A-Level Further Mathematics: Paired t-Test – Principles and Applications | 进阶数学：配对t检验的原理与应用

📚 A-Level Further Mathematics: Paired t-Test – Principles and Applications | 进阶数学：配对t检验的原理与应用

The paired t-test is one of the most commonly used statistical procedures in A-Level Further Mathematics, particularly within the Edexcel syllabus. It is designed to compare two sets of measurements taken from the same subjects or matched pairs, helping us decide whether the mean difference is statistically significant.

配对t检验是A-Level进阶数学中最常用的统计方法之一，尤其在Edexcel考纲中占有重要地位。它用于比较来自同一组对象或配对样本的两组测量值，帮助我们判断平均差异是否具有统计显著性。

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

A paired t-test is appropriate when the data consist of two related samples. This occurs in two main scenarios: repeated measurements on the same individual (before-and-after studies) and naturally matched pairs (such as twins or left-and-right hands of the same person).

当数据由两个相关样本组成时，适合使用配对t检验。这主要有两种情况：对同一对象的重复测量（前后对照研究），以及天然配对的样本（如双胞胎或同一个人的左右手）。

Before-and-after design: measuring blood pressure before and after taking a drug.
Matched pairs design: comparing exam scores of twins randomly assigned to two teaching methods.

前后对照设计：测量服药前后的血压。
配对设计：比较双胞胎分别接受两种教学方法后的考试成绩。

2. The Null and Alternative Hypotheses | 原假设与备择假设

Let dᵢ = xᵢ − yᵢ be the difference for the i-th pair. We assume these differences are normally distributed with unknown mean μd and unknown variance σd². The null hypothesis is that there is no systematic difference, i.e. μd = 0.

设 dᵢ = xᵢ − yᵢ 为第 i 对数据的差值。我们假设这些差值服从正态分布，其均值 μd 和方差 σd² 均未知。原假设为不存在系统性差异，即 μd = 0。

H₀: μd = 0 vs H₁: μd ≠ 0 (two-tailed)

The alternative hypothesis may be one-tailed depending on the research question: μd > 0 or μd < 0.

备择假设可以是单侧的，取决于研究问题：μd > 0 或 μd < 0。

3. The Test Statistic | 检验统计量

Since the population standard deviation is unknown, we use the sample standard deviation of the differences and the t-distribution with n−1 degrees of freedom.

由于总体标准差未知，我们使用差值的样本标准差以及服从 n−1 个自由度的 t 分布。

t = (d̄ − μd) / (s_d / √n)

Here d̄ is the sample mean of the differences, s_d is the sample standard deviation of the differences, and n is the number of pairs. Under H₀, this test statistic follows a t-distribution with n−1 degrees of freedom.

其中 d̄ 为差值的样本均值，s_d 为差值的样本标准差，n 为配对个数。在原假设下，该检验统计量服从自由度为 n−1 的 t 分布。

4. Step-by-Step Calculation | 分步计算流程

The calculation is performed in a systematic way, and it is essential to show all working clearly in an exam.

计算过程需要按步骤进行，考试中清晰展示所有过程至关重要。

Calculate each difference dᵢ = xᵢ − yᵢ.
Find the mean difference d̄.
Calculate the standard deviation s_d using s_d = √[Σ(dᵢ − d̄)² / (n−1)].
Compute the test statistic t = d̄ / (s_d / √n).
Compare with the critical value at the chosen significance level.

计算每个差值 dᵢ = xᵢ − yᵢ。
求差值均值 d̄。
计算标准差 s_d = √[Σ(dᵢ − d̄)² / (n−1)]。
计算检验统计量 t = d̄ / (s_d / √n)。
在选定显著性水平下与临界值比较。

5. Worked Example: Exam Performance | 例题：考试成绩前后对比

A teacher records the scores of 8 students before and after attending a revision course. Test at the 5% significance level whether the course improves scores.

一位老师记录了8名学生参加复习课程前后的成绩。在5%显著性水平下检验该课程是否提高了成绩。

Student	1	2	3	4	5	6	7	8
Before	65	70	72	60	80	75	68	74
After	70	74	75	66	83	78	72	80

We compute the differences d = After − Before: 5, 4, 3, 6, 3, 3, 4, 6.

