Hypothesis Testing in A-Level Statistics — AQA A-Level 数学假设检验完全指南

一、假设检验的本质：用样本数据对总体作出判断 | The Essence of Hypothesis Testing: Drawing Conclusions About Populations from Samples

在 A-Level 数学的统计学部分，我们经常面临这样一个问题：手里只有一小撮样本数据，却要对整个总体下结论。比如，质检员想知道一批灯泡的平均寿命是否达到了宣称的 5000 小时，但他不可能把每一只灯泡都点亮测试，因为那样灯泡就全废了。假设检验（Hypothesis Testing）就是一套规范的数学流程，它利用样本数据来评估关于总体的某个说法是否可信，并给出一个量化的决策依据。

In the statistics module of A-Level Mathematics, we often face this problem: we only have a small set of sample data, yet we must draw conclusions about an entire population. For example, a quality inspector wants to know whether a batch of light bulbs really lasts the claimed 5000 hours on average, but testing every single bulb would destroy them all. Hypothesis testing is a formal mathematical procedure that uses sample data to evaluate whether a claim about a population is credible, and it provides a quantified basis for decision-making.

假设检验的基本思路是”先假设，再检验”。我们先把想要质疑的说法作为零假设写下来，然后计算：如果这个说法真的是对的，那么出现当前样本结果（或者更极端结果）的概率有多大？如果这个概率非常小，小到不可思议，我们就有理由怀疑原假设，转而接受对立面的说法。这个过程把”信不信”的问题转化成了”概率多小”的问题，这正是统计学思维的核心。

The basic idea of hypothesis testing is “assume first, then test”. We first write down the claim we want to challenge as the null hypothesis, then calculate: if this claim were really true, how likely would it be to observe the current sample result, or something even more extreme? If this probability is extremely small, so small that it seems unbelievable, we have reason to doubt the null hypothesis and instead accept the opposite claim. This process converts the question of “what do we believe” into the question of “how small is the probability”, which is the heart of statistical thinking.

在 AQA A-Level 数学试卷中，假设检验题目通常出现在 Statistics 部分的 Paper 3 中，分值为 4 到 7 分不等。这类题目套路清晰：设定假设、计算概率、比较临界值、写出结论。只要掌握了固定的解题框架，这属于考试中”性价比”很高的得分点。

In the AQA A-Level Mathematics papers, hypothesis testing questions usually appear in the Statistics section of Paper 3, carrying between 4 and 7 marks. These questions follow a clear pattern: set up the hypotheses, calculate the probability, compare with the critical value, and write the conclusion. Once you master the fixed answering framework, these are among the highest “value for effort” marks in the exam.

二、零假设与备择假设：H0 与 H1 的正确写法 | Null and Alternative Hypotheses: How to Write H0 and H1 Correctly

任何假设检验的第一步都是写清楚两个假设。零假设 H0（Null Hypothesis）代表”现状”或”没有变化”，它总是包含等号。例如，怀疑硬币偏向正面时，H0 写为 H0: p = 0.5，意思是”正面概率仍为 0.5，硬币是公平的”。备择假设 H1（Alternative Hypothesis）代表我们想要证明的说法，它只包含不等号，可能是 p > 0.5、p < 0.5 或 p ≠ 0.5。

The first step of any hypothesis test is to state the two hypotheses clearly. The null hypothesis H0 represents the “status quo” or “no change”, and it always contains an equals sign. For example, when suspecting that a coin is biased towards heads, we write H0: p = 0.5, meaning “the probability of heads is still 0.5, the coin is fair”. The alternative hypothesis H1 represents the claim we want to prove, and it only contains an inequality: it may be p > 0.5, p < 0.5, or p ≠ 0.5.

