 97.26, 103.99, 102.44, 107.03, 103.31, 101.56, 103.16, 107.25, 105.27, 93.77, 91.66, 106.24, 107.2, 97.33, 105.87, 106.17, 112.83, 94.12, 105.68, 104.32, 104.04, 96.62, 103.7, 105.57, 99.3, 106.59, 94.18, 108.36, 101.46, 103.46, 100.37, 105.62, 98.96, 104.23, 102.45, 102.25, 106.8, 108.59, 110.74, 98.31, 105.61, 109.92, 97.41, 110.8, 101.34, 115.91, 97.49, 96.06, 100.15, 94.85, 96.26, 98.68, 109.64, 100.7, 106.01, 116.7, 100.62, 110.73, 98.17, 101.38, 93.09, 111.02, 106.63, 109.1, 104.05, 100.9, 103.24, 105.27, 93.81, 108.55, 104.73, 94.55, 106.67, 115.97, 103.99, 105.64, 109.45, 104.96, 91.92, 109.14, 101.51, 101.62, 101.05, 100.21, 110.51, 103.1, 110.38, 108.08, 94.58, 97.43, 99.65, 106.08, 106.01, 102.77, 103.3, 100.83, 101.97, 107.14, 106.23, 96.46, 106.79, 102.02, 99.59, 98.92, 103.41, 96.76, 97.99, 109.9, 101.38, 103.29, 103.39, 109.08, 95.76, 107.82, 100.14, 105.59, 102.3, 91.29, 102.27, 110.47, 104.97, 107.18, 96.25, 100.35, 100.72, 101.47, 96.97, 103.12, 93.01, 100.58, 101.41, 103.99, 98.64, 113.34, 101.76, 98.02, 100.71, 101.9, 116.14, 108.32, 108.53, 105.69, 101.39, 105.1, 102.27, 103.09, 100.87, 104.2, 107.42, 98.52, 99.7, 102.69, 91.94, 112.04, 108.79, 99.27, 102.56, 96.97, 108.19, 98.32, 106.4, 103.99, 118.41, 99.45, 92.45, 97.96, 102.33, 95.52, 106.69, 106.46, 105.72, 95.59, 105.0, 98.83, 107.48, 100.98, 101.75, 106.62, 104.01, 110.67, 108.1, 109.75, 97.5, 107.2, 95.21, 98.82, 98.59, 109.83, 102.23, 103.93, 101.86, 107.09, 104.87, 98.44, 102.54, 98.98, 108.41, 106.25, 100.75, 98.34, 104.73, 101.13, 98.55, 108.53, 101.15, 105.33, 106.13, 104.64, 110.53, 99.63, 116.45, 100.73, 97.39, 101.0, 110.66, 108.92, 105.85, 109.39, 106.53, 112.55, 105.65, 109.21, 98.35, 94.09, 109.08, 97.15, 103.68, 106.2, 98.8, 102.9, 112.18, 96.86, 110.6, 107.24, 111.54, 104.42, 92.5, 99.23, 109.46, 94.81, 104.52, 111.23, 97.59, 97.57, 101.25, 92.79, 102.6, 98.45, 103.81, 107.57, 98.58, 94.37, 109.65, 103.49, 102.98, 101.76, 100.85, 102.04, 101.32, 92.23, 101.21, 94.51, 103.91, 100.37, 105.98, 103.99, 107.86, 101.61, 105.67]；
后 n=365：[132.83, 125.08, 118.93, 130.45, 137.61, 122.63, 129.0, 126.34, 117.51, 111.69, 127.85, 119.02, 126.22, 127.35, 120.13, 123.45, 123.99, 127.29, 125.35, 118.12, 129.51, 132.1, 120.12, 127.07, 124.42, 119.26, 124.61, 117.81, 131.23, 129.2, 121.63, 125.38, 124.66, 121.13, 121.32, 129.2, 121.57, 123.84, 114.82, 124.24, 118.92, 115.25, 119.26, 126.22, 123.36, 123.62, 119.22, 124.79, 121.25, 127.62, 137.9, 123.07, 131.25, 123.93, 126.78, 118.14, 131.55, 131.18, 128.77, 121.64, 118.14, 124.38, 123.01, 119.13, 132.04, 119.06, 119.46, 121.68, 123.32, 132.39, 116.57, 117.46, 130.1, 117.29, 120.07, 124.66, 128.36, 111.33, 127.79, 123.09, 125.33, 115.64, 116.85, 122.06, 118.41, 115.74, 115.84, 132.3, 122.76, 133.16, 134.12, 122.76, 131.88, 119.95, 118.54, 123.09, 118.82, 125.06, 123.36, 127.38, 124.92, 122.63, 123.2, 128.01, 127.19, 114.63, 111.64, 127.21, 127.29, 117.65, 125.63, 126.78, 133.02, 114.6, 126.05, 125.56, 124.29, 117.64, 123.78, 125.31, 119.66, 127.14, 115.76, 128.91, 121.52, 124.31, 120.48, 126.39, 119.14, 123.55, 123.43, 123.26, 126.9, 128.7, 130.87, 118.57, 125.46, 130.5, 118.16, 131.98, 121.94, 136.86, 118.42, 115.82, 120.96, 116.08, 117.4, 119.5, 130.02, 121.0, 125.61, 137.5, 120.92, 131.4, 118.02, 121.9, 113.51, 131.3, 126.58, 129.47, 124.83, 121.38, 124.48, 126.36, 114.7, 129.74, 125.15, 114.7, 127.06, 136.3, 124.81, 126.17, 130.26, 125.26, 112.85, 128.9, 122.31, 122.64, 121.21, 120.66, 130.64, 124.64, 131.16, 129.32, 114.65, 117.93, 120.92, 125.78, 126.34, 122.92, 124.38, 120.83, 122.15, 128.52, 127.51, 116.84, 127.48, 123.03, 120.07, 119.08, 123.96, 116.81, 119.54, 130.87, 121.79, 123.95, 125.03, 130.28, 115.97, 127.89, 120.65, 126.71, 123.5, 111.62, 123.19, 130.64, 125.82, 127.19, 116.88, 120.42, 120.97, 121.42, 116.01, 124.09, 113.03, 121.35, 121.47, 125.43, 118.93, 134.7, 122.14, 118.12, 120.67, 122.84, 137.32, 129.89, 129.41, 127.45, 121.77, 126.08, 123.49, 124.02, 121.69, 124.89, 128.89, 118.48, 120.13, 123.41, 112.59, 132.59, 129.65, 119.42, 122.8, 117.77, 129.06, 118.06, 126.73, 124.46, 138.88, 119.76, 112.24, 118.2, 122.91, 116.15, 127.77, 127.13, 126.56, 115.93, 125.35, 119.45, 127.8, 121.9, 122.65, 127.11, 125.26, 130.5, 127.78, 130.07, 118.35, 128.28, 115.53, 119.58, 118.87, 129.81, 123.03, 124.37, 122.57, 127.88, 125.02, 118.82, 123.49, 119.36, 128.58, 126.84, 121.38, 117.57, 125.44, 121.18, 118.29, 129.36, 122.81, 125.37, 125.93, 125.32, 131.5, 119.55, 137.77, 122.23, 117.57, 122.15, 130.69, 129.58, 127.18, 129.94, 127.1, 133.15, 126.44, 130.16, 119.55, 114.52, 129.4, 118.58, 124.29, 127.57, 119.81, 122.81, 132.46, 116.78, 131.05, 127.6, 133.16, 124.97, 112.95, 120.59, 130.2, 114.54, 125.13, 131.62, 117.65, 117.5, 121.4, 114.09, 123.47, 120.22, 123.89, 127.63, 120.04, 115.89, 130.86, 124.24, 122.54, 121.56, 121.13, 123.24, 121.28, 111.78, 121.77, 114.64, 125.43, 120.34, 126.06, 123.66, 128.14, 122.17, 127.0]；
差值(后−前) n=365：[21.03, 21.12, 21.11, 20.4, 20.06, 20.69, 20.67, 20.17, 20.15, 19.76, 20.52, 20.36, 19.98, 20.87, 20.26, 20.65, 20.1, 20.92, 21.36, 22.17, 20.9, 21.4, 19.95, 20.9, 20.61, 20.61, 21.99, 20.04, 20.97, 20.43, 21.1, 20.5, 21.09, 20.54, 20.57, 19.95, 19.96, 20.81, 20.62, 20.67, 20.08, 21.01, 20.78, 21.48, 20.08, 20.55, 20.22, 21.44, 20.17, 19.98, 21.64, 20.07, 20.78, 20.91, 20.65, 21.27, 20.86, 20.64, 20.2, 20.06, 20.46, 19.98, 20.93, 20.62, 20.39, 20.43, 21.18, 21.2, 20.43, 20.77, 20.02, 20.9, 19.21, 20.53, 20.59, 20.17, 19.63, 20.23, 20.73, 20.19, 20.68, 20.38, 20.63, 21.16, 19.95, 21.47, 21.14, 20.89, 20.88, 21.04, 21.05, 20.8, 20.67, 20.77, 20.9, 20.52, 21.56, 21.08, 20.93, 20.35, 21.6, 21.06, 20.04, 20.76, 21.92, 20.86, 19.98, 20.97, 20.09, 20.33, 19.76, 20.61, 20.19, 20.48, 20.36, 21.25, 20.25, 21.02, 20.08, 19.73, 20.35, 20.55, 21.57, 20.56, 20.06, 20.85, 20.12, 20.77, 20.17, 19.32, 20.98, 21.01, 20.1, 20.11, 20.14, 20.26, 19.85, 20.58, 20.75, 21.18, 20.61, 20.94, 20.93, 19.76, 20.81, 21.23, 21.14, 20.82, 20.38, 20.3, 19.6, 20.8, 20.31, 20.67, 19.86, 20.52, 20.42, 20.28, 19.95, 20.37, 20.78, 20.48, 21.24, 21.09, 20.88, 21.19, 20.42, 20.15, 20.39, 20.33, 20.83, 20.53, 20.81, 20.3, 20.93, 19.77, 20.8, 21.01, 20.16, 20.45, 20.13, 21.54, 20.78, 21.24, 20.07, 20.5, 21.27, 19.69, 20.33, 20.15, 21.07, 20.0, 20.18, 21.38, 21.28, 20.38, 20.69, 21.01, 20.48, 20.16, 20.55, 20.05, 21.55, 20.97, 20.41, 20.66, 21.64, 21.21, 20.21, 20.07, 20.5, 21.12, 21.2, 20.33, 20.92, 20.17, 20.85, 20.02, 20.63, 20.07, 20.25, 19.95, 19.04, 20.97, 20.01, 20.77, 20.05, 21.44, 20.3, 21.36, 20.38, 20.1, 19.96, 20.95, 21.18, 21.57, 20.88, 21.76, 20.39, 20.98, 21.22, 20.93, 20.82, 20.68, 21.47, 19.96, 20.43, 20.72, 20.66, 20.55, 20.86, 20.14, 20.24, 20.81, 20.87, 19.74, 20.34, 20.47, 20.47, 20.31, 19.79, 20.23, 20.58, 20.63, 21.08, 20.67, 20.83, 20.34, 20.36, 20.62, 20.31, 20.92, 20.89, 20.49, 21.25, 19.84, 19.69, 20.32, 20.85, 21.08, 20.32, 20.76, 20.28, 19.97, 20.79, 20.44, 20.71, 20.79, 20.15, 20.37, 20.95, 20.38, 20.17, 20.59, 20.63, 19.23, 20.71, 20.05, 19.74, 20.83, 21.66, 20.04, 19.8, 20.68, 20.97, 19.92, 21.32, 21.5, 20.18, 21.15, 20.04, 20.66, 21.33, 20.55, 20.57, 20.6, 20.79, 20.95, 21.2, 20.42, 20.32, 21.42, 20.61, 21.37, 21.02, 19.9, 20.28, 19.92, 20.45, 20.36, 21.63, 20.56, 20.44, 21.36, 20.74, 19.73, 20.62, 20.38, 20.06, 19.93, 20.15, 21.3, 20.88, 21.77, 20.07, 20.06, 21.45, 21.53, 21.21, 20.75, 19.56, 19.8, 20.28, 21.2, 19.97, 19.54, 20.56, 20.14, 21.52, 19.98, 20.08, 19.67, 20.28, 20.57, 21.33]
后 n=365：[132.83, 125.08, 118.93, 130.45, 137.61, 122.63, 129.0, 126.34, 117.51, 111.69, 127.85, 119.02, 126.22, 127.35, 120.13, 123.45, 123.99, 127.29, 125.35, 118.12, 129.51, 132.1, 120.12, 127.07, 124.42, 119.26, 124.61, 117.81, 131.23, 129.2, 121.63, 125.38, 124.66, 121.13, 121.32, 129.2, 121.57, 123.84, 114.82, 124.24, 118.92, 115.25, 119.26, 126.22, 123.36, 123.62, 119.22, 124.79, 121.25, 127.62, 137.9, 123.07, 131.25, 123.93, 126.78, 118.14, 131.55, 131.18, 128.77, 121.64, 118.14, 124.38, 123.01, 119.13, 132.04, 119.06, 119.46, 121.68, 123.32, 132.39, 116.57, 117.46, 130.1, 117.29, 120.07, 124.66, 128.36, 111.33, 127.79, 123.09, 125.33, 115.64, 116.85, 122.06, 118.41, 115.74, 115.84, 132.3, 122.76, 133.16, 134.12, 122.76, 131.88, 119.95, 118.54, 123.09, 118.82, 125.06, 123.36, 127.38, 124.92, 122.63, 123.2, 128.01, 127.19, 114.63, 111.64, 127.21, 127.29, 117.65, 125.63, 126.78, 133.02, 114.6, 126.05, 125.56, 124.29, 117.64, 123.78, 125.31, 119.66, 127.14, 115.76, 128.91, 121.52, 124.31, 120.48, 126.39, 119.14, 123.55, 123.43, 123.26, 126.9, 128.7, 130.87, 118.57, 125.46, 130.5, 118.16, 131.98, 121.94, 136.86, 118.42, 115.82, 120.96, 116.08, 117.4, 119.5, 130.02, 121.0, 125.61, 137.5, 120.92, 131.4, 118.02, 121.9, 113.51, 131.3, 126.58, 129.47, 124.83, 121.38, 124.48, 126.36, 114.7, 129.74, 125.15, 114.7, 127.06, 136.3, 124.81, 126.17, 130.26, 125.26, 112.85, 128.9, 122.31, 122.64, 121.21, 120.66, 130.64, 124.64, 131.16, 129.32, 114.65, 117.93, 120.92, 125.78, 126.34, 122.92, 124.38, 120.83, 122.15, 128.52, 127.51, 116.84, 127.48, 123.03, 120.07, 119.08, 123.96, 116.81, 119.54, 130.87, 121.79, 123.95, 125.03, 130.28, 115.97, 127.89, 120.65, 126.71, 123.5, 111.62, 123.19, 130.64, 125.82, 127.19, 116.88, 120.42, 120.97, 121.42, 116.01, 124.09, 113.03, 121.35, 121.47, 125.43, 118.93, 134.7, 122.14, 118.12, 120.67, 122.84, 137.32, 129.89, 129.41, 127.45, 121.77, 126.08, 123.49, 124.02, 121.69, 124.89, 128.89, 118.48, 120.13, 123.41, 112.59, 132.59, 129.65, 119.42, 122.8, 117.77, 129.06, 118.06, 126.73, 124.46, 138.88, 119.76, 112.24, 118.2, 122.91, 116.15, 127.77, 127.13, 126.56, 115.93, 125.35, 119.45, 127.8, 121.9, 122.65, 127.11, 125.26, 130.5, 127.78, 130.07, 118.35, 128.28, 115.53, 119.58, 118.87, 129.81, 123.03, 124.37, 122.57, 127.88, 125.02, 118.82, 123.49, 119.36, 128.58, 126.84, 121.38, 117.57, 125.44, 121.18, 118.29, 129.36, 122.81, 125.37, 125.93, 125.32, 131.5, 119.55, 137.77, 122.23, 117.57, 122.15, 130.69, 129.58, 127.18, 129.94, 127.1, 133.15, 126.44, 130.16, 119.55, 114.52, 129.4, 118.58, 124.29, 127.57, 119.81, 122.81, 132.46, 116.78, 131.05, 127.6, 133.16, 124.97, 112.95, 120.59, 130.2, 114.54, 125.13, 131.62, 117.65, 117.5, 121.4, 114.09, 123.47, 120.22, 123.89, 127.63, 120.04, 115.89, 130.86, 124.24, 122.54, 121.56, 121.13, 123.24, 121.28, 111.78, 121.77, 114.64, 125.43, 120.34, 126.06, 123.66, 128.14, 122.17, 127.0]；
差值(后−前) n=365：[21.03, 21.12, 21.11, 20.4, 20.06, 20.69, 20.67, 20.17, 20.15, 19.76, 20.52, 20.36, 19.98, 20.87, 20.26, 20.65, 20.1, 20.92, 21.36, 22.17, 20.9, 21.4, 19.95, 20.9, 20.61, 20.61, 21.99, 20.04, 20.97, 20.43, 21.1, 20.5, 21.09, 20.54, 20.57, 19.95, 19.96, 20.81, 20.62, 20.67, 20.08, 21.01, 20.78, 21.48, 20.08, 20.55, 20.22, 21.44, 20.17, 19.98, 21.64, 20.07, 20.78, 20.91, 20.65, 21.27, 20.86, 20.64, 20.2, 20.06, 20.46, 19.98, 20.93, 20.62, 20.39, 20.43, 21.18, 21.2, 20.43, 20.77, 20.02, 20.9, 19.21, 20.53, 20.59, 20.17, 19.63, 20.23, 20.73, 20.19, 20.68, 20.38, 20.63, 21.16, 19.95, 21.47, 21.14, 20.89, 20.88, 21.04, 21.05, 20.8, 20.67, 20.77, 20.9, 20.52, 21.56, 21.08, 20.93, 20.35, 21.6, 21.06, 20.04, 20.76, 21.92, 20.86, 19.98, 20.97, 20.09, 20.33, 19.76, 20.61, 20.19, 20.48, 20.36, 21.25, 20.25, 21.02, 20.08, 19.73, 20.35, 20.55, 21.57, 20.56, 20.06, 20.85, 20.12, 20.77, 20.17, 19.32, 20.98, 21.01, 20.1, 20.11, 20.14, 20.26, 19.85, 20.58, 20.75, 21.18, 20.61, 20.94, 20.93, 19.76, 20.81, 21.23, 21.14, 20.82, 20.38, 20.3, 19.6, 20.8, 20.31, 20.67, 19.86, 20.52, 20.42, 20.28, 19.95, 20.37, 20.78, 20.48, 21.24, 21.09, 20.88, 21.19, 20.42, 20.15, 20.39, 20.33, 20.83, 20.53, 20.81, 20.3, 20.93, 19.77, 20.8, 21.01, 20.16, 20.45, 20.13, 21.54, 20.78, 21.24, 20.07, 20.5, 21.27, 19.69, 20.33, 20.15, 21.07, 20.0, 20.18, 21.38, 21.28, 20.38, 20.69, 21.01, 20.48, 20.16, 20.55, 20.05, 21.55, 20.97, 20.41, 20.66, 21.64, 21.21, 20.21, 20.07, 20.5, 21.12, 21.2, 20.33, 20.92, 20.17, 20.85, 20.02, 20.63, 20.07, 20.25, 19.95, 19.04, 20.97, 20.01, 20.77, 20.05, 21.44, 20.3, 21.36, 20.38, 20.1, 19.96, 20.95, 21.18, 21.57, 20.88, 21.76, 20.39, 20.98, 21.22, 20.93, 20.82, 20.68, 21.47, 19.96, 20.43, 20.72, 20.66, 20.55, 20.86, 20.14, 20.24, 20.81, 20.87, 19.74, 20.34, 20.47, 20.47, 20.31, 19.79, 20.23, 20.58, 20.63, 21.08, 20.67, 20.83, 20.34, 20.36, 20.62, 20.31, 20.92, 20.89, 20.49, 21.25, 19.84, 19.69, 20.32, 20.85, 21.08, 20.32, 20.76, 20.28, 19.97, 20.79, 20.44, 20.71, 20.79, 20.15, 20.37, 20.95, 20.38, 20.17, 20.59, 20.63, 19.23, 20.71, 20.05, 19.74, 20.83, 21.66, 20.04, 19.8, 20.68, 20.97, 19.92, 21.32, 21.5, 20.18, 21.15, 20.04, 20.66, 21.33, 20.55, 20.57, 20.6, 20.79, 20.95, 21.2, 20.42, 20.32, 21.42, 20.61, 21.37, 21.02, 19.9, 20.28, 19.92, 20.45, 20.36, 21.63, 20.56, 20.44, 21.36, 20.74, 19.73, 20.62, 20.38, 20.06, 19.93, 20.15, 21.3, 20.88, 21.77, 20.07, 20.06, 21.45, 21.53, 21.21, 20.75, 19.56, 19.8, 20.28, 21.2, 19.97, 19.54, 20.56, 20.14, 21.52, 19.98, 20.08, 19.67, 20.28, 20.57, 21.33]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = 20.5870，s_d = 0.5219，SE = s_d/√n = 0.0273；
t = d̄/SE = 753.6578；df = n−1 = 364；双侧 p = 0.000e+00。
d̄ = 20.5870，s_d = 0.5219，SE = s_d/√n = 0.0273；
t = d̄/SE = 753.6578；df = n−1 = 364；双侧 p = 0.000e+00。
计算结果
t = 753.6578，df = 364，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。衡量广告对销量拉动。
案例 40. 干预前后焦虑评分
原始数据
前 n=719：[30.49, 30.23, 28.41, 32.57, 33.47, 26.01, 28.94, 33.97, 29.45, 30.31, 32.32, 29.16, 31.29, 27.74, 30.51, 27.48, 29.86, 29.76, 31.64, 32.52, 29.66, 31.1, 27.43, 29.18, 31.14, 29.05, 30.72, 28.01, 30.97, 35.4, 28.4, 32.77, 34.33, 31.54, 28.41, 34.34, 29.63, 33.44, 29.34, 32.09, 26.54, 29.21, 31.34, 27.85, 29.21, 28.2, 28.98, 31.28, 30.2, 29.21, 30.4, 33.57, 30.36, 28.7, 30.54, 27.82, 27.79, 32.61, 33.58, 30.43, 29.95, 28.57, 28.66, 31.14, 29.65, 22.4, 31.43, 29.11, 29.72, 31.7, 26.89, 31.13, 29.0, 29.38, 32.34, 27.54, 27.81, 32.52, 26.65, 28.02, 32.33, 31.69, 31.81, 31.48, 33.33, 29.44, 31.45, 30.57, 32.74, 32.43, 30.56, 27.94, 27.43, 31.78, 29.87, 31.7, 29.72, 30.83, 27.7, 30.34, 30.84, 28.26, 24.11, 30.12, 31.4, 31.08, 27.95, 27.05, 33.13, 32.57, 27.64, 31.41, 27.74, 27.83, 29.23, 31.3, 30.9, 27.55, 32.35, 28.34, 28.18, 27.97, 31.58, 32.06, 27.47, 31.62, 29.53, 28.18, 29.23, 27.71, 28.99, 29.4, 30.55, 31.71, 30.23, 32.69, 30.99, 29.64, 30.29, 32.94, 28.46, 32.12, 30.94, 32.44, 27.71, 34.23, 31.55, 29.98, 29.36, 24.63, 32.07, 30.87, 32.76, 28.93, 29.43, 29.58, 31.33, 32.9, 28.96, 25.81, 32.5, 28.33, 30.17, 27.97, 28.93, 26.47, 30.87, 31.45, 32.78, 28.81, 30.23, 31.78, 31.59, 29.51, 29.22, 30.23, 30.84, 34.54, 31.01, 28.42, 29.15, 32.96, 29.31, 29.98, 31.35, 30.41, 26.92, 30.29, 30.34, 31.75, 29.13, 28.52, 33.22, 33.22, 29.56, 31.4, 33.21, 32.23, 31.52, 32.11, 29.55, 31.97, 32.41, 29.33, 30.19, 28.91, 29.95, 29.73, 28.56, 32.18, 39.06, 31.48, 31.52, 31.54, 28.1, 29.02, 28.18, 30.71, 27.85, 28.83, 29.34, 28.51, 23.63, 29.87, 27.7, 25.77, 32.6, 30.49, 32.58, 30.09, 30.88, 25.79, 28.21, 30.94, 32.03, 29.97, 27.78, 31.56, 31.83, 31.34, 26.94, 30.91, 29.34, 28.69, 29.66, 30.54, 32.58, 32.11, 30.02, 29.91, 29.81, 27.8, 31.85, 29.71, 25.65, 32.89, 29.68, 31.06, 31.4, 29.25, 29.83, 29.23, 31.89, 30.64, 30.46, 28.21, 31.83, 32.52, 27.77, 32.37, 31.83, 28.54, 31.25, 26.72, 29.83, 31.23, 31.21, 29.31, 30.64, 30.32, 31.6, 32.4, 29.78, 27.28, 28.19, 33.37, 30.53, 33.55, 30.25, 28.08, 28.36, 33.11, 31.28, 32.46, 29.34, 29.06, 31.9, 31.21, 28.96, 29.08, 34.51, 30.51, 29.69, 29.1, 29.0, 35.53, 31.04, 29.72, 24.58, 26.62, 30.03, 30.44, 28.87, 28.75, 32.54, 33.2, 28.27, 35.18, 34.3, 30.98, 27.11, 29.08, 30.32, 30.11, 35.69, 27.17, 27.55, 33.75, 31.28, 33.94, 33.48, 28.36, 31.28, 28.09, 28.36, 31.02, 30.1, 31.12, 27.39, 30.49, 29.58, 29.96, 29.54, 29.21, 31.16, 28.48, 32.06, 31.37, 32.53, 27.12, 28.84, 30.81, 29.1, 29.25, 31.18, 32.66, 27.01, 31.14, 30.8, 30.75, 28.73, 28.28, 29.86, 33.4, 30.02, 30.57, 30.55, 27.15, 25.07, 30.01, 28.73, 30.71, 29.02, 33.33, 31.55, 27.73, 33.73, 35.01, 33.2, 28.69, 33.73, 29.12, 31.02, 30.23, 29.82, 28.21, 28.91, 26.15, 30.22, 28.7, 30.66, 29.26, 27.74, 30.53, 29.01, 34.71, 31.4, 32.27, 32.34, 33.23, 28.06, 27.45, 28.38, 33.34, 27.43, 32.73, 32.67, 29.71, 28.87, 27.82, 32.84, 25.49, 31.89, 28.33, 29.52, 28.39, 34.19, 28.34, 26.34, 29.05, 32.61, 29.14, 31.81, 31.64, 30.06, 30.93, 27.91, 26.66, 31.95, 34.7, 32.9, 30.66, 27.79, 30.66, 28.06, 30.62, 22.79, 34.09, 32.26, 33.88, 30.57, 27.16, 32.33, 29.2, 29.63, 28.75, 27.0, 31.01, 28.93, 29.88, 25.33, 34.39, 30.68, 28.46, 28.11, 25.66, 28.68, 30.28, 31.88, 30.24, 29.66, 30.0, 29.93, 30.21, 28.47, 28.34, 27.93, 27.36, 32.76, 31.64, 33.28, 31.24, 28.3, 30.25, 30.2, 27.59, 33.21, 29.86, 31.38, 33.43, 29.43, 28.15, 29.19, 27.72, 29.0, 26.79, 25.72, 26.9, 29.39, 31.8, 29.64, 30.04, 32.69, 34.83, 27.13, 28.91, 33.65, 26.23, 26.66, 30.17, 29.86, 33.34, 29.23, 27.62, 26.45, 26.99, 30.75, 31.15, 29.65, 28.81, 28.38, 29.16, 27.5, 29.39, 30.75, 26.96, 34.78, 30.25, 26.99, 30.19, 30.85, 29.65, 31.61, 30.33, 30.34, 31.62, 32.72, 32.47, 32.21, 31.55, 26.46, 30.61, 31.62, 25.49, 32.2, 29.98, 28.27, 28.63, 34.13, 27.9, 25.2, 32.71, 31.88, 33.43, 30.97, 32.25, 30.79, 30.39, 32.94, 32.76, 29.68, 30.79, 29.89, 32.01, 28.88, 29.03, 31.85, 24.32, 31.32, 32.56, 26.78, 29.47, 28.98, 29.91, 31.32, 30.64, 28.09, 31.48, 34.67, 29.54, 34.41, 29.11, 29.01, 26.56, 29.32, 31.12, 28.72, 27.73, 27.92, 30.89, 29.96, 31.1, 27.79, 30.53, 30.39, 23.54, 32.07, 31.59, 29.03, 33.92, 28.08, 30.63, 33.14, 30.77, 32.7, 29.38, 26.21, 25.32, 34.99, 30.83, 27.5, 30.94, 28.06, 32.47, 29.67, 28.2, 32.14, 28.13, 26.8, 30.68, 30.26, 28.45, 25.77, 29.72, 28.73, 33.19, 32.62, 28.86, 32.7, 32.2, 27.46, 30.9, 29.85, 31.1, 32.43, 32.2, 31.51, 31.09, 31.27, 31.17, 26.53, 29.94, 28.0, 27.39, 32.71, 29.04, 29.93, 29.97, 28.64, 28.43, 29.09, 31.19, 30.59, 33.65, 32.05, 31.01, 29.89, 30.13, 31.19, 36.24, 32.82, 27.02, 33.5, 34.27, 27.34, 35.22, 30.16, 27.62, 31.02, 28.85, 31.85, 29.23, 27.92, 29.25, 31.03, 32.01, 29.91, 32.49, 33.15, 29.09, 30.31, 31.6, 28.39, 35.33, 29.41, 33.45, 30.75, 29.3, 30.17, 31.96, 28.37, 28.89, 34.32, 34.36, 28.35, 27.9, 26.8, 31.58, 34.6, 28.19, 31.31, 30.32, 29.23, 28.67, 28.12, 26.09, 31.83, 29.66, 31.39, 29.56, 28.28, 30.81, 30.11, 30.35, 29.13, 30.39, 30.58, 30.45, 28.23, 32.4, 34.91, 30.63, 32.44, 25.85, 30.25, 36.47, 31.21, 29.9, 31.91]；
后 n=719：[24.37, 24.42, 24.35, 25.87, 29.12, 21.5, 23.59, 29.1, 24.16, 25.18, 27.78, 25.42, 28.03, 22.79, 24.76, 23.49, 25.09, 25.43, 28.7, 28.73, 24.72, 27.34, 21.93, 24.21, 26.82, 23.92, 25.57, 25.16, 24.74, 31.46, 23.07, 26.16, 31.01, 26.51, 23.29, 30.71, 24.21, 29.0, 25.15, 27.85, 22.7, 23.77, 24.18, 22.56, 25.09, 22.43, 23.67, 26.36, 24.86, 25.71, 25.9, 28.82, 25.75, 22.92, 25.92, 22.69, 21.62, 28.68, 28.19, 26.18, 25.11, 23.19, 26.23, 26.59, 22.86, 18.03, 24.58, 25.54, 24.17, 27.73, 21.47, 26.13, 25.21, 24.21, 29.32, 22.11, 23.63, 28.4, 20.93, 22.56, 26.89, 28.7, 28.2, 26.63, 29.2, 23.38, 26.73, 26.84, 28.43, 29.01, 25.2, 23.76, 21.69, 24.92, 25.31, 26.13, 26.11, 24.62, 22.42, 26.31, 25.97, 23.76, 19.61, 25.58, 25.8, 26.54, 23.78, 21.02, 28.35, 27.49, 22.54, 26.35, 23.06, 23.46, 24.47, 27.78, 27.52, 23.52, 28.05, 23.98, 23.9, 23.69, 29.16, 26.9, 22.61, 26.1, 24.74, 22.95, 22.97, 22.55, 25.63, 26.08, 24.59, 27.91, 26.25, 29.16, 25.55, 23.19, 25.34, 27.79, 23.66, 28.11, 26.98, 26.61, 23.26, 27.78, 27.81, 26.09, 24.52, 20.23, 25.37, 26.2, 27.96, 23.57, 26.3, 25.25, 26.36, 28.5, 22.53, 22.02, 26.31, 24.35, 23.75, 23.73, 23.39, 20.96, 25.09, 27.26, 26.93, 24.74, 24.92, 27.77, 28.38, 23.67, 25.01, 24.65, 26.21, 29.29, 26.27, 23.65, 24.44, 27.3, 25.29, 25.99, 26.0, 25.71, 22.45, 24.49, 25.52, 26.76, 24.98, 22.92, 27.97, 26.09, 25.33, 26.73, 27.84, 26.65, 26.36, 26.31, 24.63, 29.36, 27.78, 24.64, 25.39, 23.95, 24.84, 23.12, 22.61, 27.33, 34.07, 26.76, 25.03, 27.55, 25.42, 24.52, 21.9, 24.8, 24.3, 24.33, 25.54, 22.6, 20.22, 25.65, 25.75, 20.69, 27.73, 24.74, 27.14, 25.32, 23.76, 20.1, 21.63, 25.21, 28.15, 23.67, 23.93, 27.66, 26.8, 27.31, 24.72, 25.06, 24.87, 23.67, 24.85, 25.18, 26.75, 28.5, 25.26, 24.57, 25.3, 23.0, 29.24, 25.34, 21.47, 28.47, 25.48, 26.38, 24.91, 25.75, 24.64, 23.02, 27.93, 24.96, 25.39, 22.81, 27.64, 28.33, 21.98, 27.29, 26.59, 26.93, 26.92, 21.63, 23.49, 24.94, 27.33, 25.04, 24.39, 26.44, 25.23, 27.92, 24.78, 22.67, 23.13, 27.94, 27.97, 30.15, 26.48, 22.47, 23.65, 28.09, 26.22, 28.23, 25.01, 24.26, 26.3, 26.21, 23.66, 25.12, 31.58, 25.63, 24.94, 24.82, 25.56, 30.35, 25.46, 24.61, 19.61, 20.42, 24.67, 24.79, 24.43, 25.15, 30.57, 26.48, 22.95, 30.12, 29.33, 26.97, 24.55, 23.57, 25.29, 25.01, 29.9, 23.44, 23.32, 28.19, 26.7, 28.53, 28.09, 25.16, 25.53, 23.59, 22.31, 27.73, 24.99, 25.71, 23.64, 25.53, 25.74, 23.58, 24.84, 24.14, 28.05, 22.61, 28.89, 27.17, 27.19, 23.33, 24.48, 25.12, 24.18, 24.84, 26.06, 27.64, 21.63, 26.06, 25.16, 24.45, 25.47, 24.32, 24.97, 28.7, 25.09, 25.98, 25.17, 23.8, 20.17, 25.32, 24.3, 26.65, 24.85, 28.6, 25.95, 22.59, 27.41, 31.35, 29.05, 23.89, 29.93, 23.09, 26.69, 26.11, 25.14, 23.67, 25.05, 20.65, 25.75, 25.23, 25.46, 24.43, 21.9, 26.29, 24.2, 29.69, 27.32, 26.74, 26.31, 27.39, 23.88, 21.29, 24.3, 29.8, 23.28, 27.26, 29.36, 23.51, 22.04, 23.16, 27.29, 20.14, 27.35, 22.23, 24.81, 23.1, 27.76, 24.4, 21.34, 23.74, 27.26, 23.88, 27.04, 27.45, 24.94, 27.02, 22.13, 21.85, 26.39, 30.59, 29.2, 24.55, 23.77, 25.59, 23.51, 25.56, 17.27, 29.62, 26.96, 28.24, 25.83, 21.72, 26.92, 24.07, 24.61, 25.05, 23.13, 28.7, 23.45, 24.61, 19.68, 29.48, 25.28, 22.62, 23.29, 22.1, 21.28, 26.98, 28.02, 25.4, 24.67, 26.31, 24.98, 26.6, 24.01, 24.91, 22.0, 22.62, 27.55, 27.95, 29.06, 25.73, 23.53, 25.74, 26.98, 24.23, 28.76, 23.98, 27.7, 28.48, 24.98, 21.24, 24.26, 23.57, 23.54, 22.32, 21.08, 22.51, 25.42, 25.53, 25.44, 25.83, 27.31, 30.79, 21.63, 23.92, 29.43, 21.17, 21.5, 26.12, 25.51, 29.49, 24.3, 23.46, 21.52, 21.95, 24.49, 25.4, 24.49, 22.37, 24.5, 24.82, 21.36, 24.56, 27.72, 21.71, 29.94, 25.85, 21.95, 25.67, 26.08, 25.32, 27.21, 26.3, 26.19, 26.93, 27.81, 28.38, 27.46, 26.69, 21.12, 24.48, 26.48, 19.82, 26.25, 23.77, 23.11, 24.72, 29.28, 22.39, 22.51, 27.87, 27.78, 27.61, 26.15, 27.56, 25.87, 26.69, 28.52, 28.55, 24.03, 27.17, 24.78, 26.96, 23.82, 23.51, 28.35, 19.03, 27.34, 26.39, 20.7, 24.95, 24.37, 25.32, 26.71, 25.54, 22.6, 27.62, 31.36, 25.86, 30.15, 25.12, 24.66, 21.8, 24.19, 25.48, 24.46, 23.02, 22.84, 25.66, 25.4, 25.69, 23.9, 23.92, 24.79, 19.28, 27.97, 25.89, 24.0, 28.47, 22.59, 25.27, 29.86, 26.22, 28.37, 24.44, 19.97, 22.52, 31.46, 26.0, 21.85, 26.29, 21.59, 27.4, 22.6, 22.5, 28.24, 21.76, 21.48, 27.62, 24.57, 23.77, 20.74, 24.58, 23.83, 28.14, 27.87, 24.23, 26.92, 25.0, 22.71, 24.95, 25.2, 25.91, 28.1, 27.67, 26.16, 25.55, 25.87, 26.16, 22.27, 25.78, 23.65, 24.1, 25.96, 24.38, 25.29, 22.36, 22.57, 24.24, 20.99, 27.36, 25.53, 30.52, 26.42, 26.66, 26.31, 24.56, 26.68, 31.2, 27.63, 22.14, 28.93, 28.06, 20.71, 30.02, 24.96, 21.86, 24.82, 24.52, 26.96, 24.96, 24.41, 23.82, 26.97, 26.66, 25.19, 27.86, 27.96, 24.79, 24.09, 27.11, 22.29, 31.92, 24.38, 29.72, 25.05, 23.85, 25.1, 26.8, 23.14, 22.05, 29.5, 29.04, 22.83, 24.13, 22.49, 25.75, 28.4, 23.4, 28.96, 26.04, 23.98, 23.85, 22.97, 21.71, 26.39, 25.17, 26.77, 23.74, 23.27, 24.87, 25.03, 25.75, 24.7, 26.54, 25.92, 25.42, 23.61, 29.45, 30.52, 24.09, 26.72, 21.71, 24.11, 33.02, 26.0, 26.44, 23.78]；
差值(后−前) n=719：[-6.12, -5.82, -4.06, -6.7, -4.35, -4.51, -5.35, -4.87, -5.3, -5.13, -4.54, -3.74, -3.27, -4.95, -5.76, -3.99, -4.76, -4.33, -2.94, -3.79, -4.93, -3.76, -5.5, -4.96, -4.32, -5.13, -5.15, -2.85, -6.23, -3.94, -5.33, -6.6, -3.32, -5.02, -5.12, -3.63, -5.42, -4.44, -4.19, -4.24, -3.84, -5.44, -7.16, -5.29, -4.13, -5.78, -5.31, -4.92, -5.34, -3.5, -4.5, -4.75, -4.61, -5.78, -4.62, -5.13, -6.17, -3.93, -5.39, -4.25, -4.84, -5.38, -2.43, -4.56, -6.79, -4.36, -6.84, -3.58, -5.55, -3.97, -5.41, -5.0, -3.79, -5.17, -3.02, -5.44, -4.18, -4.12, -5.71, -5.47, -5.44, -2.99, -3.62, -4.85, -4.13, -6.06, -4.72, -3.73, -4.3, -3.42, -5.36, -4.19, -5.74, -6.86, -4.56, -5.57, -3.61, -6.21, -5.28, -4.03, -4.87, -4.5, -4.5, -4.54, -5.59, -4.55, -4.18, -6.03, -4.78, -5.08, -5.1, -5.06, -4.68, -4.37, -4.76, -3.52, -3.38, -4.03, -4.3, -4.36, -4.28, -4.28, -2.42, -5.15, -4.86, -5.52, -4.79, -5.23, -6.26, -5.16, -3.36, -3.32, -5.96, -3.79, -3.98, -3.53, -5.45, -6.45, -4.96, -5.15, -4.8, -4.01, -3.96, -5.83, -4.45, -6.45, -3.73, -3.89, -4.84, -4.4, -6.7, -4.67, -4.8, -5.36, -3.13, -4.32, -4.97, -4.4, -6.42, -3.79, -6.2, -3.98, -6.42, -4.24, -5.55, -5.51, -5.78, -4.19, -5.85, -4.06, -5.32, -4.01, -3.21, -5.83, -4.21, -5.58, -4.63, -5.25, -4.74, -4.76, -4.72, -5.66, -4.02, -3.99, -5.34, -4.7, -4.47, -5.8, -4.82, -4.99, -4.15, -5.6, -5.26, -7.13, -4.23, -4.67, -5.38, -5.57, -5.17, -5.8, -4.92, -2.61, -4.63, -4.69, -4.8, -4.97, -5.11, -6.6, -5.95, -4.85, -4.99, -4.72, -6.5, -3.99, -2.68, -4.5, -6.28, -5.91, -3.55, -4.5, -3.8, -5.91, -3.4, -4.22, -1.95, -5.08, -4.87, -5.76, -5.44, -4.77, -7.12, -5.69, -6.58, -5.73, -3.88, -6.3, -3.85, -3.9, -5.03, -4.03, -2.22, -5.85, -4.47, -5.02, -4.81, -5.36, -5.84, -3.6, -4.76, -5.34, -4.51, -4.8, -2.61, -4.37, -4.18, -4.42, -4.19, -4.68, -6.49, -3.49, -5.19, -6.21, -3.96, -5.68, -5.07, -5.4, -4.19, -4.18, -5.79, -5.09, -5.24, -1.61, -4.33, -5.09, -6.35, -6.29, -3.89, -4.27, -6.25, -3.88, -6.38, -4.49, -5.0, -4.61, -5.07, -5.43, -2.56, -3.4, -3.77, -5.61, -4.71, -5.02, -5.06, -4.23, -4.33, -4.8, -5.6, -5.01, -5.31, -3.96, -2.93, -4.88, -4.75, -4.28, -3.45, -5.17, -5.59, -5.11, -4.97, -6.2, -5.36, -5.65, -4.45, -3.6, -1.97, -6.72, -5.33, -5.05, -4.97, -4.01, -2.56, -5.51, -5.03, -5.09, -5.79, -3.74, -4.23, -5.56, -4.58, -5.41, -5.39, -3.2, -5.75, -4.5, -6.05, -3.29, -5.12, -5.41, -3.75, -4.95, -3.84, -6.38, -4.7, -5.07, -3.1, -5.87, -3.17, -4.2, -5.34, -3.79, -4.36, -5.69, -4.91, -4.41, -5.12, -5.02, -5.38, -5.08, -5.64, -6.3, -3.25, -3.97, -4.89, -4.7, -4.94, -4.59, -5.38, -3.35, -4.9, -4.69, -4.42, -4.07, -4.17, -4.73, -5.6, -5.14, -6.33, -3.66, -4.15, -4.8, -3.8, -6.03, -4.32, -4.12, -4.68, -4.54, -3.86, -5.5, -4.47, -3.47, -5.2, -4.83, -5.85, -4.24, -4.81, -5.02, -4.09, -5.53, -6.03, -5.84, -4.18, -6.16, -4.08, -3.53, -4.15, -5.47, -3.31, -6.2, -6.83, -4.67, -5.55, -5.35, -4.53, -6.11, -4.71, -5.29, -6.44, -3.94, -5.01, -5.31, -5.35, -5.25, -4.77, -4.18, -5.11, -3.91, -5.78, -4.81, -5.56, -4.11, -3.69, -6.11, -4.01, -5.06, -4.56, -5.06, -5.52, -4.47, -5.29, -5.64, -4.74, -5.44, -5.41, -5.13, -5.02, -3.7, -3.87, -2.31, -5.48, -5.26, -5.66, -4.91, -5.41, -5.84, -4.82, -3.56, -7.41, -3.3, -3.86, -4.84, -4.99, -3.69, -4.95, -3.61, -4.46, -3.43, -5.93, -4.74, -5.21, -3.69, -4.21, -5.51, -4.76, -4.51, -3.23, -3.36, -4.46, -5.87, -3.68, -4.95, -4.45, -6.91, -4.93, -4.15, -5.46, -4.47, -4.65, -4.39, -3.97, -6.27, -4.2, -4.21, -5.37, -4.05, -5.5, -4.98, -4.21, -5.06, -5.15, -4.05, -4.35, -3.85, -4.93, -4.16, -4.93, -5.04, -6.26, -5.75, -5.15, -6.45, -3.87, -4.34, -6.15, -4.83, -3.03, -5.24, -4.84, -4.4, -5.05, -4.53, -4.77, -4.33, -4.4, -4.03, -4.15, -4.69, -4.91, -4.09, -4.75, -4.86, -5.33, -6.13, -5.14, -5.68, -5.94, -6.21, -5.16, -3.91, -4.85, -5.51, -2.69, -4.84, -4.1, -5.81, -4.81, -4.69, -4.92, -3.71, -4.42, -4.21, -5.65, -3.62, -5.11, -5.05, -5.06, -5.52, -3.5, -5.29, -3.98, -6.18, -6.08, -4.52, -4.61, -4.59, -4.61, -5.1, -5.49, -3.86, -3.31, -3.68, -4.26, -3.99, -4.35, -4.77, -5.13, -5.64, -4.26, -4.71, -5.08, -5.23, -4.56, -5.41, -3.89, -6.61, -5.6, -4.26, -4.1, -5.7, -5.03, -5.44, -5.49, -5.36, -3.28, -4.55, -4.32, -4.94, -6.24, -2.8, -3.54, -4.83, -5.65, -4.65, -6.47, -5.06, -7.07, -5.7, -3.9, -6.36, -5.32, -3.06, -5.69, -4.68, -5.04, -5.14, -4.9, -5.04, -4.75, -4.63, -5.78, -7.21, -4.75, -5.95, -4.65, -5.19, -4.33, -4.52, -5.34, -5.54, -5.4, -5.01, -4.25, -4.16, -4.35, -3.29, -6.75, -4.67, -4.65, -7.6, -6.07, -4.19, -8.09, -3.83, -5.06, -3.13, -5.63, -4.35, -3.58, -5.56, -4.51, -5.04, -5.19, -4.89, -4.57, -6.21, -6.63, -5.2, -5.2, -5.76, -6.19, -4.33, -4.89, -4.27, -3.51, -5.42, -4.07, -5.35, -4.72, -4.63, -5.2, -4.29, -6.22, -4.5, -6.09, -3.4, -5.03, -3.73, -5.7, -5.45, -5.07, -5.16, -5.23, -6.85, -4.82, -5.32, -5.52, -3.77, -4.31, -5.84, -6.21, -4.8, -2.35, -4.28, -5.25, -4.82, -5.15, -4.37, -5.45, -4.48, -4.62, -5.83, -5.01, -5.94, -5.09, -4.6, -4.43, -3.85, -4.66, -5.02, -4.62, -2.94, -4.39, -6.53, -5.72, -4.15, -6.14, -3.45, -5.21, -3.45, -8.13]
后 n=719：[24.37, 24.42, 24.35, 25.87, 29.12, 21.5, 23.59, 29.1, 24.16, 25.18, 27.78, 25.42, 28.03, 22.79, 24.76, 23.49, 25.09, 25.43, 28.7, 28.73, 24.72, 27.34, 21.93, 24.21, 26.82, 23.92, 25.57, 25.16, 24.74, 31.46, 23.07, 26.16, 31.01, 26.51, 23.29, 30.71, 24.21, 29.0, 25.15, 27.85, 22.7, 23.77, 24.18, 22.56, 25.09, 22.43, 23.67, 26.36, 24.86, 25.71, 25.9, 28.82, 25.75, 22.92, 25.92, 22.69, 21.62, 28.68, 28.19, 26.18, 25.11, 23.19, 26.23, 26.59, 22.86, 18.03, 24.58, 25.54, 24.17, 27.73, 21.47, 26.13, 25.21, 24.21, 29.32, 22.11, 23.63, 28.4, 20.93, 22.56, 26.89, 28.7, 28.2, 26.63, 29.2, 23.38, 26.73, 26.84, 28.43, 29.01, 25.2, 23.76, 21.69, 24.92, 25.31, 26.13, 26.11, 24.62, 22.42, 26.31, 25.97, 23.76, 19.61, 25.58, 25.8, 26.54, 23.78, 21.02, 28.35, 27.49, 22.54, 26.35, 23.06, 23.46, 24.47, 27.78, 27.52, 23.52, 28.05, 23.98, 23.9, 23.69, 29.16, 26.9, 22.61, 26.1, 24.74, 22.95, 22.97, 22.55, 25.63, 26.08, 24.59, 27.91, 26.25, 29.16, 25.55, 23.19, 25.34, 27.79, 23.66, 28.11, 26.98, 26.61, 23.26, 27.78, 27.81, 26.09, 24.52, 20.23, 25.37, 26.2, 27.96, 23.57, 26.3, 25.25, 26.36, 28.5, 22.53, 22.02, 26.31, 24.35, 23.75, 23.73, 23.39, 20.96, 25.09, 27.26, 26.93, 24.74, 24.92, 27.77, 28.38, 23.67, 25.01, 24.65, 26.21, 29.29, 26.27, 23.65, 24.44, 27.3, 25.29, 25.99, 26.0, 25.71, 22.45, 24.49, 25.52, 26.76, 24.98, 22.92, 27.97, 26.09, 25.33, 26.73, 27.84, 26.65, 26.36, 26.31, 24.63, 29.36, 27.78, 24.64, 25.39, 23.95, 24.84, 23.12, 22.61, 27.33, 34.07, 26.76, 25.03, 27.55, 25.42, 24.52, 21.9, 24.8, 24.3, 24.33, 25.54, 22.6, 20.22, 25.65, 25.75, 20.69, 27.73, 24.74, 27.14, 25.32, 23.76, 20.1, 21.63, 25.21, 28.15, 23.67, 23.93, 27.66, 26.8, 27.31, 24.72, 25.06, 24.87, 23.67, 24.85, 25.18, 26.75, 28.5, 25.26, 24.57, 25.3, 23.0, 29.24, 25.34, 21.47, 28.47, 25.48, 26.38, 24.91, 25.75, 24.64, 23.02, 27.93, 24.96, 25.39, 22.81, 27.64, 28.33, 21.98, 27.29, 26.59, 26.93, 26.92, 21.63, 23.49, 24.94, 27.33, 25.04, 24.39, 26.44, 25.23, 27.92, 24.78, 22.67, 23.13, 27.94, 27.97, 30.15, 26.48, 22.47, 23.65, 28.09, 26.22, 28.23, 25.01, 24.26, 26.3, 26.21, 23.66, 25.12, 31.58, 25.63, 24.94, 24.82, 25.56, 30.35, 25.46, 24.61, 19.61, 20.42, 24.67, 24.79, 24.43, 25.15, 30.57, 26.48, 22.95, 30.12, 29.33, 26.97, 24.55, 23.57, 25.29, 25.01, 29.9, 23.44, 23.32, 28.19, 26.7, 28.53, 28.09, 25.16, 25.53, 23.59, 22.31, 27.73, 24.99, 25.71, 23.64, 25.53, 25.74, 23.58, 24.84, 24.14, 28.05, 22.61, 28.89, 27.17, 27.19, 23.33, 24.48, 25.12, 24.18, 24.84, 26.06, 27.64, 21.63, 26.06, 25.16, 24.45, 25.47, 24.32, 24.97, 28.7, 25.09, 25.98, 25.17, 23.8, 20.17, 25.32, 24.3, 26.65, 24.85, 28.6, 25.95, 22.59, 27.41, 31.35, 29.05, 23.89, 29.93, 23.09, 26.69, 26.11, 25.14, 23.67, 25.05, 20.65, 25.75, 25.23, 25.46, 24.43, 21.9, 26.29, 24.2, 29.69, 27.32, 26.74, 26.31, 27.39, 23.88, 21.29, 24.3, 29.8, 23.28, 27.26, 29.36, 23.51, 22.04, 23.16, 27.29, 20.14, 27.35, 22.23, 24.81, 23.1, 27.76, 24.4, 21.34, 23.74, 27.26, 23.88, 27.04, 27.45, 24.94, 27.02, 22.13, 21.85, 26.39, 30.59, 29.2, 24.55, 23.77, 25.59, 23.51, 25.56, 17.27, 29.62, 26.96, 28.24, 25.83, 21.72, 26.92, 24.07, 24.61, 25.05, 23.13, 28.7, 23.45, 24.61, 19.68, 29.48, 25.28, 22.62, 23.29, 22.1, 21.28, 26.98, 28.02, 25.4, 24.67, 26.31, 24.98, 26.6, 24.01, 24.91, 22.0, 22.62, 27.55, 27.95, 29.06, 25.73, 23.53, 25.74, 26.98, 24.23, 28.76, 23.98, 27.7, 28.48, 24.98, 21.24, 24.26, 23.57, 23.54, 22.32, 21.08, 22.51, 25.42, 25.53, 25.44, 25.83, 27.31, 30.79, 21.63, 23.92, 29.43, 21.17, 21.5, 26.12, 25.51, 29.49, 24.3, 23.46, 21.52, 21.95, 24.49, 25.4, 24.49, 22.37, 24.5, 24.82, 21.36, 24.56, 27.72, 21.71, 29.94, 25.85, 21.95, 25.67, 26.08, 25.32, 27.21, 26.3, 26.19, 26.93, 27.81, 28.38, 27.46, 26.69, 21.12, 24.48, 26.48, 19.82, 26.25, 23.77, 23.11, 24.72, 29.28, 22.39, 22.51, 27.87, 27.78, 27.61, 26.15, 27.56, 25.87, 26.69, 28.52, 28.55, 24.03, 27.17, 24.78, 26.96, 23.82, 23.51, 28.35, 19.03, 27.34, 26.39, 20.7, 24.95, 24.37, 25.32, 26.71, 25.54, 22.6, 27.62, 31.36, 25.86, 30.15, 25.12, 24.66, 21.8, 24.19, 25.48, 24.46, 23.02, 22.84, 25.66, 25.4, 25.69, 23.9, 23.92, 24.79, 19.28, 27.97, 25.89, 24.0, 28.47, 22.59, 25.27, 29.86, 26.22, 28.37, 24.44, 19.97, 22.52, 31.46, 26.0, 21.85, 26.29, 21.59, 27.4, 22.6, 22.5, 28.24, 21.76, 21.48, 27.62, 24.57, 23.77, 20.74, 24.58, 23.83, 28.14, 27.87, 24.23, 26.92, 25.0, 22.71, 24.95, 25.2, 25.91, 28.1, 27.67, 26.16, 25.55, 25.87, 26.16, 22.27, 25.78, 23.65, 24.1, 25.96, 24.38, 25.29, 22.36, 22.57, 24.24, 20.99, 27.36, 25.53, 30.52, 26.42, 26.66, 26.31, 24.56, 26.68, 31.2, 27.63, 22.14, 28.93, 28.06, 20.71, 30.02, 24.96, 21.86, 24.82, 24.52, 26.96, 24.96, 24.41, 23.82, 26.97, 26.66, 25.19, 27.86, 27.96, 24.79, 24.09, 27.11, 22.29, 31.92, 24.38, 29.72, 25.05, 23.85, 25.1, 26.8, 23.14, 22.05, 29.5, 29.04, 22.83, 24.13, 22.49, 25.75, 28.4, 23.4, 28.96, 26.04, 23.98, 23.85, 22.97, 21.71, 26.39, 25.17, 26.77, 23.74, 23.27, 24.87, 25.03, 25.75, 24.7, 26.54, 25.92, 25.42, 23.61, 29.45, 30.52, 24.09, 26.72, 21.71, 24.11, 33.02, 26.0, 26.44, 23.78]；
差值(后−前) n=719：[-6.12, -5.82, -4.06, -6.7, -4.35, -4.51, -5.35, -4.87, -5.3, -5.13, -4.54, -3.74, -3.27, -4.95, -5.76, -3.99, -4.76, -4.33, -2.94, -3.79, -4.93, -3.76, -5.5, -4.96, -4.32, -5.13, -5.15, -2.85, -6.23, -3.94, -5.33, -6.6, -3.32, -5.02, -5.12, -3.63, -5.42, -4.44, -4.19, -4.24, -3.84, -5.44, -7.16, -5.29, -4.13, -5.78, -5.31, -4.92, -5.34, -3.5, -4.5, -4.75, -4.61, -5.78, -4.62, -5.13, -6.17, -3.93, -5.39, -4.25, -4.84, -5.38, -2.43, -4.56, -6.79, -4.36, -6.84, -3.58, -5.55, -3.97, -5.41, -5.0, -3.79, -5.17, -3.02, -5.44, -4.18, -4.12, -5.71, -5.47, -5.44, -2.99, -3.62, -4.85, -4.13, -6.06, -4.72, -3.73, -4.3, -3.42, -5.36, -4.19, -5.74, -6.86, -4.56, -5.57, -3.61, -6.21, -5.28, -4.03, -4.87, -4.5, -4.5, -4.54, -5.59, -4.55, -4.18, -6.03, -4.78, -5.08, -5.1, -5.06, -4.68, -4.37, -4.76, -3.52, -3.38, -4.03, -4.3, -4.36, -4.28, -4.28, -2.42, -5.15, -4.86, -5.52, -4.79, -5.23, -6.26, -5.16, -3.36, -3.32, -5.96, -3.79, -3.98, -3.53, -5.45, -6.45, -4.96, -5.15, -4.8, -4.01, -3.96, -5.83, -4.45, -6.45, -3.73, -3.89, -4.84, -4.4, -6.7, -4.67, -4.8, -5.36, -3.13, -4.32, -4.97, -4.4, -6.42, -3.79, -6.2, -3.98, -6.42, -4.24, -5.55, -5.51, -5.78, -4.19, -5.85, -4.06, -5.32, -4.01, -3.21, -5.83, -4.21, -5.58, -4.63, -5.25, -4.74, -4.76, -4.72, -5.66, -4.02, -3.99, -5.34, -4.7, -4.47, -5.8, -4.82, -4.99, -4.15, -5.6, -5.26, -7.13, -4.23, -4.67, -5.38, -5.57, -5.17, -5.8, -4.92, -2.61, -4.63, -4.69, -4.8, -4.97, -5.11, -6.6, -5.95, -4.85, -4.99, -4.72, -6.5, -3.99, -2.68, -4.5, -6.28, -5.91, -3.55, -4.5, -3.8, -5.91, -3.4, -4.22, -1.95, -5.08, -4.87, -5.76, -5.44, -4.77, -7.12, -5.69, -6.58, -5.73, -3.88, -6.3, -3.85, -3.9, -5.03, -4.03, -2.22, -5.85, -4.47, -5.02, -4.81, -5.36, -5.84, -3.6, -4.76, -5.34, -4.51, -4.8, -2.61, -4.37, -4.18, -4.42, -4.19, -4.68, -6.49, -3.49, -5.19, -6.21, -3.96, -5.68, -5.07, -5.4, -4.19, -4.18, -5.79, -5.09, -5.24, -1.61, -4.33, -5.09, -6.35, -6.29, -3.89, -4.27, -6.25, -3.88, -6.38, -4.49, -5.0, -4.61, -5.07, -5.43, -2.56, -3.4, -3.77, -5.61, -4.71, -5.02, -5.06, -4.23, -4.33, -4.8, -5.6, -5.01, -5.31, -3.96, -2.93, -4.88, -4.75, -4.28, -3.45, -5.17, -5.59, -5.11, -4.97, -6.2, -5.36, -5.65, -4.45, -3.6, -1.97, -6.72, -5.33, -5.05, -4.97, -4.01, -2.56, -5.51, -5.03, -5.09, -5.79, -3.74, -4.23, -5.56, -4.58, -5.41, -5.39, -3.2, -5.75, -4.5, -6.05, -3.29, -5.12, -5.41, -3.75, -4.95, -3.84, -6.38, -4.7, -5.07, -3.1, -5.87, -3.17, -4.2, -5.34, -3.79, -4.36, -5.69, -4.91, -4.41, -5.12, -5.02, -5.38, -5.08, -5.64, -6.3, -3.25, -3.97, -4.89, -4.7, -4.94, -4.59, -5.38, -3.35, -4.9, -4.69, -4.42, -4.07, -4.17, -4.73, -5.6, -5.14, -6.33, -3.66, -4.15, -4.8, -3.8, -6.03, -4.32, -4.12, -4.68, -4.54, -3.86, -5.5, -4.47, -3.47, -5.2, -4.83, -5.85, -4.24, -4.81, -5.02, -4.09, -5.53, -6.03, -5.84, -4.18, -6.16, -4.08, -3.53, -4.15, -5.47, -3.31, -6.2, -6.83, -4.67, -5.55, -5.35, -4.53, -6.11, -4.71, -5.29, -6.44, -3.94, -5.01, -5.31, -5.35, -5.25, -4.77, -4.18, -5.11, -3.91, -5.78, -4.81, -5.56, -4.11, -3.69, -6.11, -4.01, -5.06, -4.56, -5.06, -5.52, -4.47, -5.29, -5.64, -4.74, -5.44, -5.41, -5.13, -5.02, -3.7, -3.87, -2.31, -5.48, -5.26, -5.66, -4.91, -5.41, -5.84, -4.82, -3.56, -7.41, -3.3, -3.86, -4.84, -4.99, -3.69, -4.95, -3.61, -4.46, -3.43, -5.93, -4.74, -5.21, -3.69, -4.21, -5.51, -4.76, -4.51, -3.23, -3.36, -4.46, -5.87, -3.68, -4.95, -4.45, -6.91, -4.93, -4.15, -5.46, -4.47, -4.65, -4.39, -3.97, -6.27, -4.2, -4.21, -5.37, -4.05, -5.5, -4.98, -4.21, -5.06, -5.15, -4.05, -4.35, -3.85, -4.93, -4.16, -4.93, -5.04, -6.26, -5.75, -5.15, -6.45, -3.87, -4.34, -6.15, -4.83, -3.03, -5.24, -4.84, -4.4, -5.05, -4.53, -4.77, -4.33, -4.4, -4.03, -4.15, -4.69, -4.91, -4.09, -4.75, -4.86, -5.33, -6.13, -5.14, -5.68, -5.94, -6.21, -5.16, -3.91, -4.85, -5.51, -2.69, -4.84, -4.1, -5.81, -4.81, -4.69, -4.92, -3.71, -4.42, -4.21, -5.65, -3.62, -5.11, -5.05, -5.06, -5.52, -3.5, -5.29, -3.98, -6.18, -6.08, -4.52, -4.61, -4.59, -4.61, -5.1, -5.49, -3.86, -3.31, -3.68, -4.26, -3.99, -4.35, -4.77, -5.13, -5.64, -4.26, -4.71, -5.08, -5.23, -4.56, -5.41, -3.89, -6.61, -5.6, -4.26, -4.1, -5.7, -5.03, -5.44, -5.49, -5.36, -3.28, -4.55, -4.32, -4.94, -6.24, -2.8, -3.54, -4.83, -5.65, -4.65, -6.47, -5.06, -7.07, -5.7, -3.9, -6.36, -5.32, -3.06, -5.69, -4.68, -5.04, -5.14, -4.9, -5.04, -4.75, -4.63, -5.78, -7.21, -4.75, -5.95, -4.65, -5.19, -4.33, -4.52, -5.34, -5.54, -5.4, -5.01, -4.25, -4.16, -4.35, -3.29, -6.75, -4.67, -4.65, -7.6, -6.07, -4.19, -8.09, -3.83, -5.06, -3.13, -5.63, -4.35, -3.58, -5.56, -4.51, -5.04, -5.19, -4.89, -4.57, -6.21, -6.63, -5.2, -5.2, -5.76, -6.19, -4.33, -4.89, -4.27, -3.51, -5.42, -4.07, -5.35, -4.72, -4.63, -5.2, -4.29, -6.22, -4.5, -6.09, -3.4, -5.03, -3.73, -5.7, -5.45, -5.07, -5.16, -5.23, -6.85, -4.82, -5.32, -5.52, -3.77, -4.31, -5.84, -6.21, -4.8, -2.35, -4.28, -5.25, -4.82, -5.15, -4.37, -5.45, -4.48, -4.62, -5.83, -5.01, -5.94, -5.09, -4.6, -4.43, -3.85, -4.66, -5.02, -4.62, -2.94, -4.39, -6.53, -5.72, -4.15, -6.14, -3.45, -5.21, -3.45, -8.13]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -4.8267，s_d = 0.9441，SE = s_d/√n = 0.0352；
t = d̄/SE = -137.0871；df = n−1 = 718；双侧 p = 0.000e+00。
d̄ = -4.8267，s_d = 0.9441，SE = s_d/√n = 0.0352；
t = d̄/SE = -137.0871；df = n−1 = 718；双侧 p = 0.000e+00。
计算结果
t = -137.0871，df = 718，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估心理干预有效性。
案例 41. 两种测量仪器一致性
原始数据
前 n=929：[10.16, 9.76, 10.26, 10.24, 10.31, 10.56, 10.25, 10.19, 9.78, 10.11, 10.02, 10.07, 10.03, 10.18, 10.09, 9.89, 9.97, 9.97, 10.41, 10.2, 10.33, 9.9, 10.15, 10.03, 10.05, 10.23, 10.3, 9.95, 10.08, 10.09, 10.21, 9.99, 10.45, 10.19, 10.21, 9.8, 10.21, 10.08, 9.81, 10.08, 9.82, 10.3, 10.27, 10.14, 9.99, 9.94, 10.13, 10.26, 9.78, 10.22, 9.9, 10.31, 10.24, 10.11, 10.26, 9.94, 10.12, 10.31, 10.25, 9.99, 10.07, 10.03, 10.02, 10.22, 10.32, 10.31, 10.09, 10.19, 10.08, 10.56, 10.3, 10.05, 10.11, 9.87, 10.24, 9.84, 10.09, 10.16, 10.03, 9.82, 10.24, 9.93, 9.79, 10.2, 9.71, 9.95, 10.04, 10.19, 10.1, 9.7, 9.92, 10.32, 10.22, 10.43, 10.39, 10.05, 9.91, 10.23, 9.98, 10.23, 9.89, 10.13, 10.3, 9.95, 9.95, 9.89, 10.01, 9.99, 9.59, 10.12, 9.94, 9.68, 10.12, 10.08, 10.05, 10.31, 10.19, 9.82, 10.01, 9.95, 10.01, 9.84, 9.99, 10.01, 9.64, 9.97, 10.21, 10.06, 10.01, 9.92, 10.25, 9.83, 10.23, 10.05, 10.04, 10.02, 10.39, 10.23, 10.36, 10.06, 10.14, 9.93, 10.09, 10.08, 10.08, 10.27, 10.23, 9.76, 9.87, 9.9, 10.1, 10.41, 10.12, 10.13, 10.22, 9.97, 10.0, 10.16, 10.24, 9.71, 9.87, 10.48, 10.37, 10.07, 10.38, 9.94, 10.24, 10.13, 10.01, 10.0, 9.84, 10.02, 10.1, 9.94, 10.46, 10.09, 10.07, 10.41, 10.08, 9.78, 9.94, 10.09, 10.27, 10.29, 10.22, 10.14, 10.04, 10.17, 9.99, 10.1, 10.24, 10.39, 10.21, 10.05, 10.56, 10.14, 10.46, 9.81, 10.15, 9.9, 10.14, 10.14, 10.05, 9.82, 10.02, 10.07, 9.94, 9.83, 10.29, 9.8, 10.24, 10.44, 10.28, 10.14, 10.22, 10.23, 10.0, 10.11, 10.29, 9.94, 10.33, 10.15, 10.03, 10.19, 10.16, 10.03, 10.04, 9.99, 10.23, 10.54, 10.06, 9.85, 10.0, 10.06, 10.28, 10.34, 9.98, 10.09, 10.07, 10.18, 10.14, 10.37, 10.08, 9.94, 10.28, 9.76, 9.93, 9.87, 10.06, 10.21, 10.12, 9.99, 9.94, 10.02, 10.03, 9.98, 10.3, 10.24, 10.24, 10.48, 10.06, 10.36, 10.13, 10.05, 10.29, 10.19, 10.01, 9.96, 10.3, 10.28, 9.91, 9.98, 10.09, 10.24, 9.88, 9.97, 9.96, 10.29, 10.28, 10.17, 10.23, 10.22, 10.07, 10.12, 9.83, 9.85, 10.09, 9.86, 10.34, 10.06, 10.08, 10.04, 10.59, 9.92, 9.91, 10.28, 10.2, 9.81, 10.14, 10.04, 10.37, 10.3, 10.35, 9.98, 10.14, 10.4, 9.97, 9.84, 10.16, 10.23, 10.48, 10.07, 10.21, 10.43, 10.09, 10.1, 9.98, 9.92, 9.93, 10.21, 9.95, 9.99, 10.24, 10.35, 10.06, 10.04, 10.75, 9.99, 10.03, 10.14, 10.28, 10.16, 10.1, 10.05, 10.22, 10.19, 10.36, 10.04, 10.13, 10.42, 10.08, 10.12, 10.21, 10.26, 10.2, 9.88, 9.93, 10.13, 10.15, 10.07, 9.98, 10.02, 9.85, 10.07, 9.77, 10.27, 10.14, 10.39, 10.33, 10.11, 9.84, 10.1, 10.44, 10.06, 10.36, 10.44, 10.2, 9.73, 10.09, 10.17, 10.34, 10.27, 10.1, 10.13, 10.31, 9.92, 10.18, 10.41, 10.12, 10.2, 10.15, 10.16, 10.16, 10.14, 10.51, 10.22, 10.04, 10.14, 10.06, 10.12, 10.26, 10.35, 10.16, 10.22, 9.69, 10.15, 9.94, 10.15, 10.09, 10.06, 10.24, 10.24, 10.15, 10.07, 10.35, 10.32, 10.02, 10.06, 10.02, 10.1, 10.23, 10.07, 10.03, 10.27, 10.35, 9.9, 10.14, 9.92, 9.94, 9.92, 10.01, 9.83, 9.96, 9.75, 10.28, 9.94, 10.09, 9.91, 9.94, 10.46, 10.11, 10.07, 10.19, 9.84, 10.04, 10.36, 10.28, 10.32, 10.02, 10.23, 9.77, 10.22, 10.19, 10.33, 9.95, 9.99, 9.9, 10.27, 10.12, 10.21, 10.04, 10.54, 10.23, 10.15, 10.12, 10.18, 10.48, 10.17, 10.35, 9.98, 9.9, 10.43, 10.09, 10.17, 10.07, 10.34, 10.2, 10.05, 10.16, 9.78, 10.29, 10.17, 9.8, 10.14, 10.15, 10.29, 9.98, 10.14, 10.28, 10.21, 10.08, 10.15, 9.84, 10.02, 10.27, 10.18, 9.95, 10.1, 10.05, 9.99, 9.87, 10.55, 10.01, 9.83, 10.01, 9.89, 9.99, 9.84, 9.82, 10.1, 10.18, 10.4, 10.36, 10.55, 10.04, 10.17, 9.91, 10.04, 10.33, 10.15, 10.28, 10.25, 10.23, 10.05, 10.5, 10.26, 10.05, 9.89, 10.04, 9.74, 10.09, 10.19, 10.2, 10.18, 10.17, 10.49, 10.09, 10.33, 9.65, 10.02, 10.24, 10.25, 10.18, 10.29, 10.06, 9.94, 10.07, 9.94, 10.18, 9.96, 10.04, 10.14, 10.2, 9.77, 10.0, 10.13, 9.99, 10.33, 10.43, 10.02, 10.27, 10.41, 9.82, 10.27, 10.14, 9.79, 10.12, 10.32, 10.42, 10.01, 10.19, 10.08, 9.89, 9.81, 10.11, 10.32, 10.02, 10.0, 10.1, 10.12, 10.12, 10.04, 10.06, 10.17, 10.2, 10.25, 10.08, 10.09, 10.18, 10.26, 10.0, 9.88, 10.26, 9.73, 10.22, 10.12, 9.84, 9.97, 10.03, 9.97, 9.98, 10.18, 9.77, 10.11, 10.3, 10.31, 10.19, 10.21, 10.03, 10.51, 9.91, 10.22, 9.98, 10.28, 9.85, 9.92, 10.27, 9.78, 10.12, 10.23, 10.27, 9.99, 10.0, 10.12, 10.08, 10.14, 9.82, 10.28, 9.93, 9.94, 10.05, 9.93, 10.26, 9.75, 10.27, 10.02, 10.19, 10.08, 10.24, 10.06, 10.36, 10.0, 10.35, 10.04, 10.3, 10.13, 10.02, 10.03, 10.22, 10.32, 10.23, 10.14, 10.0, 10.07, 10.22, 10.41, 9.95, 10.0, 10.22, 9.96, 10.27, 9.66, 9.88, 9.77, 10.1, 10.49, 10.37, 10.06, 9.7, 10.08, 10.32, 9.75, 9.99, 9.98, 10.13, 10.3, 10.51, 10.02, 10.11, 10.18, 10.2, 10.2, 10.15, 10.38, 10.25, 10.27, 9.93, 9.94, 10.3, 9.97, 9.88, 10.23, 9.8, 10.33, 10.29, 10.39, 10.03, 9.74, 9.98, 10.3, 9.94, 10.11, 10.09, 10.32, 10.16, 9.92, 10.24, 10.58, 10.24, 9.98, 9.94, 10.1, 9.73, 9.91, 10.44, 10.08, 9.92, 9.89, 10.17, 10.12, 9.78, 10.11, 10.06, 10.16, 10.2, 10.38, 10.41, 10.22, 10.34, 10.17, 10.23, 10.34, 10.14, 10.03, 10.43, 10.25, 10.23, 9.67, 9.85, 10.3, 10.11, 10.36, 10.13, 10.13, 10.13, 10.24, 10.11, 10.21, 9.96, 10.25, 10.21, 9.87, 10.05, 10.03, 10.03, 10.2, 9.97, 10.28, 9.79, 10.21, 10.25, 10.17, 10.02, 9.85, 9.94, 9.93, 9.98, 9.98, 10.18, 9.88, 10.28, 10.25, 10.2, 10.27, 10.03, 10.27, 10.63, 10.15, 10.45, 10.13, 10.1, 10.21, 10.19, 9.93, 9.98, 9.95, 10.24, 10.2, 9.97, 10.04, 9.98, 10.24, 10.08, 10.21, 10.0, 10.1, 10.03, 10.15, 10.21, 10.06, 9.72, 9.94, 10.45, 9.67, 10.27, 10.23, 10.19, 10.22, 10.1, 9.81, 10.24, 10.01, 9.93, 10.48, 10.16, 10.48, 9.49, 9.67, 9.97, 10.24, 10.13, 10.25, 9.83, 10.28, 10.15, 10.06, 10.11, 10.33, 10.05, 10.06, 9.85, 10.13, 10.21, 10.28, 10.48, 10.07, 10.33, 10.07, 10.16, 10.03, 10.1, 10.23, 10.31, 9.78, 10.07, 10.25, 10.01, 9.96, 10.14, 9.98, 10.06, 10.04, 10.13, 10.38, 9.93, 10.15, 10.32, 9.87, 10.15, 10.11, 10.26, 10.01, 10.13, 9.9, 10.04, 10.08, 10.04, 10.42, 10.22, 10.06, 10.14, 10.33, 10.41, 10.22, 9.89, 10.27, 10.17, 10.19, 10.6, 10.18, 10.2, 9.97, 10.28, 9.88, 10.13, 10.18, 10.1, 9.99, 10.17, 10.48, 10.36, 10.29, 10.48, 9.86, 10.14, 10.03, 9.91, 9.74, 9.8, 9.92, 9.7, 9.9, 10.42, 10.27, 9.95, 10.04, 10.54, 10.23, 10.34, 9.9, 10.5, 10.01, 9.92, 9.68, 9.91, 9.83, 10.19, 10.14, 10.2, 10.17, 10.1, 10.0, 9.84, 10.12, 10.22, 9.89, 10.12, 10.14, 10.05, 10.11, 10.46, 10.12, 10.04, 9.89, 9.95, 10.18, 9.96, 9.88, 9.98]；
后 n=929：[10.01, 9.72, 10.2, 10.15, 10.21, 10.44, 10.19, 10.15, 9.63, 10.0, 9.93, 9.96, 9.98, 10.03, 10.01, 9.84, 9.88, 9.89, 10.35, 10.11, 10.18, 9.85, 10.06, 9.99, 9.98, 10.13, 10.29, 9.82, 10.03, 10.0, 10.11, 9.9, 10.39, 10.08, 10.1, 9.72, 10.13, 10.0, 9.7, 9.99, 9.81, 10.32, 10.24, 10.13, 9.94, 9.89, 10.0, 10.25, 9.7, 10.18, 9.82, 10.24, 10.2, 10.03, 10.13, 9.82, 10.09, 10.22, 10.21, 9.82, 9.98, 9.97, 9.95, 10.13, 10.26, 10.21, 10.04, 10.07, 9.97, 10.45, 10.27, 9.99, 9.99, 9.79, 10.19, 9.74, 9.9, 10.1, 9.95, 9.74, 10.14, 9.89, 9.65, 10.15, 9.63, 9.81, 9.89, 10.16, 9.98, 9.67, 9.81, 10.28, 10.05, 10.33, 10.29, 10.0, 9.83, 10.11, 9.93, 10.15, 9.81, 10.09, 10.16, 9.9, 9.85, 9.8, 9.92, 9.9, 9.51, 10.05, 9.84, 9.57, 10.02, 10.0, 10.07, 10.23, 10.1, 9.68, 9.98, 9.91, 9.87, 9.83, 9.85, 9.83, 9.58, 9.92, 10.11, 9.93, 9.95, 9.81, 10.23, 9.74, 10.17, 9.99, 9.94, 9.89, 10.26, 10.14, 10.3, 9.94, 10.04, 9.88, 10.06, 9.99, 10.02, 10.2, 10.1, 9.71, 9.81, 9.7, 10.01, 10.37, 10.09, 10.06, 10.13, 9.9, 10.03, 10.09, 10.19, 9.62, 9.8, 10.36, 10.25, 10.02, 10.25, 9.81, 10.19, 10.04, 10.0, 9.89, 9.78, 9.98, 10.04, 9.84, 10.4, 9.97, 10.05, 10.36, 9.96, 9.72, 9.83, 9.92, 10.22, 10.21, 10.17, 10.02, 9.99, 10.1, 9.88, 10.03, 10.15, 10.28, 10.11, 10.03, 10.51, 10.01, 10.42, 9.73, 10.04, 9.86, 10.06, 10.07, 9.99, 9.75, 9.9, 9.98, 9.88, 9.74, 10.22, 9.73, 10.17, 10.37, 10.19, 10.1, 10.12, 10.11, 9.89, 10.03, 10.18, 9.89, 10.25, 10.06, 10.01, 10.1, 10.08, 9.91, 9.97, 9.91, 10.17, 10.53, 9.93, 9.77, 9.92, 9.99, 10.23, 10.25, 9.92, 10.05, 10.01, 10.07, 10.05, 10.23, 9.94, 9.89, 10.16, 9.67, 9.83, 9.7, 10.0, 10.14, 10.03, 9.93, 9.85, 10.0, 9.92, 9.96, 10.22, 10.09, 10.13, 10.43, 9.99, 10.31, 10.08, 9.93, 10.24, 10.11, 9.93, 9.91, 10.17, 10.28, 9.89, 9.87, 10.04, 10.14, 9.81, 9.82, 9.91, 10.16, 10.21, 10.07, 10.09, 10.15, 10.01, 10.01, 9.77, 9.73, 9.97, 9.78, 10.31, 9.93, 10.05, 9.94, 10.51, 9.85, 9.87, 10.28, 10.06, 9.72, 10.09, 9.89, 10.28, 10.21, 10.33, 9.88, 10.02, 10.34, 9.87, 9.75, 10.08, 10.13, 10.41, 10.04, 10.1, 10.29, 9.95, 10.01, 9.92, 9.84, 9.91, 10.1, 9.88, 9.86, 10.18, 10.3, 10.02, 10.03, 10.65, 9.86, 9.94, 10.05, 10.18, 10.07, 10.04, 9.97, 10.11, 10.12, 10.21, 9.93, 10.06, 10.36, 10.02, 10.07, 10.07, 10.2, 10.12, 9.79, 9.9, 10.13, 10.02, 9.97, 9.96, 9.93, 9.79, 9.95, 9.6, 10.16, 10.02, 10.26, 10.27, 10.05, 9.76, 10.06, 10.37, 9.88, 10.36, 10.38, 10.16, 9.64, 10.0, 10.08, 10.19, 10.2, 10.03, 10.09, 10.17, 9.77, 10.04, 10.29, 10.09, 10.14, 10.08, 10.05, 10.03, 9.97, 10.41, 10.16, 9.94, 10.05, 9.97, 10.07, 10.16, 10.34, 10.06, 10.05, 9.61, 10.12, 9.84, 10.05, 10.09, 10.02, 10.18, 10.15, 10.12, 10.11, 10.32, 10.3, 9.91, 9.94, 9.98, 10.01, 10.12, 10.01, 10.01, 10.16, 10.29, 9.81, 10.09, 9.84, 9.83, 9.79, 9.94, 9.76, 9.86, 9.67, 10.18, 9.82, 9.97, 9.89, 9.83, 10.36, 9.98, 9.95, 10.1, 9.67, 9.93, 10.3, 10.18, 10.24, 9.95, 10.2, 9.72, 10.16, 10.1, 10.14, 9.85, 9.92, 9.84, 10.21, 10.04, 10.14, 9.89, 10.46, 10.09, 10.04, 10.01, 10.1, 10.38, 10.17, 10.23, 9.93, 9.73, 10.27, 9.97, 10.06, 9.95, 10.33, 10.12, 10.02, 10.08, 9.73, 10.15, 10.14, 9.64, 10.01, 10.07, 10.22, 9.9, 10.06, 10.27, 10.15, 9.95, 10.07, 9.71, 9.96, 10.25, 10.02, 9.9, 9.97, 10.0, 9.83, 9.81, 10.49, 9.94, 9.74, 9.87, 9.83, 9.92, 9.75, 9.71, 9.93, 10.04, 10.34, 10.27, 10.42, 9.95, 10.11, 9.85, 9.92, 10.24, 10.07, 10.13, 10.18, 10.09, 9.98, 10.42, 10.17, 9.95, 9.73, 9.93, 9.63, 10.06, 10.09, 10.06, 10.11, 10.04, 10.42, 9.98, 10.28, 9.6, 9.94, 10.19, 10.2, 10.1, 10.23, 10.03, 9.87, 10.02, 9.8, 9.99, 9.82, 9.96, 10.05, 10.11, 9.68, 9.96, 10.01, 9.92, 10.26, 10.4, 9.88, 10.16, 10.4, 9.71, 10.2, 10.05, 9.75, 10.12, 10.26, 10.32, 9.87, 10.13, 10.04, 9.77, 9.79, 9.98, 10.27, 9.96, 9.88, 9.95, 10.04, 10.02, 9.93, 9.99, 10.09, 10.08, 10.14, 10.01, 10.01, 10.12, 10.18, 9.99, 9.83, 10.14, 9.65, 10.07, 10.03, 9.72, 9.91, 10.01, 9.84, 9.9, 10.16, 9.72, 10.0, 10.19, 10.24, 10.06, 10.15, 9.9, 10.47, 9.81, 10.1, 9.91, 10.18, 9.79, 9.8, 10.23, 9.62, 10.13, 10.16, 10.25, 9.9, 9.85, 10.08, 9.92, 10.11, 9.79, 10.15, 9.86, 9.86, 9.96, 9.77, 10.2, 9.67, 10.1, 9.92, 10.11, 10.04, 10.1, 9.94, 10.24, 10.0, 10.29, 9.96, 10.24, 10.1, 9.92, 9.9, 10.09, 10.24, 10.12, 10.03, 9.92, 9.99, 10.19, 10.3, 9.88, 9.88, 10.06, 9.83, 10.25, 9.62, 9.77, 9.69, 10.11, 10.39, 10.26, 9.98, 9.62, 10.0, 10.26, 9.69, 9.93, 9.94, 9.98, 10.18, 10.41, 9.91, 10.05, 10.1, 10.17, 10.13, 10.07, 10.29, 10.1, 10.23, 9.91, 9.89, 10.14, 9.89, 9.82, 10.17, 9.68, 10.22, 10.26, 10.3, 9.95, 9.63, 9.88, 10.21, 9.93, 10.06, 10.04, 10.27, 10.12, 9.89, 10.13, 10.49, 10.17, 9.87, 9.86, 10.0, 9.62, 9.82, 10.39, 10.08, 9.87, 9.82, 10.12, 10.03, 9.69, 10.07, 9.94, 10.06, 10.05, 10.3, 10.37, 10.13, 10.33, 10.04, 10.11, 10.32, 10.0, 9.83, 10.34, 10.17, 10.13, 9.52, 9.68, 10.21, 9.99, 10.29, 10.08, 10.08, 10.06, 10.1, 10.06, 10.29, 9.82, 10.1, 10.12, 9.75, 9.95, 9.94, 9.9, 10.2, 9.84, 10.12, 9.73, 10.14, 10.18, 10.12, 9.95, 9.78, 9.84, 9.84, 9.91, 9.88, 10.15, 9.79, 10.21, 10.13, 10.09, 10.15, 9.9, 10.13, 10.59, 10.14, 10.37, 10.16, 10.05, 10.13, 10.09, 9.79, 9.96, 9.82, 10.12, 10.08, 9.83, 9.95, 9.88, 10.17, 10.1, 10.09, 9.93, 10.02, 9.94, 10.05, 10.1, 9.94, 9.73, 9.92, 10.35, 9.6, 10.14, 10.15, 10.19, 10.19, 10.13, 9.75, 10.13, 9.94, 9.84, 10.43, 10.05, 10.34, 9.4, 9.59, 9.92, 10.17, 10.08, 10.21, 9.69, 10.23, 10.1, 10.0, 10.12, 10.25, 9.92, 10.05, 9.76, 10.0, 10.1, 10.22, 10.42, 10.03, 10.19, 10.04, 10.02, 9.94, 10.04, 10.11, 10.23, 9.64, 10.0, 10.1, 9.88, 9.85, 10.09, 9.95, 10.01, 9.96, 9.99, 10.24, 9.84, 10.07, 10.27, 9.82, 10.05, 10.07, 10.16, 9.93, 10.0, 9.71, 9.96, 10.0, 9.97, 10.36, 10.08, 9.99, 10.07, 10.24, 10.3, 10.1, 9.76, 10.19, 10.11, 10.08, 10.54, 10.16, 10.13, 9.93, 10.2, 9.8, 9.99, 10.07, 10.01, 9.99, 10.07, 10.39, 10.3, 10.3, 10.42, 9.7, 10.09, 9.96, 9.84, 9.67, 9.67, 9.85, 9.58, 9.78, 10.3, 10.19, 9.91, 10.01, 10.41, 10.2, 10.25, 9.8, 10.41, 9.94, 9.85, 9.59, 9.78, 9.68, 10.08, 10.06, 10.1, 10.08, 9.98, 9.9, 9.8, 10.04, 10.17, 9.8, 10.04, 10.08, 9.93, 10.01, 10.35, 10.05, 9.93, 9.82, 9.88, 10.08, 9.87, 9.74, 9.93]；
