📚 Hypothesis Testing in A-Level WJEC Mathematics | A-Level WJEC 数学：假设检验 考点精讲
Hypothesis testing is a core part of A-Level WJEC Mathematics, allowing you to draw conclusions from sample data about a population. This revision guide covers the essential concepts, from setting up hypotheses and finding critical regions to interpreting p-values and understanding errors, all tailored to the WJEC specification.
假设检验是A-Level WJEC数学的核心内容，使你能够从样本数据中得出关于总体的结论。本复习指南涵盖了所有基本概念，从建立假设、确定临界区域到解释p值和理解错误，均针对WJEC考纲量身定制。
1. What is Hypothesis Testing? | 什么是假设检验？
Hypothesis testing is a formal procedure to decide whether to accept or reject a statement about a population parameter based on evidence from a sample. In WJEC exams, you will typically test a proportion (using the binomial distribution) or a mean (using the normal distribution).
假设检验是一个正式流程，它根据样本证据决定是否拒绝关于某个总体参数的陈述。在WJEC考试中，你通常会检验一个比例（使用二项分布）或一个均值（使用正态分布）。
The logic is indirect: we never try to prove the alternative hypothesis directly. Instead we assume the null hypothesis is true, work out how likely the observed sample result (or something more extreme) would be, and then judge whether that likelihood is small enough to cast doubt on the assumption.
其逻辑是间接的：我们从不直接证明备择假设。相反，我们先假定原假设为真，计算观测到的样本结果（或更极端结果）出现的可能性有多大，再判断这个可能性是否小到足以让该假定受到质疑。
Because the decision is based on a sample rather than the whole population, every conclusion carries a risk of error. Understanding that risk is what separates a full-mark answer from a purely mechanical one.
由于决策基于样本而非整个总体，每个结论都带有出错的风险。理解这一风险，正是满分答案与机械套用答案之间的区别。
2. The Two Hypotheses: H₀ and H₁ | 两个假设：H₀ 与 H₁
eses: the null hypothesis (H₀) and the alternative hypothesis (H₁). H₀ is the statement of "no change" or "no difference" and always contains an equality sign: p = 0.4, μ = 25, or similar.
每个假设检验都从两个相互对立的假设开始：原假设（H₀）和备择假设（H₁）。H₀ 是“无变化”或“无差异”的陈述，并且总是包含等号：p = 0.4、μ = 25 或类似形式。
H₁ is the statement you are looking for evidence to support. It contains an inequality: p > 0.4, p < 0.4 or p ≠ 0.4. The direction of the inequality determines whether the test is one-tailed or two-tailed.
H₁ 是你正在寻找证据去支持的陈述。它包含不等号：p > 0.4、p < 0.4 或 p ≠ 0.4。不等号的方向决定检验是单尾还是双尾。
You should always define the parameter in words first. For example: "Let p be the population proportion of students who pass the test." Defining the parameter before stating the hypotheses is a mark-scoring habit in WJEC mark schemes.
你应当先用文字定义参数。例如：“设 p 为通过考试的学生在总体中的比例。”在写出假设之前先定义参数，是WJEC评分标准中能拿分的习惯。
3. One-Tailed vs Two-Tailed Tests | 单尾与双尾检验
A one-tailed test is used when you are only interested in a change in one direction. H₁ takes the form p > p₀ or p < p₀, and the entire significance level sits in one tail of the distribution.
当你只关心某一方向的变化时，使用单尾检验。H₁ 采取 p > p₀ 或 p < p₀ 的形式，整个显著性水平都落在分布的一侧尾部。
A two-tailed test is used when you are interested in any change, in either direction. H₁ takes the form p ≠ p₀, and the significance level is split equally between the two tails, so 5% becomes 2.5% at each end.
当你关心任一方向的变化时，使用双尾检验。H₁ 采取 p ≠ p₀ 的形式，显著性水平被平均分配到两侧尾部，因此 5% 就变成两端各 2.5%。
| H₁ form | Tail type | 5% split | Typical wording |
| p > p₀ | Upper one-tailed | 5% in upper tail | "has increased" |
| p < p₀ | Lower one-tailed | 5% in lower tail | "has decreased" |
| p ≠ p₀ | Two-tailed | 2.5% in each tail | "has changed" |
Read the question wording carefully. Words such as "different from", "changed" or "is not equal to" signal a two-tailed test; "more than", "greater", "less than" or "reduced" signal a one-tailed test.
