📚 The Paired t-test | 配对t检验
The paired t-test is a fundamental hypothesis test in statistics used to compare the means of two related samples. Rather than treating the two sets of data as independent, it focuses on the differences between each matched pair. This approach removes variability due to differences between subjects, making it a powerful tool for before-and-after studies, repeated measures, and matched-pair designs. In the Edexcel A-Level Mathematics specification, you are expected to understand when to apply the paired t-test, how to calculate the test statistic, and how to interpret the results in context.
配对t检验是统计学中一种基本的假设检验方法，用于比较两个相关样本的均值。它并不把两组数据视为独立，而是着眼于每对配对数据之间的差值。这种方法消除了由个体间差异引起的变异，因此成为处理前后对比、重复测量及配对设计的有力工具。在 Edexcel A-Level 数学大纲中，你需要掌握何时使用配对t检验、如何计算检验统计量，以及如何在具体情境中解读结果。
1. Understanding Paired Data | 理解配对数据
-test scores for the same students, measurements on left and right eyes of the same individuals, or data collected from twins. Because the values within each pair are not independent, the standard two-sample t-test is inappropriate. Instead, we work with the differences dᵢ = xᵢ − yᵢ for each pair i = 1, 2, …, n, turning the problem into a one-sample analysis of these differences.
当一个样本中的每一个观测值都与另一个样本中的某一观测值存在对应关系时，就形成了配对数据。典型的例子有同一组学生的前测与后测成绩、同一个人的左右眼测量数据，或双胞胎的数据。由于每对内的数值并不独立，标准的双样本t检验并不适用。我们转而使用每对的差值 dᵢ = xᵢ − yᵢ (i = 1, 2, …, n)，从而将问题转化为对这些差值进行单样本分析。
The key idea is that if there is no systematic effect, the average of these differences should be close to zero. Any consistent shift away from zero suggests a treatment effect or a real difference between the two conditions.
其核心思想是，如果不存在系统性效应，那么这些差值的平均值应接近于零。任何持续偏离零的趋势都表明存在处理效应，或两种条件间存在真实差异。
2. When to Use a Paired t-test | 何时使用配对t检验
ched on important characteristics. It tests whether the mean difference μ_d between the paired observations differs from a hypothesised value, usually zero. Typical scenarios include ‘ rials, and matching individuals by age, gender, or other covariates to reduce confounding.
配对t检验专门用于对同一批统计单位进行两次测量，或者将统计单位根据重要特征进行匹配后的情况。它检验配对观测值之间的均值差 μ_d 是否异于某个假设值（通常为零）。典型的场景包括“处理前与处理后”实验、交叉试验，以及为减少混杂因素而根据年龄、性别等协变量对个体进行匹配的研究。
It is essential not to confuse the paired t-test with the two-sample t-test. If the two sets of data come from entirely independent groups – for instance, measuring the heights of boys and girls from different classes – the paired design is absent, and a two-sample t-test should be used instead. Misapplying the test can lead to incorrect conclusions and a loss of statistical power.
务必不要混淆配对t检验与双样本t检验。如果两组数据来自完全独立的群体——例如测量不同班级的男生和女生的身高——那么不存在配对设计，应采用双样本t检验。错误使用检验方法可能导致结论错误，并降低统计功效。
3. Null and Alternative Hypotheses | 原假设与备择假设
senting no change or no effect. The alternative hypothesis H₁ can take three forms depending on the direction of the suspected effect:
在配对t检验中，原假设 H₀ 认为真实的均值差 μ_d 等于某个特定值 μ₀。在大多数考题中，μ₀ = 0，表示没有变化或没有效应。备择假设 H₁ 可根据预期的效应方向取三种形式：
ce of a one-tailed or two-tailed test must be guided by the research question and should be decided before examining the data.
双侧检验：H₀: μ_d = 0 对 H₁: μ_d ≠ 0。右侧检验：H₀: μ_d = 0 对 H₁: μ_d > 0。左侧检验：H₀: μ_d = 0 对 H₁: μ_d < 0。选择单侧还是双侧检验应根据研究问题而定，并须在查看数据前确定。
4. Test Statistic and Its Distribution | 检验统计量及其分布
Once the differences dᵢ are computed, we calculate the sample mean difference d̄ and the sample standard deviation of the differences sd. The paired t-test statistic is given by:
t = (d̄ − μ₀) / (sd / √n)
计算出差值 dᵢ 之后，我们进一步计算样本均值差 d̄ 以及差值的样本标准差 sd。配对t检验的统计量为：
t = (d̄ − μ₀) / (sd / √n)
he number of pairs. The t-distribution is symmetric, bell-shaped, and has thicker tails than the standard normal distribution when n is small. As the sample size increases, it converges to the normal distribution.
在原假设下，该统计量服从自由度 ν = n − 1 的 t 分布，其中 n 为配对数目。t 分布对称、呈钟形，当 n 较小时其尾部比标准正态分布更厚。随着样本容量增大，它趋近于正态分布。
5. Computing the Differences and Summary Statistics | 差值计算与汇总统计
xᵢ − yᵢ (or yᵢ − xᵢ, as long as the direction is consistent). Then find the sample mean d̄ = (∑ dᵢ) / n. The sample standard deviation sd is calculated
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