📚 AQA Maths: Hypothesis Testing – Key Points | AQA 数学：假设检验 考点精讲
Hypothesis testing is a core part of AQA A-Level Statistics, allowing you to draw conclusions about a population from sample data. This revision guide walks you through the key concepts, procedures, and common pitfalls step by step, with worked examples for both binomial and normal distributions.
假设检验是 AQA A-Level 统计学的核心内容，使你能够根据样本数据对总体作出推断。这份考点精讲将逐步梳理关键概念、步骤和常见误区，并配有二项分布与正态分布下的例题解析。
1. Introduction to Hypothesis Testing | 假设检验简介
Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a claim about a population. It starts with a statement (the null hypothesis) and uses sample data to test its credibility.
假设检验是一种统计方法，用于判断是否有足够证据拒绝关于总体的某个声明。它从一个陈述（零假设）出发，并利用样本数据检验其可信度。
The process involves calculating the probability of the observed outcome under the assumption that the null hypothesis is true. If this probability is very low, we have reason to doubt the null hypothesis.
这个过程涉及在零假设为真的前提下，计算观察结果的概率。如果这个概率非常低，我们就有理由怀疑零假设。
2. Null and Alternative Hypotheses | 零假设与备择假设
The null hypothesis, denoted H₀, is a statement of ‘no effect’ or ‘ It is assumed true until evidence contradicts it. For example, H₀: p = 0.5 or H₀: μ = 500.
零假设，记作 H₀，是一个“没有效应”或“没有差异”的陈述，在证据反驳它之前被假定为真。例如 H₀: p = 0.5 或 H₀: μ = 500。
The alternative hypothesis, denoted H₁ or Hₐ, represents what we suspect might be true. It is the claim we test for. H₁ can be one-sided (p > 0.5) or two-sided (p ≠ 0.5).
备择假设，记作 H₁ 或 Hₐ，代表我们怀疑可能为真的情况，是我们检验的主张。H₁ 可以是单侧的（p > 0.5）或双侧的（p ≠ 0.5）。
Always state hypotheses in terms of the population parameter (p or μ), never in terms of the sample statistic. The sample provides the evidence, but the hypotheses refer to the unknown true value.
务必用总体参数（p 或 μ）来表述假设，绝不能使用样本统计量。样本提供证据，但假设指的是未知的真实值。
3. Significance Levels and Critical Regions | 显著性水平与临界区域
The significance level α is the maximum probability of making a Type I error – rejecting a true H₀. Common choices are 0.05 (5%) and 0.01 (1%), as specified in an exam question.
显著性水平 α 是发生第一类错误（拒绝一个真的 H₀）的最大概率。常见取值有 0.05 (5%) 和 0.01 (1%)，如试题中所指定。
. For a binomial test with α = 0.05, we aim to keep the probability in the critical region as close as possible to, but not exceeding, α.
临界区域（或拒绝域）是会导致拒绝 H₀ 的检验统计量取值集合。其边界称为临界值。对于 α = 0.05 的二项检验，我们力求临界区域内的概率尽可能接近 α 但不超过 α。
If the observed test statistic falls inside the critical region, we reject H₀; otherwise, we do not reject H₀. Note: we never ‘accept’ H₀, only fail to reject it.
如果观测到的检验统计量落在临界区域内，我们就拒绝 H₀；否则，我们不拒绝 H₀。注意：我们从不“接受” H₀，只能说未能拒绝 H₀。
4. One-tailed vs Two-tailed Tests | 单尾与双尾检验
A one-tailed (or one-sided) test examines whether the parameter differs in a specific direction, e.g., H₁: p > 0.5 (right-tailed) or H₁: p < 0.5 (left-tailed). All of the significance level α is placed in one tail.
