📚 Further Hypothesis Tests | 进阶假设检验
In A-Level Statistics, a hypothesis test is a formal decision rule for using sample data to judge a claim about a population parameter. This article extends the basic binomial test to Poisson and normal models, and develops the concepts of p-values, critical regions, Type I and Type II errors, and the power of a test. Understanding these ideas is essential for Edexcel exam questions that require both calculations and clear written conclusions.
在 A-Level 统计学中，假设检验是利用样本数据对总体参数的某一主张进行判断的正式决策规则。本文把基础的二项分布检验扩展到泊松分布与正态分布模型，并深入讲解 p 值、临界区域、第一类与第二类错误以及检验的功效。掌握这些内容对 Edexcel 考试中既要求计算又要求清晰文字结论的题目至关重要。
1. Hypothesis Testing Framework | 假设检验框架
A hypothesis test starts by assuming a null hypothesis H₀ is true, then checks whether the observed sample data are too unusual to be consistent with that assumption. If the data fall in a critical region, we reject H₀ in favour of the alternative hypothesis H₁; otherwise, we do not reject H₀. The conclusion must always be written in the context of the original problem.
假设检验首先假设原假设 H₀ 为真，然后判断观测到的样本数据是否“过于异常”以至于与该假设不一致。若数据落入临界区域，就拒绝 H₀ 并支持备择假设 H₁；否则不拒绝 H₀。结论必须始终结合原问题的实际背景来写。
The null hypothesis H₀ represents the default or status quo claim, such as a coin is fair or a machine is working correctly. The alternative hypothesis H₁ is the claim we are testing for, such as the proportion has increased or the mean has changed.
原假设 H₀ 表示默认或现状主张，例如硬币是均匀的或机器运行正常。备择假设 H₁ 则是我们要检验的主张，例如比例上升了或均值发生了改变。
Sample data provide evidence against H₀, but they do not prove H₁ absolutely. A test result is always a statement about likelihood under the assumed model, not a mathematical proof.
样本数据提供的是反对 H₀ 的证据，并不能绝对证明 H₁ 成立。检验结果始终是在假设模型下的可能性判断，而不是数学证明。
2. Significance Level and Critical Regions | 显著性水平与临界区域
The significance level, denoted by α, is the maximum probability of rejecting H₀ when it is actually true. Common choices in Edexcel exams are 5%, 1% and 10%. The critical region is the set of values of the test statistic for which H₀ is rejected; the critical value is the boundary of this region.
显著性水平，记作 α，是当 H₀ 实际为真时拒绝 H₀ 的最大概率。Edexcel 考试中常用的取值是 5%、1% 和 10%。临界区域是拒绝 H₀ 的检验统计量取值集合；临界值是该区域的边界。
P(Type I error) = α
For a 5% significance level, the critical region contains the most extreme 5% of outcomes under H₀. If the observed value lies inside this region, we say the result is statistically significant at the 5% level.
在 5% 显著性水平下，临界区域包含 H₀ 成立时最极端的 5% 结果。若观测值落在该区域内，就称结果在 5% 水平上具有统计显著性。
3. One-Tailed and Two-Tailed Tests | 单尾检验与双尾检验
A one-tailed test has an alternative hypothesis with a strict direction, such as H₁: p > 0.3 or H₁: μ < 100. The entire significance level α is placed in one tail of the distribution. A two-tailed test uses H₁: p ≠ 0.3 or H₁: μ ≠ 100, and the α is split equally between both tails, usually α/2 in each tail.
单尾检验的备择假设具有明确方向，例如 H₁: p > 0.3 或 H₁: μ < 100。整个显著性水平 α 放在分布的一个尾部。双尾检验使用 H₁: p ≠ 0.3 或 H₁: μ ≠ 100，α 被平均分配到两个尾部，通常每尾为 α/2。
For a two-tailed binomial test, you must find the lower and upper critical values such that the probability in each tail does not exceed α/2. This is important because the critical region is not always symmetric for discrete distributions.