差值(后−前) n=929：[-0.15, -0.04, -0.06, -0.09, -0.1, -0.12, -0.05, -0.04, -0.15, -0.1, -0.09, -0.11, -0.05, -0.16, -0.08, -0.05, -0.08, -0.08, -0.06, -0.1, -0.15, -0.04, -0.08, -0.04, -0.07, -0.1, -0.01, -0.13, -0.05, -0.1, -0.11, -0.09, -0.07, -0.12, -0.11, -0.08, -0.08, -0.08, -0.1, -0.09, -0.0, 0.02, -0.03, -0.01, -0.05, -0.04, -0.13, -0.01, -0.09, -0.04, -0.09, -0.07, -0.04, -0.08, -0.13, -0.12, -0.03, -0.08, -0.03, -0.17, -0.1, -0.06, -0.07, -0.09, -0.06, -0.09, -0.05, -0.12, -0.12, -0.11, -0.03, -0.05, -0.12, -0.07, -0.04, -0.09, -0.19, -0.06, -0.08, -0.08, -0.1, -0.05, -0.14, -0.04, -0.07, -0.15, -0.16, -0.04, -0.12, -0.03, -0.11, -0.04, -0.17, -0.1, -0.1, -0.06, -0.08, -0.12, -0.05, -0.08, -0.09, -0.04, -0.14, -0.05, -0.1, -0.08, -0.1, -0.08, -0.08, -0.07, -0.11, -0.11, -0.1, -0.08, 0.02, -0.08, -0.09, -0.15, -0.02, -0.04, -0.14, -0.01, -0.14, -0.18, -0.07, -0.05, -0.11, -0.14, -0.06, -0.11, -0.02, -0.08, -0.05, -0.05, -0.1, -0.13, -0.12, -0.09, -0.06, -0.12, -0.11, -0.05, -0.03, -0.09, -0.06, -0.07, -0.13, -0.05, -0.06, -0.2, -0.09, -0.04, -0.03, -0.06, -0.09, -0.07, 0.03, -0.07, -0.05, -0.09, -0.07, -0.12, -0.12, -0.05, -0.13, -0.12, -0.05, -0.09, -0.01, -0.12, -0.06, -0.04, -0.06, -0.1, -0.06, -0.11, -0.02, -0.06, -0.11, -0.06, -0.11, -0.17, -0.06, -0.09, -0.05, -0.11, -0.05, -0.07, -0.11, -0.07, -0.09, -0.1, -0.11, -0.02, -0.05, -0.13, -0.04, -0.08, -0.11, -0.04, -0.08, -0.07, -0.06, -0.07, -0.12, -0.09, -0.05, -0.08, -0.07, -0.07, -0.07, -0.07, -0.09, -0.04, -0.1, -0.12, -0.12, -0.08, -0.11, -0.05, -0.08, -0.09, -0.02, -0.09, -0.08, -0.11, -0.07, -0.08, -0.06, -0.01, -0.12, -0.08, -0.08, -0.07, -0.04, -0.09, -0.07, -0.04, -0.07, -0.12, -0.1, -0.14, -0.14, -0.04, -0.12, -0.1, -0.11, -0.17, -0.06, -0.07, -0.09, -0.07, -0.09, -0.01, -0.11, -0.02, -0.09, -0.14, -0.11, -0.05, -0.07, -0.05, -0.05, -0.13, -0.05, -0.08, -0.08, -0.06, -0.13, 0.0, -0.03, -0.11, -0.05, -0.1, -0.07, -0.15, -0.05, -0.13, -0.07, -0.1, -0.14, -0.07, -0.06, -0.1, -0.06, -0.12, -0.11, -0.08, -0.03, -0.13, -0.04, -0.11, -0.08, -0.07, -0.04, -0.01, -0.14, -0.09, -0.05, -0.15, -0.09, -0.09, -0.02, -0.1, -0.12, -0.06, -0.09, -0.09, -0.08, -0.1, -0.07, -0.03, -0.1, -0.14, -0.14, -0.09, -0.07, -0.07, -0.03, -0.1, -0.07, -0.13, -0.06, -0.05, -0.03, -0.01, -0.1, -0.13, -0.09, -0.09, -0.1, -0.1, -0.06, -0.08, -0.12, -0.07, -0.15, -0.11, -0.07, -0.06, -0.06, -0.05, -0.14, -0.06, -0.07, -0.09, -0.03, -0.01, -0.13, -0.1, -0.02, -0.09, -0.06, -0.12, -0.16, -0.11, -0.11, -0.14, -0.06, -0.06, -0.09, -0.04, -0.08, -0.19, 0.0, -0.06, -0.04, -0.08, -0.09, -0.1, -0.15, -0.08, -0.08, -0.04, -0.14, -0.15, -0.14, -0.11, -0.03, -0.06, -0.07, -0.11, -0.13, -0.16, -0.1, -0.07, -0.1, -0.1, -0.09, -0.06, -0.1, -0.02, -0.1, -0.16, -0.08, -0.03, -0.1, -0.1, -0.01, -0.04, -0.06, -0.09, -0.03, 0.04, -0.04, -0.02, -0.11, -0.12, -0.04, -0.1, -0.1, -0.06, -0.02, -0.11, -0.07, -0.09, -0.05, -0.08, -0.11, -0.13, -0.07, -0.07, -0.1, -0.08, -0.09, -0.11, -0.12, -0.03, -0.11, -0.1, -0.13, -0.12, -0.09, -0.17, -0.12, -0.06, -0.1, -0.09, -0.07, -0.02, -0.05, -0.06, -0.09, -0.19, -0.1, -0.07, -0.06, -0.06, -0.08, -0.07, -0.15, -0.08, -0.14, -0.11, -0.11, -0.08, -0.1, -0.0, -0.12, -0.04, -0.17, -0.16, -0.12, -0.11, -0.13, -0.0, -0.08, -0.03, -0.07, -0.05, -0.14, -0.03, -0.16, -0.13, -0.09, -0.07, -0.08, -0.08, -0.01, -0.06, -0.13, -0.08, -0.13, -0.06, -0.03, -0.15, -0.05, -0.13, -0.05, -0.15, -0.07, -0.05, -0.07, -0.09, -0.14, -0.07, -0.08, -0.09, -0.11, -0.17, -0.15, -0.06, -0.09, -0.13, -0.09, -0.06, -0.06, -0.13, -0.09, -0.08, -0.15, -0.07, -0.15, -0.07, -0.08, -0.09, -0.09, -0.16, -0.11, -0.11, -0.03, -0.1, -0.14, -0.07, -0.12, -0.07, -0.11, -0.05, -0.05, -0.08, -0.05, -0.05, -0.08, -0.06, -0.03, -0.08, -0.05, -0.14, -0.2, -0.14, -0.08, -0.09, -0.09, -0.09, -0.05, -0.12, -0.08, -0.07, -0.03, -0.14, -0.11, -0.01, -0.11, -0.06, -0.1, -0.03, 0.0, -0.05, -0.1, -0.15, -0.06, -0.04, -0.11, -0.02, -0.12, -0.05, -0.06, -0.11, -0.15, -0.08, -0.1, -0.11, -0.06, -0.08, -0.12, -0.11, -0.07, -0.09, -0.06, -0.08, -0.01, -0.05, -0.12, -0.09, -0.15, -0.09, -0.11, -0.07, -0.02, -0.13, -0.08, -0.02, -0.05, -0.1, -0.1, -0.07, -0.13, -0.06, -0.13, -0.04, -0.1, -0.12, -0.06, -0.1, -0.07, -0.12, -0.04, -0.16, 0.01, -0.07, -0.02, -0.08, -0.15, -0.03, -0.16, -0.03, -0.04, -0.13, -0.07, -0.08, -0.09, -0.16, -0.06, -0.08, -0.17, -0.1, -0.08, -0.03, -0.14, -0.13, -0.13, -0.0, -0.06, -0.08, -0.06, -0.04, -0.1, -0.13, -0.12, -0.08, -0.12, -0.11, -0.07, -0.08, -0.03, -0.12, -0.07, -0.12, -0.16, -0.13, -0.03, -0.05, -0.11, -0.08, 0.01, -0.11, -0.12, -0.07, -0.08, -0.08, -0.06, -0.06, -0.06, -0.04, -0.15, -0.12, -0.1, -0.11, -0.06, -0.08, -0.03, -0.07, -0.09, -0.09, -0.15, -0.04, -0.03, -0.05, -0.15, -0.08, -0.05, -0.06, -0.12, -0.11, -0.03, -0.1, -0.08, -0.11, -0.11, -0.1, -0.01, -0.05, -0.04, -0.06, -0.04, -0.02, -0.11, -0.09, -0.07, -0.11, -0.09, -0.09, -0.11, -0.09, -0.05, 0.0, -0.05, -0.08, -0.05, -0.09, -0.09, -0.04, -0.12, -0.1, -0.15, -0.08, -0.05, -0.09, -0.01, -0.13, -0.12, -0.02, -0.14, -0.2, -0.08, -0.08, -0.11, -0.15, -0.17, -0.09, -0.12, -0.07, -0.05, -0.05, -0.07, -0.13, -0.05, 0.08, -0.14, -0.15, -0.09, -0.12, -0.1, -0.09, -0.12, -0.01, -0.12, -0.16, -0.06, -0.07, -0.07, -0.04, -0.07, -0.08, -0.09, -0.09, -0.06, -0.1, -0.04, -0.09, -0.07, -0.12, -0.11, -0.11, -0.13, -0.14, -0.04, -0.01, -0.08, 0.03, -0.05, -0.08, -0.11, -0.14, -0.01, -0.13, -0.12, -0.12, -0.14, -0.09, -0.09, -0.07, 0.02, -0.12, -0.07, -0.08, -0.09, -0.1, -0.11, -0.11, 0.01, -0.03, -0.1, -0.07, -0.14, -0.09, 0.01, -0.04, 0.03, -0.06, -0.1, -0.07, -0.09, -0.05, -0.11, -0.13, -0.1, -0.08, -0.04, -0.07, -0.05, -0.04, -0.15, -0.05, -0.06, -0.06, 0.01, -0.08, -0.13, -0.01, -0.09, -0.13, -0.11, -0.06, -0.06, -0.04, -0.14, -0.03, -0.15, -0.09, -0.06, -0.12, -0.08, -0.14, -0.07, -0.15, -0.13, -0.11, -0.05, -0.03, -0.06, -0.08, -0.14, -0.14, -0.08, -0.08, -0.06, -0.05, -0.11, -0.04, -0.1, -0.08, -0.13, -0.2, -0.08, -0.08, -0.06, -0.06, -0.14, -0.07, -0.07, -0.08, -0.12, -0.12, -0.12, -0.08, -0.06, -0.11, -0.06, -0.01, -0.06, -0.04, -0.08, -0.08, -0.14, -0.12, -0.08, -0.0, -0.11, -0.09, -0.06, 0.0, -0.07, -0.16, -0.04, -0.07, -0.07, -0.07, -0.12, -0.07, -0.12, -0.11, -0.12, -0.08, -0.04, -0.03, -0.13, -0.03, -0.09, -0.1, -0.09, -0.07, -0.07, -0.08, -0.13, -0.15, -0.11, -0.09, -0.1, -0.09, -0.12, -0.1, -0.04, -0.08, -0.04, -0.08, -0.08, -0.06, -0.13, -0.1, -0.11, -0.07, -0.1, -0.08, -0.08, -0.1, -0.09, -0.14, -0.06]
后 n=929：[10.01, 9.72, 10.2, 10.15, 10.21, 10.44, 10.19, 10.15, 9.63, 10.0, 9.93, 9.96, 9.98, 10.03, 10.01, 9.84, 9.88, 9.89, 10.35, 10.11, 10.18, 9.85, 10.06, 9.99, 9.98, 10.13, 10.29, 9.82, 10.03, 10.0, 10.11, 9.9, 10.39, 10.08, 10.1, 9.72, 10.13, 10.0, 9.7, 9.99, 9.81, 10.32, 10.24, 10.13, 9.94, 9.89, 10.0, 10.25, 9.7, 10.18, 9.82, 10.24, 10.2, 10.03, 10.13, 9.82, 10.09, 10.22, 10.21, 9.82, 9.98, 9.97, 9.95, 10.13, 10.26, 10.21, 10.04, 10.07, 9.97, 10.45, 10.27, 9.99, 9.99, 9.79, 10.19, 9.74, 9.9, 10.1, 9.95, 9.74, 10.14, 9.89, 9.65, 10.15, 9.63, 9.81, 9.89, 10.16, 9.98, 9.67, 9.81, 10.28, 10.05, 10.33, 10.29, 10.0, 9.83, 10.11, 9.93, 10.15, 9.81, 10.09, 10.16, 9.9, 9.85, 9.8, 9.92, 9.9, 9.51, 10.05, 9.84, 9.57, 10.02, 10.0, 10.07, 10.23, 10.1, 9.68, 9.98, 9.91, 9.87, 9.83, 9.85, 9.83, 9.58, 9.92, 10.11, 9.93, 9.95, 9.81, 10.23, 9.74, 10.17, 9.99, 9.94, 9.89, 10.26, 10.14, 10.3, 9.94, 10.04, 9.88, 10.06, 9.99, 10.02, 10.2, 10.1, 9.71, 9.81, 9.7, 10.01, 10.37, 10.09, 10.06, 10.13, 9.9, 10.03, 10.09, 10.19, 9.62, 9.8, 10.36, 10.25, 10.02, 10.25, 9.81, 10.19, 10.04, 10.0, 9.89, 9.78, 9.98, 10.04, 9.84, 10.4, 9.97, 10.05, 10.36, 9.96, 9.72, 9.83, 9.92, 10.22, 10.21, 10.17, 10.02, 9.99, 10.1, 9.88, 10.03, 10.15, 10.28, 10.11, 10.03, 10.51, 10.01, 10.42, 9.73, 10.04, 9.86, 10.06, 10.07, 9.99, 9.75, 9.9, 9.98, 9.88, 9.74, 10.22, 9.73, 10.17, 10.37, 10.19, 10.1, 10.12, 10.11, 9.89, 10.03, 10.18, 9.89, 10.25, 10.06, 10.01, 10.1, 10.08, 9.91, 9.97, 9.91, 10.17, 10.53, 9.93, 9.77, 9.92, 9.99, 10.23, 10.25, 9.92, 10.05, 10.01, 10.07, 10.05, 10.23, 9.94, 9.89, 10.16, 9.67, 9.83, 9.7, 10.0, 10.14, 10.03, 9.93, 9.85, 10.0, 9.92, 9.96, 10.22, 10.09, 10.13, 10.43, 9.99, 10.31, 10.08, 9.93, 10.24, 10.11, 9.93, 9.91, 10.17, 10.28, 9.89, 9.87, 10.04, 10.14, 9.81, 9.82, 9.91, 10.16, 10.21, 10.07, 10.09, 10.15, 10.01, 10.01, 9.77, 9.73, 9.97, 9.78, 10.31, 9.93, 10.05, 9.94, 10.51, 9.85, 9.87, 10.28, 10.06, 9.72, 10.09, 9.89, 10.28, 10.21, 10.33, 9.88, 10.02, 10.34, 9.87, 9.75, 10.08, 10.13, 10.41, 10.04, 10.1, 10.29, 9.95, 10.01, 9.92, 9.84, 9.91, 10.1, 9.88, 9.86, 10.18, 10.3, 10.02, 10.03, 10.65, 9.86, 9.94, 10.05, 10.18, 10.07, 10.04, 9.97, 10.11, 10.12, 10.21, 9.93, 10.06, 10.36, 10.02, 10.07, 10.07, 10.2, 10.12, 9.79, 9.9, 10.13, 10.02, 9.97, 9.96, 9.93, 9.79, 9.95, 9.6, 10.16, 10.02, 10.26, 10.27, 10.05, 9.76, 10.06, 10.37, 9.88, 10.36, 10.38, 10.16, 9.64, 10.0, 10.08, 10.19, 10.2, 10.03, 10.09, 10.17, 9.77, 10.04, 10.29, 10.09, 10.14, 10.08, 10.05, 10.03, 9.97, 10.41, 10.16, 9.94, 10.05, 9.97, 10.07, 10.16, 10.34, 10.06, 10.05, 9.61, 10.12, 9.84, 10.05, 10.09, 10.02, 10.18, 10.15, 10.12, 10.11, 10.32, 10.3, 9.91, 9.94, 9.98, 10.01, 10.12, 10.01, 10.01, 10.16, 10.29, 9.81, 10.09, 9.84, 9.83, 9.79, 9.94, 9.76, 9.86, 9.67, 10.18, 9.82, 9.97, 9.89, 9.83, 10.36, 9.98, 9.95, 10.1, 9.67, 9.93, 10.3, 10.18, 10.24, 9.95, 10.2, 9.72, 10.16, 10.1, 10.14, 9.85, 9.92, 9.84, 10.21, 10.04, 10.14, 9.89, 10.46, 10.09, 10.04, 10.01, 10.1, 10.38, 10.17, 10.23, 9.93, 9.73, 10.27, 9.97, 10.06, 9.95, 10.33, 10.12, 10.02, 10.08, 9.73, 10.15, 10.14, 9.64, 10.01, 10.07, 10.22, 9.9, 10.06, 10.27, 10.15, 9.95, 10.07, 9.71, 9.96, 10.25, 10.02, 9.9, 9.97, 10.0, 9.83, 9.81, 10.49, 9.94, 9.74, 9.87, 9.83, 9.92, 9.75, 9.71, 9.93, 10.04, 10.34, 10.27, 10.42, 9.95, 10.11, 9.85, 9.92, 10.24, 10.07, 10.13, 10.18, 10.09, 9.98, 10.42, 10.17, 9.95, 9.73, 9.93, 9.63, 10.06, 10.09, 10.06, 10.11, 10.04, 10.42, 9.98, 10.28, 9.6, 9.94, 10.19, 10.2, 10.1, 10.23, 10.03, 9.87, 10.02, 9.8, 9.99, 9.82, 9.96, 10.05, 10.11, 9.68, 9.96, 10.01, 9.92, 10.26, 10.4, 9.88, 10.16, 10.4, 9.71, 10.2, 10.05, 9.75, 10.12, 10.26, 10.32, 9.87, 10.13, 10.04, 9.77, 9.79, 9.98, 10.27, 9.96, 9.88, 9.95, 10.04, 10.02, 9.93, 9.99, 10.09, 10.08, 10.14, 10.01, 10.01, 10.12, 10.18, 9.99, 9.83, 10.14, 9.65, 10.07, 10.03, 9.72, 9.91, 10.01, 9.84, 9.9, 10.16, 9.72, 10.0, 10.19, 10.24, 10.06, 10.15, 9.9, 10.47, 9.81, 10.1, 9.91, 10.18, 9.79, 9.8, 10.23, 9.62, 10.13, 10.16, 10.25, 9.9, 9.85, 10.08, 9.92, 10.11, 9.79, 10.15, 9.86, 9.86, 9.96, 9.77, 10.2, 9.67, 10.1, 9.92, 10.11, 10.04, 10.1, 9.94, 10.24, 10.0, 10.29, 9.96, 10.24, 10.1, 9.92, 9.9, 10.09, 10.24, 10.12, 10.03, 9.92, 9.99, 10.19, 10.3, 9.88, 9.88, 10.06, 9.83, 10.25, 9.62, 9.77, 9.69, 10.11, 10.39, 10.26, 9.98, 9.62, 10.0, 10.26, 9.69, 9.93, 9.94, 9.98, 10.18, 10.41, 9.91, 10.05, 10.1, 10.17, 10.13, 10.07, 10.29, 10.1, 10.23, 9.91, 9.89, 10.14, 9.89, 9.82, 10.17, 9.68, 10.22, 10.26, 10.3, 9.95, 9.63, 9.88, 10.21, 9.93, 10.06, 10.04, 10.27, 10.12, 9.89, 10.13, 10.49, 10.17, 9.87, 9.86, 10.0, 9.62, 9.82, 10.39, 10.08, 9.87, 9.82, 10.12, 10.03, 9.69, 10.07, 9.94, 10.06, 10.05, 10.3, 10.37, 10.13, 10.33, 10.04, 10.11, 10.32, 10.0, 9.83, 10.34, 10.17, 10.13, 9.52, 9.68, 10.21, 9.99, 10.29, 10.08, 10.08, 10.06, 10.1, 10.06, 10.29, 9.82, 10.1, 10.12, 9.75, 9.95, 9.94, 9.9, 10.2, 9.84, 10.12, 9.73, 10.14, 10.18, 10.12, 9.95, 9.78, 9.84, 9.84, 9.91, 9.88, 10.15, 9.79, 10.21, 10.13, 10.09, 10.15, 9.9, 10.13, 10.59, 10.14, 10.37, 10.16, 10.05, 10.13, 10.09, 9.79, 9.96, 9.82, 10.12, 10.08, 9.83, 9.95, 9.88, 10.17, 10.1, 10.09, 9.93, 10.02, 9.94, 10.05, 10.1, 9.94, 9.73, 9.92, 10.35, 9.6, 10.14, 10.15, 10.19, 10.19, 10.13, 9.75, 10.13, 9.94, 9.84, 10.43, 10.05, 10.34, 9.4, 9.59, 9.92, 10.17, 10.08, 10.21, 9.69, 10.23, 10.1, 10.0, 10.12, 10.25, 9.92, 10.05, 9.76, 10.0, 10.1, 10.22, 10.42, 10.03, 10.19, 10.04, 10.02, 9.94, 10.04, 10.11, 10.23, 9.64, 10.0, 10.1, 9.88, 9.85, 10.09, 9.95, 10.01, 9.96, 9.99, 10.24, 9.84, 10.07, 10.27, 9.82, 10.05, 10.07, 10.16, 9.93, 10.0, 9.71, 9.96, 10.0, 9.97, 10.36, 10.08, 9.99, 10.07, 10.24, 10.3, 10.1, 9.76, 10.19, 10.11, 10.08, 10.54, 10.16, 10.13, 9.93, 10.2, 9.8, 9.99, 10.07, 10.01, 9.99, 10.07, 10.39, 10.3, 10.3, 10.42, 9.7, 10.09, 9.96, 9.84, 9.67, 9.67, 9.85, 9.58, 9.78, 10.3, 10.19, 9.91, 10.01, 10.41, 10.2, 10.25, 9.8, 10.41, 9.94, 9.85, 9.59, 9.78, 9.68, 10.08, 10.06, 10.1, 10.08, 9.98, 9.9, 9.8, 10.04, 10.17, 9.8, 10.04, 10.08, 9.93, 10.01, 10.35, 10.05, 9.93, 9.82, 9.88, 10.08, 9.87, 9.74, 9.93]；
差值(后−前) n=929：[-0.15, -0.04, -0.06, -0.09, -0.1, -0.12, -0.05, -0.04, -0.15, -0.1, -0.09, -0.11, -0.05, -0.16, -0.08, -0.05, -0.08, -0.08, -0.06, -0.1, -0.15, -0.04, -0.08, -0.04, -0.07, -0.1, -0.01, -0.13, -0.05, -0.1, -0.11, -0.09, -0.07, -0.12, -0.11, -0.08, -0.08, -0.08, -0.1, -0.09, -0.0, 0.02, -0.03, -0.01, -0.05, -0.04, -0.13, -0.01, -0.09, -0.04, -0.09, -0.07, -0.04, -0.08, -0.13, -0.12, -0.03, -0.08, -0.03, -0.17, -0.1, -0.06, -0.07, -0.09, -0.06, -0.09, -0.05, -0.12, -0.12, -0.11, -0.03, -0.05, -0.12, -0.07, -0.04, -0.09, -0.19, -0.06, -0.08, -0.08, -0.1, -0.05, -0.14, -0.04, -0.07, -0.15, -0.16, -0.04, -0.12, -0.03, -0.11, -0.04, -0.17, -0.1, -0.1, -0.06, -0.08, -0.12, -0.05, -0.08, -0.09, -0.04, -0.14, -0.05, -0.1, -0.08, -0.1, -0.08, -0.08, -0.07, -0.11, -0.11, -0.1, -0.08, 0.02, -0.08, -0.09, -0.15, -0.02, -0.04, -0.14, -0.01, -0.14, -0.18, -0.07, -0.05, -0.11, -0.14, -0.06, -0.11, -0.02, -0.08, -0.05, -0.05, -0.1, -0.13, -0.12, -0.09, -0.06, -0.12, -0.11, -0.05, -0.03, -0.09, -0.06, -0.07, -0.13, -0.05, -0.06, -0.2, -0.09, -0.04, -0.03, -0.06, -0.09, -0.07, 0.03, -0.07, -0.05, -0.09, -0.07, -0.12, -0.12, -0.05, -0.13, -0.12, -0.05, -0.09, -0.01, -0.12, -0.06, -0.04, -0.06, -0.1, -0.06, -0.11, -0.02, -0.06, -0.11, -0.06, -0.11, -0.17, -0.06, -0.09, -0.05, -0.11, -0.05, -0.07, -0.11, -0.07, -0.09, -0.1, -0.11, -0.02, -0.05, -0.13, -0.04, -0.08, -0.11, -0.04, -0.08, -0.07, -0.06, -0.07, -0.12, -0.09, -0.05, -0.08, -0.07, -0.07, -0.07, -0.07, -0.09, -0.04, -0.1, -0.12, -0.12, -0.08, -0.11, -0.05, -0.08, -0.09, -0.02, -0.09, -0.08, -0.11, -0.07, -0.08, -0.06, -0.01, -0.12, -0.08, -0.08, -0.07, -0.04, -0.09, -0.07, -0.04, -0.07, -0.12, -0.1, -0.14, -0.14, -0.04, -0.12, -0.1, -0.11, -0.17, -0.06, -0.07, -0.09, -0.07, -0.09, -0.01, -0.11, -0.02, -0.09, -0.14, -0.11, -0.05, -0.07, -0.05, -0.05, -0.13, -0.05, -0.08, -0.08, -0.06, -0.13, 0.0, -0.03, -0.11, -0.05, -0.1, -0.07, -0.15, -0.05, -0.13, -0.07, -0.1, -0.14, -0.07, -0.06, -0.1, -0.06, -0.12, -0.11, -0.08, -0.03, -0.13, -0.04, -0.11, -0.08, -0.07, -0.04, -0.01, -0.14, -0.09, -0.05, -0.15, -0.09, -0.09, -0.02, -0.1, -0.12, -0.06, -0.09, -0.09, -0.08, -0.1, -0.07, -0.03, -0.1, -0.14, -0.14, -0.09, -0.07, -0.07, -0.03, -0.1, -0.07, -0.13, -0.06, -0.05, -0.03, -0.01, -0.1, -0.13, -0.09, -0.09, -0.1, -0.1, -0.06, -0.08, -0.12, -0.07, -0.15, -0.11, -0.07, -0.06, -0.06, -0.05, -0.14, -0.06, -0.07, -0.09, -0.03, -0.01, -0.13, -0.1, -0.02, -0.09, -0.06, -0.12, -0.16, -0.11, -0.11, -0.14, -0.06, -0.06, -0.09, -0.04, -0.08, -0.19, 0.0, -0.06, -0.04, -0.08, -0.09, -0.1, -0.15, -0.08, -0.08, -0.04, -0.14, -0.15, -0.14, -0.11, -0.03, -0.06, -0.07, -0.11, -0.13, -0.16, -0.1, -0.07, -0.1, -0.1, -0.09, -0.06, -0.1, -0.02, -0.1, -0.16, -0.08, -0.03, -0.1, -0.1, -0.01, -0.04, -0.06, -0.09, -0.03, 0.04, -0.04, -0.02, -0.11, -0.12, -0.04, -0.1, -0.1, -0.06, -0.02, -0.11, -0.07, -0.09, -0.05, -0.08, -0.11, -0.13, -0.07, -0.07, -0.1, -0.08, -0.09, -0.11, -0.12, -0.03, -0.11, -0.1, -0.13, -0.12, -0.09, -0.17, -0.12, -0.06, -0.1, -0.09, -0.07, -0.02, -0.05, -0.06, -0.09, -0.19, -0.1, -0.07, -0.06, -0.06, -0.08, -0.07, -0.15, -0.08, -0.14, -0.11, -0.11, -0.08, -0.1, -0.0, -0.12, -0.04, -0.17, -0.16, -0.12, -0.11, -0.13, -0.0, -0.08, -0.03, -0.07, -0.05, -0.14, -0.03, -0.16, -0.13, -0.09, -0.07, -0.08, -0.08, -0.01, -0.06, -0.13, -0.08, -0.13, -0.06, -0.03, -0.15, -0.05, -0.13, -0.05, -0.15, -0.07, -0.05, -0.07, -0.09, -0.14, -0.07, -0.08, -0.09, -0.11, -0.17, -0.15, -0.06, -0.09, -0.13, -0.09, -0.06, -0.06, -0.13, -0.09, -0.08, -0.15, -0.07, -0.15, -0.07, -0.08, -0.09, -0.09, -0.16, -0.11, -0.11, -0.03, -0.1, -0.14, -0.07, -0.12, -0.07, -0.11, -0.05, -0.05, -0.08, -0.05, -0.05, -0.08, -0.06, -0.03, -0.08, -0.05, -0.14, -0.2, -0.14, -0.08, -0.09, -0.09, -0.09, -0.05, -0.12, -0.08, -0.07, -0.03, -0.14, -0.11, -0.01, -0.11, -0.06, -0.1, -0.03, 0.0, -0.05, -0.1, -0.15, -0.06, -0.04, -0.11, -0.02, -0.12, -0.05, -0.06, -0.11, -0.15, -0.08, -0.1, -0.11, -0.06, -0.08, -0.12, -0.11, -0.07, -0.09, -0.06, -0.08, -0.01, -0.05, -0.12, -0.09, -0.15, -0.09, -0.11, -0.07, -0.02, -0.13, -0.08, -0.02, -0.05, -0.1, -0.1, -0.07, -0.13, -0.06, -0.13, -0.04, -0.1, -0.12, -0.06, -0.1, -0.07, -0.12, -0.04, -0.16, 0.01, -0.07, -0.02, -0.08, -0.15, -0.03, -0.16, -0.03, -0.04, -0.13, -0.07, -0.08, -0.09, -0.16, -0.06, -0.08, -0.17, -0.1, -0.08, -0.03, -0.14, -0.13, -0.13, -0.0, -0.06, -0.08, -0.06, -0.04, -0.1, -0.13, -0.12, -0.08, -0.12, -0.11, -0.07, -0.08, -0.03, -0.12, -0.07, -0.12, -0.16, -0.13, -0.03, -0.05, -0.11, -0.08, 0.01, -0.11, -0.12, -0.07, -0.08, -0.08, -0.06, -0.06, -0.06, -0.04, -0.15, -0.12, -0.1, -0.11, -0.06, -0.08, -0.03, -0.07, -0.09, -0.09, -0.15, -0.04, -0.03, -0.05, -0.15, -0.08, -0.05, -0.06, -0.12, -0.11, -0.03, -0.1, -0.08, -0.11, -0.11, -0.1, -0.01, -0.05, -0.04, -0.06, -0.04, -0.02, -0.11, -0.09, -0.07, -0.11, -0.09, -0.09, -0.11, -0.09, -0.05, 0.0, -0.05, -0.08, -0.05, -0.09, -0.09, -0.04, -0.12, -0.1, -0.15, -0.08, -0.05, -0.09, -0.01, -0.13, -0.12, -0.02, -0.14, -0.2, -0.08, -0.08, -0.11, -0.15, -0.17, -0.09, -0.12, -0.07, -0.05, -0.05, -0.07, -0.13, -0.05, 0.08, -0.14, -0.15, -0.09, -0.12, -0.1, -0.09, -0.12, -0.01, -0.12, -0.16, -0.06, -0.07, -0.07, -0.04, -0.07, -0.08, -0.09, -0.09, -0.06, -0.1, -0.04, -0.09, -0.07, -0.12, -0.11, -0.11, -0.13, -0.14, -0.04, -0.01, -0.08, 0.03, -0.05, -0.08, -0.11, -0.14, -0.01, -0.13, -0.12, -0.12, -0.14, -0.09, -0.09, -0.07, 0.02, -0.12, -0.07, -0.08, -0.09, -0.1, -0.11, -0.11, 0.01, -0.03, -0.1, -0.07, -0.14, -0.09, 0.01, -0.04, 0.03, -0.06, -0.1, -0.07, -0.09, -0.05, -0.11, -0.13, -0.1, -0.08, -0.04, -0.07, -0.05, -0.04, -0.15, -0.05, -0.06, -0.06, 0.01, -0.08, -0.13, -0.01, -0.09, -0.13, -0.11, -0.06, -0.06, -0.04, -0.14, -0.03, -0.15, -0.09, -0.06, -0.12, -0.08, -0.14, -0.07, -0.15, -0.13, -0.11, -0.05, -0.03, -0.06, -0.08, -0.14, -0.14, -0.08, -0.08, -0.06, -0.05, -0.11, -0.04, -0.1, -0.08, -0.13, -0.2, -0.08, -0.08, -0.06, -0.06, -0.14, -0.07, -0.07, -0.08, -0.12, -0.12, -0.12, -0.08, -0.06, -0.11, -0.06, -0.01, -0.06, -0.04, -0.08, -0.08, -0.14, -0.12, -0.08, -0.0, -0.11, -0.09, -0.06, 0.0, -0.07, -0.16, -0.04, -0.07, -0.07, -0.07, -0.12, -0.07, -0.12, -0.11, -0.12, -0.08, -0.04, -0.03, -0.13, -0.03, -0.09, -0.1, -0.09, -0.07, -0.07, -0.08, -0.13, -0.15, -0.11, -0.09, -0.1, -0.09, -0.12, -0.1, -0.04, -0.08, -0.04, -0.08, -0.08, -0.06, -0.13, -0.1, -0.11, -0.07, -0.1, -0.08, -0.08, -0.1, -0.09, -0.14, -0.06]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -0.0833，s_d = 0.0401，SE = s_d/√n = 0.0013；
t = d̄/SE = -63.2698；df = n−1 = 928；双侧 p = 0.000e+00。
d̄ = -0.0833，s_d = 0.0401，SE = s_d/√n = 0.0013；
t = d̄/SE = -63.2698；df = n−1 = 928；双侧 p = 0.000e+00。
计算结果
t = -63.2698，df = 928，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。检验仪器间系统偏差。
案例 42. 前后满意度
原始数据
前 n=867：[3.96, 3.82, 3.43, 3.48, 3.87, 3.05, 3.65, 3.45, 3.22, 3.55, 3.61, 3.51, 3.33, 3.64, 3.98, 3.47, 3.15, 3.6, 3.14, 3.62, 3.33, 3.5, 3.73, 3.53, 3.58, 3.53, 3.6, 4.1, 3.61, 3.39, 3.66, 3.8, 3.64, 3.96, 3.68, 3.81, 3.69, 3.94, 3.02, 3.65, 3.64, 3.67, 3.72, 3.62, 3.24, 3.69, 3.27, 4.27, 3.39, 3.22, 3.65, 3.6, 3.59, 3.58, 3.29, 3.02, 4.12, 3.83, 3.8, 3.71, 3.3, 3.32, 3.69, 3.61, 3.49, 3.66, 3.13, 3.69, 3.39, 3.47, 3.85, 3.76, 3.74, 3.31, 3.42, 3.49, 3.92, 3.38, 3.91, 3.3, 3.99, 3.6, 3.72, 2.97, 3.82, 3.31, 3.98, 3.71, 3.74, 3.45, 3.67, 3.92, 3.34, 3.66, 3.43, 3.55, 3.42, 4.19, 3.56, 3.95, 3.46, 3.51, 3.48, 3.32, 3.76, 3.47, 3.52, 3.77, 3.44, 3.61, 3.85, 3.26, 3.92, 3.41, 3.7, 3.26, 3.53, 3.97, 3.34, 3.52, 3.76, 3.73, 3.73, 3.9, 3.81, 3.4, 3.57, 3.48, 3.22, 3.62, 3.75, 3.59, 3.42, 3.36, 3.49, 3.78, 3.6, 3.53, 3.32, 3.44, 3.69, 3.85, 3.14, 3.76, 3.55, 3.54, 3.71, 3.58, 3.89, 3.4, 3.75, 3.57, 3.25, 3.38, 3.56, 3.56, 3.84, 3.45, 3.77, 3.62, 3.24, 3.45, 3.5, 3.55, 3.8, 3.58, 3.09, 3.47, 3.71, 3.33, 3.54, 3.22, 3.46, 3.61, 3.71, 3.12, 3.48, 3.46, 3.36, 3.52, 3.55, 3.43, 3.2, 3.81, 3.19, 3.92, 3.5, 3.6, 3.05, 3.66, 3.75, 4.0, 3.85, 3.41, 3.07, 3.31, 3.45, 3.5, 3.64, 3.79, 3.48, 3.47, 3.57, 3.03, 3.48, 3.77, 3.17, 3.65, 3.93, 3.97, 3.85, 3.99, 3.43, 3.79, 3.87, 3.99, 3.9, 3.62, 3.68, 3.4, 3.63, 3.67, 3.45, 3.78, 3.51, 3.66, 3.55, 3.66, 3.73, 3.81, 3.61, 3.91, 3.53, 3.33, 3.38, 3.71, 3.64, 3.28, 3.75, 3.98, 3.72, 3.84, 3.47, 3.61, 3.11, 3.72, 3.21, 3.74, 3.43, 3.65, 3.44, 3.24, 3.87, 3.39, 3.67, 3.52, 3.42, 3.98, 3.62, 3.75, 3.52, 3.79, 3.63, 3.35, 3.88, 3.62, 3.9, 3.34, 3.52, 3.71, 3.81, 3.12, 3.36, 3.53, 3.66, 3.54, 3.4, 3.56, 3.7, 3.44, 3.35, 3.67, 3.35, 3.3, 3.44, 3.55, 2.88, 3.75, 3.21, 3.79, 3.55, 3.33, 3.36, 3.41, 3.39, 3.53, 3.61, 3.28, 3.72, 3.54, 3.41, 3.37, 3.21, 3.85, 3.29, 3.73, 3.67, 3.44, 4.12, 3.81, 3.65, 3.75, 2.97, 3.52, 3.57, 3.53, 3.27, 4.13, 4.19, 3.66, 3.4, 3.75, 3.12, 3.68, 3.66, 3.62, 3.76, 4.0, 3.36, 3.48, 3.65, 3.29, 3.41, 3.47, 3.41, 3.38, 3.8, 3.65, 3.52, 3.86, 3.42, 3.74, 3.57, 3.72, 3.6, 3.47, 3.5, 3.76, 3.64, 3.35, 3.5, 3.58, 3.65, 3.1, 3.3, 3.26, 3.4, 3.87, 3.44, 3.94, 3.78, 3.3, 3.65, 3.12, 3.26, 3.59, 3.28, 3.74, 3.89, 3.05, 4.03, 3.78, 3.64, 3.31, 3.2, 3.84, 3.67, 3.14, 3.89, 3.06, 3.84, 3.64, 3.45, 3.83, 3.89, 3.65, 3.75, 3.63, 3.6, 3.53, 3.71, 3.44, 3.88, 3.35, 3.59, 3.77, 3.71, 3.47, 3.15, 3.75, 3.46, 3.84, 3.73, 3.42, 3.61, 3.69, 3.6, 3.82, 3.87, 3.46, 3.59, 3.69, 3.67, 3.98, 3.96, 3.52, 3.7, 3.7, 3.03, 3.32, 3.66, 3.56, 3.64, 3.84, 3.71, 3.53, 3.58, 3.64, 3.61, 3.78, 3.71, 3.45, 3.73, 3.64, 3.47, 3.29, 3.63, 3.72, 3.42, 3.75, 3.68, 3.4, 3.78, 3.5, 3.23, 3.32, 3.81, 3.79, 3.51, 3.12, 3.53, 3.5, 3.47, 3.4, 3.56, 3.87, 3.56, 3.54, 3.62, 3.65, 3.7, 3.4, 3.76, 3.76, 3.68, 3.44, 3.42, 4.1, 3.81, 3.51, 3.39, 3.62, 4.03, 3.78, 3.51, 3.61, 3.63, 3.55, 3.57, 3.51, 4.12, 3.45, 3.97, 4.09, 3.48, 3.44, 3.7, 3.6, 3.74, 3.73, 3.58, 3.49, 3.58, 3.95, 3.5, 3.75, 3.86, 3.44, 3.07, 3.74, 3.46, 3.69, 3.56, 3.15, 3.9, 3.23, 3.82, 3.38, 3.82, 3.6, 3.3, 3.61, 3.4, 3.45, 3.72, 3.68, 3.43, 3.86, 3.52, 3.53, 3.43, 3.79, 3.52, 3.46, 3.21, 3.6, 3.65, 3.6, 3.52, 3.31, 3.42, 3.62, 3.42, 3.89, 3.47, 3.38, 3.74, 3.44, 3.57, 3.54, 3.24, 3.2, 3.16, 3.65, 3.61, 3.42, 3.62, 3.52, 3.8, 3.86, 3.64, 3.65, 3.51, 4.13, 3.44, 3.41, 3.49, 3.88, 3.29, 3.26, 3.59, 3.73, 3.11, 3.92, 3.43, 3.88, 3.6, 3.6, 3.49, 3.65, 3.55, 3.52, 3.03, 3.67, 3.54, 3.59, 3.41, 3.4, 3.75, 3.51, 3.5, 3.52, 3.78, 3.42, 3.74, 3.38, 3.7, 3.74, 3.39, 3.51, 3.37, 3.8, 3.42, 3.56, 3.46, 3.49, 3.81, 3.54, 3.43, 3.8, 3.19, 4.0, 3.11, 3.51, 3.58, 3.48, 3.4, 3.52, 3.45, 3.66, 3.34, 3.56, 3.67, 3.8, 3.18, 3.56, 3.69, 3.71, 3.51, 3.66, 3.63, 3.41, 3.59, 3.53, 3.57, 3.82, 3.35, 3.34, 3.41, 3.42, 3.86, 3.26, 3.88, 3.33, 3.78, 3.35, 3.53, 3.35, 3.36, 3.68, 3.83, 3.13, 3.64, 3.22, 3.63, 3.83, 3.86, 3.65, 3.72, 3.51, 3.47, 3.52, 3.33, 3.16, 3.71, 3.5, 3.47, 3.67, 3.44, 3.64, 3.73, 3.16, 3.65, 3.85, 3.27, 3.77, 3.96, 3.57, 3.87, 4.07, 3.12, 3.44, 3.46, 3.88, 3.16, 3.61, 3.92, 3.56, 3.41, 3.59, 3.49, 3.31, 3.76, 3.84, 3.53, 3.41, 3.46, 3.97, 3.67, 3.54, 3.45, 4.04, 3.65, 3.49, 3.77, 3.62, 3.17, 3.41, 3.43, 3.71, 3.94, 3.96, 3.49, 3.73, 3.52, 3.35, 3.91, 3.85, 3.43, 3.18, 3.66, 3.77, 3.78, 3.96, 3.61, 3.85, 3.62, 3.82, 4.03, 3.28, 3.65, 3.8, 3.88, 3.46, 3.56, 3.41, 3.55, 3.62, 3.48, 3.76, 3.87, 3.5, 3.39, 3.31, 3.63, 3.5, 3.73, 3.51, 3.38, 3.61, 3.6, 3.57, 3.41, 3.32, 3.84, 3.65, 3.78, 3.69, 3.95, 3.57, 3.76, 3.54, 3.14, 3.86, 3.91, 3.37, 3.8, 3.58, 3.71, 3.66, 3.66, 3.37, 3.7, 3.47, 3.76, 3.94, 3.4, 4.13, 3.35, 3.45, 3.87, 3.47, 3.96, 3.86, 3.69, 3.38, 3.76, 3.73, 3.25, 3.12, 3.33, 3.55, 3.7, 3.93, 3.22, 3.46, 3.43, 3.57, 3.69, 3.23, 3.43, 3.71, 3.6, 3.31, 3.81, 3.34, 3.5, 3.49, 3.5, 3.63, 3.51, 3.52, 3.86, 3.18, 3.53, 3.83, 3.57, 3.48, 3.53, 3.78, 3.72, 2.97, 3.65, 3.49, 3.95, 3.55, 3.5, 3.53, 3.32, 3.88, 3.34, 3.94, 3.61, 3.49, 3.51, 3.55, 3.5, 3.34, 3.6, 3.35, 3.32, 3.16, 3.36, 3.67, 3.99, 3.53, 3.01, 3.41, 3.36, 3.97, 3.57, 3.9, 3.87, 3.25, 3.74, 3.58, 3.62, 3.44, 4.01, 3.26, 3.34, 3.58, 3.64, 3.17, 3.62, 3.85, 3.54, 3.94, 3.51, 3.89, 3.32, 3.29, 3.72, 3.2, 3.42, 3.41, 3.31]；
后 n=867：[4.65, 4.57, 4.14, 4.17, 4.59, 3.74, 4.33, 4.2, 3.97, 4.28, 4.3, 4.22, 4.02, 4.33, 4.72, 4.18, 3.82, 4.29, 3.82, 4.32, 4.09, 4.16, 4.43, 4.18, 4.27, 4.23, 4.35, 4.8, 4.32, 4.08, 4.35, 4.52, 4.33, 4.63, 4.38, 4.54, 4.37, 4.71, 3.72, 4.39, 4.34, 4.44, 4.44, 4.3, 3.98, 4.42, 3.96, 4.96, 4.06, 3.85, 4.36, 4.35, 4.29, 4.27, 4.04, 3.69, 4.85, 4.52, 4.48, 4.41, 4.03, 4.0, 4.42, 4.3, 4.17, 4.34, 3.87, 4.45, 4.12, 4.11, 4.61, 4.46, 4.46, 4.01, 4.09, 4.15, 4.6, 4.12, 4.66, 3.94, 4.66, 4.31, 4.46, 3.66, 4.46, 3.99, 4.67, 4.45, 4.46, 4.13, 4.38, 4.61, 4.09, 4.36, 4.15, 4.22, 4.11, 4.89, 4.28, 4.7, 4.24, 4.24, 4.16, 4.05, 4.45, 4.2, 4.2, 4.43, 4.2, 4.35, 4.54, 3.98, 4.62, 4.13, 4.41, 3.98, 4.21, 4.64, 4.1, 4.28, 4.47, 4.42, 4.43, 4.57, 4.56, 4.11, 4.32, 4.22, 3.95, 4.34, 4.49, 4.32, 4.2, 4.03, 4.2, 4.45, 4.35, 4.23, 4.02, 4.16, 4.43, 4.47, 3.88, 4.43, 4.26, 4.29, 4.37, 4.33, 4.59, 4.1, 4.45, 4.29, 3.97, 4.08, 4.27, 4.28, 4.6, 4.14, 4.5, 4.28, 3.97, 4.22, 4.2, 4.26, 4.5, 4.28, 3.8, 4.16, 4.34, 4.09, 4.29, 3.97, 4.21, 4.33, 4.39, 3.89, 4.19, 4.18, 4.05, 4.21, 4.24, 4.14, 3.9, 4.51, 3.87, 4.67, 4.22, 4.28, 3.72, 4.37, 4.47, 4.7, 4.56, 4.12, 3.79, 3.96, 4.23, 4.24, 4.4, 4.51, 4.18, 4.21, 4.22, 3.8, 4.2, 4.52, 3.84, 4.37, 4.66, 4.67, 4.59, 4.71, 4.2, 4.51, 4.59, 4.68, 4.62, 4.32, 4.4, 4.15, 4.31, 4.41, 4.12, 4.47, 4.23, 4.38, 4.36, 4.36, 4.47, 4.51, 4.3, 4.6, 4.24, 4.03, 4.13, 4.43, 4.35, 4.01, 4.53, 4.7, 4.43, 4.54, 4.16, 4.33, 3.81, 4.51, 3.94, 4.49, 4.18, 4.32, 4.14, 3.94, 4.56, 4.11, 4.4, 4.18, 4.11, 4.71, 4.37, 4.38, 4.22, 4.47, 4.33, 4.07, 4.6, 4.34, 4.59, 4.04, 4.25, 4.4, 4.54, 3.82, 4.05, 4.26, 4.37, 4.24, 4.05, 4.22, 4.41, 4.16, 4.07, 4.33, 4.08, 4.02, 4.16, 4.23, 3.61, 4.47, 3.91, 4.51, 4.25, 4.03, 4.11, 4.12, 4.12, 4.2, 4.34, 3.98, 4.4, 4.26, 4.14, 4.08, 3.93, 4.57, 4.02, 4.44, 4.43, 4.14, 4.81, 4.5, 4.35, 4.46, 3.69, 4.27, 4.24, 4.22, 4.03, 4.83, 4.9, 4.35, 4.15, 4.44, 3.81, 4.37, 4.34, 4.33, 4.46, 4.68, 4.05, 4.2, 4.33, 4.01, 4.15, 4.22, 4.08, 4.13, 4.52, 4.33, 4.19, 4.6, 4.11, 4.48, 4.31, 4.5, 4.3, 4.25, 4.21, 4.49, 4.28, 4.07, 4.24, 4.24, 4.38, 3.85, 4.03, 3.96, 4.1, 4.55, 4.09, 4.67, 4.5, 4.08, 4.44, 3.78, 4.0, 4.27, 4.0, 4.47, 4.64, 3.76, 4.79, 4.53, 4.3, 3.99, 3.93, 4.58, 4.4, 3.84, 4.61, 3.74, 4.6, 4.33, 4.17, 4.52, 4.64, 4.35, 4.45, 4.31, 4.33, 4.26, 4.45, 4.13, 4.6, 4.1, 4.28, 4.48, 4.45, 4.15, 3.84, 4.46, 4.16, 4.47, 4.43, 4.16, 4.32, 4.38, 4.33, 4.51, 4.53, 4.23, 4.29, 4.39, 4.42, 4.67, 4.66, 4.24, 4.38, 4.44, 3.78, 4.02, 4.4, 4.3, 4.38, 4.58, 4.4, 4.24, 4.26, 4.35, 4.4, 4.43, 4.48, 4.12, 4.48, 4.38, 4.17, 4.02, 4.37, 4.41, 4.08, 4.48, 4.37, 4.07, 4.46, 4.25, 3.93, 4.05, 4.56, 4.43, 4.22, 3.83, 4.21, 4.22, 4.19, 4.07, 4.22, 4.57, 4.27, 4.24, 4.29, 4.36, 4.42, 4.11, 4.44, 4.5, 4.36, 4.14, 4.13, 4.79, 4.53, 4.24, 4.04, 4.36, 4.68, 4.48, 4.27, 4.35, 4.34, 4.23, 4.32, 4.22, 4.8, 4.12, 4.71, 4.75, 4.14, 4.17, 4.43, 4.29, 4.48, 4.42, 4.32, 4.17, 4.3, 4.69, 4.24, 4.42, 4.58, 4.13, 3.78, 4.44, 4.2, 4.45, 4.23, 3.82, 4.59, 3.96, 4.51, 4.03, 4.5, 4.32, 4.01, 4.32, 4.14, 4.16, 4.41, 4.34, 4.11, 4.49, 4.23, 4.24, 4.15, 4.52, 4.22, 4.19, 3.94, 4.32, 4.35, 4.33, 4.23, 4.01, 4.2, 4.33, 4.18, 4.56, 4.16, 4.11, 4.45, 4.14, 4.29, 4.26, 3.95, 3.92, 3.85, 4.4, 4.31, 4.09, 4.36, 4.28, 4.5, 4.56, 4.38, 4.32, 4.24, 4.8, 4.16, 4.07, 4.19, 4.59, 3.96, 4.0, 4.38, 4.43, 3.84, 4.63, 4.1, 4.56, 4.32, 4.32, 4.21, 4.33, 4.28, 4.26, 3.8, 4.38, 4.23, 4.28, 4.12, 4.16, 4.46, 4.21, 4.17, 4.25, 4.46, 4.13, 4.44, 4.12, 4.43, 4.44, 4.11, 4.26, 4.07, 4.5, 4.14, 4.25, 4.18, 4.24, 4.48, 4.28, 4.2, 4.51, 3.89, 4.68, 3.79, 4.25, 4.24, 4.2, 4.1, 4.28, 4.19, 4.42, 4.07, 4.3, 4.4, 4.51, 3.88, 4.33, 4.39, 4.45, 4.29, 4.33, 4.35, 4.13, 4.36, 4.25, 4.31, 4.51, 4.1, 4.08, 4.13, 4.14, 4.57, 3.96, 4.56, 4.09, 4.45, 4.06, 4.2, 4.08, 4.1, 4.41, 4.55, 3.86, 4.33, 3.95, 4.29, 4.55, 4.56, 4.37, 4.43, 4.2, 4.18, 4.28, 3.99, 3.82, 4.41, 4.2, 4.16, 4.37, 4.13, 4.34, 4.46, 3.92, 4.31, 4.53, 3.97, 4.44, 4.65, 4.25, 4.59, 4.78, 3.81, 4.07, 4.19, 4.65, 3.86, 4.39, 4.68, 4.29, 4.12, 4.36, 4.15, 4.06, 4.44, 4.51, 4.25, 4.14, 4.14, 4.69, 4.41, 4.22, 4.17, 4.75, 4.41, 4.19, 4.51, 4.36, 3.89, 4.14, 4.12, 4.48, 4.67, 4.65, 4.18, 4.46, 4.25, 4.09, 4.59, 4.54, 4.18, 3.87, 4.35, 4.49, 4.46, 4.68, 4.27, 4.57, 4.29, 4.56, 4.77, 4.03, 4.41, 4.52, 4.59, 4.18, 4.23, 4.12, 4.3, 4.32, 4.24, 4.45, 4.55, 4.18, 4.07, 3.97, 4.34, 4.3, 4.4, 4.21, 4.11, 4.33, 4.34, 4.29, 4.12, 4.09, 4.53, 4.41, 4.5, 4.4, 4.68, 4.27, 4.41, 4.26, 3.85, 4.56, 4.61, 4.08, 4.48, 4.33, 4.4, 4.4, 4.38, 4.05, 4.43, 4.14, 4.47, 4.7, 4.07, 4.88, 4.03, 4.17, 4.53, 4.17, 4.69, 4.58, 4.46, 4.12, 4.46, 4.38, 3.98, 3.88, 4.06, 4.24, 4.44, 4.64, 3.92, 4.19, 4.15, 4.28, 4.42, 3.95, 4.13, 4.37, 4.31, 4.06, 4.47, 4.01, 4.27, 4.23, 4.2, 4.32, 4.27, 4.24, 4.59, 3.97, 4.21, 4.6, 4.26, 4.21, 4.23, 4.49, 4.4, 3.73, 4.34, 4.2, 4.71, 4.26, 4.24, 4.18, 3.98, 4.61, 3.99, 4.59, 4.28, 4.18, 4.2, 4.25, 4.19, 4.06, 4.27, 4.0, 3.97, 3.91, 4.05, 4.33, 4.68, 4.27, 3.7, 4.19, 4.07, 4.74, 4.27, 4.61, 4.56, 3.93, 4.44, 4.28, 4.32, 4.16, 4.69, 3.96, 4.08, 4.29, 4.3, 3.9, 4.4, 4.53, 4.26, 4.71, 4.25, 4.51, 4.01, 3.96, 4.43, 3.92, 4.14, 4.15, 4.04]；
差值(后−前) n=867：[0.69, 0.75, 0.71, 0.69, 0.71, 0.7, 0.68, 0.75, 0.75, 0.73, 0.69, 0.71, 0.68, 0.69, 0.74, 0.71, 0.68, 0.69, 0.68, 0.7, 0.76, 0.65, 0.69, 0.66, 0.68, 0.7, 0.75, 0.7, 0.71, 0.69, 0.69, 0.72, 0.69, 0.67, 0.7, 0.72, 0.68, 0.77, 0.7, 0.74, 0.7, 0.76, 0.72, 0.68, 0.73, 0.73, 0.7, 0.69, 0.67, 0.63, 0.71, 0.75, 0.71, 0.69, 0.75, 0.67, 0.73, 0.69, 0.68, 0.7, 0.72, 0.68, 0.73, 0.69, 0.67, 0.68, 0.74, 0.76, 0.73, 0.64, 0.76, 0.7, 0.73, 0.7, 0.67, 0.66, 0.68, 0.74, 0.75, 0.64, 0.67, 0.71, 0.74, 0.69, 0.64, 0.68, 0.69, 0.74, 0.72, 0.69, 0.72, 0.69, 0.75, 0.7, 0.72, 0.68, 0.69, 0.7, 0.71, 0.75, 0.78, 0.73, 0.67, 0.73, 0.69, 0.73, 0.68, 0.66, 0.76, 0.74, 0.7, 0.72, 0.7, 0.72, 0.72, 0.72, 0.68, 0.68, 0.76, 0.75, 0.72, 0.69, 0.7, 0.68, 0.75, 0.7, 0.75, 0.75, 0.73, 0.72, 0.74, 0.73, 0.79, 0.68, 0.71, 0.67, 0.75, 0.71, 0.71, 0.71, 0.74, 0.62, 0.74, 0.67, 0.71, 0.75, 0.66, 0.74, 0.7, 0.71, 0.7, 0.72, 0.72, 0.71, 0.71, 0.72, 0.76, 0.69, 0.73, 0.67, 0.72, 0.77, 0.7, 0.71, 0.7, 0.7, 0.7, 0.69, 0.63, 0.76, 0.75, 0.75, 0.75, 0.72, 0.68, 0.76, 0.72, 0.72, 0.69, 0.68, 0.69, 0.7, 0.7, 0.7, 0.68, 0.75, 0.72, 0.67, 0.68, 0.71, 0.73, 0.71, 0.71, 0.7, 0.71, 0.66, 0.78, 0.74, 0.76, 0.72, 0.69, 0.74, 0.65, 0.77, 0.72, 0.75, 0.67, 0.72, 0.73, 0.7, 0.74, 0.72, 0.77, 0.72, 0.73, 0.69, 0.71, 0.7, 0.72, 0.75, 0.68, 0.74, 0.67, 0.69, 0.72, 0.71, 0.81, 0.69, 0.74, 0.69, 0.69, 0.69, 0.7, 0.7, 0.75, 0.72, 0.71, 0.74, 0.78, 0.72, 0.71, 0.71, 0.69, 0.71, 0.7, 0.79, 0.73, 0.75, 0.75, 0.67, 0.7, 0.71, 0.7, 0.72, 0.73, 0.66, 0.68, 0.74, 0.75, 0.64, 0.7, 0.68, 0.7, 0.71, 0.72, 0.72, 0.69, 0.7, 0.72, 0.69, 0.73, 0.71, 0.69, 0.73, 0.71, 0.69, 0.65, 0.67, 0.71, 0.72, 0.72, 0.66, 0.73, 0.73, 0.72, 0.68, 0.72, 0.72, 0.7, 0.72, 0.7, 0.7, 0.74, 0.71, 0.72, 0.67, 0.73, 0.7, 0.68, 0.72, 0.73, 0.71, 0.72, 0.72, 0.73, 0.71, 0.76, 0.7, 0.69, 0.69, 0.7, 0.72, 0.73, 0.75, 0.67, 0.69, 0.76, 0.7, 0.71, 0.69, 0.75, 0.69, 0.68, 0.69, 0.68, 0.71, 0.7, 0.68, 0.68, 0.71, 0.68, 0.73, 0.75, 0.75, 0.67, 0.75, 0.72, 0.69, 0.67, 0.74, 0.69, 0.75, 0.74, 0.77, 0.71, 0.78, 0.71, 0.73, 0.64, 0.72, 0.74, 0.66, 0.73, 0.75, 0.73, 0.7, 0.7, 0.68, 0.65, 0.73, 0.72, 0.78, 0.79, 0.67, 0.74, 0.68, 0.73, 0.73, 0.75, 0.71, 0.76, 0.75, 0.66, 0.68, 0.74, 0.74, 0.72, 0.7, 0.71, 0.68, 0.76, 0.69, 0.72, 0.69, 0.75, 0.7, 0.7, 0.68, 0.73, 0.73, 0.74, 0.68, 0.73, 0.75, 0.69, 0.71, 0.74, 0.68, 0.7, 0.71, 0.7, 0.63, 0.71, 0.73, 0.71, 0.69, 0.73, 0.69, 0.66, 0.77, 0.7, 0.71, 0.75, 0.69, 0.69, 0.72, 0.68, 0.74, 0.75, 0.7, 0.74, 0.74, 0.74, 0.74, 0.69, 0.72, 0.68, 0.72, 0.79, 0.65, 0.77, 0.67, 0.75, 0.74, 0.7, 0.73, 0.75, 0.7, 0.66, 0.73, 0.69, 0.67, 0.68, 0.75, 0.7, 0.73, 0.75, 0.64, 0.71, 0.71, 0.68, 0.72, 0.72, 0.68, 0.66, 0.7, 0.7, 0.7, 0.67, 0.72, 0.72, 0.71, 0.68, 0.74, 0.68, 0.7, 0.71, 0.7, 0.72, 0.72, 0.65, 0.73, 0.66, 0.69, 0.76, 0.74, 0.7, 0.68, 0.74, 0.71, 0.69, 0.67, 0.73, 0.66, 0.66, 0.73, 0.73, 0.69, 0.74, 0.69, 0.75, 0.67, 0.72, 0.74, 0.74, 0.67, 0.72, 0.69, 0.71, 0.7, 0.74, 0.75, 0.67, 0.67, 0.7, 0.73, 0.69, 0.65, 0.68, 0.72, 0.71, 0.71, 0.74, 0.71, 0.68, 0.66, 0.68, 0.64, 0.71, 0.71, 0.72, 0.72, 0.7, 0.73, 0.72, 0.71, 0.69, 0.73, 0.71, 0.7, 0.78, 0.71, 0.75, 0.67, 0.69, 0.73, 0.71, 0.7, 0.71, 0.72, 0.71, 0.73, 0.69, 0.75, 0.69, 0.68, 0.74, 0.76, 0.7, 0.7, 0.74, 0.67, 0.73, 0.67, 0.72, 0.66, 0.69, 0.71, 0.67, 0.74, 0.79, 0.7, 0.73, 0.71, 0.67, 0.69, 0.72, 0.72, 0.71, 0.68, 0.73, 0.74, 0.76, 0.71, 0.69, 0.7, 0.71, 0.76, 0.71, 0.69, 0.67, 0.72, 0.69, 0.71, 0.71, 0.74, 0.73, 0.69, 0.72, 0.75, 0.7, 0.7, 0.71, 0.7, 0.73, 0.75, 0.67, 0.74, 0.77, 0.71, 0.7, 0.67, 0.69, 0.74, 0.67, 0.72, 0.7, 0.75, 0.74, 0.76, 0.73, 0.74, 0.73, 0.71, 0.7, 0.77, 0.7, 0.74, 0.78, 0.68, 0.72, 0.72, 0.77, 0.72, 0.73, 0.69, 0.75, 0.74, 0.72, 0.72, 0.71, 0.7, 0.68, 0.76, 0.66, 0.71, 0.67, 0.73, 0.74, 0.73, 0.72, 0.74, 0.69, 0.73, 0.66, 0.72, 0.7, 0.72, 0.71, 0.69, 0.71, 0.76, 0.66, 0.67, 0.71, 0.7, 0.69, 0.7, 0.69, 0.7, 0.73, 0.76, 0.66, 0.67, 0.71, 0.67, 0.7, 0.68, 0.72, 0.72, 0.69, 0.63, 0.73, 0.77, 0.69, 0.78, 0.76, 0.73, 0.71, 0.76, 0.66, 0.76, 0.68, 0.67, 0.72, 0.72, 0.68, 0.72, 0.74, 0.68, 0.72, 0.71, 0.76, 0.7, 0.74, 0.75, 0.72, 0.73, 0.69, 0.76, 0.73, 0.69, 0.69, 0.72, 0.73, 0.73, 0.68, 0.69, 0.76, 0.68, 0.69, 0.72, 0.68, 0.73, 0.66, 0.71, 0.68, 0.74, 0.74, 0.75, 0.76, 0.72, 0.71, 0.72, 0.67, 0.72, 0.75, 0.7, 0.77, 0.69, 0.68, 0.67, 0.68, 0.66, 0.71, 0.8, 0.67, 0.7, 0.73, 0.72, 0.74, 0.72, 0.71, 0.77, 0.69, 0.76, 0.72, 0.71, 0.74, 0.7, 0.65, 0.72, 0.71, 0.7, 0.7, 0.71, 0.68, 0.75, 0.69, 0.74, 0.71, 0.68, 0.73, 0.67, 0.71, 0.76, 0.66, 0.75, 0.68, 0.72, 0.66, 0.69, 0.73, 0.72, 0.77, 0.75, 0.7, 0.65, 0.73, 0.75, 0.73, 0.69, 0.74, 0.71, 0.7, 0.73, 0.72, 0.71, 0.73, 0.71, 0.7, 0.66, 0.71, 0.75, 0.66, 0.66, 0.77, 0.74, 0.7, 0.69, 0.76, 0.72, 0.73, 0.79, 0.68, 0.78, 0.7, 0.73, 0.71, 0.71, 0.68, 0.77, 0.69, 0.72, 0.76, 0.72, 0.74, 0.65, 0.67, 0.73, 0.65, 0.65, 0.67, 0.69, 0.69, 0.7, 0.69, 0.72, 0.68, 0.65, 0.64, 0.75, 0.7, 0.66, 0.69, 0.73, 0.7, 0.78, 0.71, 0.77, 0.71, 0.71, 0.69, 0.68, 0.7, 0.7, 0.69, 0.72, 0.68, 0.7, 0.74, 0.71, 0.66, 0.73, 0.78, 0.68, 0.72, 0.77, 0.74, 0.62, 0.69, 0.67, 0.7, 0.72, 0.72, 0.74, 0.73]
后 n=867：[4.65, 4.57, 4.14, 4.17, 4.59, 3.74, 4.33, 4.2, 3.97, 4.28, 4.3, 4.22, 4.02, 4.33, 4.72, 4.18, 3.82, 4.29, 3.82, 4.32, 4.09, 4.16, 4.43, 4.18, 4.27, 4.23, 4.35, 4.8, 4.32, 4.08, 4.35, 4.52, 4.33, 4.63, 4.38, 4.54, 4.37, 4.71, 3.72, 4.39, 4.34, 4.44, 4.44, 4.3, 3.98, 4.42, 3.96, 4.96, 4.06, 3.85, 4.36, 4.35, 4.29, 4.27, 4.04, 3.69, 4.85, 4.52, 4.48, 4.41, 4.03, 4.0, 4.42, 4.3, 4.17, 4.34, 3.87, 4.45, 4.12, 4.11, 4.61, 4.46, 4.46, 4.01, 4.09, 4.15, 4.6, 4.12, 4.66, 3.94, 4.66, 4.31, 4.46, 3.66, 4.46, 3.99, 4.67, 4.45, 4.46, 4.13, 4.38, 4.61, 4.09, 4.36, 4.15, 4.22, 4.11, 4.89, 4.28, 4.7, 4.24, 4.24, 4.16, 4.05, 4.45, 4.2, 4.2, 4.43, 4.2, 4.35, 4.54, 3.98, 4.62, 4.13, 4.41, 3.98, 4.21, 4.64, 4.1, 4.28, 4.47, 4.42, 4.43, 4.57, 4.56, 4.11, 4.32, 4.22, 3.95, 4.34, 4.49, 4.32, 4.2, 4.03, 4.2, 4.45, 4.35, 4.23, 4.02, 4.16, 4.43, 4.47, 3.88, 4.43, 4.26, 4.29, 4.37, 4.33, 4.59, 4.1, 4.45, 4.29, 3.97, 4.08, 4.27, 4.28, 4.6, 4.14, 4.5, 4.28, 3.97, 4.22, 4.2, 4.26, 4.5, 4.28, 3.8, 4.16, 4.34, 4.09, 4.29, 3.97, 4.21, 4.33, 4.39, 3.89, 4.19, 4.18, 4.05, 4.21, 4.24, 4.14, 3.9, 4.51, 3.87, 4.67, 4.22, 4.28, 3.72, 4.37, 4.47, 4.7, 4.56, 4.12, 3.79, 3.96, 4.23, 4.24, 4.4, 4.51, 4.18, 4.21, 4.22, 3.8, 4.2, 4.52, 3.84, 4.37, 4.66, 4.67, 4.59, 4.71, 4.2, 4.51, 4.59, 4.68, 4.62, 4.32, 4.4, 4.15, 4.31, 4.41, 4.12, 4.47, 4.23, 4.38, 4.36, 4.36, 4.47, 4.51, 4.3, 4.6, 4.24, 4.03, 4.13, 4.43, 4.35, 4.01, 4.53, 4.7, 4.43, 4.54, 4.16, 4.33, 3.81, 4.51, 3.94, 4.49, 4.18, 4.32, 4.14, 3.94, 4.56, 4.11, 4.4, 4.18, 4.11, 4.71, 4.37, 4.38, 4.22, 4.47, 4.33, 4.07, 4.6, 4.34, 4.59, 4.04, 4.25, 4.4, 4.54, 3.82, 4.05, 4.26, 4.37, 4.24, 4.05, 4.22, 4.41, 4.16, 4.07, 4.33, 4.08, 4.02, 4.16, 4.23, 3.61, 4.47, 3.91, 4.51, 4.25, 4.03, 4.11, 4.12, 4.12, 4.2, 4.34, 3.98, 4.4, 4.26, 4.14, 4.08, 3.93, 4.57, 4.02, 4.44, 4.43, 4.14, 4.81, 4.5, 4.35, 4.46, 3.69, 4.27, 4.24, 4.22, 4.03, 4.83, 4.9, 4.35, 4.15, 4.44, 3.81, 4.37, 4.34, 4.33, 4.46, 4.68, 4.05, 4.2, 4.33, 4.01, 4.15, 4.22, 4.08, 4.13, 4.52, 4.33, 4.19, 4.6, 4.11, 4.48, 4.31, 4.5, 4.3, 4.25, 4.21, 4.49, 4.28, 4.07, 4.24, 4.24, 4.38, 3.85, 4.03, 3.96, 4.1, 4.55, 4.09, 4.67, 4.5, 4.08, 4.44, 3.78, 4.0, 4.27, 4.0, 4.47, 4.64, 3.76, 4.79, 4.53, 4.3, 3.99, 3.93, 4.58, 4.4, 3.84, 4.61, 3.74, 4.6, 4.33, 4.17, 4.52, 4.64, 4.35, 4.45, 4.31, 4.33, 4.26, 4.45, 4.13, 4.6, 4.1, 4.28, 4.48, 4.45, 4.15, 3.84, 4.46, 4.16, 4.47, 4.43, 4.16, 4.32, 4.38, 4.33, 4.51, 4.53, 4.23, 4.29, 4.39, 4.42, 4.67, 4.66, 4.24, 4.38, 4.44, 3.78, 4.02, 4.4, 4.3, 4.38, 4.58, 4.4, 4.24, 4.26, 4.35, 4.4, 4.43, 4.48, 4.12, 4.48, 4.38, 4.17, 4.02, 4.37, 4.41, 4.08, 4.48, 4.37, 4.07, 4.46, 4.25, 3.93, 4.05, 4.56, 4.43, 4.22, 3.83, 4.21, 4.22, 4.19, 4.07, 4.22, 4.57, 4.27, 4.24, 4.29, 4.36, 4.42, 4.11, 4.44, 4.5, 4.36, 4.14, 4.13, 4.79, 4.53, 4.24, 4.04, 4.36, 4.68, 4.48, 4.27, 4.35, 4.34, 4.23, 4.32, 4.22, 4.8, 4.12, 4.71, 4.75, 4.14, 4.17, 4.43, 4.29, 4.48, 4.42, 4.32, 4.17, 4.3, 4.69, 4.24, 4.42, 4.58, 4.13, 3.78, 4.44, 4.2, 4.45, 4.23, 3.82, 4.59, 3.96, 4.51, 4.03, 4.5, 4.32, 4.01, 4.32, 4.14, 4.16, 4.41, 4.34, 4.11, 4.49, 4.23, 4.24, 4.15, 4.52, 4.22, 4.19, 3.94, 4.32, 4.35, 4.33, 4.23, 4.01, 4.2, 4.33, 4.18, 4.56, 4.16, 4.11, 4.45, 4.14, 4.29, 4.26, 3.95, 3.92, 3.85, 4.4, 4.31, 4.09, 4.36, 4.28, 4.5, 4.56, 4.38, 4.32, 4.24, 4.8, 4.16, 4.07, 4.19, 4.59, 3.96, 4.0, 4.38, 4.43, 3.84, 4.63, 4.1, 4.56, 4.32, 4.32, 4.21, 4.33, 4.28, 4.26, 3.8, 4.38, 4.23, 4.28, 4.12, 4.16, 4.46, 4.21, 4.17, 4.25, 4.46, 4.13, 4.44, 4.12, 4.43, 4.44, 4.11, 4.26, 4.07, 4.5, 4.14, 4.25, 4.18, 4.24, 4.48, 4.28, 4.2, 4.51, 3.89, 4.68, 3.79, 4.25, 4.24, 4.2, 4.1, 4.28, 4.19, 4.42, 4.07, 4.3, 4.4, 4.51, 3.88, 4.33, 4.39, 4.45, 4.29, 4.33, 4.35, 4.13, 4.36, 4.25, 4.31, 4.51, 4.1, 4.08, 4.13, 4.14, 4.57, 3.96, 4.56, 4.09, 4.45, 4.06, 4.2, 4.08, 4.1, 4.41, 4.55, 3.86, 4.33, 3.95, 4.29, 4.55, 4.56, 4.37, 4.43, 4.2, 4.18, 4.28, 3.99, 3.82, 4.41, 4.2, 4.16, 4.37, 4.13, 4.34, 4.46, 3.92, 4.31, 4.53, 3.97, 4.44, 4.65, 4.25, 4.59, 4.78, 3.81, 4.07, 4.19, 4.65, 3.86, 4.39, 4.68, 4.29, 4.12, 4.36, 4.15, 4.06, 4.44, 4.51, 4.25, 4.14, 4.14, 4.69, 4.41, 4.22, 4.17, 4.75, 4.41, 4.19, 4.51, 4.36, 3.89, 4.14, 4.12, 4.48, 4.67, 4.65, 4.18, 4.46, 4.25, 4.09, 4.59, 4.54, 4.18, 3.87, 4.35, 4.49, 4.46, 4.68, 4.27, 4.57, 4.29, 4.56, 4.77, 4.03, 4.41, 4.52, 4.59, 4.18, 4.23, 4.12, 4.3, 4.32, 4.24, 4.45, 4.55, 4.18, 4.07, 3.97, 4.34, 4.3, 4.4, 4.21, 4.11, 4.33, 4.34, 4.29, 4.12, 4.09, 4.53, 4.41, 4.5, 4.4, 4.68, 4.27, 4.41, 4.26, 3.85, 4.56, 4.61, 4.08, 4.48, 4.33, 4.4, 4.4, 4.38, 4.05, 4.43, 4.14, 4.47, 4.7, 4.07, 4.88, 4.03, 4.17, 4.53, 4.17, 4.69, 4.58, 4.46, 4.12, 4.46, 4.38, 3.98, 3.88, 4.06, 4.24, 4.44, 4.64, 3.92, 4.19, 4.15, 4.28, 4.42, 3.95, 4.13, 4.37, 4.31, 4.06, 4.47, 4.01, 4.27, 4.23, 4.2, 4.32, 4.27, 4.24, 4.59, 3.97, 4.21, 4.6, 4.26, 4.21, 4.23, 4.49, 4.4, 3.73, 4.34, 4.2, 4.71, 4.26, 4.24, 4.18, 3.98, 4.61, 3.99, 4.59, 4.28, 4.18, 4.2, 4.25, 4.19, 4.06, 4.27, 4.0, 3.97, 3.91, 4.05, 4.33, 4.68, 4.27, 3.7, 4.19, 4.07, 4.74, 4.27, 4.61, 4.56, 3.93, 4.44, 4.28, 4.32, 4.16, 4.69, 3.96, 4.08, 4.29, 4.3, 3.9, 4.4, 4.53, 4.26, 4.71, 4.25, 4.51, 4.01, 3.96, 4.43, 3.92, 4.14, 4.15, 4.04]；
差值(后−前) n=867：[0.69, 0.75, 0.71, 0.69, 0.71, 0.7, 0.68, 0.75, 0.75, 0.73, 0.69, 0.71, 0.68, 0.69, 0.74, 0.71, 0.68, 0.69, 0.68, 0.7, 0.76, 0.65, 0.69, 0.66, 0.68, 0.7, 0.75, 0.7, 0.71, 0.69, 0.69, 0.72, 0.69, 0.67, 0.7, 0.72, 0.68, 0.77, 0.7, 0.74, 0.7, 0.76, 0.72, 0.68, 0.73, 0.73, 0.7, 0.69, 0.67, 0.63, 0.71, 0.75, 0.71, 0.69, 0.75, 0.67, 0.73, 0.69, 0.68, 0.7, 0.72, 0.68, 0.73, 0.69, 0.67, 0.68, 0.74, 0.76, 0.73, 0.64, 0.76, 0.7, 0.73, 0.7, 0.67, 0.66, 0.68, 0.74, 0.75, 0.64, 0.67, 0.71, 0.74, 0.69, 0.64, 0.68, 0.69, 0.74, 0.72, 0.69, 0.72, 0.69, 0.75, 0.7, 0.72, 0.68, 0.69, 0.7, 0.71, 0.75, 0.78, 0.73, 0.67, 0.73, 0.69, 0.73, 0.68, 0.66, 0.76, 0.74, 0.7, 0.72, 0.7, 0.72, 0.72, 0.72, 0.68, 0.68, 0.76, 0.75, 0.72, 0.69, 0.7, 0.68, 0.75, 0.7, 0.75, 0.75, 0.73, 0.72, 0.74, 0.73, 0.79, 0.68, 0.71, 0.67, 0.75, 0.71, 0.71, 0.71, 0.74, 0.62, 0.74, 0.67, 0.71, 0.75, 0.66, 0.74, 0.7, 0.71, 0.7, 0.72, 0.72, 0.71, 0.71, 0.72, 0.76, 0.69, 0.73, 0.67, 0.72, 0.77, 0.7, 0.71, 0.7, 0.7, 0.7, 0.69, 0.63, 0.76, 0.75, 0.75, 0.75, 0.72, 0.68, 0.76, 0.72, 0.72, 0.69, 0.68, 0.69, 0.7, 0.7, 0.7, 0.68, 0.75, 0.72, 0.67, 0.68, 0.71, 0.73, 0.71, 0.71, 0.7, 0.71, 0.66, 0.78, 0.74, 0.76, 0.72, 0.69, 0.74, 0.65, 0.77, 0.72, 0.75, 0.67, 0.72, 0.73, 0.7, 0.74, 0.72, 0.77, 0.72, 0.73, 0.69, 0.71, 0.7, 0.72, 0.75, 0.68, 0.74, 0.67, 0.69, 0.72, 0.71, 0.81, 0.69, 0.74, 0.69, 0.69, 0.69, 0.7, 0.7, 0.75, 0.72, 0.71, 0.74, 0.78, 0.72, 0.71, 0.71, 0.69, 0.71, 0.7, 0.79, 0.73, 0.75, 0.75, 0.67, 0.7, 0.71, 0.7, 0.72, 0.73, 0.66, 0.68, 0.74, 0.75, 0.64, 0.7, 0.68, 0.7, 0.71, 0.72, 0.72, 0.69, 0.7, 0.72, 0.69, 0.73, 0.71, 0.69, 0.73, 0.71, 0.69, 0.65, 0.67, 0.71, 0.72, 0.72, 0.66, 0.73, 0.73, 0.72, 0.68, 0.72, 0.72, 0.7, 0.72, 0.7, 0.7, 0.74, 0.71, 0.72, 0.67, 0.73, 0.7, 0.68, 0.72, 0.73, 0.71, 0.72, 0.72, 0.73, 0.71, 0.76, 0.7, 0.69, 0.69, 0.7, 0.72, 0.73, 0.75, 0.67, 0.69, 0.76, 0.7, 0.71, 0.69, 0.75, 0.69, 0.68, 0.69, 0.68, 0.71, 0.7, 0.68, 0.68, 0.71, 0.68, 0.73, 0.75, 0.75, 0.67, 0.75, 0.72, 0.69, 0.67, 0.74, 0.69, 0.75, 0.74, 0.77, 0.71, 0.78, 0.71, 0.73, 0.64, 0.72, 0.74, 0.66, 0.73, 0.75, 0.73, 0.7, 0.7, 0.68, 0.65, 0.73, 0.72, 0.78, 0.79, 0.67, 0.74, 0.68, 0.73, 0.73, 0.75, 0.71, 0.76, 0.75, 0.66, 0.68, 0.74, 0.74, 0.72, 0.7, 0.71, 0.68, 0.76, 0.69, 0.72, 0.69, 0.75, 0.7, 0.7, 0.68, 0.73, 0.73, 0.74, 0.68, 0.73, 0.75, 0.69, 0.71, 0.74, 0.68, 0.7, 0.71, 0.7, 0.63, 0.71, 0.73, 0.71, 0.69, 0.73, 0.69, 0.66, 0.77, 0.7, 0.71, 0.75, 0.69, 0.69, 0.72, 0.68, 0.74, 0.75, 0.7, 0.74, 0.74, 0.74, 0.74, 0.69, 0.72, 0.68, 0.72, 0.79, 0.65, 0.77, 0.67, 0.75, 0.74, 0.7, 0.73, 0.75, 0.7, 0.66, 0.73, 0.69, 0.67, 0.68, 0.75, 0.7, 0.73, 0.75, 0.64, 0.71, 0.71, 0.68, 0.72, 0.72, 0.68, 0.66, 0.7, 0.7, 0.7, 0.67, 0.72, 0.72, 0.71, 0.68, 0.74, 0.68, 0.7, 0.71, 0.7, 0.72, 0.72, 0.65, 0.73, 0.66, 0.69, 0.76, 0.74, 0.7, 0.68, 0.74, 0.71, 0.69, 0.67, 0.73, 0.66, 0.66, 0.73, 0.73, 0.69, 0.74, 0.69, 0.75, 0.67, 0.72, 0.74, 0.74, 0.67, 0.72, 0.69, 0.71, 0.7, 0.74, 0.75, 0.67, 0.67, 0.7, 0.73, 0.69, 0.65, 0.68, 0.72, 0.71, 0.71, 0.74, 0.71, 0.68, 0.66, 0.68, 0.64, 0.71, 0.71, 0.72, 0.72, 0.7, 0.73, 0.72, 0.71, 0.69, 0.73, 0.71, 0.7, 0.78, 0.71, 0.75, 0.67, 0.69, 0.73, 0.71, 0.7, 0.71, 0.72, 0.71, 0.73, 0.69, 0.75, 0.69, 0.68, 0.74, 0.76, 0.7, 0.7, 0.74, 0.67, 0.73, 0.67, 0.72, 0.66, 0.69, 0.71, 0.67, 0.74, 0.79, 0.7, 0.73, 0.71, 0.67, 0.69, 0.72, 0.72, 0.71, 0.68, 0.73, 0.74, 0.76, 0.71, 0.69, 0.7, 0.71, 0.76, 0.71, 0.69, 0.67, 0.72, 0.69, 0.71, 0.71, 0.74, 0.73, 0.69, 0.72, 0.75, 0.7, 0.7, 0.71, 0.7, 0.73, 0.75, 0.67, 0.74, 0.77, 0.71, 0.7, 0.67, 0.69, 0.74, 0.67, 0.72, 0.7, 0.75, 0.74, 0.76, 0.73, 0.74, 0.73, 0.71, 0.7, 0.77, 0.7, 0.74, 0.78, 0.68, 0.72, 0.72, 0.77, 0.72, 0.73, 0.69, 0.75, 0.74, 0.72, 0.72, 0.71, 0.7, 0.68, 0.76, 0.66, 0.71, 0.67, 0.73, 0.74, 0.73, 0.72, 0.74, 0.69, 0.73, 0.66, 0.72, 0.7, 0.72, 0.71, 0.69, 0.71, 0.76, 0.66, 0.67, 0.71, 0.7, 0.69, 0.7, 0.69, 0.7, 0.73, 0.76, 0.66, 0.67, 0.71, 0.67, 0.7, 0.68, 0.72, 0.72, 0.69, 0.63, 0.73, 0.77, 0.69, 0.78, 0.76, 0.73, 0.71, 0.76, 0.66, 0.76, 0.68, 0.67, 0.72, 0.72, 0.68, 0.72, 0.74, 0.68, 0.72, 0.71, 0.76, 0.7, 0.74, 0.75, 0.72, 0.73, 0.69, 0.76, 0.73, 0.69, 0.69, 0.72, 0.73, 0.73, 0.68, 0.69, 0.76, 0.68, 0.69, 0.72, 0.68, 0.73, 0.66, 0.71, 0.68, 0.74, 0.74, 0.75, 0.76, 0.72, 0.71, 0.72, 0.67, 0.72, 0.75, 0.7, 0.77, 0.69, 0.68, 0.67, 0.68, 0.66, 0.71, 0.8, 0.67, 0.7, 0.73, 0.72, 0.74, 0.72, 0.71, 0.77, 0.69, 0.76, 0.72, 0.71, 0.74, 0.7, 0.65, 0.72, 0.71, 0.7, 0.7, 0.71, 0.68, 0.75, 0.69, 0.74, 0.71, 0.68, 0.73, 0.67, 0.71, 0.76, 0.66, 0.75, 0.68, 0.72, 0.66, 0.69, 0.73, 0.72, 0.77, 0.75, 0.7, 0.65, 0.73, 0.75, 0.73, 0.69, 0.74, 0.71, 0.7, 0.73, 0.72, 0.71, 0.73, 0.71, 0.7, 0.66, 0.71, 0.75, 0.66, 0.66, 0.77, 0.74, 0.7, 0.69, 0.76, 0.72, 0.73, 0.79, 0.68, 0.78, 0.7, 0.73, 0.71, 0.71, 0.68, 0.77, 0.69, 0.72, 0.76, 0.72, 0.74, 0.65, 0.67, 0.73, 0.65, 0.65, 0.67, 0.69, 0.69, 0.7, 0.69, 0.72, 0.68, 0.65, 0.64, 0.75, 0.7, 0.66, 0.69, 0.73, 0.7, 0.78, 0.71, 0.77, 0.71, 0.71, 0.69, 0.68, 0.7, 0.7, 0.69, 0.72, 0.68, 0.7, 0.74, 0.71, 0.66, 0.73, 0.78, 0.68, 0.72, 0.77, 0.74, 0.62, 0.69, 0.67, 0.7, 0.72, 0.72, 0.74, 0.73]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = 0.7109，s_d = 0.0312，SE = s_d/√n = 0.0011；
t = d̄/SE = 670.5592；df = n−1 = 866；双侧 p = 0.000e+00。
d̄ = 0.7109，s_d = 0.0312，SE = s_d/√n = 0.0011；
t = d̄/SE = 670.5592；df = n−1 = 866；双侧 p = 0.000e+00。
计算结果
t = 670.5592，df = 866，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估服务改进成效。
案例 43. 运动前后心率
原始数据
前 n=812：[83.02, 79.88, 75.36, 81.37, 81.43, 82.25, 82.51, 83.74, 80.22, 80.58, 80.34, 78.18, 78.18, 80.35, 84.47, 83.85, 79.76, 81.01, 81.62, 82.87, 78.59, 84.19, 78.86, 77.62, 75.12, 79.7, 81.56, 82.58, 83.53, 81.27, 85.78, 77.88, 74.98, 80.29, 80.29, 80.74, 81.79, 81.03, 79.7, 77.12, 79.08, 73.14, 87.28, 78.45, 79.23, 84.35, 78.24, 79.99, 79.76, 82.02, 82.11, 80.88, 81.51, 84.11, 82.53, 80.69, 83.31, 83.34, 81.39, 79.36, 81.53, 80.12, 79.46, 85.97, 81.89, 83.69, 81.25, 81.76, 83.56, 83.27, 79.78, 79.12, 79.94, 79.32, 77.78, 81.02, 84.42, 82.98, 84.1, 79.87, 82.05, 81.15, 82.28, 79.51, 80.04, 79.09, 78.91, 81.29, 75.88, 83.08, 77.87, 76.66, 82.1, 81.09, 83.42, 80.45, 85.14, 81.1, 80.52, 79.97, 82.43, 82.76, 80.44, 80.13, 82.45, 82.35, 82.86, 78.08, 78.19, 79.39, 82.37, 80.73, 82.22, 86.44, 79.97, 83.68, 79.64, 84.57, 82.07, 79.09, 87.03, 82.47, 84.41, 80.59, 79.0, 79.08, 78.19, 77.36, 80.18, 78.67, 83.76, 79.91, 81.85, 79.25, 79.59, 84.09, 79.94, 85.17, 82.23, 77.54, 79.32, 84.89, 81.44, 83.14, 83.57, 81.37, 84.7, 78.72, 82.71, 80.86, 82.93, 79.1, 80.28, 81.57, 78.77, 85.59, 81.07, 85.6, 83.14, 78.97, 84.88, 79.68, 83.69, 79.15, 81.87, 81.44, 79.58, 84.31, 81.18, 76.72, 81.89, 82.89, 81.85, 83.36, 80.68, 80.54, 84.05, 80.62, 86.36, 83.37, 82.49, 79.86, 82.34, 76.85, 75.94, 81.79, 82.36, 80.52, 80.82, 78.06, 82.09, 80.53, 83.22, 80.72, 82.97, 82.75, 80.06, 81.08, 79.51, 81.49, 83.17, 86.5, 83.65, 83.57, 79.92, 82.07, 80.75, 82.35, 81.23, 82.38, 78.16, 80.48, 83.58, 81.13, 80.87, 82.04, 77.35, 75.41, 82.97, 79.25, 86.09, 84.47, 76.48, 82.46, 79.47, 84.46, 80.03, 81.22, 76.47, 85.67, 81.84, 78.29, 75.15, 77.66, 77.71, 82.71, 76.98, 85.57, 81.23, 80.12, 80.43, 80.49, 81.14, 82.94, 80.2, 78.77, 83.36, 86.32, 80.11, 80.44, 85.17, 82.91, 81.09, 83.82, 83.65, 80.31, 83.59, 80.49, 80.92, 80.63, 83.83, 80.3, 84.36, 81.14, 84.2, 82.98, 85.8, 81.57, 77.64, 81.86, 87.8, 78.0, 78.91, 81.26, 79.25, 81.82, 84.7, 83.28, 81.19, 79.08, 87.29, 84.72, 84.58, 83.85, 82.06, 80.02, 81.86, 86.24, 78.11, 81.25, 80.57, 80.88, 82.31, 82.9, 82.95, 79.64, 78.56, 78.39, 82.08, 76.14, 76.02, 81.27, 88.7, 81.07, 81.42, 77.8, 82.67, 82.53, 78.54, 82.56, 79.48, 79.11, 79.83, 81.98, 80.79, 85.15, 84.2, 77.53, 80.24, 78.97, 80.72, 80.0, 82.4, 79.04, 78.36, 83.07, 87.57, 89.72, 81.48, 81.75, 80.21, 81.65, 82.89, 78.84, 76.41, 77.21, 80.39, 79.33, 81.43, 82.99, 84.47, 80.77, 82.2, 80.65, 84.67, 79.84, 84.97, 81.92, 85.07, 80.56, 80.38, 84.17, 78.68, 84.73, 82.96, 83.39, 80.42, 84.6, 79.69, 78.21, 76.25, 80.81, 84.76, 79.68, 81.1, 79.5, 78.09, 79.92, 78.39, 81.63, 76.48, 82.82, 76.28, 85.38, 78.05, 80.94, 80.64, 81.78, 81.06, 80.18, 78.1, 82.62, 81.35, 79.87, 77.98, 79.93, 84.58, 76.72, 84.22, 82.25, 81.76, 81.35, 81.92, 83.96, 82.68, 76.76, 81.67, 83.48, 78.33, 80.82, 79.64, 77.72, 81.94, 80.19, 82.52, 78.24, 79.15, 77.03, 78.19, 84.29, 84.26, 78.68, 81.72, 80.09, 85.24, 79.94, 79.3, 82.44, 82.0, 79.17, 80.86, 85.58, 83.3, 85.77, 79.26, 76.07, 78.32, 81.58, 80.41, 80.48, 81.81, 83.63, 84.04, 77.61, 78.06, 78.26, 82.68, 87.49, 77.24, 82.13, 80.45, 82.42, 81.04, 83.8, 86.01, 82.49, 78.86, 82.96, 85.84, 82.2, 81.33, 80.04, 87.08, 81.26, 78.03, 77.94, 83.49, 79.34, 76.58, 82.62, 82.07, 79.56, 76.84, 79.82, 82.35, 78.47, 76.53, 76.16, 81.15, 82.77, 84.03, 82.05, 84.36, 82.65, 82.43, 76.71, 81.33, 77.85, 82.46, 76.99, 76.04, 83.83, 85.43, 80.87, 85.81, 79.8, 80.77, 74.26, 78.63, 82.67, 84.45, 82.93, 80.92, 82.79, 77.75, 77.69, 82.37, 84.38, 83.21, 80.21, 82.53, 86.19, 80.92, 81.31, 80.19, 81.32, 81.24, 82.2, 82.91, 83.51, 81.49, 83.62, 81.13, 83.27, 83.15, 84.43, 80.9, 82.79, 80.28, 74.7, 78.06, 80.55, 83.29, 81.72, 81.77, 77.47, 81.57, 76.24, 84.79, 79.65, 81.57, 81.75, 80.54, 79.94, 82.06, 80.69, 81.48, 82.71, 81.23, 77.43, 80.02, 78.08, 80.14, 76.08, 84.61, 86.1, 80.82, 81.71, 79.2, 82.49, 79.84, 79.09, 76.97, 76.74, 81.39, 82.11, 82.18, 84.99, 79.99, 80.5, 83.79, 81.63, 81.81, 85.89, 81.93, 77.1, 83.35, 81.99, 77.01, 85.12, 80.97, 78.97, 79.96, 87.8, 76.46, 81.74, 80.85, 80.03, 82.83, 83.94, 83.71, 86.24, 80.52, 81.86, 76.58, 84.72, 87.91, 80.83, 82.75, 82.71, 76.7, 82.92, 81.27, 80.04, 79.65, 76.9, 80.13, 82.78, 79.13, 83.09, 85.03, 84.39, 78.93, 83.16, 78.11, 80.84, 80.69, 80.13, 78.65, 81.25, 78.75, 79.87, 78.99, 85.4, 80.14, 81.21, 79.17, 78.18, 88.38, 80.1, 76.08, 82.73, 80.81, 81.29, 82.65, 81.67, 82.07, 83.49, 77.35, 76.98, 78.88, 80.72, 79.96, 81.6, 80.75, 82.09, 82.14, 81.68, 83.98, 77.62, 83.4, 80.11, 82.5, 84.04, 77.43, 80.77, 83.8, 84.6, 79.89, 75.3, 83.07, 80.3, 83.74, 78.75, 81.25, 79.43, 78.74, 82.79, 82.39, 85.15, 81.51, 80.85, 78.83, 81.62, 82.24, 82.7, 76.24, 83.38, 76.87, 78.56, 81.81, 82.34, 82.94, 79.95, 81.47, 81.69, 82.48, 82.89, 79.21, 82.63, 78.8, 85.48, 83.96, 77.08, 79.59, 80.91, 82.14, 80.88, 79.54, 83.2, 83.2, 82.81, 85.21, 79.28, 81.92, 82.15, 80.33, 78.24, 80.14, 81.95, 83.02, 79.68, 81.68, 84.54, 76.01, 82.38, 78.95, 84.89, 85.06, 81.59, 81.44, 79.25, 84.19, 82.17, 81.53, 79.15, 79.72, 81.1, 85.23, 75.03, 82.12, 80.99, 78.0, 77.67, 80.88, 83.0, 81.06, 78.0, 79.23, 83.96, 83.45, 78.29, 78.69, 80.67, 82.84, 80.67, 84.59, 84.1, 78.56, 82.28, 79.09, 82.0, 81.95, 83.45, 79.11, 74.95, 82.95, 81.86, 83.81, 81.96, 80.67, 77.42, 79.21, 80.64, 76.2, 75.68, 81.08, 82.71, 82.08, 85.68, 79.35, 84.93, 82.08, 85.6, 79.58, 80.54, 86.45, 81.84, 81.52, 79.58, 81.25, 78.31, 81.29, 80.99, 82.19, 79.26, 83.36, 81.09, 81.85, 83.29, 76.06, 77.06, 76.75, 83.56, 84.21, 78.95, 80.05, 84.4, 78.19, 83.64, 83.4, 85.93, 83.02, 83.59, 86.18, 81.69, 84.35, 82.95, 79.74, 80.87, 83.91, 84.11, 83.71, 84.96, 81.49, 81.02, 80.62, 82.18, 81.09, 79.78, 77.94, 80.17]；
后 n=812：[74.43, 70.64, 66.63, 73.12, 73.96, 75.03, 74.95, 75.92, 73.2, 71.61, 72.15, 69.85, 71.08, 73.19, 75.24, 76.19, 71.92, 72.47, 73.82, 74.22, 70.81, 76.34, 70.58, 68.87, 67.57, 70.05, 73.64, 74.97, 75.69, 74.95, 76.77, 68.28, 67.24, 72.24, 72.19, 73.64, 73.55, 72.36, 71.82, 69.74, 71.09, 65.8, 79.61, 69.02, 71.02, 76.22, 70.09, 72.67, 71.23, 74.26, 74.8, 73.34, 73.41, 75.61, 74.61, 71.12, 75.29, 75.99, 71.56, 70.68, 73.07, 72.2, 71.1, 77.6, 73.4, 76.16, 73.42, 73.98, 75.34, 76.52, 70.72, 71.65, 71.51, 71.19, 69.09, 73.33, 75.93, 74.36, 76.23, 71.81, 73.88, 73.93, 73.55, 72.53, 73.48, 71.61, 71.24, 72.13, 66.91, 75.12, 71.54, 68.94, 73.71, 73.29, 75.03, 71.87, 77.84, 73.28, 72.46, 72.0, 73.91, 75.27, 72.33, 71.11, 74.58, 74.24, 75.0, 69.73, 70.13, 70.81, 74.86, 72.91, 74.07, 79.12, 71.98, 76.41, 72.47, 78.07, 72.94, 71.12, 78.96, 74.47, 76.42, 73.3, 70.76, 71.36, 70.94, 69.65, 71.42, 69.38, 76.14, 71.94, 73.53, 70.5, 71.5, 76.44, 70.58, 77.89, 73.73, 70.56, 70.83, 76.07, 73.3, 75.99, 75.42, 72.67, 76.1, 69.91, 74.53, 72.5, 74.64, 70.44, 72.34, 73.83, 71.65, 77.54, 72.78, 76.36, 75.03, 70.62, 77.36, 71.99, 76.85, 71.28, 72.98, 74.65, 71.92, 75.44, 73.02, 67.68, 74.5, 75.58, 72.26, 74.86, 71.61, 73.33, 75.87, 72.93, 78.79, 75.07, 74.55, 71.78, 75.07, 69.14, 67.63, 74.21, 74.09, 72.1, 72.02, 70.11, 73.65, 71.39, 75.7, 73.16, 74.34, 73.82, 72.18, 71.87, 71.94, 73.17, 75.8, 78.44, 76.52, 76.19, 70.79, 74.24, 72.49, 73.84, 72.64, 74.19, 70.28, 71.93, 74.75, 73.7, 73.23, 73.06, 68.37, 67.54, 74.25, 70.93, 78.41, 75.25, 68.05, 74.85, 73.05, 76.62, 72.07, 73.36, 69.09, 77.91, 74.12, 71.15, 67.8, 69.92, 70.16, 74.59, 68.75, 77.61, 74.09, 71.26, 71.36, 72.38, 72.4, 76.27, 72.57, 70.81, 75.57, 77.47, 72.13, 72.14, 77.26, 74.53, 74.19, 75.59, 75.65, 72.57, 77.39, 73.08, 71.78, 74.24, 75.66, 72.0, 76.76, 73.66, 78.07, 74.89, 79.2, 73.89, 70.42, 73.54, 79.99, 69.56, 71.17, 73.38, 72.02, 74.08, 77.52, 76.28, 73.78, 70.95, 78.85, 76.37, 77.63, 74.61, 73.93, 72.26, 74.16, 78.47, 70.45, 73.87, 72.6, 71.67, 75.22, 74.98, 74.74, 71.01, 69.5, 69.49, 74.76, 68.06, 67.66, 74.34, 80.88, 74.73, 73.38, 69.08, 74.04, 74.38, 70.64, 72.83, 71.75, 71.95, 70.26, 73.59, 72.78, 78.2, 75.17, 69.61, 71.86, 71.23, 71.8, 71.5, 74.11, 70.68, 69.54, 75.5, 79.88, 81.86, 74.1, 73.49, 72.05, 73.07, 75.54, 69.46, 67.73, 69.08, 72.47, 70.71, 71.76, 73.86, 77.64, 73.81, 73.58, 71.95, 77.09, 70.64, 77.3, 74.26, 75.89, 71.95, 71.77, 76.53, 70.49, 75.44, 75.94, 75.04, 71.71, 76.85, 70.11, 70.09, 68.1, 72.54, 76.52, 70.91, 73.38, 72.09, 69.04, 70.99, 69.17, 72.75, 69.34, 74.48, 69.08, 76.91, 69.73, 73.34, 72.54, 73.7, 73.02, 71.64, 69.14, 74.27, 72.82, 70.71, 69.76, 71.08, 76.08, 68.51, 75.72, 74.4, 72.65, 72.66, 73.26, 76.35, 75.24, 69.36, 72.93, 75.66, 71.77, 73.46, 70.88, 69.7, 73.79, 72.65, 74.29, 68.51, 70.01, 70.64, 69.49, 74.58, 76.32, 70.92, 73.54, 71.42, 75.83, 72.58, 72.05, 75.44, 74.72, 70.54, 71.45, 76.56, 75.22, 76.98, 72.13, 67.89, 70.58, 74.64, 71.82, 72.19, 73.05, 75.45, 75.36, 69.88, 71.02, 70.26, 74.31, 78.82, 70.32, 74.82, 72.32, 72.97, 73.04, 72.97, 77.29, 74.09, 70.61, 75.15, 78.28, 73.8, 72.49, 72.55, 80.06, 72.61, 70.12, 70.01, 75.65, 70.87, 68.64, 74.16, 72.7, 71.26, 68.4, 71.9, 73.76, 69.89, 67.56, 66.26, 73.49, 75.14, 76.02, 73.3, 75.42, 75.25, 74.52, 68.82, 72.91, 70.02, 74.87, 68.7, 68.56, 75.9, 78.1, 74.3, 78.1, 70.62, 73.04, 66.01, 70.65, 74.36, 75.6, 73.87, 71.77, 74.32, 69.99, 69.76, 74.34, 75.71, 74.49, 70.78, 74.28, 79.12, 72.88, 73.72, 71.58, 73.51, 72.24, 74.99, 73.93, 75.97, 74.26, 74.78, 74.75, 75.35, 75.61, 74.67, 71.78, 74.48, 71.41, 65.97, 69.85, 72.08, 74.05, 73.84, 73.75, 69.68, 72.96, 68.5, 75.75, 70.41, 74.11, 72.27, 72.84, 71.18, 74.0, 73.58, 73.06, 75.14, 72.12, 70.57, 71.62, 70.41, 72.64, 68.61, 77.91, 78.14, 72.55, 73.76, 71.37, 74.34, 71.93, 71.93, 68.63, 68.27, 74.56, 73.6, 74.16, 77.22, 71.3, 71.53, 74.98, 73.12, 73.39, 77.38, 74.45, 70.08, 74.68, 75.29, 70.81, 76.81, 73.92, 71.29, 72.04, 78.98, 68.16, 73.96, 73.42, 72.2, 75.26, 75.14, 75.53, 78.83, 71.93, 72.71, 68.43, 76.45, 79.89, 73.35, 74.76, 74.31, 69.06, 73.25, 72.76, 72.68, 71.46, 68.97, 71.7, 73.94, 70.32, 74.25, 76.98, 75.34, 70.87, 74.64, 69.41, 73.11, 71.72, 72.78, 70.83, 73.5, 71.2, 72.47, 71.15, 77.78, 71.16, 71.41, 71.22, 69.68, 81.34, 72.55, 67.53, 76.19, 73.09, 72.44, 73.85, 74.4, 74.54, 74.91, 69.35, 66.66, 70.08, 72.84, 71.41, 74.24, 72.89, 75.5, 73.32, 73.31, 75.66, 69.01, 75.7, 71.85, 75.01, 76.72, 68.76, 72.68, 75.61, 76.98, 72.27, 68.25, 75.67, 70.8, 75.37, 70.72, 73.77, 71.54, 70.05, 74.92, 75.35, 76.5, 73.88, 71.94, 70.86, 74.1, 75.95, 73.69, 68.05, 75.61, 69.4, 72.18, 73.84, 73.91, 73.89, 71.29, 74.26, 73.69, 75.35, 75.19, 70.42, 74.85, 71.99, 76.81, 76.81, 69.5, 70.33, 71.86, 74.99, 73.03, 70.79, 74.47, 75.69, 74.63, 76.82, 72.92, 74.19, 73.18, 72.19, 71.81, 71.7, 75.78, 75.38, 72.1, 73.69, 75.88, 68.17, 74.86, 72.12, 77.11, 76.82, 74.91, 74.41, 71.2, 75.74, 74.93, 73.86, 70.21, 71.82, 72.82, 76.63, 66.87, 73.64, 73.1, 68.99, 70.09, 74.08, 76.14, 71.63, 70.27, 70.99, 76.01, 76.13, 70.17, 70.81, 73.0, 75.78, 74.31, 76.75, 74.43, 69.98, 74.95, 70.64, 73.08, 75.03, 75.79, 71.01, 67.12, 74.29, 73.49, 75.84, 74.1, 72.3, 70.05, 71.41, 73.51, 67.79, 68.84, 71.1, 74.7, 74.04, 76.54, 71.22, 76.94, 73.83, 77.6, 69.98, 73.17, 78.15, 74.13, 74.05, 71.29, 73.24, 69.81, 74.54, 74.15, 74.14, 71.0, 74.23, 73.03, 73.58, 75.78, 68.37, 68.21, 70.35, 74.57, 76.14, 70.22, 71.82, 76.66, 70.99, 75.83, 75.09, 76.98, 75.47, 74.65, 77.9, 73.04, 77.71, 73.74, 71.42, 72.77, 76.12, 76.5, 76.4, 77.52, 73.55, 72.73, 71.98, 74.03, 71.55, 70.31, 70.59, 72.58]；