单尾（或单侧）检验考察参数是否在特定方向上存在差异，例如 H₁: p > 0.5（右尾）或 H₁: p < 0.5（左尾）。显著性水平 α 全部放在一侧尾部。
A two-tailed (or two-sided) test looks for any difference, with H₁: p ≠ 0.5 or μ ≠ μ₀. The α is split equally between the two tails, often α/2 in each tail. This affects critical values: for a 5% two-tailed test, we use 2.5% in each tail.
双尾（或双侧）检验则检查任意差异，例如 H₁: p ≠ 0.5 或 μ ≠ μ₀。α 被平分至两个尾部，通常每侧为 α/2。这会影响临界值的选取：对于 5% 的双尾检验，每侧使用 2.5%。
The choice of one- or two-tailed test must be justified by the context of the problem, not decided after seeing the data. Read the wording carefully: ‘increased’ suggests one-tailed, while ‘changed’ implies two-tailed.
选择单尾还是双尾检验必须以问题的实际背景为依据，而非在看到数据后才决定。仔细阅读措辞：“增加”暗示单尾检验，“改变”则暗示双尾检验。
5. Test Statistic and Binomial Distribution | 检验统计量与二项分布
For a hypothesis test concerning a population proportion p, we use the binomial distribution. The test statistic is the number of successes X ~ B(n, p₀), where p₀ is the assumed value under H₀.
对于涉及总体比例 p 的假设检验，我们使用二项分布。检验统计量为成功次数 X ~ B(n, p₀)，其中 p₀ 是 H₀ 下的假设值。
, depending on the direction of H₁. For a two-tailed test, we compare x against both tails, often by doubling the one-tailed p-value (with care for asymmetry).
将观测值 x 与 H₀ 为真时的期望情况进行比较。我们根据 H₁ 的方向计算 P(X ≤ x) 或 P(X ≥ x)。对于双尾检验，通常将单尾 p 值加倍（注意非对称性），将 x 与两个尾部进行对比。
In AQA exams, you may be asked to find the probability using binomial tables or a calculator, and to interpret the result in context. Always state clearly: assuming H₀ is true, the probability of obtaining a result as extreme as the one observed is …
在 AQA 考试中，你可能会被要求使用二项分布表或计算器求出概率，并结合实际做出解释。务必清晰表述：假定 H₀ 为真，得到至少与观测结果同样极端的结果的概率为……
6. Finding Critical Values | 寻找临界值
such that P(X ≤ c) ≤ α. For a right-tailed test, we find the smallest c such that P(X ≥ c) ≤ α.
临界值 c 将临界区域与接受区域分隔开来。对于显著性水平 α 的左尾检验，我们寻找最大的 c 使得 P(X ≤ c) ≤ α。对于右尾检验，则寻找最小的 c 使得 P(X ≥ c) ≤ α。
c₂ for the upper tail such that P(X ≥ c₂) ≤ α/2. The critical region is X ≤ c₁ or X ≥ c₂.
对于双尾二项检验，我们将 α 平分，找到两个临界值：下尾的 c₁ 满足 P(X ≤ c₁) ≤ α/2，上尾的 c₂ 满足 P(X ≥ c₂) ≤ α/2。临界区域为 X ≤ c₁ 或 X ≥ c₂。
Because the binomial distribution is discrete, the actual probability of the critical region is often less than α. This is acceptable, and you should mention the actual significance level where appropriate.
由于二项分布是离散的，临界区域的实际概率通常小于 α。这是可接受的，你应该在适当位置提及实际显著性水平。
7. p-Values | p 值
ial test with observed x, the p-value is P(X ≥ x). For a left-tailed test, it is P(X ≤ x).
p 值是在 H₀ 为真的条件下，得到至少与观测值同样极端的检验统计量的概率。对于观测值为 x 的右尾二项检验，p 值为 P(X ≥ x)；左尾检验则为 P(X ≤ x)。
-tailed p-value, but care is needed if the distribution is not symmetric. AQA typically expects this method for binomial tests.