我们计算差值 d = 后 − 前：5, 4, 3, 6, 3, 3, 4, 6。

The mean difference is d̄ = 34/8 = 4.25. The sample standard deviation can be calculated as s_d = 1.2822 (to 4 decimal places). Hence the test statistic is

差值均值为 d̄ = 34/8 = 4.25。样本标准差计算得 s_d = 1.2822（保留4位小数）。因此检验统计量为

t = 4.25 / (1.2822/√8) = 9.374

The critical value for a one-tailed test at α = 0.05 with 7 degrees of freedom is 1.895. Since 9.374 > 1.895, we reject H₀. There is strong evidence that the revision course improves scores.

在α = 0.05、自由度7的单侧检验中，临界值为1.895。因为9.374 > 1.895，我们拒绝原假设。有充分证据表明复习课程提高了成绩。

6. Assumptions of the Test | 检验的假设条件

Like all statistical tests, the paired t-test relies on several important assumptions that must be verified before drawing conclusions.

与所有统计检验一样，配对t检验依赖若干重要假设，在得出结论前必须加以验证。

Normality: the differences should be approximately normally distributed.
Independence: the pairs themselves must be independent of each other.
Measurement scale: the dependent variable should be continuous or at least interval-level.

正态性：差值应近似服从正态分布。
独立性：各配对之间必须相互独立。
测量尺度：因变量应为连续变量或至少为等距变量。

In an exam, you are usually told that the differences are normally distributed. However, in real-world applications you might need to check this with a normal probability plot or a Shapiro-Wilk test.

在考试中，通常题目会告知差值服从正态分布。但在实际应用中，您可能需要通过正态概率图或Shapiro-Wilk检验来验证这一假设。

7. Paired vs Two-Sample t-Test | 配对t检验与双样本t检验的区别

Students often confuse the paired t-test with the independent two-sample t-test. The key difference lies in the design of the data.

学生经常将配对t检验与独立双样本t检验混淆。关键区别在于数据的设计方式。

Feature	Paired t-test	Two-sample t-test
Data structure	Two related groups	Two independent groups
Degrees of freedom	n − 1	n₁ + n₂ − 2
Purpose	Removes between-subject variability	Compares separate populations

特征	配对t检验	双样本t检验
数据结构	两个相关组	两个独立组
自由度	n − 1	n₁ + n₂ − 2
目的	消除个体间变异	比较两个独立总体

8. Confidence Interval for the Mean Difference | 均值差的置信区间

Besides hypothesis testing, the paired t-test framework also allows us to construct a confidence interval for the population mean difference. This gives a range of plausible values for μd.

除假设检验外，配对t检验框架还可以构建总体均值差的置信区间，为 μd 提供一组合理的取值范围。

d̄ ± t*_{n−1, α/2} × (s_d / √n)

For the worked example above, with 95% confidence, we have:

对于上面的例题，95%置信区间为：

4.25 ± 2.365 × (1.2822/√8) = 4.25 ± 1.072

Thus the interval is (3.178, 5.322). Since zero is not included, this agrees with the hypothesis test result.

因此区间为 (3.178, 5.322)。由于不包含零，这与假设检验的结果一致。

9. Exam Tips from TutorHao | TutorHao 考试提示

In Edexcel A-Level Further Mathematics, questions on the paired t-test typically appear in Paper 3 and Paper 4 (Further Statistics). Marks are often lost through careless use of the calculator, incorrect degrees of freedom, or misstating hypotheses.

在Edexcel A-Level进阶数学中，配对t检验的题目通常出现在Paper 3和Paper 4（进阶统计）中。失分通常是由于计算器使用不当、自由度错误或假设表述不准确。

Always state H₀ and H₁ in terms of μd, not the original means.
Remember that the degrees of freedom are n−1, not n.
If using a calculator, verify that you are in the ‘paired t-test’ mode, not the ‘two-sample t-test’ mode.
When constructing a confidence interval, use the critical value t_{n−1, α/2}.

始终用 μd 来表示原假设和备择假设，而不是用原始均值。
记住自由度是 n−1，而不是 n。
使用计算器时，确认处于’配对t检验’模式，而非’双样本t检验’模式。
构建置信区间时，使用临界值 t_{n−1, α/2}。

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

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A-Level Further Mathematics: Paired t-Test – Principles and Applications | 进阶数学：配对t检验的原理与应用