差值(后−前) n=812：[-8.6, -9.25, -8.74, -8.25, -7.47, -7.22, -7.56, -7.82, -7.02, -8.97, -8.19, -8.33, -7.1, -7.15, -9.23, -7.66, -7.84, -8.54, -7.79, -8.65, -7.78, -7.84, -8.28, -8.75, -7.56, -9.65, -7.92, -7.61, -7.84, -6.32, -9.01, -9.6, -7.74, -8.06, -8.1, -7.09, -8.24, -8.67, -7.88, -7.38, -7.99, -7.34, -7.67, -9.43, -8.21, -8.13, -8.15, -7.32, -8.53, -7.77, -7.31, -7.54, -8.1, -8.5, -7.92, -9.57, -8.02, -7.35, -9.82, -8.68, -8.46, -7.93, -8.37, -8.37, -8.49, -7.53, -7.82, -7.78, -8.22, -6.74, -9.06, -7.47, -8.43, -8.13, -8.69, -7.69, -8.5, -8.62, -7.87, -8.07, -8.17, -7.22, -8.73, -6.98, -6.56, -7.48, -7.67, -9.16, -8.97, -7.97, -6.32, -7.72, -8.39, -7.8, -8.39, -8.58, -7.3, -7.82, -8.07, -7.97, -8.53, -7.5, -8.12, -9.02, -7.87, -8.11, -7.86, -8.34, -8.06, -8.59, -7.52, -7.81, -8.15, -7.32, -7.99, -7.26, -7.17, -6.5, -9.14, -7.97, -8.07, -8.0, -7.99, -7.28, -8.24, -7.72, -7.24, -7.71, -8.76, -9.29, -7.62, -7.96, -8.31, -8.75, -8.09, -7.66, -9.36, -7.28, -8.5, -6.98, -8.49, -8.82, -8.15, -7.14, -8.15, -8.7, -8.61, -8.81, -8.17, -8.37, -8.29, -8.66, -7.95, -7.74, -7.12, -8.05, -8.29, -9.24, -8.1, -8.35, -7.52, -7.69, -6.85, -7.87, -8.89, -6.78, -7.66, -8.86, -8.16, -9.03, -7.39, -7.31, -9.59, -8.49, -9.07, -7.21, -8.18, -7.69, -7.58, -8.29, -7.94, -8.08, -7.27, -7.72, -8.31, -7.58, -8.27, -8.42, -8.8, -7.95, -8.44, -9.14, -7.51, -7.55, -8.64, -8.93, -7.88, -9.21, -7.57, -8.32, -7.38, -8.06, -7.13, -7.38, -9.12, -7.84, -8.27, -8.5, -8.59, -8.19, -7.87, -8.54, -8.84, -7.42, -7.65, -8.98, -8.98, -7.86, -8.72, -8.32, -7.68, -9.23, -8.43, -7.61, -6.42, -7.84, -7.96, -7.85, -7.39, -7.76, -7.72, -7.14, -7.35, -7.75, -7.55, -8.12, -8.22, -7.96, -7.14, -8.87, -9.07, -8.11, -8.73, -6.67, -7.63, -7.95, -7.8, -8.86, -7.98, -8.29, -7.91, -8.38, -6.9, -8.23, -8.0, -7.74, -6.2, -7.41, -9.14, -6.39, -8.17, -8.29, -7.6, -7.48, -6.13, -8.09, -6.6, -7.68, -7.22, -8.32, -7.8, -8.44, -7.73, -7.89, -7.23, -7.75, -7.19, -7.0, -7.41, -8.12, -8.44, -8.34, -6.95, -9.23, -8.14, -7.76, -7.71, -7.77, -7.66, -7.38, -7.97, -9.22, -7.09, -7.91, -8.21, -8.63, -9.06, -8.9, -7.32, -8.09, -8.35, -6.93, -7.81, -6.34, -8.05, -8.72, -8.63, -8.15, -7.9, -9.73, -7.73, -7.16, -9.57, -8.39, -8.01, -6.95, -9.03, -7.92, -8.38, -7.74, -8.92, -8.49, -8.29, -8.36, -8.82, -7.57, -7.69, -7.86, -7.38, -8.26, -8.17, -8.58, -7.35, -9.38, -8.68, -8.13, -7.92, -8.62, -9.67, -9.14, -6.83, -6.96, -8.61, -8.7, -7.59, -9.19, -7.67, -7.66, -9.18, -8.61, -8.61, -7.64, -8.2, -9.29, -7.02, -8.35, -8.71, -7.75, -9.58, -8.12, -8.14, -8.27, -8.24, -8.77, -7.72, -7.41, -9.05, -8.92, -9.22, -8.88, -7.14, -8.34, -7.2, -8.47, -8.32, -7.6, -8.1, -8.08, -8.04, -8.54, -8.96, -8.34, -8.53, -9.16, -8.22, -8.85, -8.5, -8.22, -8.5, -7.85, -9.1, -8.69, -8.65, -7.61, -7.43, -7.39, -8.73, -7.82, -6.56, -7.36, -8.76, -8.01, -8.15, -7.54, -8.23, -9.73, -9.14, -6.39, -8.7, -9.71, -7.95, -7.76, -8.18, -8.67, -9.41, -7.37, -7.25, -7.0, -7.28, -8.63, -9.42, -9.01, -8.08, -8.79, -7.13, -8.17, -7.74, -6.94, -8.59, -8.29, -8.77, -8.19, -8.68, -7.72, -7.05, -7.99, -8.37, -8.68, -6.92, -7.31, -8.13, -9.45, -8.0, -10.83, -8.72, -8.41, -8.25, -7.81, -7.56, -8.4, -8.84, -7.5, -7.02, -8.65, -7.91, -7.94, -7.84, -8.47, -7.94, -8.46, -9.38, -8.31, -8.44, -7.92, -8.59, -8.58, -8.97, -9.9, -7.66, -7.63, -8.01, -8.75, -8.95, -7.4, -7.91, -7.89, -8.42, -7.83, -7.59, -8.29, -7.48, -7.93, -7.33, -6.57, -7.71, -9.18, -7.73, -8.25, -7.98, -8.31, -8.85, -9.06, -9.15, -8.47, -7.76, -7.93, -8.03, -8.67, -8.72, -9.43, -8.26, -7.07, -8.04, -7.6, -8.61, -7.82, -9.0, -7.21, -8.98, -7.54, -7.23, -8.83, -6.39, -7.92, -7.54, -9.76, -9.12, -8.31, -8.87, -8.74, -8.21, -8.47, -9.24, -7.88, -8.02, -7.79, -8.61, -7.75, -9.04, -9.25, -7.45, -9.48, -7.71, -8.77, -8.07, -7.11, -8.42, -7.57, -9.11, -6.87, -8.39, -7.67, -7.5, -7.47, -6.7, -7.96, -8.27, -7.96, -7.84, -8.15, -7.91, -7.16, -8.35, -8.47, -6.84, -8.52, -8.02, -7.77, -8.69, -8.97, -8.81, -8.52, -8.43, -8.51, -7.48, -7.02, -8.68, -6.7, -6.2, -8.3, -7.05, -7.68, -7.93, -8.83, -8.3, -7.78, -7.42, -7.82, -7.58, -8.8, -8.18, -7.41, -8.59, -9.15, -8.14, -8.27, -8.02, -7.48, -7.99, -8.4, -7.64, -9.67, -8.51, -7.35, -8.19, -7.93, -8.42, -8.84, -8.81, -8.84, -8.05, -9.04, -8.06, -8.52, -8.7, -7.73, -8.97, -7.35, -7.82, -7.75, -7.55, -7.4, -7.84, -7.62, -8.98, -9.8, -7.94, -8.5, -7.04, -7.55, -8.55, -6.54, -7.71, -8.85, -8.81, -7.28, -7.53, -8.59, -8.0, -10.32, -8.8, -7.88, -8.56, -7.36, -7.86, -6.59, -8.82, -8.38, -8.32, -8.6, -7.7, -8.25, -7.49, -7.32, -8.66, -8.09, -8.2, -7.63, -7.62, -7.05, -7.4, -9.5, -8.37, -8.03, -7.48, -7.9, -8.69, -7.87, -7.04, -8.65, -7.63, -8.91, -7.97, -7.52, -6.29, -9.01, -8.2, -7.78, -7.47, -6.38, -7.97, -8.43, -9.06, -8.66, -7.21, -8.0, -7.13, -7.7, -8.79, -7.78, -6.81, -8.67, -7.15, -7.58, -9.26, -9.04, -7.15, -7.85, -8.75, -8.73, -7.51, -8.19, -8.39, -6.36, -7.72, -8.96, -8.14, -6.43, -8.44, -6.17, -7.63, -7.57, -7.99, -8.67, -7.84, -7.51, -6.82, -7.78, -8.24, -6.68, -7.04, -8.05, -8.45, -7.24, -7.68, -8.94, -7.9, -8.28, -8.61, -8.16, -8.48, -7.89, -9.01, -7.59, -6.81, -6.87, -9.43, -7.73, -8.24, -7.95, -7.32, -8.12, -7.87, -7.66, -7.05, -6.36, -7.84, -9.67, -8.58, -7.33, -8.45, -8.92, -6.92, -7.66, -8.1, -7.82, -8.65, -8.37, -7.97, -7.86, -8.37, -7.37, -7.8, -7.12, -8.42, -6.85, -9.98, -8.02, -8.03, -9.14, -8.13, -8.0, -8.24, -8.0, -9.6, -7.37, -8.3, -7.7, -7.47, -8.29, -8.0, -8.51, -6.75, -6.85, -8.05, -8.26, -9.13, -8.06, -8.28, -7.51, -7.69, -8.85, -6.4, -8.99, -8.06, -8.73, -8.23, -7.74, -7.2, -7.81, -8.3, -8.96, -7.55, -8.94, -8.29, -8.65, -6.64, -9.2, -8.32, -8.1, -7.79, -7.62, -7.31, -7.45, -7.95, -8.28, -8.64, -8.15, -9.53, -9.47, -7.35, -7.59]
后 n=812：[74.43, 70.64, 66.63, 73.12, 73.96, 75.03, 74.95, 75.92, 73.2, 71.61, 72.15, 69.85, 71.08, 73.19, 75.24, 76.19, 71.92, 72.47, 73.82, 74.22, 70.81, 76.34, 70.58, 68.87, 67.57, 70.05, 73.64, 74.97, 75.69, 74.95, 76.77, 68.28, 67.24, 72.24, 72.19, 73.64, 73.55, 72.36, 71.82, 69.74, 71.09, 65.8, 79.61, 69.02, 71.02, 76.22, 70.09, 72.67, 71.23, 74.26, 74.8, 73.34, 73.41, 75.61, 74.61, 71.12, 75.29, 75.99, 71.56, 70.68, 73.07, 72.2, 71.1, 77.6, 73.4, 76.16, 73.42, 73.98, 75.34, 76.52, 70.72, 71.65, 71.51, 71.19, 69.09, 73.33, 75.93, 74.36, 76.23, 71.81, 73.88, 73.93, 73.55, 72.53, 73.48, 71.61, 71.24, 72.13, 66.91, 75.12, 71.54, 68.94, 73.71, 73.29, 75.03, 71.87, 77.84, 73.28, 72.46, 72.0, 73.91, 75.27, 72.33, 71.11, 74.58, 74.24, 75.0, 69.73, 70.13, 70.81, 74.86, 72.91, 74.07, 79.12, 71.98, 76.41, 72.47, 78.07, 72.94, 71.12, 78.96, 74.47, 76.42, 73.3, 70.76, 71.36, 70.94, 69.65, 71.42, 69.38, 76.14, 71.94, 73.53, 70.5, 71.5, 76.44, 70.58, 77.89, 73.73, 70.56, 70.83, 76.07, 73.3, 75.99, 75.42, 72.67, 76.1, 69.91, 74.53, 72.5, 74.64, 70.44, 72.34, 73.83, 71.65, 77.54, 72.78, 76.36, 75.03, 70.62, 77.36, 71.99, 76.85, 71.28, 72.98, 74.65, 71.92, 75.44, 73.02, 67.68, 74.5, 75.58, 72.26, 74.86, 71.61, 73.33, 75.87, 72.93, 78.79, 75.07, 74.55, 71.78, 75.07, 69.14, 67.63, 74.21, 74.09, 72.1, 72.02, 70.11, 73.65, 71.39, 75.7, 73.16, 74.34, 73.82, 72.18, 71.87, 71.94, 73.17, 75.8, 78.44, 76.52, 76.19, 70.79, 74.24, 72.49, 73.84, 72.64, 74.19, 70.28, 71.93, 74.75, 73.7, 73.23, 73.06, 68.37, 67.54, 74.25, 70.93, 78.41, 75.25, 68.05, 74.85, 73.05, 76.62, 72.07, 73.36, 69.09, 77.91, 74.12, 71.15, 67.8, 69.92, 70.16, 74.59, 68.75, 77.61, 74.09, 71.26, 71.36, 72.38, 72.4, 76.27, 72.57, 70.81, 75.57, 77.47, 72.13, 72.14, 77.26, 74.53, 74.19, 75.59, 75.65, 72.57, 77.39, 73.08, 71.78, 74.24, 75.66, 72.0, 76.76, 73.66, 78.07, 74.89, 79.2, 73.89, 70.42, 73.54, 79.99, 69.56, 71.17, 73.38, 72.02, 74.08, 77.52, 76.28, 73.78, 70.95, 78.85, 76.37, 77.63, 74.61, 73.93, 72.26, 74.16, 78.47, 70.45, 73.87, 72.6, 71.67, 75.22, 74.98, 74.74, 71.01, 69.5, 69.49, 74.76, 68.06, 67.66, 74.34, 80.88, 74.73, 73.38, 69.08, 74.04, 74.38, 70.64, 72.83, 71.75, 71.95, 70.26, 73.59, 72.78, 78.2, 75.17, 69.61, 71.86, 71.23, 71.8, 71.5, 74.11, 70.68, 69.54, 75.5, 79.88, 81.86, 74.1, 73.49, 72.05, 73.07, 75.54, 69.46, 67.73, 69.08, 72.47, 70.71, 71.76, 73.86, 77.64, 73.81, 73.58, 71.95, 77.09, 70.64, 77.3, 74.26, 75.89, 71.95, 71.77, 76.53, 70.49, 75.44, 75.94, 75.04, 71.71, 76.85, 70.11, 70.09, 68.1, 72.54, 76.52, 70.91, 73.38, 72.09, 69.04, 70.99, 69.17, 72.75, 69.34, 74.48, 69.08, 76.91, 69.73, 73.34, 72.54, 73.7, 73.02, 71.64, 69.14, 74.27, 72.82, 70.71, 69.76, 71.08, 76.08, 68.51, 75.72, 74.4, 72.65, 72.66, 73.26, 76.35, 75.24, 69.36, 72.93, 75.66, 71.77, 73.46, 70.88, 69.7, 73.79, 72.65, 74.29, 68.51, 70.01, 70.64, 69.49, 74.58, 76.32, 70.92, 73.54, 71.42, 75.83, 72.58, 72.05, 75.44, 74.72, 70.54, 71.45, 76.56, 75.22, 76.98, 72.13, 67.89, 70.58, 74.64, 71.82, 72.19, 73.05, 75.45, 75.36, 69.88, 71.02, 70.26, 74.31, 78.82, 70.32, 74.82, 72.32, 72.97, 73.04, 72.97, 77.29, 74.09, 70.61, 75.15, 78.28, 73.8, 72.49, 72.55, 80.06, 72.61, 70.12, 70.01, 75.65, 70.87, 68.64, 74.16, 72.7, 71.26, 68.4, 71.9, 73.76, 69.89, 67.56, 66.26, 73.49, 75.14, 76.02, 73.3, 75.42, 75.25, 74.52, 68.82, 72.91, 70.02, 74.87, 68.7, 68.56, 75.9, 78.1, 74.3, 78.1, 70.62, 73.04, 66.01, 70.65, 74.36, 75.6, 73.87, 71.77, 74.32, 69.99, 69.76, 74.34, 75.71, 74.49, 70.78, 74.28, 79.12, 72.88, 73.72, 71.58, 73.51, 72.24, 74.99, 73.93, 75.97, 74.26, 74.78, 74.75, 75.35, 75.61, 74.67, 71.78, 74.48, 71.41, 65.97, 69.85, 72.08, 74.05, 73.84, 73.75, 69.68, 72.96, 68.5, 75.75, 70.41, 74.11, 72.27, 72.84, 71.18, 74.0, 73.58, 73.06, 75.14, 72.12, 70.57, 71.62, 70.41, 72.64, 68.61, 77.91, 78.14, 72.55, 73.76, 71.37, 74.34, 71.93, 71.93, 68.63, 68.27, 74.56, 73.6, 74.16, 77.22, 71.3, 71.53, 74.98, 73.12, 73.39, 77.38, 74.45, 70.08, 74.68, 75.29, 70.81, 76.81, 73.92, 71.29, 72.04, 78.98, 68.16, 73.96, 73.42, 72.2, 75.26, 75.14, 75.53, 78.83, 71.93, 72.71, 68.43, 76.45, 79.89, 73.35, 74.76, 74.31, 69.06, 73.25, 72.76, 72.68, 71.46, 68.97, 71.7, 73.94, 70.32, 74.25, 76.98, 75.34, 70.87, 74.64, 69.41, 73.11, 71.72, 72.78, 70.83, 73.5, 71.2, 72.47, 71.15, 77.78, 71.16, 71.41, 71.22, 69.68, 81.34, 72.55, 67.53, 76.19, 73.09, 72.44, 73.85, 74.4, 74.54, 74.91, 69.35, 66.66, 70.08, 72.84, 71.41, 74.24, 72.89, 75.5, 73.32, 73.31, 75.66, 69.01, 75.7, 71.85, 75.01, 76.72, 68.76, 72.68, 75.61, 76.98, 72.27, 68.25, 75.67, 70.8, 75.37, 70.72, 73.77, 71.54, 70.05, 74.92, 75.35, 76.5, 73.88, 71.94, 70.86, 74.1, 75.95, 73.69, 68.05, 75.61, 69.4, 72.18, 73.84, 73.91, 73.89, 71.29, 74.26, 73.69, 75.35, 75.19, 70.42, 74.85, 71.99, 76.81, 76.81, 69.5, 70.33, 71.86, 74.99, 73.03, 70.79, 74.47, 75.69, 74.63, 76.82, 72.92, 74.19, 73.18, 72.19, 71.81, 71.7, 75.78, 75.38, 72.1, 73.69, 75.88, 68.17, 74.86, 72.12, 77.11, 76.82, 74.91, 74.41, 71.2, 75.74, 74.93, 73.86, 70.21, 71.82, 72.82, 76.63, 66.87, 73.64, 73.1, 68.99, 70.09, 74.08, 76.14, 71.63, 70.27, 70.99, 76.01, 76.13, 70.17, 70.81, 73.0, 75.78, 74.31, 76.75, 74.43, 69.98, 74.95, 70.64, 73.08, 75.03, 75.79, 71.01, 67.12, 74.29, 73.49, 75.84, 74.1, 72.3, 70.05, 71.41, 73.51, 67.79, 68.84, 71.1, 74.7, 74.04, 76.54, 71.22, 76.94, 73.83, 77.6, 69.98, 73.17, 78.15, 74.13, 74.05, 71.29, 73.24, 69.81, 74.54, 74.15, 74.14, 71.0, 74.23, 73.03, 73.58, 75.78, 68.37, 68.21, 70.35, 74.57, 76.14, 70.22, 71.82, 76.66, 70.99, 75.83, 75.09, 76.98, 75.47, 74.65, 77.9, 73.04, 77.71, 73.74, 71.42, 72.77, 76.12, 76.5, 76.4, 77.52, 73.55, 72.73, 71.98, 74.03, 71.55, 70.31, 70.59, 72.58]；
差值(后−前) n=812：[-8.6, -9.25, -8.74, -8.25, -7.47, -7.22, -7.56, -7.82, -7.02, -8.97, -8.19, -8.33, -7.1, -7.15, -9.23, -7.66, -7.84, -8.54, -7.79, -8.65, -7.78, -7.84, -8.28, -8.75, -7.56, -9.65, -7.92, -7.61, -7.84, -6.32, -9.01, -9.6, -7.74, -8.06, -8.1, -7.09, -8.24, -8.67, -7.88, -7.38, -7.99, -7.34, -7.67, -9.43, -8.21, -8.13, -8.15, -7.32, -8.53, -7.77, -7.31, -7.54, -8.1, -8.5, -7.92, -9.57, -8.02, -7.35, -9.82, -8.68, -8.46, -7.93, -8.37, -8.37, -8.49, -7.53, -7.82, -7.78, -8.22, -6.74, -9.06, -7.47, -8.43, -8.13, -8.69, -7.69, -8.5, -8.62, -7.87, -8.07, -8.17, -7.22, -8.73, -6.98, -6.56, -7.48, -7.67, -9.16, -8.97, -7.97, -6.32, -7.72, -8.39, -7.8, -8.39, -8.58, -7.3, -7.82, -8.07, -7.97, -8.53, -7.5, -8.12, -9.02, -7.87, -8.11, -7.86, -8.34, -8.06, -8.59, -7.52, -7.81, -8.15, -7.32, -7.99, -7.26, -7.17, -6.5, -9.14, -7.97, -8.07, -8.0, -7.99, -7.28, -8.24, -7.72, -7.24, -7.71, -8.76, -9.29, -7.62, -7.96, -8.31, -8.75, -8.09, -7.66, -9.36, -7.28, -8.5, -6.98, -8.49, -8.82, -8.15, -7.14, -8.15, -8.7, -8.61, -8.81, -8.17, -8.37, -8.29, -8.66, -7.95, -7.74, -7.12, -8.05, -8.29, -9.24, -8.1, -8.35, -7.52, -7.69, -6.85, -7.87, -8.89, -6.78, -7.66, -8.86, -8.16, -9.03, -7.39, -7.31, -9.59, -8.49, -9.07, -7.21, -8.18, -7.69, -7.58, -8.29, -7.94, -8.08, -7.27, -7.72, -8.31, -7.58, -8.27, -8.42, -8.8, -7.95, -8.44, -9.14, -7.51, -7.55, -8.64, -8.93, -7.88, -9.21, -7.57, -8.32, -7.38, -8.06, -7.13, -7.38, -9.12, -7.84, -8.27, -8.5, -8.59, -8.19, -7.87, -8.54, -8.84, -7.42, -7.65, -8.98, -8.98, -7.86, -8.72, -8.32, -7.68, -9.23, -8.43, -7.61, -6.42, -7.84, -7.96, -7.85, -7.39, -7.76, -7.72, -7.14, -7.35, -7.75, -7.55, -8.12, -8.22, -7.96, -7.14, -8.87, -9.07, -8.11, -8.73, -6.67, -7.63, -7.95, -7.8, -8.86, -7.98, -8.29, -7.91, -8.38, -6.9, -8.23, -8.0, -7.74, -6.2, -7.41, -9.14, -6.39, -8.17, -8.29, -7.6, -7.48, -6.13, -8.09, -6.6, -7.68, -7.22, -8.32, -7.8, -8.44, -7.73, -7.89, -7.23, -7.75, -7.19, -7.0, -7.41, -8.12, -8.44, -8.34, -6.95, -9.23, -8.14, -7.76, -7.71, -7.77, -7.66, -7.38, -7.97, -9.22, -7.09, -7.91, -8.21, -8.63, -9.06, -8.9, -7.32, -8.09, -8.35, -6.93, -7.81, -6.34, -8.05, -8.72, -8.63, -8.15, -7.9, -9.73, -7.73, -7.16, -9.57, -8.39, -8.01, -6.95, -9.03, -7.92, -8.38, -7.74, -8.92, -8.49, -8.29, -8.36, -8.82, -7.57, -7.69, -7.86, -7.38, -8.26, -8.17, -8.58, -7.35, -9.38, -8.68, -8.13, -7.92, -8.62, -9.67, -9.14, -6.83, -6.96, -8.61, -8.7, -7.59, -9.19, -7.67, -7.66, -9.18, -8.61, -8.61, -7.64, -8.2, -9.29, -7.02, -8.35, -8.71, -7.75, -9.58, -8.12, -8.14, -8.27, -8.24, -8.77, -7.72, -7.41, -9.05, -8.92, -9.22, -8.88, -7.14, -8.34, -7.2, -8.47, -8.32, -7.6, -8.1, -8.08, -8.04, -8.54, -8.96, -8.34, -8.53, -9.16, -8.22, -8.85, -8.5, -8.22, -8.5, -7.85, -9.1, -8.69, -8.65, -7.61, -7.43, -7.39, -8.73, -7.82, -6.56, -7.36, -8.76, -8.01, -8.15, -7.54, -8.23, -9.73, -9.14, -6.39, -8.7, -9.71, -7.95, -7.76, -8.18, -8.67, -9.41, -7.37, -7.25, -7.0, -7.28, -8.63, -9.42, -9.01, -8.08, -8.79, -7.13, -8.17, -7.74, -6.94, -8.59, -8.29, -8.77, -8.19, -8.68, -7.72, -7.05, -7.99, -8.37, -8.68, -6.92, -7.31, -8.13, -9.45, -8.0, -10.83, -8.72, -8.41, -8.25, -7.81, -7.56, -8.4, -8.84, -7.5, -7.02, -8.65, -7.91, -7.94, -7.84, -8.47, -7.94, -8.46, -9.38, -8.31, -8.44, -7.92, -8.59, -8.58, -8.97, -9.9, -7.66, -7.63, -8.01, -8.75, -8.95, -7.4, -7.91, -7.89, -8.42, -7.83, -7.59, -8.29, -7.48, -7.93, -7.33, -6.57, -7.71, -9.18, -7.73, -8.25, -7.98, -8.31, -8.85, -9.06, -9.15, -8.47, -7.76, -7.93, -8.03, -8.67, -8.72, -9.43, -8.26, -7.07, -8.04, -7.6, -8.61, -7.82, -9.0, -7.21, -8.98, -7.54, -7.23, -8.83, -6.39, -7.92, -7.54, -9.76, -9.12, -8.31, -8.87, -8.74, -8.21, -8.47, -9.24, -7.88, -8.02, -7.79, -8.61, -7.75, -9.04, -9.25, -7.45, -9.48, -7.71, -8.77, -8.07, -7.11, -8.42, -7.57, -9.11, -6.87, -8.39, -7.67, -7.5, -7.47, -6.7, -7.96, -8.27, -7.96, -7.84, -8.15, -7.91, -7.16, -8.35, -8.47, -6.84, -8.52, -8.02, -7.77, -8.69, -8.97, -8.81, -8.52, -8.43, -8.51, -7.48, -7.02, -8.68, -6.7, -6.2, -8.3, -7.05, -7.68, -7.93, -8.83, -8.3, -7.78, -7.42, -7.82, -7.58, -8.8, -8.18, -7.41, -8.59, -9.15, -8.14, -8.27, -8.02, -7.48, -7.99, -8.4, -7.64, -9.67, -8.51, -7.35, -8.19, -7.93, -8.42, -8.84, -8.81, -8.84, -8.05, -9.04, -8.06, -8.52, -8.7, -7.73, -8.97, -7.35, -7.82, -7.75, -7.55, -7.4, -7.84, -7.62, -8.98, -9.8, -7.94, -8.5, -7.04, -7.55, -8.55, -6.54, -7.71, -8.85, -8.81, -7.28, -7.53, -8.59, -8.0, -10.32, -8.8, -7.88, -8.56, -7.36, -7.86, -6.59, -8.82, -8.38, -8.32, -8.6, -7.7, -8.25, -7.49, -7.32, -8.66, -8.09, -8.2, -7.63, -7.62, -7.05, -7.4, -9.5, -8.37, -8.03, -7.48, -7.9, -8.69, -7.87, -7.04, -8.65, -7.63, -8.91, -7.97, -7.52, -6.29, -9.01, -8.2, -7.78, -7.47, -6.38, -7.97, -8.43, -9.06, -8.66, -7.21, -8.0, -7.13, -7.7, -8.79, -7.78, -6.81, -8.67, -7.15, -7.58, -9.26, -9.04, -7.15, -7.85, -8.75, -8.73, -7.51, -8.19, -8.39, -6.36, -7.72, -8.96, -8.14, -6.43, -8.44, -6.17, -7.63, -7.57, -7.99, -8.67, -7.84, -7.51, -6.82, -7.78, -8.24, -6.68, -7.04, -8.05, -8.45, -7.24, -7.68, -8.94, -7.9, -8.28, -8.61, -8.16, -8.48, -7.89, -9.01, -7.59, -6.81, -6.87, -9.43, -7.73, -8.24, -7.95, -7.32, -8.12, -7.87, -7.66, -7.05, -6.36, -7.84, -9.67, -8.58, -7.33, -8.45, -8.92, -6.92, -7.66, -8.1, -7.82, -8.65, -8.37, -7.97, -7.86, -8.37, -7.37, -7.8, -7.12, -8.42, -6.85, -9.98, -8.02, -8.03, -9.14, -8.13, -8.0, -8.24, -8.0, -9.6, -7.37, -8.3, -7.7, -7.47, -8.29, -8.0, -8.51, -6.75, -6.85, -8.05, -8.26, -9.13, -8.06, -8.28, -7.51, -7.69, -8.85, -6.4, -8.99, -8.06, -8.73, -8.23, -7.74, -7.2, -7.81, -8.3, -8.96, -7.55, -8.94, -8.29, -8.65, -6.64, -9.2, -8.32, -8.1, -7.79, -7.62, -7.31, -7.45, -7.95, -8.28, -8.64, -8.15, -9.53, -9.47, -7.35, -7.59]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -8.0742，s_d = 0.7225，SE = s_d/√n = 0.0254；
t = d̄/SE = -318.4633；df = n−1 = 811；双侧 p = 0.000e+00。
d̄ = -8.0742，s_d = 0.7225，SE = s_d/√n = 0.0254；
t = d̄/SE = -318.4633；df = n−1 = 811；双侧 p = 0.000e+00。
计算结果
t = -318.4633，df = 811，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。判断运动对静息心率影响。
案例 44. 治疗前后疼痛评分
原始数据
前 n=320：[7.78, 9.49, 6.99, 5.3, 9.29, 7.75, 8.32, 7.31, 8.18, 9.06, 8.98, 4.13, 8.44, 6.27, 7.14, 7.98, 8.01, 10.14, 7.78, 8.0, 8.4, 7.5, 7.74, 7.58, 7.82, 7.58, 6.79, 6.06, 4.58, 7.39, 7.8, 8.77, 6.2, 8.54, 8.02, 6.68, 7.93, 6.25, 11.08, 8.52, 7.28, 6.88, 7.52, 7.84, 8.11, 4.99, 7.01, 7.37, 6.61, 6.97, 8.67, 7.17, 7.43, 7.64, 10.82, 8.23, 5.05, 9.81, 7.71, 8.37, 8.1, 7.22, 7.14, 5.67, 6.52, 11.0, 5.25, 8.73, 6.81, 7.41, 8.37, 5.99, 8.31, 8.16, 7.25, 6.0, 6.73, 7.09, 6.57, 8.37, 8.59, 10.03, 8.32, 7.08, 7.78, 6.96, 7.69, 6.84, 6.94, 8.93, 7.32, 7.17, 7.32, 7.87, 7.45, 9.79, 8.22, 7.85, 4.84, 6.9, 7.53, 8.23, 7.62, 7.14, 7.46, 8.0, 8.53, 8.71, 8.55, 7.04, 9.33, 10.07, 7.79, 7.78, 8.32, 5.64, 6.38, 7.26, 7.98, 7.46, 6.49, 6.45, 8.66, 6.88, 6.7, 7.51, 8.72, 7.48, 8.17, 7.76, 6.92, 9.65, 6.1, 7.85, 6.88, 6.99, 8.3, 7.8, 7.86, 6.74, 8.02, 9.14, 8.72, 8.21, 6.35, 7.28, 8.82, 8.72, 7.48, 7.57, 6.29, 5.71, 6.39, 8.42, 7.4, 7.89, 9.95, 7.51, 8.31, 8.48, 5.48, 6.56, 6.13, 8.0, 8.81, 8.4, 7.33, 5.75, 8.06, 6.3, 8.49, 8.0, 9.01, 6.56, 8.82, 7.37, 7.25, 6.37, 7.33, 8.6, 8.01, 7.27, 7.13, 6.46, 6.34, 7.53, 8.23, 10.17, 6.97, 7.69, 7.2, 7.79, 9.0, 6.92, 7.52, 7.64, 6.83, 8.22, 8.23, 8.61, 7.82, 7.65, 8.71, 8.01, 8.38, 8.17, 6.51, 6.67, 8.21, 9.14, 7.63, 10.72, 8.54, 7.24, 9.3, 7.7, 7.5, 7.18, 7.85, 6.01, 6.79, 8.21, 6.13, 7.57, 7.69, 6.64, 6.88, 7.58, 7.09, 7.52, 8.39, 6.99, 8.71, 7.93, 6.67, 8.51, 7.4, 7.11, 5.29, 7.63, 9.95, 8.41, 6.43, 8.01, 7.16, 7.37, 8.42, 7.67, 9.91, 8.01, 6.6, 7.28, 9.0, 7.17, 6.67, 9.63, 8.62, 6.89, 7.05, 6.46, 6.21, 8.22, 7.07, 7.79, 6.58, 7.01, 7.98, 7.01, 9.2, 6.21, 7.2, 7.48, 7.66, 7.92, 7.36, 8.2, 5.17, 6.71, 9.01, 9.57, 5.53, 7.18, 7.18, 7.73, 6.0, 6.79, 6.35, 6.46, 6.25, 7.15, 9.33, 6.67, 8.26, 8.44, 7.18, 8.89, 7.22, 4.75, 5.83, 6.85, 9.36, 7.95, 7.63, 6.62, 7.82, 7.25, 8.58, 6.96, 10.08, 6.07, 8.37, 5.65, 6.66, 5.37, 9.2, 9.09, 6.57, 6.24, 6.32, 6.95]；
后 n=320：[4.29, 6.63, 3.55, 3.01, 5.13, 5.11, 5.27, 4.18, 5.55, 4.71, 6.15, 0.39, 5.29, 3.03, 3.12, 4.92, 4.9, 7.39, 4.71, 4.9, 5.07, 4.77, 4.14, 4.31, 4.65, 3.52, 4.23, 3.08, 0.1, 3.89, 5.11, 5.9, 3.18, 4.35, 4.85, 3.89, 5.08, 2.71, 8.15, 5.56, 3.16, 3.69, 5.2, 5.2, 4.83, 1.09, 4.55, 4.62, 2.93, 4.01, 5.49, 3.81, 3.99, 5.07, 7.73, 5.0, 1.37, 6.85, 4.18, 4.69, 5.81, 3.43, 3.57, 2.03, 4.0, 7.6, 2.59, 5.72, 3.61, 4.45, 4.48, 2.4, 4.93, 4.89, 3.6, 3.08, 3.58, 3.08, 2.98, 4.6, 5.22, 6.69, 6.05, 4.86, 3.85, 4.29, 4.6, 3.42, 3.57, 5.47, 4.43, 4.36, 4.25, 5.02, 3.36, 6.22, 6.48, 4.95, 1.75, 4.3, 3.9, 4.67, 4.34, 3.74, 5.35, 4.46, 4.82, 5.49, 4.75, 4.32, 5.55, 7.41, 5.49, 4.48, 5.02, 2.29, 2.2, 3.87, 4.13, 5.58, 3.84, 3.78, 5.67, 3.8, 4.48, 4.87, 5.66, 4.5, 4.78, 3.4, 2.96, 7.5, 3.1, 4.89, 4.25, 4.85, 5.3, 4.86, 5.06, 3.76, 4.16, 6.09, 5.97, 5.35, 2.42, 4.58, 6.94, 5.06, 3.69, 4.6, 2.25, 2.79, 3.62, 5.52, 4.31, 4.48, 6.75, 4.27, 5.72, 5.65, 3.29, 4.8, 2.67, 4.49, 6.94, 6.27, 4.66, 1.85, 5.71, 3.49, 4.14, 5.68, 4.85, 3.83, 6.18, 3.78, 4.06, 3.48, 5.23, 5.91, 6.36, 4.49, 3.58, 4.04, 2.07, 3.76, 5.55, 7.04, 3.97, 4.37, 4.23, 4.01, 5.71, 3.76, 4.69, 3.55, 2.65, 4.93, 4.97, 6.25, 4.15, 4.55, 6.46, 4.3, 5.38, 5.02, 2.72, 3.42, 4.94, 6.09, 4.89, 7.09, 6.37, 4.09, 6.25, 4.64, 4.37, 2.67, 4.35, 3.02, 3.31, 5.53, 3.84, 5.25, 4.62, 3.93, 3.5, 5.39, 3.62, 4.65, 5.11, 3.37, 4.33, 3.94, 3.39, 4.93, 4.63, 4.78, 2.31, 4.73, 6.8, 4.33, 3.77, 4.77, 3.06, 5.1, 4.96, 4.05, 7.11, 4.81, 3.76, 3.2, 5.34, 3.76, 3.48, 5.99, 5.45, 3.21, 3.42, 3.2, 3.79, 4.89, 3.38, 5.42, 3.22, 3.39, 4.5, 4.01, 5.67, 3.53, 4.64, 4.1, 4.0, 4.89, 4.6, 5.15, 2.0, 4.15, 6.15, 6.33, 2.72, 4.35, 3.76, 4.54, 3.42, 3.24, 2.7, 3.43, 3.27, 4.47, 5.96, 3.67, 5.82, 5.74, 3.81, 6.54, 4.76, 0.91, 2.1, 3.05, 5.38, 5.28, 4.36, 2.83, 4.27, 4.52, 5.67, 4.63, 6.95, 4.15, 6.07, 2.9, 2.75, 2.75, 5.78, 5.1, 3.4, 3.17, 3.73, 4.21]；
差值(后−前) n=320：[-3.49, -2.86, -3.44, -2.29, -4.16, -2.65, -3.04, -3.13, -2.63, -4.35, -2.83, -3.74, -3.15, -3.23, -4.03, -3.06, -3.11, -2.75, -3.07, -3.1, -3.33, -2.74, -3.59, -3.27, -3.17, -4.06, -2.56, -2.97, -4.48, -3.49, -2.69, -2.87, -3.02, -4.2, -3.17, -2.79, -2.85, -3.54, -2.93, -2.96, -4.12, -3.19, -2.32, -2.64, -3.28, -3.9, -2.46, -2.75, -3.68, -2.96, -3.18, -3.36, -3.44, -2.57, -3.1, -3.23, -3.68, -2.96, -3.53, -3.68, -2.29, -3.79, -3.58, -3.64, -2.52, -3.4, -2.66, -3.0, -3.2, -2.96, -3.89, -3.58, -3.38, -3.27, -3.64, -2.92, -3.15, -4.01, -3.59, -3.77, -3.38, -3.34, -2.27, -2.22, -3.93, -2.67, -3.09, -3.43, -3.37, -3.46, -2.89, -2.81, -3.07, -2.85, -4.09, -3.57, -1.75, -2.9, -3.09, -2.6, -3.63, -3.56, -3.28, -3.41, -2.11, -3.54, -3.71, -3.22, -3.8, -2.72, -3.78, -2.66, -2.3, -3.3, -3.3, -3.34, -4.18, -3.39, -3.84, -1.87, -2.65, -2.66, -2.99, -3.08, -2.22, -2.64, -3.05, -2.98, -3.39, -4.35, -3.96, -2.15, -2.99, -2.96, -2.63, -2.15, -3.01, -2.94, -2.79, -2.98, -3.87, -3.05, -2.75, -2.86, -3.93, -2.7, -1.88, -3.66, -3.79, -2.97, -4.04, -2.91, -2.78, -2.9, -3.09, -3.41, -3.19, -3.24, -2.58, -2.83, -2.18, -1.77, -3.46, -3.5, -1.87, -2.13, -2.67, -3.91, -2.35, -2.82, -4.35, -2.33, -4.16, -2.73, -2.64, -3.59, -3.19, -2.89, -2.1, -2.69, -1.65, -2.78, -3.55, -2.42, -4.27, -3.77, -2.68, -3.13, -2.99, -3.32, -2.97, -3.78, -3.29, -3.16, -2.83, -4.09, -4.18, -3.29, -3.25, -2.36, -3.68, -3.1, -2.26, -3.71, -3.01, -3.15, -3.79, -3.25, -3.26, -3.05, -2.74, -3.63, -2.16, -3.15, -3.06, -3.06, -3.12, -4.5, -3.5, -3.0, -3.48, -2.68, -2.29, -2.33, -3.08, -2.7, -3.38, -2.19, -3.47, -2.87, -3.27, -3.63, -4.38, -4.0, -3.28, -3.58, -2.77, -2.33, -2.98, -2.9, -3.14, -4.09, -2.67, -3.24, -4.1, -2.28, -3.46, -3.61, -2.81, -3.19, -2.85, -4.08, -3.65, -3.41, -3.19, -3.64, -3.18, -3.67, -3.63, -3.26, -2.42, -3.33, -3.69, -2.37, -3.36, -3.63, -3.47, -3.0, -3.53, -2.69, -2.56, -3.38, -3.65, -3.03, -2.76, -3.05, -3.17, -2.57, -2.86, -3.24, -2.82, -2.84, -3.43, -3.18, -2.58, -3.55, -3.65, -3.03, -2.98, -2.68, -3.37, -3.01, -2.45, -2.7, -3.38, -2.35, -2.46, -3.85, -3.73, -3.81, -3.98, -2.68, -3.27, -3.79, -3.55, -2.73, -2.91, -2.33, -3.13, -1.92, -2.3, -2.75, -3.91, -2.62, -3.42, -4.0, -3.17, -3.07, -2.6, -2.74]
后 n=320：[4.29, 6.63, 3.55, 3.01, 5.13, 5.11, 5.27, 4.18, 5.55, 4.71, 6.15, 0.39, 5.29, 3.03, 3.12, 4.92, 4.9, 7.39, 4.71, 4.9, 5.07, 4.77, 4.14, 4.31, 4.65, 3.52, 4.23, 3.08, 0.1, 3.89, 5.11, 5.9, 3.18, 4.35, 4.85, 3.89, 5.08, 2.71, 8.15, 5.56, 3.16, 3.69, 5.2, 5.2, 4.83, 1.09, 4.55, 4.62, 2.93, 4.01, 5.49, 3.81, 3.99, 5.07, 7.73, 5.0, 1.37, 6.85, 4.18, 4.69, 5.81, 3.43, 3.57, 2.03, 4.0, 7.6, 2.59, 5.72, 3.61, 4.45, 4.48, 2.4, 4.93, 4.89, 3.6, 3.08, 3.58, 3.08, 2.98, 4.6, 5.22, 6.69, 6.05, 4.86, 3.85, 4.29, 4.6, 3.42, 3.57, 5.47, 4.43, 4.36, 4.25, 5.02, 3.36, 6.22, 6.48, 4.95, 1.75, 4.3, 3.9, 4.67, 4.34, 3.74, 5.35, 4.46, 4.82, 5.49, 4.75, 4.32, 5.55, 7.41, 5.49, 4.48, 5.02, 2.29, 2.2, 3.87, 4.13, 5.58, 3.84, 3.78, 5.67, 3.8, 4.48, 4.87, 5.66, 4.5, 4.78, 3.4, 2.96, 7.5, 3.1, 4.89, 4.25, 4.85, 5.3, 4.86, 5.06, 3.76, 4.16, 6.09, 5.97, 5.35, 2.42, 4.58, 6.94, 5.06, 3.69, 4.6, 2.25, 2.79, 3.62, 5.52, 4.31, 4.48, 6.75, 4.27, 5.72, 5.65, 3.29, 4.8, 2.67, 4.49, 6.94, 6.27, 4.66, 1.85, 5.71, 3.49, 4.14, 5.68, 4.85, 3.83, 6.18, 3.78, 4.06, 3.48, 5.23, 5.91, 6.36, 4.49, 3.58, 4.04, 2.07, 3.76, 5.55, 7.04, 3.97, 4.37, 4.23, 4.01, 5.71, 3.76, 4.69, 3.55, 2.65, 4.93, 4.97, 6.25, 4.15, 4.55, 6.46, 4.3, 5.38, 5.02, 2.72, 3.42, 4.94, 6.09, 4.89, 7.09, 6.37, 4.09, 6.25, 4.64, 4.37, 2.67, 4.35, 3.02, 3.31, 5.53, 3.84, 5.25, 4.62, 3.93, 3.5, 5.39, 3.62, 4.65, 5.11, 3.37, 4.33, 3.94, 3.39, 4.93, 4.63, 4.78, 2.31, 4.73, 6.8, 4.33, 3.77, 4.77, 3.06, 5.1, 4.96, 4.05, 7.11, 4.81, 3.76, 3.2, 5.34, 3.76, 3.48, 5.99, 5.45, 3.21, 3.42, 3.2, 3.79, 4.89, 3.38, 5.42, 3.22, 3.39, 4.5, 4.01, 5.67, 3.53, 4.64, 4.1, 4.0, 4.89, 4.6, 5.15, 2.0, 4.15, 6.15, 6.33, 2.72, 4.35, 3.76, 4.54, 3.42, 3.24, 2.7, 3.43, 3.27, 4.47, 5.96, 3.67, 5.82, 5.74, 3.81, 6.54, 4.76, 0.91, 2.1, 3.05, 5.38, 5.28, 4.36, 2.83, 4.27, 4.52, 5.67, 4.63, 6.95, 4.15, 6.07, 2.9, 2.75, 2.75, 5.78, 5.1, 3.4, 3.17, 3.73, 4.21]；
差值(后−前) n=320：[-3.49, -2.86, -3.44, -2.29, -4.16, -2.65, -3.04, -3.13, -2.63, -4.35, -2.83, -3.74, -3.15, -3.23, -4.03, -3.06, -3.11, -2.75, -3.07, -3.1, -3.33, -2.74, -3.59, -3.27, -3.17, -4.06, -2.56, -2.97, -4.48, -3.49, -2.69, -2.87, -3.02, -4.2, -3.17, -2.79, -2.85, -3.54, -2.93, -2.96, -4.12, -3.19, -2.32, -2.64, -3.28, -3.9, -2.46, -2.75, -3.68, -2.96, -3.18, -3.36, -3.44, -2.57, -3.1, -3.23, -3.68, -2.96, -3.53, -3.68, -2.29, -3.79, -3.58, -3.64, -2.52, -3.4, -2.66, -3.0, -3.2, -2.96, -3.89, -3.58, -3.38, -3.27, -3.64, -2.92, -3.15, -4.01, -3.59, -3.77, -3.38, -3.34, -2.27, -2.22, -3.93, -2.67, -3.09, -3.43, -3.37, -3.46, -2.89, -2.81, -3.07, -2.85, -4.09, -3.57, -1.75, -2.9, -3.09, -2.6, -3.63, -3.56, -3.28, -3.41, -2.11, -3.54, -3.71, -3.22, -3.8, -2.72, -3.78, -2.66, -2.3, -3.3, -3.3, -3.34, -4.18, -3.39, -3.84, -1.87, -2.65, -2.66, -2.99, -3.08, -2.22, -2.64, -3.05, -2.98, -3.39, -4.35, -3.96, -2.15, -2.99, -2.96, -2.63, -2.15, -3.01, -2.94, -2.79, -2.98, -3.87, -3.05, -2.75, -2.86, -3.93, -2.7, -1.88, -3.66, -3.79, -2.97, -4.04, -2.91, -2.78, -2.9, -3.09, -3.41, -3.19, -3.24, -2.58, -2.83, -2.18, -1.77, -3.46, -3.5, -1.87, -2.13, -2.67, -3.91, -2.35, -2.82, -4.35, -2.33, -4.16, -2.73, -2.64, -3.59, -3.19, -2.89, -2.1, -2.69, -1.65, -2.78, -3.55, -2.42, -4.27, -3.77, -2.68, -3.13, -2.99, -3.32, -2.97, -3.78, -3.29, -3.16, -2.83, -4.09, -4.18, -3.29, -3.25, -2.36, -3.68, -3.1, -2.26, -3.71, -3.01, -3.15, -3.79, -3.25, -3.26, -3.05, -2.74, -3.63, -2.16, -3.15, -3.06, -3.06, -3.12, -4.5, -3.5, -3.0, -3.48, -2.68, -2.29, -2.33, -3.08, -2.7, -3.38, -2.19, -3.47, -2.87, -3.27, -3.63, -4.38, -4.0, -3.28, -3.58, -2.77, -2.33, -2.98, -2.9, -3.14, -4.09, -2.67, -3.24, -4.1, -2.28, -3.46, -3.61, -2.81, -3.19, -2.85, -4.08, -3.65, -3.41, -3.19, -3.64, -3.18, -3.67, -3.63, -3.26, -2.42, -3.33, -3.69, -2.37, -3.36, -3.63, -3.47, -3.0, -3.53, -2.69, -2.56, -3.38, -3.65, -3.03, -2.76, -3.05, -3.17, -2.57, -2.86, -3.24, -2.82, -2.84, -3.43, -3.18, -2.58, -3.55, -3.65, -3.03, -2.98, -2.68, -3.37, -3.01, -2.45, -2.7, -3.38, -2.35, -2.46, -3.85, -3.73, -3.81, -3.98, -2.68, -3.27, -3.79, -3.55, -2.73, -2.91, -2.33, -3.13, -1.92, -2.3, -2.75, -3.91, -2.62, -3.42, -4.0, -3.17, -3.07, -2.6, -2.74]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -3.1384，s_d = 0.5554，SE = s_d/√n = 0.0311；
t = d̄/SE = -101.0730；df = n−1 = 319；双侧 p = 2.525e-244。
d̄ = -3.1384，s_d = 0.5554，SE = s_d/√n = 0.0311；
t = d̄/SE = -101.0730；df = n−1 = 319；双侧 p = 2.525e-244。
计算结果
t = -101.0730，df = 319，p = 2.525e-244
在 α=0.05 水平下，p=2.525e-244 < 0.05，拒绝原假设 H₀。验证镇痛治疗效果。
案例 45. 前后离职意向
原始数据
前 n=832：[4.87, 2.56, 3.77, 3.18, 3.73, 2.54, 3.52, 4.4, 3.12, 4.79, 2.93, 4.01, 3.34, 3.62, 4.51, 2.45, 4.57, 5.62, 4.42, 4.01, 3.96, 4.15, 4.21, 4.6, 3.57, 4.19, 4.17, 3.13, 4.16, 4.88, 5.01, 3.34, 3.41, 2.57, 2.59, 2.92, 5.11, 3.88, 4.24, 4.47, 4.01, 3.75, 4.38, 4.31, 2.96, 4.29, 3.66, 4.3, 4.31, 3.97, 2.36, 4.79, 4.78, 3.88, 3.32, 3.55, 4.92, 3.3, 3.84, 3.32, 2.33, 4.18, 4.44, 5.17, 1.75, 4.02, 5.36, 4.64, 2.72, 4.33, 4.1, 3.69, 3.75, 2.77, 2.9, 4.85, 4.91, 4.15, 4.43, 3.85, 4.65, 4.74, 2.87, 3.61, 3.43, 4.24, 4.65, 3.59, 3.3, 3.73, 3.85, 3.39, 4.46, 4.19, 2.46, 4.17, 4.19, 4.07, 3.88, 2.99, 5.24, 3.56, 5.0, 4.22, 3.65, 4.0, 2.59, 2.66, 3.71, 4.62, 3.64, 3.36, 4.12, 5.14, 4.2, 3.22, 4.2, 1.85, 4.53, 3.55, 4.57, 4.33, 5.16, 3.9, 4.08, 5.0, 4.21, 3.39, 2.0, 3.55, 3.67, 4.71, 4.47, 4.59, 3.85, 5.99, 4.65, 3.42, 4.13, 4.73, 4.09, 6.03, 2.85, 4.48, 3.04, 4.61, 4.48, 2.21, 3.95, 4.5, 4.33, 3.64, 4.61, 3.38, 3.93, 2.59, 4.63, 4.88, 4.51, 2.12, 5.57, 3.41, 3.76, 5.04, 2.2, 2.98, 4.5, 4.56, 3.49, 4.54, 3.17, 4.87, 4.46, 3.95, 3.18, 4.34, 2.96, 4.99, 4.0, 2.84, 4.42, 4.56, 4.61, 4.46, 4.47, 1.88, 5.32, 3.4, 4.18, 4.26, 3.87, 4.85, 3.49, 5.59, 4.82, 4.69, 3.92, 3.0, 4.13, 4.29, 4.04, 4.26, 3.14, 4.87, 3.99, 4.99, 3.57, 3.63, 2.85, 4.26, 4.55, 4.4, 4.58, 3.72, 4.26, 3.26, 4.79, 3.53, 2.64, 3.99, 3.56, 3.91, 4.05, 3.02, 4.3, 4.11, 4.54, 2.6, 5.12, 2.57, 5.4, 4.65, 4.39, 3.76, 3.33, 5.73, 4.75, 3.75, 1.47, 3.75, 5.07, 3.07, 3.93, 2.17, 4.5, 3.89, 2.83, 3.25, 3.34, 4.57, 4.17, 4.86, 4.11, 4.47, 4.5, 3.92, 4.16, 2.95, 4.87, 3.75, 4.38, 3.02, 4.7, 4.28, 4.01, 2.91, 4.98, 2.97, 3.61, 3.83, 3.45, 4.42, 4.5, 4.05, 4.72, 3.76, 4.06, 6.06, 3.95, 3.99, 4.17, 4.39, 3.79, 3.57, 4.01, 4.16, 3.85, 3.5, 3.44, 3.66, 3.39, 3.79, 4.18, 3.12, 3.26, 4.17, 3.9, 4.28, 3.17, 4.46, 4.84, 4.01, 4.43, 3.06, 3.61, 4.46, 4.06, 3.54, 5.17, 5.2, 3.33, 2.23, 4.01, 3.41, 4.47, 4.86, 4.09, 3.24, 3.27, 4.96, 5.2, 3.61, 4.31, 4.19, 4.22, 4.92, 4.56, 4.2, 3.8, 4.68, 3.65, 4.84, 3.84, 4.2, 3.45, 4.28, 4.28, 3.83, 3.52, 3.07, 2.81, 3.73, 3.52, 4.14, 4.75, 3.17, 5.09, 3.09, 3.11, 4.49, 3.74, 3.85, 5.61, 4.19, 3.74, 4.34, 4.35, 3.25, 4.1, 3.2, 3.74, 4.76, 3.62, 4.05, 3.95, 5.42, 5.67, 3.64, 3.46, 2.61, 3.93, 4.28, 3.98, 4.96, 3.31, 4.65, 4.42, 4.47, 3.16, 4.28, 3.44, 3.16, 2.79, 4.32, 3.6, 4.21, 4.02, 4.46, 4.5, 3.98, 3.46, 2.66, 4.21, 4.1, 2.43, 4.48, 3.62, 3.62, 3.66, 3.2, 5.04, 4.07, 4.86, 3.73, 5.21, 4.23, 3.14, 3.43, 3.47, 2.97, 3.98, 3.68, 4.63, 3.02, 4.35, 4.05, 3.92, 5.09, 3.47, 4.37, 3.51, 3.97, 3.91, 3.47, 4.18, 5.44, 3.6, 3.14, 3.69, 2.28, 4.65, 4.62, 2.81, 5.5, 3.88, 4.1, 3.7, 3.99, 3.44, 3.63, 4.19, 4.21, 3.71, 3.9, 4.27, 4.22, 4.7, 3.94, 3.71, 4.6, 2.8, 3.9, 3.64, 5.27, 4.43, 3.12, 5.47, 4.94, 4.24, 4.86, 4.18, 3.69, 4.23, 4.07, 3.36, 2.95, 4.18, 3.55, 3.85, 4.98, 4.1, 4.52, 2.61, 4.0, 3.75, 2.65, 5.36, 5.2, 4.51, 2.91, 3.39, 3.98, 3.99, 4.59, 2.25, 3.75, 5.24, 4.23, 3.55, 4.46, 4.29, 2.63, 4.36, 3.4, 3.53, 3.37, 3.53, 3.96, 3.12, 4.47, 2.06, 2.48, 5.57, 3.24, 3.36, 4.53, 5.09, 3.17, 4.26, 5.31, 4.56, 3.62, 3.94, 4.29, 4.11, 3.33, 3.92, 3.31, 3.54, 4.05, 3.6, 5.07, 4.22, 4.25, 3.34, 3.66, 4.32, 4.0, 2.92, 4.47, 3.52, 3.91, 4.31, 3.58, 4.27, 3.45, 4.41, 3.28, 4.99, 3.33, 4.15, 3.71, 4.45, 3.75, 4.9, 4.84, 4.02, 4.03, 3.26, 3.15, 3.84, 2.18, 4.27, 4.5, 3.77, 4.67, 4.47, 2.97, 3.85, 4.92, 4.37, 4.6, 5.85, 3.91, 4.34, 4.56, 3.59, 4.04, 4.75, 3.62, 4.41, 2.86, 3.19, 3.36, 4.9, 4.36, 4.69, 3.66, 5.71, 5.7, 3.69, 3.92, 2.33, 5.4, 3.59, 5.58, 4.03, 3.86, 5.22, 5.17, 3.56, 3.91, 4.85, 4.82, 2.66, 6.7, 4.31, 3.35, 3.73, 4.05, 3.26, 3.69, 2.59, 3.57, 3.86, 5.16, 5.06, 3.47, 3.66, 4.06, 4.41, 3.85, 4.57, 4.01, 4.22, 4.77, 3.42, 5.61, 3.82, 3.49, 4.32, 3.61, 4.08, 4.78, 5.06, 4.5, 3.75, 3.51, 4.76, 2.64, 3.76, 4.68, 4.84, 4.3, 4.48, 3.84, 3.11, 3.61, 5.21, 4.47, 4.03, 2.85, 4.69, 3.18, 4.48, 4.24, 3.67, 3.58, 2.54, 3.81, 3.6, 4.42, 3.66, 4.03, 5.06, 4.21, 5.11, 5.24, 5.93, 5.26, 4.25, 4.58, 2.5, 4.1, 4.29, 4.02, 4.93, 4.0, 4.41, 4.29, 5.03, 2.63, 4.76, 5.02, 3.76, 3.14, 3.24, 3.33, 5.75, 4.39, 4.34, 3.46, 3.96, 3.56, 3.04, 3.36, 4.28, 4.15, 4.1, 4.36, 3.69, 4.7, 4.77, 4.16, 4.31, 4.52, 5.07, 2.21, 4.3, 4.39, 4.01, 3.85, 4.25, 3.93, 4.07, 4.72, 3.62, 3.67, 3.55, 3.33, 4.71, 3.02, 3.09, 5.09, 3.95, 5.48, 4.61, 5.16, 4.08, 4.5, 4.17, 2.92, 4.47, 3.09, 2.64, 2.91, 2.4, 4.5, 3.56, 4.25, 4.15, 3.89, 3.6, 2.89, 4.27, 4.6, 5.79, 4.44, 5.14, 4.04, 4.69, 3.78, 2.86, 4.4, 3.68, 3.27, 2.75, 4.55, 3.64, 2.56, 4.75, 3.88, 4.16, 3.75, 3.83, 4.1, 3.88, 4.52, 5.42, 3.99, 5.62, 3.99, 5.45, 4.87, 3.34, 3.66, 4.79, 5.25, 3.17, 3.93, 4.91, 4.16, 5.19, 4.04, 4.82, 5.42, 3.76, 3.31, 3.64, 3.75, 3.39, 3.9, 3.33, 3.1, 4.14, 3.85, 4.98, 5.21, 2.92, 3.11, 4.09, 4.96, 4.78, 3.04, 4.64, 3.23, 3.76, 4.1, 4.56, 4.31, 4.4, 3.23, 4.52, 4.09, 3.65, 3.66, 5.78, 3.17, 3.31, 3.75, 3.65, 4.22, 3.32, 4.01, 2.43, 3.63, 4.71, 3.7, 3.77, 4.48, 4.88, 4.4, 4.14, 4.93, 4.4, 4.22, 3.05, 4.0, 4.19, 3.58, 5.02, 3.43]；
后 n=832：[3.63, 1.74, 2.52, 2.2, 2.47, 1.73, 2.66, 3.52, 1.85, 2.99, 2.04, 3.14, 1.99, 2.49, 3.62, 1.79, 3.27, 5.4, 3.63, 3.45, 2.81, 2.56, 3.43, 3.76, 2.77, 2.9, 3.45, 2.03, 2.56, 3.36, 3.88, 2.22, 2.61, 1.36, 1.66, 2.04, 3.98, 3.3, 3.18, 3.2, 2.85, 3.18, 3.34, 3.65, 2.11, 3.25, 2.6, 2.9, 3.66, 3.06, 1.82, 4.33, 3.4, 2.56, 2.51, 2.43, 3.68, 2.42, 2.78, 2.09, 1.1, 2.96, 3.8, 3.75, 0.62, 2.66, 4.05, 3.7, 1.19, 3.42, 2.13, 2.96, 2.54, 2.2, 1.56, 3.49, 3.68, 2.7, 3.38, 2.6, 3.31, 3.7, 2.09, 2.29, 2.49, 3.23, 3.54, 2.77, 1.58, 2.32, 2.79, 2.82, 3.25, 2.94, 1.38, 2.81, 2.68, 2.92, 2.2, 1.97, 4.67, 2.58, 3.79, 3.41, 2.3, 3.24, 1.89, 1.72, 2.8, 3.45, 2.42, 2.25, 3.18, 4.63, 2.96, 1.94, 3.95, 1.49, 3.47, 2.11, 3.97, 3.86, 4.42, 2.57, 3.13, 4.51, 3.45, 2.33, 0.86, 2.58, 2.9, 3.42, 4.18, 3.26, 3.12, 5.23, 3.18, 2.62, 2.68, 3.4, 3.55, 5.7, 1.51, 3.26, 2.52, 3.23, 2.75, 0.91, 2.91, 3.21, 3.37, 2.23, 4.31, 1.8, 3.02, 1.14, 3.1, 4.14, 3.38, 0.55, 4.29, 1.97, 2.9, 4.35, 1.21, 1.79, 3.36, 3.28, 2.12, 3.37, 2.06, 3.73, 3.65, 3.4, 2.21, 2.93, 2.3, 3.74, 3.15, 1.85, 3.51, 3.21, 3.79, 3.31, 3.71, 0.56, 4.15, 2.6, 3.27, 3.24, 2.75, 3.47, 2.95, 4.69, 3.21, 4.23, 2.46, 1.2, 2.88, 3.19, 3.06, 2.75, 2.35, 4.07, 2.71, 3.98, 2.41, 2.22, 1.81, 3.39, 3.69, 2.84, 3.51, 2.77, 3.35, 2.19, 2.91, 2.29, 1.55, 2.62, 2.26, 2.84, 3.58, 2.23, 3.22, 3.17, 3.14, 1.59, 4.14, 1.86, 4.05, 3.35, 2.85, 2.01, 2.44, 4.69, 4.02, 2.01, 0.78, 2.65, 4.43, 1.78, 2.56, 1.07, 3.74, 2.73, 1.58, 2.29, 2.39, 3.54, 3.05, 3.76, 2.66, 3.25, 2.72, 2.89, 2.64, 2.2, 3.39, 2.82, 3.32, 2.42, 3.73, 2.98, 3.28, 1.66, 3.91, 2.18, 2.67, 2.66, 2.66, 3.4, 3.61, 2.99, 3.89, 2.1, 3.22, 4.51, 2.63, 2.79, 2.43, 3.06, 3.04, 2.99, 2.8, 2.72, 2.68, 2.19, 1.69, 2.64, 2.04, 3.15, 2.96, 2.07, 2.13, 2.52, 2.54, 3.53, 1.92, 3.3, 4.11, 3.12, 3.89, 1.26, 2.86, 3.36, 2.9, 2.29, 3.43, 3.98, 2.69, 1.51, 2.75, 1.91, 3.03, 3.87, 2.59, 2.33, 2.26, 3.95, 3.74, 2.21, 3.41, 3.38, 2.91, 4.24, 3.83, 3.68, 2.27, 3.17, 2.52, 3.68, 2.57, 3.13, 2.29, 3.1, 3.46, 2.8, 2.84, 1.71, 1.73, 2.79, 2.65, 2.87, 3.66, 1.97, 3.92, 1.61, 1.81, 3.65, 2.35, 2.42, 4.35, 3.51, 2.16, 3.38, 3.37, 1.91, 3.14, 2.19, 2.47, 4.17, 2.3, 2.83, 2.73, 4.32, 4.59, 2.81, 1.9, 1.71, 2.79, 2.81, 3.08, 3.68, 2.46, 3.31, 3.83, 3.72, 1.72, 2.99, 2.37, 1.86, 2.02, 3.05, 2.7, 3.41, 2.83, 3.41, 3.26, 2.41, 2.17, 1.84, 3.34, 2.59, 1.61, 3.22, 2.28, 2.96, 2.73, 2.5, 4.5, 3.16, 3.52, 2.69, 4.13, 2.97, 2.13, 2.75, 3.03, 2.4, 3.31, 2.59, 3.69, 2.45, 2.75, 3.06, 3.48, 3.65, 2.04, 3.33, 2.58, 3.31, 3.47, 1.54, 3.26, 4.27, 2.68, 1.87, 2.47, 1.16, 3.63, 3.68, 2.22, 4.88, 2.67, 2.86, 2.36, 2.74, 2.31, 2.74, 2.76, 3.67, 2.36, 3.2, 3.17, 3.2, 3.28, 2.93, 2.99, 3.86, 0.7, 2.56, 2.25, 4.16, 3.03, 3.05, 4.03, 3.71, 3.16, 3.43, 2.97, 2.06, 3.2, 2.72, 2.12, 1.9, 3.36, 1.89, 3.31, 3.88, 2.91, 2.97, 0.8, 2.79, 3.03, 1.63, 4.24, 4.71, 3.8, 1.48, 2.48, 2.74, 3.28, 4.01, 1.44, 2.84, 4.51, 3.76, 2.35, 3.14, 3.14, 1.53, 3.29, 2.39, 2.14, 2.36, 1.78, 3.34, 2.21, 3.01, 0.78, 1.62, 4.53, 1.83, 2.41, 3.7, 3.36, 2.32, 3.12, 4.25, 3.42, 2.42, 2.49, 3.47, 3.15, 2.16, 2.92, 1.86, 2.36, 2.97, 2.92, 3.43, 2.72, 2.63, 2.58, 2.26, 2.31, 3.34, 2.28, 4.09, 2.55, 2.61, 2.96, 2.44, 3.12, 2.82, 3.4, 2.44, 3.92, 2.21, 3.08, 2.53, 3.27, 2.31, 3.87, 3.9, 2.66, 2.53, 1.64, 2.51, 2.78, 1.31, 3.58, 3.64, 2.62, 3.54, 2.85, 1.87, 3.0, 3.69, 3.45, 2.99, 5.15, 2.69, 2.96, 3.97, 2.03, 2.89, 3.47, 3.14, 3.23, 2.03, 2.12, 2.69, 3.73, 3.66, 4.63, 2.56, 3.7, 4.4, 2.26, 2.77, 0.87, 3.9, 3.05, 4.46, 3.02, 2.86, 3.68, 4.65, 2.45, 2.69, 3.57, 3.69, 1.82, 5.57, 3.49, 2.35, 2.9, 2.48, 2.25, 2.95, 1.42, 2.34, 2.46, 4.54, 3.9, 1.93, 2.33, 2.87, 3.12, 2.69, 2.94, 3.27, 3.11, 3.45, 2.32, 4.61, 2.77, 3.12, 3.51, 2.64, 3.35, 4.0, 3.88, 3.07, 2.94, 1.96, 4.08, 1.09, 2.67, 3.92, 3.81, 2.86, 3.57, 2.43, 2.21, 2.11, 3.96, 3.15, 3.12, 1.56, 3.3, 2.36, 3.3, 2.95, 2.98, 2.1, 1.9, 2.65, 2.16, 3.49, 1.92, 3.01, 4.08, 3.61, 3.86, 3.88, 5.02, 4.91, 3.05, 3.12, 1.73, 2.99, 2.9, 2.91, 3.95, 3.0, 2.87, 3.27, 4.13, 1.56, 3.84, 3.92, 2.57, 1.83, 2.27, 1.55, 4.58, 3.33, 3.16, 2.65, 2.73, 2.07, 1.59, 1.84, 3.06, 3.47, 2.84, 3.48, 2.97, 3.57, 3.55, 2.88, 3.63, 3.47, 4.04, 0.97, 3.17, 3.28, 3.4, 2.6, 3.13, 2.9, 2.83, 3.81, 2.13, 2.18, 2.63, 2.36, 3.54, 1.41, 1.77, 4.3, 2.68, 5.03, 3.66, 3.97, 2.37, 2.88, 2.49, 1.54, 3.24, 2.05, 1.88, 1.83, 1.3, 3.01, 2.45, 2.78, 2.63, 2.56, 2.37, 1.84, 3.26, 3.42, 4.63, 3.56, 3.97, 2.97, 3.01, 2.82, 1.91, 3.56, 3.24, 1.91, 1.54, 3.54, 2.07, 1.52, 3.87, 2.39, 3.63, 2.24, 2.4, 2.72, 2.6, 3.85, 4.3, 2.68, 4.1, 2.68, 4.18, 4.0, 2.05, 2.65, 4.53, 4.28, 1.97, 2.39, 3.92, 2.92, 4.66, 2.57, 3.85, 3.71, 2.59, 1.99, 2.07, 2.86, 1.75, 3.14, 2.52, 2.05, 3.01, 2.9, 4.02, 4.26, 1.96, 2.4, 3.37, 3.75, 3.16, 1.89, 3.48, 1.62, 2.69, 3.39, 3.66, 3.15, 3.1, 1.95, 3.39, 3.19, 2.81, 2.54, 4.8, 2.22, 2.34, 3.26, 2.64, 3.5, 2.35, 2.55, 0.86, 2.61, 3.49, 2.14, 2.36, 3.92, 4.13, 3.35, 2.9, 3.65, 3.4, 2.43, 2.34, 2.93, 3.06, 2.71, 3.72, 2.72]；
差值(后−前) n=832：[-1.24, -0.82, -1.26, -0.98, -1.27, -0.81, -0.86, -0.88, -1.27, -1.8, -0.89, -0.87, -1.35, -1.14, -0.89, -0.66, -1.3, -0.22, -0.79, -0.56, -1.15, -1.6, -0.78, -0.84, -0.8, -1.29, -0.72, -1.1, -1.61, -1.52, -1.12, -1.12, -0.8, -1.2, -0.93, -0.87, -1.14, -0.58, -1.05, -1.27, -1.16, -0.57, -1.03, -0.66, -0.85, -1.04, -1.05, -1.41, -0.65, -0.91, -0.53, -0.46, -1.38, -1.32, -0.81, -1.12, -1.24, -0.88, -1.06, -1.23, -1.23, -1.22, -0.64, -1.43, -1.13, -1.36, -1.31, -0.94, -1.53, -0.91, -1.97, -0.72, -1.21, -0.57, -1.34, -1.36, -1.23, -1.45, -1.05, -1.25, -1.33, -1.05, -0.78, -1.32, -0.95, -1.02, -1.1, -0.82, -1.73, -1.41, -1.06, -0.58, -1.2, -1.26, -1.07, -1.35, -1.52, -1.14, -1.68, -1.02, -0.57, -0.97, -1.21, -0.81, -1.35, -0.77, -0.69, -0.94, -0.91, -1.17, -1.22, -1.11, -0.94, -0.51, -1.24, -1.27, -0.25, -0.37, -1.06, -1.44, -0.6, -0.47, -0.75, -1.34, -0.95, -0.49, -0.76, -1.07, -1.13, -0.97, -0.77, -1.29, -0.29, -1.33, -0.73, -0.76, -1.47, -0.8, -1.45, -1.33, -0.54, -0.34, -1.34, -1.21, -0.51, -1.38, -1.73, -1.3, -1.04, -1.28, -0.96, -1.41, -0.3, -1.58, -0.91, -1.45, -1.53, -0.74, -1.12, -1.57, -1.28, -1.45, -0.86, -0.69, -0.99, -1.19, -1.14, -1.28, -1.38, -1.17, -1.11, -1.13, -0.81, -0.55, -0.98, -1.41, -0.66, -1.25, -0.85, -0.99, -0.9, -1.35, -0.82, -1.15, -0.76, -1.31, -1.16, -0.8, -0.91, -1.02, -1.12, -1.38, -0.55, -0.9, -1.61, -0.45, -1.46, -1.79, -1.24, -1.1, -0.98, -1.51, -0.79, -0.8, -1.28, -1.0, -1.16, -1.41, -1.04, -0.87, -0.86, -1.56, -1.07, -0.96, -0.91, -1.07, -1.87, -1.24, -1.08, -1.37, -1.3, -1.07, -0.47, -0.79, -1.08, -0.94, -1.4, -1.0, -0.98, -0.71, -1.34, -1.31, -1.54, -1.75, -0.89, -1.04, -0.73, -1.75, -0.68, -1.1, -0.64, -1.29, -1.37, -1.1, -0.76, -1.16, -1.25, -0.95, -0.95, -1.02, -1.12, -1.11, -1.45, -1.22, -1.78, -1.04, -1.52, -0.75, -1.48, -0.93, -1.06, -0.6, -0.97, -1.29, -0.73, -1.25, -1.07, -0.8, -0.94, -1.17, -0.78, -1.02, -0.89, -1.06, -0.83, -1.66, -0.84, -1.55, -1.32, -1.2, -1.74, -1.32, -0.75, -0.58, -1.22, -1.45, -1.17, -1.3, -1.75, -1.03, -1.35, -0.64, -1.22, -1.05, -1.13, -1.65, -1.36, -0.75, -1.25, -1.16, -0.72, -0.89, -0.54, -1.8, -0.76, -1.09, -1.17, -1.25, -1.75, -1.22, -0.64, -0.73, -1.26, -1.5, -1.44, -0.98, -1.5, -0.92, -1.02, -1.01, -1.46, -1.4, -0.9, -0.81, -1.31, -0.68, -0.73, -0.52, -1.53, -1.5, -1.13, -1.16, -1.27, -1.07, -1.16, -1.18, -0.82, -1.03, -0.68, -1.36, -1.08, -0.94, -0.87, -1.27, -1.09, -1.2, -1.17, -1.48, -1.3, -0.85, -1.39, -1.43, -1.26, -0.68, -1.57, -0.96, -0.98, -1.34, -0.96, -1.02, -1.27, -0.59, -1.32, -1.22, -1.22, -1.1, -1.08, -0.82, -1.56, -0.9, -1.13, -1.47, -0.91, -1.28, -0.86, -1.34, -0.6, -0.75, -1.44, -1.29, -1.07, -1.31, -0.77, -1.27, -0.9, -0.8, -1.18, -1.05, -1.24, -1.57, -1.29, -0.82, -0.87, -1.51, -0.81, -1.26, -1.34, -0.65, -0.93, -0.7, -0.54, -0.91, -1.34, -1.04, -1.08, -1.26, -1.01, -0.67, -0.44, -0.57, -0.67, -1.09, -0.94, -0.58, -1.61, -0.99, -0.44, -1.44, -1.43, -1.05, -0.94, -0.65, -0.44, -1.92, -0.92, -1.17, -0.92, -1.27, -1.23, -1.12, -1.02, -0.94, -0.59, -0.62, -1.21, -1.24, -1.35, -1.25, -1.14, -0.89, -1.43, -0.54, -1.35, -0.7, -1.1, -1.02, -1.42, -1.02, -0.72, -0.74, -2.09, -1.34, -1.39, -1.11, -1.39, -0.06, -1.43, -1.23, -1.08, -1.42, -1.21, -1.62, -1.03, -1.35, -1.24, -1.05, -0.82, -1.66, -0.53, -1.1, -1.19, -1.55, -1.81, -1.21, -0.72, -1.02, -1.13, -0.5, -0.71, -1.43, -0.92, -1.24, -0.71, -0.58, -0.82, -0.91, -0.73, -0.46, -1.2, -1.32, -1.15, -1.09, -1.07, -1.01, -1.39, -1.01, -1.75, -0.62, -0.91, -1.45, -1.28, -0.85, -1.04, -1.41, -0.95, -0.83, -1.73, -0.85, -1.14, -1.06, -1.14, -1.2, -1.45, -0.82, -0.95, -1.17, -1.0, -1.45, -1.18, -1.08, -0.68, -1.64, -1.5, -1.62, -0.76, -1.4, -2.01, -0.65, -0.64, -0.38, -0.97, -1.3, -1.34, -1.14, -1.15, -0.63, -1.01, -0.84, -1.07, -1.11, -1.07, -1.18, -1.19, -1.44, -1.03, -0.94, -1.36, -1.5, -1.62, -0.64, -1.06, -0.87, -0.69, -0.86, -1.15, -1.13, -1.62, -1.1, -0.86, -1.22, -0.92, -1.62, -0.69, -1.21, -1.38, -0.59, -1.57, -1.15, -1.28, -0.48, -1.19, -0.83, -1.08, -0.67, -1.17, -0.7, -0.06, -1.1, -2.01, -1.3, -1.43, -1.15, -1.46, -1.51, -0.55, -1.12, -1.02, -0.99, -1.54, -0.51, -1.11, -1.22, -1.28, -1.13, -0.84, -1.12, -0.82, -1.0, -0.83, -1.57, -1.02, -0.74, -1.16, -1.23, -1.4, -0.62, -1.16, -1.54, -1.32, -1.19, -1.29, -1.15, -1.63, -0.74, -1.11, -1.32, -1.09, -1.01, -1.05, -0.38, -0.81, -0.97, -0.73, -0.77, -1.18, -1.43, -0.81, -1.56, -0.68, -1.56, -1.09, -0.76, -1.04, -1.44, -0.91, -1.41, -0.91, -1.5, -1.25, -1.32, -0.9, -1.29, -1.4, -0.82, -1.18, -1.29, -0.7, -1.48, -0.64, -1.16, -1.43, -0.93, -1.74, -1.03, -0.98, -0.6, -1.25, -1.36, -0.91, -0.36, -1.2, -1.46, -0.77, -1.11, -1.39, -1.11, -0.98, -1.01, -1.54, -1.02, -0.9, -1.06, -0.92, -1.1, -1.19, -1.31, -0.97, -1.78, -1.17, -1.06, -1.18, -0.81, -1.23, -1.49, -1.44, -1.52, -1.22, -0.68, -1.26, -0.88, -0.72, -1.13, -1.22, -1.29, -0.68, -1.05, -1.03, -1.25, -1.12, -1.11, -0.61, -1.25, -1.12, -1.03, -1.24, -0.91, -1.5, -1.49, -0.92, -0.97, -1.17, -1.6, -1.32, -0.8, -1.27, -0.45, -0.95, -1.18, -1.71, -1.62, -1.68, -1.38, -1.23, -1.05, -0.77, -1.08, -1.1, -1.49, -1.11, -1.46, -1.52, -1.33, -1.23, -1.05, -1.01, -1.18, -1.16, -0.88, -1.18, -1.07, -1.68, -0.96, -0.95, -0.83, -0.44, -1.36, -1.21, -1.01, -1.57, -1.04, -0.88, -1.49, -0.53, -1.51, -1.43, -1.39, -1.28, -0.67, -1.12, -1.31, -1.52, -1.31, -1.26, -0.87, -1.29, -1.01, -0.26, -0.96, -1.2, -1.54, -1.0, -1.24, -0.53, -1.47, -0.97, -1.71, -1.18, -1.33, -1.57, -0.89, -1.64, -0.76, -0.81, -1.05, -1.13, -0.95, -0.96, -0.95, -0.96, -0.7, -0.72, -1.21, -1.62, -1.15, -1.16, -1.6, -1.07, -0.71, -0.91, -1.16, -1.3, -1.28, -1.13, -0.9, -0.84, -1.12, -0.98, -0.95, -0.97, -0.49, -1.01, -0.71, -0.97, -1.46, -1.57, -1.02, -1.22, -1.56, -1.41, -0.56, -0.76, -1.05, -1.24, -1.28, -0.99, -1.79, -0.71, -1.08, -1.12, -0.87, -1.3, -0.71]
后 n=832：[3.63, 1.74, 2.52, 2.2, 2.47, 1.73, 2.66, 3.52, 1.85, 2.99, 2.04, 3.14, 1.99, 2.49, 3.62, 1.79, 3.27, 5.4, 3.63, 3.45, 2.81, 2.56, 3.43, 3.76, 2.77, 2.9, 3.45, 2.03, 2.56, 3.36, 3.88, 2.22, 2.61, 1.36, 1.66, 2.04, 3.98, 3.3, 3.18, 3.2, 2.85, 3.18, 3.34, 3.65, 2.11, 3.25, 2.6, 2.9, 3.66, 3.06, 1.82, 4.33, 3.4, 2.56, 2.51, 2.43, 3.68, 2.42, 2.78, 2.09, 1.1, 2.96, 3.8, 3.75, 0.62, 2.66, 4.05, 3.7, 1.19, 3.42, 2.13, 2.96, 2.54, 2.2, 1.56, 3.49, 3.68, 2.7, 3.38, 2.6, 3.31, 3.7, 2.09, 2.29, 2.49, 3.23, 3.54, 2.77, 1.58, 2.32, 2.79, 2.82, 3.25, 2.94, 1.38, 2.81, 2.68, 2.92, 2.2, 1.97, 4.67, 2.58, 3.79, 3.41, 2.3, 3.24, 1.89, 1.72, 2.8, 3.45, 2.42, 2.25, 3.18, 4.63, 2.96, 1.94, 3.95, 1.49, 3.47, 2.11, 3.97, 3.86, 4.42, 2.57, 3.13, 4.51, 3.45, 2.33, 0.86, 2.58, 2.9, 3.42, 4.18, 3.26, 3.12, 5.23, 3.18, 2.62, 2.68, 3.4, 3.55, 5.7, 1.51, 3.26, 2.52, 3.23, 2.75, 0.91, 2.91, 3.21, 3.37, 2.23, 4.31, 1.8, 3.02, 1.14, 3.1, 4.14, 3.38, 0.55, 4.29, 1.97, 2.9, 4.35, 1.21, 1.79, 3.36, 3.28, 2.12, 3.37, 2.06, 3.73, 3.65, 3.4, 2.21, 2.93, 2.3, 3.74, 3.15, 1.85, 3.51, 3.21, 3.79, 3.31, 3.71, 0.56, 4.15, 2.6, 3.27, 3.24, 2.75, 3.47, 2.95, 4.69, 3.21, 4.23, 2.46, 1.2, 2.88, 3.19, 3.06, 2.75, 2.35, 4.07, 2.71, 3.98, 2.41, 2.22, 1.81, 3.39, 3.69, 2.84, 3.51, 2.77, 3.35, 2.19, 2.91, 2.29, 1.55, 2.62, 2.26, 2.84, 3.58, 2.23, 3.22, 3.17, 3.14, 1.59, 4.14, 1.86, 4.05, 3.35, 2.85, 2.01, 2.44, 4.69, 4.02, 2.01, 0.78, 2.65, 4.43, 1.78, 2.56, 1.07, 3.74, 2.73, 1.58, 2.29, 2.39, 3.54, 3.05, 3.76, 2.66, 3.25, 2.72, 2.89, 2.64, 2.2, 3.39, 2.82, 3.32, 2.42, 3.73, 2.98, 3.28, 1.66, 3.91, 2.18, 2.67, 2.66, 2.66, 3.4, 3.61, 2.99, 3.89, 2.1, 3.22, 4.51, 2.63, 2.79, 2.43, 3.06, 3.04, 2.99, 2.8, 2.72, 2.68, 2.19, 1.69, 2.64, 2.04, 3.15, 2.96, 2.07, 2.13, 2.52, 2.54, 3.53, 1.92, 3.3, 4.11, 3.12, 3.89, 1.26, 2.86, 3.36, 2.9, 2.29, 3.43, 3.98, 2.69, 1.51, 2.75, 1.91, 3.03, 3.87, 2.59, 2.33, 2.26, 3.95, 3.74, 2.21, 3.41, 3.38, 2.91, 4.24, 3.83, 3.68, 2.27, 3.17, 2.52, 3.68, 2.57, 3.13, 2.29, 3.1, 3.46, 2.8, 2.84, 1.71, 1.73, 2.79, 2.65, 2.87, 3.66, 1.97, 3.92, 1.61, 1.81, 3.65, 2.35, 2.42, 4.35, 3.51, 2.16, 3.38, 3.37, 1.91, 3.14, 2.19, 2.47, 4.17, 2.3, 2.83, 2.73, 4.32, 4.59, 2.81, 1.9, 1.71, 2.79, 2.81, 3.08, 3.68, 2.46, 3.31, 3.83, 3.72, 1.72, 2.99, 2.37, 1.86, 2.02, 3.05, 2.7, 3.41, 2.83, 3.41, 3.26, 2.41, 2.17, 1.84, 3.34, 2.59, 1.61, 3.22, 2.28, 2.96, 2.73, 2.5, 4.5, 3.16, 3.52, 2.69, 4.13, 2.97, 2.13, 2.75, 3.03, 2.4, 3.31, 2.59, 3.69, 2.45, 2.75, 3.06, 3.48, 3.65, 2.04, 3.33, 2.58, 3.31, 3.47, 1.54, 3.26, 4.27, 2.68, 1.87, 2.47, 1.16, 3.63, 3.68, 2.22, 4.88, 2.67, 2.86, 2.36, 2.74, 2.31, 2.74, 2.76, 3.67, 2.36, 3.2, 3.17, 3.2, 3.28, 2.93, 2.99, 3.86, 0.7, 2.56, 2.25, 4.16, 3.03, 3.05, 4.03, 3.71, 3.16, 3.43, 2.97, 2.06, 3.2, 2.72, 2.12, 1.9, 3.36, 1.89, 3.31, 3.88, 2.91, 2.97, 0.8, 2.79, 3.03, 1.63, 4.24, 4.71, 3.8, 1.48, 2.48, 2.74, 3.28, 4.01, 1.44, 2.84, 4.51, 3.76, 2.35, 3.14, 3.14, 1.53, 3.29, 2.39, 2.14, 2.36, 1.78, 3.34, 2.21, 3.01, 0.78, 1.62, 4.53, 1.83, 2.41, 3.7, 3.36, 2.32, 3.12, 4.25, 3.42, 2.42, 2.49, 3.47, 3.15, 2.16, 2.92, 1.86, 2.36, 2.97, 2.92, 3.43, 2.72, 2.63, 2.58, 2.26, 2.31, 3.34, 2.28, 4.09, 2.55, 2.61, 2.96, 2.44, 3.12, 2.82, 3.4, 2.44, 3.92, 2.21, 3.08, 2.53, 3.27, 2.31, 3.87, 3.9, 2.66, 2.53, 1.64, 2.51, 2.78, 1.31, 3.58, 3.64, 2.62, 3.54, 2.85, 1.87, 3.0, 3.69, 3.45, 2.99, 5.15, 2.69, 2.96, 3.97, 2.03, 2.89, 3.47, 3.14, 3.23, 2.03, 2.12, 2.69, 3.73, 3.66, 4.63, 2.56, 3.7, 4.4, 2.26, 2.77, 0.87, 3.9, 3.05, 4.46, 3.02, 2.86, 3.68, 4.65, 2.45, 2.69, 3.57, 3.69, 1.82, 5.57, 3.49, 2.35, 2.9, 2.48, 2.25, 2.95, 1.42, 2.34, 2.46, 4.54, 3.9, 1.93, 2.33, 2.87, 3.12, 2.69, 2.94, 3.27, 3.11, 3.45, 2.32, 4.61, 2.77, 3.12, 3.51, 2.64, 3.35, 4.0, 3.88, 3.07, 2.94, 1.96, 4.08, 1.09, 2.67, 3.92, 3.81, 2.86, 3.57, 2.43, 2.21, 2.11, 3.96, 3.15, 3.12, 1.56, 3.3, 2.36, 3.3, 2.95, 2.98, 2.1, 1.9, 2.65, 2.16, 3.49, 1.92, 3.01, 4.08, 3.61, 3.86, 3.88, 5.02, 4.91, 3.05, 3.12, 1.73, 2.99, 2.9, 2.91, 3.95, 3.0, 2.87, 3.27, 4.13, 1.56, 3.84, 3.92, 2.57, 1.83, 2.27, 1.55, 4.58, 3.33, 3.16, 2.65, 2.73, 2.07, 1.59, 1.84, 3.06, 3.47, 2.84, 3.48, 2.97, 3.57, 3.55, 2.88, 3.63, 3.47, 4.04, 0.97, 3.17, 3.28, 3.4, 2.6, 3.13, 2.9, 2.83, 3.81, 2.13, 2.18, 2.63, 2.36, 3.54, 1.41, 1.77, 4.3, 2.68, 5.03, 3.66, 3.97, 2.37, 2.88, 2.49, 1.54, 3.24, 2.05, 1.88, 1.83, 1.3, 3.01, 2.45, 2.78, 2.63, 2.56, 2.37, 1.84, 3.26, 3.42, 4.63, 3.56, 3.97, 2.97, 3.01, 2.82, 1.91, 3.56, 3.24, 1.91, 1.54, 3.54, 2.07, 1.52, 3.87, 2.39, 3.63, 2.24, 2.4, 2.72, 2.6, 3.85, 4.3, 2.68, 4.1, 2.68, 4.18, 4.0, 2.05, 2.65, 4.53, 4.28, 1.97, 2.39, 3.92, 2.92, 4.66, 2.57, 3.85, 3.71, 2.59, 1.99, 2.07, 2.86, 1.75, 3.14, 2.52, 2.05, 3.01, 2.9, 4.02, 4.26, 1.96, 2.4, 3.37, 3.75, 3.16, 1.89, 3.48, 1.62, 2.69, 3.39, 3.66, 3.15, 3.1, 1.95, 3.39, 3.19, 2.81, 2.54, 4.8, 2.22, 2.34, 3.26, 2.64, 3.5, 2.35, 2.55, 0.86, 2.61, 3.49, 2.14, 2.36, 3.92, 4.13, 3.35, 2.9, 3.65, 3.4, 2.43, 2.34, 2.93, 3.06, 2.71, 3.72, 2.72]；
差值(后−前) n=832：[-1.24, -0.82, -1.26, -0.98, -1.27, -0.81, -0.86, -0.88, -1.27, -1.8, -0.89, -0.87, -1.35, -1.14, -0.89, -0.66, -1.3, -0.22, -0.79, -0.56, -1.15, -1.6, -0.78, -0.84, -0.8, -1.29, -0.72, -1.1, -1.61, -1.52, -1.12, -1.12, -0.8, -1.2, -0.93, -0.87, -1.14, -0.58, -1.05, -1.27, -1.16, -0.57, -1.03, -0.66, -0.85, -1.04, -1.05, -1.41, -0.65, -0.91, -0.53, -0.46, -1.38, -1.32, -0.81, -1.12, -1.24, -0.88, -1.06, -1.23, -1.23, -1.22, -0.64, -1.43, -1.13, -1.36, -1.31, -0.94, -1.53, -0.91, -1.97, -0.72, -1.21, -0.57, -1.34, -1.36, -1.23, -1.45, -1.05, -1.25, -1.33, -1.05, -0.78, -1.32, -0.95, -1.02, -1.1, -0.82, -1.73, -1.41, -1.06, -0.58, -1.2, -1.26, -1.07, -1.35, -1.52, -1.14, -1.68, -1.02, -0.57, -0.97, -1.21, -0.81, -1.35, -0.77, -0.69, -0.94, -0.91, -1.17, -1.22, -1.11, -0.94, -0.51, -1.24, -1.27, -0.25, -0.37, -1.06, -1.44, -0.6, -0.47, -0.75, -1.34, -0.95, -0.49, -0.76, -1.07, -1.13, -0.97, -0.77, -1.29, -0.29, -1.33, -0.73, -0.76, -1.47, -0.8, -1.45, -1.33, -0.54, -0.34, -1.34, -1.21, -0.51, -1.38, -1.73, -1.3, -1.04, -1.28, -0.96, -1.41, -0.3, -1.58, -0.91, -1.45, -1.53, -0.74, -1.12, -1.57, -1.28, -1.45, -0.86, -0.69, -0.99, -1.19, -1.14, -1.28, -1.38, -1.17, -1.11, -1.13, -0.81, -0.55, -0.98, -1.41, -0.66, -1.25, -0.85, -0.99, -0.9, -1.35, -0.82, -1.15, -0.76, -1.31, -1.16, -0.8, -0.91, -1.02, -1.12, -1.38, -0.55, -0.9, -1.61, -0.45, -1.46, -1.79, -1.24, -1.1, -0.98, -1.51, -0.79, -0.8, -1.28, -1.0, -1.16, -1.41, -1.04, -0.87, -0.86, -1.56, -1.07, -0.96, -0.91, -1.07, -1.87, -1.24, -1.08, -1.37, -1.3, -1.07, -0.47, -0.79, -1.08, -0.94, -1.4, -1.0, -0.98, -0.71, -1.34, -1.31, -1.54, -1.75, -0.89, -1.04, -0.73, -1.75, -0.68, -1.1, -0.64, -1.29, -1.37, -1.1, -0.76, -1.16, -1.25, -0.95, -0.95, -1.02, -1.12, -1.11, -1.45, -1.22, -1.78, -1.04, -1.52, -0.75, -1.48, -0.93, -1.06, -0.6, -0.97, -1.29, -0.73, -1.25, -1.07, -0.8, -0.94, -1.17, -0.78, -1.02, -0.89, -1.06, -0.83, -1.66, -0.84, -1.55, -1.32, -1.2, -1.74, -1.32, -0.75, -0.58, -1.22, -1.45, -1.17, -1.3, -1.75, -1.03, -1.35, -0.64, -1.22, -1.05, -1.13, -1.65, -1.36, -0.75, -1.25, -1.16, -0.72, -0.89, -0.54, -1.8, -0.76, -1.09, -1.17, -1.25, -1.75, -1.22, -0.64, -0.73, -1.26, -1.5, -1.44, -0.98, -1.5, -0.92, -1.02, -1.01, -1.46, -1.4, -0.9, -0.81, -1.31, -0.68, -0.73, -0.52, -1.53, -1.5, -1.13, -1.16, -1.27, -1.07, -1.16, -1.18, -0.82, -1.03, -0.68, -1.36, -1.08, -0.94, -0.87, -1.27, -1.09, -1.2, -1.17, -1.48, -1.3, -0.85, -1.39, -1.43, -1.26, -0.68, -1.57, -0.96, -0.98, -1.34, -0.96, -1.02, -1.27, -0.59, -1.32, -1.22, -1.22, -1.1, -1.08, -0.82, -1.56, -0.9, -1.13, -1.47, -0.91, -1.28, -0.86, -1.34, -0.6, -0.75, -1.44, -1.29, -1.07, -1.31, -0.77, -1.27, -0.9, -0.8, -1.18, -1.05, -1.24, -1.57, -1.29, -0.82, -0.87, -1.51, -0.81, -1.26, -1.34, -0.65, -0.93, -0.7, -0.54, -0.91, -1.34, -1.04, -1.08, -1.26, -1.01, -0.67, -0.44, -0.57, -0.67, -1.09, -0.94, -0.58, -1.61, -0.99, -0.44, -1.44, -1.43, -1.05, -0.94, -0.65, -0.44, -1.92, -0.92, -1.17, -0.92, -1.27, -1.23, -1.12, -1.02, -0.94, -0.59, -0.62, -1.21, -1.24, -1.35, -1.25, -1.14, -0.89, -1.43, -0.54, -1.35, -0.7, -1.1, -1.02, -1.42, -1.02, -0.72, -0.74, -2.09, -1.34, -1.39, -1.11, -1.39, -0.06, -1.43, -1.23, -1.08, -1.42, -1.21, -1.62, -1.03, -1.35, -1.24, -1.05, -0.82, -1.66, -0.53, -1.1, -1.19, -1.55, -1.81, -1.21, -0.72, -1.02, -1.13, -0.5, -0.71, -1.43, -0.92, -1.24, -0.71, -0.58, -0.82, -0.91, -0.73, -0.46, -1.2, -1.32, -1.15, -1.09, -1.07, -1.01, -1.39, -1.01, -1.75, -0.62, -0.91, -1.45, -1.28, -0.85, -1.04, -1.41, -0.95, -0.83, -1.73, -0.85, -1.14, -1.06, -1.14, -1.2, -1.45, -0.82, -0.95, -1.17, -1.0, -1.45, -1.18, -1.08, -0.68, -1.64, -1.5, -1.62, -0.76, -1.4, -2.01, -0.65, -0.64, -0.38, -0.97, -1.3, -1.34, -1.14, -1.15, -0.63, -1.01, -0.84, -1.07, -1.11, -1.07, -1.18, -1.19, -1.44, -1.03, -0.94, -1.36, -1.5, -1.62, -0.64, -1.06, -0.87, -0.69, -0.86, -1.15, -1.13, -1.62, -1.1, -0.86, -1.22, -0.92, -1.62, -0.69, -1.21, -1.38, -0.59, -1.57, -1.15, -1.28, -0.48, -1.19, -0.83, -1.08, -0.67, -1.17, -0.7, -0.06, -1.1, -2.01, -1.3, -1.43, -1.15, -1.46, -1.51, -0.55, -1.12, -1.02, -0.99, -1.54, -0.51, -1.11, -1.22, -1.28, -1.13, -0.84, -1.12, -0.82, -1.0, -0.83, -1.57, -1.02, -0.74, -1.16, -1.23, -1.4, -0.62, -1.16, -1.54, -1.32, -1.19, -1.29, -1.15, -1.63, -0.74, -1.11, -1.32, -1.09, -1.01, -1.05, -0.38, -0.81, -0.97, -0.73, -0.77, -1.18, -1.43, -0.81, -1.56, -0.68, -1.56, -1.09, -0.76, -1.04, -1.44, -0.91, -1.41, -0.91, -1.5, -1.25, -1.32, -0.9, -1.29, -1.4, -0.82, -1.18, -1.29, -0.7, -1.48, -0.64, -1.16, -1.43, -0.93, -1.74, -1.03, -0.98, -0.6, -1.25, -1.36, -0.91, -0.36, -1.2, -1.46, -0.77, -1.11, -1.39, -1.11, -0.98, -1.01, -1.54, -1.02, -0.9, -1.06, -0.92, -1.1, -1.19, -1.31, -0.97, -1.78, -1.17, -1.06, -1.18, -0.81, -1.23, -1.49, -1.44, -1.52, -1.22, -0.68, -1.26, -0.88, -0.72, -1.13, -1.22, -1.29, -0.68, -1.05, -1.03, -1.25, -1.12, -1.11, -0.61, -1.25, -1.12, -1.03, -1.24, -0.91, -1.5, -1.49, -0.92, -0.97, -1.17, -1.6, -1.32, -0.8, -1.27, -0.45, -0.95, -1.18, -1.71, -1.62, -1.68, -1.38, -1.23, -1.05, -0.77, -1.08, -1.1, -1.49, -1.11, -1.46, -1.52, -1.33, -1.23, -1.05, -1.01, -1.18, -1.16, -0.88, -1.18, -1.07, -1.68, -0.96, -0.95, -0.83, -0.44, -1.36, -1.21, -1.01, -1.57, -1.04, -0.88, -1.49, -0.53, -1.51, -1.43, -1.39, -1.28, -0.67, -1.12, -1.31, -1.52, -1.31, -1.26, -0.87, -1.29, -1.01, -0.26, -0.96, -1.2, -1.54, -1.0, -1.24, -0.53, -1.47, -0.97, -1.71, -1.18, -1.33, -1.57, -0.89, -1.64, -0.76, -0.81, -1.05, -1.13, -0.95, -0.96, -0.95, -0.96, -0.7, -0.72, -1.21, -1.62, -1.15, -1.16, -1.6, -1.07, -0.71, -0.91, -1.16, -1.3, -1.28, -1.13, -0.9, -0.84, -1.12, -0.98, -0.95, -0.97, -0.49, -1.01, -0.71, -0.97, -1.46, -1.57, -1.02, -1.22, -1.56, -1.41, -0.56, -0.76, -1.05, -1.24, -1.28, -0.99, -1.79, -0.71, -1.08, -1.12, -0.87, -1.3, -0.71]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -1.0923，s_d = 0.3206，SE = s_d/√n = 0.0111；
