The Paired t-Test in Edexcel A-Level Maths | Edexcel A-Level 数学中的配对 t 检验

📚 The Paired t-Test in Edexcel A-Level Maths | Edexcel A-Level 数学中的配对 t 检验

In the Edexcel A-Level Further Mathematics Statistics specification, the paired t-test is used when two samples are naturally linked in pairs, such as ‘before and after’ measurements on the same individuals. It tests whether the mean difference between paired observations is significantly different from zero.

在 Edexcel A-Level 进阶数学统计部分中，当两个样本天然成对出现时，例如同一个体的“前后”测量，就会使用配对 t 检验。它用于检验配对观测值之间的平均差是否显著不同于零。

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1. When to Use a Paired t-Test | 何时使用配对 t 检验

Use a paired t-test when each data value in sample A is linked to exactly one data value in sample B, so the differences are meaningful. Common designs include a before-and-after test on the same subject, left-right measurements, or twins matched on a factor.

当样本 A 中的每个数据值与样本 B 中的恰好一个数据值相对应，并且差值具有实际意义时，应使用配对 t 检验。常见的设计包括同一对象的测试前后对比、左右侧测量，或按某一因素匹配的双胞胎。

• Before and after training scores | 训练前后的成绩
• Left and right eye measurements | 左右眼测量
• Husband and wife income in a matched survey | 匹配调查中夫妻的收入

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2. Hypotheses | 假设设定

For a two-tailed test, the null and alternative hypotheses are H₀: μ_d = 0 and H₁: μ_d ≠ 0. For a one-tailed test, use H₁: μ_d > 0 or H₁: μ_d < 0, where μ_d is the population mean difference.

对于双尾检验，原假设和备择假设为 H₀: μ_d = 0 和 H₁: μ_d ≠ 0。对于单尾检验，则使用 H₁: μ_d > 0 或 H₁: μ_d < 0，其中 μ_d 是总体平均差。

The choice between one-tailed and two-tailed depends on the research question. If you are only interested in whether there is a change, use two-tailed; if you specifically expect an increase or decrease, use one-tailed.

单尾与双尾的选择取决于研究问题。如果只关心是否存在变化，使用双尾；如果明确预期增加或减少，则使用单尾。

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3. Calculating Differences | 计算差值

Define d_i = x_i − y_i for each pair. Keep the order consistent across all pairs. Then compute the mean difference d̄ = Σd_i / n. The sign of d̄ matters for one-tailed tests.

对每一对数据定义 d_i = x_i − y_i，并在所有配对中保持顺序一致。然后计算平均差 d̄ = Σd_i / n。在单尾检验中，d̄ 的正负号非常重要。

For example, if d = after − before, a negative d̄ means the measurement decreased on average. If d = before − after, the interpretation reverses.

例如，如果 d = 后 − 前，那么 d̄ 为负表示测量值平均下降。如果 d = 前 − 后，解释则相反。

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4. Sample Variance and Standard Error | 差值的样本方差与标准误

Use s_d² = Σ(d_i − d̄)² / (n − 1). This is the unbiased estimate of variance because we are working with a sample and using the t distribution. Then the standard error is s_d / √n.

使用 s_d² = Σ(d_i − d̄)² / (n − 1)。这是方差的无偏估计，因为我们是基于样本并使用 t 分布。然后标准误为 s_d / √n。

s_d² = Σ(d_i − d̄)² / (n − 1)

Do not divide by n here, because the sample variance must be unbiased for the t-test.

这里不要除以 n，因为 t 检验中的样本方差必须是无偏的。

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5. Test Statistic and Distribution | 检验统计量及其分布

The paired t-test statistic is t = (d̄ − μ_d) / (s_d / √n). Under H₀, μ_d = 0, and the statistic has a t-distribution with ν = n − 1 degrees of freedom.

配对 t 检验的统计量为 t = (d̄ − μ_d) / (s_d / √n)。在原假设 H₀ 下，μ_d = 0，该统计量服从自由度为 ν = n − 1 的 t 分布。

t = (d̄ − μ_d) / (s_d / √n), ν = n − 1

Replacing μ_d with 0 gives t = d̄ / (s_d / √n) in most exam questions.