要仔细读题干措辞。“不同于”、“发生了变化”或“不等于”这类词提示双尾检验；“多于”、“大于”、“少于”或“降低了”这类词提示单尾检验。
4. Testing a Proportion with the Binomial Distribution | 用二项分布检验比例
When the data consist of a count of successes out of a fixed number of independent trials, the binomial distribution is the natural model. You write X ~ B(n, p) under the null hypothesis, where p is the value stated in H₀.
当数据是在固定次数的独立试验中成功次数的计数时，二项分布就是天然的模型。在原假设下你写出 X ~ B(n, p)，其中 p 是 H₀ 中给出的取值。
biased towards heads and is tossed 20 times. If H₀ states p = 0.5, then under H₀ we have X ~ 0, 0.5), and we calculate the probability of getting the observed number of heads or more.
假设怀疑一枚硬币偏向正面，抛掷 20 次。如果 H₀ 陈述 p = 0.5，则在 H₀ 下 X ~ B(20, 0.5)，我们计算得到观测到的正面次数或更多次的概率。
d test we consider both tails and compare the smaller relevant probability with half the significance level.
对于上尾，我们需要 P(X ≥ 观测值)。对于下尾，我们需要 P(X ≤ 观测值)。对于双尾检验，我们考虑两侧尾部，并把较小的相关概率与显著性水平的一半作比较。
You must state clearly that you are assuming H₀ is true, and you must give the distribution you are using. Both of these carry marks in WJEC papers.
你必须明确说明你是在假设 H₀ 为真的前提下计算，并且必须给出你所使用的分布。这两点在WJEC试卷中都能得分。
5. Finding Critical Regions and Critical Values | 求临界域与临界值
ial test, the critical region has the form X ≥ c for an upper-tailed test or X ≤ c for a lower-tailed test.
临界域是使 H₀ 被拒绝的检验统计量取值的集合。它的边界就是临界值，通常记为 c。在二项检验中，临界域在上尾检验里形如 X ≥ c，在下尾检验里形如 X ≤ c。
To find c at the 5% c) ≤ 0.05. For a lower-tailed test, find the largest integer c such that P(X ≤ c) ≤ 0.05.
要在 5% 水平下为上尾检验求 c，就找出满足 P(X ≥ c) ≤ 0.05 的最小整数 c。对于下尾检验，则找出满足 P(X ≤ c) ≤ 0.05 的最大整数 c。
For example, if X ~ B(20, 0.3) and we test H₀: p = 0.3 against H₁: p > 0.3 at the 5% we find the smallest c with P(X ≥ c) ≤ 0.05. That value of c defines the critical region.
例如，若 X ~ B(20, 0.3)，我们要在 5% 水平下检验 H₀: p = 0.3 对 H₁: p > 0.3，就从表中读取累积二项概率，直到找到满足 P(X ≥ c) ≤ 0.05 的最小 c。该 c 值就定义了临界域。
The actual significance level is the probability of falling in the critical region when H₀ is true. Because the binomial distribution is discrete, this is often slightly less than the nominal 5% or 10%, and WJEC may ask you to state it.
实际显著性水平是 H₀ 为真时落入临界域的概率。由于二项分布是离散的，这个值往往略小于名义上的 5% 或 10%，WJEC 可能会要求你写出它。
Actual significance level = P(X in critical region | H₀ true)
6. Testing a Mean with the Normal Distribution | 用正态分布检验均值
When the test concerns a population mean and the sample size is reasonably large, or the population is known to be normal, the normal distribution is used. If the sample mean is X̄ and the population standard deviation is σ, then under H₀ the sample mean is modelled as:
当检验涉及总体均值，且样本量足够大，或已知总体服从正态分布时，就使用正态分布。若样本均值为 X̄、总体标准差为 σ，则在 H₀ 下，样本均值被建模为：
X̄ ~ N(μ, σ² ÷ n)
The standardised test statistic is found by subtracting the hypothesised mean and dividing by the standard error:
标准化检验统计量通过减去假设的均值、再除以标准误得到：
z = (X̄ − μ) ÷ (σ ÷ √n)
For a lower-tailed test at the 5% re z with −1.645; for an upper-tailed test compare with +1.645; for a two-tailed test at 5% ritical values come from the standard normal distribution and should be quoted accurately.