对于双尾二项分布检验，必须求出下临界值和上临界值，使得每个尾部的概率不超过 α/2。这一点很重要，因为对于离散分布，临界区域并不总是对称的。
Always check the wording of the question: phrases such as “has increased”, “is lower than”, or “differs from” tell you whether the test is one-tailed or two-tailed.
一定要仔细读题：“增加了”“低于”或“与……不同”等措辞可以告诉你该用单尾还是双尾检验。
4. p-Value Method | p 值法
The p-value is the probability of obtaining a sample result at least as extreme as the observed value, assuming H₀ is true. If the p-value is less than or equal to α, we reject H₀. If it is greater than α, we do not reject H₀.
p 值是在 H₀ 为真的条件下，得到至少与观测值同样极端的样本结果的概率。若 p 值小于或等于 α，则拒绝 H₀；若 p 值大于 α，则不拒绝 H₀。
For a one-tailed test H₁: p > 0.3, the p-value is P(X ≥ x). For a two-tailed test, you often calculate one tail and double it, or sum the probability of both tails, ensuring the calculation matches the alternative hypothesis.
对于单尾检验 H₁: p > 0.3，p 值为 P(X ≥ x)。对于双尾检验，通常计算一个尾部的概率再乘以 2，或计算两个尾部概率之和，以确保计算与备择假设一致。
The p-value method is particularly useful when a question asks “find the p-value and state your conclusion” rather than specifying a critical region. It gives a direct measure of strength of evidence against H₀.
当题目要求“求 p 值并给出结论”而不是指定临界区域时，p 值法特别有用。它直接衡量了反对 H₀ 的证据强度。
5. Binomial Hypothesis Tests | 二项分布假设检验
A binomial test is used when there is a fixed number of independent trials n, each with two possible outcomes, and the parameter under test is the probability of success p. The test statistic is X ~ B(n, p₀), where p₀ is the value of p stated in H₀.
二项分布检验适用于固定次数的独立试验 n，每次试验只有两种可能结果，且待检验的参数是成功概率 p。检验统计量为 X ~ B(n, p₀)，其中 p₀ 是 H₀ 中给出的 p 值。
Example: A company claims that 40% of customers are satisfied. A sample of 20 customers is taken and 12 say they are satisfied. Test at the 5% significance level whether the proportion is greater than 0.4.
示例：某公司声称 40% 的客户满意。随机抽取 20 名客户，其中 12 人表示满意。在 5% 显著性水平下检验满意比例是否大于 0.4。
Let X be the number of satisfied customers, so X ~ B(20, 0.4). The hypotheses are H₀: p = 0.4 and H₁: p > 0.4. The observed value is x = 12, so the p-value is P(X ≥ 12) = 1 − P(X ≤ 11). Using a calculator, P(X ≤ 11) = 0.9435, so p-value = 0.0565.
设 X 为满意客户人数，则 X ~ B(20, 0.4)。假设为 H₀: p = 0.4，H₁: p > 0.4。观测值为 x = 12，因此 p 值为 P(X ≥ 12) = 1 − P(X ≤ 11)。使用计算器可得 P(X ≤ 11) = 0.9435，所以 p 值 = 0.0565。
Since 0.0565 > 0.05, we do not reject H₀. There is insufficient evidence at the 5% level to suggest that the proportion of satisfied customers is greater than 0.4.
因为 0.0565 > 0.05，我们不拒绝 H₀。在 5% 水平上，没有充分证据表明满意客户比例大于 0.4。
6. Poisson Hypothesis Tests | 泊松分布假设检验
A Poisson test is used when the data are counts of events occurring in a fixed interval of time, space, or volume, and events occur independently at a constant average rate λ. Under H₀, we assume X ~ Po(λ₀), where λ₀ is the claimed rate.
泊松分布检验适用于对固定时间、空间或体积间隔内事件发生次数进行计数，且事件以恒定的平均速率 λ 独立发生的情形。在 H₀ 下，我们假设 X ~ Po(λ₀)，其中 λ₀ 是声称的速率。
P(X = k) = e^(−λ) × λ^k ÷ k!