在大多数考试题中，将 μ_d 替换为 0，得到 t = d̄ / (s_d / √n)。

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6. Critical Values and Rejection Region | 临界值与拒绝域

Find the critical value t_crit from t-tables using ν = n − 1 and the significance level α. For a two-tailed test, split α equally into two tails. Reject H₀ if |t| > t_crit, or if the p-value is less than α.

使用 ν = n − 1 和显著性水平 α 从 t 分布表中查找临界值 t_crit。对于双尾检验，将 α 平均分配到两个尾部。如果 |t| > t_crit，或者 p 值小于 α，则拒绝 H₀。

ν	tcrit (5% two-tailed)	tcrit (1% two-tailed)
7	2.365	3.499
8	2.306	3.355

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7. p-Value Method | p 值方法

The p-value is the probability of observing a test statistic at least as extreme as the calculated t, assuming H₀ is true. If the p-value is less than α, reject H₀. Calculators or statistical tables give approximate p-values.

p 值是在 H₀ 为真的前提下，观察到与计算所得 t 统计量同样极端或更极端的概率。如果 p 值小于 α，则拒绝 H₀。计算器或统计表可给出近似 p 值。

For a two-tailed test with t = −4.00 and ν = 7, the p-value is approximately 0.005, which is much less than 0.05, so we reject H₀.

对于 t = −4.00、ν = 7 的双尾检验，p 值约为 0.005，远小于 0.05，因此拒绝 H₀。

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8. Confidence Interval for Mean Difference | 平均差的置信区间

A 100(1 − α)% confidence interval for μ_d is d̄ ± t_{ν, α/2} × (s_d / √n). If the interval does not contain 0, this is consistent with rejecting H₀ at level α for a two-tailed test.

μ_d 的 100(1 − α)% 置信区间为 d̄ ± t_{ν, α/2} × (s_d / √n)。如果该区间不包含 0，则与在显著性水平 α 下拒绝双尾检验的 H₀ 一致。

d̄ ± t_{ν, α/2} × (s_d / √n)

This interval also gives a range of plausible values for the true mean difference, which is useful in practical reports.

该区间还给出了真实平均差的合理数值范围，在实际报告中也很有用。

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9. Worked Example | 例题解析

A coach records the 100 m times, in seconds, of 8 athletes before and after a training programme. The differences d = after − before are:

一名教练记录了 8 名运动员在训练项目前后的 100 米跑时间（单位：秒）。差值 d = 后 − 前 为：

• -0.4, -0.2, -0.6, 0.0, -0.3, -0.5, -0.1, -0.2

Test at the 5% significance level whether the training programme has changed the mean time.

在 5% 显著性水平下检验该训练项目是否改变了平均时间。

Step 1: H₀: μ_d = 0, H₁: μ_d ≠ 0, α = 0.05.

步骤 1：H₀: μ_d = 0，H₁: μ_d ≠ 0，α = 0.05。

Step 2: Compute the mean difference. Σd_i = -2.3, so d̄ = -2.3 / 8 = -0.2875 s.

步骤 2：计算平均差。Σd_i = -2.3，因此 d̄ = -2.3 / 8 = -0.2875 秒。

Step 3: Compute the sample variance of differences. Σ(d_i − d̄)² = 0.28875, so s_d² = 0.28875 / 7 = 0.04125 and s_d ≈ 0.2031.

步骤 3：计算差值的样本方差。Σ(d_i − d̄)² = 0.28875，因此 s_d² = 0.28875 / 7 = 0.04125，s_d ≈ 0.2031。

Step 4: Standard error = 0.2031 / √8 ≈ 0.0718. Test statistic t = -0.2875 / 0.0718 ≈ -4.00.

步骤 4：标准误 = 0.2031 / √8 ≈ 0.0718。检验统计量 t = -0.2875 / 0.0718 ≈ -4.00。

Step 5: Degrees of freedom ν = 7. The critical value for a 5% two-tailed test is 2.365. Since |-4.00| > 2.365, reject H₀.

步骤 5：自由度 ν = 7。5% 双尾检验的临界值为 2.365。因为 |-4.00| > 2.365，所以拒绝 H₀

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