Hypothesis Test for the Difference Between Means | 两均值之差的假设检验

📚 Hypothesis Test for the Difference Between Means | 两均值之差的假设检验

Hypothesis testing for the difference between two means is a core Edexcel A-Level statistics skill. It compares two independent population means by examining the difference between two sample means. This article walks through the null and alternative hypotheses, the normal two-sample z-test, standard error, critical values, worked examples, p-values, errors, and exam tips.

对两个均值之差进行假设检验是 Edexcel A-Level 统计部分的核心技能。它通过考察两个样本均值之差来比较两个独立总体的均值。本文将讲解原假设与备择假设、正态双样本 Z 检验、标准误、临界值、例题、p 值、两类错误以及考试提示。

1. What the Test Is For | 检验目的

This test decides whether the means of two independent populations are significantly different. It uses the difference between two sample means, x̄₁ – x̄₂, as evidence. Typical applications include comparing test scores between two classes, output from two machines, or effectiveness of two medical treatments.

该检验用于判断两个独立总体的均值是否存在显著差异。它以两个样本均值之差 x̄₁ – x̄₂ 作为证据。典型应用包括比较两个班级的考试成绩、两台机器的产量或两种医疗方案的效果。

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

The null hypothesis normally states that the two population means are equal: H₀: μ₁ = μ₂, or equivalently H₀: μ₁ – μ₂ = 0. The alternative hypothesis can be two-tailed, H₁: μ₁ ≠ μ₂, or one-tailed, H₁: μ₁ > μ₂ or H₁: μ₁ < μ₂. The direction of H₁ must reflect the claim being investigated.

原假设通常表示两个总体均值相等：H₀: μ₁ = μ₂，或等价地 H₀: μ₁ – μ₂ = 0。备择假设可以是双尾的 H₁: μ₁ ≠ μ₂，也可以是单尾的 H₁: μ₁ > μ₂ 或 H₁: μ₁ < μ₂。H₁ 的方向必须反映正在研究的论断。

3. Conditions for the Normal Two-Sample Z-Test | 正态双样本 Z 检验的条件

The normal two-sample z-test is valid when the samples are independent, both populations are normally distributed, and the population standard deviations σ₁ and σ₂ are known. If the population standard deviations are unknown but both sample sizes are large (typically n₁ > 30 and n₂ > 30), the sample standard deviations may be used as estimates.

当样本独立、两个总体均服从正态分布且总体标准差 σ₁ 和 σ₂ 已知时，正态双样本 Z 检验有效。如果总体标准差未知但两个样本量都较大（通常 n₁ > 30 且 n₂ > 30），可以用样本标准差作为估计值。

4. The Test Statistic | 检验统计量

For a null hypothesis H₀: μ₁ – μ₂ = d₀, the test statistic is calculated as:

对于原假设 H₀: μ₁ – μ₂ = d₀，检验统计量的计算公式为：

z = (x̄₁ – x̄₂ – d₀) / √(σ₁²/n₁ + σ₂²/n₂)

In most Edexcel questions d₀ = 0, so the statistic simplifies to z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂). If d₀ is not zero, it represents the hypothesised difference under H₀.

在大多数 Edexcel 试题中 d₀ = 0，因此统计量简化为 z = (x̄₁ – x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)。若 d₀ 不为零，它表示 H₀ 中假设的差值。

5. Standard Error of the Difference | 均值差的标准误

The denominator is the standard error of the difference between two sample means. For independent samples, the variance of x̄₁ – x̄₂ is the sum of the two individual variances:

分母是两个样本均值之差的标准误。对于独立样本，x̄₁ – x̄₂ 的方差是两个单独方差之和：

Var(x̄₁ – x̄₂) = σ₁²/n₁ + σ₂²/n₂

A very common mistake is to subtract the variances. Since the samples are independent, the standard errors always add, giving a larger denominator than either individual standard error.

一个非常常见的错误是对方差做减法。由于样本独立，标准误总是相加，所得分母大于任意一个单独的标准误。

6. Critical Values and Significance Levels | 临界值与显著性水平

For a two-tailed test at the 5% significance level, the critical values are ±1.96. For a one-tailed test at 5%, the critical value is 1.645 for H₁: μ₁ > μ₂, or -1.645 for H₁: μ₁ < μ₂. At the 1% level, the two-tailed critical value is ±2.576.

在 5% 显著性水平下，双尾检验的临界值为 ±1.96。5% 单尾检验中，H₁: μ₁ > μ₂ 的临界值为 1.645，H₁: μ₁ < μ₂ 的临界值为 -1.645。在 1% 水平下，双尾临界值为 ±2.576。

7. Worked Example: Two-Tailed Test | 例题：双尾检验

Example: Two machines fill bottles. A sample of 50 bottles from machine 1 has mean 501 ml, and a sample of 40 bottles from machine 2 has mean 499 ml. The population standard deviations are σ₁ = 4 ml and σ₂ = 5 ml. Test at the 5% significance level whether the mean fill volumes differ.

例：两台机器装瓶。机器 1 的 50

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）