Example: A call centre claims it receives an average of 3 calls per minute. In a particular minute, 7 calls are received. Test at the 5% level whether the rate has increased.
示例：某呼叫中心声称平均每分钟接听 3 个电话。在某一分钟内接到了 7 个电话。在 5% 显著性水平下检验接听率是否上升。
Let X ~ Po(3). H₀: λ = 3 and H₁: λ > 3. The observed value is x = 7. The p-value is P(X ≥ 7) = 1 − P(X ≤ 6). Using a calculator or tables, P(X ≤ 6) = 0.9665, so p-value = 0.0335.
设 X ~ Po(3)。H₀: λ = 3，H₁: λ > 3。观测值为 x = 7。p 值为 P(X ≥ 7) = 1 − P(X ≤ 6)。使用计算器或查表可得 P(X ≤ 6) = 0.9665，因此 p 值 = 0.0335。
Since 0.0335 < 0.05, we reject H₀. There is sufficient evidence at the 5% level to suggest that the mean number of calls per minute has increased above 3.
因为 0.0335 < 0.05，我们拒绝 H₀。在 5% 水平上，有充分证据表明每分钟平均电话数已增加到 3 以上。
7. Normal Hypothesis Tests for the Mean | 正态分布均值假设检验
When the population is normally distributed and the population standard deviation σ is known, the sample mean x̄ follows a normal distribution with mean μ and standard deviation σ/√n. The test statistic is calculated as:
当总体服从正态分布且总体标准差 σ 已知时，样本均值 x̄ 服从均值为 μ、标准差为 σ/√n 的正态分布。检验统计量计算如下：
Z = (x̄ − μ₀) / (σ / √n)
Compare the calculated Z value with the critical value from the standard normal distribution. For a 5% one-tailed test, the critical value is 1.645; for a 5% two-tailed test, the critical values are ±1.96.
将计算出的 Z 值与标准正态分布的临界值进行比较。对于 5% 单尾检验，临界值为 1.645；对于 5% 双尾检验，临界值为 ±1.96。
Example: A machine fills packets with a mean mass expected to be 100 g. The standard deviation σ is known to be 15 g. A sample of 25 packets has mean mass 94 g. Test at the 5% level whether the mean mass is less than 100 g.
示例：一台机器包装的袋装食品平均质量预期为 100 g。已知标准差 σ = 15 g。随机抽取 25 袋，样本均值为 94 g。在 5% 水平下检验平均质量是否小于 100 g。
H₀: μ = 100, H₁: μ < 100. The test statistic is Z = (94 − 100) / (15 / √25) = −6 / 3 = −2.0. The critical value for a one-tailed 5% test is −1.645. Since −2.0 < −1.645, we reject H₀.
H₀: μ = 100，H₁: μ < 100。检验统计量为 Z = (94 − 100) / (15 / √25) = −6 / 3 = −2.0。5% 单尾检验的临界值为 −1.645。因为 −2.0 < −1.645，我们拒绝 H₀。
There is sufficient evidence at the 5% significance level to suggest that the mean mass is less than 100 g.
在 5% 显著性水平下，有充分证据表明平均质量小于 100 g。
8. Type I and Type II Errors | 第一类错误与第二类错误
A Type I error occurs when H₀ is true but the test rejects it. The probability of a Type I error is α, the significance level. A Type II error occurs when H₀ is false but the test fails to reject it. The probability of a Type II error is denoted by β.
第一类错误发生在 H₀ 为真但检验拒绝了它的情形。第一类错误的概率就是显著性水平 α。第二类错误发生在 H₀ 为假但检验未能拒绝它的情形。第二类错误的概率记作 β。
| Decision | H₀ is true | H₀ is false |
|---|---|---|
| Reject H₀ | Type I error (α) | Correct decision |
| Do not reject H₀ | Correct decision | Type II error (β) |
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