在双尾检验中，p 值通常计算为 2 × 较小的单尾 p 值，但如果分布不对称则需要谨慎。AQA 对二项检验一般期望使用这种方法。
p > α, we do not reject H₀. Reporting the p-value gives more information than a simple ‘reject/do not reject’ and is increasingly common.
若 p 值 ≤ α，则拒绝 H₀；若 p > α，则不拒绝 H₀。报告 p 值比单纯的“拒绝/不拒绝”能提供更多信息，这一做法日益普遍。
8. Hypothesis Testing with Normal Distribution | 正态分布下的假设检验
When the population is normal, or the sample size is large (Central Limit Theorem), and the standard deviation σ is known, we use the standard normal distribution. The test statistic is:
当总体服从正态分布，或样本量足够大（中心极限定理）且标准差 σ 已知时，我们使用标准正态分布。检验统计量为：
Z = (x̄ − μ₀) / (σ/√n)
where x̄ is the sample mean, μ₀ is the population mean under H₀, σ is the population standard deviation, and n is the sample size.
其中 x̄ 为样本均值，μ₀ 为 H₀ 下的总体均值，σ 为总体标准差，n 为样本量。
the calculated Z-value with critical values from the standard normal distribution. For a 5% right-tailed test, the critical value is 1.6449 (often 1.645), and for a two-tailed 5% test, the critical values are ±1.96.
我们将计算得到的 Z 值与标准正态分布的临界值进行比较。对于 5% 右尾检验，临界值为 1.6449（通常取 1.645）；对于 5% 双尾检验，临界值为 ±1.96。
If σ is unknown and the sample is large, we may use the sample standard deviation s and proceed similarly, though at A-Level this normally uses the normal approximation. For small samples with unknown σ, the t-distribution is used, but that is less common in basic hypothesis testing sections.
如果 σ 未知且样本量较大，我们可以使用样本标准差 s 进行类似处理，尽管在 A-Level 中这通常属于正态近似。对于 σ 未知的小样本，应使用 t 分布，但这在基础假设检验章节中较少出现。
9. Type I and Type II Errors | 第一类错误与第二类错误
A Type I error occurs when we wrongly reject a true H₀, a 'false positive'. Its probability is controlled by α. A Type II error occurs when we fail to reject a false H₀, a 'false negative', with probability β.
当 H₀ 为真却被错误拒绝时，发生第一类错误（“假阳性”），其概率由 α 控制。当 H₀ 为假却未能拒绝时，发生第二类错误（“假阴性”），概率为 β。
The power of a test is 1 − β, the probability of correctly rejecting a false H₀. Power increases with larger sample size, larger effect size, and higher α.
检验的功效为 1 − β，即正确拒绝一个错误 H₀ 的概率。样本量越大、效应量越大、α 越高，功效越高。
| Decision | H₀ True | H₀ False |
|---|---|---|
| Reject H₀ | Type I error (α) | Correct decision (Power) |
| Do not reject H₀ | Correct decision | Type II error (β) |
10. Common Mistakes and Tips | 常见错误与技巧
Always define the test statistic and state hypotheses clearly before performing calculations. Confusing H₀ and H₁, or writing them in terms of sample rather than population quantities, is a frequent error.
务必在进行计算前明确检验统计量并清晰陈述假设。混淆 H₀ 与 H₁，或用样本量而非总体量表述假设，是常见错误。
observed value (e.g., P(X ≥ 15), not P(X > 15) unless specified). Check the direction: ‘at least as extreme’ includes the observed value.
使用二项分布时，确保概率计算包含观测值本身（例如 P(X ≥ 15)，而非 P(X > 15)，除非另有说明）。检查方向性：“至少同样极端” 包含观测值。
For normal distribution tests, always standardise correctly and use the correct critical Z-value for one-tailed or two-tailed tests. Misreading the alternative hypothesis can lead to selecting the wrong critical region.