t = d̄/SE = -98.2797；df = n−1 = 831；双侧 p = 0.000e+00。
d̄ = -1.0923，s_d = 0.3206，SE = s_d/√n = 0.0111；
t = d̄/SE = -98.2797；df = n−1 = 831；双侧 p = 0.000e+00。
计算结果
t = -98.2797，df = 831，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估留任措施效果。
案例 46. 软件更新前后崩溃数
原始数据
前 n=606：[4.87, 5.74, 3.33, 5.81, 5.77, 3.68, 5.32, 5.85, 6.48, 4.81, 4.05, 4.32, 5.96, 6.45, 4.87, 5.39, 7.27, 7.37, 4.33, 6.49, 7.27, 6.64, 5.62, 5.11, 7.33, 4.86, 6.97, 5.84, 7.06, 6.36, 5.8, 6.03, 6.32, 5.6, 5.54, 5.86, 4.48, 6.33, 5.05, 5.79, 3.99, 3.23, 6.55, 5.65, 4.79, 5.74, 6.08, 6.17, 6.87, 4.84, 4.4, 4.27, 6.24, 5.45, 6.36, 6.11, 4.91, 6.46, 6.12, 4.02, 4.84, 4.97, 4.13, 4.48, 4.7, 4.96, 5.57, 5.35, 6.12, 5.23, 4.74, 6.56, 5.99, 5.55, 4.92, 5.27, 5.16, 4.91, 5.49, 4.88, 4.24, 4.97, 4.9, 4.16, 4.23, 5.02, 4.28, 5.84, 5.27, 4.17, 4.98, 5.31, 5.2, 4.57, 4.05, 3.97, 5.39, 5.36, 5.54, 5.21, 5.1, 4.96, 5.49, 5.0, 5.85, 5.53, 4.42, 6.46, 6.8, 5.18, 5.35, 4.23, 4.19, 7.08, 4.17, 3.97, 4.27, 6.49, 4.97, 3.9, 3.75, 5.34, 4.82, 6.22, 5.21, 5.37, 5.52, 4.36, 4.75, 3.55, 6.67, 6.67, 5.5, 5.74, 5.49, 5.56, 5.6, 7.88, 4.82, 6.07, 5.22, 5.02, 5.7, 6.29, 6.22, 5.62, 5.31, 4.76, 4.08, 5.94, 8.33, 6.4, 5.21, 5.76, 5.52, 4.52, 5.6, 5.69, 4.95, 6.07, 6.08, 4.68, 5.72, 5.81, 3.74, 6.94, 4.83, 6.07, 5.66, 5.78, 4.03, 5.61, 5.02, 5.9, 5.43, 5.09, 4.65, 4.19, 5.38, 6.13, 5.71, 4.18, 6.1, 3.59, 5.89, 5.59, 4.17, 6.06, 5.09, 5.53, 5.21, 6.22, 5.5, 4.23, 5.07, 7.0, 6.03, 4.41, 6.04, 6.7, 3.78, 4.82, 4.85, 5.04, 4.7, 5.21, 5.36, 6.77, 6.3, 4.88, 5.9, 4.6, 4.77, 5.75, 4.74, 5.73, 4.41, 5.16, 5.63, 4.48, 5.14, 6.12, 5.11, 6.32, 5.46, 5.55, 4.63, 6.49, 5.37, 5.38, 4.23, 4.19, 4.64, 4.07, 6.21, 5.88, 5.85, 6.01, 3.96, 5.41, 7.15, 6.32, 6.95, 5.72, 5.63, 4.68, 6.4, 6.33, 7.14, 4.42, 4.96, 6.07, 6.31, 5.11, 3.38, 5.01, 5.92, 5.42, 5.56, 5.42, 3.77, 3.06, 4.16, 4.26, 4.7, 3.74, 5.63, 4.09, 8.04, 6.37, 4.91, 4.39, 5.06, 5.12, 6.26, 5.28, 6.13, 6.21, 6.09, 5.34, 6.2, 4.42, 5.58, 5.69, 5.94, 5.57, 5.02, 4.6, 5.33, 4.52, 5.39, 6.69, 6.14, 6.17, 4.13, 3.11, 4.77, 5.1, 6.27, 6.68, 5.85, 4.87, 4.79, 5.23, 4.91, 5.19, 5.97, 7.59, 4.82, 6.39, 4.56, 4.9, 5.22, 4.14, 6.7, 5.11, 6.5, 5.54, 5.93, 6.34, 3.33, 5.66, 3.76, 5.91, 4.52, 5.38, 5.07, 5.0, 4.86, 5.27, 3.85, 5.55, 2.63, 4.77, 5.28, 4.34, 4.89, 6.1, 4.64, 4.42, 5.03, 5.0, 6.08, 4.57, 3.24, 5.92, 7.27, 5.04, 6.83, 6.62, 5.22, 4.39, 7.34, 5.27, 5.98, 6.14, 6.27, 6.94, 6.58, 3.09, 6.12, 5.04, 5.76, 5.27, 7.04, 3.41, 5.9, 5.23, 4.38, 5.35, 4.43, 6.15, 5.84, 7.27, 5.47, 4.58, 4.8, 4.16, 6.06, 5.14, 6.44, 5.72, 2.62, 5.97, 5.32, 5.69, 5.1, 5.33, 6.98, 7.27, 5.24, 2.84, 5.18, 6.52, 5.1, 4.48, 6.11, 5.62, 5.4, 4.77, 5.1, 5.89, 5.97, 5.32, 4.22, 3.87, 5.15, 4.09, 4.98, 6.04, 6.57, 6.22, 7.44, 4.73, 5.18, 5.87, 5.3, 3.27, 5.24, 6.06, 5.59, 4.93, 5.68, 5.64, 5.4, 4.66, 5.05, 5.61, 5.93, 3.7, 6.3, 3.72, 3.97, 4.93, 4.73, 4.82, 7.2, 5.6, 4.73, 5.42, 5.99, 5.12, 4.45, 5.27, 6.93, 3.43, 5.48, 5.08, 5.67, 6.12, 6.16, 4.98, 4.85, 5.71, 5.43, 5.81, 5.44, 6.47, 5.75, 6.85, 4.8, 4.58, 5.61, 6.34, 5.66, 6.17, 5.2, 5.07, 5.6, 5.91, 5.32, 5.03, 4.82, 4.78, 5.54, 5.05, 6.78, 5.11, 6.09, 4.25, 5.52, 6.68, 5.61, 5.82, 5.53, 5.37, 4.3, 4.28, 4.13, 5.56, 5.87, 3.47, 5.11, 6.1, 4.14, 4.4, 4.67, 4.95, 6.76, 4.86, 4.98, 5.53, 6.05, 5.24, 4.26, 6.98, 5.27, 4.95, 7.35, 4.41, 5.32, 6.64, 5.1, 4.17, 3.29, 5.92, 5.49, 4.74, 5.54, 4.75, 6.69, 4.83, 4.77, 5.41, 5.33, 7.23, 4.89, 6.36, 5.49, 6.62, 4.21, 6.23, 5.54, 7.12, 5.14, 5.85, 4.36, 4.59, 5.81, 4.31, 3.76, 4.62, 4.42, 6.28, 7.21, 6.68, 5.25, 5.45, 4.81, 4.8, 4.84, 6.15, 3.75, 5.25, 5.26, 6.37, 6.07, 6.14, 4.61, 5.13, 5.39, 3.89, 4.3, 6.58, 4.09, 5.22, 5.95, 6.44, 4.45, 5.65, 6.42, 5.57, 2.01, 5.45, 4.7, 3.53, 6.08, 5.74, 6.2, 4.07, 4.56, 3.85, 4.57, 4.98, 6.84, 5.93, 4.86, 4.96, 6.56, 4.52, 6.15, 5.23, 5.78, 5.11, 4.4, 5.39, 5.65, 5.42, 6.05, 5.4, 4.18, 5.65, 5.42, 4.98, 5.53, 5.79]；
后 n=606：[1.35, 2.96, 1.59, 2.7, 2.68, -0.24, 2.24, 3.32, 3.07, 1.64, 1.76, 1.23, 3.11, 4.27, 1.69, 1.91, 4.08, 4.24, 1.39, 3.84, 3.75, 4.02, 1.94, 1.32, 4.42, 2.2, 3.92, 2.55, 4.67, 2.55, 2.75, 2.83, 3.58, 2.86, 2.52, 3.54, 2.13, 2.71, 1.88, 3.15, 1.3, 0.08, 4.2, 3.21, 2.47, 2.73, 3.38, 3.51, 3.39, 2.31, 1.55, 1.29, 2.78, 2.89, 3.51, 3.25, 2.28, 3.18, 3.85, 1.83, 1.25, 2.08, 0.68, 1.37, 2.04, 1.52, 2.44, 2.59, 3.9, 2.34, 2.77, 3.86, 3.35, 2.84, 1.34, 2.29, 2.71, 1.39, 2.76, 1.32, 0.83, 1.8, 1.47, 0.87, 1.92, 2.1, -0.07, 3.32, 1.78, 1.02, 1.2, 1.95, 2.33, 1.56, 0.85, 0.52, 2.75, 2.52, 2.84, 1.62, 2.05, 1.44, 3.04, 1.51, 3.4, 2.55, 2.15, 3.98, 3.69, 2.27, 2.2, 1.08, 1.51, 4.63, 1.96, 0.89, 1.39, 4.03, 1.27, 0.8, 0.88, 3.15, 1.3, 2.99, 2.24, 2.44, 2.88, 0.9, 1.56, 1.67, 3.48, 4.1, 2.43, 2.1, 2.24, 1.87, 2.67, 4.49, 1.88, 2.17, 2.67, 2.58, 3.09, 3.63, 3.06, 3.07, 2.05, 1.81, 1.75, 2.9, 5.2, 3.07, 2.19, 2.63, 2.74, 1.0, 2.81, 2.92, 1.13, 2.3, 2.43, 1.72, 3.45, 2.32, 0.29, 3.64, 1.94, 2.96, 3.0, 3.1, 0.17, 2.18, 2.38, 1.66, 3.48, 1.97, 1.53, 1.05, 2.16, 3.61, 1.99, 1.4, 2.72, -0.03, 3.46, 1.55, 0.61, 2.92, 2.33, 2.06, 2.43, 3.14, 2.65, 0.21, 2.09, 3.47, 2.48, 1.57, 3.0, 4.14, 1.62, 1.6, 1.91, 1.76, 0.72, 2.1, 2.83, 4.59, 3.22, 2.3, 3.37, 1.72, 1.51, 2.47, 0.95, 2.08, 1.48, 1.85, 2.55, 2.01, 2.62, 3.62, 2.3, 2.89, 2.8, 1.95, 1.73, 3.57, 1.59, 1.84, 1.72, 0.98, 2.08, 1.29, 3.17, 3.06, 2.64, 3.0, 0.65, 2.73, 5.16, 3.06, 3.56, 3.08, 2.86, 1.88, 2.01, 4.95, 4.14, 1.86, 1.81, 3.2, 3.75, 2.35, -0.05, 1.69, 2.87, 1.59, 2.58, 2.42, 0.09, 1.12, 1.75, 1.83, 1.37, 1.11, 2.15, -0.28, 5.59, 3.69, 2.4, 1.62, 2.81, 2.34, 3.95, 2.85, 2.95, 2.93, 2.63, 1.98, 3.4, 1.65, 3.18, 3.16, 2.49, 2.88, 1.93, 1.33, 1.55, 0.9, 2.56, 4.0, 3.03, 3.0, 0.65, 0.28, 2.14, 1.83, 3.39, 3.66, 3.42, 2.11, 2.36, 1.98, 1.58, 2.39, 1.7, 5.1, 1.67, 3.51, 1.64, 1.81, 2.15, 0.95, 3.71, 2.09, 2.79, 3.2, 3.2, 3.23, -0.26, 2.41, 1.12, 2.76, 2.32, 3.13, 2.17, 2.0, 1.94, 2.19, 1.41, 2.65, -0.06, 1.38, 2.51, 0.21, 2.17, 3.0, 1.79, 0.82, 2.19, 2.48, 2.94, 1.34, 0.41, 2.93, 4.88, 1.85, 3.52, 3.15, 3.12, 1.62, 4.52, 2.99, 3.13, 3.54, 3.02, 3.75, 3.42, 0.09, 2.37, 1.34, 2.63, 2.74, 4.16, -0.04, 2.71, 2.37, 1.68, 2.22, 1.45, 3.38, 3.58, 3.9, 1.85, 1.88, 2.27, 1.03, 3.51, 2.8, 3.47, 2.17, -0.12, 2.86, 1.82, 2.51, 2.54, 2.54, 4.72, 4.75, 1.92, -0.64, 1.49, 3.63, 1.9, 1.18, 3.41, 2.25, 2.4, 1.65, 1.78, 2.77, 3.45, 2.18, 1.63, 0.79, 1.72, 1.16, 1.27, 2.64, 3.75, 2.84, 4.36, 1.59, 1.41, 3.49, 2.17, 1.3, 2.48, 3.53, 2.52, 2.95, 2.65, 2.73, 2.86, 2.23, 2.43, 2.25, 2.77, 0.37, 2.9, 0.19, 0.95, 2.44, 1.46, 1.52, 4.15, 2.65, 1.72, 1.98, 1.95, 2.41, 0.85, 2.72, 4.45, 0.31, 2.04, 1.17, 2.45, 2.6, 3.91, 1.21, 3.01, 3.2, 2.73, 2.38, 2.74, 2.91, 3.0, 4.08, 1.42, 2.16, 2.45, 3.91, 1.98, 2.85, 2.94, 1.72, 2.54, 2.52, 2.48, 1.59, 1.93, 1.86, 1.99, 1.49, 3.94, 2.58, 3.4, 1.3, 2.91, 4.22, 2.23, 3.1, 2.85, 2.77, 1.14, 1.16, 0.52, 1.55, 2.77, 0.74, 0.89, 3.41, 0.87, 1.47, 1.51, 2.33, 3.93, 1.48, 2.18, 2.53, 2.8, 1.94, 0.14, 3.83, 2.4, 1.94, 5.31, 2.68, 2.34, 3.39, 2.2, 0.71, 1.03, 2.68, 2.76, 2.68, 3.19, 2.1, 3.9, 1.95, 0.67, 2.01, 2.26, 3.67, 1.86, 2.34, 2.49, 3.71, 1.51, 3.37, 2.09, 3.27, 2.3, 2.69, 1.14, 2.04, 2.31, 0.7, 1.09, 1.75, 2.02, 3.67, 4.44, 4.24, 2.56, 2.4, 2.32, 1.74, 1.67, 2.45, 0.91, 2.44, 1.72, 3.68, 2.95, 2.63, 0.89, 2.62, 2.56, 1.44, 0.99, 4.07, 1.04, 1.91, 2.98, 2.68, 1.16, 2.91, 3.37, 1.34, -0.53, 2.27, 1.84, -0.4, 2.9, 2.28, 3.38, 1.5, 1.59, 0.94, 1.98, 2.07, 3.74, 2.81, 2.41, 1.4, 4.36, 2.14, 2.61, 2.95, 2.76, 2.32, 1.79, 2.17, 2.97, 2.42, 3.09, 2.61, 0.89, 3.48, 2.96, 1.28, 2.32, 3.14]；
差值(后−前) n=606：[-3.52, -2.77, -1.74, -3.11, -3.09, -3.92, -3.08, -2.53, -3.41, -3.18, -2.28, -3.08, -2.85, -2.17, -3.18, -3.48, -3.19, -3.13, -2.94, -2.65, -3.52, -2.62, -3.69, -3.79, -2.91, -2.66, -3.05, -3.28, -2.39, -3.81, -3.05, -3.2, -2.74, -2.74, -3.02, -2.32, -2.35, -3.62, -3.17, -2.64, -2.69, -3.15, -2.36, -2.44, -2.32, -3.01, -2.71, -2.66, -3.49, -2.53, -2.85, -2.98, -3.46, -2.57, -2.86, -2.86, -2.63, -3.28, -2.27, -2.19, -3.59, -2.89, -3.46, -3.11, -2.66, -3.45, -3.13, -2.76, -2.23, -2.89, -1.97, -2.7, -2.64, -2.71, -3.58, -2.98, -2.45, -3.52, -2.72, -3.56, -3.41, -3.17, -3.43, -3.29, -2.31, -2.92, -4.35, -2.51, -3.49, -3.16, -3.78, -3.36, -2.87, -3.01, -3.2, -3.45, -2.64, -2.84, -2.7, -3.6, -3.06, -3.51, -2.44, -3.5, -2.45, -2.98, -2.27, -2.48, -3.12, -2.91, -3.15, -3.14, -2.69, -2.45, -2.21, -3.07, -2.89, -2.47, -3.7, -3.09, -2.87, -2.19, -3.53, -3.24, -2.96, -2.93, -2.64, -3.46, -3.2, -1.88, -3.19, -2.57, -3.08, -3.64, -3.25, -3.7, -2.93, -3.39, -2.94, -3.9, -2.55, -2.44, -2.61, -2.66, -3.16, -2.55, -3.26, -2.94, -2.33, -3.04, -3.14, -3.33, -3.02, -3.14, -2.78, -3.53, -2.79, -2.77, -3.82, -3.77, -3.65, -2.96, -2.27, -3.49, -3.45, -3.3, -2.89, -3.12, -2.66, -2.68, -3.86, -3.43, -2.64, -4.24, -1.94, -3.12, -3.12, -3.14, -3.21, -2.52, -3.72, -2.78, -3.38, -3.62, -2.43, -4.03, -3.56, -3.14, -2.77, -3.47, -2.78, -3.09, -2.84, -4.02, -2.97, -3.53, -3.54, -2.84, -3.04, -2.56, -2.16, -3.22, -2.95, -3.28, -3.98, -3.11, -2.52, -2.19, -3.08, -2.58, -2.53, -2.88, -3.25, -3.28, -3.79, -3.64, -2.93, -3.31, -3.08, -2.48, -2.52, -2.5, -2.81, -3.42, -2.66, -3.6, -2.89, -2.92, -3.78, -3.55, -2.51, -3.21, -2.56, -2.79, -3.04, -2.82, -3.21, -3.0, -3.31, -2.68, -1.99, -3.26, -3.39, -2.64, -2.77, -2.8, -4.38, -1.38, -3.0, -2.56, -3.15, -2.88, -2.56, -2.76, -3.43, -3.33, -3.05, -3.83, -2.98, -3.0, -3.69, -1.94, -2.41, -2.43, -3.33, -2.63, -3.48, -4.37, -2.45, -2.68, -2.51, -2.77, -2.25, -2.78, -2.31, -2.43, -3.18, -3.28, -3.45, -3.36, -2.8, -2.77, -2.39, -2.54, -3.45, -2.7, -3.08, -3.27, -3.78, -3.62, -2.83, -2.69, -3.12, -3.17, -3.48, -2.82, -2.62, -3.27, -2.89, -3.02, -2.43, -2.75, -2.44, -3.25, -3.33, -2.8, -4.27, -2.5, -3.15, -2.89, -2.92, -3.09, -3.06, -3.18, -2.99, -3.02, -3.71, -2.34, -2.72, -3.1, -3.6, -3.26, -2.64, -3.15, -2.2, -2.25, -2.91, -3.0, -2.91, -3.08, -2.44, -2.91, -2.68, -3.39, -2.77, -4.13, -2.72, -3.1, -2.85, -3.59, -2.84, -2.52, -3.13, -3.23, -2.84, -2.99, -2.4, -3.19, -3.31, -3.47, -2.1, -2.76, -2.83, -2.28, -2.85, -2.6, -3.25, -3.18, -3.16, -3.0, -3.75, -3.7, -3.13, -2.54, -2.89, -3.44, -3.19, -2.86, -2.69, -3.14, -2.98, -2.77, -2.26, -3.37, -3.61, -2.71, -2.53, -3.14, -2.55, -2.34, -2.97, -3.55, -2.74, -3.11, -3.51, -3.18, -2.55, -2.79, -2.26, -2.52, -3.31, -3.48, -3.68, -2.89, -3.19, -3.3, -2.7, -3.37, -3.0, -3.12, -3.32, -3.13, -2.51, -3.14, -2.59, -3.08, -3.43, -2.93, -3.71, -3.4, -2.81, -3.38, -3.08, -3.14, -3.77, -2.39, -3.12, -1.98, -2.76, -2.53, -3.07, -1.98, -3.03, -2.91, -2.54, -2.43, -2.62, -3.36, -3.16, -3.33, -3.4, -3.53, -3.02, -2.49, -3.27, -3.3, -3.04, -2.95, -3.0, -3.45, -4.04, -2.71, -3.59, -2.54, -2.48, -3.12, -3.44, -3.9, -3.22, -3.52, -2.25, -3.77, -1.85, -2.51, -2.7, -3.43, -2.69, -3.56, -2.75, -2.78, -3.38, -2.43, -3.16, -2.43, -3.68, -3.32, -2.26, -3.34, -3.07, -3.4, -2.84, -3.44, -2.89, -2.92, -3.55, -3.56, -2.84, -2.53, -2.69, -2.95, -2.61, -2.46, -3.38, -2.72, -2.69, -2.6, -3.17, -3.13, -3.61, -4.01, -3.1, -2.74, -4.23, -2.69, -3.27, -2.93, -3.16, -2.62, -2.83, -3.39, -2.8, -3.0, -3.25, -3.3, -4.12, -3.15, -2.87, -3.01, -2.05, -1.73, -2.98, -3.25, -2.9, -3.46, -2.26, -3.24, -2.73, -2.06, -2.35, -2.65, -2.78, -2.87, -4.1, -3.41, -3.06, -3.57, -3.03, -4.02, -3.0, -2.91, -2.7, -2.85, -3.45, -3.85, -2.84, -3.16, -3.22, -2.54, -3.5, -3.61, -2.67, -2.87, -2.4, -2.61, -2.77, -2.44, -2.69, -3.05, -2.49, -3.06, -3.17, -3.7, -2.83, -2.82, -3.54, -2.7, -3.12, -3.51, -3.72, -2.51, -2.83, -2.45, -3.31, -2.51, -3.05, -3.31, -2.97, -3.76, -3.28, -2.75, -3.05, -4.24, -2.54, -3.18, -2.85, -3.93, -3.18, -3.46, -2.82, -2.58, -2.97, -2.91, -2.58, -2.91, -3.1, -3.12, -2.45, -3.56, -2.2, -2.38, -3.54, -2.29, -3.02, -2.79, -2.61, -3.23, -2.68, -3.0, -2.97, -2.78, -3.29, -2.18, -2.45, -3.69, -3.21, -2.65]
后 n=606：[1.35, 2.96, 1.59, 2.7, 2.68, -0.24, 2.24, 3.32, 3.07, 1.64, 1.76, 1.23, 3.11, 4.27, 1.69, 1.91, 4.08, 4.24, 1.39, 3.84, 3.75, 4.02, 1.94, 1.32, 4.42, 2.2, 3.92, 2.55, 4.67, 2.55, 2.75, 2.83, 3.58, 2.86, 2.52, 3.54, 2.13, 2.71, 1.88, 3.15, 1.3, 0.08, 4.2, 3.21, 2.47, 2.73, 3.38, 3.51, 3.39, 2.31, 1.55, 1.29, 2.78, 2.89, 3.51, 3.25, 2.28, 3.18, 3.85, 1.83, 1.25, 2.08, 0.68, 1.37, 2.04, 1.52, 2.44, 2.59, 3.9, 2.34, 2.77, 3.86, 3.35, 2.84, 1.34, 2.29, 2.71, 1.39, 2.76, 1.32, 0.83, 1.8, 1.47, 0.87, 1.92, 2.1, -0.07, 3.32, 1.78, 1.02, 1.2, 1.95, 2.33, 1.56, 0.85, 0.52, 2.75, 2.52, 2.84, 1.62, 2.05, 1.44, 3.04, 1.51, 3.4, 2.55, 2.15, 3.98, 3.69, 2.27, 2.2, 1.08, 1.51, 4.63, 1.96, 0.89, 1.39, 4.03, 1.27, 0.8, 0.88, 3.15, 1.3, 2.99, 2.24, 2.44, 2.88, 0.9, 1.56, 1.67, 3.48, 4.1, 2.43, 2.1, 2.24, 1.87, 2.67, 4.49, 1.88, 2.17, 2.67, 2.58, 3.09, 3.63, 3.06, 3.07, 2.05, 1.81, 1.75, 2.9, 5.2, 3.07, 2.19, 2.63, 2.74, 1.0, 2.81, 2.92, 1.13, 2.3, 2.43, 1.72, 3.45, 2.32, 0.29, 3.64, 1.94, 2.96, 3.0, 3.1, 0.17, 2.18, 2.38, 1.66, 3.48, 1.97, 1.53, 1.05, 2.16, 3.61, 1.99, 1.4, 2.72, -0.03, 3.46, 1.55, 0.61, 2.92, 2.33, 2.06, 2.43, 3.14, 2.65, 0.21, 2.09, 3.47, 2.48, 1.57, 3.0, 4.14, 1.62, 1.6, 1.91, 1.76, 0.72, 2.1, 2.83, 4.59, 3.22, 2.3, 3.37, 1.72, 1.51, 2.47, 0.95, 2.08, 1.48, 1.85, 2.55, 2.01, 2.62, 3.62, 2.3, 2.89, 2.8, 1.95, 1.73, 3.57, 1.59, 1.84, 1.72, 0.98, 2.08, 1.29, 3.17, 3.06, 2.64, 3.0, 0.65, 2.73, 5.16, 3.06, 3.56, 3.08, 2.86, 1.88, 2.01, 4.95, 4.14, 1.86, 1.81, 3.2, 3.75, 2.35, -0.05, 1.69, 2.87, 1.59, 2.58, 2.42, 0.09, 1.12, 1.75, 1.83, 1.37, 1.11, 2.15, -0.28, 5.59, 3.69, 2.4, 1.62, 2.81, 2.34, 3.95, 2.85, 2.95, 2.93, 2.63, 1.98, 3.4, 1.65, 3.18, 3.16, 2.49, 2.88, 1.93, 1.33, 1.55, 0.9, 2.56, 4.0, 3.03, 3.0, 0.65, 0.28, 2.14, 1.83, 3.39, 3.66, 3.42, 2.11, 2.36, 1.98, 1.58, 2.39, 1.7, 5.1, 1.67, 3.51, 1.64, 1.81, 2.15, 0.95, 3.71, 2.09, 2.79, 3.2, 3.2, 3.23, -0.26, 2.41, 1.12, 2.76, 2.32, 3.13, 2.17, 2.0, 1.94, 2.19, 1.41, 2.65, -0.06, 1.38, 2.51, 0.21, 2.17, 3.0, 1.79, 0.82, 2.19, 2.48, 2.94, 1.34, 0.41, 2.93, 4.88, 1.85, 3.52, 3.15, 3.12, 1.62, 4.52, 2.99, 3.13, 3.54, 3.02, 3.75, 3.42, 0.09, 2.37, 1.34, 2.63, 2.74, 4.16, -0.04, 2.71, 2.37, 1.68, 2.22, 1.45, 3.38, 3.58, 3.9, 1.85, 1.88, 2.27, 1.03, 3.51, 2.8, 3.47, 2.17, -0.12, 2.86, 1.82, 2.51, 2.54, 2.54, 4.72, 4.75, 1.92, -0.64, 1.49, 3.63, 1.9, 1.18, 3.41, 2.25, 2.4, 1.65, 1.78, 2.77, 3.45, 2.18, 1.63, 0.79, 1.72, 1.16, 1.27, 2.64, 3.75, 2.84, 4.36, 1.59, 1.41, 3.49, 2.17, 1.3, 2.48, 3.53, 2.52, 2.95, 2.65, 2.73, 2.86, 2.23, 2.43, 2.25, 2.77, 0.37, 2.9, 0.19, 0.95, 2.44, 1.46, 1.52, 4.15, 2.65, 1.72, 1.98, 1.95, 2.41, 0.85, 2.72, 4.45, 0.31, 2.04, 1.17, 2.45, 2.6, 3.91, 1.21, 3.01, 3.2, 2.73, 2.38, 2.74, 2.91, 3.0, 4.08, 1.42, 2.16, 2.45, 3.91, 1.98, 2.85, 2.94, 1.72, 2.54, 2.52, 2.48, 1.59, 1.93, 1.86, 1.99, 1.49, 3.94, 2.58, 3.4, 1.3, 2.91, 4.22, 2.23, 3.1, 2.85, 2.77, 1.14, 1.16, 0.52, 1.55, 2.77, 0.74, 0.89, 3.41, 0.87, 1.47, 1.51, 2.33, 3.93, 1.48, 2.18, 2.53, 2.8, 1.94, 0.14, 3.83, 2.4, 1.94, 5.31, 2.68, 2.34, 3.39, 2.2, 0.71, 1.03, 2.68, 2.76, 2.68, 3.19, 2.1, 3.9, 1.95, 0.67, 2.01, 2.26, 3.67, 1.86, 2.34, 2.49, 3.71, 1.51, 3.37, 2.09, 3.27, 2.3, 2.69, 1.14, 2.04, 2.31, 0.7, 1.09, 1.75, 2.02, 3.67, 4.44, 4.24, 2.56, 2.4, 2.32, 1.74, 1.67, 2.45, 0.91, 2.44, 1.72, 3.68, 2.95, 2.63, 0.89, 2.62, 2.56, 1.44, 0.99, 4.07, 1.04, 1.91, 2.98, 2.68, 1.16, 2.91, 3.37, 1.34, -0.53, 2.27, 1.84, -0.4, 2.9, 2.28, 3.38, 1.5, 1.59, 0.94, 1.98, 2.07, 3.74, 2.81, 2.41, 1.4, 4.36, 2.14, 2.61, 2.95, 2.76, 2.32, 1.79, 2.17, 2.97, 2.42, 3.09, 2.61, 0.89, 3.48, 2.96, 1.28, 2.32, 3.14]；
差值(后−前) n=606：[-3.52, -2.77, -1.74, -3.11, -3.09, -3.92, -3.08, -2.53, -3.41, -3.18, -2.28, -3.08, -2.85, -2.17, -3.18, -3.48, -3.19, -3.13, -2.94, -2.65, -3.52, -2.62, -3.69, -3.79, -2.91, -2.66, -3.05, -3.28, -2.39, -3.81, -3.05, -3.2, -2.74, -2.74, -3.02, -2.32, -2.35, -3.62, -3.17, -2.64, -2.69, -3.15, -2.36, -2.44, -2.32, -3.01, -2.71, -2.66, -3.49, -2.53, -2.85, -2.98, -3.46, -2.57, -2.86, -2.86, -2.63, -3.28, -2.27, -2.19, -3.59, -2.89, -3.46, -3.11, -2.66, -3.45, -3.13, -2.76, -2.23, -2.89, -1.97, -2.7, -2.64, -2.71, -3.58, -2.98, -2.45, -3.52, -2.72, -3.56, -3.41, -3.17, -3.43, -3.29, -2.31, -2.92, -4.35, -2.51, -3.49, -3.16, -3.78, -3.36, -2.87, -3.01, -3.2, -3.45, -2.64, -2.84, -2.7, -3.6, -3.06, -3.51, -2.44, -3.5, -2.45, -2.98, -2.27, -2.48, -3.12, -2.91, -3.15, -3.14, -2.69, -2.45, -2.21, -3.07, -2.89, -2.47, -3.7, -3.09, -2.87, -2.19, -3.53, -3.24, -2.96, -2.93, -2.64, -3.46, -3.2, -1.88, -3.19, -2.57, -3.08, -3.64, -3.25, -3.7, -2.93, -3.39, -2.94, -3.9, -2.55, -2.44, -2.61, -2.66, -3.16, -2.55, -3.26, -2.94, -2.33, -3.04, -3.14, -3.33, -3.02, -3.14, -2.78, -3.53, -2.79, -2.77, -3.82, -3.77, -3.65, -2.96, -2.27, -3.49, -3.45, -3.3, -2.89, -3.12, -2.66, -2.68, -3.86, -3.43, -2.64, -4.24, -1.94, -3.12, -3.12, -3.14, -3.21, -2.52, -3.72, -2.78, -3.38, -3.62, -2.43, -4.03, -3.56, -3.14, -2.77, -3.47, -2.78, -3.09, -2.84, -4.02, -2.97, -3.53, -3.54, -2.84, -3.04, -2.56, -2.16, -3.22, -2.95, -3.28, -3.98, -3.11, -2.52, -2.19, -3.08, -2.58, -2.53, -2.88, -3.25, -3.28, -3.79, -3.64, -2.93, -3.31, -3.08, -2.48, -2.52, -2.5, -2.81, -3.42, -2.66, -3.6, -2.89, -2.92, -3.78, -3.55, -2.51, -3.21, -2.56, -2.79, -3.04, -2.82, -3.21, -3.0, -3.31, -2.68, -1.99, -3.26, -3.39, -2.64, -2.77, -2.8, -4.38, -1.38, -3.0, -2.56, -3.15, -2.88, -2.56, -2.76, -3.43, -3.33, -3.05, -3.83, -2.98, -3.0, -3.69, -1.94, -2.41, -2.43, -3.33, -2.63, -3.48, -4.37, -2.45, -2.68, -2.51, -2.77, -2.25, -2.78, -2.31, -2.43, -3.18, -3.28, -3.45, -3.36, -2.8, -2.77, -2.39, -2.54, -3.45, -2.7, -3.08, -3.27, -3.78, -3.62, -2.83, -2.69, -3.12, -3.17, -3.48, -2.82, -2.62, -3.27, -2.89, -3.02, -2.43, -2.75, -2.44, -3.25, -3.33, -2.8, -4.27, -2.5, -3.15, -2.89, -2.92, -3.09, -3.06, -3.18, -2.99, -3.02, -3.71, -2.34, -2.72, -3.1, -3.6, -3.26, -2.64, -3.15, -2.2, -2.25, -2.91, -3.0, -2.91, -3.08, -2.44, -2.91, -2.68, -3.39, -2.77, -4.13, -2.72, -3.1, -2.85, -3.59, -2.84, -2.52, -3.13, -3.23, -2.84, -2.99, -2.4, -3.19, -3.31, -3.47, -2.1, -2.76, -2.83, -2.28, -2.85, -2.6, -3.25, -3.18, -3.16, -3.0, -3.75, -3.7, -3.13, -2.54, -2.89, -3.44, -3.19, -2.86, -2.69, -3.14, -2.98, -2.77, -2.26, -3.37, -3.61, -2.71, -2.53, -3.14, -2.55, -2.34, -2.97, -3.55, -2.74, -3.11, -3.51, -3.18, -2.55, -2.79, -2.26, -2.52, -3.31, -3.48, -3.68, -2.89, -3.19, -3.3, -2.7, -3.37, -3.0, -3.12, -3.32, -3.13, -2.51, -3.14, -2.59, -3.08, -3.43, -2.93, -3.71, -3.4, -2.81, -3.38, -3.08, -3.14, -3.77, -2.39, -3.12, -1.98, -2.76, -2.53, -3.07, -1.98, -3.03, -2.91, -2.54, -2.43, -2.62, -3.36, -3.16, -3.33, -3.4, -3.53, -3.02, -2.49, -3.27, -3.3, -3.04, -2.95, -3.0, -3.45, -4.04, -2.71, -3.59, -2.54, -2.48, -3.12, -3.44, -3.9, -3.22, -3.52, -2.25, -3.77, -1.85, -2.51, -2.7, -3.43, -2.69, -3.56, -2.75, -2.78, -3.38, -2.43, -3.16, -2.43, -3.68, -3.32, -2.26, -3.34, -3.07, -3.4, -2.84, -3.44, -2.89, -2.92, -3.55, -3.56, -2.84, -2.53, -2.69, -2.95, -2.61, -2.46, -3.38, -2.72, -2.69, -2.6, -3.17, -3.13, -3.61, -4.01, -3.1, -2.74, -4.23, -2.69, -3.27, -2.93, -3.16, -2.62, -2.83, -3.39, -2.8, -3.0, -3.25, -3.3, -4.12, -3.15, -2.87, -3.01, -2.05, -1.73, -2.98, -3.25, -2.9, -3.46, -2.26, -3.24, -2.73, -2.06, -2.35, -2.65, -2.78, -2.87, -4.1, -3.41, -3.06, -3.57, -3.03, -4.02, -3.0, -2.91, -2.7, -2.85, -3.45, -3.85, -2.84, -3.16, -3.22, -2.54, -3.5, -3.61, -2.67, -2.87, -2.4, -2.61, -2.77, -2.44, -2.69, -3.05, -2.49, -3.06, -3.17, -3.7, -2.83, -2.82, -3.54, -2.7, -3.12, -3.51, -3.72, -2.51, -2.83, -2.45, -3.31, -2.51, -3.05, -3.31, -2.97, -3.76, -3.28, -2.75, -3.05, -4.24, -2.54, -3.18, -2.85, -3.93, -3.18, -3.46, -2.82, -2.58, -2.97, -2.91, -2.58, -2.91, -3.1, -3.12, -2.45, -3.56, -2.2, -2.38, -3.54, -2.29, -3.02, -2.79, -2.61, -3.23, -2.68, -3.0, -2.97, -2.78, -3.29, -2.18, -2.45, -3.69, -3.21, -2.65]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -2.9989，s_d = 0.4687，SE = s_d/√n = 0.0190；
t = d̄/SE = -157.5046；df = n−1 = 605；双侧 p = 0.000e+00。
d̄ = -2.9989，s_d = 0.4687，SE = s_d/√n = 0.0190；
t = d̄/SE = -157.5046；df = n−1 = 605；双侧 p = 0.000e+00。
计算结果
t = -157.5046，df = 605，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。判断更新是否降低崩溃。
案例 47. 前后阅读速度
原始数据
前 n=374：[193.29, 198.21, 199.19, 198.63, 204.13, 201.54, 224.25, 209.36, 207.59, 197.96, 204.2, 192.12, 202.0, 209.68, 212.3, 209.02, 196.85, 204.68, 201.07, 203.4, 193.56, 202.2, 204.34, 201.03, 184.45, 205.98, 206.69, 200.92, 202.52, 197.79, 205.65, 200.48, 201.22, 192.16, 202.52, 196.89, 196.56, 209.82, 214.31, 199.58, 203.75, 196.3, 208.11, 212.32, 198.29, 188.43, 203.46, 212.95, 219.06, 201.55, 201.22, 209.94, 198.08, 184.2, 209.98, 208.08, 190.69, 208.67, 205.78, 195.51, 206.94, 208.84, 211.35, 203.85, 214.53, 195.95, 213.75, 210.05, 188.97, 216.21, 200.3, 207.59, 205.9, 195.22, 190.49, 212.94, 199.18, 205.73, 218.17, 201.12, 205.61, 204.55, 200.45, 191.2, 198.12, 199.86, 209.41, 201.47, 200.45, 185.36, 195.84, 206.43, 190.64, 197.89, 204.41, 206.81, 196.95, 207.78, 206.96, 202.15, 191.56, 217.41, 186.11, 212.52, 194.0, 202.68, 209.7, 208.55, 192.84, 205.52, 211.95, 208.58, 200.37, 203.45, 203.82, 188.29, 199.38, 198.76, 193.95, 210.26, 193.67, 195.91, 213.66, 194.24, 200.58, 202.28, 197.43, 209.68, 192.78, 213.91, 198.04, 203.77, 202.68, 196.79, 207.4, 209.75, 182.96, 200.09, 198.49, 200.2, 209.58, 206.67, 212.6, 210.53, 204.06, 190.8, 198.23, 218.48, 200.94, 195.79, 202.75, 196.26, 211.6, 181.78, 196.07, 215.76, 210.24, 196.78, 203.6, 186.37, 207.99, 199.0, 213.32, 201.0, 201.57, 206.96, 198.57, 203.45, 217.08, 209.09, 208.12, 201.92, 195.77, 203.73, 210.81, 193.23, 207.44, 205.1, 208.18, 206.22, 197.78, 193.89, 209.45, 204.93, 213.8, 204.87, 203.81, 203.07, 204.09, 190.31, 209.78, 217.59, 211.08, 198.17, 187.4, 195.58, 204.22, 210.15, 206.29, 190.26, 198.19, 200.56, 221.02, 196.02, 197.66, 209.22, 196.24, 205.01, 204.39, 213.67, 201.47, 206.06, 201.2, 209.36, 192.08, 198.1, 194.77, 204.8, 204.67, 208.31, 196.46, 201.76, 196.7, 195.83, 198.9, 204.93, 198.49, 215.61, 201.28, 211.42, 216.29, 208.86, 192.39, 212.52, 203.78, 205.76, 223.97, 211.95, 195.77, 204.44, 208.56, 218.63, 198.74, 208.36, 209.86, 218.18, 211.11, 207.65, 206.96, 203.58, 188.39, 208.1, 209.1, 208.07, 198.36, 208.17, 211.19, 207.73, 192.49, 202.14, 205.05, 204.15, 211.05, 211.06, 204.48, 216.88, 212.74, 199.78, 192.81, 206.09, 210.6, 211.17, 203.33, 195.67, 193.53, 197.7, 202.49, 198.62, 196.2, 198.3, 195.99, 207.0, 221.39, 199.31, 211.22, 206.52, 196.14, 205.13, 211.65, 192.73, 214.85, 220.56, 191.99, 187.69, 207.8, 203.48, 210.51, 208.89, 201.07, 198.35, 193.95, 203.78, 198.21, 200.26, 200.3, 204.55, 203.02, 208.32, 188.69, 211.86, 194.06, 190.39, 201.69, 203.74, 206.68, 207.39, 193.78, 197.33, 196.84, 198.9, 208.26, 195.95, 215.49, 199.07, 191.7, 203.5, 192.1, 201.85, 207.56, 198.83, 204.46, 205.95, 190.21, 197.38, 190.68, 208.24, 196.79, 200.83, 185.51, 205.97, 204.65, 216.05, 197.96, 191.83, 213.01, 200.42, 194.45, 220.74, 215.99, 207.97, 206.85, 205.96, 198.59, 196.3, 208.46, 206.5, 200.64, 189.81, 193.35, 212.53, 213.78, 197.21, 199.01, 206.58, 205.44, 204.47, 186.98, 209.51, 204.32, 204.31, 196.23, 184.95, 200.11, 189.37]；
后 n=374：[224.56, 227.99, 228.98, 228.24, 235.57, 233.13, 254.1, 240.98, 238.98, 228.57, 234.3, 222.39, 232.39, 240.62, 243.54, 239.62, 227.97, 237.07, 231.91, 233.13, 225.3, 233.36, 235.47, 233.16, 216.48, 237.15, 238.15, 233.16, 233.02, 229.56, 235.74, 231.75, 231.59, 223.94, 233.45, 226.99, 227.9, 240.19, 244.4, 231.41, 234.62, 226.28, 238.66, 242.72, 228.35, 217.84, 235.7, 244.69, 249.53, 232.7, 232.13, 241.61, 228.46, 215.19, 240.44, 238.27, 221.15, 239.33, 237.69, 227.79, 238.59, 240.56, 242.93, 234.65, 245.26, 226.96, 243.62, 241.15, 220.53, 247.19, 230.96, 238.32, 236.82, 225.97, 222.04, 243.36, 230.06, 237.76, 249.51, 233.99, 237.46, 234.88, 230.82, 222.13, 229.66, 230.3, 239.86, 233.15, 231.4, 215.23, 228.29, 236.77, 221.74, 228.43, 235.43, 237.74, 227.64, 238.32, 236.42, 233.39, 221.93, 247.91, 217.03, 243.65, 224.1, 233.53, 241.62, 238.89, 224.21, 236.57, 242.55, 239.32, 231.74, 233.45, 234.17, 219.3, 229.7, 229.31, 225.47, 242.45, 224.49, 226.09, 244.84, 224.6, 231.14, 233.05, 227.02, 240.1, 223.41, 244.09, 228.24, 234.74, 233.26, 227.72, 238.13, 239.99, 214.21, 231.47, 228.56, 231.57, 241.15, 237.22, 244.18, 241.6, 234.44, 223.01, 229.13, 249.95, 231.82, 226.82, 234.14, 227.63, 242.23, 213.05, 227.2, 246.08, 240.72, 228.4, 234.48, 216.82, 238.4, 229.75, 243.55, 231.75, 234.41, 237.23, 229.57, 233.92, 248.09, 240.5, 240.16, 232.19, 227.67, 235.0, 241.55, 224.54, 238.47, 235.88, 239.39, 237.2, 229.36, 224.28, 240.06, 236.16, 243.55, 236.64, 234.18, 234.02, 233.93, 221.7, 240.97, 248.06, 242.13, 228.8, 217.74, 225.37, 235.02, 240.73, 237.47, 221.02, 228.95, 232.66, 252.29, 227.7, 228.41, 239.75, 226.76, 235.29, 234.64, 245.08, 231.54, 237.57, 231.71, 241.15, 223.15, 229.74, 224.42, 234.57, 236.89, 239.13, 227.72, 232.34, 228.37, 227.66, 229.42, 234.99, 230.03, 247.42, 232.4, 240.93, 246.54, 239.7, 224.0, 242.77, 235.38, 237.39, 254.66, 241.84, 226.07, 235.76, 240.48, 249.43, 228.73, 240.94, 240.92, 248.12, 241.63, 238.97, 238.85, 235.94, 220.47, 239.41, 239.63, 237.96, 229.5, 237.53, 241.96, 239.18, 224.21, 232.9, 236.06, 235.06, 242.43, 242.64, 236.59, 246.88, 243.07, 230.87, 223.44, 236.95, 240.98, 241.92, 234.11, 225.68, 225.1, 229.77, 233.18, 229.56, 226.52, 228.27, 228.01, 237.75, 253.34, 229.79, 241.67, 235.98, 227.33, 235.44, 242.11, 223.59, 244.55, 251.35, 222.45, 218.78, 238.91, 234.21, 241.59, 240.62, 231.85, 230.0, 223.88, 233.72, 228.23, 231.03, 232.48, 234.81, 235.05, 238.86, 219.33, 243.04, 224.62, 221.72, 233.0, 235.8, 237.59, 237.25, 225.02, 229.42, 227.57, 229.71, 239.57, 227.03, 246.72, 229.63, 222.58, 236.33, 224.36, 233.66, 238.22, 230.68, 235.27, 237.97, 220.22, 229.36, 221.39, 238.73, 227.32, 230.69, 215.98, 237.39, 235.53, 246.12, 229.0, 223.2, 243.75, 232.02, 225.02, 251.26, 247.54, 239.86, 237.92, 236.07, 229.52, 225.53, 238.73, 238.1, 231.66, 219.94, 225.39, 243.91, 242.93, 228.26, 231.03, 238.62, 235.22, 234.49, 217.31, 240.08, 236.24, 235.0, 227.48, 215.46, 230.72, 219.87]；
差值(后−前) n=374：[31.26, 29.78, 29.79, 29.6, 31.44, 31.59, 29.85, 31.62, 31.39, 30.61, 30.1, 30.27, 30.39, 30.94, 31.24, 30.6, 31.12, 32.4, 30.84, 29.73, 31.74, 31.16, 31.13, 32.13, 32.03, 31.17, 31.46, 32.24, 30.5, 31.77, 30.09, 31.27, 30.37, 31.79, 30.92, 30.1, 31.34, 30.36, 30.09, 31.82, 30.87, 29.98, 30.56, 30.4, 30.05, 29.41, 32.24, 31.74, 30.48, 31.15, 30.91, 31.67, 30.38, 31.0, 30.46, 30.2, 30.45, 30.66, 31.92, 32.28, 31.65, 31.72, 31.59, 30.8, 30.74, 31.0, 29.87, 31.1, 31.56, 30.98, 30.66, 30.73, 30.92, 30.75, 31.55, 30.42, 30.88, 32.03, 31.34, 32.87, 31.85, 30.33, 30.37, 30.93, 31.53, 30.44, 30.45, 31.67, 30.95, 29.87, 32.46, 30.34, 31.1, 30.54, 31.03, 30.93, 30.69, 30.54, 29.46, 31.25, 30.37, 30.5, 30.93, 31.13, 30.1, 30.85, 31.92, 30.34, 31.38, 31.05, 30.61, 30.74, 31.37, 30.0, 30.35, 31.01, 30.32, 30.54, 31.52, 32.2, 30.82, 30.18, 31.17, 30.36, 30.56, 30.77, 29.59, 30.42, 30.63, 30.18, 30.2, 30.97, 30.57, 30.93, 30.73, 30.24, 31.24, 31.38, 30.07, 31.37, 31.57, 30.55, 31.58, 31.07, 30.38, 32.21, 30.91, 31.46, 30.87, 31.03, 31.39, 31.37, 30.63, 31.27, 31.12, 30.32, 30.47, 31.62, 30.88, 30.45, 30.41, 30.74, 30.23, 30.74, 32.84, 30.27, 31.0, 30.47, 31.01, 31.41, 32.04, 30.27, 31.9, 31.28, 30.74, 31.31, 31.04, 30.78, 31.21, 30.98, 31.59, 30.4, 30.61, 31.23, 29.75, 31.78, 30.37, 30.95, 29.84, 31.39, 31.18, 30.47, 31.05, 30.62, 30.34, 29.79, 30.8, 30.58, 31.17, 30.75, 30.76, 32.1, 31.27, 31.68, 30.76, 30.53, 30.52, 30.27, 30.25, 31.41, 30.07, 31.51, 30.51, 31.79, 31.07, 31.64, 29.65, 29.77, 32.21, 30.82, 31.25, 30.59, 31.67, 31.84, 30.52, 30.06, 31.54, 31.81, 31.12, 29.5, 30.24, 30.84, 31.61, 30.25, 31.6, 31.63, 30.69, 29.9, 30.29, 31.32, 31.93, 30.8, 29.98, 32.58, 31.06, 29.94, 30.52, 31.32, 31.89, 32.35, 32.08, 31.31, 30.53, 29.89, 31.14, 29.36, 30.77, 31.45, 31.71, 30.76, 31.02, 30.92, 31.38, 31.58, 32.11, 30.0, 30.33, 31.09, 30.63, 30.86, 30.37, 30.75, 30.78, 30.01, 31.57, 32.06, 30.69, 30.95, 30.33, 29.96, 32.01, 30.76, 31.95, 30.49, 30.45, 29.46, 31.2, 30.31, 30.46, 30.86, 29.7, 30.79, 30.46, 31.09, 31.11, 30.72, 31.08, 31.72, 30.78, 31.65, 29.94, 29.94, 30.03, 30.77, 32.17, 30.27, 32.03, 30.53, 30.64, 31.18, 30.55, 31.32, 31.31, 32.06, 30.92, 29.85, 31.24, 32.09, 30.73, 30.81, 31.31, 31.07, 31.23, 30.56, 30.88, 32.83, 32.25, 31.81, 30.67, 31.85, 30.81, 32.03, 30.01, 31.98, 30.71, 30.49, 30.53, 29.87, 30.47, 31.42, 30.88, 30.07, 31.04, 31.37, 30.74, 31.6, 30.57, 30.51, 31.54, 31.9, 31.07, 30.11, 30.93, 29.23, 30.27, 31.59, 31.02, 30.13, 32.03, 31.38, 29.15, 31.05, 32.02, 32.04, 29.78, 30.02, 30.33, 30.57, 31.93, 30.69, 31.26, 30.51, 30.61, 30.5]
后 n=374：[224.56, 227.99, 228.98, 228.24, 235.57, 233.13, 254.1, 240.98, 238.98, 228.57, 234.3, 222.39, 232.39, 240.62, 243.54, 239.62, 227.97, 237.07, 231.91, 233.13, 225.3, 233.36, 235.47, 233.16, 216.48, 237.15, 238.15, 233.16, 233.02, 229.56, 235.74, 231.75, 231.59, 223.94, 233.45, 226.99, 227.9, 240.19, 244.4, 231.41, 234.62, 226.28, 238.66, 242.72, 228.35, 217.84, 235.7, 244.69, 249.53, 232.7, 232.13, 241.61, 228.46, 215.19, 240.44, 238.27, 221.15, 239.33, 237.69, 227.79, 238.59, 240.56, 242.93, 234.65, 245.26, 226.96, 243.62, 241.15, 220.53, 247.19, 230.96, 238.32, 236.82, 225.97, 222.04, 243.36, 230.06, 237.76, 249.51, 233.99, 237.46, 234.88, 230.82, 222.13, 229.66, 230.3, 239.86, 233.15, 231.4, 215.23, 228.29, 236.77, 221.74, 228.43, 235.43, 237.74, 227.64, 238.32, 236.42, 233.39, 221.93, 247.91, 217.03, 243.65, 224.1, 233.53, 241.62, 238.89, 224.21, 236.57, 242.55, 239.32, 231.74, 233.45, 234.17, 219.3, 229.7, 229.31, 225.47, 242.45, 224.49, 226.09, 244.84, 224.6, 231.14, 233.05, 227.02, 240.1, 223.41, 244.09, 228.24, 234.74, 233.26, 227.72, 238.13, 239.99, 214.21, 231.47, 228.56, 231.57, 241.15, 237.22, 244.18, 241.6, 234.44, 223.01, 229.13, 249.95, 231.82, 226.82, 234.14, 227.63, 242.23, 213.05, 227.2, 246.08, 240.72, 228.4, 234.48, 216.82, 238.4, 229.75, 243.55, 231.75, 234.41, 237.23, 229.57, 233.92, 248.09, 240.5, 240.16, 232.19, 227.67, 235.0, 241.55, 224.54, 238.47, 235.88, 239.39, 237.2, 229.36, 224.28, 240.06, 236.16, 243.55, 236.64, 234.18, 234.02, 233.93, 221.7, 240.97, 248.06, 242.13, 228.8, 217.74, 225.37, 235.02, 240.73, 237.47, 221.02, 228.95, 232.66, 252.29, 227.7, 228.41, 239.75, 226.76, 235.29, 234.64, 245.08, 231.54, 237.57, 231.71, 241.15, 223.15, 229.74, 224.42, 234.57, 236.89, 239.13, 227.72, 232.34, 228.37, 227.66, 229.42, 234.99, 230.03, 247.42, 232.4, 240.93, 246.54, 239.7, 224.0, 242.77, 235.38, 237.39, 254.66, 241.84, 226.07, 235.76, 240.48, 249.43, 228.73, 240.94, 240.92, 248.12, 241.63, 238.97, 238.85, 235.94, 220.47, 239.41, 239.63, 237.96, 229.5, 237.53, 241.96, 239.18, 224.21, 232.9, 236.06, 235.06, 242.43, 242.64, 236.59, 246.88, 243.07, 230.87, 223.44, 236.95, 240.98, 241.92, 234.11, 225.68, 225.1, 229.77, 233.18, 229.56, 226.52, 228.27, 228.01, 237.75, 253.34, 229.79, 241.67, 235.98, 227.33, 235.44, 242.11, 223.59, 244.55, 251.35, 222.45, 218.78, 238.91, 234.21, 241.59, 240.62, 231.85, 230.0, 223.88, 233.72, 228.23, 231.03, 232.48, 234.81, 235.05, 238.86, 219.33, 243.04, 224.62, 221.72, 233.0, 235.8, 237.59, 237.25, 225.02, 229.42, 227.57, 229.71, 239.57, 227.03, 246.72, 229.63, 222.58, 236.33, 224.36, 233.66, 238.22, 230.68, 235.27, 237.97, 220.22, 229.36, 221.39, 238.73, 227.32, 230.69, 215.98, 237.39, 235.53, 246.12, 229.0, 223.2, 243.75, 232.02, 225.02, 251.26, 247.54, 239.86, 237.92, 236.07, 229.52, 225.53, 238.73, 238.1, 231.66, 219.94, 225.39, 243.91, 242.93, 228.26, 231.03, 238.62, 235.22, 234.49, 217.31, 240.08, 236.24, 235.0, 227.48, 215.46, 230.72, 219.87]；
差值(后−前) n=374：[31.26, 29.78, 29.79, 29.6, 31.44, 31.59, 29.85, 31.62, 31.39, 30.61, 30.1, 30.27, 30.39, 30.94, 31.24, 30.6, 31.12, 32.4, 30.84, 29.73, 31.74, 31.16, 31.13, 32.13, 32.03, 31.17, 31.46, 32.24, 30.5, 31.77, 30.09, 31.27, 30.37, 31.79, 30.92, 30.1, 31.34, 30.36, 30.09, 31.82, 30.87, 29.98, 30.56, 30.4, 30.05, 29.41, 32.24, 31.74, 30.48, 31.15, 30.91, 31.67, 30.38, 31.0, 30.46, 30.2, 30.45, 30.66, 31.92, 32.28, 31.65, 31.72, 31.59, 30.8, 30.74, 31.0, 29.87, 31.1, 31.56, 30.98, 30.66, 30.73, 30.92, 30.75, 31.55, 30.42, 30.88, 32.03, 31.34, 32.87, 31.85, 30.33, 30.37, 30.93, 31.53, 30.44, 30.45, 31.67, 30.95, 29.87, 32.46, 30.34, 31.1, 30.54, 31.03, 30.93, 30.69, 30.54, 29.46, 31.25, 30.37, 30.5, 30.93, 31.13, 30.1, 30.85, 31.92, 30.34, 31.38, 31.05, 30.61, 30.74, 31.37, 30.0, 30.35, 31.01, 30.32, 30.54, 31.52, 32.2, 30.82, 30.18, 31.17, 30.36, 30.56, 30.77, 29.59, 30.42, 30.63, 30.18, 30.2, 30.97, 30.57, 30.93, 30.73, 30.24, 31.24, 31.38, 30.07, 31.37, 31.57, 30.55, 31.58, 31.07, 30.38, 32.21, 30.91, 31.46, 30.87, 31.03, 31.39, 31.37, 30.63, 31.27, 31.12, 30.32, 30.47, 31.62, 30.88, 30.45, 30.41, 30.74, 30.23, 30.74, 32.84, 30.27, 31.0, 30.47, 31.01, 31.41, 32.04, 30.27, 31.9, 31.28, 30.74, 31.31, 31.04, 30.78, 31.21, 30.98, 31.59, 30.4, 30.61, 31.23, 29.75, 31.78, 30.37, 30.95, 29.84, 31.39, 31.18, 30.47, 31.05, 30.62, 30.34, 29.79, 30.8, 30.58, 31.17, 30.75, 30.76, 32.1, 31.27, 31.68, 30.76, 30.53, 30.52, 30.27, 30.25, 31.41, 30.07, 31.51, 30.51, 31.79, 31.07, 31.64, 29.65, 29.77, 32.21, 30.82, 31.25, 30.59, 31.67, 31.84, 30.52, 30.06, 31.54, 31.81, 31.12, 29.5, 30.24, 30.84, 31.61, 30.25, 31.6, 31.63, 30.69, 29.9, 30.29, 31.32, 31.93, 30.8, 29.98, 32.58, 31.06, 29.94, 30.52, 31.32, 31.89, 32.35, 32.08, 31.31, 30.53, 29.89, 31.14, 29.36, 30.77, 31.45, 31.71, 30.76, 31.02, 30.92, 31.38, 31.58, 32.11, 30.0, 30.33, 31.09, 30.63, 30.86, 30.37, 30.75, 30.78, 30.01, 31.57, 32.06, 30.69, 30.95, 30.33, 29.96, 32.01, 30.76, 31.95, 30.49, 30.45, 29.46, 31.2, 30.31, 30.46, 30.86, 29.7, 30.79, 30.46, 31.09, 31.11, 30.72, 31.08, 31.72, 30.78, 31.65, 29.94, 29.94, 30.03, 30.77, 32.17, 30.27, 32.03, 30.53, 30.64, 31.18, 30.55, 31.32, 31.31, 32.06, 30.92, 29.85, 31.24, 32.09, 30.73, 30.81, 31.31, 31.07, 31.23, 30.56, 30.88, 32.83, 32.25, 31.81, 30.67, 31.85, 30.81, 32.03, 30.01, 31.98, 30.71, 30.49, 30.53, 29.87, 30.47, 31.42, 30.88, 30.07, 31.04, 31.37, 30.74, 31.6, 30.57, 30.51, 31.54, 31.9, 31.07, 30.11, 30.93, 29.23, 30.27, 31.59, 31.02, 30.13, 32.03, 31.38, 29.15, 31.05, 32.02, 32.04, 29.78, 30.02, 30.33, 30.57, 31.93, 30.69, 31.26, 30.51, 30.61, 30.5]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = 30.9175，s_d = 0.6974，SE = s_d/√n = 0.0361；
t = d̄/SE = 857.3583；df = n−1 = 373；双侧 p = 0.000e+00。
d̄ = 30.9175，s_d = 0.6974，SE = s_d/√n = 0.0361；
t = d̄/SE = 857.3583；df = n−1 = 373；双侧 p = 0.000e+00。
计算结果
t = 857.3583，df = 373，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估训练对阅读提速。
案例 48. 干预前后 BMI
原始数据
前 n=980：[27.86, 28.07, 25.75, 25.28, 28.4, 25.17, 26.33, 26.64, 29.31, 26.18, 28.16, 27.9, 25.37, 23.64, 28.11, 26.16, 25.28, 25.72, 24.98, 27.91, 27.18, 28.35, 26.95, 24.76, 26.32, 27.05, 27.4, 28.1, 27.44, 28.0, 26.21, 26.49, 26.66, 25.27, 26.75, 24.7, 27.42, 26.1, 26.9, 23.92, 25.16, 26.31, 25.67, 25.69, 26.83, 26.94, 26.74, 25.66, 27.6, 26.31, 27.06, 27.06, 26.86, 26.79, 27.34, 27.37, 25.8, 25.18, 27.53, 26.47, 25.4, 27.5, 26.4, 26.15, 23.53, 27.97, 26.99, 25.67, 25.29, 28.51, 26.63, 26.66, 26.85, 25.98, 26.52, 26.09, 24.97, 26.9, 25.86, 25.69, 26.35, 25.89, 27.65, 26.75, 26.32, 27.77, 26.07, 26.72, 25.93, 25.22, 25.42, 25.06, 25.95, 27.42, 25.9, 26.99, 27.1, 25.53, 25.16, 25.8, 25.37, 27.03, 26.87, 24.8, 26.23, 26.38, 27.96, 27.24, 26.55, 26.21, 25.42, 28.01, 27.25, 26.76, 25.0, 25.43, 26.6, 26.2, 26.0, 26.97, 26.96, 27.16, 27.14, 25.35, 27.49, 27.99, 24.23, 26.75, 26.36, 28.7, 27.41, 26.78, 26.92, 27.78, 27.78, 26.48, 26.24, 27.28, 26.98, 25.02, 26.0, 27.26, 25.37, 25.55, 26.1, 27.96, 27.09, 26.98, 25.31, 25.86, 26.29, 27.71, 27.87, 25.92, 26.45, 26.18, 26.19, 28.39, 27.25, 25.76, 26.8, 25.28, 25.27, 26.5, 27.54, 24.51, 26.11, 27.87, 24.97, 25.99, 26.01, 26.28, 25.64, 26.43, 26.63, 25.9, 25.59, 27.52, 25.21, 26.49, 26.27, 24.95, 25.22, 25.72, 25.41, 27.26, 26.95, 26.64, 26.7, 27.42, 27.97, 25.19, 24.87, 26.76, 24.53, 24.9, 27.03, 25.89, 26.04, 25.91, 26.89, 26.82, 25.55, 24.74, 28.56, 26.18, 23.97, 26.23, 26.46, 25.51, 25.83, 24.79, 26.17, 27.14, 24.88, 27.8, 26.63, 26.03, 27.84, 27.56, 26.91, 27.11, 25.54, 25.94, 28.28, 27.7, 28.04, 25.68, 25.65, 25.91, 27.12, 25.36, 26.38, 28.1, 26.1, 26.48, 25.97, 27.22, 25.83, 25.35, 26.53, 26.68, 26.84, 27.22, 26.27, 27.89, 26.01, 25.96, 25.19, 27.12, 27.01, 27.01, 24.79, 27.84, 26.21, 23.02, 26.44, 27.45, 26.4, 25.24, 26.65, 26.1, 25.89, 25.09, 24.64, 25.39, 26.71, 24.69, 26.26, 26.19, 26.17, 25.64, 27.79, 28.25, 27.79, 28.32, 27.09, 28.12, 28.0, 27.11, 24.72, 24.97, 26.94, 27.36, 27.93, 27.48, 26.52, 25.36, 27.25, 25.27, 25.86, 25.64, 25.67, 25.03, 26.58, 25.7, 25.52, 25.66, 26.72, 25.66, 26.34, 24.64, 27.3, 25.71, 26.3, 28.98, 27.5, 26.36, 27.39, 26.44, 28.82, 25.81, 26.96, 27.11, 26.52, 27.42, 25.35, 24.55, 25.33, 25.71, 25.45, 27.29, 25.4, 27.81, 26.24, 27.87, 26.28, 26.25, 25.97, 26.73, 28.21, 27.62, 27.22, 27.66, 26.24, 25.37, 27.7, 26.81, 26.05, 25.2, 25.47, 27.15, 26.48, 27.92, 26.38, 23.92, 26.2, 28.71, 26.11, 26.64, 25.1, 24.94, 25.35, 25.9, 25.32, 27.49, 24.79, 25.76, 27.88, 25.13, 25.51, 26.32, 26.18, 26.7, 26.13, 25.28, 25.9, 27.26, 26.1, 25.2, 25.63, 25.45, 27.34, 26.95, 26.33, 28.35, 26.5, 28.31, 27.2, 27.91, 28.25, 26.39, 24.6, 27.85, 25.95, 25.64, 26.59, 24.79, 25.17, 27.43, 26.73, 26.37, 24.94, 26.19, 28.22, 27.52, 26.33, 27.44, 25.94, 27.25, 24.83, 25.29, 27.8, 26.06, 24.73, 26.55, 26.81, 28.45, 25.32, 26.77, 25.94, 25.15, 25.55, 25.62, 27.82, 27.48, 25.7, 27.71, 26.48, 25.97, 26.16, 27.13, 27.45, 25.0, 26.08, 26.28, 25.67, 25.57, 25.69, 26.82, 26.57, 25.03, 26.51, 25.6, 25.11, 26.06, 27.43, 26.73, 27.93, 26.2, 26.14, 26.58, 26.62, 26.41, 27.48, 25.94, 26.53, 27.35, 26.19, 27.09, 26.32, 25.3, 27.53, 26.76, 25.85, 26.67, 27.15, 27.2, 25.9, 27.11, 25.75, 27.04, 27.26, 27.45, 28.62, 25.89, 25.0, 28.62, 26.98, 26.0, 27.4, 28.97, 24.26, 24.96, 28.0, 26.81, 24.84, 26.41, 25.52, 26.14, 26.77, 25.07, 27.87, 25.48, 26.45, 27.27, 25.58, 25.43, 28.08, 25.83, 27.61, 27.25, 26.93, 27.0, 24.16, 25.73, 27.71, 24.65, 27.35, 27.15, 25.84, 26.68, 25.73, 27.87, 27.66, 26.66, 27.08, 26.02, 27.09, 26.97, 26.19, 25.45, 26.08, 26.45, 26.71, 28.61, 26.4, 25.63, 25.53, 26.72, 26.45, 25.43, 25.4, 25.64, 25.85, 26.51, 25.42, 27.83, 25.97, 24.9, 25.55, 26.78, 26.26, 25.59, 27.27, 28.3, 26.24, 27.37, 26.93, 26.03, 28.18, 27.61, 25.75, 27.13, 27.74, 23.3, 25.92, 26.04, 26.5, 27.26, 27.07, 26.45, 25.45, 26.19, 25.32, 27.45, 26.18, 26.24, 26.3, 25.6, 23.99, 24.91, 24.58, 24.48, 26.67, 27.08, 24.96, 25.57, 25.8, 25.72, 25.39, 27.25, 25.05, 26.79, 26.56, 28.78, 26.94, 25.93, 25.75, 26.74, 27.73, 27.77, 27.78, 25.68, 26.84, 26.69, 26.78, 25.07, 25.5, 27.03, 27.22, 26.09, 27.03, 26.89, 27.64, 26.3, 27.07, 26.88, 26.13, 27.62, 27.33, 25.84, 27.16, 26.98, 26.4, 27.99, 27.07, 24.97, 24.31, 26.92, 26.16, 26.9, 26.06, 25.75, 25.68, 27.6, 25.48, 27.19, 26.81, 27.23, 26.61, 26.41, 25.86, 26.22, 26.8, 26.84, 26.23, 27.98, 26.5, 27.04, 26.05, 25.84, 27.7, 28.46, 26.38, 26.55, 27.51, 25.36, 25.28, 26.93, 27.28, 24.18, 24.48, 26.79, 24.3, 26.12, 25.57, 25.76, 27.02, 28.1, 25.96, 25.92, 27.0, 26.83, 26.32, 25.99, 25.89, 26.83, 26.4, 26.5, 25.89, 26.71, 25.23, 26.75, 25.71, 26.96, 24.31, 27.07, 25.73, 26.56, 23.56, 24.53, 28.33, 26.83, 27.17, 27.46, 27.52, 23.5, 27.24, 28.17, 25.08, 25.73, 28.38, 26.82, 25.98, 27.89, 25.91, 25.61, 27.05, 25.99, 24.28, 26.08, 25.23, 25.47, 26.82, 27.91, 27.77, 24.93, 26.07, 25.68, 26.45, 27.58, 26.74, 26.38, 29.39, 26.17, 26.64, 27.29, 27.69, 26.53, 25.61, 25.14, 25.14, 26.74, 28.04, 26.69, 28.29, 27.12, 25.32, 27.23, 27.8, 24.74, 26.67, 25.83, 27.7, 26.26, 26.73, 26.77, 28.06, 27.08, 25.85, 27.29, 26.77, 24.65, 26.48, 26.04, 25.57, 28.01, 25.48, 24.72, 26.54, 28.72, 27.34, 25.59, 27.64, 27.76, 27.48, 28.0, 26.44, 27.18, 25.47, 26.25, 25.55, 25.29, 27.0, 26.98, 25.92, 26.11, 27.92, 23.71, 26.42, 27.62, 25.32, 25.11, 25.44, 26.6, 27.01, 29.12, 25.27, 25.63, 28.48, 25.58, 27.88, 25.97, 26.04, 25.97, 25.99, 26.03, 25.19, 26.5, 24.8, 26.64, 26.78, 28.08, 26.77, 25.36, 25.45, 28.5, 24.85, 25.82, 26.88, 27.17, 25.87, 25.37, 26.14, 25.97, 29.09, 24.93, 26.19, 27.47, 27.41, 26.58, 26.79, 27.0, 26.4, 28.18, 27.01, 27.67, 26.35, 25.45, 25.81, 27.9, 24.72, 25.72, 25.13, 28.3, 26.82, 25.75, 25.81, 26.45, 24.57, 26.81, 27.01, 27.0, 27.02, 26.03, 26.88, 25.22, 27.17, 25.38, 26.19, 28.65, 25.87, 25.77, 27.93, 26.06, 27.61, 24.86, 26.32, 26.47, 24.96, 27.61, 24.54, 26.16, 26.21, 25.53, 27.27, 26.41, 25.81, 25.93, 24.85, 24.47, 26.67, 28.05, 27.39, 25.6, 27.43, 25.43, 26.78, 25.94, 24.67, 27.09, 26.29, 24.7, 24.9, 27.29, 28.07, 25.65, 25.63, 26.18, 26.65, 26.57, 26.95, 25.12, 25.46, 25.8, 26.19, 23.88, 27.67, 25.22, 26.71, 27.12, 24.67, 26.75, 26.7, 27.79, 25.79, 25.54, 25.22, 26.45, 25.43, 25.24, 27.53, 26.78, 25.58, 25.74, 26.93, 27.86, 27.01, 25.39, 25.66, 26.86, 28.17, 26.5, 26.4, 25.58, 25.97, 26.15, 26.88, 26.7, 25.3, 25.13, 26.36, 26.42, 26.95, 26.43, 27.96, 22.38, 24.64, 25.59, 26.26, 24.51, 27.17, 25.61, 25.76, 26.4, 26.99, 27.3, 27.57, 26.02, 26.2, 26.79, 27.35, 26.18, 28.02, 24.95, 24.76, 25.49, 27.44, 25.77, 27.66, 26.11, 23.86, 26.61, 27.12, 26.55, 25.83, 29.12, 26.16, 27.36, 24.9, 28.39, 25.82, 26.45, 26.58, 27.06, 28.58, 26.66, 26.94, 27.16, 26.71, 26.91, 26.65, 26.49, 25.79, 26.04, 26.69, 26.43, 26.8, 25.24, 26.6, 24.34, 26.14, 27.81, 26.99, 26.0, 26.29, 25.72, 26.53, 28.2]；
后 n=980：[25.93, 26.57, 23.88, 22.61, 26.27, 22.61, 23.72, 23.91, 28.02, 23.06, 25.45, 25.28, 22.4, 22.13, 26.08, 24.54, 22.36, 23.85, 22.43, 26.03, 25.1, 26.08, 24.96, 22.6, 24.25, 24.43, 25.4, 26.31, 25.95, 25.97, 24.57, 24.37, 23.7, 23.98, 24.25, 23.49, 26.52, 23.78, 23.5, 22.17, 22.66, 23.32, 23.11, 23.77, 25.3, 24.86, 26.25, 23.23, 25.3, 24.57, 23.93, 24.56, 25.19, 24.29, 26.97, 25.32, 24.22, 23.56, 24.52, 24.72, 24.52, 24.79, 24.46, 24.67, 22.71, 26.53, 25.83, 23.58, 21.77, 27.7, 24.75, 24.13, 24.52, 23.16, 24.0, 25.15, 22.58, 24.58, 25.39, 24.35, 25.3, 24.03, 24.76, 23.3, 23.74, 25.15, 23.29, 25.21, 23.75, 21.18, 23.69, 22.31, 25.81, 25.63, 24.27, 24.97, 25.83, 23.32, 23.88, 24.54, 22.68, 25.39, 24.03, 23.52, 23.6, 23.96, 25.06, 25.1, 24.13, 24.21, 23.41, 25.8, 25.53, 24.79, 21.89, 22.49, 25.93, 24.03, 24.34, 25.17, 25.35, 27.0, 24.46, 24.77, 26.05, 25.63, 22.25, 23.99, 24.57, 27.1, 25.93, 24.47, 24.42, 26.0, 25.08, 24.26, 25.0, 25.75, 24.59, 23.46, 23.24, 26.31, 23.36, 24.83, 23.21, 25.89, 24.32, 24.17, 23.91, 23.52, 24.34, 25.63, 24.7, 23.4, 23.69, 24.74, 24.35, 26.68, 24.99, 23.37, 23.92, 24.27, 23.17, 24.82, 25.22, 22.62, 23.53, 26.45, 22.99, 23.31, 23.87, 23.9, 23.62, 24.22, 25.76, 23.86, 23.64, 26.31, 23.0, 24.66, 23.65, 23.34, 22.87, 23.78, 22.9, 25.56, 25.51, 24.97, 23.63, 25.51, 25.43, 22.56, 22.66, 24.59, 22.25, 22.54, 25.7, 22.72, 24.28, 24.66, 24.38, 24.76, 22.25, 23.0, 26.89, 22.64, 22.65, 23.68, 23.43, 23.69, 22.77, 23.41, 24.18, 26.7, 23.2, 26.25, 24.5, 24.91, 26.27, 24.38, 24.4, 25.86, 24.12, 24.96, 26.17, 25.38, 25.83, 23.21, 23.06, 23.77, 24.49, 23.66, 24.88, 26.03, 23.55, 24.43, 24.98, 26.8, 23.51, 22.17, 23.64, 25.26, 24.63, 25.36, 24.94, 25.78, 23.56, 23.55, 22.85, 26.07, 24.57, 25.79, 22.45, 26.46, 23.8, 23.13, 24.54, 24.93, 24.84, 23.18, 24.67, 23.46, 23.88, 23.47, 23.02, 23.55, 23.98, 23.53, 23.45, 23.53, 24.4, 23.62, 25.87, 26.77, 25.66, 26.55, 25.2, 25.83, 26.2, 24.41, 23.02, 22.64, 25.9, 26.07, 26.73, 25.17, 23.49, 24.15, 25.38, 23.43, 24.47, 22.55, 24.64, 23.42, 24.88, 23.03, 23.67, 23.69, 25.02, 24.23, 23.86, 21.69, 26.53, 23.15, 23.92, 26.86, 26.05, 25.48, 26.06, 22.81, 27.03, 24.28, 24.9, 26.46, 24.85, 25.09, 22.46, 23.2, 23.46, 22.93, 23.98, 25.82, 23.35, 24.95, 22.95, 24.95, 25.26, 24.59, 25.07, 24.13, 25.66, 25.32, 25.3, 24.79, 25.2, 24.03, 26.3, 23.99, 24.52, 22.75, 23.59, 25.43, 24.23, 25.87, 24.55, 21.66, 22.79, 27.12, 24.16, 23.98, 23.2, 22.09, 22.88, 23.86, 24.34, 25.98, 23.18, 24.99, 26.62, 23.51, 24.55, 24.32, 23.29, 25.55, 24.48, 23.02, 24.29, 26.23, 23.86, 23.65, 22.5, 23.03, 24.84, 25.14, 24.07, 27.48, 24.56, 26.33, 26.03, 25.31, 25.32, 24.26, 22.49, 26.55, 25.16, 22.64, 24.07, 24.53, 22.02, 24.56, 24.96, 24.96, 22.34, 25.41, 27.26, 25.64, 23.96, 26.72, 23.64, 25.45, 21.71, 23.58, 25.03, 23.33, 23.49, 23.86, 24.74, 25.74, 22.87, 23.9, 23.87, 23.29, 23.01, 22.24, 24.91, 24.57, 24.07, 25.22, 24.43, 24.2, 23.81, 25.33, 25.61, 22.55, 24.1, 23.44, 23.89, 23.57, 23.78, 24.61, 25.0, 22.75, 23.31, 22.92, 24.49, 23.82, 24.37, 25.45, 26.16, 24.47, 24.63, 25.61, 25.09, 24.52, 25.4, 23.07, 24.98, 24.33, 23.19, 24.06, 23.75, 22.93, 24.52, 25.05, 24.86, 24.99, 25.8, 23.62, 24.55, 24.67, 24.18, 25.43, 24.53, 25.9, 26.87, 24.47, 22.57, 26.79, 25.17, 24.33, 24.71, 27.51, 22.44, 22.35, 25.25, 24.69, 22.81, 25.38, 22.99, 23.63, 25.13, 23.39, 25.26, 22.51, 25.21, 24.37, 24.97, 23.85, 25.76, 23.98, 25.97, 25.1, 24.1, 24.46, 22.66, 23.14, 25.18, 23.02, 25.81, 25.54, 23.83, 24.5, 23.16, 25.13, 26.29, 23.97, 24.92, 23.84, 24.05, 24.73, 24.49, 23.52, 24.09, 24.78, 24.7, 26.18, 24.2, 23.8, 23.83, 24.82, 24.68, 23.0, 22.74, 23.71, 23.23, 23.68, 23.29, 26.47, 24.11, 22.48, 24.68, 24.65, 25.17, 23.92, 26.07, 25.88, 23.43, 24.21, 24.57, 23.38, 26.11, 26.48, 23.63, 25.12, 24.38, 21.26, 23.02, 25.32, 24.19, 25.52, 25.4, 24.39, 23.82, 24.36, 23.0, 26.34, 23.7, 24.47, 23.9, 23.22, 22.04, 24.11, 23.18, 21.54, 24.52, 24.46, 20.7, 23.37, 23.0, 22.55, 24.21, 25.62, 22.71, 25.47, 25.48, 26.11, 24.53, 22.76, 23.06, 25.01, 25.61, 25.64, 25.74, 25.11, 24.57, 24.46, 23.52, 23.75, 24.46, 25.25, 24.9, 23.53, 24.57, 25.64, 25.68, 24.94, 24.69, 25.55, 24.75, 25.52, 24.33, 23.79, 25.16, 25.67, 24.37, 27.1, 25.41, 22.76, 22.82, 24.48, 23.86, 25.4, 23.62, 23.45, 23.87, 25.61, 23.31, 26.02, 26.32, 24.92, 24.55, 23.54, 23.81, 24.76, 25.03, 24.83, 24.67, 25.47, 25.01, 24.67, 24.26, 23.24, 26.04, 26.79, 24.56, 24.45, 25.59, 22.61, 23.7, 23.94, 26.88, 22.02, 22.9, 24.81, 22.53, 23.33, 23.22, 23.18, 25.18, 25.43, 23.37, 23.54, 24.4, 23.91, 23.52, 24.69, 24.35, 24.36, 24.88, 24.12, 24.41, 24.92, 22.85, 25.28, 23.24, 25.34, 22.44, 23.92, 24.46, 24.5, 21.98, 23.0, 25.58, 24.12, 26.0, 26.91, 24.58, 21.33, 24.48, 26.44, 23.09, 23.14, 25.98, 23.98, 24.22, 25.71, 25.26, 24.56, 25.22, 23.89, 23.74, 23.26, 22.99, 24.1, 24.37, 26.47, 26.15, 23.21, 24.36, 23.27, 25.49, 26.17, 25.02, 24.84, 27.98, 24.27, 24.2, 25.44, 25.33, 25.31, 24.05, 23.03, 22.81, 24.69, 26.7, 24.56, 25.31, 25.85, 23.12, 24.69, 26.61, 22.62, 25.77, 24.96, 26.79, 24.59, 24.63, 24.39, 25.58, 24.96, 22.87, 25.93, 24.24, 22.63, 24.56, 24.11, 24.22, 26.45, 22.95, 23.63, 23.91, 25.8, 26.02, 23.15, 26.05, 25.2, 25.73, 25.82, 24.16, 25.7, 21.72, 24.31, 24.33, 22.72, 23.97, 24.63, 23.51, 23.72, 25.17, 21.28, 22.58, 25.04, 22.71, 23.07, 22.88, 25.41, 25.35, 27.44, 23.24, 24.39, 26.22, 23.53, 25.09, 23.52, 22.99, 24.85, 24.57, 25.03, 24.35, 24.9, 23.11, 24.14, 24.05, 27.22, 25.75, 22.94, 23.09, 25.93, 22.35, 22.78, 24.84, 26.3, 23.88, 24.03, 24.13, 23.34, 27.67, 23.08, 24.38, 25.7, 25.24, 24.46, 25.92, 24.91, 24.42, 25.69, 24.69, 25.8, 24.77, 23.52, 23.81, 25.92, 23.44, 24.71, 22.87, 26.05, 24.47, 23.82, 24.87, 22.86, 22.57, 24.68, 24.64, 26.38, 23.74, 23.32, 24.01, 22.88, 25.75, 22.79, 25.44, 26.56, 23.18, 24.23, 27.14, 24.75, 25.36, 22.25, 24.08, 24.47, 23.7, 25.69, 22.57, 24.59, 24.56, 23.69, 24.4, 23.66, 24.59, 23.82, 23.0, 22.04, 24.69, 26.27, 26.29, 25.49, 25.54, 24.15, 23.51, 23.97, 23.05, 24.71, 24.3, 21.86, 21.86, 26.31, 25.76, 22.55, 23.74, 24.49, 25.07, 24.25, 24.53, 22.33, 22.99, 23.98, 23.54, 22.2, 25.92, 24.88, 24.49, 24.94, 22.44, 23.65, 25.29, 26.32, 23.7, 23.51, 23.34, 24.45, 23.28, 22.72, 26.34, 24.53, 23.45, 23.61, 24.89, 24.71, 24.94, 23.48, 23.57, 23.95, 25.66, 24.44, 24.23, 24.36, 23.8, 24.0, 25.56, 24.47, 22.29, 23.03, 23.08, 23.71, 26.43, 25.35, 25.29, 19.46, 23.08, 23.3, 23.79, 22.26, 26.19, 23.49, 23.61, 24.64, 25.44, 25.65, 25.21, 23.92, 23.96, 24.26, 24.32, 23.92, 26.65, 22.97, 22.85, 23.47, 24.88, 23.01, 25.16, 23.98, 21.64, 24.68, 25.39, 24.92, 24.81, 26.93, 25.17, 23.62, 22.97, 26.06, 23.49, 24.51, 25.11, 26.3, 27.0, 23.74, 24.98, 25.35, 24.76, 25.51, 25.77, 24.83, 22.99, 24.27, 23.42, 24.1, 24.24, 22.58, 24.7, 22.6, 24.52, 25.78, 24.42, 23.82, 24.63, 23.15, 24.49, 26.57]；
差值(后−前) n=980：[-1.93, -1.5, -1.87, -2.67, -2.13, -2.56, -2.6, -2.73, -1.29, -3.12, -2.71, -2.62, -2.98, -1.51, -2.03, -1.62, -2.92, -1.87, -2.54, -1.89, -2.08, -2.27, -1.99, -2.16, -2.07, -2.62, -2.0, -1.79, -1.5, -2.02, -1.64, -2.12, -2.96, -1.3, -2.5, -1.2, -0.9, -2.32, -3.39, -1.75, -2.5, -2.99, -2.56, -1.92, -1.54, -2.08, -0.49, -2.43, -2.3, -1.74, -3.13, -2.49, -1.67, -2.51, -0.37, -2.05, -1.58, -1.63, -3.02, -1.75, -0.88, -2.71, -1.94, -1.49, -0.82, -1.44, -1.17, -2.09, -3.52, -0.81, -1.88, -2.52, -2.33, -2.82, -2.52, -0.94, -2.39, -2.32, -0.47, -1.33, -1.04, -1.86, -2.89, -3.45, -2.57, -2.61, -2.78, -1.51, -2.18, -4.04, -1.73, -2.75, -0.15, -1.79, -1.63, -2.02, -1.27, -2.21, -1.29, -1.26, -2.69, -1.63, -2.84, -1.28, -2.63, -2.42, -2.9, -2.14, -2.42, -2.0, -2.01, -2.21, -1.72, -1.97, -3.1, -2.94, -0.67, -2.18, -1.66, -1.79, -1.61, -0.16, -2.69, -0.57, -1.44, -2.36, -1.98, -2.77, -1.78, -1.6, -1.47, -2.31, -2.49, -1.78, -2.7, -2.23, -1.24, -1.53, -2.38, -1.56, -2.76, -0.95, -2.01, -0.71, -2.89, -2.07, -2.78, -2.81, -1.4, -2.34, -1.95, -2.08, -3.17, -2.52, -2.76, -1.44, -1.84, -1.71, -2.26, -2.4, -2.88, -1.01, -2.11, -1.67, -2.32, -1.89, -2.58, -1.42, -1.98, -2.69, -2.14, -2.38, -2.01, -2.21, -0.87, -2.05, -1.95, -1.21, -2.21, -1.84, -2.62, -1.62, -2.35, -1.94, -2.51, -1.7, -1.44, -1.67, -3.07, -1.92, -2.54, -2.64, -2.21, -2.17, -2.28, -2.37, -1.33, -3.17, -1.76, -1.25, -2.51, -2.06, -3.31, -1.74, -1.67, -3.54, -1.32, -2.55, -3.03, -1.82, -3.06, -1.39, -1.99, -0.44, -1.68, -1.55, -2.13, -1.12, -1.57, -3.18, -2.51, -1.26, -1.42, -0.98, -2.11, -2.32, -2.21, -2.47, -2.59, -2.14, -2.63, -1.7, -1.5, -2.06, -2.55, -2.05, -1.0, -0.42, -2.31, -3.18, -2.89, -1.42, -2.21, -1.86, -1.34, -2.11, -2.45, -2.42, -2.35, -1.05, -2.45, -1.22, -2.34, -1.38, -2.42, 0.11, -1.9, -2.52, -1.56, -2.07, -1.97, -2.64, -2.01, -1.62, -1.62, -1.84, -2.73, -1.16, -2.81, -2.67, -1.76, -2.02, -1.92, -1.48, -2.13, -1.77, -1.89, -2.29, -1.79, -2.7, -1.71, -2.33, -1.04, -1.29, -1.2, -2.3, -3.03, -1.22, -1.86, -1.84, -1.4, -3.09, -1.04, -1.61, -1.7, -2.67, -1.85, -1.97, -1.7, -1.43, -2.48, -2.95, -0.76, -2.55, -2.38, -2.12, -1.45, -0.88, -1.33, -3.63, -1.8, -1.53, -2.06, -0.65, -1.67, -2.33, -2.88, -1.35, -1.87, -2.78, -1.47, -1.47, -2.05, -2.86, -3.29, -2.92, -1.03, -1.66, -0.9, -2.6, -2.55, -2.29, -1.93, -2.87, -1.03, -1.34, -1.4, -2.82, -1.53, -2.45, -1.88, -1.72, -2.25, -2.05, -1.83, -2.26, -3.4, -1.58, -1.96, -2.67, -1.91, -2.85, -2.47, -2.04, -0.98, -1.52, -1.61, -0.77, -1.26, -1.62, -0.97, -2.0, -2.89, -1.14, -1.65, -2.26, -1.62, -1.03, -2.24, -1.56, -3.13, -2.42, -2.5, -1.81, -2.26, -0.87, -1.94, -1.99, -1.17, -2.6, -2.92, -2.13, -2.11, -1.3, -0.8, -3.0, -2.52, -0.26, -3.15, -2.87, -1.77, -1.41, -2.61, -0.77, -0.97, -1.88, -2.37, -0.72, -2.3, -1.8, -3.12, -1.71, -2.77, -2.72, -1.24, -2.69, -2.07, -2.71, -2.45, -2.87, -2.07, -1.86, -2.54, -3.38, -2.91, -2.91, -1.63, -2.49, -2.06, -1.77, -2.34, -1.8, -1.84, -2.44, -1.98, -2.84, -1.78, -2.0, -1.91, -2.21, -1.57, -2.29, -3.2, -2.68, -0.62, -2.24, -3.06, -1.28, -1.77, -1.73, -1.51, -0.97, -1.53, -1.88, -2.09, -2.87, -1.56, -3.03, -3.0, -3.04, -2.57, -2.36, -3.01, -1.71, -0.98, -1.68, -1.35, -3.58, -1.35, -2.43, -1.57, -1.6, -2.73, -1.56, -1.75, -1.41, -2.43, -1.83, -1.8, -1.68, -2.69, -1.46, -1.82, -2.61, -2.75, -2.12, -2.03, -1.03, -2.53, -2.51, -1.63, -1.69, -2.6, -2.97, -1.24, -2.9, -0.61, -1.58, -2.31, -1.85, -1.64, -2.15, -2.83, -2.54, -1.51, -2.59, -2.53, -1.62, -1.55, -1.61, -2.01, -2.18, -2.57, -2.74, -1.36, -2.69, -2.16, -2.18, -3.04, -2.24, -1.71, -1.94, -1.99, -1.67, -2.02, -2.43, -2.2, -1.83, -1.7, -1.9, -1.77, -2.43, -2.66, -1.93, -2.62, -2.83, -2.13, -1.37, -1.86, -2.42, -0.87, -2.13, -1.08, -1.66, -1.2, -2.43, -2.81, -3.16, -2.36, -2.65, -2.07, -1.13, -2.12, -2.02, -3.36, -2.05, -2.89, -0.72, -2.31, -1.74, -1.68, -2.05, -1.63, -1.83, -2.31, -1.12, -2.47, -1.77, -2.39, -2.39, -1.95, -0.8, -1.4, -2.94, -2.16, -2.62, -4.26, -2.2, -2.8, -3.17, -1.19, -1.64, -2.34, -1.32, -1.08, -2.67, -2.41, -3.17, -2.69, -1.73, -2.11, -2.12, -2.04, -0.57, -2.27, -2.23, -3.25, -1.32, -1.04, -1.78, -2.32, -2.55, -2.46, -1.25, -1.97, -1.36, -2.38, -1.33, -1.38, -2.1, -3.01, -2.05, -1.99, -1.31, -2.03, -0.89, -1.66, -2.21, -1.49, -2.44, -2.29, -1.5, -2.45, -2.29, -1.81, -1.99, -2.17, -1.17, -0.49, -2.32, -2.06, -2.87, -2.05, -1.46, -1.77, -2.02, -1.55, -2.51, -1.5, -2.37, -1.79, -2.61, -1.66, -1.67, -1.81, -2.1, -1.92, -2.75, -1.58, -3.0, -0.4, -2.16, -1.58, -1.98, -1.77, -2.79, -2.35, -2.58, -1.85, -2.66, -2.58, -2.38, -2.6, -2.92, -2.8, -1.31, -1.54, -2.47, -1.52, -2.39, -1.48, -1.78, -2.39, -1.47, -2.47, -1.63, -1.87, -3.15, -1.27, -2.06, -1.58, -1.53, -2.75, -2.7, -1.17, -0.55, -2.94, -2.18, -2.76, -1.73, -1.99, -2.59, -2.4, -2.84, -1.75, -2.18, -0.65, -1.04, -1.83, -2.1, -0.54, -2.83, -2.24, -1.38, -2.44, -1.45, -1.62, -1.71, -1.71, -2.41, -0.96, -1.42, -1.72, -1.54, -1.4, -1.89, -2.44, -1.85, -2.37, -1.22, -1.56, -2.11, -2.34, -2.05, -1.35, -2.12, -2.98, -1.27, -2.2, -2.54, -1.19, -2.12, -0.9, -0.88, -0.92, -1.67, -2.1, -2.38, -2.48, -2.12, -2.98, -1.37, -2.53, -2.02, -1.92, -1.93, -1.35, -1.56, -2.53, -1.09, -2.63, -2.91, -1.31, -2.44, -1.59, -2.56, -1.75, -2.19, -2.28, -1.49, -3.76, -1.94, -1.22, -2.58, -3.03, -2.34, -2.4, -2.39, -2.74, -2.43, -3.84, -2.58, -2.61, -2.03, -2.55, -1.19, -1.66, -1.68, -2.03, -1.25, -2.26, -2.04, -2.79, -2.45, -3.04, -1.13, -1.42, -1.0, -0.84, -1.6, -1.69, -2.49, -2.73, -0.87, -1.02, -2.42, -2.36, -2.57, -2.5, -3.03, -2.04, -0.87, -1.99, -1.34, -2.01, -2.64, -1.42, -1.85, -1.81, -1.77, -2.17, -2.11, -0.87, -2.09, -1.97, -2.49, -2.32, -1.87, -1.58, -1.94, -1.99, -1.98, -1.28, -1.01, -2.26, -2.25, -2.35, -1.93, -0.95, -3.59, -2.0, -2.13, -2.38, -0.62, -3.28, -2.72, -2.87, -2.34, -1.43, -2.59, -0.75, -2.09, -2.69, -1.54, -0.78, -1.31, -2.25, -2.61, -2.24, -2.0, -1.26, -1.92, -1.97, -1.57, -1.65, -1.83, -2.87, -2.75, -1.22, -2.11, -1.85, -2.43, -1.98, -1.78, -1.09, -0.11, -1.88, -1.28, -3.27, -1.97, -1.63, -2.38, -1.99, -2.84, -3.04, -0.98, -2.32, -3.1, -1.9, -1.69, -1.58, -2.32, -2.42, -2.79, -2.47, -1.82, -2.65, -1.68, -1.75, -0.34, -2.22, -2.18, -2.24, -3.1, -1.42, -1.47, -2.09, -2.03, -1.88, -1.99, -2.15, -2.52, -1.18, -2.26, -2.14, -2.12, -2.04, -3.15, -2.07, -1.91, -2.09, -2.91, -2.51, -2.06, -2.17, -1.22, -2.17, -2.15, -1.32, -2.23, -3.01, -2.1, -3.28, -2.72, -0.52, -1.07, -2.68, -2.92, -1.56, -2.29, -2.46, -2.25, -0.98, -2.12, -2.15, -1.76, -1.55, -1.65, -2.36, -2.1, -2.24, -2.52, -3.03, -2.26, -1.37, -1.98, -1.91, -2.03, -2.55, -2.76, -2.49, -2.12, -2.22, -1.93, -1.73, -1.62, -1.02, -2.2, -0.98, -3.74, -1.93, -2.33, -2.33, -1.94, -1.48, -0.76, -1.59, -2.93, -1.96, -1.8, -1.95, -1.4, -0.88, -1.66, -2.8, -1.77, -3.27, -2.33, -2.56, -2.66, -1.9, -1.75, -1.61, -2.04, -2.56, -2.18, -1.66, -2.57, -2.04, -1.63]
