Hypothesis Testing: Rejection of the Null Hypothesis | 假设检验：原假设的拒绝

📚 Hypothesis Testing: Rejection of the Null Hypothesis | 假设检验：原假设的拒绝

In Edexcel A Level Mathematics, hypothesis testing is a key statistical tool. The phrase “rejection of the null hypothesis” refers to the decision that the sample data provide sufficient evidence against the assumed value of a population parameter, such as p in a binomial distribution or μ in a normal distribution. This article explains when and how we reject H₀, how to set up rejection regions, and how to avoid common errors.

在 Edexcel A Level 数学中，假设检验是一个核心统计工具。”拒绝原假设” 这一说法指的是：样本数据提供了足够的证据反对总体参数的假设值，比如二项分布中的 p 或正态分布中的 μ。本文将解释我们何时以及如何拒绝 H₀、如何建立拒绝域以及如何避免常见错误。

1. What is Hypothesis Testing? | 什么是假设检验？

Hypothesis testing begins with a claim about a population parameter. For example, a manufacturer may claim that a coin is fair, so p = 0.5. We collect sample data and ask whether the observed result is too unlikely under this claim. If it is too unlikely, we reject the null hypothesis.

假设检验从关于总体参数的一个声明开始。例如，制造商可能声称一枚硬币是公平的，因此 p = 0.5。我们收集样本数据，并考察在该声明下观测结果是否过于不可能。如果它过于不可能，我们就拒绝原假设。

In Edexcel exams, you will usually test a proportion using the binomial distribution or a mean using the normal distribution. The logic is the same: assume H₀ is true, then measure how extreme the sample result is.

在 Edexcel 考试中，你通常使用二项分布检验比例，或使用正态分布检验均值。逻辑是相同的：先假定 H₀ 为真，然后衡量样本结果有多极端。

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

The null hypothesis H₀ is a statement of no change or no effect, and it always contains an equality such as p = 0.5, p = 0.2 or μ = 100. The alternative hypothesis H₁ states what we are trying to find evidence for, using >, < or ≠.

原假设 H₀ 是一种无变化或无效果的陈述，它总是包含等式，如 p = 0.5、p = 0.2 或 μ = 100。备择假设 H₁ 说明我们试图寻找证据支持的内容，使用 >、< 或 ≠。

For a binomial test, H₀: p = 0.5 and H₁: p > 0.5 is a one-tailed test. If H₁: p ≠ 0.5, it is two-tailed. You must state both hypotheses clearly before carrying out the test.

对于二项检验，H₀: p = 0.5 且 H₁: p > 0.5 是单尾检验。如果 H₁: p ≠ 0.5，则为双尾检验。你在进行检验之前必须清楚地写出两个假设。

3. Significance Level and Rejection Region | 显著性水平与拒绝域

The significance level α is the maximum probability of rejecting H₀ when it is actually true. Common values are 5% and 1%. The rejection region, also called the critical region, is the set of sample results that lead to rejection of H₀.

显著性水平 α 是当 H₀ 实际为真时拒绝 H₀ 的最大概率。常用值是 5% 和 1%。拒绝域（也称为临界域）是导致拒绝 H₀ 的样本结果集合。

For example, if X ~ B(20, 0.5) and we test H₁: p > 0.5 at the 5% level, the rejection region might be X ≥ 15 because P(X ≥ 15) = 0.0207 ≤ 0.05.

例如，如果 X ~ B(20, 0.5) 且我们在 5% 水平下检验 H₁: p > 0.5，拒绝域可能是 X ≥ 15，因为 P(X ≥ 15) = 0.0207 ≤ 0.05。

4. Test Statistic and P-Value | 检验统计量与P值

A test statistic is a value calculated from sample data. In a binomial test, the test statistic is the observed number of successes X. In a normal test, it is often the standardised value z = (x̄ − μ₀) / (σ / √n), where x̄ is the sample mean, μ₀ is the hypothesised population mean, σ is the population standard deviation and n is the sample size.

检验统计量是从样本数据计算出的一个值。在二项检验中，检验统计量是观测到的成功次数 X。在正态检验中，它通常是标准化值 z = (x̄ − μ₀) / (σ / √n)，其中 x̄ 是样本均值，μ₀ 是假设的总体均值，σ 是总体标准差，n 是样本容量。

The p-value is the probability of obtaining a result at least as extreme as the observed one, assuming H₀ is true. If the p-value is less than or equal to α, we reject H₀.

P 值是在 H₀ 为真的前提下，得到至少与观测结果同样极端的结果的概率。如果 P 值小于或等于 α，我们就拒绝 H₀。

p-value ≤ α ⟹ reject H₀

5. One-Tailed vs Two-Tailed Tests | 单尾与双尾检验

A one-tailed test is used when H₁ specifies a direction, such as p > 0.5 or μ < 100. In this case, the whole significance level α is placed in one tail of the distribution.

当 H₁ 指明方向时，如 p > 0.5 或 μ < 100，使用单尾检验。此时整个显著性水平 α 被放在分布的一个尾部。

A two-tailed test is used when H₁ is p ≠ 0.5 or μ ≠ 100. The significance level is split equally between both tails, so each tail has α/2. Alternatively, you can compute a two-tailed p-value by doubling the smaller tail probability.

当 H₁ 为 p ≠ 0

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