Calculating the Size and Power of a Test | 计算检验的显著性水平与功效

📚 Calculating the Size and Power of a Test | 计算检验的显著性水平与功效

Hypothesis testing is a cornerstone of A-Level statistics. When we perform a test, we do not simply decide “reject H₀” or “not reject H₀”; we must also quantify how reliable that decision is. Two numerical measures, the size of a test and the power of a test, tell us exactly how likely the test is to make a correct or incorrect conclusion. This article explains both concepts, shows how to calculate them for binomial tests, and applies them to the style of questions found in Edexcel examinations.

假设检验是 A-Level 统计学的核心内容。当我们进行检验时，不仅仅是判断”拒绝 H₀”或”不拒绝 H₀”；还必须量化该结论的可靠性。检验的显著性水平（size）与检验的功效（power）这两个数值度量，能够准确告诉我们检验作出正确或错误结论的概率。本文将解释这两个概念，演示如何对二项检验进行计算，并结合 Edexcel 考试题型进行应用。

1. Introduction to Hypothesis Testing | 假设检验简介

In a hypothesis test, we start with a null hypothesis H₀, which usually describes the value of a population parameter p under the assumption of no change or no effect. The alternative hypothesis H₁ describes the claim we suspect to be true, and may be one-sided (p > p₀ or p < p₀) or two-sided (p ≠ p₀). A test statistic X, often a binomial count, is observed from the sample. We compare X with a critical region C, chosen so that the probability of X falling in C under H₀ is small. If X falls in C, we reject H₀ in favour of H₁.

在假设检验中，我们首先设定原假设 H₀，它通常描述在”无变化”或”无效”假设下总体参数 p 的取值。备择假设 H₁ 描述我们所怀疑的主张，可以是单侧的（p > p₀ 或 p < p₀），也可以是双侧的（p ≠ p₀）。检验统计量 X（通常是二项计数）由样本观测得到。我们将 X 与临界区域 C 进行比较，C 的选取使得在 H₀ 成立时 X 落入 C 的概率很小。若 X 落入 C，则拒绝 H₀，转而接受 H₁。

2. Type I and Type II Errors | 第一类与第二类错误

A Type I error occurs when we reject H₀ even though H₀ is actually true. Its probability is

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