后 n=980：[25.93, 26.57, 23.88, 22.61, 26.27, 22.61, 23.72, 23.91, 28.02, 23.06, 25.45, 25.28, 22.4, 22.13, 26.08, 24.54, 22.36, 23.85, 22.43, 26.03, 25.1, 26.08, 24.96, 22.6, 24.25, 24.43, 25.4, 26.31, 25.95, 25.97, 24.57, 24.37, 23.7, 23.98, 24.25, 23.49, 26.52, 23.78, 23.5, 22.17, 22.66, 23.32, 23.11, 23.77, 25.3, 24.86, 26.25, 23.23, 25.3, 24.57, 23.93, 24.56, 25.19, 24.29, 26.97, 25.32, 24.22, 23.56, 24.52, 24.72, 24.52, 24.79, 24.46, 24.67, 22.71, 26.53, 25.83, 23.58, 21.77, 27.7, 24.75, 24.13, 24.52, 23.16, 24.0, 25.15, 22.58, 24.58, 25.39, 24.35, 25.3, 24.03, 24.76, 23.3, 23.74, 25.15, 23.29, 25.21, 23.75, 21.18, 23.69, 22.31, 25.81, 25.63, 24.27, 24.97, 25.83, 23.32, 23.88, 24.54, 22.68, 25.39, 24.03, 23.52, 23.6, 23.96, 25.06, 25.1, 24.13, 24.21, 23.41, 25.8, 25.53, 24.79, 21.89, 22.49, 25.93, 24.03, 24.34, 25.17, 25.35, 27.0, 24.46, 24.77, 26.05, 25.63, 22.25, 23.99, 24.57, 27.1, 25.93, 24.47, 24.42, 26.0, 25.08, 24.26, 25.0, 25.75, 24.59, 23.46, 23.24, 26.31, 23.36, 24.83, 23.21, 25.89, 24.32, 24.17, 23.91, 23.52, 24.34, 25.63, 24.7, 23.4, 23.69, 24.74, 24.35, 26.68, 24.99, 23.37, 23.92, 24.27, 23.17, 24.82, 25.22, 22.62, 23.53, 26.45, 22.99, 23.31, 23.87, 23.9, 23.62, 24.22, 25.76, 23.86, 23.64, 26.31, 23.0, 24.66, 23.65, 23.34, 22.87, 23.78, 22.9, 25.56, 25.51, 24.97, 23.63, 25.51, 25.43, 22.56, 22.66, 24.59, 22.25, 22.54, 25.7, 22.72, 24.28, 24.66, 24.38, 24.76, 22.25, 23.0, 26.89, 22.64, 22.65, 23.68, 23.43, 23.69, 22.77, 23.41, 24.18, 26.7, 23.2, 26.25, 24.5, 24.91, 26.27, 24.38, 24.4, 25.86, 24.12, 24.96, 26.17, 25.38, 25.83, 23.21, 23.06, 23.77, 24.49, 23.66, 24.88, 26.03, 23.55, 24.43, 24.98, 26.8, 23.51, 22.17, 23.64, 25.26, 24.63, 25.36, 24.94, 25.78, 23.56, 23.55, 22.85, 26.07, 24.57, 25.79, 22.45, 26.46, 23.8, 23.13, 24.54, 24.93, 24.84, 23.18, 24.67, 23.46, 23.88, 23.47, 23.02, 23.55, 23.98, 23.53, 23.45, 23.53, 24.4, 23.62, 25.87, 26.77, 25.66, 26.55, 25.2, 25.83, 26.2, 24.41, 23.02, 22.64, 25.9, 26.07, 26.73, 25.17, 23.49, 24.15, 25.38, 23.43, 24.47, 22.55, 24.64, 23.42, 24.88, 23.03, 23.67, 23.69, 25.02, 24.23, 23.86, 21.69, 26.53, 23.15, 23.92, 26.86, 26.05, 25.48, 26.06, 22.81, 27.03, 24.28, 24.9, 26.46, 24.85, 25.09, 22.46, 23.2, 23.46, 22.93, 23.98, 25.82, 23.35, 24.95, 22.95, 24.95, 25.26, 24.59, 25.07, 24.13, 25.66, 25.32, 25.3, 24.79, 25.2, 24.03, 26.3, 23.99, 24.52, 22.75, 23.59, 25.43, 24.23, 25.87, 24.55, 21.66, 22.79, 27.12, 24.16, 23.98, 23.2, 22.09, 22.88, 23.86, 24.34, 25.98, 23.18, 24.99, 26.62, 23.51, 24.55, 24.32, 23.29, 25.55, 24.48, 23.02, 24.29, 26.23, 23.86, 23.65, 22.5, 23.03, 24.84, 25.14, 24.07, 27.48, 24.56, 26.33, 26.03, 25.31, 25.32, 24.26, 22.49, 26.55, 25.16, 22.64, 24.07, 24.53, 22.02, 24.56, 24.96, 24.96, 22.34, 25.41, 27.26, 25.64, 23.96, 26.72, 23.64, 25.45, 21.71, 23.58, 25.03, 23.33, 23.49, 23.86, 24.74, 25.74, 22.87, 23.9, 23.87, 23.29, 23.01, 22.24, 24.91, 24.57, 24.07, 25.22, 24.43, 24.2, 23.81, 25.33, 25.61, 22.55, 24.1, 23.44, 23.89, 23.57, 23.78, 24.61, 25.0, 22.75, 23.31, 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差值(后−前) n=980：[-1.93, -1.5, -1.87, -2.67, -2.13, -2.56, -2.6, -2.73, -1.29, -3.12, -2.71, -2.62, -2.98, -1.51, -2.03, -1.62, -2.92, -1.87, -2.54, -1.89, -2.08, -2.27, -1.99, -2.16, -2.07, -2.62, -2.0, -1.79, -1.5, -2.02, -1.64, -2.12, -2.96, -1.3, -2.5, -1.2, -0.9, -2.32, -3.39, -1.75, -2.5, -2.99, -2.56, -1.92, -1.54, -2.08, -0.49, -2.43, -2.3, -1.74, -3.13, -2.49, -1.67, -2.51, -0.37, -2.05, -1.58, -1.63, -3.02, -1.75, -0.88, -2.71, -1.94, -1.49, -0.82, -1.44, -1.17, -2.09, -3.52, -0.81, -1.88, -2.52, -2.33, -2.82, -2.52, -0.94, -2.39, -2.32, -0.47, -1.33, -1.04, -1.86, -2.89, -3.45, -2.57, -2.61, -2.78, -1.51, -2.18, -4.04, -1.73, -2.75, -0.15, -1.79, -1.63, -2.02, -1.27, -2.21, -1.29, -1.26, -2.69, -1.63, -2.84, -1.28, -2.63, -2.42, -2.9, -2.14, -2.42, -2.0, -2.01, -2.21, -1.72, -1.97, -3.1, -2.94, -0.67, -2.18, -1.66, -1.79, -1.61, -0.16, -2.69, -0.57, -1.44, -2.36, -1.98, -2.77, -1.78, -1.6, -1.47, -2.31, -2.49, -1.78, -2.7, -2.23, -1.24, -1.53, -2.38, -1.56, -2.76, -0.95, -2.01, 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-3.17, -2.69, -1.73, -2.11, -2.12, -2.04, -0.57, -2.27, -2.23, -3.25, -1.32, -1.04, -1.78, -2.32, -2.55, -2.46, -1.25, -1.97, -1.36, -2.38, -1.33, -1.38, -2.1, -3.01, -2.05, -1.99, -1.31, -2.03, -0.89, -1.66, -2.21, -1.49, -2.44, -2.29, -1.5, -2.45, -2.29, -1.81, -1.99, -2.17, -1.17, -0.49, -2.32, -2.06, -2.87, -2.05, -1.46, -1.77, -2.02, -1.55, -2.51, -1.5, -2.37, -1.79, -2.61, -1.66, -1.67, -1.81, -2.1, -1.92, -2.75, -1.58, -3.0, -0.4, -2.16, -1.58, -1.98, -1.77, -2.79, -2.35, -2.58, -1.85, -2.66, -2.58, -2.38, -2.6, -2.92, -2.8, -1.31, -1.54, -2.47, -1.52, -2.39, -1.48, -1.78, -2.39, -1.47, -2.47, -1.63, -1.87, -3.15, -1.27, -2.06, -1.58, -1.53, -2.75, -2.7, -1.17, -0.55, -2.94, -2.18, -2.76, -1.73, -1.99, -2.59, -2.4, -2.84, -1.75, -2.18, -0.65, -1.04, -1.83, -2.1, -0.54, -2.83, -2.24, -1.38, -2.44, -1.45, -1.62, -1.71, -1.71, -2.41, -0.96, -1.42, -1.72, -1.54, -1.4, -1.89, -2.44, -1.85, -2.37, -1.22, -1.56, -2.11, -2.34, -2.05, -1.35, -2.12, -2.98, -1.27, -2.2, -2.54, -1.19, -2.12, -0.9, -0.88, -0.92, -1.67, -2.1, -2.38, -2.48, -2.12, -2.98, -1.37, -2.53, -2.02, -1.92, -1.93, -1.35, -1.56, -2.53, -1.09, -2.63, -2.91, -1.31, -2.44, -1.59, -2.56, -1.75, -2.19, -2.28, -1.49, -3.76, -1.94, -1.22, -2.58, -3.03, -2.34, -2.4, -2.39, -2.74, -2.43, -3.84, -2.58, -2.61, -2.03, -2.55, -1.19, -1.66, -1.68, -2.03, -1.25, -2.26, -2.04, -2.79, -2.45, -3.04, -1.13, -1.42, -1.0, -0.84, -1.6, -1.69, -2.49, -2.73, -0.87, -1.02, -2.42, -2.36, -2.57, -2.5, -3.03, -2.04, -0.87, -1.99, -1.34, -2.01, -2.64, -1.42, -1.85, -1.81, -1.77, -2.17, -2.11, -0.87, -2.09, -1.97, -2.49, -2.32, -1.87, -1.58, -1.94, -1.99, -1.98, -1.28, -1.01, -2.26, -2.25, -2.35, -1.93, -0.95, -3.59, -2.0, -2.13, -2.38, -0.62, -3.28, -2.72, -2.87, -2.34, -1.43, -2.59, -0.75, -2.09, -2.69, -1.54, -0.78, -1.31, -2.25, -2.61, -2.24, -2.0, -1.26, -1.92, -1.97, -1.57, -1.65, -1.83, -2.87, -2.75, -1.22, -2.11, -1.85, -2.43, -1.98, -1.78, -1.09, -0.11, -1.88, -1.28, -3.27, -1.97, -1.63, -2.38, -1.99, -2.84, -3.04, -0.98, -2.32, -3.1, -1.9, -1.69, -1.58, -2.32, -2.42, -2.79, -2.47, -1.82, -2.65, -1.68, -1.75, -0.34, -2.22, -2.18, -2.24, -3.1, -1.42, -1.47, -2.09, -2.03, -1.88, -1.99, -2.15, -2.52, -1.18, -2.26, -2.14, -2.12, -2.04, -3.15, -2.07, -1.91, -2.09, -2.91, -2.51, -2.06, -2.17, -1.22, -2.17, -2.15, -1.32, -2.23, -3.01, -2.1, -3.28, -2.72, -0.52, -1.07, -2.68, -2.92, -1.56, -2.29, -2.46, -2.25, -0.98, -2.12, -2.15, -1.76, -1.55, -1.65, -2.36, -2.1, -2.24, -2.52, -3.03, -2.26, -1.37, -1.98, -1.91, -2.03, -2.55, -2.76, -2.49, -2.12, -2.22, -1.93, -1.73, -1.62, -1.02, -2.2, -0.98, -3.74, -1.93, -2.33, -2.33, -1.94, -1.48, -0.76, -1.59, -2.93, -1.96, -1.8, -1.95, -1.4, -0.88, -1.66, -2.8, -1.77, -3.27, -2.33, -2.56, -2.66, -1.9, -1.75, -1.61, -2.04, -2.56, -2.18, -1.66, -2.57, -2.04, -1.63]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -2.0211，s_d = 0.6526，SE = s_d/√n = 0.0208；
t = d̄/SE = -96.9529；df = n−1 = 979；双侧 p = 0.000e+00。
d̄ = -2.0211，s_d = 0.6526，SE = s_d/√n = 0.0208；
t = d̄/SE = -96.9529；df = n−1 = 979；双侧 p = 0.000e+00。
计算结果
t = -96.9529，df = 979，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。检验健康管理效果。
案例 49. 前后生产效率
原始数据
前 n=557：[94.25, 90.03, 94.42, 93.23, 94.43, 86.12, 94.68, 93.42, 94.9, 94.55, 94.36, 88.26, 98.34, 91.57, 90.65, 89.01, 91.58, 92.21, 95.15, 90.09, 89.84, 88.98, 87.87, 94.88, 94.08, 91.58, 87.98, 89.91, 89.52, 92.77, 86.7, 94.25, 90.48, 94.22, 90.88, 93.24, 92.0, 90.14, 90.55, 95.63, 90.8, 87.79, 97.49, 91.16, 93.2, 88.0, 88.68, 88.96, 93.24, 88.96, 94.94, 91.13, 89.06, 91.89, 94.28, 93.44, 98.16, 93.56, 93.37, 97.45, 93.71, 96.61, 89.52, 92.64, 94.94, 95.45, 91.91, 91.17, 85.2, 85.98, 90.25, 92.7, 96.0, 85.97, 94.78, 96.43, 93.72, 95.09, 95.74, 88.42, 89.66, 94.76, 96.68, 90.95, 88.35, 90.57, 91.53, 93.98, 91.84, 90.08, 89.57, 87.65, 91.11, 90.51, 94.21, 93.58, 89.56, 97.63, 86.74, 90.74, 93.48, 88.01, 92.92, 91.78, 93.08, 96.96, 93.08, 95.38, 96.01, 98.74, 91.52, 93.05, 89.17, 94.53, 88.72, 88.55, 87.9, 94.19, 93.76, 94.24, 95.44, 97.17, 94.46, 95.06, 93.08, 92.59, 91.87, 100.82, 95.48, 98.88, 91.07, 92.66, 94.46, 90.9, 95.4, 90.53, 82.82, 91.77, 93.55, 91.56, 91.22, 92.96, 89.56, 90.26, 87.62, 93.8, 88.91, 88.65, 93.67, 90.93, 86.93, 89.2, 92.32, 88.75, 90.99, 95.85, 90.98, 96.42, 94.27, 89.23, 100.12, 87.97, 90.67, 91.04, 92.77, 94.46, 95.81, 90.17, 92.77, 94.19, 96.06, 87.83, 89.96, 91.26, 97.83, 91.67, 91.24, 91.25, 90.41, 94.6, 92.52, 92.11, 86.99, 90.84, 91.69, 91.33, 94.18, 95.93, 89.24, 90.16, 97.52, 92.39, 92.81, 94.24, 87.57, 100.73, 90.02, 88.87, 91.52, 88.24, 93.81, 93.91, 89.32, 89.66, 95.57, 92.88, 93.5, 90.84, 94.87, 90.81, 94.59, 86.61, 91.47, 97.48, 85.57, 93.22, 90.36, 94.12, 92.19, 90.73, 92.56, 92.77, 93.01, 96.26, 93.51, 89.3, 88.38, 92.02, 91.12, 95.44, 86.98, 93.53, 91.56, 93.42, 93.75, 91.97, 94.71, 92.11, 94.83, 99.69, 96.02, 93.47, 88.8, 92.9, 89.85, 90.79, 91.87, 93.21, 89.14, 90.83, 93.0, 89.76, 93.52, 90.91, 93.5, 91.92, 96.36, 98.28, 91.24, 95.06, 87.93, 94.08, 94.69, 88.07, 91.68, 98.59, 92.7, 95.65, 93.26, 90.55, 91.91, 94.32, 90.59, 87.27, 95.93, 86.33, 99.3, 93.71, 91.08, 90.9, 89.83, 90.14, 97.85, 88.97, 92.24, 96.75, 93.1, 93.65, 89.93, 92.81, 87.42, 93.9, 89.13, 94.11, 89.67, 94.03, 91.56, 93.89, 92.61, 93.22, 92.47, 96.85, 87.09, 89.32, 91.83, 92.36, 85.99, 90.18, 87.03, 89.35, 92.35, 96.67, 95.92, 95.58, 94.25, 87.0, 92.77, 94.14, 88.1, 92.21, 91.57, 93.6, 91.65, 90.42, 91.88, 96.44, 97.34, 90.03, 95.4, 87.8, 94.01, 85.46, 95.3, 88.08, 88.13, 90.58, 88.71, 96.2, 95.57, 97.46, 92.92, 91.57, 99.38, 92.99, 93.81, 92.66, 92.92, 91.78, 85.79, 87.64, 85.25, 94.42, 89.81, 89.2, 92.87, 94.59, 90.1, 91.47, 92.5, 91.3, 90.73, 92.68, 92.88, 94.37, 90.18, 93.96, 87.86, 87.56, 91.96, 100.15, 90.64, 94.31, 91.09, 89.08, 92.0, 91.12, 92.06, 92.86, 94.53, 91.23, 92.27, 95.55, 90.3, 91.4, 94.16, 91.23, 88.96, 93.47, 91.31, 91.55, 96.37, 97.1, 95.41, 89.2, 90.98, 96.34, 91.55, 93.08, 96.03, 91.64, 99.34, 95.14, 90.12, 97.74, 90.78, 87.99, 88.33, 87.49, 90.92, 93.03, 93.98, 91.42, 95.52, 90.74, 91.12, 94.44, 92.45, 92.43, 91.48, 92.92, 89.98, 97.82, 89.02, 96.1, 93.84, 94.31, 93.35, 93.36, 94.02, 91.43, 96.84, 88.11, 92.55, 91.47, 92.49, 92.82, 93.53, 93.93, 92.98, 89.22, 94.33, 92.97, 91.1, 91.27, 90.18, 92.63, 90.5, 93.49, 89.48, 95.98, 95.67, 91.03, 94.07, 94.51, 86.87, 93.46, 94.26, 95.47, 90.88, 88.03, 92.44, 91.71, 94.24, 94.27, 92.95, 95.59, 97.38, 95.45, 94.33, 94.01, 92.92, 92.57, 91.51, 94.88, 89.96, 91.81, 97.46, 94.4, 91.85, 92.3, 92.12, 95.66, 90.51, 93.68, 91.92, 91.72, 89.38, 92.42, 95.26, 96.39, 88.93, 94.58, 93.4, 86.92, 86.37, 93.66, 88.03, 91.01, 90.17, 92.62, 88.81, 85.45, 91.53, 93.93, 94.61, 88.23, 92.36, 94.86, 92.52, 91.79, 93.24, 90.84, 91.29, 90.63, 98.73, 94.08, 90.76, 97.09, 90.04, 96.74, 87.74, 93.32, 92.41, 94.97, 90.63, 90.3, 90.72, 89.97, 94.16, 95.26, 91.68, 95.34, 91.09, 92.86, 89.52, 93.16, 91.01, 88.08, 93.04, 89.8, 84.75, 93.33, 94.96, 95.56, 95.79, 96.78, 91.58, 97.85, 92.13, 84.08, 91.33, 97.15, 96.27, 89.86, 91.57, 95.24, 97.28]；
后 n=557：[105.86, 100.63, 105.45, 102.89, 104.58, 96.82, 105.81, 102.89, 105.24, 106.03, 104.88, 100.58, 108.22, 102.34, 101.65, 99.12, 101.4, 102.59, 106.07, 101.11, 100.87, 99.23, 98.25, 105.95, 105.24, 100.58, 100.47, 101.18, 100.45, 103.44, 96.41, 105.17, 101.71, 105.69, 101.71, 104.44, 102.43, 100.82, 101.23, 105.22, 102.03, 99.11, 108.55, 103.31, 104.23, 98.81, 100.43, 100.28, 103.16, 99.93, 106.27, 102.36, 99.45, 103.01, 105.08, 103.52, 109.56, 103.96, 103.87, 108.63, 105.23, 107.58, 100.25, 103.95, 104.72, 106.26, 103.0, 101.84, 96.28, 96.8, 100.53, 104.03, 106.26, 97.47, 106.78, 107.07, 103.9, 106.05, 105.52, 98.85, 100.62, 105.21, 107.37, 101.91, 99.62, 100.69, 102.48, 105.54, 103.44, 101.58, 100.15, 98.16, 101.69, 101.07, 104.58, 104.3, 100.54, 107.9, 96.55, 101.68, 103.43, 99.24, 103.56, 103.91, 104.72, 108.07, 102.09, 106.49, 106.6, 109.35, 102.72, 104.61, 101.01, 105.03, 100.19, 99.43, 99.04, 104.35, 105.01, 105.92, 105.99, 107.58, 106.23, 105.4, 103.7, 102.4, 102.51, 111.54, 107.07, 110.26, 101.65, 103.22, 105.21, 101.21, 106.51, 102.67, 94.41, 102.67, 104.37, 102.03, 102.77, 103.41, 101.36, 100.66, 98.57, 105.47, 100.38, 100.29, 104.29, 102.35, 97.23, 100.05, 104.03, 98.73, 102.48, 107.06, 101.78, 106.31, 105.34, 100.56, 111.9, 98.58, 102.06, 100.95, 103.77, 105.04, 106.99, 100.56, 104.39, 105.47, 105.99, 98.78, 100.57, 102.56, 109.88, 102.36, 102.67, 99.89, 100.25, 105.25, 103.51, 103.02, 96.89, 101.9, 102.28, 100.65, 104.44, 106.21, 99.74, 101.36, 108.69, 103.03, 103.49, 105.6, 98.4, 111.4, 100.47, 99.37, 102.11, 97.87, 104.46, 105.27, 100.6, 99.44, 106.25, 103.14, 104.12, 102.26, 104.94, 102.04, 106.32, 96.46, 102.28, 108.73, 97.07, 103.3, 101.44, 105.62, 103.06, 100.63, 104.21, 102.96, 104.58, 106.84, 103.87, 99.22, 98.97, 103.48, 102.98, 105.22, 97.75, 104.21, 101.48, 104.8, 104.48, 102.2, 106.8, 102.39, 105.93, 111.07, 106.32, 103.77, 99.68, 103.49, 101.44, 102.06, 102.24, 104.5, 100.66, 102.44, 103.49, 100.83, 103.53, 102.25, 103.3, 103.0, 106.69, 110.96, 101.02, 106.01, 98.77, 105.65, 105.51, 98.53, 102.26, 108.64, 104.06, 106.81, 103.83, 101.27, 103.55, 105.13, 101.0, 98.58, 106.65, 96.77, 110.11, 104.56, 102.07, 101.08, 100.6, 100.72, 109.24, 100.39, 102.78, 107.8, 103.32, 104.03, 100.75, 104.13, 98.18, 104.51, 99.6, 105.78, 100.21, 103.31, 102.33, 103.35, 103.34, 103.31, 103.04, 107.77, 96.43, 99.75, 102.93, 102.05, 96.92, 100.95, 97.83, 100.43, 102.76, 107.62, 105.28, 107.52, 105.86, 96.8, 103.54, 104.43, 98.75, 103.49, 102.52, 105.31, 102.8, 101.25, 102.24, 106.89, 108.91, 100.49, 106.47, 99.16, 105.19, 95.44, 106.86, 99.03, 98.93, 101.23, 99.12, 106.56, 106.58, 108.6, 103.86, 101.8, 110.42, 104.1, 103.99, 103.5, 103.49, 101.81, 96.2, 97.35, 97.26, 105.25, 100.62, 100.57, 103.6, 105.64, 100.32, 103.38, 102.44, 102.7, 101.87, 103.26, 103.21, 105.09, 100.55, 104.77, 97.9, 97.96, 102.7, 110.69, 102.38, 104.16, 101.1, 99.3, 103.26, 102.24, 102.84, 103.93, 106.11, 101.51, 103.23, 105.8, 100.51, 102.34, 105.43, 101.79, 99.47, 104.91, 101.86, 101.14, 107.9, 108.1, 106.21, 99.04, 102.71, 107.4, 102.98, 103.98, 105.79, 101.96, 109.87, 105.32, 99.94, 108.66, 101.42, 99.25, 99.2, 98.05, 102.24, 104.55, 104.19, 102.53, 106.29, 100.62, 100.94, 104.95, 102.73, 102.47, 103.13, 103.98, 101.59, 108.18, 99.76, 106.52, 104.63, 105.28, 103.84, 105.05, 104.17, 101.36, 108.51, 100.21, 103.8, 102.3, 103.41, 103.47, 105.34, 105.53, 102.89, 99.66, 105.0, 103.19, 101.49, 101.85, 100.51, 103.87, 101.26, 105.85, 100.17, 106.31, 106.43, 101.79, 105.11, 105.51, 97.55, 104.48, 104.6, 106.33, 101.81, 99.48, 102.39, 103.22, 104.54, 104.42, 105.21, 106.23, 109.33, 105.94, 104.87, 104.55, 103.57, 103.12, 102.3, 104.16, 100.89, 102.31, 107.88, 105.16, 102.74, 102.71, 103.23, 105.67, 101.09, 103.97, 103.18, 102.48, 99.74, 104.01, 106.22, 106.99, 100.09, 105.43, 105.79, 98.16, 97.76, 104.16, 98.72, 102.3, 102.01, 103.35, 98.87, 96.36, 102.27, 105.24, 106.16, 98.45, 102.94, 104.94, 103.52, 102.73, 104.35, 100.83, 101.93, 100.73, 109.91, 105.1, 101.27, 108.01, 100.8, 106.78, 98.6, 104.06, 102.98, 105.56, 100.37, 101.11, 101.2, 100.68, 105.83, 105.29, 102.23, 105.24, 102.13, 104.24, 99.91, 104.12, 101.84, 99.08, 103.3, 99.94, 96.61, 104.25, 105.54, 106.75, 106.77, 107.52, 102.63, 108.02, 102.63, 94.7, 102.07, 108.17, 106.21, 100.42, 102.5, 105.61, 108.8]；
差值(后−前) n=557：[11.6, 10.6, 11.02, 9.66, 10.14, 10.7, 11.13, 9.47, 10.34, 11.48, 10.53, 12.32, 9.88, 10.77, 11.0, 10.11, 9.82, 10.38, 10.92, 11.02, 11.02, 10.25, 10.38, 11.07, 11.17, 9.0, 12.49, 11.28, 10.94, 10.67, 9.71, 10.93, 11.23, 11.46, 10.83, 11.2, 10.44, 10.68, 10.68, 9.58, 11.24, 11.33, 11.06, 12.15, 11.03, 10.81, 11.75, 11.32, 9.92, 10.97, 11.33, 11.24, 10.39, 11.12, 10.8, 10.08, 11.4, 10.4, 10.5, 11.18, 11.52, 10.97, 10.73, 11.31, 9.78, 10.81, 11.09, 10.67, 11.08, 10.82, 10.29, 11.33, 10.25, 11.5, 12.0, 10.64, 10.18, 10.96, 9.78, 10.43, 10.97, 10.45, 10.7, 10.96, 11.27, 10.12, 10.95, 11.56, 11.6, 11.5, 10.59, 10.51, 10.58, 10.56, 10.37, 10.72, 10.98, 10.27, 9.81, 10.94, 9.95, 11.23, 10.64, 12.13, 11.63, 11.1, 9.02, 11.11, 10.59, 10.6, 11.2, 11.56, 11.84, 10.5, 11.47, 10.87, 11.14, 10.16, 11.25, 11.68, 10.55, 10.41, 11.77, 10.34, 10.62, 9.81, 10.64, 10.71, 11.58, 11.38, 10.59, 10.56, 10.74, 10.31, 11.12, 12.14, 11.59, 10.9, 10.83, 10.47, 11.54, 10.45, 11.8, 10.4, 10.95, 11.66, 11.47, 11.64, 10.62, 11.42, 10.3, 10.85, 11.71, 9.98, 11.49, 11.21, 10.79, 9.89, 11.07, 11.34, 11.79, 10.61, 11.39, 9.9, 11.0, 10.59, 11.18, 10.39, 11.62, 11.28, 9.93, 10.95, 10.61, 11.3, 12.05, 10.7, 11.43, 8.64, 9.84, 10.65, 10.99, 10.91, 9.89, 11.06, 10.59, 9.33, 10.26, 10.28, 10.5, 11.21, 11.17, 10.64, 10.68, 11.36, 10.83, 10.66, 10.45, 10.5, 10.59, 9.63, 10.65, 11.36, 11.28, 9.78, 10.68, 10.26, 10.62, 11.42, 10.07, 11.23, 11.72, 9.84, 10.81, 11.25, 11.5, 10.08, 11.09, 11.5, 10.87, 9.9, 11.66, 10.19, 11.57, 10.59, 10.36, 9.92, 10.59, 11.46, 11.86, 9.78, 10.78, 10.68, 9.92, 11.38, 10.73, 10.22, 12.09, 10.27, 11.11, 11.38, 10.3, 10.3, 10.88, 10.59, 11.59, 11.27, 10.37, 11.28, 11.52, 11.61, 10.49, 11.06, 10.0, 11.34, 9.81, 11.07, 10.33, 12.68, 9.77, 10.95, 10.85, 11.57, 10.81, 10.45, 10.58, 10.05, 11.37, 11.15, 10.57, 10.71, 11.64, 10.82, 10.4, 11.31, 10.72, 10.44, 10.81, 10.84, 10.99, 10.18, 10.77, 10.58, 11.39, 11.42, 10.53, 11.05, 10.22, 10.38, 10.82, 11.32, 10.76, 10.61, 10.47, 11.67, 10.54, 9.28, 10.77, 9.45, 10.72, 10.09, 10.57, 10.92, 9.34, 10.44, 11.1, 9.69, 10.92, 10.77, 10.81, 11.08, 10.42, 10.95, 9.35, 11.95, 11.61, 9.8, 10.77, 10.29, 10.65, 11.29, 10.95, 11.71, 11.16, 10.83, 10.37, 10.45, 11.57, 10.46, 11.08, 11.37, 11.18, 9.98, 11.56, 10.95, 10.8, 10.66, 10.41, 10.36, 11.01, 11.13, 10.94, 10.23, 11.04, 11.12, 10.17, 10.84, 10.57, 10.03, 10.41, 9.71, 12.0, 10.82, 10.81, 11.37, 10.73, 11.05, 10.21, 11.91, 9.94, 11.4, 11.14, 10.58, 10.33, 10.72, 10.37, 10.81, 10.04, 10.4, 10.74, 10.53, 11.74, 9.85, 10.01, 10.22, 11.27, 11.13, 10.78, 11.07, 11.58, 10.28, 10.95, 10.25, 10.21, 10.94, 11.27, 10.56, 10.51, 11.44, 10.55, 9.6, 11.53, 11.0, 10.81, 9.84, 11.73, 11.06, 11.43, 10.9, 9.77, 10.32, 10.54, 10.17, 9.82, 10.92, 10.65, 11.26, 10.87, 10.56, 11.32, 11.52, 10.2, 11.11, 10.77, 9.89, 9.82, 10.51, 10.29, 10.04, 11.65, 11.06, 11.61, 10.36, 10.74, 10.42, 10.78, 10.96, 10.49, 11.69, 10.15, 9.93, 11.67, 12.1, 11.26, 10.84, 10.92, 10.66, 11.81, 11.61, 9.91, 10.43, 10.67, 10.22, 10.39, 10.58, 10.33, 11.24, 10.76, 12.36, 10.69, 10.33, 10.76, 10.76, 11.04, 11.0, 10.69, 11.02, 10.34, 10.85, 10.93, 11.46, 9.95, 11.51, 10.3, 10.16, 12.26, 10.64, 11.95, 10.49, 10.54, 10.53, 10.65, 10.55, 10.79, 9.28, 10.92, 10.5, 10.43, 10.76, 10.89, 10.41, 11.1, 10.01, 10.58, 10.29, 11.26, 10.76, 10.36, 11.59, 10.96, 10.6, 11.16, 10.85, 12.39, 11.23, 11.39, 10.5, 10.68, 11.29, 11.83, 10.73, 10.06, 10.91, 10.74, 11.31, 11.55, 10.22, 10.58, 10.09, 11.0, 10.93, 11.12, 9.99, 10.63, 10.09, 11.18, 11.01, 10.51, 10.92, 10.76, 10.04, 10.86, 10.74, 10.56, 10.59, 9.74, 10.81, 10.47, 10.7, 11.67, 10.04, 10.55, 9.9, 11.03, 11.37, 10.39, 10.96, 10.84, 10.99, 10.27, 10.14, 11.86, 10.92, 10.57, 11.19, 10.98, 10.74, 11.05, 10.16, 10.5, 10.62, 10.74, 11.02, 9.95, 10.55, 10.93, 10.38, 11.52]
后 n=557：[105.86, 100.63, 105.45, 102.89, 104.58, 96.82, 105.81, 102.89, 105.24, 106.03, 104.88, 100.58, 108.22, 102.34, 101.65, 99.12, 101.4, 102.59, 106.07, 101.11, 100.87, 99.23, 98.25, 105.95, 105.24, 100.58, 100.47, 101.18, 100.45, 103.44, 96.41, 105.17, 101.71, 105.69, 101.71, 104.44, 102.43, 100.82, 101.23, 105.22, 102.03, 99.11, 108.55, 103.31, 104.23, 98.81, 100.43, 100.28, 103.16, 99.93, 106.27, 102.36, 99.45, 103.01, 105.08, 103.52, 109.56, 103.96, 103.87, 108.63, 105.23, 107.58, 100.25, 103.95, 104.72, 106.26, 103.0, 101.84, 96.28, 96.8, 100.53, 104.03, 106.26, 97.47, 106.78, 107.07, 103.9, 106.05, 105.52, 98.85, 100.62, 105.21, 107.37, 101.91, 99.62, 100.69, 102.48, 105.54, 103.44, 101.58, 100.15, 98.16, 101.69, 101.07, 104.58, 104.3, 100.54, 107.9, 96.55, 101.68, 103.43, 99.24, 103.56, 103.91, 104.72, 108.07, 102.09, 106.49, 106.6, 109.35, 102.72, 104.61, 101.01, 105.03, 100.19, 99.43, 99.04, 104.35, 105.01, 105.92, 105.99, 107.58, 106.23, 105.4, 103.7, 102.4, 102.51, 111.54, 107.07, 110.26, 101.65, 103.22, 105.21, 101.21, 106.51, 102.67, 94.41, 102.67, 104.37, 102.03, 102.77, 103.41, 101.36, 100.66, 98.57, 105.47, 100.38, 100.29, 104.29, 102.35, 97.23, 100.05, 104.03, 98.73, 102.48, 107.06, 101.78, 106.31, 105.34, 100.56, 111.9, 98.58, 102.06, 100.95, 103.77, 105.04, 106.99, 100.56, 104.39, 105.47, 105.99, 98.78, 100.57, 102.56, 109.88, 102.36, 102.67, 99.89, 100.25, 105.25, 103.51, 103.02, 96.89, 101.9, 102.28, 100.65, 104.44, 106.21, 99.74, 101.36, 108.69, 103.03, 103.49, 105.6, 98.4, 111.4, 100.47, 99.37, 102.11, 97.87, 104.46, 105.27, 100.6, 99.44, 106.25, 103.14, 104.12, 102.26, 104.94, 102.04, 106.32, 96.46, 102.28, 108.73, 97.07, 103.3, 101.44, 105.62, 103.06, 100.63, 104.21, 102.96, 104.58, 106.84, 103.87, 99.22, 98.97, 103.48, 102.98, 105.22, 97.75, 104.21, 101.48, 104.8, 104.48, 102.2, 106.8, 102.39, 105.93, 111.07, 106.32, 103.77, 99.68, 103.49, 101.44, 102.06, 102.24, 104.5, 100.66, 102.44, 103.49, 100.83, 103.53, 102.25, 103.3, 103.0, 106.69, 110.96, 101.02, 106.01, 98.77, 105.65, 105.51, 98.53, 102.26, 108.64, 104.06, 106.81, 103.83, 101.27, 103.55, 105.13, 101.0, 98.58, 106.65, 96.77, 110.11, 104.56, 102.07, 101.08, 100.6, 100.72, 109.24, 100.39, 102.78, 107.8, 103.32, 104.03, 100.75, 104.13, 98.18, 104.51, 99.6, 105.78, 100.21, 103.31, 102.33, 103.35, 103.34, 103.31, 103.04, 107.77, 96.43, 99.75, 102.93, 102.05, 96.92, 100.95, 97.83, 100.43, 102.76, 107.62, 105.28, 107.52, 105.86, 96.8, 103.54, 104.43, 98.75, 103.49, 102.52, 105.31, 102.8, 101.25, 102.24, 106.89, 108.91, 100.49, 106.47, 99.16, 105.19, 95.44, 106.86, 99.03, 98.93, 101.23, 99.12, 106.56, 106.58, 108.6, 103.86, 101.8, 110.42, 104.1, 103.99, 103.5, 103.49, 101.81, 96.2, 97.35, 97.26, 105.25, 100.62, 100.57, 103.6, 105.64, 100.32, 103.38, 102.44, 102.7, 101.87, 103.26, 103.21, 105.09, 100.55, 104.77, 97.9, 97.96, 102.7, 110.69, 102.38, 104.16, 101.1, 99.3, 103.26, 102.24, 102.84, 103.93, 106.11, 101.51, 103.23, 105.8, 100.51, 102.34, 105.43, 101.79, 99.47, 104.91, 101.86, 101.14, 107.9, 108.1, 106.21, 99.04, 102.71, 107.4, 102.98, 103.98, 105.79, 101.96, 109.87, 105.32, 99.94, 108.66, 101.42, 99.25, 99.2, 98.05, 102.24, 104.55, 104.19, 102.53, 106.29, 100.62, 100.94, 104.95, 102.73, 102.47, 103.13, 103.98, 101.59, 108.18, 99.76, 106.52, 104.63, 105.28, 103.84, 105.05, 104.17, 101.36, 108.51, 100.21, 103.8, 102.3, 103.41, 103.47, 105.34, 105.53, 102.89, 99.66, 105.0, 103.19, 101.49, 101.85, 100.51, 103.87, 101.26, 105.85, 100.17, 106.31, 106.43, 101.79, 105.11, 105.51, 97.55, 104.48, 104.6, 106.33, 101.81, 99.48, 102.39, 103.22, 104.54, 104.42, 105.21, 106.23, 109.33, 105.94, 104.87, 104.55, 103.57, 103.12, 102.3, 104.16, 100.89, 102.31, 107.88, 105.16, 102.74, 102.71, 103.23, 105.67, 101.09, 103.97, 103.18, 102.48, 99.74, 104.01, 106.22, 106.99, 100.09, 105.43, 105.79, 98.16, 97.76, 104.16, 98.72, 102.3, 102.01, 103.35, 98.87, 96.36, 102.27, 105.24, 106.16, 98.45, 102.94, 104.94, 103.52, 102.73, 104.35, 100.83, 101.93, 100.73, 109.91, 105.1, 101.27, 108.01, 100.8, 106.78, 98.6, 104.06, 102.98, 105.56, 100.37, 101.11, 101.2, 100.68, 105.83, 105.29, 102.23, 105.24, 102.13, 104.24, 99.91, 104.12, 101.84, 99.08, 103.3, 99.94, 96.61, 104.25, 105.54, 106.75, 106.77, 107.52, 102.63, 108.02, 102.63, 94.7, 102.07, 108.17, 106.21, 100.42, 102.5, 105.61, 108.8]；
差值(后−前) n=557：[11.6, 10.6, 11.02, 9.66, 10.14, 10.7, 11.13, 9.47, 10.34, 11.48, 10.53, 12.32, 9.88, 10.77, 11.0, 10.11, 9.82, 10.38, 10.92, 11.02, 11.02, 10.25, 10.38, 11.07, 11.17, 9.0, 12.49, 11.28, 10.94, 10.67, 9.71, 10.93, 11.23, 11.46, 10.83, 11.2, 10.44, 10.68, 10.68, 9.58, 11.24, 11.33, 11.06, 12.15, 11.03, 10.81, 11.75, 11.32, 9.92, 10.97, 11.33, 11.24, 10.39, 11.12, 10.8, 10.08, 11.4, 10.4, 10.5, 11.18, 11.52, 10.97, 10.73, 11.31, 9.78, 10.81, 11.09, 10.67, 11.08, 10.82, 10.29, 11.33, 10.25, 11.5, 12.0, 10.64, 10.18, 10.96, 9.78, 10.43, 10.97, 10.45, 10.7, 10.96, 11.27, 10.12, 10.95, 11.56, 11.6, 11.5, 10.59, 10.51, 10.58, 10.56, 10.37, 10.72, 10.98, 10.27, 9.81, 10.94, 9.95, 11.23, 10.64, 12.13, 11.63, 11.1, 9.02, 11.11, 10.59, 10.6, 11.2, 11.56, 11.84, 10.5, 11.47, 10.87, 11.14, 10.16, 11.25, 11.68, 10.55, 10.41, 11.77, 10.34, 10.62, 9.81, 10.64, 10.71, 11.58, 11.38, 10.59, 10.56, 10.74, 10.31, 11.12, 12.14, 11.59, 10.9, 10.83, 10.47, 11.54, 10.45, 11.8, 10.4, 10.95, 11.66, 11.47, 11.64, 10.62, 11.42, 10.3, 10.85, 11.71, 9.98, 11.49, 11.21, 10.79, 9.89, 11.07, 11.34, 11.79, 10.61, 11.39, 9.9, 11.0, 10.59, 11.18, 10.39, 11.62, 11.28, 9.93, 10.95, 10.61, 11.3, 12.05, 10.7, 11.43, 8.64, 9.84, 10.65, 10.99, 10.91, 9.89, 11.06, 10.59, 9.33, 10.26, 10.28, 10.5, 11.21, 11.17, 10.64, 10.68, 11.36, 10.83, 10.66, 10.45, 10.5, 10.59, 9.63, 10.65, 11.36, 11.28, 9.78, 10.68, 10.26, 10.62, 11.42, 10.07, 11.23, 11.72, 9.84, 10.81, 11.25, 11.5, 10.08, 11.09, 11.5, 10.87, 9.9, 11.66, 10.19, 11.57, 10.59, 10.36, 9.92, 10.59, 11.46, 11.86, 9.78, 10.78, 10.68, 9.92, 11.38, 10.73, 10.22, 12.09, 10.27, 11.11, 11.38, 10.3, 10.3, 10.88, 10.59, 11.59, 11.27, 10.37, 11.28, 11.52, 11.61, 10.49, 11.06, 10.0, 11.34, 9.81, 11.07, 10.33, 12.68, 9.77, 10.95, 10.85, 11.57, 10.81, 10.45, 10.58, 10.05, 11.37, 11.15, 10.57, 10.71, 11.64, 10.82, 10.4, 11.31, 10.72, 10.44, 10.81, 10.84, 10.99, 10.18, 10.77, 10.58, 11.39, 11.42, 10.53, 11.05, 10.22, 10.38, 10.82, 11.32, 10.76, 10.61, 10.47, 11.67, 10.54, 9.28, 10.77, 9.45, 10.72, 10.09, 10.57, 10.92, 9.34, 10.44, 11.1, 9.69, 10.92, 10.77, 10.81, 11.08, 10.42, 10.95, 9.35, 11.95, 11.61, 9.8, 10.77, 10.29, 10.65, 11.29, 10.95, 11.71, 11.16, 10.83, 10.37, 10.45, 11.57, 10.46, 11.08, 11.37, 11.18, 9.98, 11.56, 10.95, 10.8, 10.66, 10.41, 10.36, 11.01, 11.13, 10.94, 10.23, 11.04, 11.12, 10.17, 10.84, 10.57, 10.03, 10.41, 9.71, 12.0, 10.82, 10.81, 11.37, 10.73, 11.05, 10.21, 11.91, 9.94, 11.4, 11.14, 10.58, 10.33, 10.72, 10.37, 10.81, 10.04, 10.4, 10.74, 10.53, 11.74, 9.85, 10.01, 10.22, 11.27, 11.13, 10.78, 11.07, 11.58, 10.28, 10.95, 10.25, 10.21, 10.94, 11.27, 10.56, 10.51, 11.44, 10.55, 9.6, 11.53, 11.0, 10.81, 9.84, 11.73, 11.06, 11.43, 10.9, 9.77, 10.32, 10.54, 10.17, 9.82, 10.92, 10.65, 11.26, 10.87, 10.56, 11.32, 11.52, 10.2, 11.11, 10.77, 9.89, 9.82, 10.51, 10.29, 10.04, 11.65, 11.06, 11.61, 10.36, 10.74, 10.42, 10.78, 10.96, 10.49, 11.69, 10.15, 9.93, 11.67, 12.1, 11.26, 10.84, 10.92, 10.66, 11.81, 11.61, 9.91, 10.43, 10.67, 10.22, 10.39, 10.58, 10.33, 11.24, 10.76, 12.36, 10.69, 10.33, 10.76, 10.76, 11.04, 11.0, 10.69, 11.02, 10.34, 10.85, 10.93, 11.46, 9.95, 11.51, 10.3, 10.16, 12.26, 10.64, 11.95, 10.49, 10.54, 10.53, 10.65, 10.55, 10.79, 9.28, 10.92, 10.5, 10.43, 10.76, 10.89, 10.41, 11.1, 10.01, 10.58, 10.29, 11.26, 10.76, 10.36, 11.59, 10.96, 10.6, 11.16, 10.85, 12.39, 11.23, 11.39, 10.5, 10.68, 11.29, 11.83, 10.73, 10.06, 10.91, 10.74, 11.31, 11.55, 10.22, 10.58, 10.09, 11.0, 10.93, 11.12, 9.99, 10.63, 10.09, 11.18, 11.01, 10.51, 10.92, 10.76, 10.04, 10.86, 10.74, 10.56, 10.59, 9.74, 10.81, 10.47, 10.7, 11.67, 10.04, 10.55, 9.9, 11.03, 11.37, 10.39, 10.96, 10.84, 10.99, 10.27, 10.14, 11.86, 10.92, 10.57, 11.19, 10.98, 10.74, 11.05, 10.16, 10.5, 10.62, 10.74, 11.02, 9.95, 10.55, 10.93, 10.38, 11.52]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = 10.7863，s_d = 0.5989，SE = s_d/√n = 0.0254；
t = d̄/SE = 425.0713；df = n−1 = 556；双侧 p = 0.000e+00。
d̄ = 10.7863，s_d = 0.5989，SE = s_d/√n = 0.0254；
t = d̄/SE = 425.0713；df = n−1 = 556；双侧 p = 0.000e+00。
计算结果
t = 425.0713，df = 556，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。评估流程优化成效。
案例 50. 术前术后指标
原始数据
前 n=538：[118.98, 117.14, 130.47, 122.41, 129.89, 120.58, 121.11, 123.15, 125.1, 118.66, 122.87, 125.08, 122.64, 123.83, 119.22, 123.88, 121.08, 126.04, 125.78, 119.55, 121.36, 118.91, 122.41, 120.25, 122.7, 121.44, 123.72, 122.79, 123.91, 116.58, 128.78, 121.14, 116.99, 121.25, 125.21, 125.39, 123.16, 123.52, 124.3, 123.49, 118.56, 121.13, 117.52, 118.28, 128.94, 123.93, 120.44, 127.65, 126.13, 126.88, 121.73, 125.2, 119.6, 122.67, 122.93, 120.26, 120.44, 125.07, 127.54, 128.56, 119.75, 113.84, 124.15, 124.27, 121.05, 124.21, 122.15, 121.45, 123.65, 122.22, 120.6, 121.19, 121.97, 124.18, 121.72, 120.97, 120.93, 124.57, 120.99, 117.92, 117.78, 127.2, 125.02, 122.02, 120.46, 118.14, 117.4, 119.44, 123.75, 122.85, 122.35, 119.48, 120.66, 119.51, 120.64, 120.42, 122.83, 125.26, 129.46, 126.21, 125.24, 120.71, 118.62, 115.12, 121.61, 121.68, 119.13, 126.06, 122.79, 124.52, 130.12, 125.6, 126.49, 120.96, 121.01, 123.67, 126.97, 126.18, 121.37, 125.98, 124.38, 126.2, 119.64, 119.69, 118.61, 120.77, 127.09, 121.72, 123.41, 123.81, 125.08, 124.41, 125.69, 124.23, 121.66, 124.28, 117.98, 115.68, 119.41, 127.76, 120.52, 126.79, 124.33, 127.01, 112.45, 122.05, 119.21, 126.6, 122.44, 118.67, 120.86, 122.46, 124.98, 117.91, 118.22, 121.58, 125.54, 121.72, 121.49, 124.99, 125.51, 121.67, 124.53, 121.72, 122.23, 122.64, 118.97, 124.2, 123.1, 122.0, 123.73, 122.43, 127.99, 123.71, 117.04, 121.91, 123.63, 126.69, 124.32, 119.52, 120.41, 117.57, 122.69, 116.54, 121.48, 126.57, 127.4, 127.2, 118.31, 122.56, 118.72, 120.33, 121.68, 119.36, 123.48, 118.45, 119.04, 117.95, 118.28, 123.04, 124.96, 118.52, 123.25, 126.76, 119.91, 125.7, 128.37, 125.07, 123.25, 122.29, 122.2, 122.54, 119.52, 125.7, 121.03, 124.25, 120.94, 128.58, 125.38, 123.09, 122.01, 117.3, 123.75, 125.04, 123.15, 114.55, 124.23, 119.69, 120.14, 126.12, 122.31, 126.47, 123.05, 120.25, 122.59, 120.09, 121.56, 121.94, 119.76, 118.89, 121.74, 120.27, 128.68, 123.02, 116.81, 129.39, 123.45, 121.09, 126.73, 125.17, 122.57, 119.03, 119.67, 122.57, 122.62, 121.21, 121.68, 120.77, 126.97, 121.84, 122.24, 122.58, 123.95, 120.16, 127.22, 122.98, 119.03, 124.81, 122.01, 119.06, 124.85, 121.71, 119.76, 122.95, 124.4, 119.3, 122.89, 123.4, 119.63, 117.45, 127.34, 121.19, 121.75, 121.59, 120.5, 124.87, 122.62, 120.64, 120.92, 123.28, 124.94, 126.69, 121.95, 125.41, 119.28, 122.56, 120.98, 123.03, 121.47, 125.05, 119.58, 121.55, 121.48, 122.44, 124.54, 124.42, 119.38, 122.89, 120.58, 121.67, 119.72, 124.5, 117.32, 126.87, 127.08, 126.03, 120.1, 119.08, 126.53, 128.42, 121.88, 123.98, 114.12, 119.39, 119.39, 121.7, 125.33, 121.4, 124.83, 119.53, 118.06, 125.31, 125.22, 118.98, 117.84, 121.21, 125.62, 124.4, 116.11, 121.81, 117.74, 119.57, 125.48, 120.01, 119.5, 117.36, 118.39, 126.31, 122.6, 127.83, 125.64, 122.38, 122.4, 125.66, 119.66, 120.52, 120.67, 119.47, 123.42, 124.43, 122.51, 125.33, 122.26, 127.73, 118.54, 126.88, 119.27, 124.93, 117.49, 119.56, 123.71, 122.52, 124.11, 121.5, 123.93, 123.12, 124.44, 122.76, 120.06, 118.64, 125.5, 125.4, 126.64, 118.19, 123.17, 122.42, 121.9, 125.13, 128.46, 119.78, 122.68, 119.48, 121.27, 129.54, 121.81, 123.03, 120.53, 115.01, 121.82, 119.84, 122.7, 120.84, 126.19, 122.96, 122.43, 122.65, 117.68, 120.43, 130.23, 121.27, 113.8, 121.49, 119.87, 124.04, 123.47, 127.02, 117.95, 121.9, 128.34, 120.68, 120.39, 123.44, 121.64, 122.98, 122.45, 123.65, 117.03, 119.27, 124.94, 124.45, 125.39, 126.0, 119.77, 120.97, 119.43, 116.01, 119.54, 123.69, 120.35, 119.65, 122.97, 115.57, 128.61, 125.71, 122.73, 122.1, 126.97, 118.35, 122.78, 117.11, 118.1, 119.67, 119.38, 126.31, 122.97, 121.36, 122.61, 120.64, 118.63, 123.78, 122.91, 132.96, 120.42, 118.72, 124.71, 124.76, 120.37, 122.28, 125.15, 127.72, 122.02, 125.98, 120.06, 124.91, 119.81, 120.88, 118.36, 121.95, 126.66, 120.27, 117.71, 119.1, 122.33, 119.51, 124.42, 121.97, 122.56, 118.49, 118.2, 116.5, 122.09, 115.51, 120.67, 126.16, 119.21, 122.54, 122.26, 122.61, 121.78, 123.4, 119.19, 120.57, 120.36, 121.97, 122.38, 115.92, 123.94, 125.4, 125.3, 124.74, 124.61, 124.07, 127.96, 115.91, 120.71, 123.43, 128.23, 124.17, 124.87, 121.03, 126.42, 124.14, 121.17, 123.31, 119.56, 120.12, 119.51, 130.76, 123.05, 121.22, 123.1, 118.65, 119.21, 122.71, 114.17, 113.54, 123.68, 121.31]；
后 n=538：[110.72, 108.2, 119.05, 113.92, 119.19, 108.94, 111.56, 114.27, 114.7, 108.4, 112.63, 114.57, 114.29, 114.67, 110.28, 113.61, 109.33, 116.18, 117.11, 112.58, 112.02, 108.92, 113.44, 111.14, 112.92, 110.46, 114.31, 114.33, 114.07, 105.75, 117.97, 111.11, 108.6, 112.09, 116.54, 115.28, 113.58, 113.67, 114.64, 113.46, 108.66, 113.7, 105.95, 109.85, 121.86, 114.79, 111.31, 118.56, 115.83, 118.43, 111.84, 115.93, 109.61, 113.34, 113.07, 110.7, 110.53, 115.12, 117.11, 118.24, 110.84, 104.54, 113.67, 114.94, 109.81, 112.8, 112.81, 111.26, 113.23, 113.55, 110.37, 110.4, 112.86, 114.26, 112.25, 110.54, 110.5, 115.14, 112.17, 107.88, 109.09, 117.88, 114.16, 113.45, 110.54, 107.49, 107.56, 109.43, 113.12, 113.96, 110.26, 109.96, 111.43, 109.09, 111.6, 109.51, 114.93, 115.86, 118.81, 116.51, 116.68, 110.45, 108.59, 105.53, 110.05, 111.17, 108.76, 115.67, 113.76, 114.96, 119.5, 117.46, 116.36, 112.48, 110.9, 114.09, 117.45, 116.8, 111.36, 116.93, 112.99, 116.68, 111.0, 109.96, 109.54, 110.52, 117.03, 112.94, 114.63, 114.88, 114.43, 115.93, 117.52, 113.44, 113.38, 112.2, 107.98, 106.53, 109.8, 118.73, 110.57, 117.34, 112.86, 117.64, 104.82, 113.03, 109.13, 118.1, 112.19, 110.21, 112.04, 111.77, 115.79, 107.61, 105.9, 113.33, 117.2, 112.9, 110.82, 116.5, 117.03, 112.95, 113.34, 112.31, 112.06, 113.71, 108.74, 114.32, 112.66, 111.98, 114.69, 112.22, 116.99, 115.06, 108.22, 111.14, 113.4, 117.94, 115.55, 110.02, 110.53, 109.65, 114.16, 106.86, 110.54, 117.18, 118.83, 118.45, 107.78, 116.36, 109.14, 109.79, 111.06, 110.56, 115.17, 107.96, 109.58, 108.7, 106.99, 112.57, 115.04, 107.43, 114.03, 116.7, 111.14, 116.52, 118.29, 114.84, 114.01, 114.4, 112.03, 113.16, 110.68, 115.96, 112.16, 113.52, 111.28, 119.28, 114.61, 114.6, 113.63, 106.29, 113.33, 116.19, 113.74, 105.22, 114.84, 109.32, 110.78, 116.7, 111.47, 116.88, 116.77, 110.22, 113.7, 109.9, 112.93, 111.72, 110.23, 108.1, 113.22, 111.22, 120.16, 115.2, 108.93, 120.47, 114.13, 109.66, 118.02, 114.79, 113.82, 110.64, 110.13, 111.48, 113.81, 112.73, 111.83, 112.04, 116.73, 113.3, 111.95, 114.06, 114.42, 111.06, 117.4, 113.37, 109.54, 114.4, 112.37, 110.43, 116.01, 112.59, 113.02, 112.38, 115.26, 107.82, 113.43, 113.96, 109.69, 108.1, 118.12, 111.52, 111.16, 111.48, 110.58, 114.96, 115.21, 111.2, 110.93, 114.41, 115.77, 116.83, 112.8, 114.38, 110.89, 114.28, 111.92, 113.69, 111.81, 115.49, 108.24, 112.26, 111.91, 112.43, 113.88, 114.84, 109.95, 116.42, 111.0, 112.29, 107.95, 113.34, 108.13, 115.73, 118.29, 115.44, 110.9, 109.44, 118.27, 120.03, 112.88, 113.91, 104.07, 109.16, 108.99, 111.69, 116.35, 110.87, 117.21, 110.36, 107.34, 116.12, 115.12, 108.04, 108.44, 113.04, 117.15, 113.81, 106.91, 112.14, 107.22, 110.42, 116.36, 109.38, 110.66, 106.61, 108.43, 117.02, 113.8, 118.25, 116.26, 112.65, 113.06, 117.41, 111.6, 111.3, 111.7, 110.32, 114.62, 113.9, 113.52, 114.31, 113.18, 117.97, 109.88, 118.94, 110.15, 116.11, 107.81, 108.37, 113.65, 112.17, 112.74, 111.34, 114.89, 112.59, 113.87, 113.09, 110.35, 110.01, 116.82, 113.85, 117.74, 111.08, 112.95, 113.22, 112.53, 113.77, 118.2, 110.05, 113.38, 109.28, 111.62, 120.47, 112.23, 112.42, 109.0, 105.64, 112.97, 110.36, 113.25, 111.26, 115.98, 114.57, 112.68, 113.47, 108.44, 111.35, 120.16, 111.43, 106.29, 112.74, 111.25, 112.98, 112.78, 118.51, 109.67, 112.94, 118.8, 112.48, 110.29, 113.75, 111.63, 112.25, 112.49, 113.55, 108.06, 111.7, 116.31, 115.34, 116.61, 117.37, 111.8, 112.12, 108.46, 105.15, 111.14, 113.23, 111.98, 110.61, 113.49, 104.85, 117.19, 116.29, 113.27, 113.4, 118.18, 108.48, 113.2, 106.17, 107.61, 106.91, 110.11, 116.2, 112.35, 111.3, 111.41, 110.19, 109.18, 114.78, 113.46, 122.83, 110.63, 110.94, 114.49, 115.75, 111.73, 114.44, 115.81, 117.36, 113.05, 116.45, 110.84, 113.99, 111.41, 111.33, 109.77, 111.53, 115.21, 111.76, 108.44, 109.8, 111.67, 110.92, 113.88, 112.73, 111.5, 108.89, 108.07, 107.25, 112.13, 105.86, 110.74, 116.11, 110.11, 112.56, 114.58, 113.45, 112.01, 111.88, 110.81, 109.57, 111.61, 110.47, 114.86, 106.08, 114.72, 117.1, 115.65, 114.83, 115.43, 113.63, 119.89, 104.48, 111.14, 113.39, 118.68, 114.55, 116.34, 110.46, 116.8, 112.96, 110.46, 115.36, 110.75, 111.1, 109.27, 122.01, 113.28, 110.92, 112.01, 110.55, 110.91, 112.43, 104.64, 103.69, 114.21, 111.69]；
差值(后−前) n=538：[-8.26, -8.94, -11.42, -8.49, -10.7, -11.64, -9.55, -8.88, -10.39, -10.26, -10.24, -10.51, -8.35, -9.15, -8.94, -10.27, -11.74, -9.86, -8.67, -6.97, -9.34, -9.99, -8.96, -9.11, -9.78, -10.98, -9.41, -8.46, -9.84, -10.83, -10.81, -10.03, -8.4, -9.16, -8.67, -10.11, -9.58, -9.86, -9.67, -10.02, -9.9, -7.42, -11.57, -8.43, -7.07, -9.14, -9.13, -9.1, -10.3, -8.45, -9.89, -9.26, -9.99, -9.33, -9.85, -9.57, -9.9, -9.94, -10.43, -10.32, -8.91, -9.31, -10.48, -9.32, -11.24, -11.41, -9.34, -10.19, -10.42, -8.67, -10.23, -10.78, -9.1, -9.92, -9.46, -10.43, -10.43, -9.43, -8.82, -10.04, -8.7, -9.33, -10.86, -8.56, -9.92, -10.65, -9.84, -10.01, -10.64, -8.9, -12.09, -9.52, -9.23, -10.42, -9.04, -10.91, -7.9, -9.4, -10.65, -9.7, -8.56, -10.26, -10.03, -9.59, -11.56, -10.51, -10.37, -10.39, -9.04, -9.56, -10.62, -8.14, -10.13, -8.49, -10.1, -9.58, -9.52, -9.38, -10.01, -9.06, -11.39, -9.52, -8.64, -9.73, -9.07, -10.25, -10.06, -8.78, -8.79, -8.93, -10.65, -8.48, -8.18, -10.79, -8.29, -12.08, -10.0, -9.15, -9.61, -9.03, -9.95, -9.44, -11.47, -9.37, -7.62, -9.02, -10.08, -8.5, -10.26, -8.46, -8.82, -10.69, -9.19, -10.3, -12.32, -8.25, -8.33, -8.82, -10.67, -8.48, -8.48, -8.72, -11.19, -9.42, -10.17, -8.93, -10.23, -9.88, -10.45, -10.02, -9.03, -10.21, -11.0, -8.65, -8.81, -10.78, -10.23, -8.75, -8.77, -9.5, -9.88, -7.92, -8.53, -9.67, -10.94, -9.39, -8.57, -8.75, -10.53, -6.2, -9.58, -10.53, -10.63, -8.8, -8.32, -10.49, -9.47, -9.26, -11.29, -10.47, -9.92, -11.09, -9.22, -10.06, -8.77, -9.18, -10.08, -10.23, -9.24, -7.88, -10.16, -9.38, -8.84, -9.74, -8.87, -10.74, -9.66, -9.3, -10.77, -8.48, -8.38, -11.01, -10.42, -8.85, -9.41, -9.33, -9.39, -10.37, -9.36, -9.42, -10.84, -9.59, -6.28, -10.03, -8.89, -10.2, -8.64, -10.22, -9.53, -10.79, -8.52, -9.05, -8.51, -7.82, -7.88, -8.92, -9.32, -11.44, -8.71, -10.37, -8.75, -8.39, -9.54, -11.09, -8.82, -8.48, -9.84, -8.73, -10.24, -8.54, -10.29, -8.52, -9.53, -9.1, -9.81, -9.61, -9.49, -10.41, -9.64, -8.63, -8.83, -9.12, -6.74, -10.57, -9.13, -11.49, -9.45, -9.44, -9.94, -9.34, -9.22, -9.66, -10.59, -10.11, -9.92, -9.91, -7.41, -9.44, -10.0, -8.87, -9.16, -9.86, -9.15, -11.03, -8.39, -8.28, -9.07, -9.34, -9.66, -9.56, -11.34, -9.29, -9.56, -10.0, -10.66, -9.59, -9.44, -6.47, -9.58, -9.38, -11.78, -11.15, -9.18, -11.14, -8.79, -10.59, -9.2, -9.64, -8.26, -8.39, -9.0, -10.07, -10.05, -10.23, -10.4, -10.01, -8.98, -10.52, -7.61, -9.18, -10.73, -9.19, -10.1, -10.94, -9.39, -8.17, -8.46, -10.59, -9.2, -9.67, -10.52, -9.15, -9.12, -10.64, -8.84, -10.75, -9.96, -9.29, -8.81, -9.57, -9.38, -9.73, -9.34, -8.24, -8.06, -9.22, -8.97, -9.15, -8.8, -10.53, -8.99, -11.01, -9.09, -9.76, -8.65, -7.93, -9.12, -8.82, -9.68, -11.19, -10.06, -10.35, -11.37, -10.16, -9.05, -10.53, -10.56, -9.67, -9.71, -8.63, -8.68, -11.54, -8.9, -7.11, -10.22, -9.2, -9.37, -11.36, -10.26, -9.73, -9.3, -10.2, -9.64, -9.07, -9.58, -10.62, -11.53, -9.37, -8.84, -9.48, -9.45, -9.57, -10.21, -8.39, -9.75, -9.18, -9.23, -9.08, -10.07, -9.83, -7.51, -8.75, -8.61, -11.06, -10.69, -8.51, -8.28, -8.96, -9.55, -8.2, -10.1, -9.69, -10.01, -10.73, -9.96, -10.09, -8.97, -7.56, -8.63, -9.12, -8.78, -8.63, -7.96, -8.85, -10.97, -10.86, -8.4, -10.46, -8.37, -9.04, -9.49, -10.72, -11.42, -9.42, -9.46, -8.7, -8.8, -9.87, -9.58, -10.94, -10.49, -12.76, -9.27, -10.1, -10.62, -10.06, -11.2, -10.44, -9.45, -8.99, -9.45, -10.13, -9.79, -7.77, -10.22, -9.01, -8.65, -7.84, -9.34, -10.35, -8.97, -9.53, -9.21, -10.92, -8.4, -9.55, -8.59, -10.42, -11.45, -8.51, -9.27, -9.3, -10.66, -8.59, -10.54, -9.24, -11.07, -9.59, -10.13, -9.25, -9.97, -9.65, -9.94, -10.05, -9.1, -9.98, -7.68, -9.16, -9.77, -11.53, -8.38, -11.0, -8.75, -11.5, -7.53, -9.84, -9.22, -8.29, -9.65, -9.91, -9.18, -10.43, -8.07, -11.43, -9.57, -10.04, -9.55, -9.62, -8.54, -10.56, -9.61, -11.18, -10.71, -7.95, -8.81, -9.03, -10.24, -8.74, -9.76, -10.3, -11.09, -8.11, -8.3, -10.28, -9.53, -9.85, -9.47, -9.62]
后 n=538：[110.72, 108.2, 119.05, 113.92, 119.19, 108.94, 111.56, 114.27, 114.7, 108.4, 112.63, 114.57, 114.29, 114.67, 110.28, 113.61, 109.33, 116.18, 117.11, 112.58, 112.02, 108.92, 113.44, 111.14, 112.92, 110.46, 114.31, 114.33, 114.07, 105.75, 117.97, 111.11, 108.6, 112.09, 116.54, 115.28, 113.58, 113.67, 114.64, 113.46, 108.66, 113.7, 105.95, 109.85, 121.86, 114.79, 111.31, 118.56, 115.83, 118.43, 111.84, 115.93, 109.61, 113.34, 113.07, 110.7, 110.53, 115.12, 117.11, 118.24, 110.84, 104.54, 113.67, 114.94, 109.81, 112.8, 112.81, 111.26, 113.23, 113.55, 110.37, 110.4, 112.86, 114.26, 112.25, 110.54, 110.5, 115.14, 112.17, 107.88, 109.09, 117.88, 114.16, 113.45, 110.54, 107.49, 107.56, 109.43, 113.12, 113.96, 110.26, 109.96, 111.43, 109.09, 111.6, 109.51, 114.93, 115.86, 118.81, 116.51, 116.68, 110.45, 108.59, 105.53, 110.05, 111.17, 108.76, 115.67, 113.76, 114.96, 119.5, 117.46, 116.36, 112.48, 110.9, 114.09, 117.45, 116.8, 111.36, 116.93, 112.99, 116.68, 111.0, 109.96, 109.54, 110.52, 117.03, 112.94, 114.63, 114.88, 114.43, 115.93, 117.52, 113.44, 113.38, 112.2, 107.98, 106.53, 109.8, 118.73, 110.57, 117.34, 112.86, 117.64, 104.82, 113.03, 109.13, 118.1, 112.19, 110.21, 112.04, 111.77, 115.79, 107.61, 105.9, 113.33, 117.2, 112.9, 110.82, 116.5, 117.03, 112.95, 113.34, 112.31, 112.06, 113.71, 108.74, 114.32, 112.66, 111.98, 114.69, 112.22, 116.99, 115.06, 108.22, 111.14, 113.4, 117.94, 115.55, 110.02, 110.53, 109.65, 114.16, 106.86, 110.54, 117.18, 118.83, 118.45, 107.78, 116.36, 109.14, 109.79, 111.06, 110.56, 115.17, 107.96, 109.58, 108.7, 106.99, 112.57, 115.04, 107.43, 114.03, 116.7, 111.14, 116.52, 118.29, 114.84, 114.01, 114.4, 112.03, 113.16, 110.68, 115.96, 112.16, 113.52, 111.28, 119.28, 114.61, 114.6, 113.63, 106.29, 113.33, 116.19, 113.74, 105.22, 114.84, 109.32, 110.78, 116.7, 111.47, 116.88, 116.77, 110.22, 113.7, 109.9, 112.93, 111.72, 110.23, 108.1, 113.22, 111.22, 120.16, 115.2, 108.93, 120.47, 114.13, 109.66, 118.02, 114.79, 113.82, 110.64, 110.13, 111.48, 113.81, 112.73, 111.83, 112.04, 116.73, 113.3, 111.95, 114.06, 114.42, 111.06, 117.4, 113.37, 109.54, 114.4, 112.37, 110.43, 116.01, 112.59, 113.02, 112.38, 115.26, 107.82, 113.43, 113.96, 109.69, 108.1, 118.12, 111.52, 111.16, 111.48, 110.58, 114.96, 115.21, 111.2, 110.93, 114.41, 115.77, 116.83, 112.8, 114.38, 110.89, 114.28, 111.92, 113.69, 111.81, 115.49, 108.24, 112.26, 111.91, 112.43, 113.88, 114.84, 109.95, 116.42, 111.0, 112.29, 107.95, 113.34, 108.13, 115.73, 118.29, 115.44, 110.9, 109.44, 118.27, 120.03, 112.88, 113.91, 104.07, 109.16, 108.99, 111.69, 116.35, 110.87, 117.21, 110.36, 107.34, 116.12, 115.12, 108.04, 108.44, 113.04, 117.15, 113.81, 106.91, 112.14, 107.22, 110.42, 116.36, 109.38, 110.66, 106.61, 108.43, 117.02, 113.8, 118.25, 116.26, 112.65, 113.06, 117.41, 111.6, 111.3, 111.7, 110.32, 114.62, 113.9, 113.52, 114.31, 113.18, 117.97, 109.88, 118.94, 110.15, 116.11, 107.81, 108.37, 113.65, 112.17, 112.74, 111.34, 114.89, 112.59, 113.87, 113.09, 110.35, 110.01, 116.82, 113.85, 117.74, 111.08, 112.95, 113.22, 112.53, 113.77, 118.2, 110.05, 113.38, 109.28, 111.62, 120.47, 112.23, 112.42, 109.0, 105.64, 112.97, 110.36, 113.25, 111.26, 115.98, 114.57, 112.68, 113.47, 108.44, 111.35, 120.16, 111.43, 106.29, 112.74, 111.25, 112.98, 112.78, 118.51, 109.67, 112.94, 118.8, 112.48, 110.29, 113.75, 111.63, 112.25, 112.49, 113.55, 108.06, 111.7, 116.31, 115.34, 116.61, 117.37, 111.8, 112.12, 108.46, 105.15, 111.14, 113.23, 111.98, 110.61, 113.49, 104.85, 117.19, 116.29, 113.27, 113.4, 118.18, 108.48, 113.2, 106.17, 107.61, 106.91, 110.11, 116.2, 112.35, 111.3, 111.41, 110.19, 109.18, 114.78, 113.46, 122.83, 110.63, 110.94, 114.49, 115.75, 111.73, 114.44, 115.81, 117.36, 113.05, 116.45, 110.84, 113.99, 111.41, 111.33, 109.77, 111.53, 115.21, 111.76, 108.44, 109.8, 111.67, 110.92, 113.88, 112.73, 111.5, 108.89, 108.07, 107.25, 112.13, 105.86, 110.74, 116.11, 110.11, 112.56, 114.58, 113.45, 112.01, 111.88, 110.81, 109.57, 111.61, 110.47, 114.86, 106.08, 114.72, 117.1, 115.65, 114.83, 115.43, 113.63, 119.89, 104.48, 111.14, 113.39, 118.68, 114.55, 116.34, 110.46, 116.8, 112.96, 110.46, 115.36, 110.75, 111.1, 109.27, 122.01, 113.28, 110.92, 112.01, 110.55, 110.91, 112.43, 104.64, 103.69, 114.21, 111.69]；
差值(后−前) n=538：[-8.26, -8.94, -11.42, -8.49, -10.7, -11.64, -9.55, -8.88, -10.39, -10.26, -10.24, -10.51, -8.35, -9.15, -8.94, -10.27, -11.74, -9.86, -8.67, -6.97, -9.34, -9.99, -8.96, -9.11, -9.78, -10.98, -9.41, -8.46, -9.84, -10.83, -10.81, -10.03, -8.4, -9.16, -8.67, -10.11, -9.58, -9.86, -9.67, -10.02, -9.9, -7.42, -11.57, -8.43, -7.07, -9.14, -9.13, -9.1, -10.3, -8.45, -9.89, -9.26, -9.99, -9.33, -9.85, -9.57, -9.9, -9.94, -10.43, -10.32, -8.91, -9.31, -10.48, -9.32, -11.24, -11.41, -9.34, -10.19, -10.42, -8.67, -10.23, -10.78, -9.1, -9.92, -9.46, -10.43, -10.43, -9.43, -8.82, -10.04, -8.7, -9.33, -10.86, -8.56, -9.92, -10.65, -9.84, -10.01, -10.64, -8.9, -12.09, -9.52, -9.23, -10.42, -9.04, -10.91, -7.9, -9.4, -10.65, -9.7, -8.56, -10.26, -10.03, -9.59, -11.56, -10.51, -10.37, -10.39, -9.04, -9.56, -10.62, -8.14, -10.13, -8.49, -10.1, -9.58, -9.52, -9.38, -10.01, -9.06, -11.39, -9.52, -8.64, -9.73, -9.07, -10.25, -10.06, -8.78, -8.79, -8.93, -10.65, -8.48, -8.18, -10.79, -8.29, -12.08, -10.0, -9.15, -9.61, -9.03, -9.95, -9.44, -11.47, -9.37, -7.62, -9.02, -10.08, -8.5, -10.26, -8.46, -8.82, -10.69, -9.19, -10.3, -12.32, -8.25, -8.33, -8.82, -10.67, -8.48, -8.48, -8.72, -11.19, -9.42, -10.17, -8.93, -10.23, -9.88, -10.45, -10.02, -9.03, -10.21, -11.0, -8.65, -8.81, -10.78, -10.23, -8.75, -8.77, -9.5, -9.88, -7.92, -8.53, -9.67, -10.94, -9.39, -8.57, -8.75, -10.53, -6.2, -9.58, -10.53, -10.63, -8.8, -8.32, -10.49, -9.47, -9.26, -11.29, -10.47, -9.92, -11.09, -9.22, -10.06, -8.77, -9.18, -10.08, -10.23, -9.24, -7.88, -10.16, -9.38, -8.84, -9.74, -8.87, -10.74, -9.66, -9.3, -10.77, -8.48, -8.38, -11.01, -10.42, -8.85, -9.41, -9.33, -9.39, -10.37, -9.36, -9.42, -10.84, -9.59, -6.28, -10.03, -8.89, -10.2, -8.64, -10.22, -9.53, -10.79, -8.52, -9.05, -8.51, -7.82, -7.88, -8.92, -9.32, -11.44, -8.71, -10.37, -8.75, -8.39, -9.54, -11.09, -8.82, -8.48, -9.84, -8.73, -10.24, -8.54, -10.29, -8.52, -9.53, -9.1, -9.81, -9.61, -9.49, -10.41, -9.64, -8.63, -8.83, -9.12, -6.74, -10.57, -9.13, -11.49, -9.45, -9.44, -9.94, -9.34, -9.22, -9.66, -10.59, -10.11, -9.92, -9.91, -7.41, -9.44, -10.0, -8.87, -9.16, -9.86, -9.15, -11.03, -8.39, -8.28, -9.07, -9.34, -9.66, -9.56, -11.34, -9.29, -9.56, -10.0, -10.66, -9.59, -9.44, -6.47, -9.58, -9.38, -11.78, -11.15, -9.18, -11.14, -8.79, -10.59, -9.2, -9.64, -8.26, -8.39, -9.0, -10.07, -10.05, -10.23, -10.4, -10.01, -8.98, -10.52, -7.61, -9.18, -10.73, -9.19, -10.1, -10.94, -9.39, -8.17, -8.46, -10.59, -9.2, -9.67, -10.52, -9.15, -9.12, -10.64, -8.84, -10.75, -9.96, -9.29, -8.81, -9.57, -9.38, -9.73, -9.34, -8.24, -8.06, -9.22, -8.97, -9.15, -8.8, -10.53, -8.99, -11.01, -9.09, -9.76, -8.65, -7.93, -9.12, -8.82, -9.68, -11.19, -10.06, -10.35, -11.37, -10.16, -9.05, -10.53, -10.56, -9.67, -9.71, -8.63, -8.68, -11.54, -8.9, -7.11, -10.22, -9.2, -9.37, -11.36, -10.26, -9.73, -9.3, -10.2, -9.64, -9.07, -9.58, -10.62, -11.53, -9.37, -8.84, -9.48, -9.45, -9.57, -10.21, -8.39, -9.75, -9.18, -9.23, -9.08, -10.07, -9.83, -7.51, -8.75, -8.61, -11.06, -10.69, -8.51, -8.28, -8.96, -9.55, -8.2, -10.1, -9.69, -10.01, -10.73, -9.96, -10.09, -8.97, -7.56, -8.63, -9.12, -8.78, -8.63, -7.96, -8.85, -10.97, -10.86, -8.4, -10.46, -8.37, -9.04, -9.49, -10.72, -11.42, -9.42, -9.46, -8.7, -8.8, -9.87, -9.58, -10.94, -10.49, -12.76, -9.27, -10.1, -10.62, -10.06, -11.2, -10.44, -9.45, -8.99, -9.45, -10.13, -9.79, -7.77, -10.22, -9.01, -8.65, -7.84, -9.34, -10.35, -8.97, -9.53, -9.21, -10.92, -8.4, -9.55, -8.59, -10.42, -11.45, -8.51, -9.27, -9.3, -10.66, -8.59, -10.54, -9.24, -11.07, -9.59, -10.13, -9.25, -9.97, -9.65, -9.94, -10.05, -9.1, -9.98, -7.68, -9.16, -9.77, -11.53, -8.38, -11.0, -8.75, -11.5, -7.53, -9.84, -9.22, -8.29, -9.65, -9.91, -9.18, -10.43, -8.07, -11.43, -9.57, -10.04, -9.55, -9.62, -8.54, -10.56, -9.61, -11.18, -10.71, -7.95, -8.81, -9.03, -10.24, -8.74, -9.76, -10.3, -11.09, -8.11, -8.3, -10.28, -9.53, -9.85, -9.47, -9.62]
分析过程
H₀: μ_d=0，H₁: μ_d≠0。
d̄ = -9.5797，s_d = 0.9800，SE = s_d/√n = 0.0423；
t = d̄/SE = -226.7262；df = n−1 = 537；双侧 p = 0.000e+00。
d̄ = -9.5797，s_d = 0.9800，SE = s_d/√n = 0.0423；
t = d̄/SE = -226.7262；df = n−1 = 537；双侧 p = 0.000e+00。
计算结果
t = -226.7262，df = 537，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。验证手术改善效果。
卡方 拟合优度
案例 51. 骰子是否均匀(6面)
原始数据
观测频数(共 n=847)：[128, 143, 132, 143, 150, 151]；期望比例：[0.167, 0.167, 0.167, 0.167, 0.167, 0.167]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 847；期望 E = 总数×比例 = [141.17, 141.17, 141.17, 141.17, 141.17, 141.17]；
χ² = Σ(O−E)²/E = 3.1086；df = k−1 = 5；p = 0.6832。
总数 = 847；期望 E = 总数×比例 = [141.17, 141.17, 141.17, 141.17, 141.17, 141.17]；
χ² = Σ(O−E)²/E = 3.1086；df = k−1 = 5；p = 0.6832。
计算结果
χ² = 3.1086，df = 5，p = 0.6832
在 α=0.05 水平下，p=0.6832 ≥ 0.05，不拒绝原假设 H₀。检验骰子是否存在偏面。
案例 52. 血型分布是否符合理论
原始数据
观测频数(共 n=937)：[362, 301, 195, 79]；期望比例：[0.4, 0.3, 0.2, 0.1]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 937；期望 E = 总数×比例 = [374.8, 281.1, 187.4, 93.7]；
χ² = Σ(O−E)²/E = 4.4603；df = k−1 = 3；p = 0.2159。
总数 = 937；期望 E = 总数×比例 = [374.8, 281.1, 187.4, 93.7]；
χ² = Σ(O−E)²/E = 4.4603；df = k−1 = 3；p = 0.2159。
计算结果
χ² = 4.4603，df = 3，p = 0.2159
在 α=0.05 水平下，p=0.2159 ≥ 0.05，不拒绝原假设 H₀。验证血型比例是否偏离理论。
案例 53. 一周到店量是否均匀
原始数据
观测频数(共 n=414)：[45, 47, 62, 55, 52, 89, 64]；期望比例：[0.143, 0.143, 0.143, 0.143, 0.143, 0.143, 0.143]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 414；期望 E = 总数×比例 = [59.14, 59.14, 59.14, 59.14, 59.14, 59.14, 59.14]；
χ² = Σ(O−E)²/E = 22.6377；df = k−1 = 6；p = 9.274e-04。
总数 = 414；期望 E = 总数×比例 = [59.14, 59.14, 59.14, 59.14, 59.14, 59.14, 59.14]；
χ² = Σ(O−E)²/E = 22.6377；df = k−1 = 6；p = 9.274e-04。
计算结果
χ² = 22.6377，df = 6，p = 9.274e-04
在 α=0.05 水平下，p=9.274e-04 < 0.05，拒绝原假设 H₀。判断客流是否均匀分布。
案例 54. 颜色偏好是否符合预期
原始数据
观测频数(共 n=497)：[159, 135, 82, 72, 49]；期望比例：[0.3, 0.25, 0.2, 0.15, 0.1]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 497；期望 E = 总数×比例 = [149.1, 124.25, 99.4, 74.55, 49.7]；
χ² = Σ(O−E)²/E = 4.7304；df = k−1 = 4；p = 0.3161。
总数 = 497；期望 E = 总数×比例 = [149.1, 124.25, 99.4, 74.55, 49.7]；
χ² = Σ(O−E)²/E = 4.7304；df = k−1 = 4；p = 0.3161。
计算结果
χ² = 4.7304，df = 4，p = 0.3161
在 α=0.05 水平下，p=0.3161 ≥ 0.05，不拒绝原假设 H₀。检验颜色偏好结构。
案例 55. 手机品牌占比
原始数据
观测频数(共 n=624)：[190, 160, 113, 105, 56]；期望比例：[0.3, 0.25, 0.2, 0.15, 0.1]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 624；期望 E = 总数×比例 = [187.2, 156.0, 124.8, 93.6, 62.4]；
χ² = Σ(O−E)²/E = 3.3050；df = k−1 = 4；p = 0.5081。
总数 = 624；期望 E = 总数×比例 = [187.2, 156.0, 124.8, 93.6, 62.4]；
χ² = Σ(O−E)²/E = 3.3050；df = k−1 = 4；p = 0.5081。
计算结果
χ² = 3.3050，df = 4，p = 0.5081
在 α=0.05 水平下，p=0.5081 ≥ 0.05，不拒绝原假设 H₀。验证市场份额结构。
案例 56. 四季出生是否均匀
原始数据
观测频数(共 n=780)：[201, 185, 198, 196]；期望比例：[0.25, 0.25, 0.25, 0.25]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 780；期望 E = 总数×比例 = [195.0, 195.0, 195.0, 195.0]；
χ² = Σ(O−E)²/E = 0.7487；df = k−1 = 3；p = 0.8617。
总数 = 780；期望 E = 总数×比例 = [195.0, 195.0, 195.0, 195.0]；
χ² = Σ(O−E)²/E = 0.7487；df = k−1 = 3；p = 0.8617。
计算结果
χ² = 0.7487，df = 3，p = 0.8617
在 α=0.05 水平下，p=0.8617 ≥ 0.05，不拒绝原假设 H₀。判断出生季节性。
案例 57. 用户等级分布
原始数据
观测频数(共 n=713)：[435, 215, 63]；期望比例：[0.6, 0.3, 0.1]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 713；期望 E = 总数×比例 = [427.8, 213.9, 71.3]；