写假设时有一个关键细节：H0 和 H1 中的参数必须是总体的参数（population parameter），而不是样本统计量。如果是比例问题用 p 表示总体比例，如果是均值问题用 μ 表示总体均值。同时，H1 的方向完全由题目语言决定：”是否大于””是否增加”对应 >，”是否小于””是否下降”对应 <，”是否不同””是否改变”对应 ≠。

There is a key detail when writing hypotheses: the parameter in H0 and H1 must be a population parameter, not a sample statistic. Use p for a population proportion and μ for a population mean. At the same time, the direction of H1 is entirely determined by the language of the question: “is it greater than” or “has it increased” gives >, “is it less than” or “has it decreased” gives <, and “is it different” or “has it changed” gives ≠.

AQA 评分时，假设写错方向（比如该用单尾却写成双尾）通常会直接扣掉后续所有比较步骤的分数，因为后面所有的计算都建立在错误的假设之上。因此，动笔计算之前，务必花十秒钟从题目原文中找出决定方向的关键词。

When AQA marks your work, writing the hypothesis in the wrong direction (for example, using a two-tailed test when a one-tailed test is required) usually costs all the marks for the subsequent comparison steps, because every later calculation is built on the wrong hypothesis. Therefore, before you start calculating, always spend ten seconds finding the keyword in the question that decides the direction.

题目关键词 Keyword	H1 方向 Direction
greater than / increased / more than（大于/增加）	p > p0 或 μ > μ0（单尾右）
less than / decreased / fewer（小于/减少）	p < p0 或 μ < μ0（单尾左）
different / changed / not equal（不同/改变）	p ≠ p0 或 μ ≠ μ0（双尾）

三、显著性水平：5% 检验意味着什么 | Significance Levels: What a 5% Test Really Means

显著性水平（Significance Level）用希腊字母 α 表示，是假设检验中最重要的预设参数。它定义了”小到不可思议”的门槛：如果零假设 H0 为真，我们愿意承受多大的错误拒绝风险。AQA 题目中最常见的是 5% 显著性水平，其次是 1% 和 10%。例如”以 5% 的显著性水平检验”，意思是：如果 H0 为真，而我们仍错误地拒绝了它，这种错误的概率被控制在 5% 以内。

The significance level, denoted by the Greek letter α, is the most important preset parameter in hypothesis testing. It defines the threshold of “too unlikely to believe”: if the null hypothesis H0 is true, it sets how much risk of wrongly rejecting it we are willing to accept. The most common significance level in AQA questions is 5%, followed by 1% and 10%. For example, “test at the 5% significance level” means: if H0 were true and we still wrongly rejected it, the probability of that error is capped at 5%.

为什么不用更小的显著性水平呢？因为显著性水平越小，拒绝 H0 的门槛越高，我们越不容易拒绝；但代价是，当 H0 确实是错误的时候，我们也不容易发现它。这就像安检：安检越严格，误伤好人的概率越低（第一类错误小），但漏掉坏人的概率越高（第二类错误大）。所以显著性水平的选择是两类错误之间的权衡。

Why not use an even smaller significance level? Because the smaller the significance level, the higher the bar for rejecting H0, and the less likely we are to reject it; but the price is that when H0 is genuinely wrong, we are also less likely to detect it. This is like airport security: the stricter the screening, the lower the chance of wrongly stopping an innocent passenger (small Type I error), but the higher the chance of letting a real threat through (large Type II error). Choosing a significance level is therefore a trade-off between the two types of error.

在 AQA 考试中，显著性水平通常直接写在题目里，不需要你自己选择。但你必须理解它的含义，因为结论句要体现它：”由于 p 值 0.0207 小于 5% 的显著性水平，我们拒绝 H0″。如果题目要求 1% 显著性水平而你没有重新计算临界值，就会出错 – 显著性水平改变，临界值必须跟着变。

In the AQA exam, the significance level is usually stated directly in the question, so you do not choose it yourself. But you must understand what it means, because the conclusion sentence must reflect it: “Since the p-value 0.0207 is less than the 5% significance level, we reject H0”. If a question requires the 1% significance level and you do not recalculate the critical value, you will make an error: when the significance level changes, the critical value must change with it.

