Type I and Type II Errors in Hypothesis Testing | 假设检验中的第一类与第二类错误

📚 Type I and Type II Errors in Hypothesis Testing | 假设检验中的第一类与第二类错误

In Edexcel A Level Mathematics, hypothesis testing asks you to make decisions using sample data. Because decisions are based on samples, they can be wrong. The two formal errors are called Type I and Type II errors. Understanding these errors is essential for interpreting significance levels, critical regions and the power of a test. This article explains both types, gives Edexcel-style examples, and shows how to write precise exam answers.

在爱德思 A Level 数学中，假设检验要求你利用样本数据作出决策。由于决策基于样本，因此可能出现错误。两类正式错误称为第一类错误和第二类错误。理解这些错误对于解释显著性水平、拒绝域以及检验功效至关重要。本文将解释两类错误，给出爱德思风格例题，并展示如何写出精确的考试答案。

1. Hypothesis Testing Refresher | 假设检验回顾

Hypothesis testing starts with two competing statements about a population parameter, such as a binomial probability p or a population mean μ. You collect a random sample, calculate a test statistic, and compare it with a critical value or a p-value.

假设检验首先对总体参数（例如二项概率 p 或总体均值 μ）提出两个相互竞争的陈述。你收集随机样本，计算检验统计量，并将其与临界值或 p 值进行比较。

If the sample result falls in the critical region, you reject the null hypothesis H₀. If it does not, you do not reject H₀. This decision rule controls the probability of rejecting H₀ when H₀ is actually true.

如果样本结果落在拒绝域中，则拒绝原假设 H₀；如果没有落在拒绝域中，则不拒绝 H₀。该决策规则控制了当 H₀ 实际上为真时拒绝 H₀ 的概率。

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

The null hypothesis H₀ is the default position: for example, H₀: p = 0.5 or H₀: μ = 20. The alternative hypothesis H₁ states the change or difference you are testing for: H₁: p > 0.5, H₁: μ ≠ 20, and so on.

原假设 H₀ 是默认立场：例如 H₀: p = 0.5 或 H₀: μ = 20。备择假设 H₁ 说明你正在检验的变化或差异：H₁: p > 0.5、H₁: μ ≠ 20 等。

• H₀: null hypothesis — the claim assumed true unless strong evidence contradicts it | 原假设 H₀：除非有强证据反对，否则假定为真的陈述
• H₁: alternative hypothesis — the claim you are trying to find evidence for | 备择假设 H₁：你试图寻找证据支持的陈述

A one-tailed test uses H₁ with > or <, while a two-tailed test uses ≠. The type of alternative affects where the critical region is placed, but it does not change the meaning of Type I and Type II errors.

单尾检验使用带 > 或 < 的 H₁，而双尾检验使用 ≠。备择假设的类型会影响拒绝域的位置，但不会改变第一类错误和第二类错误的含义。

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

The significance level α is the maximum probability of rejecting H₀ when H₀ is true. In Edexcel questions, common levels are 0.05, 0.01, and occasionally 0.10. The critical region is the set of sample statistics that lead to rejecting H₀.

显著性水平 α 是当 H₀ 为真时拒绝 H₀ 的最大概率。在爱德思考试题中，常见水平为 0.05、0.01，偶尔为 0.10。拒绝域是导致拒绝 H₀ 的样本统计量的集合。

For a one-tailed test with H₁: p > 0.5, the critical region might be X ≥ c, where X is the number of successes. For a two-tailed test, the critical region is split between both tails, often with α/2 in each tail.

对于 H₁: p > 0.5 的单尾检验，拒绝域可能是 X ≥ c，其中 X 是成功次数。对于双尾检验，拒绝域被分配到两个尾部，通常每个尾部为 α/2。

4. What Is a Type I Error? | 什么是第一类错误

A Type I error occurs when you reject the null hypothesis H₀ when H₀ is actually true. It is a false positive: the test says there is an effect or difference, but in reality there is none.

第一类错误发生在原假设 H₀ 实际上为真时，你却拒绝了它。这是一个假阳性：检验表明有影响或差异，但实际上并不存在。

P(Type I error) = P(reject H₀ | H₀ is true) = α

The vertical bar | means “given that”. So the probability of a Type I error is exactly the significance level α. If you use a 5% significance level, then P(Type I error) = 0.05.

竖线 | 表示“在……条件下”。因此，第一类错误的概率恰好等于显著性水平 α。如果使用 5% 的显著性水平，那么 P(第一类错误) = 0.05。

In a binomial test of H₀: p = 0.5 with α = 0.05, rejecting H₀ when X is in the critical region would be a Type I error if the coin is actually fair.

在 H₀: p = 0.5、α = 0.05 的二项检验中，如果硬币实际上是公平的，而 X 落在拒绝域中并拒绝 H₀，就会犯第一类错误。

5. What Is a Type II Error? | 什么是第二类错误

A Type II error occurs when you fail to reject H₀ when H₀ is actually false. It is a false negative: the test misses a real effect or difference.

第二类错误发生在原假设 H₀ 实际上为假时，你却没有拒绝它。这是一个假阴性：检验错过了一个真实存在的效应或差异。

P(Type II error) = P(do not reject H₀ | H₀ is false) = β

The probability β is not fixed by the significance level. It depends on the true value of the parameter, the sample size n, and the position of the critical region.

概率 β 不由显著性水平固定。它取决于参数的真实值、样本量 n 以及拒绝域的位置。

For example, if H₀: μ = 20 is false because μ really equals 22, failing to reject H₀ would be a Type II error. Its probability β is the chance that the sample

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