χ² = Σ(O−E)²/E = 1.0930；df = k−1 = 2；p = 0.5790。
总数 = 713；期望 E = 总数×比例 = [427.8, 213.9, 71.3]；
χ² = Σ(O−E)²/E = 1.0930；df = k−1 = 2；p = 0.5790。
计算结果
χ² = 1.0930，df = 2，p = 0.5790
在 α=0.05 水平下，p=0.5790 ≥ 0.05，不拒绝原假设 H₀。检验会员结构是否合理。
案例 58. 投诉类型分布
原始数据
观测频数(共 n=500)：[188, 158, 99, 55]；期望比例：[0.4, 0.3, 0.2, 0.1]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 500；期望 E = 总数×比例 = [200.0, 150.0, 100.0, 50.0]；
χ² = Σ(O−E)²/E = 1.6567；df = k−1 = 3；p = 0.6466。
总数 = 500；期望 E = 总数×比例 = [200.0, 150.0, 100.0, 50.0]；
χ² = Σ(O−E)²/E = 1.6567；df = k−1 = 3；p = 0.6466。
计算结果
χ² = 1.6567，df = 3，p = 0.6466
在 α=0.05 水平下，p=0.6466 ≥ 0.05，不拒绝原假设 H₀。验证投诉归因结构。
案例 59. 偏好程度分布
原始数据
观测频数(共 n=755)：[367, 219, 169]；期望比例：[0.5, 0.3, 0.2]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 755；期望 E = 总数×比例 = [377.5, 226.5, 151.0]；
χ² = Σ(O−E)²/E = 2.6861；df = k−1 = 2；p = 0.2610。
总数 = 755；期望 E = 总数×比例 = [377.5, 226.5, 151.0]；
χ² = Σ(O−E)²/E = 2.6861；df = k−1 = 2；p = 0.2610。
计算结果
χ² = 2.6861，df = 2，p = 0.2610
在 α=0.05 水平下，p=0.2610 ≥ 0.05，不拒绝原假设 H₀。检查态度分布。
案例 60. 性别比是否 1:1
原始数据
观测频数(共 n=478)：[224, 254]；期望比例：[0.5, 0.5]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 478；期望 E = 总数×比例 = [239.0, 239.0]；
χ² = Σ(O−E)²/E = 1.8828；df = k−1 = 1；p = 0.1700。
总数 = 478；期望 E = 总数×比例 = [239.0, 239.0]；
χ² = Σ(O−E)²/E = 1.8828；df = k−1 = 1；p = 0.1700。
计算结果
χ² = 1.8828，df = 1，p = 0.1700
在 α=0.05 水平下，p=0.1700 ≥ 0.05，不拒绝原假设 H₀。检验样本性别平衡。
案例 61. 渠道订单占比
原始数据
观测频数(共 n=740)：[305, 249, 186]；期望比例：[0.4, 0.35, 0.25]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 740；期望 E = 总数×比例 = [296.0, 259.0, 185.0]；
χ² = Σ(O−E)²/E = 0.6652；df = k−1 = 2；p = 0.7171。
总数 = 740；期望 E = 总数×比例 = [296.0, 259.0, 185.0]；
χ² = Σ(O−E)²/E = 0.6652；df = k−1 = 2；p = 0.7171。
计算结果
χ² = 0.6652，df = 2，p = 0.7171
在 α=0.05 水平下，p=0.7171 ≥ 0.05，不拒绝原假设 H₀。验证渠道结构。
案例 62. 产品评级分布
原始数据
观测频数(共 n=548)：[114, 274, 129, 31]；期望比例：[0.2, 0.5, 0.25, 0.05]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 548；期望 E = 总数×比例 = [109.6, 274.0, 137.0, 27.4]；
χ² = Σ(O−E)²/E = 1.1168；df = k−1 = 3；p = 0.7730。
总数 = 548；期望 E = 总数×比例 = [109.6, 274.0, 137.0, 27.4]；
χ² = Σ(O−E)²/E = 1.1168；df = k−1 = 3；p = 0.7730。
计算结果
χ² = 1.1168，df = 3，p = 0.7730
在 α=0.05 水平下，p=0.7730 ≥ 0.05，不拒绝原假设 H₀。检查质量评级结构。
案例 63. 交通方式占比
原始数据
观测频数(共 n=830)：[226, 349, 176, 79]；期望比例：[0.3, 0.4, 0.2, 0.1]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 830；期望 E = 总数×比例 = [249.0, 332.0, 166.0, 83.0]；
χ² = Σ(O−E)²/E = 3.7902；df = k−1 = 3；p = 0.2850。
总数 = 830；期望 E = 总数×比例 = [249.0, 332.0, 166.0, 83.0]；
χ² = Σ(O−E)²/E = 3.7902；df = k−1 = 3；p = 0.2850。
计算结果
χ² = 3.7902，df = 3，p = 0.2850
在 α=0.05 水平下，p=0.2850 ≥ 0.05，不拒绝原假设 H₀。验证通勤结构。
案例 64. 阅读偏好分布
原始数据
观测频数(共 n=514)：[178, 151, 115, 70]；期望比例：[0.35, 0.3, 0.2, 0.15]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 514；期望 E = 总数×比例 = [179.9, 154.2, 102.8, 77.1]；
χ² = Σ(O−E)²/E = 2.1882；df = k−1 = 3；p = 0.5343。
总数 = 514；期望 E = 总数×比例 = [179.9, 154.2, 102.8, 77.1]；
χ² = Σ(O−E)²/E = 2.1882；df = k−1 = 3；p = 0.5343。
计算结果
χ² = 2.1882，df = 3，p = 0.5343
在 α=0.05 水平下，p=0.5343 ≥ 0.05，不拒绝原假设 H₀。检查阅读取向。
案例 65. 购买时段分布
原始数据
观测频数(共 n=558)：[158, 219, 181]；期望比例：[0.3, 0.4, 0.3]
分析过程
H₀: 分布符合期望比例，H₁: 不符合。
总数 = 558；期望 E = 总数×比例 = [167.4, 223.2, 167.4]；
χ² = Σ(O−E)²/E = 1.7118；df = k−1 = 2；p = 0.4249。
总数 = 558；期望 E = 总数×比例 = [167.4, 223.2, 167.4]；
χ² = Σ(O−E)²/E = 1.7118；df = k−1 = 2；p = 0.4249。
计算结果
χ² = 1.7118，df = 2，p = 0.4249
在 α=0.05 水平下，p=0.4249 ≥ 0.05，不拒绝原假设 H₀。验证购物时段结构。
卡方 独立性
案例 66. 性别与购买意愿
原始数据
列联表(行×列)：[[244, 123], [170, 208]]；总样本量 n=745
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[367, 378]，列和=[414, 331]，总数=745；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=34.9000；df=(r−1)(c−1)=1；p=3.471e-09。
行和=[367, 378]，列和=[414, 331]，总数=745；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=34.9000；df=(r−1)(c−1)=1；p=3.471e-09。
计算结果
χ² = 34.9000，df = 1，p = 3.471e-09
在 α=0.05 水平下，p=3.471e-09 < 0.05，拒绝原假设 H₀。检验购买意愿是否与性别相关。
案例 67. 吸烟与肺癌
原始数据
列联表(行×列)：[[232, 147], [93, 484]]；总样本量 n=956
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[379, 577]，列和=[325, 631]，总数=956；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=207.3164；df=(r−1)(c−1)=1；p=1.003e-12。
行和=[379, 577]，列和=[325, 631]，总数=956；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=207.3164；df=(r−1)(c−1)=1；p=1.003e-12。
计算结果
χ² = 207.3164，df = 1，p = 1.003e-12
在 α=0.05 水平下，p=1.003e-12 < 0.05，拒绝原假设 H₀。验证吸烟与患病是否关联。
案例 68. 学历与高收入
原始数据
列联表(行×列)：[[46, 72], [93, 106], [69, 53]]；总样本量 n=439
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[118, 199, 122]，列和=[208, 231]，总数=439；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=7.4920；df=(r−1)(c−1)=2；p=0.0236。
行和=[118, 199, 122]，列和=[208, 231]，总数=439；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=7.4920；df=(r−1)(c−1)=2；p=0.0236。
计算结果
χ² = 7.4920，df = 2，p = 0.0236
在 α=0.05 水平下，p=0.0236 < 0.05，拒绝原假设 H₀。检验学历与收入关联。
案例 69. 地区与满意度
原始数据
列联表(行×列)：[[112, 79, 46], [84, 88, 57], [89, 57, 59]]；总样本量 n=671
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[237, 229, 205]，列和=[285, 224, 162]，总数=671；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=10.8643；df=(r−1)(c−1)=4；p=0.0281。
行和=[237, 229, 205]，列和=[285, 224, 162]，总数=671；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=10.8643；df=(r−1)(c−1)=4；p=0.0281。
计算结果
χ² = 10.8643，df = 4，p = 0.0281
在 α=0.05 水平下，p=0.0281 < 0.05，拒绝原假设 H₀。判断满意度是否因地区而异。
案例 70. 广告与购买
原始数据
列联表(行×列)：[[61, 147], [45, 377]]；总样本量 n=630
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[208, 422]，列和=[106, 524]，总数=630；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=34.6786；df=(r−1)(c−1)=1；p=3.889e-09。
行和=[208, 422]，列和=[106, 524]，总数=630；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=34.6786；df=(r−1)(c−1)=1；p=3.889e-09。
计算结果
χ² = 34.6786，df = 1，p = 3.889e-09
在 α=0.05 水平下，p=3.889e-09 < 0.05，拒绝原假设 H₀。检验广告是否促进购买。
案例 71. 运动与健康
原始数据
列联表(行×列)：[[237, 66], [161, 225]]；总样本量 n=689
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[303, 386]，列和=[398, 291]，总数=689；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=92.7361；df=(r−1)(c−1)=1；p=4.119e-13。
行和=[303, 386]，列和=[398, 291]，总数=689；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=92.7361；df=(r−1)(c−1)=1；p=4.119e-13。
计算结果
χ² = 92.7361，df = 1，p = 4.119e-13
在 α=0.05 水平下，p=4.119e-13 < 0.05，拒绝原假设 H₀。验证运动与健康关联。
案例 72. 性别与专业选择
原始数据
列联表(行×列)：[[65, 91, 112], [122, 115, 100]]；总样本量 n=605
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[268, 337]，列和=[187, 206, 212]，总数=605；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=13.1513；df=(r−1)(c−1)=2；p=0.0014。
行和=[268, 337]，列和=[187, 206, 212]，总数=605；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=13.1513；df=(r−1)(c−1)=2；p=0.0014。
计算结果
χ² = 13.1513，df = 2，p = 0.0014
在 α=0.05 水平下，p=0.0014 < 0.05，拒绝原假设 H₀。检验专业选择是否与性别相关。
案例 73. 年龄与品牌偏好
原始数据
列联表(行×列)：[[211, 107], [167, 159], [83, 210]]；总样本量 n=937
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[318, 326, 293]，列和=[461, 476]，总数=937；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=89.0394；df=(r−1)(c−1)=2；p=6.277e-13。
行和=[318, 326, 293]，列和=[461, 476]，总数=937；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=89.0394；df=(r−1)(c−1)=2；p=6.277e-13。
计算结果
χ² = 89.0394，df = 2，p = 6.277e-13
在 α=0.05 水平下，p=6.277e-13 < 0.05，拒绝原假设 H₀。判断品牌偏好是否随年龄变化。
案例 74. 包装与回购
原始数据
列联表(行×列)：[[297, 181], [208, 294]]；总样本量 n=980
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[478, 502]，列和=[505, 475]，总数=980；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=42.0047；df=(r−1)(c−1)=1；p=9.130e-11。
行和=[478, 502]，列和=[505, 475]，总数=980；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=42.0047；df=(r−1)(c−1)=1；p=9.130e-11。
计算结果
χ² = 42.0047，df = 1，p = 9.130e-11
在 α=0.05 水平下，p=9.130e-11 < 0.05，拒绝原假设 H₀。检验包装颜色对回购影响。
案例 75. 培训与晋升
原始数据
列联表(行×列)：[[59, 112], [35, 294]]；总样本量 n=500
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[171, 329]，列和=[94, 406]，总数=500；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=41.9776；df=(r−1)(c−1)=1；p=9.256e-11。
行和=[171, 329]，列和=[94, 406]，总数=500；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=41.9776；df=(r−1)(c−1)=1；p=9.256e-11。
计算结果
χ² = 41.9776，df = 1，p = 9.256e-11
在 α=0.05 水平下，p=9.256e-11 < 0.05，拒绝原假设 H₀。验证培训是否提升晋升。
案例 76. 城市与网购
原始数据
列联表(行×列)：[[116, 43], [73, 103]]；总样本量 n=335
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[159, 176]，列和=[189, 146]，总数=335；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=33.6646；df=(r−1)(c−1)=1；p=6.548e-09。
行和=[159, 176]，列和=[189, 146]，总数=335；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=33.6646；df=(r−1)(c−1)=1；p=6.548e-09。
计算结果
χ² = 33.6646，df = 1，p = 6.548e-09
在 α=0.05 水平下，p=6.548e-09 < 0.05，拒绝原假设 H₀。检验城市等级与网购关联。
案例 77. 饮食与疾病
原始数据
列联表(行×列)：[[198, 122], [99, 324]]；总样本量 n=743
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[320, 423]，列和=[297, 446]，总数=743；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=112.3697；df=(r−1)(c−1)=1；p=7.097e-13。
行和=[320, 423]，列和=[297, 446]，总数=743；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=112.3697；df=(r−1)(c−1)=1；p=7.097e-13。
计算结果
χ² = 112.3697，df = 1，p = 7.097e-13
在 α=0.05 水平下，p=7.097e-13 < 0.05，拒绝原假设 H₀。验证高脂饮食与疾病关联。
案例 78. 学历与就业
原始数据
列联表(行×列)：[[111, 40], [166, 28], [92, 18]]；总样本量 n=455
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[151, 194, 110]，列和=[369, 86]，总数=455；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=8.6621；df=(r−1)(c−1)=2；p=0.0132。
行和=[151, 194, 110]，列和=[369, 86]，总数=455；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=8.6621；df=(r−1)(c−1)=2；p=0.0132。
计算结果
χ² = 8.6621，df = 2，p = 0.0132
在 α=0.05 水平下，p=0.0132 < 0.05，拒绝原假设 H₀。检验学历对就业影响。
案例 79. 季节与感冒
原始数据
列联表(行×列)：[[20, 101], [14, 127], [29, 110], [44, 81]]；总样本量 n=526
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[121, 141, 139, 125]，列和=[107, 419]，总数=526；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=27.5735；df=(r−1)(c−1)=3；p=4.463e-06。
行和=[121, 141, 139, 125]，列和=[107, 419]，总数=526；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=27.5735；df=(r−1)(c−1)=3；p=4.463e-06。
计算结果
χ² = 27.5735，df = 3，p = 4.463e-06
在 α=0.05 水平下，p=4.463e-06 < 0.05，拒绝原假设 H₀。判断感冒是否具季节性。
案例 80. 促销与销量
原始数据
列联表(行×列)：[[287, 143], [181, 280]]；总样本量 n=891
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[430, 461]，列和=[468, 423]，总数=891；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=67.3827；df=(r−1)(c−1)=1；p=4.257e-13。
行和=[430, 461]，列和=[468, 423]，总数=891；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=67.3827；df=(r−1)(c−1)=1；p=4.257e-13。
计算结果
χ² = 67.3827，df = 1，p = 4.257e-13
在 α=0.05 水平下，p=4.257e-13 < 0.05，拒绝原假设 H₀。检验促销对高销量影响。
案例 81. 车型与满意度
原始数据
列联表(行×列)：[[161, 126, 75], [91, 129, 167]]；总样本量 n=749
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[362, 387]，列和=[252, 255, 242]，总数=749；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=53.6803；df=(r−1)(c−1)=2；p=2.626e-12。
行和=[362, 387]，列和=[252, 255, 242]，总数=749；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=53.6803；df=(r−1)(c−1)=2；p=2.626e-12。
计算结果
χ² = 53.6803，df = 2，p = 2.626e-12
在 α=0.05 水平下，p=2.626e-12 < 0.05，拒绝原假设 H₀。验证车型与满意度关联。
案例 82. 性别与运动偏好
原始数据
列联表(行×列)：[[69, 56, 52], [46, 34, 90]]；总样本量 n=347
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[177, 170]，列和=[115, 90, 142]，总数=347；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=20.0137；df=(r−1)(c−1)=2；p=4.509e-05。
行和=[177, 170]，列和=[115, 90, 142]，总数=347；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=20.0137；df=(r−1)(c−1)=2；p=4.509e-05。
计算结果
χ² = 20.0137，df = 2，p = 4.509e-05
在 α=0.05 水平下，p=4.509e-05 < 0.05，拒绝原假设 H₀。检验运动偏好是否与性别相关。
案例 83. 地区与品牌
原始数据
列联表(行×列)：[[72, 52, 49], [75, 76, 39], [40, 60, 72]]；总样本量 n=535
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[173, 190, 172]，列和=[187, 188, 160]，总数=535；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=26.5480；df=(r−1)(c−1)=4；p=2.453e-05。
行和=[173, 190, 172]，列和=[187, 188, 160]，总数=535；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=26.5480；df=(r−1)(c−1)=4；p=2.453e-05。
计算结果
χ² = 26.5480，df = 4，p = 2.453e-05
在 α=0.05 水平下，p=2.453e-05 < 0.05，拒绝原假设 H₀。判断品牌选择是否因地区而异。
案例 84. 年龄与理财
原始数据
列联表(行×列)：[[187, 115], [201, 51], [107, 178]]；总样本量 n=839
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[302, 252, 285]，列和=[495, 344]，总数=839；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=100.2085；df=(r−1)(c−1)=2；p=4.250e-13。
行和=[302, 252, 285]，列和=[495, 344]，总数=839；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=100.2085；df=(r−1)(c−1)=2；p=4.250e-13。
计算结果
χ² = 100.2085，df = 2，p = 4.250e-13
在 α=0.05 水平下，p=4.250e-13 < 0.05，拒绝原假设 H₀。验证理财参与是否随年龄变化。
案例 85. 渠道与流失
原始数据
列联表(行×列)：[[176, 396], [78, 202]]；总样本量 n=852
分析过程
H₀: 两变量独立，H₁: 相关。
行和=[572, 280]，列和=[254, 598]，总数=852；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=0.7618；df=(r−1)(c−1)=1；p=0.3828。
行和=[572, 280]，列和=[254, 598]，总数=852；
期望 Eᵢⱼ=行和×列和/总数；χ²=Σ(O−E)²/E=0.7618；df=(r−1)(c−1)=1；p=0.3828。
计算结果
χ² = 0.7618，df = 1，p = 0.3828
在 α=0.05 水平下，p=0.3828 ≥ 0.05，不拒绝原假设 H₀。检验渠道对流失影响。
单样本 z 检验
案例 86. 灯泡寿命是否 1000h(已知σ)
原始数据
样本 n=413：[1068.17, 972.39, 947.62, 978.76, 1091.93, 1006.06, 995.35, 990.85, 994.93, 905.15, 988.54, 1000.54, 1022.63, 1043.23, 986.68, 925.52, 1068.34, 999.85, 1018.86, 989.03, 1040.07, 1037.13, 962.83, 954.46, 1085.32, 882.13, 1036.62, 954.39, 993.47, 957.82, 875.58, 1032.19, 1033.39, 1010.77, 976.66, 1041.58, 1076.39, 980.9, 1078.2, 1018.1, 967.22, 950.46, 977.65, 1006.89, 1061.99, 991.86, 1012.46, 993.23, 1023.54, 1026.01, 995.42, 1062.57, 913.72, 1001.49, 975.52, 1004.1, 1037.8, 1050.56, 1013.56, 1055.09, 1029.74, 1011.05, 958.45, 1033.25, 1094.69, 1062.92, 936.49, 1026.97, 1047.78, 1049.2, 972.05, 967.24, 976.9, 995.82, 972.22, 954.08, 985.49, 1085.87, 999.21, 1066.08, 963.54, 1047.51, 938.68, 1048.86, 998.43, 1011.82, 1090.72, 1052.65, 1062.2, 1001.2, 941.06, 1052.65, 986.69, 945.77, 987.19, 998.27, 1060.7, 1076.0, 982.49, 1103.32, 988.2, 1063.73, 937.98, 995.06, 1109.88, 1001.83, 960.37, 951.23, 1017.08, 926.99, 1077.94, 1062.45, 1070.55, 1021.33, 1014.2, 963.66, 944.23, 1068.03, 1027.97, 1040.02, 976.66, 1036.84, 927.07, 999.82, 1003.58, 1083.3, 1070.73, 1001.12, 951.56, 1029.91, 1038.14, 990.14, 1082.52, 1022.56, 976.21, 1042.25, 1039.54, 1012.23, 1063.9, 1029.56, 1062.98, 1119.1, 1049.42, 1041.57, 1070.68, 1043.97, 961.56, 970.37, 1021.35, 1077.56, 1051.21, 988.96, 950.35, 1056.57, 974.02, 1012.7, 982.58, 1008.41, 980.9, 1000.0, 1032.31, 1049.25, 960.98, 970.8, 952.04, 963.94, 939.93, 950.14, 959.59, 1005.08, 1018.57, 997.38, 986.43, 925.97, 1062.11, 903.86, 1017.71, 1047.85, 1034.18, 1019.99, 974.17, 1058.8, 1036.01, 1060.97, 1047.83, 1028.06, 1122.65, 1023.43, 1007.2, 1094.15, 1027.5, 967.5, 1091.15, 1034.21, 1072.63, 1022.15, 903.95, 992.29, 953.64, 966.41, 987.8, 1035.03, 1065.76, 969.46, 968.99, 997.37, 1088.82, 1021.54, 1040.7, 1050.32, 988.0, 1004.12, 1050.97, 1003.23, 1029.85, 994.98, 1004.85, 1029.61, 1011.61, 978.05, 1032.84, 1017.56, 953.64, 995.26, 985.0, 986.72, 1048.42, 1011.97, 1000.04, 1153.16, 953.75, 1068.01, 904.16, 1028.88, 1079.18, 1057.69, 965.68, 919.06, 1092.27, 950.22, 1036.89, 999.06, 979.83, 1050.67, 1027.4, 980.17, 1000.84, 1046.9, 932.04, 1006.85, 979.54, 1079.47, 942.23, 966.66, 1008.75, 1034.61, 1053.97, 1091.47, 994.44, 1040.96, 1106.9, 1049.17, 950.79, 957.89, 986.2, 954.51, 934.84, 945.32, 926.04, 963.47, 944.27, 961.19, 1080.51, 1052.37, 1008.37, 1048.95, 1004.17, 988.97, 1018.67, 967.6, 986.66, 1002.68, 991.14, 991.89, 964.08, 1052.18, 942.17, 1008.13, 957.03, 1010.35, 947.43, 1012.87, 996.77, 1074.51, 967.59, 1045.95, 1001.81, 1017.77, 1063.08, 987.16, 1051.54, 999.15, 1009.19, 970.15, 958.27, 1021.4, 969.92, 985.73, 1035.59, 933.91, 1125.75, 1022.77, 993.28, 926.55, 980.77, 961.8, 1125.09, 989.97, 950.17, 985.73, 1012.77, 963.89, 1016.17, 1064.22, 1059.32, 1111.61, 1008.64, 1013.4, 1075.58, 935.88, 910.48, 1053.95, 1015.66, 998.38, 1049.96, 1047.52, 886.51, 1082.04, 997.83, 978.99, 1006.89, 1082.57, 1033.57, 960.62, 993.71, 992.58, 1069.92, 992.9, 944.93, 1013.28, 991.07, 942.11, 965.65, 976.45, 1051.98, 997.87, 955.35, 1090.74, 1001.57, 1014.23, 1034.39, 1073.74, 979.82, 1114.66, 961.11, 1030.33, 1057.99, 924.79, 943.15, 1001.82, 984.46, 1033.81, 1011.21, 1059.51, 1031.62, 1052.17, 1051.86, 1068.32, 986.89, 995.08, 987.65, 1030.99, 1021.68, 1019.81, 996.44, 961.05, 1007.79, 969.4, 992.09, 1113.44, 979.97, 974.47, 973.76, 989.21, 1023.44, 1062.39, 1018.33, 1040.11, 1036.22, 936.03, 1109.88, 1062.42, 993.0, 1053.38, 1088.63, 1030.29, 1026.47, 1040.73, 1042.61, 1072.7, 986.05, 974.93, 990.28]；已知 σ=50，μ₀=1000，x̄=1009.8182
分析过程
H₀: μ=1000，H₁: μ≠1000（已知σ用z检验）。
SE=σ/√n=50/√413=2.4603；z=(x̄−μ₀)/SE=3.9906；双侧 p=6.591e-05。
SE=σ/√n=50/√413=2.4603；z=(x̄−μ₀)/SE=3.9906；双侧 p=6.591e-05。
计算结果
z = 3.9906，p = 6.591e-05
在 α=0.05 水平下，p=6.591e-05 < 0.05，拒绝原假设 H₀。已知总体标准差时检验均值。
案例 87. 零件长度是否 50mm(已知σ)
原始数据
样本 n=474：[49.46, 50.43, 49.01, 49.95, 50.46, 49.75, 49.01, 50.47, 50.91, 50.6, 49.6, 50.5, 50.01, 50.23, 50.38, 51.33, 50.22, 49.92, 49.22, 49.42, 50.64, 49.55, 49.96, 49.95, 50.53, 49.98, 49.95, 49.74, 50.5, 50.39, 49.94, 49.47, 49.94, 50.34, 50.12, 49.45, 50.3, 50.0, 49.6, 49.29, 50.25, 49.52, 49.02, 49.94, 49.62, 49.33, 50.38, 49.91, 50.41, 50.07, 49.64, 50.25, 49.22, 50.16, 50.19, 49.27, 49.89, 49.32, 50.38, 49.79, 49.28, 50.27, 49.46, 50.23, 49.51, 50.16, 50.37, 50.19, 49.15, 50.1, 50.07, 49.59, 48.92, 50.28, 50.13, 50.24, 50.39, 50.83, 49.67, 49.8, 50.62, 50.78, 49.35, 49.68, 50.1, 49.26, 48.63, 49.83, 50.05, 49.7, 49.47, 49.1, 50.26, 49.89, 50.11, 50.14, 49.71, 49.45, 49.81, 50.18, 48.97, 50.6, 50.2, 50.18, 49.98, 50.0, 49.87, 49.91, 48.67, 50.35, 50.28, 50.59, 50.19, 49.57, 51.11, 49.87, 50.57, 49.79, 50.46, 49.61, 50.25, 51.01, 50.0, 49.14, 49.48, 50.26, 49.83, 49.34, 49.7, 50.12, 49.19, 49.88, 50.25, 50.22, 48.99, 50.11, 50.89, 49.59, 50.25, 49.44, 49.91, 50.42, 49.97, 49.86, 49.68, 49.77, 50.41, 49.16, 49.66, 49.45, 50.36, 49.3, 50.49, 49.88, 49.76, 50.39, 50.08, 49.41, 49.73, 49.99, 49.99, 50.13, 50.27, 50.05, 49.89, 49.69, 50.38, 49.58, 50.61, 50.93, 50.02, 50.16, 49.91, 50.58, 49.18, 50.83, 50.4, 49.89, 50.3, 50.46, 49.81, 49.85, 49.37, 49.88, 49.85, 49.94, 50.17, 50.7, 49.85, 50.29, 50.63, 50.55, 50.34, 49.63, 49.6, 50.33, 50.51, 49.72, 50.06, 49.94, 49.9, 50.61, 50.94, 49.56, 50.08, 49.76, 49.16, 50.35, 49.68, 49.94, 50.28, 49.38, 51.38, 49.44, 50.2, 49.63, 50.74, 50.04, 50.19, 49.92, 49.96, 49.54, 50.38, 50.61, 49.45, 50.3, 49.59, 49.94, 49.66, 49.75, 49.51, 49.76, 49.87, 50.65, 50.0, 50.19, 50.11, 49.79, 50.28, 50.53, 49.22, 49.21, 49.86, 50.54, 49.6, 49.38, 49.61, 50.15, 49.57, 49.7, 50.27, 49.81, 49.3, 49.68, 48.75, 50.25, 49.38, 50.56, 50.81, 49.57, 50.52, 49.68, 49.46, 50.35, 50.17, 49.87, 50.84, 50.68, 50.25, 50.01, 50.13, 49.81, 49.77, 49.95, 49.36, 49.75, 50.01, 50.57, 49.93, 50.21, 50.82, 48.96, 50.46, 50.2, 50.21, 50.3, 49.85, 49.48, 49.64, 50.18, 50.15, 49.61, 50.18, 49.86, 49.65, 50.28, 50.14, 50.01, 50.9, 50.18, 49.89, 49.63, 49.39, 49.89, 50.38, 50.27, 50.5, 49.86, 50.36, 48.83, 49.36, 49.97, 51.14, 50.06, 50.69, 50.36, 48.95, 50.24, 50.11, 50.27, 50.01, 49.63, 50.35, 50.34, 50.67, 50.34, 49.67, 50.48, 50.73, 49.47, 50.57, 49.34, 49.51, 49.65, 50.36, 50.77, 49.62, 49.97, 49.82, 48.97, 49.93, 50.75, 50.02, 50.56, 49.55, 50.83, 49.96, 50.21, 49.38, 48.87, 50.86, 50.22, 49.48, 50.47, 50.37, 50.12, 49.74, 48.96, 49.99, 50.09, 50.53, 50.28, 49.81, 50.32, 49.95, 50.05, 50.14, 50.21, 50.02, 50.49, 49.23, 49.65, 49.49, 50.18, 50.25, 49.67, 50.08, 50.4, 49.45, 50.01, 50.42, 50.52, 50.47, 50.23, 49.31, 49.73, 49.5, 49.89, 50.09, 49.13, 50.35, 49.98, 50.67, 50.58, 49.63, 50.44, 49.91, 50.14, 50.5, 50.52, 50.34, 49.72, 49.54, 49.99, 49.63, 50.08, 49.82, 49.89, 50.77, 49.92, 50.37, 50.05, 50.19, 50.6, 50.17, 50.18, 50.11, 50.04, 49.69, 49.73, 48.92, 50.16, 50.04, 49.88, 50.16, 49.55, 50.64, 49.2, 49.73, 49.32, 50.69, 50.09, 49.55, 49.87, 49.42, 49.89, 49.57, 49.54, 49.38, 50.67, 49.12, 50.16, 49.85, 50.3, 50.28, 49.35, 49.73, 50.46, 49.03, 50.23, 50.4, 50.34, 50.58, 50.26, 49.98, 49.65, 49.81, 49.62, 50.44, 50.64, 49.33, 49.95, 49.98, 50.27, 50.01, 50.24, 50.43, 50.29, 50.08, 50.68, 50.3, 49.65, 50.48, 49.35]；已知 σ=0.5，μ₀=50，x̄=49.9841
分析过程
H₀: μ=50，H₁: μ≠50（已知σ用z检验）。
SE=σ/√n=0.5/√474=0.0230；z=(x̄−μ₀)/SE=-0.6905；双侧 p=0.4899。
SE=σ/√n=0.5/√474=0.0230；z=(x̄−μ₀)/SE=-0.6905；双侧 p=0.4899。
计算结果
z = -0.6905，p = 0.4899
在 α=0.05 水平下，p=0.4899 ≥ 0.05，不拒绝原假设 H₀。大样本已知σ的精度检验。
案例 88. IQ 是否 100(已知σ)
原始数据
样本 n=823：[107.45, 108.37, 103.47, 111.2, 116.92, 93.34, 124.31, 106.43, 110.83, 88.18, 114.01, 97.45, 116.14, 102.2, 101.28, 124.65, 126.42, 127.32, 136.27, 94.75, 88.53, 97.81, 108.43, 106.9, 112.17, 123.29, 122.68, 106.15, 131.02, 120.51, 121.45, 78.34, 121.27, 118.66, 108.78, 106.63, 105.99, 93.13, 118.03, 98.47, 104.35, 99.04, 95.36, 95.5, 97.39, 80.45, 109.85, 79.56, 93.24, 120.59, 105.71, 97.9, 66.37, 71.31, 74.35, 102.08, 104.81, 105.56, 97.63, 88.78, 109.07, 109.77, 85.46, 116.43, 97.42, 101.92, 94.81, 100.73, 77.66, 112.46, 81.04, 79.01, 112.78, 99.88, 97.82, 95.69, 82.83, 87.88, 77.01, 97.66, 116.42, 90.5, 99.75, 97.11, 72.11, 127.03, 119.43, 84.01, 98.78, 114.3, 118.01, 113.99, 76.49, 127.94, 96.38, 81.49, 91.44, 87.95, 95.4, 119.99, 98.74, 111.06, 64.23, 120.89, 118.89, 101.3, 97.75, 122.35, 121.85, 108.09, 105.24, 92.52, 87.93, 97.01, 140.71, 109.78, 105.23, 104.09, 107.87, 70.09, 99.75, 96.85, 101.42, 105.92, 61.44, 96.4, 110.37, 123.21, 76.33, 95.41, 123.51, 104.53, 124.13, 96.72, 90.1, 94.83, 93.27, 100.8, 101.17, 94.02, 67.42, 89.3, 119.76, 120.79, 100.27, 98.21, 85.68, 109.67, 111.65, 112.17, 105.92, 95.57, 130.06, 104.35, 95.72, 95.78, 113.61, 100.31, 81.94, 106.35, 106.39, 107.71, 97.28, 124.82, 95.8, 110.7, 119.72, 113.41, 117.15, 112.36, 111.67, 102.52, 102.46, 108.23, 77.06, 114.42, 94.96, 97.6, 85.15, 111.86, 106.36, 86.64, 101.69, 88.31, 89.49, 104.69, 87.08, 98.03, 103.75, 109.64, 94.47, 94.72, 90.87, 113.84, 111.87, 103.75, 101.05, 128.91, 106.34, 104.06, 95.51, 105.34, 101.83, 96.14, 101.83, 108.79, 92.66, 108.96, 81.08, 84.89, 96.39, 83.23, 81.07, 107.49, 75.45, 86.41, 90.07, 87.7, 114.37, 63.29, 110.44, 86.45, 110.99, 101.96, 106.72, 96.41, 94.43, 94.74, 90.63, 97.32, 86.39, 131.67, 115.45, 100.73, 90.6, 112.07, 112.67, 80.96, 86.16, 114.36, 110.15, 102.36, 115.87, 110.03, 108.54, 80.85, 102.09, 127.21, 106.2, 79.39, 110.08, 85.53, 74.08, 101.04, 95.57, 91.91, 95.74, 108.53, 83.39, 107.05, 106.93, 118.9, 104.99, 100.08, 103.65, 85.59, 108.91, 102.31, 91.64, 108.07, 86.53, 98.64, 86.37, 102.67, 99.98, 116.78, 89.25, 115.87, 94.96, 132.92, 100.3, 91.51, 113.01, 114.79, 107.81, 88.36, 89.5, 93.45, 107.78, 93.59, 100.77, 85.05, 57.74, 115.14, 107.37, 114.44, 96.98, 81.86, 96.33, 107.63, 105.59, 110.85, 84.95, 102.37, 106.72, 85.86, 94.21, 117.3, 105.33, 120.06, 77.89, 129.02, 107.99, 98.25, 95.98, 102.02, 103.97, 129.35, 117.63, 94.41, 106.97, 114.58, 127.13, 92.14, 88.16, 104.74, 105.71, 94.24, 105.36, 111.91, 135.99, 108.02, 113.7, 101.79, 81.68, 92.37, 104.08, 131.19, 105.83, 101.58, 94.11, 93.39, 124.15, 77.98, 98.1, 100.98, 113.69, 97.48, 129.52, 110.43, 124.76, 87.74, 103.64, 84.46, 102.47, 83.84, 101.97, 91.92, 89.81, 91.3, 89.97, 103.3, 104.32, 71.81, 84.22, 104.93, 109.82, 112.9, 85.75, 118.52, 103.58, 91.63, 105.27, 84.6, 78.45, 104.43, 105.53, 99.23, 112.39, 107.29, 112.61, 122.68, 83.73, 98.73, 109.26, 113.66, 118.98, 109.37, 108.79, 110.3, 100.93, 106.38, 116.73, 100.45, 116.46, 89.69, 128.44, 108.5, 128.39, 82.6, 66.12, 93.76, 95.89, 118.84, 107.82, 92.72, 82.13, 88.95, 102.88, 71.01, 84.9, 85.56, 96.54, 117.21, 118.39, 107.86, 119.35, 99.3, 96.19, 118.15, 70.07, 101.25, 100.94, 99.02, 126.96, 108.78, 119.5, 91.86, 100.72, 118.79, 89.55, 83.62, 93.93, 92.42, 107.5, 90.55, 78.7, 89.85, 114.91, 86.76, 107.84, 85.95, 79.42, 100.7, 93.97, 80.97, 91.85, 102.8, 82.84, 108.65, 113.28, 95.9, 94.26, 84.26, 95.63, 103.85, 94.86, 82.32, 132.28, 117.03, 107.86, 95.4, 95.87, 118.62, 91.1, 90.83, 106.88, 114.89, 119.38, 106.97, 83.33, 100.72, 82.08, 90.08, 116.54, 89.87, 104.31, 107.67, 84.71, 91.64, 113.0, 106.95, 83.56, 99.45, 127.3, 121.54, 113.27, 89.84, 78.85, 102.04, 111.52, 112.97, 104.79, 140.32, 128.44, 94.58, 100.96, 94.59, 112.53, 97.46, 84.96, 96.81, 121.96, 68.22, 102.78, 102.15, 119.3, 106.35, 107.95, 89.88, 122.17, 84.28, 122.57, 110.31, 106.39, 84.6, 103.04, 115.43, 99.67, 107.97, 106.45, 103.59, 123.48, 96.04, 105.95, 89.6, 91.94, 128.99, 122.92, 99.81, 97.58, 106.34, 98.4, 96.37, 93.59, 104.58, 84.31, 66.12, 113.31, 106.56, 87.12, 113.58, 87.2, 83.55, 104.45, 98.14, 116.44, 93.17, 126.36, 124.64, 92.12, 99.55, 84.41, 94.13, 119.03, 67.49, 112.8, 84.52, 82.32, 118.2, 114.56, 99.35, 117.94, 81.25, 119.14, 107.94, 118.97, 98.21, 121.44, 118.62, 100.44, 128.35, 106.35, 126.36, 106.71, 92.71, 87.6, 121.7, 101.54, 106.05, 115.01, 98.22, 97.42, 98.64, 75.96, 96.76, 82.21, 117.16, 103.79, 93.31, 69.52, 115.74, 94.9, 91.86, 102.51, 136.52, 108.89, 120.67, 82.74, 111.0, 71.56, 115.93, 104.45, 72.4, 92.12, 80.74, 88.67, 115.93, 109.58, 96.44, 88.63, 63.68, 119.49, 99.01, 92.85, 81.67, 118.26, 88.25, 72.19, 100.02, 109.41, 114.89, 113.57, 103.33, 89.86, 104.15, 86.32, 110.17, 117.87, 67.87, 121.85, 86.37, 77.32, 114.61, 87.82, 98.5, 133.46, 105.48, 111.59, 102.05, 74.7, 110.5, 100.27, 92.72, 122.46, 94.49, 86.12, 107.95, 115.39, 105.01, 94.57, 89.97, 109.69, 92.57, 127.3, 132.92, 106.3, 69.33, 108.07, 76.38, 115.47, 101.83, 105.43, 103.91, 91.3, 100.81, 83.32, 108.3, 113.89, 118.68, 63.03, 91.28, 117.96, 102.64, 116.14, 103.96, 99.85, 81.07, 104.62, 112.45, 127.01, 122.34, 91.84, 108.87, 100.98, 96.68, 111.57, 103.09, 116.4, 97.09, 94.24, 116.36, 127.77, 136.42, 116.98, 112.4, 96.06, 108.76, 93.41, 117.22, 103.15, 77.27, 79.14, 88.49, 108.8, 98.07, 106.75, 94.12, 79.16, 96.57, 105.89, 135.53, 100.24, 102.4, 133.82, 98.54, 95.07, 89.86, 121.73, 83.88, 109.05, 94.53, 91.38, 95.77, 105.39, 89.46, 93.1, 104.32, 131.91, 108.49, 101.57, 114.41, 114.12, 98.71, 103.59, 128.78, 104.58, 105.87, 108.46, 99.46, 86.54, 114.95, 100.86, 76.14, 81.34, 70.87, 127.0, 137.96, 101.0, 96.27, 112.76, 91.22, 101.95, 105.88, 65.56, 68.85, 100.07, 91.22, 107.46, 101.04, 102.78, 118.04, 90.58, 101.36, 93.66, 105.81, 94.68, 101.07, 96.23, 95.97, 127.06, 92.39, 89.06, 91.02, 77.25, 99.65, 83.55, 109.07, 124.53, 69.94, 84.32, 107.74, 74.28, 104.15, 94.25, 117.64, 129.41, 87.51, 93.59, 87.86, 90.93, 127.38, 103.36, 109.73, 79.6, 99.19, 112.82, 75.17, 77.75, 103.81, 91.56, 99.3, 94.48, 79.98, 89.69, 96.42, 85.65, 133.94, 102.78, 103.32, 113.16, 92.41, 118.95, 84.14, 98.94, 118.06, 59.62, 105.83, 120.1, 100.08, 78.61, 114.65, 104.48]；已知 σ=15，μ₀=100，x̄=101.4106
分析过程
H₀: μ=100，H₁: μ≠100（已知σ用z检验）。
SE=σ/√n=15/√823=0.5229；z=(x̄−μ₀)/SE=2.6978；双侧 p=0.0070。
SE=σ/√n=15/√823=0.5229；z=(x̄−μ₀)/SE=2.6978；双侧 p=0.0070。
计算结果
z = 2.6978，p = 0.0070
在 α=0.05 水平下，p=0.0070 < 0.05，拒绝原假设 H₀。标准化测试均值检验。
案例 89. 包装重量是否 200g(已知σ)
原始数据
样本 n=642：[200.66, 200.81, 201.32, 201.36, 203.69, 202.45, 200.29, 203.6, 200.27, 203.14, 201.96, 201.54, 201.88, 201.37, 197.28, 200.9, 200.74, 201.71, 200.81, 197.58, 202.32, 205.7, 198.6, 202.13, 198.08, 203.12, 201.1, 200.22, 201.21, 201.06, 198.6, 201.4, 199.63, 204.39, 200.4, 200.02, 198.27, 198.06, 201.31, 202.4, 200.61, 199.65, 201.69, 203.46, 201.18, 201.04, 199.86, 200.18, 198.18, 197.88, 203.23, 201.03, 198.46, 199.43, 201.03, 204.27, 202.24, 198.43, 202.93, 196.98, 199.69, 198.96, 198.16, 197.59, 199.62, 200.39, 200.96, 199.0, 198.59, 200.67, 202.1, 201.3, 202.16, 204.18, 199.62, 202.01, 196.97, 197.83, 199.59, 200.91, 203.83, 200.77, 201.08, 201.34, 201.54, 199.42, 199.05, 198.49, 203.21, 198.95, 200.96, 195.36, 203.23, 198.17, 202.65, 200.4, 203.12, 198.38, 199.86, 200.34, 201.58, 197.97, 197.63, 201.09, 202.04, 198.68, 201.02, 199.85, 197.71, 205.37, 198.21, 197.65, 202.42, 199.59, 201.95, 200.87, 203.05, 201.52, 205.98, 198.1, 200.88, 203.5, 205.42, 203.11, 200.72, 202.5, 199.94, 201.72, 200.04, 201.04, 199.57, 202.9, 199.54, 198.26, 198.67, 201.98, 198.2, 201.49, 203.23, 201.55, 202.5, 202.84, 202.11, 200.99, 202.11, 202.46, 200.44, 201.43, 202.69, 198.84, 197.04, 202.68, 203.33, 200.1, 203.32, 203.72, 201.0, 201.07, 203.36, 196.45, 201.32, 202.16, 201.89, 198.97, 201.43, 200.27, 201.09, 202.08, 202.0, 198.17, 203.89, 201.19, 206.49, 204.91, 202.99, 201.51, 197.13, 200.31, 199.22, 200.62, 199.55, 201.06, 200.9, 196.14, 201.01, 199.15, 200.35, 200.95, 200.16, 199.89, 198.86, 201.0, 199.88, 205.04, 201.76, 198.46, 200.09, 197.49, 201.88, 200.37, 201.6, 199.04, 200.53, 202.14, 203.05, 200.83, 196.93, 202.23, 197.8, 200.9, 200.75, 201.75, 202.63, 201.45, 202.02, 200.35, 198.66, 199.8, 199.8, 201.02, 201.53, 205.6, 200.7, 201.01, 202.17, 203.65, 198.24, 203.07, 199.41, 201.22, 199.56, 199.57, 197.41, 200.2, 204.64, 200.1, 200.68, 198.29, 202.58, 201.0, 200.51, 204.09, 201.4, 203.34, 200.25, 198.71, 200.59, 198.95, 200.78, 201.1, 199.73, 199.47, 203.38, 202.83, 202.77, 203.51, 201.76, 201.78, 200.67, 200.4, 199.79, 202.22, 201.71, 199.95, 199.09, 197.7, 201.24, 201.0, 198.53, 198.32, 201.98, 202.71, 200.0, 200.69, 202.26, 195.92, 200.11, 200.51, 201.83, 198.88, 201.54, 202.92, 201.05, 201.9, 200.25, 205.45, 202.43, 204.51, 202.66, 201.01, 202.34, 200.36, 198.3, 200.48, 202.29, 200.25, 201.2, 200.18, 202.85, 200.84, 197.49, 202.75, 199.08, 204.94, 202.34, 200.71, 203.31, 197.96, 203.68, 203.73, 203.77, 200.27, 199.43, 200.68, 202.72, 202.21, 198.95, 202.55, 200.8, 199.32, 201.51, 199.85, 201.59, 201.58, 201.57, 202.27, 199.74, 199.46, 200.15, 203.59, 198.35, 204.0, 198.14, 199.52, 199.97, 200.34, 198.9, 204.45, 199.76, 203.54, 199.41, 198.81, 197.87, 204.42, 204.39, 203.53, 199.9, 199.89, 198.88, 201.8, 200.72, 199.02, 203.87, 202.67, 201.28, 201.83, 202.02, 204.05, 203.43, 202.25, 200.6, 199.77, 201.76, 202.81, 205.28, 200.73, 200.74, 200.6, 201.27, 203.13, 200.15, 204.37, 201.27, 200.47, 200.77, 201.68, 200.79, 203.13, 199.09, 199.36, 196.54, 201.01, 201.09, 199.58, 200.2, 205.62, 203.49, 201.19, 199.05, 201.56, 200.48, 203.3, 201.44, 202.34, 201.38, 200.19, 200.74, 197.57, 198.73, 202.68, 202.53, 202.76, 200.38, 202.96, 202.0, 202.95, 198.08, 199.16, 197.87, 196.86, 202.2, 202.52, 202.27, 203.18, 202.8, 202.54, 198.04, 200.05, 199.67, 200.85, 200.27, 202.99, 199.28, 201.96, 205.01, 201.22, 203.78, 206.22, 200.5, 201.73, 202.78, 198.18, 200.22, 201.0, 200.06, 202.58, 202.39, 197.47, 204.97, 199.33, 200.3, 206.32, 202.86, 200.51, 202.41, 196.51, 197.8, 200.35, 202.69, 204.43, 200.58, 199.33, 200.69, 202.21, 198.51, 198.99, 206.45, 197.96, 203.8, 201.43, 200.45, 201.78, 203.85, 201.53, 204.74, 201.33, 200.68, 202.68, 200.78, 197.55, 202.83, 199.78, 200.01, 204.73, 202.85, 203.54, 204.14, 200.76, 201.33, 200.32, 202.18, 197.92, 200.14, 203.16, 200.33, 200.92, 203.72, 203.7, 202.36, 201.03, 201.08, 204.93, 199.84, 200.09, 200.15, 201.8, 201.35, 199.18, 199.75, 201.43, 201.74, 198.01, 200.55, 204.47, 201.13, 200.89, 201.37, 201.17, 199.33, 201.68, 205.2, 201.86, 197.03, 202.27, 198.62, 199.65, 200.6, 201.65, 199.97, 199.18, 200.23, 199.14, 199.06, 200.0, 202.72, 201.05, 199.36, 201.43, 200.72, 200.78, 199.94, 199.45, 202.95, 206.57, 201.6, 199.33, 197.93, 199.04, 203.48, 200.47, 204.7, 200.85, 198.99, 200.54, 198.96, 202.62, 201.3, 199.71, 201.35, 198.85, 204.16, 199.84, 201.11, 197.39, 199.02, 200.6, 200.43, 201.49, 200.29, 203.4, 201.52, 199.45, 202.73, 204.02, 200.85, 201.22, 203.27, 204.06, 203.12, 201.95, 205.56, 202.01, 200.36, 200.29, 202.21, 200.73, 203.04, 199.36, 202.0, 198.05, 201.19, 201.3, 200.96, 199.05, 200.45, 199.53, 201.13, 199.64, 201.84, 198.48, 199.09, 203.25, 196.72, 202.67, 200.94, 200.4, 200.72, 203.75, 201.75, 203.74, 199.89, 203.39, 201.08, 200.05, 200.95, 201.29, 203.36, 204.0, 201.96, 200.98, 201.62, 199.38, 201.69, 200.06, 204.15, 203.44, 200.66, 201.27, 202.2, 202.48, 203.11, 197.72, 202.32, 198.39, 201.67, 201.64, 203.16, 204.01, 198.95, 201.11, 200.28, 201.79, 200.54, 196.58, 202.31, 200.74, 199.73, 197.48, 204.54, 203.23, 202.7, 200.88]；已知 σ=2，μ₀=200，x̄=201.0456
分析过程
H₀: μ=200，H₁: μ≠200（已知σ用z检验）。
SE=σ/√n=2/√642=0.0789；z=(x̄−μ₀)/SE=13.2463；双侧 p=0.000e+00。
SE=σ/√n=2/√642=0.0789；z=(x̄−μ₀)/SE=13.2463；双侧 p=0.000e+00。
计算结果
z = 13.2463，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。合规净重检验。
案例 90. 电池续航是否 10h(已知σ)
原始数据
样本 n=652：[11.38, 10.96, 10.51, 9.2, 11.96, 8.78, 10.67, 11.07, 10.03, 10.43, 9.84, 9.95, 10.04, 10.38, 11.03, 10.31, 11.12, 10.51, 11.38, 10.48, 9.66, 11.18, 11.37, 9.6, 11.57, 10.09, 9.88, 10.09, 9.69, 9.26, 10.19, 9.35, 9.98, 9.87, 9.19, 10.05, 12.07, 9.77, 10.08, 9.81, 10.73, 10.34, 11.54, 9.15, 11.12, 10.04, 9.33, 9.48, 9.89, 11.25, 9.71, 9.76, 10.56, 9.58, 12.06, 8.81, 11.02, 10.64, 10.12, 9.89, 10.49, 9.69, 9.76, 10.41, 10.24, 10.14, 8.04, 11.31, 9.93, 10.26, 10.78, 11.48, 10.47, 10.72, 10.5, 11.26, 11.01, 11.36, 11.83, 9.85, 10.93, 9.91, 10.05, 10.28, 9.42, 11.02, 11.03, 9.79, 8.37, 10.16, 11.62, 10.57, 8.67, 9.23, 10.1, 11.76, 8.78, 8.62, 10.24, 11.54, 9.95, 10.7, 9.41, 9.73, 11.03, 8.21, 8.96, 10.37, 11.8, 10.12, 10.41, 10.96, 8.59, 11.24, 10.58, 8.65, 10.4, 9.56, 10.69, 9.79, 10.25, 10.44, 9.78, 10.14, 10.93, 10.16, 10.45, 9.53, 11.02, 8.31, 9.21, 10.58, 10.19, 11.62, 9.58, 9.85, 11.1, 10.27, 9.43, 9.29, 9.55, 10.1, 9.91, 10.73, 10.5, 11.06, 10.14, 11.04, 9.8, 8.58, 8.96, 9.67, 10.45, 8.83, 11.57, 10.25, 9.96, 9.43, 9.89, 10.05, 10.15, 8.46, 9.56, 10.11, 10.89, 8.91, 8.72, 8.75, 10.44, 8.81, 10.77, 12.04, 11.6, 11.25, 10.92, 10.7, 9.54, 8.68, 9.71, 11.31, 10.19, 10.01, 9.9, 10.96, 8.62, 10.63, 10.64, 10.46, 9.35, 9.78, 8.84, 10.32, 12.55, 9.37, 12.16, 9.17, 10.45, 9.15, 9.31, 10.43, 10.67, 10.09, 8.82, 10.2, 10.88, 11.32, 9.05, 9.68, 10.57, 9.56, 10.05, 9.78, 10.2, 9.44, 9.73, 10.2, 11.2, 10.6, 11.07, 10.01, 9.53, 10.65, 10.85, 10.51, 10.0, 9.36, 10.21, 8.87, 9.09, 8.99, 11.44, 11.52, 10.06, 10.34, 10.03, 9.95, 10.04, 9.11, 10.05, 10.26, 10.32, 10.59, 10.38, 9.84, 9.79, 11.09, 10.28, 9.99, 9.51, 9.34, 9.65, 8.64, 10.6, 10.95, 10.11, 10.93, 10.84, 11.11, 11.25, 9.15, 10.24, 10.84, 11.16, 9.66, 10.11, 11.07, 10.92, 10.82, 10.7, 10.15, 10.82, 12.36, 11.25, 10.13, 9.96, 9.55, 10.87, 10.78, 10.15, 10.1, 10.74, 10.69, 9.62, 8.76, 11.19, 10.51, 10.18, 8.58, 9.84, 10.64, 10.26, 10.93, 9.24, 9.47, 10.66, 10.88, 10.62, 10.05, 11.06, 9.61, 10.37, 10.24, 10.19, 10.63, 10.75, 9.93, 9.75, 10.58, 9.25, 9.37, 10.06, 11.25, 10.11, 8.89, 11.15, 10.41, 9.69, 10.68, 11.42, 9.99, 11.49, 10.37, 8.29, 9.18, 10.49, 9.0, 11.5, 10.51, 9.17, 10.35, 9.99, 8.55, 10.9, 10.0, 9.86, 10.4, 11.08, 10.44, 10.34, 11.37, 10.48, 9.64, 9.86, 8.59, 10.07, 9.69, 9.34, 10.37, 8.94, 8.49, 7.85, 11.49, 10.28, 11.14, 9.5, 10.43, 11.18, 10.42, 9.95, 9.64, 10.3, 10.57, 10.18, 9.86, 11.26, 10.45, 11.1, 10.24, 10.96, 10.3, 11.67, 10.83, 10.78, 9.96, 10.5, 9.37, 9.41, 10.02, 10.48, 11.39, 9.93, 10.17, 10.46, 10.82, 10.37, 9.02, 10.97, 10.51, 11.7, 9.96, 9.79, 10.39, 11.0, 10.6, 9.45, 10.09, 9.93, 9.57, 10.01, 10.49, 11.15, 10.13, 10.86, 9.26, 9.9, 12.08, 10.09, 10.3, 8.99, 10.33, 10.73, 8.72, 8.86, 9.25, 10.5, 9.46, 11.19, 10.93, 9.41, 9.76, 11.43, 10.5, 10.08, 9.63, 9.76, 10.51, 10.2, 10.13, 10.27, 10.23, 8.57, 10.2, 10.9, 9.94, 9.82, 10.37, 10.39, 9.78, 9.74, 9.64, 11.28, 11.07, 9.09, 9.73, 10.2, 9.79, 11.5, 11.01, 9.68, 10.22, 8.92, 8.52, 10.93, 10.29, 9.72, 10.87, 10.45, 9.56, 10.26, 10.17, 9.91, 10.91, 10.3, 9.37, 9.62, 9.22, 10.43, 10.66, 8.95, 10.15, 10.35, 10.24, 10.17, 9.14, 9.31, 10.46, 10.98, 10.52, 9.13, 10.29, 10.83, 9.44, 10.79, 11.36, 10.8, 10.14, 11.13, 9.35, 9.36, 9.07, 11.0, 10.71, 10.9, 10.49, 10.79, 10.44, 10.63, 11.13, 10.56, 11.75, 9.78, 9.02, 9.91, 10.56, 10.05, 9.88, 9.55, 10.22, 10.27, 8.76, 10.04, 9.98, 10.32, 9.76, 9.99, 9.99, 10.43, 10.18, 10.62, 11.19, 11.15, 11.34, 11.38, 9.2, 9.54, 10.16, 10.04, 10.11, 10.88, 10.42, 10.16, 9.7, 10.79, 9.52, 11.74, 10.63, 10.09, 11.41, 10.56, 10.09, 10.98, 10.93, 10.06, 10.04, 8.89, 10.31, 10.48, 10.4, 10.46, 10.65, 12.25, 11.41, 9.6, 10.5, 8.77, 11.24, 10.47, 10.91, 10.11, 10.51, 11.12, 11.34, 11.54, 10.53, 9.71, 10.69, 8.64, 10.64, 10.64, 10.51, 9.49, 9.69, 9.66, 9.92, 10.65, 9.84, 9.8, 10.2, 9.41, 9.71, 10.0, 11.34, 11.22, 10.64, 9.9, 11.52, 9.68, 10.37, 11.81, 10.84, 11.17, 10.31, 10.09, 9.65, 10.33, 11.02, 10.78, 11.82, 9.53, 9.54, 10.43, 10.64, 9.86, 8.7, 9.2, 9.98, 10.52, 10.06, 11.22, 10.0, 10.6, 10.17, 10.02, 10.37, 8.97, 9.49, 9.96, 11.51, 10.19, 10.98, 10.43, 9.85, 10.54, 9.57, 11.37, 12.35, 12.12, 9.93, 10.46, 9.2, 9.88, 9.35, 11.81, 9.45, 9.67, 9.76, 9.98, 10.05, 10.18, 10.22, 10.88, 9.9, 11.41, 10.42, 10.46, 9.8, 9.89, 11.67, 10.0, 9.86, 10.35, 9.82]；已知 σ=0.8，μ₀=10，x̄=10.2234
分析过程
H₀: μ=10，H₁: μ≠10（已知σ用z检验）。
SE=σ/√n=0.8/√652=0.0313；z=(x̄−μ₀)/SE=7.1307；双侧 p=9.983e-13。
SE=σ/√n=0.8/√652=0.0313；z=(x̄−μ₀)/SE=7.1307；双侧 p=9.983e-13。
计算结果
z = 7.1307，p = 9.983e-13
在 α=0.05 水平下，p=9.983e-13 < 0.05，拒绝原假设 H₀。性能达标检验。
案例 91. 考试均分是否 75(已知σ)
原始数据
样本 n=338：[81.1, 88.32, 91.71, 65.84, 69.95, 81.45, 88.15, 66.41, 65.44, 84.42, 92.96, 99.01, 77.55, 91.54, 71.81, 85.29, 71.23, 80.08, 85.25, 82.28, 76.82, 54.34, 71.48, 79.45, 77.96, 84.33, 68.98, 64.79, 74.08, 82.69, 68.09, 76.57, 72.7, 89.21, 77.59, 57.34, 73.31, 65.41, 80.91, 81.56, 57.57, 72.89, 92.7, 68.18, 86.94, 74.95, 91.3, 69.23, 73.15, 75.56, 92.08, 92.79, 85.75, 49.12, 73.34, 66.64, 75.26, 82.05, 54.47, 96.84, 74.43, 73.44, 90.19, 93.74, 76.29, 75.97, 102.45, 69.99, 80.86, 83.68, 63.48, 78.74, 56.79, 75.62, 72.38, 70.43, 83.11, 99.09, 73.72, 78.72, 72.99, 75.93, 83.31, 65.69, 76.82, 91.27, 86.2, 86.84, 82.31, 83.46, 88.12, 85.54, 66.8, 96.72, 83.85, 90.65, 67.54, 81.49, 68.11, 90.34, 85.11, 93.6, 73.18, 63.37, 77.65, 79.53, 86.91, 74.02, 72.9, 83.76, 65.92, 82.34, 76.88, 92.47, 93.5, 78.54, 85.41, 79.38, 77.57, 86.95, 91.71, 68.95, 76.27, 101.27, 70.32, 70.96, 76.18, 89.7, 81.27, 85.1, 70.73, 77.46, 72.95, 72.31, 80.51, 60.45, 63.31, 85.37, 71.5, 73.97, 72.2, 76.82, 72.62, 67.86, 78.26, 86.32, 84.42, 73.97, 63.72, 81.15, 66.4, 106.31, 74.95, 70.0, 75.87, 86.18, 77.87, 77.85, 72.18, 94.05, 78.15, 94.1, 67.49, 70.36, 80.94, 57.51, 75.29, 91.98, 74.49, 74.02, 77.72, 82.42, 84.13, 76.37, 68.24, 63.27, 69.19, 97.43, 74.35, 70.71, 88.19, 83.68, 84.19, 93.08, 74.55, 75.98, 75.32, 78.14, 84.91, 68.12, 72.53, 74.79, 82.68, 61.03, 77.57, 81.8, 69.62, 78.67, 61.58, 79.2, 77.27, 55.12, 66.29, 76.45, 71.18, 82.61, 71.96, 92.13, 59.84, 97.0, 73.67, 70.9, 96.13, 91.33, 83.87, 56.05, 65.63, 61.1, 67.62, 75.54, 79.45, 75.86, 95.72, 84.82, 71.68, 75.82, 75.29, 84.54, 73.85, 90.65, 48.1, 85.71, 71.87, 60.3, 80.87, 71.43, 62.62, 78.22, 73.87, 93.58, 80.69, 83.96, 81.63, 70.74, 80.28, 73.51, 77.34, 88.09, 81.51, 91.04, 70.96, 94.03, 63.16, 86.12, 84.89, 84.88, 92.39, 83.69, 72.95, 83.88, 73.94, 91.47, 87.02, 70.33, 63.55, 72.19, 93.79, 82.0, 90.99, 77.0, 88.32, 82.53, 81.26, 71.69, 76.94, 75.46, 70.23, 78.49, 80.9, 85.59, 65.3, 70.0, 68.31, 79.03, 86.33, 63.47, 70.38, 83.65, 92.93, 84.54, 76.89, 77.43, 70.17, 72.58, 77.4, 85.2, 60.87, 75.76, 100.52, 77.49, 92.64, 84.43, 78.58, 86.1, 82.71, 85.05, 76.43, 67.86, 82.1, 56.9, 56.08, 84.89, 67.99, 77.41, 64.95, 72.81, 71.88, 85.3, 73.46, 58.19, 64.51, 97.62, 100.89, 80.02, 86.97, 89.64, 79.66, 71.97, 62.94, 80.86, 86.83, 71.31, 68.34, 77.67, 62.23, 74.54, 80.09, 63.73]；已知 σ=10，μ₀=75，x̄=77.8006
分析过程
H₀: μ=75，H₁: μ≠75（已知σ用z检验）。
SE=σ/√n=10/√338=0.5439；z=(x̄−μ₀)/SE=5.1488；双侧 p=2.621e-07。
SE=σ/√n=10/√338=0.5439；z=(x̄−μ₀)/SE=5.1488；双侧 p=2.621e-07。
计算结果
z = 5.1488，p = 2.621e-07
在 α=0.05 水平下，p=2.621e-07 < 0.05，拒绝原假设 H₀。大样本成绩检验。
案例 92. 平均反应时是否 300ms(已知σ)
原始数据
样本 n=774：[297.69, 296.05, 299.82, 305.72, 298.52, 298.1, 255.05, 278.37, 280.34, 262.32, 282.12, 312.79, 308.86, 302.5, 294.29, 340.59, 280.31, 333.8, 321.71, 341.11, 276.21, 325.39, 295.89, 303.99, 291.34, 300.97, 302.21, 290.53, 303.9, 261.17, 303.37, 274.37, 295.17, 264.27, 309.2, 336.32, 313.28, 301.26, 287.5, 299.81, 302.33, 271.95, 279.98, 258.85, 299.53, 310.84, 290.29, 344.43, 291.37, 305.01, 297.07, 314.26, 312.87, 291.82, 305.96, 265.25, 302.06, 305.63, 313.25, 317.32, 301.63, 306.54, 284.24, 296.83, 316.15, 288.07, 323.77, 310.55, 277.37, 285.8, 288.85, 291.6, 285.27, 256.68, 293.96, 254.58, 286.42, 322.87, 294.16, 303.64, 283.84, 264.45, 281.69, 331.05, 274.15, 282.32, 312.84, 256.38, 277.65, 267.25, 267.42, 305.11, 292.87, 290.43, 269.35, 254.47, 309.8, 299.13, 320.7, 299.25, 320.62, 275.67, 257.5, 290.37, 301.42, 282.23, 328.56, 251.33, 306.61, 293.87, 302.35, 317.16, 291.17, 310.02, 286.21, 299.77, 321.83, 293.44, 287.11, 305.77, 284.72, 288.94, 287.61, 327.75, 273.97, 289.83, 295.9, 303.09, 260.64, 305.41, 282.51, 280.14, 307.93, 339.23, 267.98, 314.93, 281.18, 304.16, 279.35, 289.71, 301.82, 276.49, 327.57, 314.2, 290.12, 293.07, 260.25, 301.16, 330.68, 230.4, 306.37, 303.79, 311.38, 317.98, 267.5, 307.05, 307.72, 285.41, 288.96, 284.17, 294.96, 312.37, 323.23, 251.84, 294.79, 292.03, 285.18, 308.91, 296.81, 272.76, 306.48, 261.03, 310.61, 325.14, 293.46, 277.26, 333.04, 273.13, 275.66, 324.79, 272.32, 291.86, 287.27, 285.7, 286.42, 314.89, 295.02, 320.85, 300.66, 318.37, 259.46, 307.58, 313.24, 322.21, 290.57, 302.68, 293.29, 300.09, 324.87, 269.67, 289.94, 317.48, 299.48, 293.87, 309.89, 286.45, 295.92, 282.08, 278.2, 287.91, 300.08, 302.77, 290.17, 294.52, 302.71, 303.01, 289.08, 298.06, 318.19, 268.57, 307.86, 306.54, 279.26, 301.92, 273.88, 303.76, 298.82, 283.72, 312.84, 301.05, 322.9, 275.42, 264.71, 261.39, 311.41, 310.62, 321.8, 326.36, 283.22, 292.16, 285.42, 299.72, 282.73, 314.14, 278.89, 281.92, 308.76, 332.76, 332.72, 307.82, 285.83, 300.42, 318.86, 312.93, 274.65, 262.83, 283.27, 314.47, 298.09, 291.37, 313.87, 272.01, 316.21, 325.86, 292.13, 299.29, 344.09, 293.48, 289.08, 309.8, 311.73, 258.98, 305.58, 252.63, 245.61, 307.44, 308.09, 303.93, 280.88, 284.52, 306.96, 255.36, 337.67, 331.79, 312.23, 267.54, 306.95, 269.57, 287.08, 312.16, 294.69, 295.84, 302.53, 275.08, 304.57, 292.69, 290.47, 332.25, 302.23, 324.02, 304.69, 295.32, 280.44, 346.65, 271.91, 257.01, 295.51, 300.34, 326.48, 245.04, 309.44, 292.18, 289.38, 286.17, 259.89, 247.56, 304.32, 300.24, 295.68, 295.94, 318.98, 315.51, 291.08, 267.34, 266.44, 272.52, 268.66, 273.12, 328.46, 299.07, 293.16, 302.01, 283.12, 281.75, 346.44, 297.05, 308.13, 300.39, 271.6, 308.55, 282.44, 296.58, 282.43, 316.03, 314.26, 313.38, 292.32, 297.35, 298.17, 266.56, 290.04, 257.52, 274.36, 258.02, 288.55, 311.51, 301.56, 304.13, 267.86, 305.74, 305.62, 307.12, 258.16, 301.88, 311.56, 291.08, 314.59, 285.31, 263.78, 299.43, 296.09, 323.03, 288.23, 290.95, 316.47, 271.62, 287.82, 287.39, 321.57, 294.59, 288.61, 335.4, 276.58, 318.01, 282.98, 294.66, 297.21, 302.65, 290.46, 269.72, 316.61, 325.75, 323.05, 320.98, 339.59, 292.79, 297.73, 292.03, 277.35, 294.17, 293.74, 276.18, 291.48, 321.75, 300.3, 263.93, 315.17, 326.97, 301.29, 305.78, 312.45, 280.8, 303.36, 285.38, 265.47, 305.66, 256.06, 303.13, 275.87, 318.6, 278.32, 280.46, 327.86, 274.98, 287.68, 301.23, 302.75, 278.73, 340.83, 254.42, 279.1, 295.87, 301.35, 297.49, 319.18, 300.9, 307.83, 302.3, 306.11, 291.36, 296.55, 286.72, 308.22, 303.7, 296.99, 314.81, 300.99, 284.4, 309.83, 271.87, 273.51, 315.74, 287.37, 311.92, 315.93, 292.27, 305.56, 292.27, 303.31, 306.47, 332.22, 311.76, 285.88, 289.38, 304.65, 318.24, 311.06, 289.65, 304.59, 299.87, 312.57, 299.96, 308.1, 287.58, 273.13, 299.72, 332.32, 292.49, 298.87, 313.34, 284.35, 305.39, 264.78, 249.47, 312.53, 302.25, 255.47, 309.18, 303.3, 312.36, 310.19, 293.44, 260.13, 315.48, 327.09, 286.7, 332.33, 321.48, 291.51, 317.98, 275.58, 297.25, 330.96, 263.81, 308.71, 327.68, 318.1, 266.14, 278.41, 328.92, 295.57, 312.58, 271.58, 311.22, 307.22, 310.44, 283.93, 312.37, 287.35, 291.78, 290.13, 311.76, 260.1, 336.9, 274.21, 314.35, 319.93, 304.92, 274.59, 272.91, 317.77, 284.07, 327.89, 304.62, 318.24, 254.43, 303.76, 323.2, 253.48, 315.21, 309.64, 295.29, 281.91, 304.45, 280.33, 269.43, 283.04, 307.41, 281.36, 275.59, 302.38, 311.33, 313.73, 261.68, 292.99, 276.75, 283.67, 291.91, 333.8, 323.27, 292.96, 276.67, 265.65, 311.0, 321.66, 274.52, 308.26, 327.41, 300.65, 330.68, 264.22, 310.79, 257.61, 294.16, 267.87, 286.19, 297.13, 318.25, 302.1, 278.61, 293.08, 322.53, 302.06, 299.61, 313.18, 299.64, 288.37, 268.0, 243.05, 237.57, 283.37, 330.13, 320.12, 265.25, 315.38, 300.84, 291.05, 300.75, 281.05, 291.55, 313.9, 297.63, 281.77, 326.28, 290.14, 285.43, 288.82, 290.95, 267.19, 292.76, 330.01, 298.13, 291.43, 325.63, 310.69, 282.36, 303.55, 316.35, 276.59, 273.28, 276.98, 301.79, 306.1, 298.19, 296.65, 271.31, 288.71, 286.35, 273.28, 262.03, 287.81, 304.17, 284.19, 286.24, 286.51, 315.37, 301.68, 278.57, 305.45, 288.93, 275.27, 274.99, 268.16, 320.8, 295.42, 285.86, 281.89, 279.75, 279.79, 333.69, 289.5, 283.11, 295.93, 265.05, 296.5, 309.45, 282.42, 315.69, 301.48, 329.9, 315.37, 268.83, 296.33, 306.23, 304.17, 317.03, 274.14, 295.7, 289.18, 267.95, 291.05, 293.61, 335.21, 266.38, 294.65, 290.21, 348.53, 289.98, 302.86, 261.57, 302.85, 334.72, 308.66, 304.44, 301.36, 310.61, 294.34, 308.26, 297.34, 291.66, 276.77, 295.77, 265.06, 279.54, 278.53, 324.62, 297.63, 261.56, 306.91, 306.18, 289.45, 336.75, 307.13, 287.4, 301.26, 290.79, 289.96, 309.2, 298.48, 300.04, 322.26, 303.75, 298.81, 289.68, 293.59, 271.36, 256.47, 326.39, 281.63, 322.07, 302.8, 286.1, 326.3, 300.51, 298.6, 333.61, 306.83, 289.32, 321.41, 289.19, 261.68, 296.37, 286.72, 285.07, 294.38, 290.5, 315.25, 304.12, 304.14, 291.35, 310.99, 311.95, 309.64, 280.62, 256.38, 238.73, 296.52, 319.8, 323.49, 325.59, 293.88, 307.99, 303.45, 319.28, 276.28, 320.75, 345.48, 275.97, 286.97, 288.6, 340.36, 309.37, 294.71, 289.34, 293.44, 295.77, 305.44, 329.05, 248.92, 258.47, 273.22, 253.52, 284.78, 300.89]；已知 σ=20，μ₀=300，x̄=296.0749
分析过程
H₀: μ=300，H₁: μ≠300（已知σ用z检验）。
SE=σ/√n=20/√774=0.7189；z=(x̄−μ₀)/SE=-5.4600；双侧 p=4.760e-08。
SE=σ/√n=20/√774=0.7189；z=(x̄−μ₀)/SE=-5.4600；双侧 p=4.760e-08。
计算结果
z = -5.4600，p = 4.760e-08
在 α=0.05 水平下，p=4.760e-08 < 0.05，拒绝原假设 H₀。反应能力基准检验。
双样本 z 检验
案例 93. 两厂产品均值差异
原始数据
组1 n=557：[496.62, 505.81, 492.77, 501.57, 515.17, 508.06, 491.8, 508.93, 508.61, 496.22, 495.44, 508.07, 497.16, 503.47, 495.28, 506.38, 503.79, 513.27, 514.32, 492.04, 496.29, 521.83, 513.21, 511.94, 495.97, 501.39, 513.86, 472.75, 498.68, 495.84, 503.15, 475.36, 518.61, 511.83, 500.94, 520.16, 508.0, 497.25, 500.43, 500.45, 519.64, 493.39, 507.05, 502.37, 497.54, 511.14, 504.7, 488.85, 484.57, 491.37, 503.98, 501.03, 515.2, 499.89, 501.7, 492.83, 496.3, 493.27, 512.99, 498.76, 504.58, 491.64, 492.66, 492.87, 519.53, 512.3, 492.22, 502.64, 500.74, 519.34, 493.7, 512.34, 519.5, 503.77, 510.61, 509.66, 510.83, 496.74, 503.23, 494.53, 517.01, 493.84, 502.32, 502.79, 517.75, 520.42, 498.16, 522.32, 514.64, 478.87, 500.59, 499.83, 474.01, 509.78, 500.64, 492.82, 494.45, 514.18, 498.1, 486.85, 503.1, 506.03, 491.14, 532.18, 496.41, 512.79, 513.56, 495.62, 520.73, 488.05, 525.95, 489.94, 487.86, 523.63, 512.43, 506.79, 520.05, 503.56, 504.36, 503.36, 499.87, 504.09, 514.42, 508.49, 497.09, 518.99, 527.97, 496.9, 505.02, 509.26, 516.32, 497.89, 485.46, 500.39, 511.99, 521.31, 496.15, 496.99, 508.61, 505.47, 496.43, 500.1, 500.26, 521.42, 512.12, 505.35, 499.37, 504.91, 513.6, 498.05, 490.38, 511.42, 496.27, 519.35, 496.62, 519.65, 505.12, 512.56, 495.98, 515.28, 517.37, 488.98, 500.52, 504.13, 510.18, 505.67, 491.92, 506.79, 504.68, 505.32, 502.49, 482.35, 497.51, 501.94, 488.54, 495.42, 496.13, 505.47, 492.26, 496.78, 512.41, 517.33, 511.58, 511.77, 516.65, 498.13, 514.31, 516.62, 503.13, 502.18, 502.13, 486.63, 490.11, 492.56, 498.09, 495.39, 506.47, 498.0, 503.42, 525.21, 473.08, 495.76, 503.04, 502.16, 499.5, 503.48, 501.58, 494.17, 518.02, 500.57, 488.49, 495.53, 489.44, 515.08, 499.96, 503.15, 493.28, 508.61, 491.18, 516.47, 499.34, 497.81, 499.68, 504.4, 526.92, 505.37, 492.38, 494.06, 505.16, 510.11, 510.4, 505.24, 499.28, 514.66, 503.01, 504.16, 512.67, 493.24, 507.66, 517.88, 491.99, 502.02, 510.28, 497.03, 506.89, 498.57, 499.79, 527.9, 495.56, 495.25, 507.61, 503.85, 482.95, 477.96, 498.09, 494.22, 499.89, 512.03, 514.39, 495.78, 499.95, 505.83, 507.77, 524.16, 478.56, 497.1, 493.73, 500.62, 503.04, 486.73, 496.76, 487.27, 490.33, 491.58, 493.44, 515.46, 506.39, 500.31, 493.98, 507.07, 505.32, 509.27, 515.76, 494.26, 497.2, 491.97, 494.6, 516.9, 497.95, 502.29, 508.64, 512.48, 499.14, 495.48, 505.81, 524.82, 526.37, 509.19, 488.76, 490.15, 506.57, 489.85, 503.05, 489.93, 497.55, 501.5, 507.23, 485.91, 519.53, 516.11, 503.42, 490.47, 495.9, 510.04, 491.97, 506.45, 514.83, 495.29, 497.92, 488.54, 507.99, 497.83, 507.78, 505.97, 513.75, 493.91, 492.83, 485.84, 508.86, 504.01, 510.67, 504.46, 519.72, 507.63, 495.46, 488.16, 510.41, 500.28, 506.71, 510.67, 486.09, 488.72, 498.99, 511.61, 496.07, 484.05, 500.01, 502.86, 499.35, 499.08, 490.33, 484.82, 501.6, 495.07, 501.33, 503.47, 491.53, 510.58, 506.45, 485.86, 485.79, 500.41, 495.05, 509.79, 500.92, 496.97, 507.16, 495.62, 488.41, 509.01, 513.01, 490.63, 513.52, 516.51, 509.54, 486.81, 496.84, 501.45, 501.28, 493.11, 502.25, 502.63, 490.37, 506.09, 512.11, 494.85, 509.69, 528.55, 496.21, 486.62, 485.22, 497.89, 494.31, 499.23, 484.06, 506.75, 488.68, 507.65, 498.14, 499.2, 514.92, 514.38, 515.57, 498.83, 510.93, 496.41, 506.13, 496.53, 515.03, 507.44, 503.01, 505.69, 503.64, 502.49, 508.79, 482.28, 497.27, 496.86, 488.85, 527.59, 490.28, 510.2, 499.25, 484.88, 484.47, 509.23, 501.65, 502.97, 498.17, 493.59, 512.33, 495.9, 509.4, 510.27, 504.92, 501.71, 492.51, 498.7, 493.43, 507.05, 525.68, 514.48, 502.04, 513.58, 508.47, 507.43, 501.42, 500.45, 491.61, 498.66, 505.14, 490.74, 503.9, 502.21, 492.98, 496.75, 489.96, 510.38, 519.41, 484.35, 504.49, 505.92, 488.64, 505.66, 495.87, 496.95, 491.73, 494.4, 496.28, 516.32, 506.02, 491.81, 497.06, 493.67, 518.46, 505.68, 498.68, 500.27, 518.75, 517.16, 491.27, 502.06, 509.03, 501.1, 508.9, 497.46, 515.29, 505.78, 515.73, 497.09, 495.78, 491.97, 491.2, 512.19, 512.99, 511.83, 514.63, 512.58, 502.91, 493.26, 496.92, 505.46, 500.45, 499.28, 504.28, 516.55, 495.97, 480.15, 504.88, 497.44, 494.48, 482.28, 500.49, 494.0, 495.06, 500.28, 490.89, 496.27, 510.55, 492.42, 501.91, 494.43, 502.07, 505.8, 484.88, 503.11, 496.12, 509.06, 495.66, 495.95, 509.54, 518.03, 490.6, 523.24, 498.53, 494.99, 502.82, 502.08, 501.5, 499.4, 513.78, 473.66, 526.26, 490.16, 489.99, 502.38, 485.08, 504.29, 497.87, 489.14, 508.91, 505.06, 498.37, 493.83, 498.37, 496.12, 482.22]；x̄₁=502.0580，已知 σ₁=10；
组2 n=364：[506.63, 471.13, 499.83, 493.98, 499.43, 485.05, 524.01, 495.63, 496.15, 513.18, 504.16, 494.34, 500.02, 503.49, 500.96, 500.36, 511.86, 493.38, 485.49, 503.85, 490.71, 496.73, 478.94, 482.09, 499.53, 501.94, 500.39, 486.86, 496.4, 472.17, 493.77, 484.53, 505.71, 485.69, 485.87, 498.57, 484.74, 493.58, 483.33, 498.62, 493.12, 487.31, 505.5, 483.53, 513.74, 509.25, 491.96, 484.73, 504.39, 511.06, 483.78, 502.42, 514.1, 486.68, 505.4, 505.44, 495.54, 487.02, 491.95, 495.79, 502.81, 482.84, 503.96, 491.32, 498.65, 485.05, 490.51, 512.9, 497.31, 486.81, 496.94, 487.43, 476.49, 495.35, 479.18, 490.9, 509.2, 512.09, 505.24, 503.91, 500.96, 509.12, 508.75, 511.38, 484.29, 509.05, 499.75, 505.59, 497.02, 492.99, 509.56, 490.16, 511.38, 505.83, 496.94, 494.33, 505.03, 473.94, 493.11, 508.8, 502.65, 497.31, 499.96, 503.11, 480.45, 490.36, 503.27, 493.72, 498.4, 505.4, 479.38, 485.59, 516.64, 495.34, 479.19, 496.55, 514.37, 500.62, 494.52, 522.72, 516.56, 493.75, 504.35, 493.94, 487.26, 481.3, 490.69, 495.93, 485.36, 507.99, 495.95, 492.31, 482.74, 509.14, 503.74, 500.88, 494.9, 510.32, 497.2, 495.13, 494.98, 503.09, 479.1, 506.95, 487.51, 517.11, 493.05, 491.2, 479.01, 506.22, 504.93, 496.79, 490.45, 505.61, 502.27, 492.91, 513.68, 506.42, 497.66, 490.44, 514.52, 505.47, 503.61, 479.59, 506.79, 500.63, 495.63, 494.87, 487.68, 498.73, 485.68, 509.73, 500.21, 488.58, 497.3, 489.04, 507.71, 496.2, 505.17, 496.75, 507.8, 502.41, 487.29, 512.42, 496.94, 496.34, 497.56, 499.22, 499.36, 493.65, 488.27, 502.9, 490.06, 478.46, 497.27, 501.4, 500.77, 502.9, 493.6, 492.97, 489.88, 494.31, 509.34, 510.16, 501.02, 500.71, 494.04, 499.83, 505.56, 492.08, 502.61, 505.46, 501.42, 491.82, 495.56, 489.59, 495.71, 500.64, 493.11, 505.0, 496.07, 490.63, 507.08, 501.89, 491.42, 495.47, 509.91, 502.65, 512.63, 509.01, 519.4, 491.34, 495.87, 499.87, 491.95, 492.01, 506.14, 475.88, 509.71, 477.97, 495.44, 488.29, 511.92, 477.13, 499.59, 486.36, 495.29, 501.18, 502.42, 491.21, 501.85, 503.7, 485.31, 501.87, 499.07, 483.08, 510.61, 498.78, 491.57, 491.19, 506.23, 504.98, 493.02, 485.29, 490.31, 492.38, 511.3, 489.8, 498.05, 508.0, 500.92, 474.63, 497.46, 491.94, 494.29, 495.4, 492.84, 498.37, 505.86, 504.45, 495.21, 486.78, 477.9, 504.3, 494.81, 486.33, 499.11, 497.55, 511.68, 505.5, 487.04, 511.28, 487.55, 491.72, 501.41, 490.2, 500.1, 494.98, 499.74, 484.38, 501.89, 498.66, 486.1, 493.99, 503.05, 482.1, 513.79, 475.54, 498.93, 492.75, 488.57, 486.35, 528.3, 486.35, 493.32, 491.97, 491.21, 491.76, 500.31, 484.86, 520.29, 505.53, 514.98, 521.88, 489.65, 489.89, 494.56, 505.45, 507.32, 487.5, 494.58, 515.69, 506.9, 502.24, 489.64, 489.4, 510.93, 514.73, 486.05, 498.57, 478.85, 501.11, 487.42, 507.45, 507.15, 475.61, 490.4, 493.63, 509.57, 493.58, 494.44, 513.82, 491.61, 495.5, 510.89, 488.22, 497.76, 506.0, 506.68, 492.19, 488.57, 489.32, 497.9, 492.7]；x̄₂=497.2422，已知 σ₂=10