对于正态分布检验，务必正确标准化，并针对单尾或双尾检验使用正确的临界 Z 值。误读备择假设可能导致选错临界区域。
Remember to state a conclusion in context: ‘There is sufficient evidence at the 5% level to reject the null hypothesis and conclude that …’ or ‘There is insufficient evidence …’. Never say ‘prove’.
记得结合情境给出结论：“在 5% 的显著性水平下，有充分证据拒绝零假设，从而认为……”或“没有充分证据……”。切勿使用“证明”一词。
11. Worked Examples (Binomial) | 例题解析（二项分布）
A manufacturer claims that 30% of customers prefer a new design. In a random sample of 20 customers, 10 prefer the new design. Test at the 5% significance level whether the true proportion is greater than 30%.
某制造商声称 30% 的顾客偏爱新设计。随机抽取 20 名顾客，有 10 名偏爱新设计。在 5% 显著性水平下检验真实比例是否大于 30%。
H₀: p = 0.3, H₁: p > 0.3 (right-tailed). Test statistic X ~ 10) = 1 − P(X ≤ 9). From binomial tables, P(X ≤ 9) = 0.9520 (approx), so p-value ≈ 0.0480.
H₀: p = 0.3，H₁: p > 0.3（右尾检验）。在 H₀ 下检验统计量 X ~ B(20, 0.3)。观测值 x = 10。p 值 = P(X ≥ 10) = 1 − P(X ≤ 9)。查二项分布表，P(X ≤ 9) ≈ 0.9520，故 p 值 ≈ 0.0480。
Since 0.0480 < 0.05, we reject H₀. There is sufficient evidence at the 5% level to suggest that more than 30% of customers prefer the new design.
由于 0.0480 < 0.05，我们拒绝 H₀。在 5% 水平下有充分证据表明超过 30% 的顾客偏爱新设计。
so the critical region is X ≥ 10. Observed 10 falls in the critical region, so reject H₀.
另一种方法，寻找临界值：满足 P(X ≥ c) ≤ 0.05 的最小 c。P(X ≥ 11) ≈ 0.0171，P(X ≥ 10) ≈ 0.0480 ≤ 0.05，因此临界区域为 X ≥ 10。观测值 10 落入临界区域，故拒绝 H₀。
12. Worked Examples (Normal) | 例题解析（正态分布）
A factory’s light bulbs have a stated mean lifetime of 800 hours with standard deviation 40 hours. A sample of 30 bulbs gives a mean of 788 hours. Test at the 1% significance level whether the mean lifetime has decreased.
某工厂的灯泡声称平均寿命为 800 小时，标准差为 40 小时。随机抽取 30 只灯泡，平均寿命为 788 小时。在 1% 显著性水平下检验平均寿命是否下降。
H₀: μ = 800, H₁: μ < 800 (left-tailed). Under H₀, Z = (x̄ − 800) / (40/√30) ~ N(0,1). Calculated Z = (788 − 800) / (40/√30) ≈ −12 / 7.303 = −1.643.
H₀: μ = 800，H₁: μ < 800（左尾检验）。在 H₀ 下，Z = (x̄ − 800) / (40/√30) ~ N(0,1)。计算 Z = (788 − 800) / (40/√30) ≈ −1.643。
For a 1% calculated Z = −1.643 is greater than −2.326, so it does not fall in the critical region.
对于 1% 左尾检验，临界值为 −2.326（因 Φ⁻¹(0.01) ≈ −2.326）。计算得到的 Z = −1.643 大于 −2.326，因此未落入临界区域。
We do not reject H₀. There is insufficient evidence at the 1% level to conclude that the mean lifetime has decreased.
我们不拒绝 H₀。在 1% 水平下没有充分证据表明平均寿命已下降。
ould be ±2.576; still not significant. Always align the test direction with the suspicion.
如果问题要求进行 α = 0.01 的双尾检验，临界值将为 ±2.576；结果仍不显著。务必使检验方向与怀疑的效应一致。
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