在 5% 水平下，下尾检验将 z 与 −1.645 比较；上尾检验与 +1.645 比较；5% 双尾检验与 ±1.96 比较。这些临界值来自标准正态分布，应当准确引用。
Alternatively, use the p-value from your calculator or tables: if the p-value is less than the significance level, reject H₀. Both routes are accepted, but the comparison method must be stated clearly.
或者，使用计算器或统计表给出的 p 值：若 p 值小于显著性水平，则拒绝 H₀。两种方法都被接受，但必须清楚说明所采用的比较方式。
7. Understanding p-values | 理解 p 值
A p-value is the probability of obtaining a result at least as extreme as the observed result, assuming that H₀ is true. It is not the probability that H₀ is true, a very common misconception.
p 值是在假定 H₀ 为真的条件下，得到至少与观测结果同样极端的结果的概率。它并不是 H₀ 为真的概率，这是一个非常常见的误解。
A small p-value means the observed data would be unlikely if H₀ were true, so we have evidence against H₀. A large p-value means the data are consistent with H₀, so we do not have sufficient evidence to reject it.
p 值很小意味着若 H₀ 为真，观测数据不太可能出现，因此我们有反对 H₀ 的证据。p 值较大意味着数据与 H₀ 相容，因此我们没有足够证据拒绝它。
Remember that failing to reject H₀ is never proof that H₀ is true. It simply means the sample did not provide strong enough evidence against it, possibly because the sample was too small.
记住，未能拒绝 H₀ 绝不等于证明 H₀ 为真。它只是说明样本没有提供足够强的反证，也可能是因为样本量太小。
8. Significance Levels and the 5% Convention | 显著性水平与 5% 惯例
The significance level, usually denoted α, is the threshold probability chosen before the test is carried out. The most common choice in WJEC papers is 5%, though 1% and 10% also appear.
显著性水平通常记为 α，是在进行检验之前选定的阈值概率。WJEC 试卷中最常见的选择是 5%，但 1% 和 10% 也会出现。
A 5% significance level means that, if H₀ is true, there is a 5% chance of incorrectly rejecting it. Lowering α to 1% makes it harder to reject H₀, reducing the chance of a false positive but increasing the chance of missing a genuine effect.
5% 的显著性水平意味着，若 H₀ 为真，有 5% 的概率错误地拒绝它。把 α 降到 1% 会使拒绝 H₀ 更难，从而降低假阳性概率，但会增加漏掉真实效应的概率。
The significance level must be chosen before seeing the data. Choosing it afterwards to obtain a desired conclusion is a serious methodological error.
显著性水平必须在看到数据之前选定。事后为了得到想要的结论才去挑选它，是严重的方法学错误。
9. 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 we reject it. Its probability is exactly the significance level α, so at the 5% level a Type I error happens 5% of the time in the long run.
第一类错误发生在 H₀ 为真而我们却拒绝它的时候。它的概率恰好等于显著性水平 α，所以在 5% 水平下，长期来看第一类错误有 5% 的发生率。
A Type II error occurs when H₀ is false but we fail to reject it. Its probability is usually denoted β, and the quantity 1 − β is called the power of the test.
第二类错误发生在 H₀ 为假而我们却未能拒绝它的时候。其概率通常记为 β，而量 1 − β 被称为检验的功效。
| Reality / Decision | Reject H₀ | Do not reject H₀ |
| H₀ true | Type I error (prob = α) | Correct decision |
| H₀ false | Correct decision | Type II error (prob = β) |
In WJEC questions you are often asked to write these errors in the context of the problem, for example: "A Type I error would be concluding that the proportion of defective items has decreased when in fact it has not."
