A-Level Further Mathematics: Solving Probabilities of Type I and Type II Errors with the Normal Distribution | A-Level 进阶数学：用正态分布求解两类错误概率

📚 A-Level Further Mathematics: Solving Probabilities of Type I and Type II Errors with the Normal Distribution | A-Level 进阶数学：用正态分布求解两类错误概率

Hypothesis testing is one of the most examined topics in Edexcel A-Level Further Mathematics. When the test statistic follows a normal distribution, the probabilities of Type I and Type II errors can be calculated exactly using the sampling distribution of the mean. This article breaks that process into clear stages and shows worked examples that mirror past-paper questions.

假设检验是 Edexcel 进阶数学中最常考的专题之一。当检验统计量服从正态分布时，我们可以利用样本均值的抽样分布精确计算出第一类错误与第二类错误的概率。本文将这一过程拆解成清晰的步骤，并通过与真题风格一致的例题加以演示。

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1. The Logic of Hypothesis Testing | 假设检验的逻辑框架

Every hypothesis test begins with a null hypothesis H₀ and an alternative hypothesis H₁. H₀ is the claim we assume to be true until the data provides sufficient evidence against it. H₁ represents the claim we want to establish, and it may be one-tailed or two-tailed. A one-tailed test uses either < or >; a two-tailed test uses ≠.

每一次假设检验都从原假设 H₀ 和备择假设 H₁ 开始。H₀ 是我们假定成立的陈述，除非数据提供足够证据反对它；H₁ 则是我们希望证明的陈述，它可以是单尾的，也可以是双尾的。单尾检验使用 < 或 >，双尾检验使用 ≠。

The decision to reject or not reject H₀ is made by comparing a test statistic with a critical region. The critical region is determined by the significance level α, usually 5% or 1%. If the test statistic falls inside the critical region, we reject H₀; if it falls outside, we do not reject H₀. Both decisions carry a risk of being wrong, which leads to the two types of error.

是否拒绝 H₀ 的判断，需要将检验统计量与拒绝域进行比较。拒绝域由显著性水平 α 决定，通常取 5% 或 1%。若检验统计量落入拒绝域，则拒绝 H₀；若落在拒绝域之外，则不能拒绝 H₀。这两种判断都有可能出错，于是便产生了这两类错误。

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2. The Engine: The Sampling Distribution of the Mean | 核心引擎：样本均值分布

When a random sample of size n is drawn from a normal distribution N(μ, σ²), the sample mean x̄ follows a normal distribution with the same mean μ but a smaller variance. This is the key theorem that powers all normal-based hypothesis tests.

当从正态分布 N(μ, σ²) 中抽取容量为 n 的随机样本时，样本均值 x̄ 服从均值仍为 μ、但方差更小的正态分布。这是所有基于正态分布的假设检验所依赖的核心定理。

x̄ ~ N(μ, σ²/n)

To calculate probabilities, we standardise x̄ using the standard error σ/√n. The standardised statistic is:

为了计算概率，我们需要利用标准误 σ/√n 对 x̄ 进行标准化。标准化统计量为：

z = (x̄ - μ) / (σ/√n)

In an exam, always check two conditions before using this formula: the sample must be random, and either the population is normal or n is large enough for the Central Limit Theorem to apply. Edexcel questions normally state that the population standard deviation is known, or that the sample size is sufficiently large.

在考试中使用该公式前，务必检查两个条件：样本必须随机；总体需为正态分布，或者 n 足够大从而可以应用中心极限定理。Edexcel 题目通常会明确说明总体标准差已知，或样本量足够大。

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3. Step 1 - Locate the Critical Region | 第一步：确定拒绝域

The critical region is the set of values of x̄ that would cause us to reject H₀. To find it, we use the null distribution x̄ ~ N(μ₀, σ²/n) and the significance level α.

拒绝域是使我们会拒绝 H₀ 的 x̄ 取值集合。要找到它，需要利用原假设下的分布 x̄ ~ N(μ₀, σ²/n) 以及显著性水平 α。

For a lower-tailed test H₁: μ < μ₀, the critical value is:

对于左尾检验 H₁: μ < μ₀，临界值为：

x̄_crit = μ₀ - z(0.05) × σ/√n

For an upper-tailed test H₁: μ > μ₀, the critical value is:

对于右尾检验 H₁: μ > μ₀，临界值为：

x̄_crit = μ₀ + z(0.05) × σ/√n

For a two-tailed test H₁: μ ≠ μ₀, we split α equally between the two tails and use z(0.025) on each side:

对于双尾检验 H₁: μ ≠ μ₀，将 α 平均分配到两个尾端，并在两侧使用 z(0.025)：

x̄_crit = μ₀ ± z(0.025) × σ/√n

ten as an inequality, not just as a single number. For example, a lower-tailed test with x̄_crit = 984.4 has the rejection region x̄ < 984.4.

许多学生会犯一个错误：只写出临界值而不写成不等式。例如，左尾检验若 x̄_crit = 984.4，则拒绝域应写成 x̄ < 984.4。

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4. Step 2 - Probability of Type I Error | 第二步：第一类错误概率

A Type I error occurs when H₀ is true but we reject it. Its probability is denoted by α. By construction, α is simply the significance level chosen at the start of the test.

第一类错误发生在 H₀ 为真时我们却拒绝了它。其概率用 α 表示。根据构造方式，α 就是在检验开始时选定的显著性水平。

error) = 0.05. You do not need to recalculate it from the distribution in most questions.

如果题目要求写出 P(Type I error)，答案通常就是 α。例如，若 α = 0.05，则 P(Type I error) = 0.05。大多数题目都不需要你再从分布中重新计算它。

However, understanding this link is useful because it tells you that the critical region is chosen so that the area under the H₀ curve inside the rejection region equals exactly α. This idea connects Step 1 and Step 2 in a single chain of reasoning.

然而，理解这一联系十分重要，因为它说明拒绝域的选择应使得 H₀ 曲线下落入拒绝域的面积恰好等于 α。这个想法将第一步和第二步连接成一条完整的推理链。

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5. Step 3 - Probability of Type II Error | 第三步：第二类错误概率

A Type II error occurs when H₀ is false but we fail to reject it. Its probability is denoted by β. Unlike α, β is not fixed at the start of the test; it depends on the true value of the population mean under H₁.

第二类错误发生在 H₀ 为假时我们却未能拒绝它。其概率用 β 表示。与 α 不同，β 在检验开始时并不是

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