Hypothesis Testing | 假设检验

📚 Hypothesis Testing | 假设检验

Hypothesis testing is a core topic in Edexcel A Level Mathematics. It gives a formal framework for using sample data to judge whether a claim about a binomial probability p is supported by the evidence, and it appears in almost every past paper on the binomial distribution.

假设检验是 Edexcel A-Level 数学的核心主题。它提供了一个正式框架，用于根据样本数据判断关于二项概率 p 的论断是否得到证据支持，并且在几乎每一份涉及二项分布的真题中都会出现。

This article covers null and alternative hypotheses, significance levels, one-tailed and two-tailed tests, critical regions, p-values, the assumptions behind the binomial model, and fully worked examples with the binomial distribution.

本文涵盖原假设与备择假设、显著性水平、单尾与双尾检验、临界区域、p 值、二项模型的前提假设，以及基于二项分布的完整例题。

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1. What is Hypothesis Testing? | 什么是假设检验？

Hypothesis testing is a formal statistical method for deciding whether there is enough evidence in a sample to reject a stated belief about a population parameter, such as the probability p in a binomial model. Instead of proving a claim is true, the test asks whether the observed data would be surprisingly unlikely if the claim were true.

假设检验是一种正式的统计方法，用于判断样本中是否有足够证据来拒绝关于总体参数的某个既定论断，例如二项模型中的概率 p。它并不是去证明某个论断为真，而是问：如果该论断成立，观测到的数据是否罕见得令人意外。

In Edexcel A Level Mathematics, hypothesis testing is introduced through the binomial distribution, where you test claims about the probability of success p using the observed number of successes from n independent trials. Later, the same logic is applied to the normal distribution and to correlation coefficients.

在 Edexcel A-Level 数学中，假设检验通过二项分布引入，即利用 n 次独立试验中的成功次数来检验关于成功概率 p 的命题。之后，同样的逻辑会被应用到正态分布和相关系数上。

The key idea is the sampling distribution. If H₀ is true and we repeat the experiment many times, the number of successes follows a known distribution. A result far out in the tail of that distribution casts doubt on H₀.

核心思想是抽样分布。如果 H₀ 为真，并且我们把实验重复很多次，成功次数会服从一个已知分布。若结果落在该分布尾部很远处，就会让人们对 H₀ 产生怀疑。

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2. The Binomial Model and Its Conditions | 二项模型及其条件

Before any test can be carried out, the random variable must be defined. In a binomial hypothesis test we write X ~ B(n, p), where n is the fixed number of trials and p is the probability of success on a single trial.

在进行任何检验之前，必须先定义随机变量。在二项假设检验中，我们写成 X ~ B(n, p)，其中 n 是固定的试验次数，p 是单次试验成功的概率。

Four conditions must be satisfied for the binomial model to be valid: a fixed number of trials n; each trial has exactly two outcomes, usually called success and failure; the trials are independent of one another; and the probability of success p is constant across all trials.

二项模型成立需要满足四个条件：试验次数 n 固定；每次试验只有两种结果，通常称为成功与失败；各次试验彼此独立；成功概率 p 在所有试验中保持恒定。

If a question involves sampling without replacement from a small population, the trials are not independent and the binomial model is not appropriate. Similarly, if the probability changes from trial to trial, a different model is needed. Examiners often award a mark simply for stating the distribution correctly.

如果题目涉及从较小总体中无放回抽样，各次试验就不独立，二项模型并不适用。同样，如果每次试验的概率会变化，就需要换用其他模型。考官常常仅因为正确写出分布就给分。

The probability of exactly x successes is given by the binomial formula:

在 n 次试验中恰好取得 x 次成功的概率由二项公式给出：

P(X = x) = C(n, x) × pˣ × (1 − p)ⁿ⁻ˣ

In practice you will rarely use this formula by hand. Instead you use cumulative binomial tables, or the binomial cumulative distribution function on a calculator.

实际做题时你很少手动使用这个公式，而是查累积二项分布表，或者使用计算器上的二项累积分布函数。

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

₀ is the claimed probability. It is the hypothesis that is assumed true until the evidence suggests otherwise.

原假设记为 H₀，是默认立场或现状，通常写作 H₀: p = p₀，其中 p₀ 是声称的概率。在没有相反证据之前，我们假定它成立。

The alternative hypothesis, written H₁, is what you are trying to find evidence for. It takes one of three forms: p > p₀ for an increase, p < p₀ for a decrease, or p ≠ p₀ for any change at all.

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备择假设记为 H₁，是你试图寻找证据支持的主张。它有三种形式：p > p₀ 表示增大，p < p₀ 表示减小，p ≠ p₀ 表示发生了任意方向的变化。

tion parameter p, never about the sample statistic. So you write H₀: p = 0.3 and not H₀: p̂ n though 0.3 came from a problem statement about a large population or a long-run process.