组2 n=364：[506.63, 471.13, 499.83, 493.98, 499.43, 485.05, 524.01, 495.63, 496.15, 513.18, 504.16, 494.34, 500.02, 503.49, 500.96, 500.36, 511.86, 493.38, 485.49, 503.85, 490.71, 496.73, 478.94, 482.09, 499.53, 501.94, 500.39, 486.86, 496.4, 472.17, 493.77, 484.53, 505.71, 485.69, 485.87, 498.57, 484.74, 493.58, 483.33, 498.62, 493.12, 487.31, 505.5, 483.53, 513.74, 509.25, 491.96, 484.73, 504.39, 511.06, 483.78, 502.42, 514.1, 486.68, 505.4, 505.44, 495.54, 487.02, 491.95, 495.79, 502.81, 482.84, 503.96, 491.32, 498.65, 485.05, 490.51, 512.9, 497.31, 486.81, 496.94, 487.43, 476.49, 495.35, 479.18, 490.9, 509.2, 512.09, 505.24, 503.91, 500.96, 509.12, 508.75, 511.38, 484.29, 509.05, 499.75, 505.59, 497.02, 492.99, 509.56, 490.16, 511.38, 505.83, 496.94, 494.33, 505.03, 473.94, 493.11, 508.8, 502.65, 497.31, 499.96, 503.11, 480.45, 490.36, 503.27, 493.72, 498.4, 505.4, 479.38, 485.59, 516.64, 495.34, 479.19, 496.55, 514.37, 500.62, 494.52, 522.72, 516.56, 493.75, 504.35, 493.94, 487.26, 481.3, 490.69, 495.93, 485.36, 507.99, 495.95, 492.31, 482.74, 509.14, 503.74, 500.88, 494.9, 510.32, 497.2, 495.13, 494.98, 503.09, 479.1, 506.95, 487.51, 517.11, 493.05, 491.2, 479.01, 506.22, 504.93, 496.79, 490.45, 505.61, 502.27, 492.91, 513.68, 506.42, 497.66, 490.44, 514.52, 505.47, 503.61, 479.59, 506.79, 500.63, 495.63, 494.87, 487.68, 498.73, 485.68, 509.73, 500.21, 488.58, 497.3, 489.04, 507.71, 496.2, 505.17, 496.75, 507.8, 502.41, 487.29, 512.42, 496.94, 496.34, 497.56, 499.22, 499.36, 493.65, 488.27, 502.9, 490.06, 478.46, 497.27, 501.4, 500.77, 502.9, 493.6, 492.97, 489.88, 494.31, 509.34, 510.16, 501.02, 500.71, 494.04, 499.83, 505.56, 492.08, 502.61, 505.46, 501.42, 491.82, 495.56, 489.59, 495.71, 500.64, 493.11, 505.0, 496.07, 490.63, 507.08, 501.89, 491.42, 495.47, 509.91, 502.65, 512.63, 509.01, 519.4, 491.34, 495.87, 499.87, 491.95, 492.01, 506.14, 475.88, 509.71, 477.97, 495.44, 488.29, 511.92, 477.13, 499.59, 486.36, 495.29, 501.18, 502.42, 491.21, 501.85, 503.7, 485.31, 501.87, 499.07, 483.08, 510.61, 498.78, 491.57, 491.19, 506.23, 504.98, 493.02, 485.29, 490.31, 492.38, 511.3, 489.8, 498.05, 508.0, 500.92, 474.63, 497.46, 491.94, 494.29, 495.4, 492.84, 498.37, 505.86, 504.45, 495.21, 486.78, 477.9, 504.3, 494.81, 486.33, 499.11, 497.55, 511.68, 505.5, 487.04, 511.28, 487.55, 491.72, 501.41, 490.2, 500.1, 494.98, 499.74, 484.38, 501.89, 498.66, 486.1, 493.99, 503.05, 482.1, 513.79, 475.54, 498.93, 492.75, 488.57, 486.35, 528.3, 486.35, 493.32, 491.97, 491.21, 491.76, 500.31, 484.86, 520.29, 505.53, 514.98, 521.88, 489.65, 489.89, 494.56, 505.45, 507.32, 487.5, 494.58, 515.69, 506.9, 502.24, 489.64, 489.4, 510.93, 514.73, 486.05, 498.57, 478.85, 501.11, 487.42, 507.45, 507.15, 475.61, 490.4, 493.63, 509.57, 493.58, 494.44, 513.82, 491.61, 495.5, 510.89, 488.22, 497.76, 506.0, 506.68, 492.19, 488.57, 489.32, 497.9, 492.7]；x̄₂=497.2422，已知 σ₂=10
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.6740；z=(x̄₁−x̄₂)/SE=7.1452；双侧 p=8.988e-13。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.6740；z=(x̄₁−x̄₂)/SE=7.1452；双侧 p=8.988e-13。
计算结果
z = 7.1452，p = 8.988e-13
在 α=0.05 水平下，p=8.988e-13 < 0.05，拒绝原假设 H₀。已知σ比较两厂均值。
案例 94. 两城市收入差异
原始数据
组1 n=484：[4261.6, 4247.45, 5116.61, 4635.52, 3149.12, 6543.41, 3955.88, 5181.08, 6101.71, 5716.11, 4622.68, 5025.75, 5178.03, 4555.59, 5057.85, 6209.54, 3345.68, 6234.65, 4565.94, 6658.62, 4673.08, 4227.17, 5808.03, 3308.93, 5587.87, 4528.59, 3722.5, 5628.48, 4707.94, 5403.94, 5268.53, 5753.64, 4683.83, 5218.31, 4996.06, 4291.92, 5560.67, 5918.24, 4588.48, 5661.18, 5845.8, 5561.65, 3626.08, 4937.98, 3780.42, 7493.17, 3727.53, 4894.09, 2947.73, 5433.96, 6087.28, 5629.45, 5702.51, 4870.26, 4256.41, 4525.72, 4933.46, 5724.52, 4626.9, 4926.63, 4657.25, 4081.81, 5334.24, 4389.67, 3458.41, 4262.62, 5723.13, 7482.71, 5515.32, 4114.06, 4425.74, 5100.7, 4714.39, 4483.18, 3741.3, 6190.04, 6513.74, 6542.78, 6437.05, 4845.5, 5794.86, 5687.62, 5181.01, 4510.85, 5857.65, 5159.28, 5139.53, 4166.11, 4846.91, 6473.51, 6087.3, 5983.75, 5371.81, 5298.83, 5836.71, 5616.5, 4398.06, 3970.54, 5870.79, 6278.57, 6325.84, 5030.71, 4220.88, 3907.97, 5503.93, 6842.57, 3735.56, 5486.34, 3436.2, 7194.71, 3960.27, 3841.65, 4876.13, 5164.04, 5371.08, 4474.65, 4814.33, 6149.02, 4824.86, 5501.41, 3909.26, 5079.32, 5544.84, 4178.98, 5194.61, 4381.9, 5410.64, 5090.5, 6855.36, 4459.02, 4051.63, 4922.42, 5729.86, 4998.26, 4614.32, 5316.54, 5008.15, 4554.91, 4230.24, 5068.49, 3431.32, 4333.29, 4509.09, 5204.23, 4574.66, 5063.18, 5300.15, 4334.83, 5515.51, 5262.62, 5878.18, 3048.37, 4551.43, 6217.32, 4346.1, 4395.35, 3196.21, 4151.27, 4612.86, 5266.55, 4797.43, 3800.88, 5196.27, 4571.86, 6281.07, 4002.71, 4032.05, 4291.71, 5103.76, 5345.33, 6006.53, 5271.19, 4839.96, 5177.92, 5219.34, 3155.72, 4758.69, 4009.89, 5547.89, 4740.12, 4739.96, 5055.62, 4321.09, 4470.13, 4143.17, 5235.38, 6122.0, 6149.97, 5163.52, 4062.46, 6110.26, 3346.87, 4994.62, 4933.27, 3954.3, 4479.46, 4218.02, 6100.64, 3923.94, 6110.69, 4539.24, 4877.84, 4446.46, 5962.64, 4000.27, 4895.5, 4159.15, 4231.44, 3385.35, 5743.03, 5141.02, 5113.45, 3861.68, 4690.12, 5429.61, 6927.38, 4877.3, 3946.51, 3338.19, 5668.49, 5378.99, 4899.1, 5625.84, 5612.38, 6995.26, 4189.48, 6012.82, 4070.59, 4259.2, 5634.46, 3498.96, 5373.57, 4753.37, 5224.91, 4433.11, 5455.43, 4517.5, 4435.41, 5894.84, 4307.37, 4930.51, 5200.02, 4979.52, 5421.5, 4233.21, 5781.72, 4252.93, 7219.87, 3575.59, 5507.38, 3430.93, 3969.74, 4278.16, 5409.65, 5260.28, 5516.18, 5302.77, 5102.5, 5326.04, 5460.0, 4608.62, 5134.45, 5759.34, 6871.7, 4910.14, 3586.03, 5178.68, 4798.97, 4575.85, 5504.02, 5688.39, 5617.09, 4041.26, 5358.89, 4925.97, 4408.59, 5618.08, 4882.33, 5942.35, 4954.11, 4182.47, 4432.04, 4936.33, 4205.89, 5872.05, 3530.35, 4541.83, 6589.88, 5405.63, 5226.28, 4602.96, 6706.74, 5144.68, 4873.71, 5683.8, 5295.31, 6000.5, 5364.44, 5193.48, 4329.53, 3737.95, 5391.83, 4562.26, 4569.49, 6372.96, 6534.81, 5680.62, 4920.51, 4063.03, 5732.32, 4207.22, 3764.17, 6162.12, 4956.28, 4389.42, 3727.09, 4219.23, 4653.21, 5139.58, 6646.16, 5418.32, 5783.24, 4644.59, 5655.14, 4900.76, 3480.31, 4496.25, 5337.03, 3668.93, 5076.89, 4649.75, 6023.83, 6491.78, 4491.73, 5643.58, 5276.09, 4723.63, 5889.85, 5451.18, 4531.81, 4657.88, 4935.39, 6793.58, 4251.98, 4414.89, 3431.41, 6266.33, 5011.71, 5354.67, 6456.41, 5099.01, 5809.42, 5468.03, 4384.77, 5323.84, 6938.55, 4215.53, 3949.37, 4771.37, 4474.11, 4833.18, 5023.53, 4152.94, 5167.95, 5823.18, 5295.81, 6015.55, 5007.98, 3423.77, 3644.71, 4821.61, 3928.37, 4415.18, 4300.11, 4429.53, 5527.18, 5087.01, 5018.69, 4186.08, 5238.76, 4045.59, 6240.35, 3495.64, 4844.99, 4734.62, 4054.75, 4765.12, 4440.93, 4768.81, 5105.15, 3755.71, 4849.88, 4180.95, 3898.67, 4796.01, 4270.44, 4359.15, 3551.09, 4474.44, 7325.35, 4426.8, 5568.79, 4922.63, 6569.57, 6570.48, 5956.91, 5233.15, 4478.38, 4412.16, 5076.96, 5514.92, 5126.79, 4571.54, 4720.84, 6125.61, 4872.09, 4663.32, 4624.45, 6269.13, 5684.0, 5229.95, 5445.45, 5645.27, 4724.83, 5507.48, 4964.46, 4707.75, 3774.34, 4124.42, 5662.08, 5198.43, 5406.63, 4217.01, 4946.53, 6198.26, 4205.43, 6111.15, 4648.33, 6131.0, 6529.52, 5079.63, 4382.53, 4319.78, 4642.43, 4785.77, 5625.91, 4961.29, 4939.01, 4163.54, 5103.24, 4913.52, 4944.32, 4567.61, 4939.05, 5475.4, 5981.43, 5183.71, 3822.98, 6893.42, 5084.03, 4773.59, 3944.86, 4013.28, 4733.76, 6736.76, 5483.09, 6265.08, 5510.08, 3817.02, 5657.24, 5603.03, 4583.79, 3928.07, 4172.94, 5674.11, 4630.19, 4509.08, 6578.21, 5607.39, 5059.78, 5124.97, 5361.18, 5029.88, 2913.14]；x̄₁=4988.2678，已知 σ₁=800；
组2 n=895：[4571.43, 3261.4, 3332.39, 3870.53, 4560.55, 4397.92, 2433.37, 4540.66, 5007.58, 3539.1, 5019.27, 5766.92, 4263.35, 4999.54, 3911.91, 4259.28, 5276.12, 4183.59, 5043.52, 6574.7, 3940.76, 4584.86, 4023.62, 3624.48, 4449.72, 6125.28, 4945.96, 5109.28, 4567.71, 3416.46, 3591.53, 5890.02, 4289.64, 5411.59, 4774.78, 3985.62, 3460.88, 5229.84, 4810.03, 3901.68, 6545.3, 3992.29, 4846.59, 4461.81, 4553.63, 6278.64, 4250.7, 6046.3, 4235.99, 4386.69, 3044.72, 4329.08, 4222.95, 5018.86, 5084.69, 4452.78, 3533.35, 5486.31, 5872.94, 5049.01, 4685.11, 5187.46, 4255.33, 5737.4, 4055.31, 5303.51, 4542.75, 3251.6, 4244.71, 5150.74, 2992.88, 4636.43, 4595.92, 5953.2, 4263.15, 5151.97, 4855.85, 4720.41, 4664.02, 3532.69, 4893.13, 5366.23, 4641.58, 5019.44, 4216.89, 5577.92, 5514.26, 5005.5, 5466.76, 5122.01, 4877.76, 4758.58, 5778.97, 6328.69, 4976.31, 5080.02, 2912.92, 5114.65, 4787.82, 4730.16, 5552.65, 5315.48, 2958.82, 5326.77, 5482.0, 4960.64, 4003.69, 4016.14, 5289.16, 4362.81, 4180.87, 4655.37, 4378.86, 5735.07, 3428.97, 5897.05, 4474.24, 4647.46, 5082.69, 3908.56, 4205.94, 5722.0, 4688.05, 6452.18, 4329.14, 4420.56, 4950.12, 4972.64, 4128.14, 3955.08, 5501.28, 4577.12, 4649.09, 4265.49, 5281.66, 6674.1, 3982.48, 4471.26, 5411.07, 4845.12, 4644.31, 4713.67, 5734.48, 3439.32, 4361.29, 4216.05, 3014.03, 4490.38, 5098.83, 4612.58, 4888.41, 5999.2, 6009.0, 5902.41, 4135.0, 4801.25, 3435.14, 4193.1, 4144.93, 5294.75, 4520.99, 3808.26, 5842.3, 5574.0, 4905.85, 3574.36, 4787.13, 5372.63, 4497.69, 4443.61, 4007.78, 5683.61, 4449.46, 3925.82, 4254.52, 4849.24, 5092.45, 6126.26, 4628.41, 4570.76, 3944.52, 4609.08, 3391.54, 4478.86, 4230.86, 4543.94, 5771.12, 5078.02, 3674.57, 4893.23, 4030.66, 5053.68, 4751.2, 3647.67, 4184.06, 5889.35, 4418.71, 6254.84, 4048.26, 6407.89, 5298.51, 5237.07, 4739.04, 5020.5, 5904.44, 6301.9, 5244.2, 4667.42, 5729.5, 3984.59, 5179.69, 3041.08, 4208.68, 4969.19, 4876.53, 3966.16, 4894.49, 5732.9, 4568.14, 4889.0, 4292.34, 3982.03, 4743.57, 4219.48, 5089.61, 4736.25, 4585.38, 5008.87, 6054.51, 4131.53, 3838.11, 6251.53, 5093.48, 4238.45, 3430.08, 3722.35, 5548.08, 4078.36, 4434.74, 4257.3, 5243.61, 4517.24, 5176.11, 4596.61, 6001.56, 5848.29, 5449.55, 5043.7, 4651.31, 5256.14, 5323.29, 5446.18, 4044.66, 4556.96, 3993.98, 4675.68, 4999.8, 5251.18, 5249.45, 5692.73, 4393.07, 3853.78, 4637.61, 4674.74, 6194.77, 4469.61, 2164.87, 5671.74, 4938.53, 5806.28, 4273.63, 4395.3, 5205.06, 3622.16, 4540.53, 4106.7, 4905.88, 7092.52, 5520.84, 4848.67, 3742.9, 4226.14, 3905.06, 4866.32, 5225.32, 4853.17, 3527.82, 4848.92, 4611.95, 3975.49, 5330.36, 4599.93, 6043.86, 4213.24, 4435.9, 4378.38, 5967.41, 6003.27, 3391.26, 4327.41, 5749.57, 4036.99, 4455.97, 3879.94, 4677.59, 5786.48, 5407.72, 5133.7, 5963.43, 5984.82, 4082.26, 3639.82, 4073.56, 3943.93, 5280.72, 4841.01, 5838.89, 4687.12, 5474.4, 4843.67, 4212.61, 5564.73, 5205.85, 5260.24, 2705.91, 4522.86, 4800.36, 3826.56, 4858.13, 4771.12, 4917.46, 4356.73, 4954.14, 4651.78, 3662.58, 5237.47, 5455.03, 6165.88, 5083.52, 5457.99, 4469.34, 4927.7, 5096.09, 5417.06, 4195.26, 3859.92, 3625.39, 5483.49, 4934.92, 4328.45, 5730.45, 4900.34, 5306.56, 4540.94, 6071.29, 6238.91, 5544.78, 3953.01, 4794.06, 3930.47, 5150.2, 6449.44, 4601.74, 4922.46, 5070.32, 5898.3, 3291.16, 5135.38, 5063.87, 4516.98, 4755.87, 6011.06, 5190.11, 4995.9, 5032.31, 5909.65, 5023.86, 4902.43, 4340.05, 4688.45, 4277.7, 4734.32, 6754.49, 4683.28, 3796.47, 3966.47, 5273.9, 4808.69, 5790.77, 5305.7, 5735.55, 4318.29, 4780.52, 3425.29, 4354.27, 4190.45, 5048.02, 2691.17, 5144.01, 4837.81, 4300.38, 5159.16, 5348.47, 4834.18, 4777.2, 5793.62, 4842.4, 5096.11, 5036.97, 4187.54, 5655.42, 5963.78, 6312.4, 6021.83, 6406.84, 3962.81, 3445.03, 3855.08, 4971.0, 4183.37, 4886.5, 4934.98, 3124.99, 3577.75, 6091.3, 4125.82, 4454.69, 4557.2, 3011.59, 4847.35, 4945.79, 4862.2, 4209.96, 4583.28, 4590.22, 4789.12, 5244.19, 4543.45, 4131.71, 4718.36, 4728.34, 4102.96, 3381.69, 3590.57, 3922.35, 4961.1, 3976.45, 3894.19, 4163.85, 4789.15, 3652.93, 6017.86, 5516.54, 5459.78, 3612.89, 4700.04, 4771.43, 4812.96, 5983.56, 6767.27, 5626.32, 5215.66, 5574.16, 3489.89, 5943.28, 5127.49, 5424.97, 4830.13, 5106.15, 4352.39, 4767.38, 4748.21, 4641.43, 3878.1, 5093.8, 4063.01, 4227.08, 3966.52, 4907.13, 5077.1, 5347.43, 4603.04, 3607.84, 4966.12, 3580.26, 4622.49, 5489.33, 4960.93, 5652.14, 5257.36, 4853.07, 5400.36, 4196.04, 5541.61, 4633.09, 5320.67, 5208.43, 4529.77, 2356.83, 6087.83, 5031.68, 4366.68, 4705.58, 6293.18, 5740.12, 5250.54, 3378.15, 4353.7, 3858.14, 4659.01, 3367.61, 6147.97, 4672.11, 4677.76, 3676.13, 4216.3, 4301.33, 4874.77, 3887.65, 5287.69, 4964.65, 4068.33, 6999.51, 5117.97, 4274.57, 5615.1, 4962.11, 4434.14, 5088.31, 3103.06, 4889.57, 6309.13, 5169.79, 4276.0, 4590.16, 4611.8, 4704.15, 3773.2, 4591.9, 6121.46, 3422.55, 1903.24, 6236.0, 5147.19, 7341.35, 5009.09, 4672.49, 4529.19, 4803.85, 4659.34, 5208.03, 6340.12, 5613.26, 5137.4, 5282.84, 5281.14, 4544.15, 3782.62, 5222.94, 4472.3, 4758.06, 5292.56, 4703.63, 5516.99, 4632.5, 4444.56, 5669.02, 5048.29, 3419.33, 5739.14, 4997.9, 4259.11, 6077.6, 4749.99, 6447.3, 5479.13, 4352.93, 5089.83, 5410.88, 4823.99, 5085.19, 4561.58, 4645.7, 5605.78, 4036.56, 3756.28, 5740.84, 4896.6, 3931.52, 4580.1, 5822.25, 4492.23, 5244.5, 5522.74, 5622.93, 5841.77, 5254.57, 5194.71, 5554.4, 4832.26, 4081.89, 4035.53, 6205.61, 3462.06, 6017.94, 4528.09, 3700.08, 5217.21, 4781.71, 5860.78, 5780.61, 5124.97, 4344.66, 5273.04, 5461.0, 4233.65, 4284.07, 4799.27, 4552.95, 4954.97, 3666.52, 4560.05, 5675.43, 3420.76, 4435.51, 3594.12, 5380.75, 4804.6, 4649.41, 5379.92, 5451.59, 4290.64, 4600.31, 4737.2, 4435.87, 4904.87, 6167.3, 4916.43, 3930.13, 4799.97, 3474.85, 4601.02, 5508.09, 4635.29, 5281.28, 3977.48, 4907.04, 4786.48, 3144.08, 4966.8, 5481.4, 3913.61, 4292.56, 5634.69, 4709.16, 3419.47, 2607.28, 4231.34, 5657.76, 5487.98, 5582.88, 4173.74, 3926.2, 4720.11, 5450.17, 5275.06, 4651.17, 6414.29, 5219.56, 4695.12, 4231.17, 4436.53, 4086.94, 4467.25, 6331.84, 5994.99, 4168.03, 4712.88, 5522.51, 3960.01, 6248.07, 5153.78, 5210.56, 3497.94, 4949.53, 4870.27, 5534.55, 4741.65, 6020.74, 4743.68, 4653.77, 4940.27, 5346.13, 5753.83, 4715.07, 2679.35, 4997.78, 5190.24, 4594.05, 2726.53, 3804.01, 5363.95, 5667.2, 5723.39, 5822.4, 5806.58, 5223.96, 4214.37, 6573.65, 5420.51, 4613.68, 4402.89, 3695.95, 5805.58, 5475.54, 5010.99, 4297.75, 4857.49, 3437.54, 4951.02, 6828.46, 5193.36, 5282.31, 5265.82, 4341.73, 6255.12, 4117.8, 4499.05, 3307.46, 5937.54, 4556.89, 5333.0, 5388.55, 4272.96, 6474.26, 4501.63, 6477.88, 5549.02, 5333.09, 4332.33, 3759.58, 4715.76, 5281.11, 4227.06, 3357.81, 4918.55, 5524.9, 5041.53, 4999.31, 4603.72, 4547.53, 5565.6, 5082.61, 4697.76, 4588.96, 5728.18, 5791.01, 5491.52, 4182.25, 5306.87, 4574.96, 4828.65, 6113.15, 4781.93, 4699.67, 4493.48, 4993.9, 5072.44, 4983.43, 5017.99, 5479.55, 5196.89, 4193.24, 3707.3, 4294.54, 5487.43, 4929.08, 5910.31, 3762.5, 4408.6, 4561.05, 6108.21, 4945.47, 3826.04, 4098.83, 4977.25, 5013.62, 5129.48, 5819.53, 3333.11, 4758.19, 6237.96, 4776.8, 5073.93, 4800.66, 5536.9, 5683.99, 5821.38, 4593.48, 5171.02, 4707.67, 3459.86, 4701.98, 4219.42, 4010.85, 4107.83, 4743.69, 4702.62, 4625.15, 5228.99, 4502.56, 4881.97, 4592.77, 4351.5, 5116.55, 3906.01, 3389.38, 4824.85, 4538.55, 4850.53, 4090.05, 4967.5, 4758.57, 5234.07, 5314.2, 4453.78, 6096.98, 4585.59, 4879.87, 4337.62, 5181.32, 4548.07, 4339.55, 3392.08, 5639.35, 4389.88, 4081.47, 5440.93, 5324.25, 3746.71, 4777.25, 3148.24, 4953.12, 2952.84, 4398.52, 4438.17, 5006.17, 4890.93, 5085.57, 5488.53, 5053.29, 5871.21, 4309.82, 4282.4, 3235.51, 5655.0, 5917.68, 3325.23, 5542.41, 4900.38, 5503.81, 5504.17, 3866.43, 4997.53, 5336.2, 3801.02, 4428.45, 4728.67, 5866.09, 4674.5, 3752.36, 4461.76, 4670.3, 5006.82, 3701.58, 5002.97, 4256.75, 5954.24, 4523.56, 4058.48, 4921.15, 5247.84, 4234.87, 5241.1, 5927.26, 5965.99, 4655.43, 5087.05, 5088.48, 5576.99, 5900.92, 5114.36, 5657.23, 3779.26, 5334.72]；x̄₂=4796.1083，已知 σ₂=800
组2 n=895：[4571.43, 3261.4, 3332.39, 3870.53, 4560.55, 4397.92, 2433.37, 4540.66, 5007.58, 3539.1, 5019.27, 5766.92, 4263.35, 4999.54, 3911.91, 4259.28, 5276.12, 4183.59, 5043.52, 6574.7, 3940.76, 4584.86, 4023.62, 3624.48, 4449.72, 6125.28, 4945.96, 5109.28, 4567.71, 3416.46, 3591.53, 5890.02, 4289.64, 5411.59, 4774.78, 3985.62, 3460.88, 5229.84, 4810.03, 3901.68, 6545.3, 3992.29, 4846.59, 4461.81, 4553.63, 6278.64, 4250.7, 6046.3, 4235.99, 4386.69, 3044.72, 4329.08, 4222.95, 5018.86, 5084.69, 4452.78, 3533.35, 5486.31, 5872.94, 5049.01, 4685.11, 5187.46, 4255.33, 5737.4, 4055.31, 5303.51, 4542.75, 3251.6, 4244.71, 5150.74, 2992.88, 4636.43, 4595.92, 5953.2, 4263.15, 5151.97, 4855.85, 4720.41, 4664.02, 3532.69, 4893.13, 5366.23, 4641.58, 5019.44, 4216.89, 5577.92, 5514.26, 5005.5, 5466.76, 5122.01, 4877.76, 4758.58, 5778.97, 6328.69, 4976.31, 5080.02, 2912.92, 5114.65, 4787.82, 4730.16, 5552.65, 5315.48, 2958.82, 5326.77, 5482.0, 4960.64, 4003.69, 4016.14, 5289.16, 4362.81, 4180.87, 4655.37, 4378.86, 5735.07, 3428.97, 5897.05, 4474.24, 4647.46, 5082.69, 3908.56, 4205.94, 5722.0, 4688.05, 6452.18, 4329.14, 4420.56, 4950.12, 4972.64, 4128.14, 3955.08, 5501.28, 4577.12, 4649.09, 4265.49, 5281.66, 6674.1, 3982.48, 4471.26, 5411.07, 4845.12, 4644.31, 4713.67, 5734.48, 3439.32, 4361.29, 4216.05, 3014.03, 4490.38, 5098.83, 4612.58, 4888.41, 5999.2, 6009.0, 5902.41, 4135.0, 4801.25, 3435.14, 4193.1, 4144.93, 5294.75, 4520.99, 3808.26, 5842.3, 5574.0, 4905.85, 3574.36, 4787.13, 5372.63, 4497.69, 4443.61, 4007.78, 5683.61, 4449.46, 3925.82, 4254.52, 4849.24, 5092.45, 6126.26, 4628.41, 4570.76, 3944.52, 4609.08, 3391.54, 4478.86, 4230.86, 4543.94, 5771.12, 5078.02, 3674.57, 4893.23, 4030.66, 5053.68, 4751.2, 3647.67, 4184.06, 5889.35, 4418.71, 6254.84, 4048.26, 6407.89, 5298.51, 5237.07, 4739.04, 5020.5, 5904.44, 6301.9, 5244.2, 4667.42, 5729.5, 3984.59, 5179.69, 3041.08, 4208.68, 4969.19, 4876.53, 3966.16, 4894.49, 5732.9, 4568.14, 4889.0, 4292.34, 3982.03, 4743.57, 4219.48, 5089.61, 4736.25, 4585.38, 5008.87, 6054.51, 4131.53, 3838.11, 6251.53, 5093.48, 4238.45, 3430.08, 3722.35, 5548.08, 4078.36, 4434.74, 4257.3, 5243.61, 4517.24, 5176.11, 4596.61, 6001.56, 5848.29, 5449.55, 5043.7, 4651.31, 5256.14, 5323.29, 5446.18, 4044.66, 4556.96, 3993.98, 4675.68, 4999.8, 5251.18, 5249.45, 5692.73, 4393.07, 3853.78, 4637.61, 4674.74, 6194.77, 4469.61, 2164.87, 5671.74, 4938.53, 5806.28, 4273.63, 4395.3, 5205.06, 3622.16, 4540.53, 4106.7, 4905.88, 7092.52, 5520.84, 4848.67, 3742.9, 4226.14, 3905.06, 4866.32, 5225.32, 4853.17, 3527.82, 4848.92, 4611.95, 3975.49, 5330.36, 4599.93, 6043.86, 4213.24, 4435.9, 4378.38, 5967.41, 6003.27, 3391.26, 4327.41, 5749.57, 4036.99, 4455.97, 3879.94, 4677.59, 5786.48, 5407.72, 5133.7, 5963.43, 5984.82, 4082.26, 3639.82, 4073.56, 3943.93, 5280.72, 4841.01, 5838.89, 4687.12, 5474.4, 4843.67, 4212.61, 5564.73, 5205.85, 5260.24, 2705.91, 4522.86, 4800.36, 3826.56, 4858.13, 4771.12, 4917.46, 4356.73, 4954.14, 4651.78, 3662.58, 5237.47, 5455.03, 6165.88, 5083.52, 5457.99, 4469.34, 4927.7, 5096.09, 5417.06, 4195.26, 3859.92, 3625.39, 5483.49, 4934.92, 4328.45, 5730.45, 4900.34, 5306.56, 4540.94, 6071.29, 6238.91, 5544.78, 3953.01, 4794.06, 3930.47, 5150.2, 6449.44, 4601.74, 4922.46, 5070.32, 5898.3, 3291.16, 5135.38, 5063.87, 4516.98, 4755.87, 6011.06, 5190.11, 4995.9, 5032.31, 5909.65, 5023.86, 4902.43, 4340.05, 4688.45, 4277.7, 4734.32, 6754.49, 4683.28, 3796.47, 3966.47, 5273.9, 4808.69, 5790.77, 5305.7, 5735.55, 4318.29, 4780.52, 3425.29, 4354.27, 4190.45, 5048.02, 2691.17, 5144.01, 4837.81, 4300.38, 5159.16, 5348.47, 4834.18, 4777.2, 5793.62, 4842.4, 5096.11, 5036.97, 4187.54, 5655.42, 5963.78, 6312.4, 6021.83, 6406.84, 3962.81, 3445.03, 3855.08, 4971.0, 4183.37, 4886.5, 4934.98, 3124.99, 3577.75, 6091.3, 4125.82, 4454.69, 4557.2, 3011.59, 4847.35, 4945.79, 4862.2, 4209.96, 4583.28, 4590.22, 4789.12, 5244.19, 4543.45, 4131.71, 4718.36, 4728.34, 4102.96, 3381.69, 3590.57, 3922.35, 4961.1, 3976.45, 3894.19, 4163.85, 4789.15, 3652.93, 6017.86, 5516.54, 5459.78, 3612.89, 4700.04, 4771.43, 4812.96, 5983.56, 6767.27, 5626.32, 5215.66, 5574.16, 3489.89, 5943.28, 5127.49, 5424.97, 4830.13, 5106.15, 4352.39, 4767.38, 4748.21, 4641.43, 3878.1, 5093.8, 4063.01, 4227.08, 3966.52, 4907.13, 5077.1, 5347.43, 4603.04, 3607.84, 4966.12, 3580.26, 4622.49, 5489.33, 4960.93, 5652.14, 5257.36, 4853.07, 5400.36, 4196.04, 5541.61, 4633.09, 5320.67, 5208.43, 4529.77, 2356.83, 6087.83, 5031.68, 4366.68, 4705.58, 6293.18, 5740.12, 5250.54, 3378.15, 4353.7, 3858.14, 4659.01, 3367.61, 6147.97, 4672.11, 4677.76, 3676.13, 4216.3, 4301.33, 4874.77, 3887.65, 5287.69, 4964.65, 4068.33, 6999.51, 5117.97, 4274.57, 5615.1, 4962.11, 4434.14, 5088.31, 3103.06, 4889.57, 6309.13, 5169.79, 4276.0, 4590.16, 4611.8, 4704.15, 3773.2, 4591.9, 6121.46, 3422.55, 1903.24, 6236.0, 5147.19, 7341.35, 5009.09, 4672.49, 4529.19, 4803.85, 4659.34, 5208.03, 6340.12, 5613.26, 5137.4, 5282.84, 5281.14, 4544.15, 3782.62, 5222.94, 4472.3, 4758.06, 5292.56, 4703.63, 5516.99, 4632.5, 4444.56, 5669.02, 5048.29, 3419.33, 5739.14, 4997.9, 4259.11, 6077.6, 4749.99, 6447.3, 5479.13, 4352.93, 5089.83, 5410.88, 4823.99, 5085.19, 4561.58, 4645.7, 5605.78, 4036.56, 3756.28, 5740.84, 4896.6, 3931.52, 4580.1, 5822.25, 4492.23, 5244.5, 5522.74, 5622.93, 5841.77, 5254.57, 5194.71, 5554.4, 4832.26, 4081.89, 4035.53, 6205.61, 3462.06, 6017.94, 4528.09, 3700.08, 5217.21, 4781.71, 5860.78, 5780.61, 5124.97, 4344.66, 5273.04, 5461.0, 4233.65, 4284.07, 4799.27, 4552.95, 4954.97, 3666.52, 4560.05, 5675.43, 3420.76, 4435.51, 3594.12, 5380.75, 4804.6, 4649.41, 5379.92, 5451.59, 4290.64, 4600.31, 4737.2, 4435.87, 4904.87, 6167.3, 4916.43, 3930.13, 4799.97, 3474.85, 4601.02, 5508.09, 4635.29, 5281.28, 3977.48, 4907.04, 4786.48, 3144.08, 4966.8, 5481.4, 3913.61, 4292.56, 5634.69, 4709.16, 3419.47, 2607.28, 4231.34, 5657.76, 5487.98, 5582.88, 4173.74, 3926.2, 4720.11, 5450.17, 5275.06, 4651.17, 6414.29, 5219.56, 4695.12, 4231.17, 4436.53, 4086.94, 4467.25, 6331.84, 5994.99, 4168.03, 4712.88, 5522.51, 3960.01, 6248.07, 5153.78, 5210.56, 3497.94, 4949.53, 4870.27, 5534.55, 4741.65, 6020.74, 4743.68, 4653.77, 4940.27, 5346.13, 5753.83, 4715.07, 2679.35, 4997.78, 5190.24, 4594.05, 2726.53, 3804.01, 5363.95, 5667.2, 5723.39, 5822.4, 5806.58, 5223.96, 4214.37, 6573.65, 5420.51, 4613.68, 4402.89, 3695.95, 5805.58, 5475.54, 5010.99, 4297.75, 4857.49, 3437.54, 4951.02, 6828.46, 5193.36, 5282.31, 5265.82, 4341.73, 6255.12, 4117.8, 4499.05, 3307.46, 5937.54, 4556.89, 5333.0, 5388.55, 4272.96, 6474.26, 4501.63, 6477.88, 5549.02, 5333.09, 4332.33, 3759.58, 4715.76, 5281.11, 4227.06, 3357.81, 4918.55, 5524.9, 5041.53, 4999.31, 4603.72, 4547.53, 5565.6, 5082.61, 4697.76, 4588.96, 5728.18, 5791.01, 5491.52, 4182.25, 5306.87, 4574.96, 4828.65, 6113.15, 4781.93, 4699.67, 4493.48, 4993.9, 5072.44, 4983.43, 5017.99, 5479.55, 5196.89, 4193.24, 3707.3, 4294.54, 5487.43, 4929.08, 5910.31, 3762.5, 4408.6, 4561.05, 6108.21, 4945.47, 3826.04, 4098.83, 4977.25, 5013.62, 5129.48, 5819.53, 3333.11, 4758.19, 6237.96, 4776.8, 5073.93, 4800.66, 5536.9, 5683.99, 5821.38, 4593.48, 5171.02, 4707.67, 3459.86, 4701.98, 4219.42, 4010.85, 4107.83, 4743.69, 4702.62, 4625.15, 5228.99, 4502.56, 4881.97, 4592.77, 4351.5, 5116.55, 3906.01, 3389.38, 4824.85, 4538.55, 4850.53, 4090.05, 4967.5, 4758.57, 5234.07, 5314.2, 4453.78, 6096.98, 4585.59, 4879.87, 4337.62, 5181.32, 4548.07, 4339.55, 3392.08, 5639.35, 4389.88, 4081.47, 5440.93, 5324.25, 3746.71, 4777.25, 3148.24, 4953.12, 2952.84, 4398.52, 4438.17, 5006.17, 4890.93, 5085.57, 5488.53, 5053.29, 5871.21, 4309.82, 4282.4, 3235.51, 5655.0, 5917.68, 3325.23, 5542.41, 4900.38, 5503.81, 5504.17, 3866.43, 4997.53, 5336.2, 3801.02, 4428.45, 4728.67, 5866.09, 4674.5, 3752.36, 4461.76, 4670.3, 5006.82, 3701.58, 5002.97, 4256.75, 5954.24, 4523.56, 4058.48, 4921.15, 5247.84, 4234.87, 5241.1, 5927.26, 5965.99, 4655.43, 5087.05, 5088.48, 5576.99, 5900.92, 5114.36, 5657.23, 3779.26, 5334.72]；x̄₂=4796.1083，已知 σ₂=800
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=45.1375；z=(x̄₁−x̄₂)/SE=4.2572；双侧 p=2.070e-05。
SE=√(σ₁²/n₁+σ₂²/n₂)=45.1375；z=(x̄₁−x̄₂)/SE=4.2572；双侧 p=2.070e-05。
计算结果
z = 4.2572，p = 2.070e-05
在 α=0.05 水平下，p=2.070e-05 < 0.05，拒绝原假设 H₀。比较城市收入水平。
案例 95. 两种药疗效差异
原始数据
组1 n=921：[89.87, 91.51, 87.1, 73.1, 98.38, 83.52, 82.33, 74.39, 69.54, 63.63, 80.48, 91.79, 83.48, 86.32, 86.5, 90.62, 97.63, 81.63, 79.24, 101.35, 89.21, 75.12, 93.95, 93.33, 88.59, 87.09, 97.83, 96.06, 91.04, 85.39, 81.77, 77.89, 93.39, 84.66, 66.86, 74.57, 79.91, 95.82, 81.32, 79.51, 78.22, 101.53, 94.19, 89.81, 100.03, 77.6, 105.68, 82.51, 78.51, 97.16, 90.91, 82.81, 86.78, 83.93, 79.76, 96.07, 86.74, 87.12, 87.28, 84.75, 86.55, 84.96, 89.62, 86.55, 91.92, 92.15, 84.83, 80.19, 88.68, 96.64, 84.5, 86.7, 87.65, 91.57, 86.01, 95.43, 80.97, 95.65, 72.66, 96.89, 79.93, 88.24, 88.44, 69.48, 81.8, 72.87, 89.62, 78.09, 87.69, 83.45, 85.86, 83.91, 81.25, 72.39, 81.89, 85.6, 81.75, 77.14, 85.68, 84.77, 80.29, 91.25, 95.39, 87.92, 82.65, 92.1, 107.29, 68.12, 94.94, 81.53, 70.36, 95.79, 82.82, 89.02, 96.18, 92.84, 69.05, 75.8, 89.07, 84.6, 89.39, 68.73, 85.19, 85.35, 79.73, 88.67, 87.85, 86.99, 76.81, 84.17, 90.39, 95.44, 87.14, 86.15, 78.61, 81.57, 70.2, 92.02, 74.47, 95.02, 79.78, 94.98, 76.86, 78.57, 76.66, 100.56, 81.61, 81.34, 85.3, 73.87, 83.38, 96.39, 86.95, 88.04, 77.88, 75.21, 75.42, 73.4, 81.8, 79.99, 93.9, 93.72, 84.27, 80.31, 69.53, 77.23, 91.71, 75.29, 103.03, 82.77, 81.17, 88.57, 91.94, 89.56, 73.19, 85.26, 96.94, 85.88, 87.05, 89.29, 85.16, 101.63, 78.13, 84.35, 76.23, 72.07, 93.73, 74.44, 73.27, 84.81, 73.14, 79.75, 77.35, 99.37, 72.21, 85.59, 81.9, 89.32, 89.81, 103.98, 94.0, 86.27, 73.39, 93.49, 74.39, 92.72, 77.82, 81.28, 101.53, 92.59, 72.32, 81.77, 82.49, 82.08, 87.0, 86.43, 83.7, 87.86, 73.93, 79.31, 99.53, 91.2, 88.89, 86.94, 72.41, 76.42, 87.27, 88.83, 77.28, 103.45, 85.21, 74.47, 87.98, 69.97, 82.15, 85.13, 105.41, 74.68, 85.01, 82.76, 89.53, 82.85, 80.81, 83.09, 74.43, 99.93, 97.24, 85.46, 91.78, 99.16, 64.81, 79.74, 79.31, 85.79, 86.54, 78.04, 85.76, 85.3, 94.46, 94.72, 78.57, 88.59, 91.39, 82.03, 89.92, 90.62, 88.29, 92.88, 83.02, 82.34, 98.97, 86.55, 81.65, 85.09, 79.49, 78.2, 85.87, 84.57, 92.55, 86.58, 79.47, 86.36, 84.95, 75.08, 79.79, 89.58, 82.58, 82.53, 74.24, 80.75, 82.63, 77.53, 70.62, 92.37, 81.77, 93.44, 98.08, 81.87, 78.8, 87.74, 73.73, 87.5, 86.79, 76.29, 80.98, 89.84, 76.82, 83.42, 85.55, 84.34, 74.26, 78.89, 87.64, 77.21, 76.14, 77.84, 82.62, 80.64, 79.62, 88.34, 87.88, 85.44, 89.44, 80.61, 93.76, 91.95, 81.88, 92.91, 92.01, 75.68, 83.7, 88.43, 97.73, 80.41, 91.86, 86.21, 76.01, 86.41, 96.6, 85.93, 80.13, 87.92, 96.58, 94.63, 74.94, 92.93, 85.58, 77.06, 86.94, 91.13, 93.28, 72.27, 85.61, 72.31, 67.26, 97.12, 92.62, 79.75, 97.34, 88.86, 82.18, 88.89, 92.23, 83.08, 75.24, 82.31, 85.92, 79.86, 83.4, 67.93, 88.36, 67.28, 79.11, 95.23, 79.67, 91.54, 79.19, 89.47, 90.46, 86.71, 95.73, 85.33, 96.73, 78.04, 92.23, 90.52, 95.54, 75.6, 76.72, 82.52, 91.55, 108.11, 66.78, 88.1, 91.31, 78.98, 75.65, 101.04, 84.36, 87.76, 85.79, 89.17, 79.62, 109.67, 75.05, 96.7, 81.09, 91.26, 83.77, 74.76, 91.13, 91.49, 89.75, 94.93, 66.65, 103.85, 75.51, 81.42, 93.26, 93.12, 79.9, 98.56, 89.08, 90.48, 77.12, 89.91, 83.27, 75.91, 82.97, 68.18, 69.02, 95.36, 81.21, 95.69, 84.95, 71.98, 74.77, 96.87, 73.17, 86.61, 77.54, 94.47, 90.73, 82.77, 94.05, 94.23, 96.49, 82.31, 85.16, 102.15, 87.53, 93.45, 81.49, 82.21, 96.33, 80.02, 88.64, 77.19, 85.83, 85.18, 83.29, 86.71, 80.24, 86.53, 94.44, 97.11, 83.6, 83.6, 83.6, 76.8, 105.51, 72.21, 91.11, 85.33, 84.74, 88.85, 78.96, 100.46, 81.44, 68.34, 90.13, 97.71, 90.52, 87.34, 76.49, 70.1, 73.51, 82.96, 82.29, 70.13, 85.7, 64.07, 82.46, 84.41, 84.78, 64.77, 91.19, 85.89, 78.4, 76.4, 88.98, 81.39, 83.05, 98.52, 84.43, 84.71, 93.06, 79.99, 82.46, 94.12, 105.69, 92.65, 82.19, 85.07, 99.55, 71.88, 64.78, 86.33, 92.09, 82.83, 74.9, 85.99, 82.76, 89.54, 90.84, 87.4, 90.94, 88.55, 85.58, 88.71, 72.76, 77.15, 82.03, 86.42, 85.45, 82.07, 80.3, 93.88, 92.96, 92.23, 89.02, 81.28, 82.61, 68.71, 96.62, 76.36, 75.82, 85.16, 93.15, 85.72, 88.37, 98.38, 79.64, 95.25, 88.52, 91.97, 90.29, 88.45, 74.68, 88.88, 84.58, 83.09, 82.6, 75.54, 88.22, 90.66, 82.86, 78.24, 98.76, 102.76, 77.87, 85.48, 86.64, 87.91, 92.6, 77.05, 78.68, 91.51, 90.03, 78.39, 88.38, 80.75, 76.48, 89.56, 81.68, 85.56, 92.23, 98.79, 80.87, 93.21, 93.7, 87.86, 98.32, 93.77, 88.2, 99.28, 84.84, 83.77, 80.31, 86.68, 84.84, 90.97, 76.47, 82.17, 66.16, 90.05, 70.82, 84.37, 90.0, 84.54, 77.85, 84.5, 85.1, 80.75, 94.27, 81.13, 94.77, 91.85, 86.76, 80.83, 95.25, 63.29, 64.99, 75.75, 86.25, 75.13, 78.95, 80.26, 76.07, 80.54, 82.91, 84.15, 82.98, 75.08, 78.21, 88.76, 81.27, 72.76, 94.93, 92.53, 81.03, 68.67, 89.52, 84.87, 78.85, 85.31, 96.7, 100.3, 81.25, 84.96, 92.16, 96.54, 71.72, 82.24, 90.98, 97.6, 77.98, 92.99, 80.36, 78.85, 88.07, 95.18, 86.89, 97.37, 87.28, 82.4, 90.64, 80.96, 81.56, 85.05, 83.01, 84.79, 82.7, 83.68, 87.26, 86.0, 76.03, 85.88, 90.65, 74.52, 78.46, 84.1, 98.18, 87.11, 91.52, 77.96, 78.92, 85.08, 93.02, 96.96, 84.84, 75.37, 85.1, 94.55, 83.26, 74.01, 82.28, 80.1, 69.53, 86.35, 86.1, 91.47, 95.66, 82.25, 80.69, 60.76, 93.17, 93.13, 79.33, 86.72, 81.84, 88.73, 94.89, 88.08, 85.21, 87.1, 90.94, 71.63, 93.67, 67.17, 83.06, 107.59, 82.84, 77.6, 90.81, 89.16, 97.1, 70.34, 84.17, 74.45, 83.95, 84.52, 84.84, 79.52, 77.53, 77.33, 95.32, 64.91, 90.18, 83.27, 85.75, 81.82, 80.52, 89.67, 89.82, 93.18, 98.59, 95.92, 94.76, 74.83, 78.13, 92.51, 88.1, 86.04, 74.07, 82.79, 77.65, 94.34, 72.17, 92.94, 79.55, 91.01, 82.2, 87.22, 85.63, 79.18, 82.81, 73.62, 82.56, 91.1, 84.36, 85.53, 76.81, 92.57, 79.4, 71.34, 87.4, 93.7, 91.87, 78.58, 92.77, 84.67, 87.57, 81.57, 94.16, 88.63, 95.26, 79.91, 76.9, 86.14, 84.41, 91.31, 90.29, 80.66, 88.66, 70.0, 89.6, 85.89, 78.85, 86.74, 97.09, 103.11, 76.5, 94.13, 89.29, 81.65, 81.11, 82.38, 79.32, 83.57, 84.21, 79.16, 86.21, 79.86, 79.68, 79.44, 90.73, 95.99, 91.21, 77.91, 77.24, 84.86, 86.78, 82.14, 80.19, 80.43, 85.08, 74.62, 83.83, 78.23, 90.01, 91.26, 87.59, 72.24, 100.94, 98.32, 81.89, 75.93, 82.19, 82.68, 81.49, 93.91, 82.88, 85.45, 80.88, 99.43, 79.63, 90.45, 84.56, 87.01, 89.83, 101.19, 89.75, 94.29, 74.32, 94.44, 81.31, 86.88, 77.34, 88.82, 78.13, 74.78, 90.81, 72.79, 84.73, 87.33, 80.99, 79.59, 81.31, 80.74, 70.28, 79.1, 88.66, 88.11, 99.88, 75.59, 85.17, 82.82, 84.53, 86.62, 93.83, 77.61, 94.72, 78.47, 92.35, 85.23, 87.57, 75.92, 81.45, 67.89, 77.59, 82.9, 97.45, 87.03, 72.27, 73.64, 79.55, 71.86, 87.29, 84.42, 94.18, 91.72, 77.75, 84.72, 88.58, 74.99, 93.51, 73.12, 101.16, 90.34, 100.05, 99.38, 73.58, 89.01, 102.32, 73.6, 82.48, 94.07, 86.45, 104.63, 85.04]；x̄₁=85.0746，已知 σ₁=8；
组2 n=978：[92.91, 90.02, 74.06, 70.03, 87.59, 73.31, 76.92, 78.15, 74.04, 85.57, 91.31, 90.26, 86.25, 81.37, 95.4, 66.73, 78.91, 84.94, 81.65, 80.17, 79.14, 90.02, 79.21, 84.69, 85.96, 80.3, 75.16, 97.27, 87.94, 71.85, 84.75, 92.17, 100.78, 91.27, 80.2, 103.05, 89.62, 94.33, 81.64, 81.71, 76.01, 94.88, 93.09, 84.19, 84.28, 88.04, 80.26, 70.09, 83.77, 83.26, 88.6, 72.08, 78.46, 94.16, 80.05, 72.46, 76.15, 75.75, 84.0, 78.84, 84.64, 87.94, 93.06, 84.98, 84.96, 81.43, 69.22, 80.92, 91.6, 66.1, 84.67, 80.17, 95.97, 78.79, 86.0, 83.67, 83.8, 82.54, 78.1, 75.31, 81.8, 99.01, 89.99, 81.39, 72.29, 78.63, 85.43, 92.08, 87.9, 68.87, 83.1, 78.46, 79.8, 80.61, 81.94, 95.73, 83.03, 87.38, 93.25, 83.61, 68.09, 81.03, 93.48, 72.21, 100.82, 84.85, 68.72, 86.82, 93.72, 93.9, 79.72, 71.7, 86.54, 85.22, 84.96, 75.81, 90.81, 82.38, 89.23, 95.37, 82.6, 85.98, 87.19, 77.12, 86.0, 86.13, 85.38, 77.03, 83.05, 89.62, 85.43, 93.37, 87.93, 74.32, 73.4, 82.94, 88.06, 76.59, 92.35, 79.48, 81.67, 79.5, 68.63, 87.87, 77.19, 76.22, 87.35, 88.66, 80.59, 74.09, 93.62, 80.01, 86.38, 96.67, 87.21, 87.22, 81.07, 85.82, 91.8, 92.5, 74.95, 80.39, 76.45, 89.77, 77.93, 80.29, 84.87, 73.66, 90.68, 92.58, 75.3, 74.34, 74.27, 87.67, 80.15, 77.23, 84.97, 73.07, 82.19, 92.23, 87.82, 71.09, 78.93, 91.46, 87.53, 74.17, 70.09, 80.28, 90.13, 73.27, 90.06, 72.21, 82.07, 83.02, 83.74, 79.19, 85.34, 90.18, 82.55, 82.5, 75.92, 69.48, 75.85, 77.01, 74.46, 85.86, 90.69, 75.83, 79.59, 82.43, 90.22, 80.64, 68.64, 93.13, 84.35, 78.78, 80.35, 79.93, 79.51, 82.24, 95.41, 83.62, 78.31, 75.25, 81.6, 84.61, 91.39, 78.22, 76.64, 84.94, 65.7, 84.38, 72.5, 73.84, 80.94, 82.46, 92.74, 70.03, 87.02, 83.16, 82.77, 84.44, 83.49, 85.65, 79.89, 95.5, 70.78, 72.66, 78.27, 91.92, 86.89, 76.17, 81.69, 77.54, 78.55, 82.09, 76.95, 76.61, 73.91, 90.1, 76.52, 73.55, 86.79, 77.55, 75.49, 67.7, 85.22, 76.83, 84.56, 86.68, 81.56, 77.36, 84.62, 83.41, 81.13, 82.95, 82.43, 86.18, 78.27, 86.71, 77.05, 78.41, 68.2, 82.62, 97.15, 76.03, 75.38, 76.38, 79.23, 84.1, 79.81, 80.93, 68.16, 83.3, 63.97, 85.73, 88.77, 79.64, 69.96, 91.15, 78.52, 85.33, 73.47, 83.21, 77.67, 85.19, 92.35, 84.62, 91.08, 82.06, 85.79, 77.47, 88.15, 91.0, 82.21, 78.72, 70.98, 85.88, 78.04, 86.28, 88.39, 97.59, 95.6, 84.36, 81.9, 84.26, 85.03, 86.89, 88.78, 82.62, 73.88, 78.0, 68.43, 80.83, 85.93, 83.14, 73.2, 87.6, 82.19, 76.72, 76.19, 70.47, 89.01, 89.21, 99.06, 82.69, 94.22, 84.32, 88.9, 97.17, 81.05, 78.32, 86.73, 74.77, 84.77, 76.83, 92.52, 63.39, 76.52, 88.35, 96.87, 70.67, 76.79, 72.49, 77.96, 80.59, 89.08, 87.76, 89.9, 78.38, 84.42, 68.43, 87.52, 91.74, 70.36, 91.24, 78.82, 85.0, 80.77, 94.77, 82.98, 62.03, 89.54, 66.84, 80.74, 86.23, 81.2, 80.63, 65.47, 74.68, 84.61, 97.67, 66.17, 68.5, 82.89, 75.51, 80.33, 73.3, 72.81, 77.55, 91.21, 87.27, 83.54, 81.84, 88.43, 81.33, 97.32, 85.26, 82.87, 79.85, 76.49, 78.16, 73.91, 89.35, 80.71, 79.07, 75.73, 96.27, 86.57, 84.57, 78.35, 79.21, 93.46, 72.24, 81.9, 82.33, 93.19, 86.47, 80.4, 92.01, 93.36, 85.71, 85.51, 84.03, 73.54, 68.31, 85.07, 82.99, 77.28, 61.63, 95.77, 70.9, 66.39, 78.93, 75.56, 79.45, 84.58, 66.33, 76.49, 80.16, 76.07, 91.61, 87.67, 75.37, 87.53, 80.85, 79.27, 86.14, 74.48, 78.97, 83.65, 84.04, 66.28, 68.35, 89.34, 83.09, 77.47, 96.58, 66.75, 81.54, 72.91, 75.77, 85.67, 79.56, 89.97, 80.76, 84.97, 82.3, 84.33, 90.71, 92.13, 74.26, 86.89, 84.41, 84.28, 78.6, 69.57, 81.96, 81.15, 78.91, 92.32, 67.3, 82.54, 86.23, 83.24, 85.2, 96.31, 89.53, 88.94, 81.06, 88.98, 82.94, 70.56, 83.2, 70.48, 85.51, 77.14, 98.38, 83.93, 74.33, 79.2, 79.06, 79.52, 79.87, 67.77, 81.74, 67.39, 76.76, 80.21, 77.53, 87.87, 88.4, 80.46, 81.03, 73.99, 84.47, 90.66, 91.33, 75.12, 76.59, 84.3, 78.3, 86.93, 85.77, 78.17, 81.73, 93.02, 83.98, 79.39, 69.07, 73.6, 68.15, 79.0, 76.41, 76.49, 82.01, 78.81, 78.41, 89.93, 79.83, 76.59, 83.08, 80.01, 71.58, 82.08, 82.86, 72.71, 76.44, 79.97, 75.75, 85.46, 69.07, 72.56, 84.46, 65.45, 69.58, 88.0, 83.07, 93.05, 73.04, 80.24, 82.55, 74.24, 74.01, 64.05, 68.43, 85.48, 76.27, 74.37, 72.6, 84.34, 83.42, 80.7, 71.06, 90.6, 84.31, 79.63, 76.71, 81.38, 91.81, 84.79, 63.78, 94.28, 91.15, 93.22, 87.58, 76.62, 71.54, 87.12, 75.01, 94.5, 77.56, 85.43, 64.14, 78.87, 81.27, 99.23, 84.78, 78.12, 87.53, 83.35, 90.34, 69.24, 66.74, 90.79, 80.88, 82.75, 77.82, 85.27, 78.17, 95.13, 71.18, 93.86, 90.34, 81.82, 78.31, 93.25, 86.97, 69.64, 74.98, 76.08, 88.66, 71.61, 86.95, 88.62, 85.18, 68.29, 90.14, 82.26, 66.81, 81.37, 65.8, 81.22, 80.09, 92.52, 78.98, 84.33, 93.81, 92.64, 79.1, 95.75, 90.11, 90.58, 75.16, 80.24, 83.92, 78.69, 90.7, 96.97, 69.06, 88.07, 90.25, 77.19, 75.94, 87.59, 86.68, 71.62, 81.35, 79.72, 82.05, 77.65, 86.65, 96.22, 68.36, 79.5, 79.81, 85.18, 73.17, 84.12, 94.54, 75.27, 81.41, 74.4, 89.12, 61.2, 78.0, 69.25, 87.52, 83.0, 87.72, 95.86, 94.15, 66.77, 76.65, 72.25, 74.33, 70.91, 82.18, 68.51, 97.03, 90.84, 80.99, 79.07, 80.49, 90.36, 72.39, 85.85, 88.32, 83.09, 94.26, 88.28, 82.24, 85.43, 76.66, 74.81, 78.46, 84.22, 77.28, 80.69, 74.64, 60.81, 79.28, 89.61, 69.65, 81.27, 72.85, 70.73, 73.48, 77.14, 88.81, 79.94, 100.38, 89.57, 91.14, 91.74, 72.56, 77.37, 64.4, 80.77, 81.87, 71.76, 81.67, 78.06, 81.11, 74.89, 82.99, 98.16, 73.36, 83.2, 85.44, 82.59, 82.73, 84.13, 86.64, 74.15, 73.71, 90.62, 79.12, 93.52, 89.07, 73.94, 93.22, 88.32, 86.08, 87.02, 82.96, 83.75, 63.67, 87.91, 97.87, 86.97, 77.16, 87.05, 70.62, 94.05, 80.26, 102.41, 77.89, 97.16, 86.89, 81.19, 76.87, 94.49, 76.32, 73.52, 73.63, 75.67, 82.24, 82.8, 88.88, 92.36, 82.89, 76.6, 70.35, 66.3, 90.81, 77.32, 90.26, 80.6, 79.03, 76.37, 65.02, 71.16, 79.49, 92.72, 83.91, 73.62, 81.16, 81.69, 81.14, 75.45, 87.82, 78.53, 68.0, 79.7, 83.79, 73.24, 81.42, 98.8, 89.84, 79.91, 87.35, 89.23, 67.86, 60.13, 75.1, 84.94, 71.84, 75.98, 72.77, 88.95, 73.21, 78.91, 94.6, 91.9, 86.5, 69.67, 77.11, 77.75, 91.49, 79.94, 80.22, 90.13, 83.49, 91.84, 93.36, 75.6, 84.44, 88.7, 74.59, 91.43, 75.08, 75.96, 82.5, 77.77, 81.4, 82.59, 87.23, 91.77, 79.95, 76.44, 81.57, 78.63, 88.99, 87.05, 90.8, 81.32, 76.64, 85.0, 93.16, 78.79, 74.05, 82.07, 78.76, 82.57, 82.51, 76.04, 89.4, 94.87, 78.26, 78.74, 69.81, 88.93, 80.36, 97.19, 81.44, 87.1, 97.25, 87.41, 88.95, 87.04, 86.9, 101.8, 71.8, 85.42, 84.46, 75.49, 95.05, 71.11, 72.93, 76.01, 86.06, 82.66, 74.63, 100.8, 61.36, 89.38, 67.13, 76.15, 80.88, 85.64, 69.02, 73.1, 81.01, 91.25, 76.22, 78.96, 94.52, 84.58, 88.65, 78.54, 82.58, 87.94, 79.4, 78.22, 75.15, 77.18, 73.46, 86.9, 68.24, 88.63, 66.18, 83.09, 76.66, 75.89, 90.19, 86.84, 81.48, 78.65, 90.56, 82.51, 98.28, 74.05, 74.87, 84.29, 78.83, 94.72, 86.79, 89.04, 92.47, 77.12, 73.6, 83.77, 75.58, 87.47, 87.75, 93.74, 80.12, 83.96, 93.74, 87.14, 74.45, 80.39, 79.62, 81.58, 73.77, 66.7, 76.67, 69.85, 81.17, 92.84, 71.34, 98.76, 71.86, 74.19, 84.0, 76.56, 81.0]；x̄₂=81.8332，已知 σ₂=8
组2 n=978：[92.91, 90.02, 74.06, 70.03, 87.59, 73.31, 76.92, 78.15, 74.04, 85.57, 91.31, 90.26, 86.25, 81.37, 95.4, 66.73, 78.91, 84.94, 81.65, 80.17, 79.14, 90.02, 79.21, 84.69, 85.96, 80.3, 75.16, 97.27, 87.94, 71.85, 84.75, 92.17, 100.78, 91.27, 80.2, 103.05, 89.62, 94.33, 81.64, 81.71, 76.01, 94.88, 93.09, 84.19, 84.28, 88.04, 80.26, 70.09, 83.77, 83.26, 88.6, 72.08, 78.46, 94.16, 80.05, 72.46, 76.15, 75.75, 84.0, 78.84, 84.64, 87.94, 93.06, 84.98, 84.96, 81.43, 69.22, 80.92, 91.6, 66.1, 84.67, 80.17, 95.97, 78.79, 86.0, 83.67, 83.8, 82.54, 78.1, 75.31, 81.8, 99.01, 89.99, 81.39, 72.29, 78.63, 85.43, 92.08, 87.9, 68.87, 83.1, 78.46, 79.8, 80.61, 81.94, 95.73, 83.03, 87.38, 93.25, 83.61, 68.09, 81.03, 93.48, 72.21, 100.82, 84.85, 68.72, 86.82, 93.72, 93.9, 79.72, 71.7, 86.54, 85.22, 84.96, 75.81, 90.81, 82.38, 89.23, 95.37, 82.6, 85.98, 87.19, 77.12, 86.0, 86.13, 85.38, 77.03, 83.05, 89.62, 85.43, 93.37, 87.93, 74.32, 73.4, 82.94, 88.06, 76.59, 92.35, 79.48, 81.67, 79.5, 68.63, 87.87, 77.19, 76.22, 87.35, 88.66, 80.59, 74.09, 93.62, 80.01, 86.38, 96.67, 87.21, 87.22, 81.07, 85.82, 91.8, 92.5, 74.95, 80.39, 76.45, 89.77, 77.93, 80.29, 84.87, 73.66, 90.68, 92.58, 75.3, 74.34, 74.27, 87.67, 80.15, 77.23, 84.97, 73.07, 82.19, 92.23, 87.82, 71.09, 78.93, 91.46, 87.53, 74.17, 70.09, 80.28, 90.13, 73.27, 90.06, 72.21, 82.07, 83.02, 83.74, 79.19, 85.34, 90.18, 82.55, 82.5, 75.92, 69.48, 75.85, 77.01, 74.46, 85.86, 90.69, 75.83, 79.59, 82.43, 90.22, 80.64, 68.64, 93.13, 84.35, 78.78, 80.35, 79.93, 79.51, 82.24, 95.41, 83.62, 78.31, 75.25, 81.6, 84.61, 91.39, 78.22, 76.64, 84.94, 65.7, 84.38, 72.5, 73.84, 80.94, 82.46, 92.74, 70.03, 87.02, 83.16, 82.77, 84.44, 83.49, 85.65, 79.89, 95.5, 70.78, 72.66, 78.27, 91.92, 86.89, 76.17, 81.69, 77.54, 78.55, 82.09, 76.95, 76.61, 73.91, 90.1, 76.52, 73.55, 86.79, 77.55, 75.49, 67.7, 85.22, 76.83, 84.56, 86.68, 81.56, 77.36, 84.62, 83.41, 81.13, 82.95, 82.43, 86.18, 78.27, 86.71, 77.05, 78.41, 68.2, 82.62, 97.15, 76.03, 75.38, 76.38, 79.23, 84.1, 79.81, 80.93, 68.16, 83.3, 63.97, 85.73, 88.77, 79.64, 69.96, 91.15, 78.52, 85.33, 73.47, 83.21, 77.67, 85.19, 92.35, 84.62, 91.08, 82.06, 85.79, 77.47, 88.15, 91.0, 82.21, 78.72, 70.98, 85.88, 78.04, 86.28, 88.39, 97.59, 95.6, 84.36, 81.9, 84.26, 85.03, 86.89, 88.78, 82.62, 73.88, 78.0, 68.43, 80.83, 85.93, 83.14, 73.2, 87.6, 82.19, 76.72, 76.19, 70.47, 89.01, 89.21, 99.06, 82.69, 94.22, 84.32, 88.9, 97.17, 81.05, 78.32, 86.73, 74.77, 84.77, 76.83, 92.52, 63.39, 76.52, 88.35, 96.87, 70.67, 76.79, 72.49, 77.96, 80.59, 89.08, 87.76, 89.9, 78.38, 84.42, 68.43, 87.52, 91.74, 70.36, 91.24, 78.82, 85.0, 80.77, 94.77, 82.98, 62.03, 89.54, 66.84, 80.74, 86.23, 81.2, 80.63, 65.47, 74.68, 84.61, 97.67, 66.17, 68.5, 82.89, 75.51, 80.33, 73.3, 72.81, 77.55, 91.21, 87.27, 83.54, 81.84, 88.43, 81.33, 97.32, 85.26, 82.87, 79.85, 76.49, 78.16, 73.91, 89.35, 80.71, 79.07, 75.73, 96.27, 86.57, 84.57, 78.35, 79.21, 93.46, 72.24, 81.9, 82.33, 93.19, 86.47, 80.4, 92.01, 93.36, 85.71, 85.51, 84.03, 73.54, 68.31, 85.07, 82.99, 77.28, 61.63, 95.77, 70.9, 66.39, 78.93, 75.56, 79.45, 84.58, 66.33, 76.49, 80.16, 76.07, 91.61, 87.67, 75.37, 87.53, 80.85, 79.27, 86.14, 74.48, 78.97, 83.65, 84.04, 66.28, 68.35, 89.34, 83.09, 77.47, 96.58, 66.75, 81.54, 72.91, 75.77, 85.67, 79.56, 89.97, 80.76, 84.97, 82.3, 84.33, 90.71, 92.13, 74.26, 86.89, 84.41, 84.28, 78.6, 69.57, 81.96, 81.15, 78.91, 92.32, 67.3, 82.54, 86.23, 83.24, 85.2, 96.31, 89.53, 88.94, 81.06, 88.98, 82.94, 70.56, 83.2, 70.48, 85.51, 77.14, 98.38, 83.93, 74.33, 79.2, 79.06, 79.52, 79.87, 67.77, 81.74, 67.39, 76.76, 80.21, 77.53, 87.87, 88.4, 80.46, 81.03, 73.99, 84.47, 90.66, 91.33, 75.12, 76.59, 84.3, 78.3, 86.93, 85.77, 78.17, 81.73, 93.02, 83.98, 79.39, 69.07, 73.6, 68.15, 79.0, 76.41, 76.49, 82.01, 78.81, 78.41, 89.93, 79.83, 76.59, 83.08, 80.01, 71.58, 82.08, 82.86, 72.71, 76.44, 79.97, 75.75, 85.46, 69.07, 72.56, 84.46, 65.45, 69.58, 88.0, 83.07, 93.05, 73.04, 80.24, 82.55, 74.24, 74.01, 64.05, 68.43, 85.48, 76.27, 74.37, 72.6, 84.34, 83.42, 80.7, 71.06, 90.6, 84.31, 79.63, 76.71, 81.38, 91.81, 84.79, 63.78, 94.28, 91.15, 93.22, 87.58, 76.62, 71.54, 87.12, 75.01, 94.5, 77.56, 85.43, 64.14, 78.87, 81.27, 99.23, 84.78, 78.12, 87.53, 83.35, 90.34, 69.24, 66.74, 90.79, 80.88, 82.75, 77.82, 85.27, 78.17, 95.13, 71.18, 93.86, 90.34, 81.82, 78.31, 93.25, 86.97, 69.64, 74.98, 76.08, 88.66, 71.61, 86.95, 88.62, 85.18, 68.29, 90.14, 82.26, 66.81, 81.37, 65.8, 81.22, 80.09, 92.52, 78.98, 84.33, 93.81, 92.64, 79.1, 95.75, 90.11, 90.58, 75.16, 80.24, 83.92, 78.69, 90.7, 96.97, 69.06, 88.07, 90.25, 77.19, 75.94, 87.59, 86.68, 71.62, 81.35, 79.72, 82.05, 77.65, 86.65, 96.22, 68.36, 79.5, 79.81, 85.18, 73.17, 84.12, 94.54, 75.27, 81.41, 74.4, 89.12, 61.2, 78.0, 69.25, 87.52, 83.0, 87.72, 95.86, 94.15, 66.77, 76.65, 72.25, 74.33, 70.91, 82.18, 68.51, 97.03, 90.84, 80.99, 79.07, 80.49, 90.36, 72.39, 85.85, 88.32, 83.09, 94.26, 88.28, 82.24, 85.43, 76.66, 74.81, 78.46, 84.22, 77.28, 80.69, 74.64, 60.81, 79.28, 89.61, 69.65, 81.27, 72.85, 70.73, 73.48, 77.14, 88.81, 79.94, 100.38, 89.57, 91.14, 91.74, 72.56, 77.37, 64.4, 80.77, 81.87, 71.76, 81.67, 78.06, 81.11, 74.89, 82.99, 98.16, 73.36, 83.2, 85.44, 82.59, 82.73, 84.13, 86.64, 74.15, 73.71, 90.62, 79.12, 93.52, 89.07, 73.94, 93.22, 88.32, 86.08, 87.02, 82.96, 83.75, 63.67, 87.91, 97.87, 86.97, 77.16, 87.05, 70.62, 94.05, 80.26, 102.41, 77.89, 97.16, 86.89, 81.19, 76.87, 94.49, 76.32, 73.52, 73.63, 75.67, 82.24, 82.8, 88.88, 92.36, 82.89, 76.6, 70.35, 66.3, 90.81, 77.32, 90.26, 80.6, 79.03, 76.37, 65.02, 71.16, 79.49, 92.72, 83.91, 73.62, 81.16, 81.69, 81.14, 75.45, 87.82, 78.53, 68.0, 79.7, 83.79, 73.24, 81.42, 98.8, 89.84, 79.91, 87.35, 89.23, 67.86, 60.13, 75.1, 84.94, 71.84, 75.98, 72.77, 88.95, 73.21, 78.91, 94.6, 91.9, 86.5, 69.67, 77.11, 77.75, 91.49, 79.94, 80.22, 90.13, 83.49, 91.84, 93.36, 75.6, 84.44, 88.7, 74.59, 91.43, 75.08, 75.96, 82.5, 77.77, 81.4, 82.59, 87.23, 91.77, 79.95, 76.44, 81.57, 78.63, 88.99, 87.05, 90.8, 81.32, 76.64, 85.0, 93.16, 78.79, 74.05, 82.07, 78.76, 82.57, 82.51, 76.04, 89.4, 94.87, 78.26, 78.74, 69.81, 88.93, 80.36, 97.19, 81.44, 87.1, 97.25, 87.41, 88.95, 87.04, 86.9, 101.8, 71.8, 85.42, 84.46, 75.49, 95.05, 71.11, 72.93, 76.01, 86.06, 82.66, 74.63, 100.8, 61.36, 89.38, 67.13, 76.15, 80.88, 85.64, 69.02, 73.1, 81.01, 91.25, 76.22, 78.96, 94.52, 84.58, 88.65, 78.54, 82.58, 87.94, 79.4, 78.22, 75.15, 77.18, 73.46, 86.9, 68.24, 88.63, 66.18, 83.09, 76.66, 75.89, 90.19, 86.84, 81.48, 78.65, 90.56, 82.51, 98.28, 74.05, 74.87, 84.29, 78.83, 94.72, 86.79, 89.04, 92.47, 77.12, 73.6, 83.77, 75.58, 87.47, 87.75, 93.74, 80.12, 83.96, 93.74, 87.14, 74.45, 80.39, 79.62, 81.58, 73.77, 66.7, 76.67, 69.85, 81.17, 92.84, 71.34, 98.76, 71.86, 74.19, 84.0, 76.56, 81.0]；x̄₂=81.8332，已知 σ₂=8
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.3673；z=(x̄₁−x̄₂)/SE=8.8243；双侧 p=0.000e+00。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.3673；z=(x̄₁−x̄₂)/SE=8.8243；双侧 p=0.000e+00。
计算结果
z = 8.8243，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。比较药物疗效均值。
案例 96. 两班成绩差异
原始数据
组1 n=778：[65.03, 66.9, 66.7, 78.98, 87.35, 79.01, 69.82, 77.81, 85.29, 80.05, 84.8, 68.0, 73.89, 80.79, 80.91, 68.52, 74.83, 78.8, 87.64, 71.74, 73.28, 85.89, 61.97, 87.1, 80.14, 67.53, 81.61, 87.36, 73.33, 83.1, 71.96, 75.2, 83.02, 94.99, 69.27, 87.2, 73.48, 79.19, 82.92, 62.84, 79.23, 82.84, 76.95, 67.76, 73.52, 74.19, 90.54, 84.68, 68.77, 79.83, 102.75, 77.16, 83.39, 80.88, 65.67, 71.45, 92.6, 89.19, 78.72, 79.21, 76.42, 78.3, 71.67, 86.84, 61.66, 68.02, 72.53, 89.25, 96.69, 78.42, 81.61, 95.14, 79.51, 82.0, 90.85, 65.28, 83.93, 90.59, 83.12, 87.27, 95.98, 73.77, 87.94, 79.92, 85.49, 85.49, 63.43, 60.92, 79.12, 79.37, 86.97, 77.66, 68.24, 86.03, 68.31, 88.2, 83.01, 83.44, 87.86, 76.68, 88.98, 80.01, 74.46, 92.78, 88.28, 79.6, 80.09, 73.18, 75.25, 69.43, 71.76, 71.65, 89.04, 76.6, 83.41, 97.54, 77.85, 76.48, 70.18, 76.58, 90.83, 81.48, 67.0, 67.04, 87.08, 74.58, 74.75, 71.98, 97.47, 93.38, 80.95, 83.18, 89.1, 91.17, 86.97, 85.88, 79.51, 65.08, 71.92, 72.94, 82.99, 84.05, 94.63, 63.54, 88.08, 73.65, 74.06, 75.03, 81.52, 78.61, 87.35, 85.74, 94.8, 89.56, 74.6, 73.18, 77.66, 98.2, 77.96, 97.29, 90.33, 92.38, 86.71, 84.07, 91.0, 93.09, 72.02, 79.78, 71.65, 84.88, 68.07, 60.59, 72.13, 85.45, 87.38, 60.35, 84.21, 86.27, 81.23, 92.05, 83.29, 75.36, 72.63, 84.96, 89.18, 62.56, 76.07, 68.11, 88.9, 96.25, 92.8, 90.95, 75.39, 97.62, 72.19, 73.03, 81.53, 78.24, 91.61, 79.84, 94.66, 79.19, 91.3, 81.68, 65.98, 70.51, 87.65, 80.52, 77.3, 87.1, 84.84, 73.73, 78.88, 82.17, 63.87, 76.37, 72.48, 74.88, 66.64, 82.3, 89.17, 89.53, 91.87, 64.34, 70.75, 86.62, 76.61, 77.58, 62.51, 109.52, 79.55, 78.09, 89.94, 80.04, 76.72, 91.8, 83.41, 67.39, 74.75, 71.9, 74.98, 88.58, 74.44, 71.26, 78.84, 105.3, 83.81, 77.63, 80.26, 84.08, 88.48, 89.01, 91.91, 75.72, 72.78, 68.38, 65.35, 85.35, 73.85, 89.88, 85.33, 92.29, 77.54, 89.95, 64.98, 86.81, 67.54, 75.59, 67.72, 86.96, 87.52, 87.81, 74.56, 71.38, 89.92, 77.9, 80.09, 77.84, 77.25, 70.45, 80.48, 77.29, 94.82, 80.68, 68.02, 79.96, 78.47, 91.07, 82.49, 88.09, 88.97, 63.35, 68.23, 80.93, 84.32, 83.99, 89.5, 81.97, 83.23, 80.27, 68.81, 73.7, 57.06, 71.29, 97.96, 82.07, 72.81, 88.64, 64.75, 79.13, 80.19, 84.48, 84.32, 80.13, 96.65, 86.84, 80.18, 78.96, 76.86, 91.14, 70.74, 77.72, 84.59, 72.43, 90.59, 78.49, 88.36, 73.18, 73.19, 78.43, 75.86, 84.24, 63.32, 76.48, 81.43, 78.3, 69.43, 88.27, 83.36, 77.49, 87.54, 73.46, 77.52, 87.47, 86.92, 65.55, 87.1, 55.17, 62.55, 81.12, 85.8, 82.41, 103.54, 77.96, 82.39, 104.31, 94.94, 71.24, 83.72, 60.02, 53.58, 84.91, 87.7, 77.84, 75.09, 80.29, 78.08, 82.39, 96.34, 115.88, 84.19, 61.8, 78.79, 81.32, 94.51, 85.04, 86.69, 84.75, 87.32, 79.91, 90.72, 67.82, 82.19, 77.24, 74.16, 88.35, 80.9, 88.5, 71.14, 60.72, 63.62, 62.46, 68.92, 89.01, 79.48, 75.69, 76.59, 76.35, 82.81, 89.95, 88.54, 87.56, 91.08, 83.27, 87.88, 93.49, 66.22, 91.99, 79.93, 77.3, 76.79, 83.23, 77.57, 79.43, 68.93, 105.23, 66.97, 66.95, 86.62, 91.43, 80.5, 88.05, 69.97, 72.68, 84.73, 87.17, 79.74, 88.13, 75.42, 72.82, 86.12, 81.01, 72.92, 89.13, 83.52, 74.74, 82.73, 76.53, 71.01, 72.51, 87.93, 83.08, 98.78, 69.71, 86.3, 70.61, 86.98, 95.5, 74.01, 94.57, 71.76, 74.32, 87.09, 84.17, 65.88, 100.82, 86.11, 76.28, 79.49, 91.85, 82.23, 83.37, 69.36, 54.67, 99.38, 80.79, 75.59, 91.22, 87.59, 69.71, 79.81, 74.16, 77.4, 84.03, 85.24, 75.06, 74.55, 82.94, 91.49, 85.05, 73.19, 85.03, 82.21, 78.68, 91.47, 85.45, 80.53, 75.41, 63.86, 82.99, 83.07, 67.46, 72.9, 54.03, 82.9, 72.0, 72.41, 81.34, 77.08, 97.72, 81.31, 67.47, 74.34, 82.9, 83.02, 101.34, 83.44, 66.72, 82.01, 73.91, 80.22, 83.81, 67.96, 78.76, 70.25, 85.36, 84.25, 91.4, 66.42, 64.13, 68.12, 84.75, 70.86, 73.88, 90.37, 91.94, 93.39, 85.3, 91.62, 75.58, 79.78, 86.2, 84.35, 87.24, 78.26, 76.13, 88.83, 94.57, 86.37, 72.96, 74.59, 69.03, 74.71, 78.74, 59.63, 81.25, 91.2, 80.46, 80.98, 84.99, 90.12, 72.69, 95.74, 65.24, 93.08, 80.36, 59.08, 82.99, 75.76, 82.71, 65.26, 66.51, 86.4, 70.73, 83.59, 74.66, 82.19, 80.54, 70.89, 71.97, 84.23, 76.36, 88.93, 84.88, 80.82, 78.17, 89.98, 86.05, 77.28, 79.38, 106.6, 84.05, 67.03, 82.62, 70.68, 90.22, 77.69, 81.79, 92.26, 85.07, 99.99, 66.5, 92.74, 72.63, 79.93, 77.83, 72.9, 90.45, 93.26, 82.1, 83.01, 82.37, 89.2, 89.98, 83.9, 81.66, 101.17, 83.84, 74.37, 78.71, 91.2, 84.78, 85.24, 91.37, 89.01, 77.08, 88.53, 79.41, 65.28, 73.12, 67.63, 65.33, 59.3, 84.25, 86.93, 61.01, 89.11, 89.41, 96.78, 77.44, 69.09, 70.57, 92.42, 77.77, 85.94, 81.03, 85.1, 91.19, 92.62, 84.88, 99.66, 68.69, 89.5, 84.97, 80.39, 80.41, 94.01, 77.72, 66.38, 93.75, 80.13, 71.06, 75.21, 75.44, 101.22, 85.48, 68.81, 73.36, 108.38, 63.65, 75.93, 91.21, 89.61, 89.54, 81.42, 79.09, 82.03, 97.53, 78.9, 67.88, 72.11, 79.87, 69.65, 86.34, 92.03, 70.88, 72.06, 81.82, 80.55, 70.15, 78.73, 93.78, 88.4, 79.7, 77.3, 56.43, 63.6, 82.11, 79.52, 73.57, 73.69, 81.88, 73.71, 67.96, 77.42, 68.38, 75.65, 83.46, 85.37, 70.91, 80.13, 82.26, 77.92, 68.51, 78.06, 85.48, 63.97, 97.52, 71.04, 83.45, 67.33, 58.62, 67.6, 84.61, 93.12, 66.62, 92.58, 73.74, 80.18, 86.38, 61.92, 78.48, 78.58, 84.19, 71.91, 82.1, 97.92, 81.84, 71.95, 80.47, 60.07, 79.51, 82.53, 83.42, 86.19, 83.33, 96.94, 82.4, 74.36, 75.04, 70.97, 71.41, 84.89, 85.59, 85.07, 75.23, 77.57, 65.75, 88.04, 86.73, 61.37, 73.74, 86.69, 101.83, 91.56, 66.81, 93.47, 81.75, 86.41, 86.06, 82.48, 84.9, 74.95, 79.43, 96.29, 75.3, 61.88, 75.34, 80.1, 69.27, 74.02, 71.26, 86.14, 92.65, 79.13, 82.69, 62.85, 92.09]；x̄₁=80.1820，已知 σ₁=10；
组2 n=838：[92.31, 75.89, 74.81, 54.22, 66.35, 91.06, 82.91, 54.29, 81.43, 65.76, 86.44, 71.45, 77.95, 82.65, 75.68, 55.0, 75.55, 74.85, 62.63, 80.85, 66.66, 70.52, 71.5, 78.74, 87.36, 69.95, 69.35, 82.78, 85.96, 83.62, 92.77, 79.73, 90.79, 67.78, 68.76, 82.48, 78.11, 91.3, 87.36, 61.6, 71.71, 77.24, 59.48, 74.19, 74.6, 66.96, 80.42, 62.26, 75.07, 80.79, 97.18, 77.97, 85.98, 76.31, 77.78, 74.03, 69.75, 81.75, 72.73, 79.07, 88.8, 69.16, 78.39, 70.44, 89.29, 77.59, 75.66, 68.74, 72.12, 77.33, 66.79, 69.71, 77.75, 77.25, 91.52, 76.56, 78.35, 69.09, 61.8, 83.78, 63.2, 78.95, 71.83, 78.78, 75.11, 85.4, 80.62, 85.76, 59.12, 80.63, 70.4, 81.98, 68.39, 79.86, 73.9, 56.36, 74.25, 73.34, 79.72, 80.08, 99.07, 74.32, 66.79, 77.38, 97.72, 74.84, 77.01, 69.77, 75.54, 59.9, 75.97, 83.62, 81.14, 68.46, 79.98, 67.34, 62.32, 82.75, 96.73, 81.82, 78.73, 76.94, 57.25, 92.19, 89.42, 67.3, 86.51, 70.48, 92.67, 74.61, 81.04, 80.42, 76.08, 87.79, 77.2, 76.62, 83.47, 73.83, 85.67, 66.72, 79.01, 95.32, 84.75, 78.99, 72.87, 68.1, 69.22, 78.61, 68.89, 106.66, 88.04, 84.58, 64.33, 82.62, 74.04, 64.35, 80.26, 87.12, 78.87, 77.37, 80.41, 79.18, 64.55, 76.76, 78.17, 76.47, 82.57, 69.51, 73.85, 77.84, 69.48, 73.66, 68.36, 59.64, 75.87, 65.03, 65.46, 72.32, 69.06, 91.22, 85.31, 82.8, 66.72, 68.08, 80.86, 70.82, 78.53, 79.95, 72.76, 81.29, 81.76, 97.88, 65.18, 79.13, 64.25, 72.03, 74.9, 69.87, 76.37, 63.53, 77.7, 75.38, 70.8, 61.5, 64.7, 63.8, 74.55, 68.55, 62.35, 79.74, 74.0, 88.6, 75.6, 88.46, 79.93, 82.97, 66.43, 75.95, 70.2, 90.08, 76.58, 95.1, 75.0, 76.32, 82.42, 71.67, 76.83, 65.95, 75.95, 76.18, 85.99, 79.29, 69.02, 65.27, 85.94, 90.03, 68.46, 76.17, 51.35, 57.24, 81.56, 90.17, 69.41, 77.61, 76.26, 75.67, 59.03, 103.75, 73.06, 66.91, 69.16, 73.07, 65.76, 66.8, 86.77, 71.1, 76.34, 82.28, 73.51, 77.99, 75.52, 92.65, 85.96, 75.7, 86.39, 70.4, 81.19, 77.91, 64.97, 79.19, 81.01, 64.24, 76.0, 74.63, 76.82, 76.27, 81.38, 60.86, 74.84, 89.19, 87.81, 83.6, 83.34, 93.05, 72.56, 72.19, 67.55, 84.54, 63.72, 92.21, 85.1, 64.78, 72.28, 86.26, 61.23, 75.94, 78.52, 77.06, 96.23, 64.26, 70.25, 71.8, 72.1, 60.68, 69.02, 95.29, 87.19, 74.03, 67.7, 77.53, 97.07, 89.84, 73.48, 76.1, 99.77, 62.05, 71.82, 64.6, 75.54, 67.53, 60.11, 59.61, 60.6, 72.25, 75.16, 79.17, 90.49, 75.86, 92.97, 94.06, 67.44, 85.39, 79.11, 83.93, 66.42, 81.09, 72.97, 65.01, 75.66, 60.25, 81.41, 88.02, 78.64, 40.1, 102.09, 62.89, 84.32, 66.19, 74.89, 77.23, 68.78, 74.12, 74.82, 74.87, 79.98, 67.33, 67.83, 72.87, 75.42, 75.87, 77.2, 86.69, 81.17, 73.6, 95.63, 77.96, 92.75, 71.51, 75.54, 76.04, 77.88, 93.83, 78.75, 71.17, 79.3, 69.61, 60.9, 81.19, 80.38, 77.39, 74.82, 76.14, 80.9, 85.05, 82.62, 80.29, 77.03, 97.74, 78.28, 59.9, 62.89, 88.4, 84.46, 75.79, 72.97, 56.86, 64.99, 99.15, 75.51, 81.76, 67.09, 97.16, 78.33, 67.31, 69.17, 80.05, 82.12, 76.14, 54.37, 61.35, 80.98, 67.15, 59.86, 71.0, 80.9, 77.57, 72.46, 82.23, 76.18, 75.74, 78.65, 66.09, 68.61, 77.19, 73.6, 81.92, 87.25, 71.11, 69.61, 67.04, 84.96, 79.25, 88.69, 69.85, 60.82, 73.25, 95.69, 55.86, 88.54, 87.96, 72.44, 73.1, 63.34, 73.25, 69.35, 67.98, 92.16, 77.24, 85.22, 79.34, 73.18, 79.44, 70.11, 86.56, 80.25, 93.22, 78.75, 88.22, 71.11, 66.34, 80.17, 77.58, 57.79, 67.75, 83.48, 97.54, 86.13, 83.59, 92.46, 80.48, 80.64, 82.05, 76.95, 73.36, 72.47, 80.98, 83.33, 65.39, 90.11, 91.75, 69.8, 77.55, 99.41, 57.65, 70.99, 73.86, 64.78, 65.69, 78.11, 87.77, 84.56, 73.49, 79.82, 59.98, 81.69, 87.72, 51.89, 85.29, 58.31, 65.86, 58.84, 86.5, 104.73, 80.98, 78.31, 83.43, 79.11, 64.23, 90.04, 83.64, 68.53, 95.18, 78.34, 93.11, 89.33, 73.47, 72.54, 73.22, 82.59, 70.91, 79.1, 92.07, 94.19, 69.89, 83.62, 55.23, 85.22, 64.73, 56.75, 79.62, 64.09, 76.3, 84.44, 74.21, 63.47, 86.52, 107.73, 86.27, 96.74, 80.08, 76.74, 75.86, 61.07, 71.26, 72.09, 72.34, 80.55, 92.05, 69.32, 72.8, 74.6, 79.17, 86.05, 58.43, 95.12, 84.1, 82.7, 96.21, 74.22, 81.21, 71.36, 92.02, 68.22, 76.34, 62.35, 89.22, 79.29, 79.36, 83.67, 92.67, 64.13, 53.47, 97.77, 75.64, 91.45, 76.84, 78.33, 76.7, 77.65, 67.76, 93.27, 84.32, 61.83, 68.5, 66.04, 66.15, 69.18, 62.3, 72.9, 76.37, 81.45, 79.06, 72.56, 70.85, 85.74, 76.84, 73.1, 72.37, 77.46, 73.59, 76.42, 73.98, 69.57, 84.91, 78.09, 86.44, 85.91, 68.86, 80.66, 81.02, 70.4, 70.82, 83.03, 73.35, 74.73, 83.34, 83.27, 91.26, 71.01, 92.48, 70.04, 84.28, 89.37, 68.46, 68.34, 78.45, 75.33, 86.76, 89.84, 70.3, 77.39, 77.75, 59.1, 83.59, 70.27, 74.1, 75.26, 69.84, 93.49, 67.36, 64.59, 96.12, 97.15, 78.35, 78.76, 77.28, 69.92, 103.62, 77.27, 83.64, 65.61, 71.75, 62.34, 73.65, 92.77, 82.4, 72.28, 71.08, 65.2, 106.47, 76.62, 68.43, 83.55, 72.92, 95.05, 71.8, 62.02, 86.73, 72.95, 74.36, 62.92, 81.26, 101.34, 79.94, 88.81, 60.8, 77.87, 67.89, 80.56, 56.5, 80.98, 81.13, 51.44, 50.74, 92.81, 54.1, 77.78, 67.9, 82.81, 91.15, 71.45, 74.67, 85.05, 78.74, 87.84, 87.1, 81.51, 70.99, 78.58, 77.5, 78.99, 66.23, 93.68, 82.41, 73.59, 61.06, 65.71, 81.75, 71.33, 73.35, 99.55, 63.67, 94.64, 68.43, 78.2, 65.3, 88.27, 72.73, 69.49, 87.94, 101.06, 80.16, 69.48, 57.47, 94.89, 83.29, 63.73, 82.17, 88.73, 69.66, 73.43, 91.97, 64.71, 74.04, 82.49, 80.87, 76.56, 70.84, 87.56, 64.18, 77.47, 66.46, 92.49, 79.39, 86.99, 80.82, 81.36, 74.15, 71.85, 82.91, 65.52, 68.2, 67.16, 65.26, 67.33, 74.99, 93.73, 84.43, 61.67, 82.05, 74.64, 75.98, 85.87, 84.3, 76.86, 91.15, 87.47, 87.99, 66.6, 77.32, 76.38, 69.78, 84.97, 81.3, 77.36, 105.03, 64.63, 63.07, 77.96, 68.17, 73.48, 65.92, 72.67, 73.44, 84.98, 46.57, 52.73, 78.36, 81.45, 62.46, 93.03, 70.62, 92.16, 68.65, 74.67, 79.67, 73.96, 72.97, 86.61, 83.08, 71.32, 59.91, 81.09, 51.12, 67.29, 78.76, 81.35, 56.38, 73.28, 85.48, 64.22, 70.66, 84.61, 84.1, 57.04, 90.83, 69.11, 62.99, 102.49, 87.99, 80.74, 88.68, 67.06, 63.42, 67.39, 71.63, 71.3, 72.42, 73.49, 69.14, 55.34, 90.69, 70.94, 86.9, 80.16, 78.88, 59.67, 89.6]；x̄₂=76.5895，已知 σ₂=10