在 WJEC 题目中，常常要求你把这两类错误写在题目情境里，例如：“第一类错误是指实际上次品比例并未下降，却得出结论说它下降了。”
Increasing the sample size reduces β while keeping α fixed, which is why larger samples give more reliable tests.
在保持 α 不变的情况下增大样本量可以降低 β，这就是样本量越大检验越可靠的原因。
10. Writing Conclusions in WJEC Style | 按 WJEC 风格书写结论
A full-mark conclusion has three parts: the comparison, the decision, and the context. First state the comparison, for example "0.032 < 0.05". Then state the decision, "reject H₀". Then interpret it in context, "there is sufficient evidence that the proportion has increased".
满分结论包含三部分：比较、决策和情境。首先说明比较结果，例如“0.032 < 0.05”。然后说明决策：“拒绝 H₀”。最后结合情境解释：“有充分证据表明该比例上升了”。
Never write "accept H₀". The correct phrasing is "there is insufficient evidence to reject H₀", or "the evidence does not support H₁". H₀ is not proved; it is simply not contradicted.
绝不要写“接受 H₀”。正确的表述是“没有足够证据拒绝 H₀”，或“证据不支持 H₁”。H₀ 并未被证明，它只是没有被推翻而已。
Always refer back to the original real-world setting. Answers that stop at "reject H₀" without a contextual sentence typically lose the final interpretation mark.
始终要回到原本的现实情境。只写到“拒绝 H₀”而没有情境句的答案，通常会丢掉最后的解释分。
11. Common Exam Pitfalls | 常见考试陷阱
The most frequent mistake is using the wrong tail. Check whether the question says "increase", "decrease" or "change", and check whether the observed value lies above or below the hypothesised value.
最常见的错误是用错尾部。要检查题目说的是“增加”、“减少”还是“变化”，并检查观测值位于假设值的上方还是下方。
Other common errors include forgetting to halve the significance level for a two-tailed test, using σ instead of σ ÷ √n as the standard error, and mixing up the conditions for a Type I and Type II error.
其他常见错误包括：双尾检验时忘记把显著性水平除以 2，把标准误误用为 σ 而不是 σ ÷ √n，以及混淆第一类错误和第二类错误的条件。
- Do not state H₁ before defining the parameter in words. | 不要在没有用文字定义参数之前就写出 H₁。
- Do not compare the observed value itself with the significance level. | 不要把观测值本身与显著性水平作比较。
- Do not write "prove" or "accept" in a conclusion. | 不要在结论中使用“证明”或“接受”这类字眼。
- Do not forget to state the distribution used under H₀. | 不要忘记说明在 H₀ 下所使用的分布。
12. Worked Example and Revision Strategy | 例题与复习策略
Consider a claim that a machine produces 20% defective items. A sample of 15 items contains 7 defectives. Test at the 5% level whether the defect rate has increased. Here H₀: p = 0.2 and H₁: p > 0.2, with X ~ B(15, 0.2) under H₀.
假设某机器生产的次品率为 20%。抽取 15 件样品，其中有 7 件次品。在 5% 水平下检验次品率是否上升。这里 H₀: p = 0.2，H₁: p > 0.2，在 H₀ 下 X ~ B(15, 0.2)。
we reject H₀ and conclude there is sufficient evidence that the defect rate has increased.
我们需要计算 H₀ 下的 P(X ≥ 7)。利用累积二项分布表，该概率较小。若 P(X ≥ 7) ≤ 0.05，则拒绝 H₀，并得出结论：有充分证据表明次品率上升了。
For revision, practise writing complete solutions: define the parameter, state both hypotheses, name the distribution, compute the probability or test statistic, compare with the significance level, and write a contextual conclusion.
复习时，要练习写出完整解答：定义参数、写出两个假设、指明分布、计算概率或检验统计量、与显著性水平比较，最后写出带情境的结论。
Define → Hypothesise → Model → Compare → Conclude
Mastering this five-step structure will make hypothesis testing one of the most reliable sources of marks in your WJEC examination.
掌握这个五步结构，假设检验将成为你 WJEC 考试中最稳定的得分点之一。
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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