四、单尾与双尾检验：何时用大于号，何时用不等号 | One-Tailed vs Two-Tailed Tests: When to Use Greater-Than and When to Use Not-Equal

单尾检验（One-Tailed Test）只在分布的一侧寻找证据。当题目说”检验硬币是否偏向正面”时，我们只关心正面概率是否大于 0.5，反面概率是否小于 0.5 根本不重要，所以用 H1: p > 0.5，检验只看右尾。反之”检验是否偏向反面”用 H1: p < 0.5，只看左尾。单尾检验的优点是门槛更低、更容易拒绝 H0，因为它把全部显著性水平 α 集中在一侧。

A one-tailed test looks for evidence on only one side of the distribution. When a question says “test whether the coin is biased towards heads”, we only care whether the probability of heads is greater than 0.5; whether the probability of tails is less than 0.5 is irrelevant, so we use H1: p > 0.5 and examine only the right tail. Conversely, “test whether it is biased towards tails” gives H1: p < 0.5, examining only the left tail. The advantage of a one-tailed test is that the bar is lower and rejecting H0 is easier, because the entire significance level α is concentrated on one side.

双尾检验（Two-Tailed Test）用于没有任何方向提示的情况。比如”检验这枚硬币是否公平” – 不公平可能意味着偏向正面，也可能意味着偏向反面，两种方向都要考虑，所以写 H1: p ≠ 0.5。双尾检验的关键陷阱是：5% 的显著性水平要平均分到两条尾巴上，每条尾巴只有 2.5%。很多同学在双尾检验中仍然在单侧用完整的 5% 找临界值，导致临界区域偏大、结论错误。

A two-tailed test is used when there is no directional hint at all. For example, “test whether this coin is fair”: unfair could mean biased towards heads or towards tails, and both directions must be considered, so we write H1: p ≠ 0.5. The key trap in a two-tailed test is that the 5% significance level must be split evenly between the two tails, giving only 2.5% in each tail. Many students still look up the critical value using the full 5% on one side in a two-tailed test, making the critical region too large and the conclusion wrong.

判断单尾还是双尾，最可靠的方法是回到题目原文找方向词。”increase、greater、more than、exceed”都指向单尾右；”decrease、less than、fewer、below”指向单尾左；而”different、changed、fair、consistent with”这类中性的说法指向双尾。如果题目同时给了方向词和”检验是否公平”这种双尾表述，以更具体的那个为准。

The most reliable way to decide between one-tailed and two-tailed is to return to the exact wording of the question. “Increase, greater, more than, exceed” all point to the right tail; “decrease, less than, fewer, below” point to the left tail; and neutral phrasing such as “different, changed, fair, consistent with” points to a two-tailed test. If the question contains both a directional word and a two-tailed phrase such as “test whether it is fair”, follow the more specific one.

五、二项分布检验：从抛硬币到产品合格率 | Binomial Distribution Tests: From Coin Tossing to Quality Control

二项分布检验是 AQA A-Level 数学中最常考的假设检验类型。它的适用条件是：试验结果只有成功与失败两种；每次试验相互独立；成功概率 p 在每次试验中保持不变。模型写作 X ~ B(n, p)，其中 n 是试验次数，X 是成功次数。考试中最经典的例子是抛硬币：一枚硬币被抛 20 次，出现 15 次正面，问这枚硬币是否在 5% 显著性水平下偏向正面。

The binomial distribution test is the most frequently examined type of hypothesis test in AQA A-Level Mathematics. Its conditions are: each trial has only two outcomes, success and failure; the trials are independent; and the success probability p stays the same in every trial. The model is written as X ~ B(n, p), where n is the number of trials and X is the number of successes. The classic exam example is coin tossing: a coin is tossed 20 times and lands heads 15 times; test at the 5% significance level whether the coin is biased towards heads.

