Difference between means of two independent normal distributions | 两个独立正态分布均值之差的推断

📚 Difference between means of two independent normal distributions | 两个独立正态分布均值之差的推断

In A-Level Statistics, comparing two population means is a core skill. When we take independent random samples from two normal populations, the difference between the sample means is itself a normal random variable. This allows us to construct confidence intervals and carry out hypothesis tests for the difference between the two population means.

在 A-Level 统计学中，比较两个总体的均值是一项核心技能。当我们从两个正态总体中独立抽取随机样本时，样本均值之差本身也是一个正态随机变量。这使得我们能够为两个总体均值之差构造置信区间并进行假设检验。

1. Key Assumptions and Notation | 关键假设与符号

We consider two independent normal populations with means μ₁ and μ₂ and variances σ₁² and σ₂². Independent random samples of sizes n₁ and n₂ are drawn from these populations. The sample means are denoted by X̄₁ and X̄₂, and their observed values by x̄₁ and x̄₂.

我们考虑两个独立的正态总体，均值分别为 μ₁ 和 μ₂，方差分别为 σ₁² 和 σ₂²。从这两个总体中分别抽取容量为 n₁ 和 n₂ 的独立随机样本。样本均值记为 X̄₁ 和 X̄₂，其观测值记为 x̄₁ 和 x̄₂。

The key assumptions are independence of the two samples, normality of both populations, and known population variances. When variances are unknown, large sample sizes allow us to replace them with sample variances.

关键假设包括两个样本相互独立、两个总体均服从正态分布，以及总体方差已知。当方差未知时，在大样本条件下可以用样本方差替代。

2. Distribution of the Difference in Sample Means | 样本均值之差的分布

Since X̄₁ and X̄₂ are normally distributed and independent, their difference X̄₁ – X̄₂ is also normally distributed. The expected value is μ₁ – μ₂, and the variance is the sum of the individual variances: σ₁²/n₁ + σ₂²/n₂.

由于 X̄₁ 和 X̄₂ 均服从正态分布且相互独立，它们的差 X̄₁ – X̄₂ 也服从正态分布。其期望为 μ₁ – μ₂，方差为各自方差之和：σ₁²/n₁ + σ₂²/n₂。

This result follows from the linearity of expectation and the independence assumption which makes the covariance zero.

这一结果源于期望的线性性质以及独立性假设使得协方差为零。

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

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