组2 n=838：[92.31, 75.89, 74.81, 54.22, 66.35, 91.06, 82.91, 54.29, 81.43, 65.76, 86.44, 71.45, 77.95, 82.65, 75.68, 55.0, 75.55, 74.85, 62.63, 80.85, 66.66, 70.52, 71.5, 78.74, 87.36, 69.95, 69.35, 82.78, 85.96, 83.62, 92.77, 79.73, 90.79, 67.78, 68.76, 82.48, 78.11, 91.3, 87.36, 61.6, 71.71, 77.24, 59.48, 74.19, 74.6, 66.96, 80.42, 62.26, 75.07, 80.79, 97.18, 77.97, 85.98, 76.31, 77.78, 74.03, 69.75, 81.75, 72.73, 79.07, 88.8, 69.16, 78.39, 70.44, 89.29, 77.59, 75.66, 68.74, 72.12, 77.33, 66.79, 69.71, 77.75, 77.25, 91.52, 76.56, 78.35, 69.09, 61.8, 83.78, 63.2, 78.95, 71.83, 78.78, 75.11, 85.4, 80.62, 85.76, 59.12, 80.63, 70.4, 81.98, 68.39, 79.86, 73.9, 56.36, 74.25, 73.34, 79.72, 80.08, 99.07, 74.32, 66.79, 77.38, 97.72, 74.84, 77.01, 69.77, 75.54, 59.9, 75.97, 83.62, 81.14, 68.46, 79.98, 67.34, 62.32, 82.75, 96.73, 81.82, 78.73, 76.94, 57.25, 92.19, 89.42, 67.3, 86.51, 70.48, 92.67, 74.61, 81.04, 80.42, 76.08, 87.79, 77.2, 76.62, 83.47, 73.83, 85.67, 66.72, 79.01, 95.32, 84.75, 78.99, 72.87, 68.1, 69.22, 78.61, 68.89, 106.66, 88.04, 84.58, 64.33, 82.62, 74.04, 64.35, 80.26, 87.12, 78.87, 77.37, 80.41, 79.18, 64.55, 76.76, 78.17, 76.47, 82.57, 69.51, 73.85, 77.84, 69.48, 73.66, 68.36, 59.64, 75.87, 65.03, 65.46, 72.32, 69.06, 91.22, 85.31, 82.8, 66.72, 68.08, 80.86, 70.82, 78.53, 79.95, 72.76, 81.29, 81.76, 97.88, 65.18, 79.13, 64.25, 72.03, 74.9, 69.87, 76.37, 63.53, 77.7, 75.38, 70.8, 61.5, 64.7, 63.8, 74.55, 68.55, 62.35, 79.74, 74.0, 88.6, 75.6, 88.46, 79.93, 82.97, 66.43, 75.95, 70.2, 90.08, 76.58, 95.1, 75.0, 76.32, 82.42, 71.67, 76.83, 65.95, 75.95, 76.18, 85.99, 79.29, 69.02, 65.27, 85.94, 90.03, 68.46, 76.17, 51.35, 57.24, 81.56, 90.17, 69.41, 77.61, 76.26, 75.67, 59.03, 103.75, 73.06, 66.91, 69.16, 73.07, 65.76, 66.8, 86.77, 71.1, 76.34, 82.28, 73.51, 77.99, 75.52, 92.65, 85.96, 75.7, 86.39, 70.4, 81.19, 77.91, 64.97, 79.19, 81.01, 64.24, 76.0, 74.63, 76.82, 76.27, 81.38, 60.86, 74.84, 89.19, 87.81, 83.6, 83.34, 93.05, 72.56, 72.19, 67.55, 84.54, 63.72, 92.21, 85.1, 64.78, 72.28, 86.26, 61.23, 75.94, 78.52, 77.06, 96.23, 64.26, 70.25, 71.8, 72.1, 60.68, 69.02, 95.29, 87.19, 74.03, 67.7, 77.53, 97.07, 89.84, 73.48, 76.1, 99.77, 62.05, 71.82, 64.6, 75.54, 67.53, 60.11, 59.61, 60.6, 72.25, 75.16, 79.17, 90.49, 75.86, 92.97, 94.06, 67.44, 85.39, 79.11, 83.93, 66.42, 81.09, 72.97, 65.01, 75.66, 60.25, 81.41, 88.02, 78.64, 40.1, 102.09, 62.89, 84.32, 66.19, 74.89, 77.23, 68.78, 74.12, 74.82, 74.87, 79.98, 67.33, 67.83, 72.87, 75.42, 75.87, 77.2, 86.69, 81.17, 73.6, 95.63, 77.96, 92.75, 71.51, 75.54, 76.04, 77.88, 93.83, 78.75, 71.17, 79.3, 69.61, 60.9, 81.19, 80.38, 77.39, 74.82, 76.14, 80.9, 85.05, 82.62, 80.29, 77.03, 97.74, 78.28, 59.9, 62.89, 88.4, 84.46, 75.79, 72.97, 56.86, 64.99, 99.15, 75.51, 81.76, 67.09, 97.16, 78.33, 67.31, 69.17, 80.05, 82.12, 76.14, 54.37, 61.35, 80.98, 67.15, 59.86, 71.0, 80.9, 77.57, 72.46, 82.23, 76.18, 75.74, 78.65, 66.09, 68.61, 77.19, 73.6, 81.92, 87.25, 71.11, 69.61, 67.04, 84.96, 79.25, 88.69, 69.85, 60.82, 73.25, 95.69, 55.86, 88.54, 87.96, 72.44, 73.1, 63.34, 73.25, 69.35, 67.98, 92.16, 77.24, 85.22, 79.34, 73.18, 79.44, 70.11, 86.56, 80.25, 93.22, 78.75, 88.22, 71.11, 66.34, 80.17, 77.58, 57.79, 67.75, 83.48, 97.54, 86.13, 83.59, 92.46, 80.48, 80.64, 82.05, 76.95, 73.36, 72.47, 80.98, 83.33, 65.39, 90.11, 91.75, 69.8, 77.55, 99.41, 57.65, 70.99, 73.86, 64.78, 65.69, 78.11, 87.77, 84.56, 73.49, 79.82, 59.98, 81.69, 87.72, 51.89, 85.29, 58.31, 65.86, 58.84, 86.5, 104.73, 80.98, 78.31, 83.43, 79.11, 64.23, 90.04, 83.64, 68.53, 95.18, 78.34, 93.11, 89.33, 73.47, 72.54, 73.22, 82.59, 70.91, 79.1, 92.07, 94.19, 69.89, 83.62, 55.23, 85.22, 64.73, 56.75, 79.62, 64.09, 76.3, 84.44, 74.21, 63.47, 86.52, 107.73, 86.27, 96.74, 80.08, 76.74, 75.86, 61.07, 71.26, 72.09, 72.34, 80.55, 92.05, 69.32, 72.8, 74.6, 79.17, 86.05, 58.43, 95.12, 84.1, 82.7, 96.21, 74.22, 81.21, 71.36, 92.02, 68.22, 76.34, 62.35, 89.22, 79.29, 79.36, 83.67, 92.67, 64.13, 53.47, 97.77, 75.64, 91.45, 76.84, 78.33, 76.7, 77.65, 67.76, 93.27, 84.32, 61.83, 68.5, 66.04, 66.15, 69.18, 62.3, 72.9, 76.37, 81.45, 79.06, 72.56, 70.85, 85.74, 76.84, 73.1, 72.37, 77.46, 73.59, 76.42, 73.98, 69.57, 84.91, 78.09, 86.44, 85.91, 68.86, 80.66, 81.02, 70.4, 70.82, 83.03, 73.35, 74.73, 83.34, 83.27, 91.26, 71.01, 92.48, 70.04, 84.28, 89.37, 68.46, 68.34, 78.45, 75.33, 86.76, 89.84, 70.3, 77.39, 77.75, 59.1, 83.59, 70.27, 74.1, 75.26, 69.84, 93.49, 67.36, 64.59, 96.12, 97.15, 78.35, 78.76, 77.28, 69.92, 103.62, 77.27, 83.64, 65.61, 71.75, 62.34, 73.65, 92.77, 82.4, 72.28, 71.08, 65.2, 106.47, 76.62, 68.43, 83.55, 72.92, 95.05, 71.8, 62.02, 86.73, 72.95, 74.36, 62.92, 81.26, 101.34, 79.94, 88.81, 60.8, 77.87, 67.89, 80.56, 56.5, 80.98, 81.13, 51.44, 50.74, 92.81, 54.1, 77.78, 67.9, 82.81, 91.15, 71.45, 74.67, 85.05, 78.74, 87.84, 87.1, 81.51, 70.99, 78.58, 77.5, 78.99, 66.23, 93.68, 82.41, 73.59, 61.06, 65.71, 81.75, 71.33, 73.35, 99.55, 63.67, 94.64, 68.43, 78.2, 65.3, 88.27, 72.73, 69.49, 87.94, 101.06, 80.16, 69.48, 57.47, 94.89, 83.29, 63.73, 82.17, 88.73, 69.66, 73.43, 91.97, 64.71, 74.04, 82.49, 80.87, 76.56, 70.84, 87.56, 64.18, 77.47, 66.46, 92.49, 79.39, 86.99, 80.82, 81.36, 74.15, 71.85, 82.91, 65.52, 68.2, 67.16, 65.26, 67.33, 74.99, 93.73, 84.43, 61.67, 82.05, 74.64, 75.98, 85.87, 84.3, 76.86, 91.15, 87.47, 87.99, 66.6, 77.32, 76.38, 69.78, 84.97, 81.3, 77.36, 105.03, 64.63, 63.07, 77.96, 68.17, 73.48, 65.92, 72.67, 73.44, 84.98, 46.57, 52.73, 78.36, 81.45, 62.46, 93.03, 70.62, 92.16, 68.65, 74.67, 79.67, 73.96, 72.97, 86.61, 83.08, 71.32, 59.91, 81.09, 51.12, 67.29, 78.76, 81.35, 56.38, 73.28, 85.48, 64.22, 70.66, 84.61, 84.1, 57.04, 90.83, 69.11, 62.99, 102.49, 87.99, 80.74, 88.68, 67.06, 63.42, 67.39, 71.63, 71.3, 72.42, 73.49, 69.14, 55.34, 90.69, 70.94, 86.9, 80.16, 78.88, 59.67, 89.6]；x̄₂=76.5895，已知 σ₂=10
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.4979；z=(x̄₁−x̄₂)/SE=7.2158；双侧 p=5.365e-13。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.4979；z=(x̄₁−x̄₂)/SE=7.2158；双侧 p=5.365e-13。
计算结果
z = 7.2158，p = 5.365e-13
在 α=0.05 水平下，p=5.365e-13 < 0.05，拒绝原假设 H₀。比较班级平均成绩。
案例 97. 两工艺尺寸差异
原始数据
组1 n=416：[51.34, 49.88, 49.62, 50.48, 49.4, 49.7, 49.86, 50.68, 49.86, 50.37, 50.19, 49.45, 50.82, 49.48, 49.87, 49.56, 50.04, 50.23, 49.64, 50.05, 48.85, 50.11, 50.75, 49.62, 50.41, 50.28, 50.29, 49.39, 49.59, 49.42, 50.56, 51.03, 50.08, 49.89, 49.74, 50.62, 50.49, 50.32, 50.29, 50.36, 50.13, 50.85, 50.74, 50.19, 49.56, 50.61, 49.99, 49.25, 49.93, 50.17, 51.7, 51.1, 49.99, 50.85, 49.48, 50.5, 50.1, 51.11, 50.06, 49.8, 51.09, 49.48, 50.3, 50.05, 50.65, 49.7, 49.53, 50.31, 50.6, 50.54, 50.48, 50.6, 49.78, 51.06, 50.17, 50.41, 50.63, 51.05, 50.25, 51.71, 49.51, 49.83, 49.7, 49.26, 50.98, 50.61, 50.96, 49.92, 49.94, 50.33, 50.48, 50.54, 50.76, 49.59, 50.31, 49.57, 50.5, 49.57, 50.77, 50.41, 49.77, 49.68, 50.92, 50.66, 50.0, 49.96, 49.47, 49.73, 49.07, 50.46, 50.65, 50.81, 50.11, 50.35, 49.89, 49.56, 50.02, 50.88, 51.48, 50.29, 50.53, 50.93, 50.02, 50.94, 49.69, 50.61, 50.7, 49.97, 50.73, 50.26, 49.78, 50.01, 50.6, 49.95, 49.92, 50.04, 49.84, 50.54, 50.49, 49.87, 49.13, 50.26, 49.62, 49.39, 50.63, 50.09, 49.96, 50.61, 50.53, 49.22, 50.4, 51.01, 50.82, 51.05, 49.73, 50.66, 49.79, 49.43, 49.85, 50.68, 50.7, 49.69, 49.75, 50.36, 49.37, 50.17, 49.64, 51.29, 50.22, 49.64, 50.87, 49.68, 49.83, 50.22, 49.8, 50.12, 49.56, 50.21, 49.78, 51.18, 50.86, 50.62, 50.23, 50.19, 50.07, 49.53, 50.54, 49.99, 50.51, 49.88, 50.61, 49.95, 50.38, 49.58, 50.59, 50.59, 50.24, 50.48, 49.49, 50.06, 50.65, 49.9, 49.8, 50.8, 50.68, 49.08, 49.88, 51.28, 51.47, 50.41, 49.68, 49.68, 49.1, 50.38, 50.24, 50.21, 50.12, 51.08, 51.04, 49.74, 49.22, 49.79, 50.78, 49.8, 50.35, 50.25, 49.55, 49.59, 50.48, 49.88, 49.67, 51.05, 50.17, 49.87, 49.29, 50.37, 49.94, 49.7, 50.93, 49.62, 49.69, 49.99, 50.7, 50.43, 50.25, 50.55, 49.7, 50.63, 49.72, 49.79, 50.01, 49.42, 50.56, 49.3, 50.06, 50.5, 50.79, 50.68, 49.76, 50.17, 50.07, 50.19, 50.23, 49.73, 50.25, 50.34, 50.05, 50.51, 50.33, 50.25, 49.57, 50.56, 50.41, 49.94, 50.43, 49.89, 49.51, 49.84, 50.42, 50.29, 49.92, 49.57, 49.36, 50.49, 50.69, 50.29, 50.09, 50.09, 51.15, 49.76, 50.92, 50.81, 50.61, 50.15, 49.62, 49.83, 49.36, 50.07, 49.6, 50.23, 49.87, 50.56, 50.65, 50.47, 49.85, 51.2, 50.6, 49.94, 50.46, 50.23, 50.33, 49.71, 50.74, 50.97, 49.74, 50.11, 50.13, 49.07, 50.86, 50.23, 50.08, 49.77, 49.77, 49.25, 49.7, 50.23, 49.88, 50.58, 49.38, 50.82, 49.65, 49.15, 50.07, 49.58, 50.66, 50.96, 50.62, 49.85, 49.56, 50.43, 50.46, 50.12, 49.99, 50.58, 49.89, 50.76, 49.18, 49.25, 50.71, 50.82, 50.22, 50.42, 50.06, 49.96, 50.78, 50.47, 49.74, 49.65, 50.27, 50.59, 49.86, 49.56, 50.32, 50.56, 50.73, 51.29, 49.58, 49.67, 50.03, 49.07, 50.4, 50.57, 49.47, 51.31, 50.83, 49.79, 50.57, 49.12, 50.51, 49.63, 50.4, 50.43, 50.09, 49.91, 50.74, 50.11, 50.23, 50.34, 50.06, 49.62, 50.02, 50.59, 50.21, 49.16, 50.12, 49.96, 50.47, 50.16, 50.45, 50.57, 50.17, 50.36, 49.87, 50.41, 51.28, 50.34, 49.81, 49.49, 51.17, 50.07, 50.83, 49.95, 50.39, 50.38, 50.66, 50.86]；x̄₁=50.1802，已知 σ₁=0.5；
组2 n=396：[50.17, 49.52, 50.27, 49.79, 49.44, 49.72, 49.77, 50.47, 49.3, 50.0, 49.39, 50.66, 49.8, 50.17, 49.98, 50.54, 49.99, 50.39, 49.34, 49.16, 49.96, 49.78, 50.24, 49.94, 49.95, 49.7, 50.58, 49.27, 49.85, 49.53, 50.42, 49.09, 49.99, 49.59, 50.38, 49.86, 49.48, 49.47, 49.73, 50.66, 49.9, 50.27, 49.31, 50.67, 49.97, 50.38, 49.91, 49.63, 49.85, 50.47, 50.48, 49.83, 50.38, 49.24, 49.81, 49.71, 49.96, 49.64, 49.45, 48.55, 50.76, 50.5, 49.71, 50.5, 49.68, 49.88, 51.08, 49.45, 50.07, 49.67, 50.2, 50.38, 50.18, 49.66, 49.7, 49.36, 50.14, 50.27, 49.66, 49.65, 50.86, 50.04, 49.75, 50.04, 50.09, 50.13, 49.66, 50.74, 50.03, 49.77, 50.08, 49.4, 50.69, 50.05, 49.92, 50.81, 49.66, 49.88, 49.17, 50.56, 50.13, 49.49, 50.15, 51.59, 49.06, 49.59, 50.46, 50.64, 49.81, 49.85, 50.56, 49.46, 49.98, 49.55, 49.95, 48.85, 50.24, 49.47, 50.36, 48.63, 50.27, 50.01, 49.89, 50.1, 49.75, 49.96, 50.83, 50.59, 50.65, 50.01, 49.98, 50.82, 50.0, 50.22, 49.34, 49.94, 49.06, 49.86, 50.79, 49.3, 49.86, 49.77, 50.59, 50.01, 49.68, 49.62, 49.44, 50.31, 50.51, 49.81, 50.38, 50.3, 49.64, 49.71, 50.19, 50.41, 49.83, 50.07, 50.12, 50.45, 50.1, 50.73, 49.45, 50.07, 49.86, 49.52, 50.33, 50.49, 50.33, 50.7, 49.45, 49.77, 49.81, 50.39, 50.25, 49.88, 49.72, 49.75, 49.03, 50.15, 49.97, 49.41, 50.81, 49.75, 50.39, 49.69, 50.51, 50.23, 49.48, 50.05, 49.91, 49.04, 49.8, 49.6, 50.27, 49.67, 49.4, 50.44, 49.85, 49.54, 49.58, 50.22, 49.86, 50.99, 50.54, 48.81, 49.52, 49.85, 49.47, 49.23, 49.64, 49.84, 50.0, 49.91, 50.45, 50.4, 49.58, 49.87, 50.25, 49.23, 50.43, 50.2, 49.64, 50.9, 49.64, 50.04, 49.92, 50.3, 49.74, 49.54, 51.06, 49.84, 50.14, 50.24, 49.39, 51.55, 50.68, 49.21, 50.18, 49.32, 50.73, 49.92, 50.63, 50.18, 50.57, 49.76, 50.2, 51.27, 49.69, 49.86, 50.14, 50.25, 49.97, 50.99, 49.38, 50.13, 49.57, 49.55, 49.79, 49.61, 51.05, 49.53, 49.53, 49.87, 49.96, 48.94, 49.51, 50.01, 49.71, 50.39, 50.28, 49.8, 50.0, 50.11, 50.15, 49.99, 49.31, 50.01, 49.49, 49.6, 49.89, 50.07, 50.19, 49.96, 49.38, 49.81, 50.67, 50.46, 49.23, 49.77, 49.57, 49.3, 50.07, 49.11, 50.01, 49.79, 50.29, 49.7, 49.98, 50.21, 50.55, 50.19, 50.45, 50.16, 50.36, 49.64, 49.71, 49.38, 48.58, 50.12, 49.99, 49.37, 49.89, 50.23, 50.21, 50.89, 50.24, 49.7, 49.46, 50.1, 50.21, 49.78, 50.77, 50.56, 50.3, 49.73, 49.99, 49.86, 50.42, 49.79, 49.19, 49.92, 49.94, 49.89, 49.23, 50.22, 50.09, 50.25, 50.63, 50.24, 49.46, 50.28, 49.91, 50.7, 49.87, 49.68, 50.6, 50.48, 49.97, 49.59, 49.84, 51.03, 49.83, 50.22, 49.99, 49.47, 49.89, 49.84, 49.68, 48.99, 50.38, 50.03, 50.75, 50.53, 50.08, 50.13, 50.15, 49.62, 50.57, 48.84, 50.67, 50.36, 50.41, 50.35, 49.96, 50.16, 49.87, 49.95, 49.89, 50.44, 50.34, 49.89, 50.66, 49.74, 49.92, 50.63, 49.35, 50.93, 49.28, 50.62, 49.39, 50.61, 49.77, 49.8, 50.65, 49.96]；x̄₂=49.9809，已知 σ₂=0.5
组2 n=396：[50.17, 49.52, 50.27, 49.79, 49.44, 49.72, 49.77, 50.47, 49.3, 50.0, 49.39, 50.66, 49.8, 50.17, 49.98, 50.54, 49.99, 50.39, 49.34, 49.16, 49.96, 49.78, 50.24, 49.94, 49.95, 49.7, 50.58, 49.27, 49.85, 49.53, 50.42, 49.09, 49.99, 49.59, 50.38, 49.86, 49.48, 49.47, 49.73, 50.66, 49.9, 50.27, 49.31, 50.67, 49.97, 50.38, 49.91, 49.63, 49.85, 50.47, 50.48, 49.83, 50.38, 49.24, 49.81, 49.71, 49.96, 49.64, 49.45, 48.55, 50.76, 50.5, 49.71, 50.5, 49.68, 49.88, 51.08, 49.45, 50.07, 49.67, 50.2, 50.38, 50.18, 49.66, 49.7, 49.36, 50.14, 50.27, 49.66, 49.65, 50.86, 50.04, 49.75, 50.04, 50.09, 50.13, 49.66, 50.74, 50.03, 49.77, 50.08, 49.4, 50.69, 50.05, 49.92, 50.81, 49.66, 49.88, 49.17, 50.56, 50.13, 49.49, 50.15, 51.59, 49.06, 49.59, 50.46, 50.64, 49.81, 49.85, 50.56, 49.46, 49.98, 49.55, 49.95, 48.85, 50.24, 49.47, 50.36, 48.63, 50.27, 50.01, 49.89, 50.1, 49.75, 49.96, 50.83, 50.59, 50.65, 50.01, 49.98, 50.82, 50.0, 50.22, 49.34, 49.94, 49.06, 49.86, 50.79, 49.3, 49.86, 49.77, 50.59, 50.01, 49.68, 49.62, 49.44, 50.31, 50.51, 49.81, 50.38, 50.3, 49.64, 49.71, 50.19, 50.41, 49.83, 50.07, 50.12, 50.45, 50.1, 50.73, 49.45, 50.07, 49.86, 49.52, 50.33, 50.49, 50.33, 50.7, 49.45, 49.77, 49.81, 50.39, 50.25, 49.88, 49.72, 49.75, 49.03, 50.15, 49.97, 49.41, 50.81, 49.75, 50.39, 49.69, 50.51, 50.23, 49.48, 50.05, 49.91, 49.04, 49.8, 49.6, 50.27, 49.67, 49.4, 50.44, 49.85, 49.54, 49.58, 50.22, 49.86, 50.99, 50.54, 48.81, 49.52, 49.85, 49.47, 49.23, 49.64, 49.84, 50.0, 49.91, 50.45, 50.4, 49.58, 49.87, 50.25, 49.23, 50.43, 50.2, 49.64, 50.9, 49.64, 50.04, 49.92, 50.3, 49.74, 49.54, 51.06, 49.84, 50.14, 50.24, 49.39, 51.55, 50.68, 49.21, 50.18, 49.32, 50.73, 49.92, 50.63, 50.18, 50.57, 49.76, 50.2, 51.27, 49.69, 49.86, 50.14, 50.25, 49.97, 50.99, 49.38, 50.13, 49.57, 49.55, 49.79, 49.61, 51.05, 49.53, 49.53, 49.87, 49.96, 48.94, 49.51, 50.01, 49.71, 50.39, 50.28, 49.8, 50.0, 50.11, 50.15, 49.99, 49.31, 50.01, 49.49, 49.6, 49.89, 50.07, 50.19, 49.96, 49.38, 49.81, 50.67, 50.46, 49.23, 49.77, 49.57, 49.3, 50.07, 49.11, 50.01, 49.79, 50.29, 49.7, 49.98, 50.21, 50.55, 50.19, 50.45, 50.16, 50.36, 49.64, 49.71, 49.38, 48.58, 50.12, 49.99, 49.37, 49.89, 50.23, 50.21, 50.89, 50.24, 49.7, 49.46, 50.1, 50.21, 49.78, 50.77, 50.56, 50.3, 49.73, 49.99, 49.86, 50.42, 49.79, 49.19, 49.92, 49.94, 49.89, 49.23, 50.22, 50.09, 50.25, 50.63, 50.24, 49.46, 50.28, 49.91, 50.7, 49.87, 49.68, 50.6, 50.48, 49.97, 49.59, 49.84, 51.03, 49.83, 50.22, 49.99, 49.47, 49.89, 49.84, 49.68, 48.99, 50.38, 50.03, 50.75, 50.53, 50.08, 50.13, 50.15, 49.62, 50.57, 48.84, 50.67, 50.36, 50.41, 50.35, 49.96, 50.16, 49.87, 49.95, 49.89, 50.44, 50.34, 49.89, 50.66, 49.74, 49.92, 50.63, 49.35, 50.93, 49.28, 50.62, 49.39, 50.61, 49.77, 49.8, 50.65, 49.96]；x̄₂=49.9809，已知 σ₂=0.5
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.0351；z=(x̄₁−x̄₂)/SE=5.6773；双侧 p=1.368e-08。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.0351；z=(x̄₁−x̄₂)/SE=5.6773；双侧 p=1.368e-08。
计算结果
z = 5.6773，p = 1.368e-08
在 α=0.05 水平下，p=1.368e-08 < 0.05，拒绝原假设 H₀。比较工艺精度。
案例 98. 两地区销量差异
原始数据
组1 n=315：[889.99, 1053.03, 1527.48, 1433.22, 871.8, 1368.19, 1286.47, 1111.76, 1265.85, 1136.73, 1008.2, 1324.56, 1169.36, 1295.89, 1091.99, 1129.85, 1380.13, 966.57, 1108.74, 1217.97, 1072.19, 1550.75, 1338.95, 1313.62, 1357.89, 1059.96, 1416.37, 1371.51, 971.04, 966.51, 982.75, 1137.78, 1205.59, 789.38, 1009.82, 856.09, 1079.26, 1455.09, 956.48, 1375.92, 811.05, 906.35, 1275.66, 1231.98, 1382.72, 1267.66, 1081.6, 1148.03, 1297.97, 759.95, 972.39, 1339.94, 1130.92, 1208.9, 1212.59, 1172.9, 1197.12, 1043.57, 1397.87, 1125.3, 1223.57, 1136.01, 970.39, 1061.97, 1284.71, 1173.53, 1209.84, 1507.81, 1395.03, 946.82, 1349.65, 1118.07, 1519.41, 1578.73, 1187.23, 923.04, 767.18, 1460.61, 1362.88, 1091.61, 1221.58, 976.19, 957.07, 1198.93, 1013.22, 1447.13, 1110.43, 1163.19, 1094.35, 1251.16, 1011.44, 1508.11, 816.83, 1163.42, 1342.48, 1147.4, 926.46, 1306.22, 1301.61, 1048.1, 1140.99, 948.26, 1224.64, 1096.2, 1049.76, 1104.6, 953.39, 1090.77, 1193.77, 858.58, 1147.41, 1183.56, 1106.4, 1128.35, 1493.83, 1150.42, 961.22, 1240.9, 1192.98, 1140.39, 1191.45, 814.51, 1107.05, 1353.7, 892.4, 1027.62, 1186.51, 1125.49, 1230.92, 1076.95, 1006.7, 777.21, 934.07, 1176.93, 1440.88, 1020.72, 1264.56, 1001.03, 1182.98, 883.52, 1007.17, 824.01, 1356.73, 1245.06, 1426.94, 1385.25, 1196.41, 1057.62, 1391.8, 1291.45, 1199.12, 1291.28, 974.51, 1358.48, 1404.11, 755.19, 1057.67, 1422.06, 1312.33, 1350.84, 784.53, 1004.07, 1124.23, 1179.13, 927.92, 973.78, 1273.15, 1806.79, 1488.94, 1045.71, 1098.05, 955.72, 1381.31, 1194.9, 1295.99, 941.68, 998.49, 882.82, 1037.96, 976.48, 1078.31, 1195.86, 854.47, 1167.09, 862.66, 1013.3, 1078.17, 1063.26, 995.86, 963.89, 1090.91, 1352.44, 1059.59, 857.48, 1305.45, 1266.79, 1365.78, 1316.82, 854.32, 987.85, 1488.09, 1277.74, 828.29, 1384.19, 1041.86, 995.36, 1106.62, 1218.54, 1386.72, 866.69, 1316.24, 1172.7, 813.74, 968.4, 945.12, 1310.47, 1209.01, 1073.77, 1188.76, 1299.0, 1703.34, 792.59, 1124.2, 1388.25, 1040.85, 1624.04, 1078.79, 1656.11, 1262.9, 828.97, 1075.35, 923.7, 837.63, 1406.69, 1172.2, 1187.59, 1260.75, 1151.75, 1067.69, 1195.96, 1113.83, 1109.65, 908.17, 1290.02, 683.64, 1070.87, 1149.13, 1287.46, 858.76, 1298.72, 1053.33, 1198.6, 1155.17, 1609.09, 1232.36, 1161.93, 1341.68, 960.49, 977.84, 1003.6, 1149.05, 1069.62, 1489.23, 1392.43, 968.38, 1485.22, 1487.1, 1261.12, 1287.9, 1260.91, 1776.01, 1372.97, 861.09, 1376.13, 860.31, 1061.81, 963.95, 1065.04, 1166.77, 1384.23, 1056.0, 1254.41, 1345.96, 1132.82, 1391.27, 1092.55, 1144.29, 1276.93, 1117.23, 1324.82, 996.31, 925.52, 1231.31, 1037.35, 1501.02, 1331.28, 1156.97, 1147.48, 1121.59, 828.81, 1213.03, 1366.63, 1202.83, 836.65, 1412.97, 1236.28, 908.87, 1307.8, 1326.34, 1213.62, 1111.08, 1037.46, 1051.22, 1068.43, 1646.08]；x̄₁=1158.3547，已知 σ₁=200；
组2 n=465：[1016.2, 1004.33, 921.13, 1143.62, 1494.42, 851.41, 1257.09, 1101.26, 911.05, 1243.0, 789.89, 962.75, 1679.1, 1238.32, 1187.3, 1341.24, 1227.36, 935.95, 954.95, 1514.15, 1028.65, 1352.29, 1325.39, 1126.99, 935.59, 1434.02, 1189.91, 1212.16, 1271.58, 946.86, 812.28, 1359.85, 974.65, 1200.19, 974.19, 1448.47, 1173.57, 1280.02, 1089.21, 1186.99, 1318.72, 1078.11, 1453.77, 1105.75, 1493.07, 783.35, 1211.81, 1073.88, 849.19, 1344.89, 980.74, 1258.79, 1254.8, 1252.55, 964.72, 1212.09, 1097.07, 1191.05, 1191.51, 1040.42, 1002.2, 1047.09, 1087.77, 1052.7, 902.61, 1113.33, 1494.66, 1469.24, 1310.86, 915.17, 944.58, 1361.26, 1247.92, 1264.55, 1113.25, 1105.2, 1105.75, 1110.33, 1300.81, 1441.12, 1086.2, 1143.97, 1102.12, 1315.05, 1502.47, 1414.71, 967.14, 1312.5, 1534.55, 1022.64, 1224.03, 860.03, 1200.07, 1259.76, 1622.8, 1171.71, 999.82, 1191.74, 1302.43, 1172.32, 1042.94, 1135.69, 996.86, 1336.68, 1061.51, 1316.53, 730.08, 1224.1, 1067.58, 981.83, 1166.84, 983.83, 1340.63, 1404.8, 1361.31, 1181.89, 861.41, 1154.18, 996.85, 1466.76, 1104.25, 687.72, 1222.61, 1369.17, 1088.85, 1278.6, 1351.37, 1242.07, 1488.26, 1116.75, 1417.55, 1093.97, 1161.09, 1264.02, 1246.76, 936.93, 663.6, 948.59, 1103.49, 1316.43, 1227.9, 1431.01, 1232.33, 1055.25, 1133.23, 694.23, 1252.15, 811.85, 1311.46, 972.31, 1386.76, 1165.06, 1251.19, 1534.19, 1138.96, 750.59, 1072.86, 997.51, 790.62, 1344.92, 1311.95, 1391.76, 949.71, 1082.33, 1146.7, 1200.19, 1246.11, 1128.97, 1485.1, 1135.95, 1008.23, 1472.66, 942.95, 1362.31, 1099.04, 1135.32, 1099.53, 1138.72, 1537.62, 1027.58, 1271.82, 1318.94, 1119.48, 902.72, 1103.03, 1200.61, 1388.92, 1264.78, 1046.33, 1353.55, 1189.51, 1203.86, 1515.61, 1349.46, 1220.17, 1084.2, 1532.62, 1393.38, 1214.42, 1459.33, 1328.5, 1193.94, 1100.9, 948.17, 908.09, 1197.19, 1156.02, 1108.66, 1488.49, 1240.36, 1004.34, 1538.14, 1357.31, 1339.35, 1039.93, 938.28, 956.63, 713.57, 1116.45, 973.68, 1328.85, 1271.12, 1231.96, 1288.79, 1462.44, 981.1, 1177.09, 1074.9, 1800.68, 1210.75, 1238.48, 1274.82, 1274.24, 1420.0, 1647.78, 967.71, 1294.78, 969.24, 889.75, 1085.34, 1133.38, 1210.04, 854.85, 1344.52, 1001.55, 1165.9, 1279.35, 1399.9, 734.95, 1268.81, 1067.76, 915.32, 1302.89, 1143.73, 1107.91, 848.59, 1013.68, 1225.05, 1052.23, 1554.03, 1015.86, 1137.4, 1254.31, 1210.06, 1261.27, 1230.58, 1053.98, 887.99, 929.24, 1004.23, 1095.86, 1281.36, 720.71, 1200.57, 1141.88, 1085.18, 918.03, 891.32, 933.7, 1207.44, 1280.75, 979.85, 1263.11, 1267.68, 1028.75, 1255.89, 1186.24, 1151.77, 910.87, 955.0, 865.68, 1077.01, 1092.5, 875.37, 1154.86, 1333.51, 878.7, 1130.36, 1240.0, 842.07, 1397.12, 1121.12, 1099.55, 1156.96, 1177.17, 1135.09, 1355.08, 1259.88, 1219.88, 1202.84, 912.22, 997.37, 1286.41, 812.62, 1318.59, 608.05, 1526.32, 1120.96, 1268.07, 1149.68, 1220.48, 991.66, 873.19, 1188.44, 1105.34, 1200.84, 1278.7, 1335.13, 1294.66, 1248.0, 1341.9, 1553.5, 1217.42, 1080.8, 908.64, 1603.26, 1308.35, 911.88, 1387.66, 811.89, 1059.73, 1382.68, 1157.21, 1489.88, 1402.47, 1252.58, 1187.24, 1255.78, 1554.17, 1134.02, 1344.09, 1045.55, 1174.21, 1029.91, 1043.78, 1139.66, 1064.76, 1420.32, 929.27, 1032.03, 1149.89, 1317.68, 1223.97, 920.54, 1175.03, 842.77, 1479.95, 951.16, 1031.14, 1253.78, 868.17, 1050.92, 945.28, 1114.92, 1165.73, 1247.88, 1208.14, 1119.33, 1338.81, 1066.48, 1058.07, 1150.84, 1383.87, 1466.33, 1194.96, 996.33, 1100.5, 1502.03, 1629.55, 1326.63, 1187.55, 1424.19, 1256.82, 1100.26, 1321.03, 1082.16, 1407.65, 758.72, 1093.5, 994.99, 1395.97, 995.77, 1206.16, 1505.23, 1130.56, 951.47, 1013.2, 948.72, 947.76, 1313.76, 1309.75, 1742.76, 870.3, 1149.11, 1039.46, 1086.87, 724.79, 1065.18, 668.05, 889.19, 1048.61, 1113.71, 1342.69, 923.27, 885.9, 1099.3, 1024.21, 969.81, 851.78, 1357.87, 1021.99, 987.69, 1657.26, 1298.22, 1168.86, 1365.18, 1198.88, 983.05, 1108.12, 1490.25, 1692.2, 1017.75, 1202.21, 1313.28, 1526.31, 1594.79, 1014.69, 885.51, 1155.23, 833.91, 1170.32, 1186.7, 905.46, 953.34, 1017.69, 985.43, 1286.2, 1093.07, 1216.69, 1264.42, 1197.52, 1009.62, 1426.41, 1215.55, 1139.18]；x̄₂=1161.4954，已知 σ₂=200
组2 n=465：[1016.2, 1004.33, 921.13, 1143.62, 1494.42, 851.41, 1257.09, 1101.26, 911.05, 1243.0, 789.89, 962.75, 1679.1, 1238.32, 1187.3, 1341.24, 1227.36, 935.95, 954.95, 1514.15, 1028.65, 1352.29, 1325.39, 1126.99, 935.59, 1434.02, 1189.91, 1212.16, 1271.58, 946.86, 812.28, 1359.85, 974.65, 1200.19, 974.19, 1448.47, 1173.57, 1280.02, 1089.21, 1186.99, 1318.72, 1078.11, 1453.77, 1105.75, 1493.07, 783.35, 1211.81, 1073.88, 849.19, 1344.89, 980.74, 1258.79, 1254.8, 1252.55, 964.72, 1212.09, 1097.07, 1191.05, 1191.51, 1040.42, 1002.2, 1047.09, 1087.77, 1052.7, 902.61, 1113.33, 1494.66, 1469.24, 1310.86, 915.17, 944.58, 1361.26, 1247.92, 1264.55, 1113.25, 1105.2, 1105.75, 1110.33, 1300.81, 1441.12, 1086.2, 1143.97, 1102.12, 1315.05, 1502.47, 1414.71, 967.14, 1312.5, 1534.55, 1022.64, 1224.03, 860.03, 1200.07, 1259.76, 1622.8, 1171.71, 999.82, 1191.74, 1302.43, 1172.32, 1042.94, 1135.69, 996.86, 1336.68, 1061.51, 1316.53, 730.08, 1224.1, 1067.58, 981.83, 1166.84, 983.83, 1340.63, 1404.8, 1361.31, 1181.89, 861.41, 1154.18, 996.85, 1466.76, 1104.25, 687.72, 1222.61, 1369.17, 1088.85, 1278.6, 1351.37, 1242.07, 1488.26, 1116.75, 1417.55, 1093.97, 1161.09, 1264.02, 1246.76, 936.93, 663.6, 948.59, 1103.49, 1316.43, 1227.9, 1431.01, 1232.33, 1055.25, 1133.23, 694.23, 1252.15, 811.85, 1311.46, 972.31, 1386.76, 1165.06, 1251.19, 1534.19, 1138.96, 750.59, 1072.86, 997.51, 790.62, 1344.92, 1311.95, 1391.76, 949.71, 1082.33, 1146.7, 1200.19, 1246.11, 1128.97, 1485.1, 1135.95, 1008.23, 1472.66, 942.95, 1362.31, 1099.04, 1135.32, 1099.53, 1138.72, 1537.62, 1027.58, 1271.82, 1318.94, 1119.48, 902.72, 1103.03, 1200.61, 1388.92, 1264.78, 1046.33, 1353.55, 1189.51, 1203.86, 1515.61, 1349.46, 1220.17, 1084.2, 1532.62, 1393.38, 1214.42, 1459.33, 1328.5, 1193.94, 1100.9, 948.17, 908.09, 1197.19, 1156.02, 1108.66, 1488.49, 1240.36, 1004.34, 1538.14, 1357.31, 1339.35, 1039.93, 938.28, 956.63, 713.57, 1116.45, 973.68, 1328.85, 1271.12, 1231.96, 1288.79, 1462.44, 981.1, 1177.09, 1074.9, 1800.68, 1210.75, 1238.48, 1274.82, 1274.24, 1420.0, 1647.78, 967.71, 1294.78, 969.24, 889.75, 1085.34, 1133.38, 1210.04, 854.85, 1344.52, 1001.55, 1165.9, 1279.35, 1399.9, 734.95, 1268.81, 1067.76, 915.32, 1302.89, 1143.73, 1107.91, 848.59, 1013.68, 1225.05, 1052.23, 1554.03, 1015.86, 1137.4, 1254.31, 1210.06, 1261.27, 1230.58, 1053.98, 887.99, 929.24, 1004.23, 1095.86, 1281.36, 720.71, 1200.57, 1141.88, 1085.18, 918.03, 891.32, 933.7, 1207.44, 1280.75, 979.85, 1263.11, 1267.68, 1028.75, 1255.89, 1186.24, 1151.77, 910.87, 955.0, 865.68, 1077.01, 1092.5, 875.37, 1154.86, 1333.51, 878.7, 1130.36, 1240.0, 842.07, 1397.12, 1121.12, 1099.55, 1156.96, 1177.17, 1135.09, 1355.08, 1259.88, 1219.88, 1202.84, 912.22, 997.37, 1286.41, 812.62, 1318.59, 608.05, 1526.32, 1120.96, 1268.07, 1149.68, 1220.48, 991.66, 873.19, 1188.44, 1105.34, 1200.84, 1278.7, 1335.13, 1294.66, 1248.0, 1341.9, 1553.5, 1217.42, 1080.8, 908.64, 1603.26, 1308.35, 911.88, 1387.66, 811.89, 1059.73, 1382.68, 1157.21, 1489.88, 1402.47, 1252.58, 1187.24, 1255.78, 1554.17, 1134.02, 1344.09, 1045.55, 1174.21, 1029.91, 1043.78, 1139.66, 1064.76, 1420.32, 929.27, 1032.03, 1149.89, 1317.68, 1223.97, 920.54, 1175.03, 842.77, 1479.95, 951.16, 1031.14, 1253.78, 868.17, 1050.92, 945.28, 1114.92, 1165.73, 1247.88, 1208.14, 1119.33, 1338.81, 1066.48, 1058.07, 1150.84, 1383.87, 1466.33, 1194.96, 996.33, 1100.5, 1502.03, 1629.55, 1326.63, 1187.55, 1424.19, 1256.82, 1100.26, 1321.03, 1082.16, 1407.65, 758.72, 1093.5, 994.99, 1395.97, 995.77, 1206.16, 1505.23, 1130.56, 951.47, 1013.2, 948.72, 947.76, 1313.76, 1309.75, 1742.76, 870.3, 1149.11, 1039.46, 1086.87, 724.79, 1065.18, 668.05, 889.19, 1048.61, 1113.71, 1342.69, 923.27, 885.9, 1099.3, 1024.21, 969.81, 851.78, 1357.87, 1021.99, 987.69, 1657.26, 1298.22, 1168.86, 1365.18, 1198.88, 983.05, 1108.12, 1490.25, 1692.2, 1017.75, 1202.21, 1313.28, 1526.31, 1594.79, 1014.69, 885.51, 1155.23, 833.91, 1170.32, 1186.7, 905.46, 953.34, 1017.69, 985.43, 1286.2, 1093.07, 1216.69, 1264.42, 1197.52, 1009.62, 1426.41, 1215.55, 1139.18]；x̄₂=1161.4954，已知 σ₂=200
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=14.5947；z=(x̄₁−x̄₂)/SE=-0.2152；双侧 p=0.8296。
SE=√(σ₁²/n₁+σ₂²/n₂)=14.5947；z=(x̄₁−x̄₂)/SE=-0.2152；双侧 p=0.8296。
计算结果
z = -0.2152，p = 0.8296
在 α=0.05 水平下，p=0.8296 ≥ 0.05，不拒绝原假设 H₀。比较区域销量。
案例 99. 两版本转化率差异
原始数据
组1 n=933：[0.33, 0.29, 0.31, 0.31, 0.23, 0.34, 0.37, 0.19, 0.34, 0.35, 0.26, 0.36, 0.27, 0.24, 0.34, 0.38, 0.31, 0.35, 0.32, 0.35, 0.33, 0.37, 0.29, 0.34, 0.32, 0.33, 0.25, 0.31, 0.32, 0.27, 0.33, 0.33, 0.28, 0.34, 0.27, 0.37, 0.29, 0.31, 0.32, 0.31, 0.24, 0.36, 0.35, 0.41, 0.32, 0.38, 0.34, 0.26, 0.27, 0.33, 0.24, 0.35, 0.36, 0.25, 0.43, 0.32, 0.3, 0.38, 0.24, 0.37, 0.26, 0.29, 0.32, 0.36, 0.39, 0.34, 0.35, 0.3, 0.27, 0.36, 0.24, 0.27, 0.35, 0.3, 0.32, 0.3, 0.25, 0.29, 0.38, 0.32, 0.32, 0.26, 0.34, 0.36, 0.32, 0.4, 0.32, 0.35, 0.36, 0.32, 0.29, 0.36, 0.39, 0.38, 0.34, 0.27, 0.26, 0.3, 0.25, 0.34, 0.35, 0.42, 0.36, 0.27, 0.33, 0.33, 0.35, 0.37, 0.24, 0.41, 0.27, 0.31, 0.27, 0.33, 0.27, 0.36, 0.34, 0.43, 0.33, 0.36, 0.27, 0.26, 0.25, 0.32, 0.31, 0.36, 0.28, 0.39, 0.28, 0.31, 0.37, 0.3, 0.31, 0.27, 0.37, 0.37, 0.35, 0.24, 0.37, 0.34, 0.28, 0.31, 0.32, 0.37, 0.35, 0.38, 0.28, 0.28, 0.29, 0.28, 0.27, 0.31, 0.32, 0.26, 0.31, 0.34, 0.29, 0.38, 0.37, 0.32, 0.21, 0.14, 0.3, 0.37, 0.33, 0.31, 0.35, 0.27, 0.37, 0.25, 0.3, 0.35, 0.31, 0.32, 0.36, 0.36, 0.33, 0.32, 0.27, 0.36, 0.39, 0.33, 0.35, 0.34, 0.27, 0.31, 0.31, 0.2, 0.31, 0.28, 0.3, 0.32, 0.31, 0.4, 0.31, 0.31, 0.28, 0.35, 0.28, 0.35, 0.28, 0.33, 0.27, 0.35, 0.22, 0.4, 0.36, 0.35, 0.34, 0.28, 0.33, 0.32, 0.43, 0.36, 0.34, 0.32, 0.36, 0.28, 0.29, 0.3, 0.35, 0.34, 0.31, 0.29, 0.33, 0.34, 0.26, 0.32, 0.29, 0.35, 0.43, 0.33, 0.27, 0.36, 0.3, 0.33, 0.2, 0.22, 0.28, 0.3, 0.3, 0.25, 0.3, 0.32, 0.37, 0.25, 0.29, 0.26, 0.3, 0.29, 0.39, 0.33, 0.4, 0.31, 0.34, 0.33, 0.3, 0.28, 0.28, 0.33, 0.33, 0.26, 0.38, 0.26, 0.34, 0.29, 0.33, 0.28, 0.26, 0.4, 0.35, 0.31, 0.22, 0.21, 0.3, 0.3, 0.42, 0.34, 0.31, 0.31, 0.36, 0.3, 0.4, 0.36, 0.27, 0.24, 0.33, 0.29, 0.29, 0.27, 0.28, 0.3, 0.35, 0.33, 0.36, 0.31, 0.33, 0.25, 0.29, 0.33, 0.25, 0.37, 0.29, 0.28, 0.25, 0.29, 0.28, 0.44, 0.29, 0.36, 0.24, 0.29, 0.29, 0.23, 0.31, 0.3, 0.33, 0.21, 0.28, 0.25, 0.37, 0.32, 0.31, 0.29, 0.33, 0.3, 0.34, 0.35, 0.35, 0.23, 0.28, 0.32, 0.32, 0.27, 0.29, 0.34, 0.38, 0.28, 0.32, 0.32, 0.38, 0.33, 0.25, 0.25, 0.28, 0.3, 0.37, 0.36, 0.23, 0.37, 0.41, 0.3, 0.35, 0.28, 0.36, 0.36, 0.36, 0.26, 0.28, 0.3, 0.33, 0.22, 0.26, 0.26, 0.27, 0.25, 0.29, 0.4, 0.36, 0.23, 0.34, 0.32, 0.31, 0.36, 0.34, 0.28, 0.4, 0.37, 0.23, 0.35, 0.29, 0.39, 0.37, 0.32, 0.29, 0.36, 0.34, 0.32, 0.33, 0.31, 0.33, 0.22, 0.36, 0.21, 0.3, 0.34, 0.44, 0.33, 0.37, 0.28, 0.29, 0.33, 0.22, 0.2, 0.28, 0.36, 0.26, 0.33, 0.41, 0.24, 0.24, 0.42, 0.3, 0.23, 0.31, 0.35, 0.41, 0.34, 0.34, 0.27, 0.31, 0.19, 0.31, 0.32, 0.29, 0.34, 0.28, 0.37, 0.4, 0.31, 0.35, 0.34, 0.29, 0.35, 0.33, 0.27, 0.32, 0.42, 0.29, 0.38, 0.22, 0.33, 0.3, 0.26, 0.27, 0.41, 0.32, 0.34, 0.4, 0.38, 0.3, 0.3, 0.29, 0.29, 0.28, 0.35, 0.33, 0.3, 0.27, 0.29, 0.28, 0.32, 0.28, 0.37, 0.23, 0.38, 0.4, 0.28, 0.32, 0.3, 0.29, 0.34, 0.33, 0.29, 0.33, 0.33, 0.38, 0.35, 0.36, 0.3, 0.33, 0.34, 0.23, 0.4, 0.43, 0.23, 0.21, 0.34, 0.27, 0.3, 0.32, 0.34, 0.37, 0.29, 0.33, 0.32, 0.32, 0.3, 0.32, 0.36, 0.31, 0.34, 0.33, 0.35, 0.32, 0.3, 0.36, 0.34, 0.3, 0.41, 0.24, 0.31, 0.31, 0.33, 0.3, 0.31, 0.21, 0.31, 0.23, 0.33, 0.28, 0.39, 0.32, 0.33, 0.29, 0.34, 0.28, 0.3, 0.27, 0.41, 0.36, 0.29, 0.28, 0.27, 0.37, 0.36, 0.39, 0.26, 0.33, 0.21, 0.26, 0.32, 0.35, 0.33, 0.27, 0.4, 0.43, 0.36, 0.35, 0.25, 0.34, 0.35, 0.31, 0.28, 0.26, 0.28, 0.41, 0.27, 0.23, 0.41, 0.3, 0.3, 0.33, 0.27, 0.3, 0.32, 0.33, 0.33, 0.37, 0.28, 0.36, 0.27, 0.32, 0.31, 0.29, 0.34, 0.34, 0.38, 0.26, 0.31, 0.29, 0.33, 0.39, 0.45, 0.37, 0.23, 0.33, 0.3, 0.35, 0.38, 0.28, 0.37, 0.35, 0.32, 0.3, 0.32, 0.31, 0.33, 0.23, 0.29, 0.3, 0.27, 0.34, 0.22, 0.28, 0.36, 0.32, 0.26, 0.25, 0.42, 0.29, 0.32, 0.31, 0.38, 0.36, 0.28, 0.21, 0.32, 0.46, 0.44, 0.34, 0.37, 0.39, 0.27, 0.38, 0.25, 0.34, 0.35, 0.28, 0.29, 0.37, 0.4, 0.37, 0.32, 0.38, 0.32, 0.28, 0.33, 0.32, 0.3, 0.32, 0.21, 0.3, 0.23, 0.37, 0.21, 0.29, 0.26, 0.3, 0.29, 0.38, 0.32, 0.34, 0.4, 0.4, 0.36, 0.39, 0.37, 0.38, 0.22, 0.22, 0.3, 0.33, 0.38, 0.38, 0.28, 0.18, 0.36, 0.33, 0.34, 0.35, 0.32, 0.39, 0.39, 0.32, 0.33, 0.39, 0.36, 0.32, 0.42, 0.38, 0.29, 0.39, 0.34, 0.3, 0.31, 0.32, 0.34, 0.23, 0.29, 0.28, 0.24, 0.42, 0.29, 0.32, 0.27, 0.31, 0.3, 0.39, 0.38, 0.29, 0.35, 0.32, 0.39, 0.4, 0.32, 0.33, 0.26, 0.4, 0.3, 0.35, 0.29, 0.29, 0.29, 0.38, 0.29, 0.29, 0.28, 0.33, 0.43, 0.38, 0.26, 0.32, 0.32, 0.29, 0.28, 0.32, 0.32, 0.38, 0.34, 0.36, 0.37, 0.31, 0.26, 0.27, 0.26, 0.33, 0.34, 0.35, 0.32, 0.27, 0.39, 0.34, 0.34, 0.31, 0.35, 0.36, 0.35, 0.16, 0.24, 0.31, 0.25, 0.34, 0.34, 0.31, 0.3, 0.32, 0.37, 0.31, 0.3, 0.29, 0.29, 0.32, 0.3, 0.34, 0.35, 0.37, 0.3, 0.38, 0.35, 0.29, 0.29, 0.35, 0.3, 0.31, 0.31, 0.29, 0.35, 0.32, 0.21, 0.32, 0.35, 0.29, 0.37, 0.41, 0.31, 0.36, 0.4, 0.38, 0.3, 0.34, 0.24, 0.39, 0.26, 0.3, 0.35, 0.27, 0.37, 0.34, 0.36, 0.3, 0.42, 0.3, 0.21, 0.38, 0.32, 0.22, 0.17, 0.3, 0.31, 0.31, 0.27, 0.27, 0.37, 0.29, 0.37, 0.39, 0.21, 0.32, 0.29, 0.27, 0.32, 0.35, 0.38, 0.32, 0.26, 0.32, 0.32, 0.35, 0.4, 0.27, 0.33, 0.42, 0.37, 0.24, 0.39, 0.27, 0.31, 0.32, 0.4, 0.22, 0.27, 0.34, 0.3, 0.35, 0.34, 0.25, 0.37, 0.4, 0.39, 0.33, 0.31, 0.37, 0.39, 0.28, 0.41, 0.37, 0.4, 0.29, 0.24, 0.32, 0.35, 0.44, 0.4, 0.3, 0.36, 0.33, 0.34, 0.22, 0.34, 0.23, 0.34, 0.27, 0.23, 0.38, 0.37, 0.36, 0.35, 0.34, 0.3, 0.37, 0.37, 0.38, 0.26, 0.36, 0.36, 0.29, 0.33, 0.25, 0.36, 0.33, 0.33, 0.33, 0.26, 0.33, 0.33, 0.31, 0.29, 0.32, 0.35, 0.41, 0.33, 0.29, 0.28, 0.27, 0.33, 0.3, 0.34, 0.38, 0.37, 0.35, 0.43, 0.31, 0.29, 0.34, 0.41, 0.32, 0.31, 0.32, 0.41, 0.31, 0.26, 0.31, 0.34, 0.38, 0.33, 0.43, 0.27, 0.3, 0.31, 0.34, 0.33, 0.42]；x̄₁=0.3192，已知 σ₁=0.05；
组2 n=441：[0.3, 0.27, 0.31, 0.27, 0.22, 0.25, 0.3, 0.33, 0.31, 0.28, 0.31, 0.24, 0.28, 0.24, 0.27, 0.23, 0.26, 0.35, 0.28, 0.32, 0.38, 0.24, 0.29, 0.28, 0.2, 0.31, 0.39, 0.23, 0.33, 0.28, 0.37, 0.37, 0.3, 0.27, 0.31, 0.25, 0.25, 0.29, 0.29, 0.33, 0.31, 0.25, 0.36, 0.29, 0.33, 0.35, 0.28, 0.32, 0.38, 0.28, 0.31, 0.26, 0.25, 0.38, 0.32, 0.35, 0.24, 0.24, 0.23, 0.33, 0.33, 0.31, 0.36, 0.33, 0.33, 0.4, 0.27, 0.3, 0.37, 0.26, 0.32, 0.26, 0.31, 0.32, 0.23, 0.34, 0.27, 0.23, 0.3, 0.28, 0.3, 0.35, 0.25, 0.27, 0.22, 0.28, 0.25, 0.28, 0.3, 0.27, 0.25, 0.31, 0.31, 0.28, 0.29, 0.36, 0.44, 0.34, 0.32, 0.31, 0.25, 0.26, 0.3, 0.33, 0.22, 0.27, 0.3, 0.34, 0.26, 0.28, 0.26, 0.29, 0.29, 0.23, 0.26, 0.29, 0.37, 0.31, 0.32, 0.3, 0.31, 0.28, 0.28, 0.28, 0.25, 0.36, 0.32, 0.26, 0.3, 0.38, 0.31, 0.29, 0.37, 0.31, 0.24, 0.24, 0.33, 0.24, 0.36, 0.23, 0.15, 0.33, 0.31, 0.23, 0.27, 0.17, 0.26, 0.42, 0.26, 0.31, 0.31, 0.32, 0.31, 0.34, 0.35, 0.28, 0.28, 0.27, 0.34, 0.32, 0.28, 0.35, 0.33, 0.27, 0.26, 0.32, 0.32, 0.27, 0.29, 0.29, 0.29, 0.24, 0.26, 0.28, 0.3, 0.23, 0.26, 0.33, 0.28, 0.3, 0.31, 0.29, 0.36, 0.31, 0.29, 0.3, 0.25, 0.18, 0.28, 0.36, 0.3, 0.29, 0.33, 0.36, 0.26, 0.3, 0.27, 0.3, 0.22, 0.36, 0.28, 0.36, 0.29, 0.22, 0.33, 0.34, 0.35, 0.3, 0.36, 0.31, 0.27, 0.31, 0.3, 0.37, 0.28, 0.26, 0.23, 0.27, 0.33, 0.17, 0.24, 0.27, 0.27, 0.25, 0.29, 0.28, 0.33, 0.28, 0.23, 0.32, 0.35, 0.29, 0.38, 0.32, 0.35, 0.27, 0.31, 0.26, 0.33, 0.38, 0.3, 0.29, 0.3, 0.21, 0.34, 0.35, 0.29, 0.29, 0.27, 0.31, 0.31, 0.34, 0.32, 0.35, 0.27, 0.33, 0.3, 0.34, 0.27, 0.3, 0.25, 0.3, 0.37, 0.32, 0.29, 0.26, 0.37, 0.25, 0.29, 0.25, 0.23, 0.29, 0.29, 0.34, 0.24, 0.34, 0.15, 0.29, 0.36, 0.3, 0.29, 0.26, 0.3, 0.31, 0.24, 0.31, 0.29, 0.32, 0.24, 0.3, 0.33, 0.24, 0.24, 0.43, 0.26, 0.29, 0.27, 0.23, 0.19, 0.32, 0.4, 0.35, 0.33, 0.35, 0.3, 0.28, 0.29, 0.21, 0.31, 0.3, 0.24, 0.3, 0.26, 0.37, 0.32, 0.32, 0.35, 0.27, 0.31, 0.34, 0.36, 0.32, 0.28, 0.31, 0.23, 0.26, 0.28, 0.21, 0.28, 0.33, 0.28, 0.34, 0.33, 0.35, 0.34, 0.39, 0.4, 0.27, 0.28, 0.3, 0.31, 0.38, 0.28, 0.29, 0.29, 0.24, 0.41, 0.26, 0.3, 0.31, 0.33, 0.29, 0.28, 0.23, 0.16, 0.37, 0.24, 0.33, 0.3, 0.3, 0.34, 0.32, 0.31, 0.38, 0.25, 0.36, 0.31, 0.23, 0.39, 0.25, 0.19, 0.21, 0.28, 0.37, 0.21, 0.38, 0.44, 0.26, 0.27, 0.34, 0.3, 0.34, 0.23, 0.27, 0.37, 0.36, 0.16, 0.37, 0.25, 0.28, 0.34, 0.33, 0.34, 0.33, 0.26, 0.29, 0.27, 0.33, 0.32, 0.35, 0.33, 0.26, 0.35, 0.36, 0.28, 0.26, 0.22, 0.29, 0.39, 0.24, 0.34, 0.33, 0.39, 0.28, 0.23, 0.24, 0.36, 0.32, 0.31, 0.25, 0.28, 0.3, 0.34, 0.26, 0.3, 0.28, 0.2, 0.22, 0.22, 0.31, 0.22, 0.22, 0.27, 0.23, 0.31, 0.22, 0.34, 0.29, 0.29, 0.36, 0.4]；x̄₂=0.2963，已知 σ₂=0.05
组2 n=441：[0.3, 0.27, 0.31, 0.27, 0.22, 0.25, 0.3, 0.33, 0.31, 0.28, 0.31, 0.24, 0.28, 0.24, 0.27, 0.23, 0.26, 0.35, 0.28, 0.32, 0.38, 0.24, 0.29, 0.28, 0.2, 0.31, 0.39, 0.23, 0.33, 0.28, 0.37, 0.37, 0.3, 0.27, 0.31, 0.25, 0.25, 0.29, 0.29, 0.33, 0.31, 0.25, 0.36, 0.29, 0.33, 0.35, 0.28, 0.32, 0.38, 0.28, 0.31, 0.26, 0.25, 0.38, 0.32, 0.35, 0.24, 0.24, 0.23, 0.33, 0.33, 0.31, 0.36, 0.33, 0.33, 0.4, 0.27, 0.3, 0.37, 0.26, 0.32, 0.26, 0.31, 0.32, 0.23, 0.34, 0.27, 0.23, 0.3, 0.28, 0.3, 0.35, 0.25, 0.27, 0.22, 0.28, 0.25, 0.28, 0.3, 0.27, 0.25, 0.31, 0.31, 0.28, 0.29, 0.36, 0.44, 0.34, 0.32, 0.31, 0.25, 0.26, 0.3, 0.33, 0.22, 0.27, 0.3, 0.34, 0.26, 0.28, 0.26, 0.29, 0.29, 0.23, 0.26, 0.29, 0.37, 0.31, 0.32, 0.3, 0.31, 0.28, 0.28, 0.28, 0.25, 0.36, 0.32, 0.26, 0.3, 0.38, 0.31, 0.29, 0.37, 0.31, 0.24, 0.24, 0.33, 0.24, 0.36, 0.23, 0.15, 0.33, 0.31, 0.23, 0.27, 0.17, 0.26, 0.42, 0.26, 0.31, 0.31, 0.32, 0.31, 0.34, 0.35, 0.28, 0.28, 0.27, 0.34, 0.32, 0.28, 0.35, 0.33, 0.27, 0.26, 0.32, 0.32, 0.27, 0.29, 0.29, 0.29, 0.24, 0.26, 0.28, 0.3, 0.23, 0.26, 0.33, 0.28, 0.3, 0.31, 0.29, 0.36, 0.31, 0.29, 0.3, 0.25, 0.18, 0.28, 0.36, 0.3, 0.29, 0.33, 0.36, 0.26, 0.3, 0.27, 0.3, 0.22, 0.36, 0.28, 0.36, 0.29, 0.22, 0.33, 0.34, 0.35, 0.3, 0.36, 0.31, 0.27, 0.31, 0.3, 0.37, 0.28, 0.26, 0.23, 0.27, 0.33, 0.17, 0.24, 0.27, 0.27, 0.25, 0.29, 0.28, 0.33, 0.28, 0.23, 0.32, 0.35, 0.29, 0.38, 0.32, 0.35, 0.27, 0.31, 0.26, 0.33, 0.38, 0.3, 0.29, 0.3, 0.21, 0.34, 0.35, 0.29, 0.29, 0.27, 0.31, 0.31, 0.34, 0.32, 0.35, 0.27, 0.33, 0.3, 0.34, 0.27, 0.3, 0.25, 0.3, 0.37, 0.32, 0.29, 0.26, 0.37, 0.25, 0.29, 0.25, 0.23, 0.29, 0.29, 0.34, 0.24, 0.34, 0.15, 0.29, 0.36, 0.3, 0.29, 0.26, 0.3, 0.31, 0.24, 0.31, 0.29, 0.32, 0.24, 0.3, 0.33, 0.24, 0.24, 0.43, 0.26, 0.29, 0.27, 0.23, 0.19, 0.32, 0.4, 0.35, 0.33, 0.35, 0.3, 0.28, 0.29, 0.21, 0.31, 0.3, 0.24, 0.3, 0.26, 0.37, 0.32, 0.32, 0.35, 0.27, 0.31, 0.34, 0.36, 0.32, 0.28, 0.31, 0.23, 0.26, 0.28, 0.21, 0.28, 0.33, 0.28, 0.34, 0.33, 0.35, 0.34, 0.39, 0.4, 0.27, 0.28, 0.3, 0.31, 0.38, 0.28, 0.29, 0.29, 0.24, 0.41, 0.26, 0.3, 0.31, 0.33, 0.29, 0.28, 0.23, 0.16, 0.37, 0.24, 0.33, 0.3, 0.3, 0.34, 0.32, 0.31, 0.38, 0.25, 0.36, 0.31, 0.23, 0.39, 0.25, 0.19, 0.21, 0.28, 0.37, 0.21, 0.38, 0.44, 0.26, 0.27, 0.34, 0.3, 0.34, 0.23, 0.27, 0.37, 0.36, 0.16, 0.37, 0.25, 0.28, 0.34, 0.33, 0.34, 0.33, 0.26, 0.29, 0.27, 0.33, 0.32, 0.35, 0.33, 0.26, 0.35, 0.36, 0.28, 0.26, 0.22, 0.29, 0.39, 0.24, 0.34, 0.33, 0.39, 0.28, 0.23, 0.24, 0.36, 0.32, 0.31, 0.25, 0.28, 0.3, 0.34, 0.26, 0.3, 0.28, 0.2, 0.22, 0.22, 0.31, 0.22, 0.22, 0.27, 0.23, 0.31, 0.22, 0.34, 0.29, 0.29, 0.36, 0.4]；x̄₂=0.2963，已知 σ₂=0.05
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.0029；z=(x̄₁−x̄₂)/SE=7.9259；双侧 p=2.220e-15。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.0029；z=(x̄₁−x̄₂)/SE=7.9259；双侧 p=2.220e-15。
计算结果
z = 7.9259，p = 2.220e-15
在 α=0.05 水平下，p=2.220e-15 < 0.05，拒绝原假设 H₀。比较转化率均值。
案例 100. 两批次纯度差异
原始数据
组1 n=891：[99.15, 98.96, 99.68, 98.88, 100.12, 99.3, 99.0, 98.64, 99.57, 99.11, 99.4, 99.08, 99.02, 99.39, 100.05, 99.37, 99.19, 99.45, 98.86, 99.04, 99.4, 98.45, 99.23, 99.77, 99.98, 98.37, 99.48, 99.22, 99.0, 99.65, 99.69, 98.97, 99.46, 98.5, 99.61, 98.63, 98.92, 99.01, 99.39, 100.11, 99.01, 99.05, 99.08, 99.33, 99.36, 99.78, 99.52, 99.13, 98.81, 99.11, 100.2, 99.86, 99.2, 99.52, 99.22, 99.34, 99.69, 99.52, 100.14, 99.14, 98.97, 98.9, 99.19, 99.22, 99.97, 99.59, 99.76, 98.73, 98.23, 99.56, 99.46, 99.54, 99.26, 98.9, 98.07, 100.62, 99.26, 98.4, 99.28, 99.53, 99.21, 98.57, 99.19, 98.7, 99.05, 98.42, 98.94, 99.07, 99.45, 99.35, 99.58, 99.43, 99.61, 98.82, 98.28, 100.71, 99.27, 99.15, 99.54, 99.1, 98.05, 98.71, 99.23, 99.26, 99.13, 99.01, 99.06, 99.47, 99.77, 100.06, 99.25, 100.23, 99.86, 99.81, 99.1, 99.47, 98.54, 99.57, 99.86, 99.46, 99.15, 99.25, 99.85, 99.13, 98.64, 99.04, 98.83, 98.92, 97.77, 100.05, 99.0, 98.9, 99.03, 99.38, 99.08, 98.34, 99.35, 99.23, 98.91, 100.16, 99.83, 98.45, 99.65, 99.04, 99.7, 98.91, 98.35, 99.03, 99.49, 98.8, 98.87, 99.51, 99.26, 98.57, 99.78, 99.79, 99.67, 98.6, 99.92, 98.77, 98.98, 99.61, 99.19, 99.4, 99.03, 98.96, 98.5, 99.65, 98.92, 99.21, 99.18, 99.12, 99.19, 99.31, 98.76, 98.17, 99.21, 99.0, 98.92, 99.53, 99.53, 99.43, 98.66, 99.88, 98.53, 98.47, 98.49, 98.68, 98.53, 99.36, 98.97, 99.51, 99.79, 98.51, 99.65, 98.73, 99.49, 99.17, 97.9, 99.6, 99.01, 99.32, 98.85, 99.05, 99.23, 99.57, 99.66, 98.22, 99.47, 99.14, 98.85, 98.73, 98.72, 98.8, 98.19, 99.98, 99.67, 99.7, 99.28, 98.7, 98.95, 98.5, 98.6, 98.58, 98.72, 99.01, 98.68, 98.59, 99.9, 98.91, 98.82, 98.81, 99.61, 98.69, 98.74, 98.2, 99.47, 98.83, 99.49, 98.72, 99.08, 99.46, 99.21, 99.18, 98.57, 99.14, 99.4, 98.28, 99.65, 98.93, 98.73, 100.16, 98.83, 99.89, 99.93, 98.62, 98.84, 98.41, 98.58, 99.02, 99.92, 99.46, 99.61, 98.56, 99.2, 99.25, 98.31, 99.38, 99.55, 99.3, 98.14, 99.02, 99.25, 99.37, 99.12, 98.81, 97.46, 99.35, 99.37, 99.48, 97.78, 98.83, 99.85, 99.62, 99.58, 99.18, 98.85, 98.27, 98.37, 99.2, 98.9, 98.67, 98.22, 99.6, 99.57, 99.04, 99.71, 98.88, 100.0, 99.37, 99.56, 99.16, 99.15, 98.95, 99.07, 98.88, 99.36, 98.79, 99.5, 99.54, 99.44, 98.94, 97.59, 99.44, 99.01, 99.03, 99.46, 99.05, 99.44, 99.17, 99.93, 100.04, 99.58, 98.42, 98.23, 99.44, 99.15, 98.48, 99.53, 99.49, 99.27, 99.78, 99.25, 99.17, 100.53, 100.31, 99.31, 99.84, 98.86, 99.3, 99.25, 99.62, 99.8, 99.42, 98.14, 98.77, 99.73, 98.87, 98.93, 99.47, 99.97, 98.89, 99.64, 99.14, 98.71, 99.37, 98.83, 99.44, 99.58, 99.38, 99.01, 99.49, 99.66, 98.55, 99.03, 99.43, 98.86, 99.49, 99.71, 99.62, 99.85, 99.31, 98.82, 98.63, 99.11, 98.56, 99.41, 98.62, 99.66, 99.13, 98.62, 99.46, 98.09, 99.48, 98.41, 99.83, 100.53, 99.66, 99.01, 99.32, 99.95, 99.1, 99.25, 99.14, 99.14, 99.94, 98.48, 99.49, 98.38, 100.33, 99.17, 99.77, 99.06, 99.64, 98.65, 99.26, 99.14, 99.96, 99.08, 99.14, 99.9, 98.93, 100.36, 100.49, 99.39, 99.36, 99.95, 99.4, 99.58, 99.46, 98.93, 99.52, 98.75, 97.81, 98.72, 99.36, 99.5, 99.42, 99.12, 99.29, 98.76, 98.97, 99.21, 99.39, 98.29, 98.6, 99.13, 98.88, 98.3, 100.08, 99.63, 99.28, 99.45, 98.61, 98.91, 98.45, 99.21, 99.02, 98.93, 99.21, 99.16, 99.23, 98.41, 98.73, 99.11, 99.21, 98.54, 98.71, 99.59, 98.56, 99.07, 99.61, 99.16, 99.23, 99.52, 99.29, 99.34, 99.5, 99.42, 99.2, 99.04, 99.65, 99.95, 98.0, 98.65, 100.82, 98.61, 98.96, 99.07, 99.69, 99.12, 99.21, 98.51, 99.25, 99.01, 99.77, 98.91, 99.48, 98.58, 98.79, 99.73, 99.03, 100.44, 99.56, 99.73, 98.52, 99.22, 99.35, 99.14, 99.02, 98.52, 100.08, 99.71, 99.2, 99.31, 98.71, 99.66, 99.04, 98.21, 99.43, 99.69, 99.06, 99.57, 98.66, 98.86, 99.47, 98.77, 98.96, 98.39, 98.91, 99.27, 99.19, 99.52, 98.46, 99.39, 99.02, 99.38, 98.29, 98.13, 99.74, 98.75, 100.06, 98.97, 98.48, 99.05, 98.86, 99.13, 99.36, 100.03, 99.12, 99.04, 99.74, 99.82, 99.66, 99.53, 99.86, 99.63, 98.38, 98.86, 97.96, 99.54, 99.62, 98.29, 98.97, 98.87, 99.29, 99.38, 98.75, 98.49, 99.72, 99.31, 98.65, 99.6, 99.43, 99.06, 99.58, 99.73, 100.13, 99.24, 98.51, 100.77, 99.62, 99.23, 99.32, 99.13, 99.48, 99.16, 98.86, 99.32, 98.66, 99.19, 98.94, 99.88, 99.4, 100.22, 98.18, 98.74, 99.2, 98.98, 98.95, 98.72, 100.45, 99.27, 98.3, 99.42, 99.98, 99.32, 99.83, 98.96, 98.66, 98.67, 99.35, 99.55, 99.66, 99.76, 98.68, 100.5, 98.89, 99.06, 98.89, 100.11, 98.5, 98.81, 98.8, 99.15, 99.69, 99.32, 98.86, 99.54, 98.63, 99.2, 99.77, 99.22, 99.66, 99.64, 99.13, 99.79, 99.34, 99.95, 98.55, 100.03, 99.1, 99.51, 98.93, 99.66, 99.31, 99.57, 98.34, 99.21, 99.99, 98.88, 99.84, 99.48, 99.16, 99.67, 99.55, 99.06, 99.47, 99.68, 99.52, 99.5, 98.68, 99.68, 99.76, 99.59, 99.16, 99.39, 99.0, 98.35, 99.25, 99.05, 99.06, 99.26, 98.37, 99.35, 98.77, 98.91, 100.19, 99.1, 98.36, 99.85, 98.45, 98.79, 99.47, 99.33, 99.48, 99.57, 99.53, 99.44, 98.75, 99.84, 98.37, 99.11, 100.19, 99.32, 98.99, 98.07, 98.48, 99.44, 99.11, 98.99, 99.2, 100.11, 98.44, 100.0, 99.01, 99.25, 99.04, 98.4, 99.35, 99.08, 98.74, 99.28, 99.1, 99.49, 98.71, 98.13, 98.48, 99.71, 98.85, 99.85, 99.8, 99.22, 99.35, 99.29, 98.67, 100.42, 99.3, 98.92, 99.28, 99.31, 99.88, 98.81, 99.01, 98.17, 99.3, 99.26, 99.2, 99.31, 100.25, 99.98, 99.47, 99.08, 99.01, 99.55, 98.98, 98.52, 100.31, 98.6, 98.96, 99.13, 99.35, 98.41, 99.46, 99.13, 99.21, 99.41, 98.92, 98.16, 98.94, 99.54, 99.38, 98.93, 99.18, 98.72, 100.01, 98.28, 98.41, 99.46, 99.1, 99.25, 99.28, 99.28, 98.98, 98.4, 99.68, 98.99, 98.45, 98.7, 99.52, 99.91, 99.89, 99.06, 99.01, 98.47, 98.8, 99.56, 99.51, 100.15, 99.42, 99.91, 98.78, 99.68, 98.63, 99.43, 98.49, 99.45, 100.01, 98.84, 98.68, 99.45, 99.21, 99.57, 99.19, 99.39, 99.36, 98.96, 98.79, 99.11, 100.24, 99.62, 99.52, 99.1, 99.01, 99.16, 99.95, 99.21, 98.42, 99.38, 99.91, 99.64, 99.64, 99.75, 99.24, 98.18, 98.52, 98.74, 99.66, 99.48, 99.19, 99.66, 99.21, 99.35, 99.23, 99.57, 100.42, 98.66, 99.21, 99.2, 99.58, 98.23, 100.22, 98.81, 99.55, 99.02, 98.59, 99.73, 99.05, 99.3, 98.82, 100.01, 99.1, 99.78, 98.54, 98.98, 98.57, 98.76, 99.17, 98.8, 98.38, 99.43, 99.55, 98.99, 98.83, 99.28, 99.04, 98.35, 98.51, 99.47, 99.43, 98.53, 99.65, 99.31, 99.42, 98.96, 99.31, 99.63, 99.04, 98.78, 99.26, 100.11, 98.73, 99.37, 99.68, 99.92, 99.27, 99.37, 99.65, 98.44, 99.09, 99.49, 99.82, 98.59, 98.79, 99.51, 98.71, 99.16, 99.2, 99.82, 99.22, 99.58]；x̄₁=99.1963，已知 σ₁=0.5；
组2 n=605：[98.86, 98.51, 98.99, 99.33, 98.34, 98.89, 99.34, 98.49, 97.91, 98.98, 98.51, 99.1, 98.69, 99.11, 99.39, 98.17, 99.35, 98.51, 99.02, 99.11, 98.72, 98.18, 98.53, 99.24, 98.03, 99.51, 98.61, 98.89, 98.93, 99.23, 99.45, 98.97, 99.12, 98.54, 98.87, 99.69, 98.92, 98.79, 98.33, 99.38, 99.13, 98.89, 99.97, 98.03, 99.03, 98.82, 98.41, 99.28, 98.62, 98.74, 98.86, 98.6, 98.26, 99.98, 98.44, 98.81, 99.15, 98.43, 99.22, 97.92, 99.17, 98.83, 99.77, 98.78, 98.41, 99.34, 100.02, 98.85, 99.33, 99.12, 98.24, 98.16, 98.76, 99.48, 98.89, 98.64, 98.79, 98.9, 98.18, 98.47, 98.35, 98.93, 99.54, 98.27, 98.08, 98.48, 100.12, 99.34, 98.06, 98.97, 98.01, 99.35, 98.64, 98.24, 99.13, 98.7, 99.69, 99.32, 98.8, 98.15, 99.09, 98.43, 97.71, 98.37, 98.77, 98.53, 98.6, 98.86, 98.59, 99.35, 99.4, 98.67, 99.36, 98.98, 98.92, 99.5, 99.21, 99.41, 98.97, 99.67, 99.29, 99.36, 98.8, 98.56, 98.56, 99.78, 98.99, 98.77, 98.3, 99.28, 98.69, 99.45, 98.56, 97.73, 99.22, 99.26, 98.08, 97.76, 99.4, 99.58, 98.91, 99.0, 98.7, 98.89, 98.41, 98.6, 98.49, 100.03, 98.71, 98.39, 99.25, 99.2, 99.16, 98.91, 99.34, 98.7, 98.02, 98.88, 98.67, 98.67, 98.78, 99.42, 98.87, 98.71, 99.35, 97.55, 99.37, 98.59, 98.29, 98.4, 98.73, 99.25, 98.68, 99.07, 98.6, 98.89, 99.3, 99.17, 98.77, 98.36, 99.48, 98.73, 98.31, 99.04, 99.56, 98.69, 98.54, 97.85, 98.42, 98.55, 98.92, 98.58, 98.56, 98.8, 97.69, 98.57, 98.9, 98.56, 99.41, 98.79, 98.86, 98.36, 98.56, 99.77, 99.49, 98.99, 98.98, 99.64, 99.87, 99.85, 99.21, 98.99, 98.54, 99.97, 98.64, 98.56, 98.56, 98.37, 99.49, 99.26, 99.74, 99.01, 99.04, 99.19, 98.13, 98.02, 99.07, 98.77, 99.09, 97.61, 97.88, 99.21, 98.88, 99.59, 99.26, 98.85, 98.05, 99.63, 98.56, 98.85, 98.32, 98.38, 98.39, 99.3, 100.41, 98.59, 98.33, 98.46, 98.8, 98.8, 98.55, 98.72, 99.01, 98.32, 98.45, 98.84, 99.09, 99.26, 99.31, 98.49, 99.1, 98.43, 98.11, 98.87, 98.37, 99.15, 99.23, 98.09, 98.36, 98.89, 98.44, 99.56, 98.55, 98.69, 99.11, 98.52, 98.79, 98.94, 99.85, 98.32, 98.78, 98.23, 99.15, 97.83, 98.59, 98.41, 98.82, 99.31, 98.79, 99.4, 98.57, 99.33, 98.64, 97.62, 99.84, 98.67, 98.29, 99.2, 98.13, 99.51, 98.46, 98.78, 99.93, 99.19, 98.41, 99.04, 98.69, 98.67, 98.32, 98.23, 98.12, 99.15, 98.26, 99.26, 99.01, 98.53, 99.59, 99.11, 98.64, 99.45, 98.82, 98.71, 99.29, 98.56, 98.29, 98.46, 99.87, 98.56, 98.82, 99.0, 98.41, 98.67, 98.56, 98.19, 98.29, 98.02, 98.39, 98.87, 100.16, 98.44, 99.27, 98.71, 97.89, 98.49, 99.8, 99.42, 98.62, 99.39, 98.34, 98.61, 98.43, 98.87, 98.94, 98.76, 99.17, 98.99, 99.06, 98.81, 99.59, 98.81, 99.08, 98.4, 98.02, 98.38, 98.92, 97.69, 98.65, 99.71, 99.49, 98.79, 99.38, 98.74, 98.77, 98.83, 99.04, 98.1, 99.51, 99.11, 98.77, 98.76, 99.07, 98.84, 98.3, 98.73, 99.2, 98.63, 98.84, 99.12, 99.76, 98.84, 98.94, 97.53, 98.35, 99.32, 99.23, 98.05, 99.84, 99.34, 99.39, 98.88, 98.65, 98.8, 99.2, 98.89, 98.2, 98.56, 98.24, 98.67, 98.87, 98.81, 98.78, 99.21, 98.8, 98.68, 98.4, 98.65, 99.04, 99.42, 98.41, 98.4, 97.86, 98.96, 98.6, 98.2, 99.21, 98.59, 98.65, 98.8, 98.89, 98.92, 99.04, 99.18, 98.58, 99.21, 99.13, 99.69, 99.49, 99.11, 98.44, 97.72, 98.04, 98.76, 98.91, 99.34, 97.86, 99.0, 99.19, 98.61, 98.63, 99.42, 98.24, 99.53, 98.05, 98.32, 98.6, 98.58, 98.97, 98.03, 98.54, 98.57, 98.69, 99.28, 98.9, 99.02, 99.83, 98.38, 98.04, 98.58, 98.18, 97.41, 98.49, 98.58, 98.91, 99.17, 98.35, 98.15, 99.1, 98.69, 99.42, 98.67, 98.97, 99.02, 98.38, 99.35, 99.0, 98.12, 98.76, 99.3, 100.04, 98.9, 99.85, 99.23, 98.81, 99.3, 98.22, 98.94, 98.5, 98.52, 98.76, 98.73, 99.04, 99.38, 98.81, 98.06, 99.14, 99.53, 98.56, 99.02, 97.99, 98.84, 99.29, 98.33, 98.95, 98.59, 98.6, 98.58, 98.37, 98.67, 98.3, 99.02, 98.85, 98.11, 98.82, 98.59, 99.87, 98.56, 98.3, 99.2, 99.31, 99.11, 98.39, 99.15, 99.0, 98.69, 99.04, 99.54, 98.04, 99.45, 98.43, 99.18, 98.74, 98.89, 99.18, 98.73, 98.81, 99.0, 99.7, 99.45, 98.84, 99.42, 98.88, 99.97, 98.7, 99.51, 99.57, 98.71, 98.19, 99.0, 98.83, 99.92, 99.03, 97.89, 98.58, 98.67, 98.34, 98.71, 99.1, 99.49, 98.91, 98.47, 98.08, 98.44, 98.87, 98.96, 98.44, 99.48, 99.22, 98.71, 99.15, 98.91, 99.38, 98.21, 97.78, 99.21, 99.13, 99.29, 98.19, 98.11, 98.66, 98.77, 98.26, 98.55, 98.8, 98.12, 98.74, 98.22, 98.68, 98.66, 98.69, 99.33, 99.3, 99.0, 98.07, 98.97, 98.35]；x̄₂=98.8262，已知 σ₂=0.5
组2 n=605：[98.86, 98.51, 98.99, 99.33, 98.34, 98.89, 99.34, 98.49, 97.91, 98.98, 98.51, 99.1, 98.69, 99.11, 99.39, 98.17, 99.35, 98.51, 99.02, 99.11, 98.72, 98.18, 98.53, 99.24, 98.03, 99.51, 98.61, 98.89, 98.93, 99.23, 99.45, 98.97, 99.12, 98.54, 98.87, 99.69, 98.92, 98.79, 98.33, 99.38, 99.13, 98.89, 99.97, 98.03, 99.03, 98.82, 98.41, 99.28, 98.62, 98.74, 98.86, 98.6, 98.26, 99.98, 98.44, 98.81, 99.15, 98.43, 99.22, 97.92, 99.17, 98.83, 99.77, 98.78, 98.41, 99.34, 100.02, 98.85, 99.33, 99.12, 98.24, 98.16, 98.76, 99.48, 98.89, 98.64, 98.79, 98.9, 98.18, 98.47, 98.35, 98.93, 99.54, 98.27, 98.08, 98.48, 100.12, 99.34, 98.06, 98.97, 98.01, 99.35, 98.64, 98.24, 99.13, 98.7, 99.69, 99.32, 98.8, 98.15, 99.09, 98.43, 97.71, 98.37, 98.77, 98.53, 98.6, 98.86, 98.59, 99.35, 99.4, 98.67, 99.36, 98.98, 98.92, 99.5, 99.21, 99.41, 98.97, 99.67, 99.29, 99.36, 98.8, 98.56, 98.56, 99.78, 98.99, 98.77, 98.3, 99.28, 98.69, 99.45, 98.56, 97.73, 99.22, 99.26, 98.08, 97.76, 99.4, 99.58, 98.91, 99.0, 98.7, 98.89, 98.41, 98.6, 98.49, 100.03, 98.71, 98.39, 99.25, 99.2, 99.16, 98.91, 99.34, 98.7, 98.02, 98.88, 98.67, 98.67, 98.78, 99.42, 98.87, 98.71, 99.35, 97.55, 99.37, 98.59, 98.29, 98.4, 98.73, 99.25, 98.68, 99.07, 98.6, 98.89, 99.3, 99.17, 98.77, 98.36, 99.48, 98.73, 98.31, 99.04, 99.56, 98.69, 98.54, 97.85, 98.42, 98.55, 98.92, 98.58, 98.56, 98.8, 97.69, 98.57, 98.9, 98.56, 99.41, 98.79, 98.86, 98.36, 98.56, 99.77, 99.49, 98.99, 98.98, 99.64, 99.87, 99.85, 99.21, 98.99, 98.54, 99.97, 98.64, 98.56, 98.56, 98.37, 99.49, 99.26, 99.74, 99.01, 99.04, 99.19, 98.13, 98.02, 99.07, 98.77, 99.09, 97.61, 97.88, 99.21, 98.88, 99.59, 99.26, 98.85, 98.05, 99.63, 98.56, 98.85, 98.32, 98.38, 98.39, 99.3, 100.41, 98.59, 98.33, 98.46, 98.8, 98.8, 98.55, 98.72, 99.01, 98.32, 98.45, 98.84, 99.09, 99.26, 99.31, 98.49, 99.1, 98.43, 98.11, 98.87, 98.37, 99.15, 99.23, 98.09, 98.36, 98.89, 98.44, 99.56, 98.55, 98.69, 99.11, 98.52, 98.79, 98.94, 99.85, 98.32, 98.78, 98.23, 99.15, 97.83, 98.59, 98.41, 98.82, 99.31, 98.79, 99.4, 98.57, 99.33, 98.64, 97.62, 99.84, 98.67, 98.29, 99.2, 98.13, 99.51, 98.46, 98.78, 99.93, 99.19, 98.41, 99.04, 98.69, 98.67, 98.32, 98.23, 98.12, 99.15, 98.26, 99.26, 99.01, 98.53, 99.59, 99.11, 98.64, 99.45, 98.82, 98.71, 99.29, 98.56, 98.29, 98.46, 99.87, 98.56, 98.82, 99.0, 98.41, 98.67, 98.56, 98.19, 98.29, 98.02, 98.39, 98.87, 100.16, 98.44, 99.27, 98.71, 97.89, 98.49, 99.8, 99.42, 98.62, 99.39, 98.34, 98.61, 98.43, 98.87, 98.94, 98.76, 99.17, 98.99, 99.06, 98.81, 99.59, 98.81, 99.08, 98.4, 98.02, 98.38, 98.92, 97.69, 98.65, 99.71, 99.49, 98.79, 99.38, 98.74, 98.77, 98.83, 99.04, 98.1, 99.51, 99.11, 98.77, 98.76, 99.07, 98.84, 98.3, 98.73, 99.2, 98.63, 98.84, 99.12, 99.76, 98.84, 98.94, 97.53, 98.35, 99.32, 99.23, 98.05, 99.84, 99.34, 99.39, 98.88, 98.65, 98.8, 99.2, 98.89, 98.2, 98.56, 98.24, 98.67, 98.87, 98.81, 98.78, 99.21, 98.8, 98.68, 98.4, 98.65, 99.04, 99.42, 98.41, 98.4, 97.86, 98.96, 98.6, 98.2, 99.21, 98.59, 98.65, 98.8, 98.89, 98.92, 99.04, 99.18, 98.58, 99.21, 99.13, 99.69, 99.49, 99.11, 98.44, 97.72, 98.04, 98.76, 98.91, 99.34, 97.86, 99.0, 99.19, 98.61, 98.63, 99.42, 98.24, 99.53, 98.05, 98.32, 98.6, 98.58, 98.97, 98.03, 98.54, 98.57, 98.69, 99.28, 98.9, 99.02, 99.83, 98.38, 98.04, 98.58, 98.18, 97.41, 98.49, 98.58, 98.91, 99.17, 98.35, 98.15, 99.1, 98.69, 99.42, 98.67, 98.97, 99.02, 98.38, 99.35, 99.0, 98.12, 98.76, 99.3, 100.04, 98.9, 99.85, 99.23, 98.81, 99.3, 98.22, 98.94, 98.5, 98.52, 98.76, 98.73, 99.04, 99.38, 98.81, 98.06, 99.14, 99.53, 98.56, 99.02, 97.99, 98.84, 99.29, 98.33, 98.95, 98.59, 98.6, 98.58, 98.37, 98.67, 98.3, 99.02, 98.85, 98.11, 98.82, 98.59, 99.87, 98.56, 98.3, 99.2, 99.31, 99.11, 98.39, 99.15, 99.0, 98.69, 99.04, 99.54, 98.04, 99.45, 98.43, 99.18, 98.74, 98.89, 99.18, 98.73, 98.81, 99.0, 99.7, 99.45, 98.84, 99.42, 98.88, 99.97, 98.7, 99.51, 99.57, 98.71, 98.19, 99.0, 98.83, 99.92, 99.03, 97.89, 98.58, 98.67, 98.34, 98.71, 99.1, 99.49, 98.91, 98.47, 98.08, 98.44, 98.87, 98.96, 98.44, 99.48, 99.22, 98.71, 99.15, 98.91, 99.38, 98.21, 97.78, 99.21, 99.13, 99.29, 98.19, 98.11, 98.66, 98.77, 98.26, 98.55, 98.8, 98.12, 98.74, 98.22, 98.68, 98.66, 98.69, 99.33, 99.3, 99.0, 98.07, 98.97, 98.35]；x̄₂=98.8262，已知 σ₂=0.5
分析过程
H₀: μ₁=μ₂，H₁: μ₁≠μ₂（已知σ用z检验）。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.0263；z=(x̄₁−x̄₂)/SE=14.0510；双侧 p=0.000e+00。
SE=√(σ₁²/n₁+σ₂²/n₂)=0.0263；z=(x̄₁−x̄₂)/SE=14.0510；双侧 p=0.000e+00。
计算结果
z = 14.0510，p = 0.000e+00
在 α=0.05 水平下，p=0.000e+00 < 0.05，拒绝原假设 H₀。比较产品纯度。
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