假设永远是关于总体参数 p 的，而不是关于样本统计量的。所以你写 H₀: p = 0.3，而不是 H₀: p̂ = 0.3，即使 0.3 是从题目中对一个较大总体或长期过程描述里得来的。

The wording of the question tells you which alternative to use. Phrases such as "has increased", "is more likely" or "has improved" point to a greater-than alternative, while "has decreased", "is less likely" or "has reduced" point to a less-than alternative. Phrases like "has changed" or "is different from" point to a two-tailed alternative.

题目的措辞决定了该用哪种备择假设。“增加了”“更可能”“有所改善”指向大于型的备择假设；“减少了”“可能性更低”“有所降低”指向小于型的备择假设；而“发生了变化”“与……不同”则指向双尾备择假设。

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4. The Significance Level | 显著性水平

The significance level, denoted α, is the probability of rejecting H₀ when H₀ is actually true. It is chosen by the person carrying out the test before any data is collected, and it controls how strict the test is.

显著性水平记为 α，是在 H₀ 实际上为真时却拒绝 H₀ 的概率。它由进行检验的人在收集数据之前选定，用来控制检验的严格程度。

The most common value in A Level questions is 5%, written α = 0.05. Other frequently used values are 10% (α = 0.10), 2% (α = 0.02) and 1% 0.01). A smaller significance level means you need stronger evidence before rejecting H₀.

A-Level 题目中最常用的值是 5%，写作 α = 0.05。其他常见取值有 10%（α = 0.10）、2%（α = 0.02）和 1%（α = 0.01）。显著性水平越小，表明拒绝 H₀ 所需的证据就越强。

It is important to understand what the significance level does not mean. A 5% significance level does not mean there is a 5% chance that H₀ is true, and it does not mean the result is unimportant. It is a fixed threshold against which the tail probability is compared.

要注意显著性水平不代表什么。5% 的显著性水平并不意味着 H₀ 为真的概率是 5%，也不意味着结果不重要。它只是一个固定的阈值，用来与尾部概率作比较。

Reject H₀ if the tail probability ≤ α

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5. One-Tailed and Two-Tailed Tests | 单尾检验与双尾检验

A one-tailed test is used when the alternative hypothesis specifies a direction. If H₁: p > p₀ you look only at the upper tail of the distribution, and if H₁: p < p₀ you look only at the lower tail. The whole of α is placed in that one tail.

当备择假设指明了方向时，就使用单尾检验。如果 H₁: p > p₀，只需查看分布的上尾；如果 H₁: p < p₀，只需查看下尾。整个 α 全部放在这一侧尾部。

A two-tailed test is used when the alternative hypothesis is H₁: p ≠ p₀. Here the significance level is shared between the two tails, so each tail receives α ÷ 2. For a 5% two-tailed test, each tail is compared against 0.025.

当备择假设为 H₁: p ≠ p₀ 时，就使用双尾检验。此时显著性水平要在两侧尾部之间分配，每侧分到 α ÷ 2。对于 5% 的双尾检验，每一侧尾部都要与 0.025 作比较。

A useful check when deciding which test to use is to read the final sentence of the question. The direction of the claim in that sentence, not the direction of the observed data, determines whether the test is one-tailed or two-tailed. Choosing the wrong tail is one of the most common errors in exams.

判断该用哪种检验的一个实用方法是读题干的最后一句话。决定单尾还是双尾的，是这句话中论断的方向，而不是观测数据的方向。选错尾部是考试中最常见的错误之一。

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6. Critical Regions and Critical Values | 临界区域与临界值

The critical region is the set of values of the test statistic for which H₀ is rejected. The boundary values of the critical region are called critical values. Together they define the rejection region of the test.

临界区域是使 H₀ 被拒绝的那些检验统计量取值的集合。临界区域的边界值称为临界值。它们共同定义了检验的拒绝域。

t you find the largest integer c such that P(X ≤ c) ≤ α, giving the critical region X ≤ c.

对于上尾检验，你要找出满足 P(X ≥ c) ≤ α 的最小整数 c，临界区域就是 X ≥ c。对于下尾检验，你要找出满足 P(X ≤ c) ≤ α 的最大整数 c，临界区域就是 X ≤ c。

ause the binomial is discrete, the two tails usually contain slightly less than α in total, and the critical region may not be perfectly symmetric.

对于水平为 α 的双尾检验，你要找出满足 P(X ≤ c) ≤ α ÷ 2 的最大 c，以及满足 P(X ≥ d) ≤ α ÷ 2 的最小 d，于是临界区域为 X ≤ c 或 X ≥ d。由于二项分布是离散的，两侧尾部合起来通常略小于 α，临界区域也可能不完全对称。

The acceptance region is simply everything not

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