📚 State Power Classifications | 统计功效分类
In Edexcel A-Level Mathematics, hypothesis testing is a core statistical topic. The term “power” is not a single fixed property; it can be classified by its level, by the direction of the test, and by the factors that determine it. This article sets out the main power classifications and how they arise in binomial and normal hypothesis tests.
在 Edexcel A-Level 数学中，假设检验是统计学的核心主题。“功效”并不是一个单一的固定属性；它可以按水平、按检验方向以及按决定它的因素进行分类。本文阐述主要的功效分类，以及它们在二项检验和正态检验中如何产生。
1. Hypothesis Testing Basics | 假设检验基础
A hypothesis test uses sample data to decide whether to reject a claim about a population parameter. The null hypothesis, written H₀, is the default position; the alternative hypothesis, written H₁, is the claim we are testing for.
假设检验使用样本数据来决定是否拒绝关于总体参数的某个主张。原假设写作 H₀，是默认立场；备择假设写作 H₁，是我们试图支持的断言。
For binomial tests, H₀ has the form p = p₀ and H₁ has the form p > p₀, p < p₀ or p ≠ p₀. For normal tests, H₀ often states μ = μ₀ against a one-sided or two-sided alternative.
对于二项检验，H₀ 的形式为 p = p₀，H₁ 的形式为 p > p₀、p < p₀ 或 p ≠ p₀。对于正态检验，H₀ 通常陈述 μ = μ₀，备择假设为单边或双边。
H₀: p = p₀ and H₁: p > p₀
2. Type I and Type II Errors | 第一类与第二类错误
A Type I error occurs when H₀ is true but the test rejects it. Its probability is the significance level α. A Type II error occurs when H₁ is true but the test fails to reject H₀; its probability is β.
当 H₀ 为真但检验拒绝了它时，发生第一类错误。其概率是显著性水平 α。当 H₁ 为真但检验未能拒绝 H₀ 时，发生第二类错误；其概率是 β。
α = P(reject H₀ | H₀ is true)
β = P(do not reject H₀ | H₁ is true)
These two error probabilities are linked: reducing one usually increases the other unless the sample size or effect size changes.
这两种错误概率是相互关联的：通常减少一种会增加另一种，除非样本量或效应量发生变化。
3. Defining Statistical Power | 定义统计功效
Power is the probability of correctly rejecting H₀ when H₁ is true. It is the complement of the Type II error probability, so a powerful test is one that is likely to detect a real effect or difference when it exists.
功效是当 H₁ 为真时正确拒绝 H₀ 的概率。它是第二类错误概率的补数，因此功效高的检验在真实效应或差异存在时更有可能检测到它。
Power = 1 − β = P(reject H₀ | H₁ is true)
In Edexcel Statistics questions, you should always define β first and then state that power is 1 − β, because this shows the logical link between the two probabilities.
在 Edexcel 统计学题目中，你应该始终先定义 β，然后说明功效为 1 − β，因为这展示了两者之间的逻辑联系。
4. Power as
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