完整的解题过程如下。第一步，定义变量：设 X 为 20 次抛掷中正面的次数，X ~ B(20, p)。第二步，写假设：H0: p = 0.5，H1: p > 0.5（单尾右）。第三步，计算在 H0 成立的前提下出现 15 次或更多正面的概率：P(X ≥ 15) = 1 – P(X ≤ 14)。查二项分布累积概率表，当 n = 20、p = 0.5 时 P(X ≤ 14) = 0.9793，所以 P(X ≥ 15) = 1 – 0.9793 = 0.0207。

The complete procedure is as follows. Step one, define the variable: let X be the number of heads in 20 tosses, X ~ B(20, p). Step two, state the hypotheses: H0: p = 0.5, H1: p > 0.5 (right one-tailed). Step three, calculate the probability of observing 15 or more heads assuming H0 is true: P(X ≥ 15) = 1 – P(X ≤ 14). Looking up the binomial cumulative table, with n = 20 and p = 0.5 we have P(X ≤ 14) = 0.9793, so P(X ≥ 15) = 1 – 0.9793 = 0.0207.

第四步，比较：0.0207 < 0.05，小于显著性水平。第五步，下结论：在 5% 显著性水平下，我们有充分证据拒绝 H0，即硬币确实偏向正面。注意结论必须用”in context”（结合题目背景）的语言写出来，不能只说”拒绝零假设”，而要说”有证据表明这枚硬币抛得正面偏多”。

Step four, compare: 0.0207 < 0.05, which is smaller than the significance level. Step five, conclude: at the 5% significance level, there is sufficient evidence to reject H0, meaning the coin is indeed biased towards heads. Note that the conclusion must be written “in context” (linked to the background of the question); you cannot just say “reject the null hypothesis”, you must say “there is evidence that this coin produces more heads than tails”.

六、临界值与临界区域：拒绝边界的计算 | Critical Values and Critical Regions: Calculating the Rejection Boundary

除了直接计算概率，AQA 考试还经常要求你求出临界值（Critical Value）和临界区域（Critical Region）。临界值是临界区域的边界：对于右尾检验，临界值是满足 P(X ≥ c) ≤ α 的最小的 c。回到抛硬币的例子，我们已知 P(X ≥ 15) = 0.0207 ≤ 0.05，再算 P(X ≥ 14)：查表得 P(X ≤ 13) = 0.9423，所以 P(X ≥ 14) = 1 – 0.9423 = 0.0577 > 0.05。因此最小的满足条件的 c 是 15，临界区域为 X ≥ 15。

Besides calculating probabilities directly, AQA exams often ask you to find the critical value and the critical region. The critical value is the boundary of the critical region: for a right-tailed test, it is the smallest c such that P(X ≥ c) ≤ α. Returning to the coin example, we already know P(X ≥ 15) = 0.0207 ≤ 0.05; now calculate P(X ≥ 14): from the table, P(X ≤ 13) = 0.9423, so P(X ≥ 14) = 1 – 0.9423 = 0.0577 > 0.05. Therefore the smallest c satisfying the condition is 15, and the critical region is X ≥ 15.

临界区域把样本结果的所有可能取值分成两部分：落在临界区域内的值会导致拒绝 H0，落在临界区域外的值（称为接受域，Acceptance Region）则不足以拒绝 H0。注意”接受 H0″这个说法其实不太严谨 – 更准确的说法是”没有足够证据拒绝 H0″，因为不拒绝不等于证明 H0 为真，只是样本证据不够强。

The critical region divides all possible sample outcomes into two parts: values inside the critical region lead to rejection of H0, while values outside it (called the acceptance region) do not provide enough evidence to reject H0. Note that the phrase “accept H0” is not quite rigorous; the more accurate phrasing is “there is insufficient evidence to reject H0”, because failing to reject does not prove H0 is true, it only means the sample evidence is not strong enough.

