📚 Finding Type I and Type II Errors Using the Normal Distribution | 利用正态分布求第一类与第二类错误
In Edexcel A-Level Statistics, hypothesis tests about a population mean often assume that the sample mean follows a normal distribution. This topic shows how to calculate the probabilities of the two possible decision errors: rejecting a true null hypothesis, and failing to reject a false null hypothesis.
在 Edexcel A-Level 统计学中，关于总体均值的假设检验通常假定样本均值服从正态分布。本主题讲解如何计算两种可能决策错误的概率：拒绝为真的原假设，以及未拒绝为假的原假设。
The skills examined are: reading a significance level as a Type I error probability, finding critical values on the X̄ scale, and standardising under a stated alternative mean to obtain a Type II error probability and the power of the test.
考查的核心技能是：把显著性水平直接读作第一类错误的概率、在 X̄ 的尺度上求临界值，以及在给定的备择均值下做标准化，从而得到第二类错误的概率和检验的功效。
1. Hypothesis Testing Recap | 假设检验回顾
, and an alternative hypothesis H₁ which may be one-tailed (H₁: μ > μ₀ or H₁: μ < μ₀) or two-tailed (H₁: μ ≠ μ₀).
显著性检验从原假设 H₀ 开始，例如 H₀: μ = μ₀，以及备择假设 H₁，它可以是单尾的（H₁: μ > μ₀ 或 H₁: μ < μ₀），也可以是双尾的（H₁: μ ≠ μ₀）。
When the population standard deviation σ is known, and either the population itself is normal or the sample size n is large, the test statistic is
当总体标准差 σ 已知，且总体本身服从正态分布或样本量 n 较大时，检验统计量为
Z = (X̄ − μ₀) / (σ / √n) ~ N(0, 1) under H₀
Under H₀ the sample mean satisfies X̄ ~ N(μ₀, σ²/n), so its standard deviation is the standard error σ/√n, not σ.
在 H₀ 下，样本均值满足 X̄ ~ N(μ₀, σ²/n)，因此其标准差是标准误 σ/√n，而不是 σ。
The decision rule compares the observed value of Z with critical values such as 1.645 for a 5% one-tailed test or 1.960 for a 5% two-tailed test. If the statistic falls in the critical region, we reject H₀.
决策规则把观测到的 Z 值与临界值比较，例如 5% 单尾检验的 1.645，或 5% 双尾检验的 1.960。若统计量落入拒绝域，就拒绝 H₀。
2. The Two Error Types | 两类错误
A Type I error occurs when H₀ is true but the test rejects it. A Type II error occurs when H₀ is false but the test does not reject it.
第一类错误发生在 H₀ 为真但检验拒绝了它。第二类错误发生在 H₀ 为假但检验没有拒绝它。
These errors are mutually exclusive in a single decision, but their probabilities trade off against each other when the sample size is fixed.
两类错误在一次决策中互斥，但在样本量固定时，它们的概率相互制约。
| Reality / 真实情况 | Reject H₀ / 拒绝 H₀ | Do not reject H₀ / 不拒绝 H₀ |
| H₀ is true / H₀ 为真 | Type I error, probability α / 第一类错误，概率 α | Correct decision / 正确决策 |
| H₀ is false / H₀ 为假 | Correct decision (power) / 正确决策（功效） | Type II error, probability β / 第二类错误，概率 β |
α = P(reject H₀ | H₀ true) β = P(do not reject H₀ | H₀ false)
Because the two events "reject" and "do not reject" partition the sample space, α and β never occur in the same decision, yet they are linked through the position of the critical region.
由于"拒绝"与"不拒绝"这两个事件把样本空间分割开来，α 与 β 不会出现在同一次决策中，但它们通过拒绝域的位置相互关联。
3. Type I Error and the Significance Level | 第一类错误与显著性水平
The probability of a Type I error is exactly the significance level of the test, because the critical region is chosen so that its area under H₀ equals α.
第一类错误的概率恰好等于检验的显著性水平，因为拒绝域正是按"在 H₀ 下其面积为 α"来选取的。
P(Type I error) = P(reject H₀ | μ = μ₀) = α
For example, with H₀: μ = 20, H₁: μ > 20, σ = 4 and n = 25 at the 5% level, the critical value on the X̄ scale is 20 + 1.645 × (4/√25) = 21.316, and P(X̄ > 21.316 | μ = 20) = 0.05 exactly.
例如，对 H₀: μ = 20、H₁: μ > 20、σ = 4、n = 25 的 5% 水平检验，X̄ 尺度上的临界值为 20 + 1.645 × (4/√25) = 21.316，且 P(X̄ > 21.316 | μ = 20) = 0.05，完全相等。
In a two-tailed 5% test using 1.96, the total area of 0.05 splits into 0.025 in each tail, so the Type I error probability is still 0.05.
在采用 1.96 的 5% 双尾检验中，总面积 0.05 分为两尾各 0.025，因此第一类错误的概率仍是 0.05。
Consequence for exams: when a question asks for the probability of a Type I error, you usually read it straight off the significance level; you only compute it if the critical region has been rounded or defined by an unusual rule.