当显著性水平变化时，临界值也会变化。如果上面的硬币例子改用 1% 显著性水平，我们需要找满足 P(X ≥ c) ≤ 0.01 的最小 c。P(X ≥ 17) = 1 – P(X ≤ 16) = 1 – 0.9987 = 0.0013 ≤ 0.01，而 P(X ≥ 16) = 1 – 0.9941 = 0.0059 > 0.01，所以新的临界值是 17，临界区域变为 X ≥ 17。此时观察到 15 次正面就不足以拒绝 H0 了。

When the significance level changes, the critical value changes too. If the coin example above used the 1% significance level, we would need the smallest c with P(X ≥ c) ≤ 0.01. We find P(X ≥ 17) = 1 – P(X ≤ 16) = 1 – 0.9987 = 0.0013 ≤ 0.01, while P(X ≥ 16) = 1 – 0.9941 = 0.0059 > 0.01, so the new critical value is 17 and the critical region becomes X ≥ 17. In this case, observing 15 heads would no longer be enough to reject H0.

七、p 值法：另一种决策路径 | The p-Value Method: An Alternative Decision Path

p 值（p-value）的定义是：在 H0 成立的条件下，观察到当前样本结果或比它更极端的结果的概率。在上面的例子中，观察到 15 次正面，p 值就是 P(X ≥ 15) = 0.0207。p 值法（p-Value Method）的决策规则极其简洁：如果 p 值 < 显著性水平 α，拒绝 H0；如果 p 值 ≥ α，不拒绝 H0。p 值越小，证据越强。

The p-value is defined as: assuming H0 is true, the probability of observing the current sample result or something even more extreme. In the example above, having observed 15 heads, the p-value is P(X ≥ 15) = 0.0207. The decision rule of the p-value method is extremely concise: if the p-value < the significance level α, reject H0; if the p-value ≥ α, do not reject H0. The smaller the p-value, the stronger the evidence.

p 值法和临界值法在逻辑上是完全等价的：p 值 < α 当且仅当样本结果落在临界区域内。两者的区别只是呈现方式不同。在 AQA 考试中，两种方法都被接受，但很多学生觉得 p 值法更直观，因为它只需要一次概率计算和一次比较，而临界值法需要额外的查表步骤。不过要注意：p 值法写结论时仍然要明确写出 p 值与显著性水平的比较过程。

The p-value method and the critical value method are logically equivalent: the p-value < α if and only if the sample result lies inside the critical region. The only difference is the way they are presented. In the AQA exam, both methods are accepted, but many students find the p-value method more intuitive because it needs only one probability calculation and one comparison, whereas the critical value method requires an extra table-lookup step. However, note that when using the p-value method you must still clearly show the comparison between the p-value and the significance level in your conclusion.

一个常见的丢分点：只写出”p 值 = 0.0207″却不写它与 0.05 的比较，或者比较方向写反（写成 0.05 < 0.0207）。AQA 的评分标准通常要求三个要素齐全：p 值的计算、与显著性水平的比较、基于比较的结论。缺任何一个都会扣分。

A common mark-losing mistake: writing only “p-value = 0.0207” without stating the comparison with 0.05, or writing the comparison in the wrong direction (such as 0.05 < 0.0207). AQA marking schemes usually require three elements: the calculation of the p-value, the comparison with the significance level, and a conclusion based on that comparison. Missing any one of them costs marks.

八、正态分布检验：已知方差下的 z 检验 | Normal Distribution Tests: The z-Test with Known Variance

当研究对象是连续型数据且总体服从正态分布时，我们使用基于正态分布的检验。最常见的情形是：总体方差已知（或标准差已知），要对总体均值 μ 做检验。设 X ~ N(μ, σ²)，样本容量为 n，样本均值为 x̄，则检验统计量为 z = (x̄ – μ0) / (σ / √n)，其中 μ0 是 H0 中的假设均值。这个 z 统计量服从标准正态分布 N(0, 1)。

When the data are continuous and the population follows a normal distribution, we use tests based on the normal distribution. The most common situation is: the population variance (or standard deviation) is known, and we want to test the population mean μ. Let X ~ N(μ, σ²), with sample size n and sample mean x̄; then the test statistic is z = (x̄ – μ0) / (σ / √n), where μ0 is the assumed mean in H0. This z statistic follows the standard normal distribution N(0, 1).

标准正态分布的关键临界值必须背熟：单尾 5% 检验对应 z = 1.645；双尾 5% 检验对应 z = ±1.96；单尾 1% 检验对应 z = 2.326；双尾 1% 检验对应 z = ±2.576。这些数值在公式册中有表可查，但考试时间有限，熟练记忆能省下宝贵的查表时间。

You must memorise the key critical values of the standard normal distribution: a one-tailed 5% test corresponds to z = 1.645; a two-tailed 5% test corresponds to z = ±1.96; a one-tailed 1% test corresponds to z = 2.326; and a two-tailed 1% test corresponds to z = ±2.576. These values are available in the formula booklet, but exam time is limited, so memorising them saves precious table-lookup time.

完整例题：某品牌薯片宣称每包净重均值为 150 克，标准差 8 克。质检员随机抽取 50 包，测得平均净重 147 克。在 5% 显著性水平下检验薯片是否装量不足。设 X ~ N(μ, 64)，H0: μ = 150，H1: μ < 150（单尾左）。计算 z = (147 – 150) / (8 / √50) = -3 / 1.131 = -2.65。查表得单尾 5% 临界值为 -1.645，而 -2.65 < -1.645，落在拒绝域内，因此拒绝 H0，有充分证据表明薯片装量确实不足。

Complete worked example: a brand of crisps claims that the mean net weight per bag is 150 grams, with a standard deviation of 8 grams. An inspector randomly selects 50 bags and finds a mean net weight of 147 grams. Test at the 5% significance level whether the bags are underfilled. Let X ~ N(μ, 64), H0: μ = 150, H1: μ < 150 (left one-tailed). Compute z = (147 – 150) / (8 / √50) = -3 / 1.131 = -2.65. From the table, the one-tailed 5% critical value is -1.645; since -2.65 < -1.645, the statistic falls in the rejection region. We therefore reject H0 and conclude there is sufficient evidence that the bags are indeed underfilled.

检验类型 Test Type	5% 临界值 Critical Value	1% 临界值 Critical Value
单尾 One-tailed	1.645	2.326
双尾 Two-tailed	1.960	2.576

九、第一类错误与第二类错误：检验的风险 | Type I and Type II Errors: The Risks of Testing

假设检验不可能永远正确，它存在两类本质不同的错误。第一类错误（Type I Error）：H0 实际上是正确的，但我们错误地拒绝了它。这类错误的概率正好等于显著性水平 α – 这正是 α 的定义。第二类错误（Type II Error）：H0 实际上是错误的，但我们没有拒绝它，接受了错误。第二类错误的概率记作 β，它没有固定数值，需要针对具体的备择参数值单独计算。

Hypothesis testing cannot always be correct; it is subject to two fundamentally different kinds of error. A Type I Error occurs when H0 is actually true but we wrongly reject it. The probability of this error is exactly the significance level α – this is precisely what α means. A Type II Error occurs when H0 is actually false but we fail to reject it, accepting a wrong claim. The probability of a Type II error is denoted β; it has no fixed value and must be calculated separately for each specific alternative parameter value.

两类错误此消彼长：显著性水平 α 越小，第一类错误越少，但第二类错误 β 越多；反之亦然。要想同时减小两类错误，唯一的办法是增大样本容量 n – 样本越大，检验统计量的方差越小，分布越集中，两类错误都会下降。这也是为什么严格的科学实验总是追求大样本。

The two types of error trade off against each other: the smaller the significance level α, the fewer Type I errors but the more Type II errors β, and vice versa. The only way to reduce both types of error simultaneously is to increase the sample size n: the larger the sample, the smaller the variance of the test statistic, the more concentrated the distribution, and the lower both errors become. This is why rigorous scientific experiments always pursue large samples.

	H0 为真 H0 True	H0 为假 H0 False
拒绝 H0 Reject H0	第一类错误（概率 α）Type I Error	正确决策 Correct
不拒绝 H0 Do Not Reject	正确决策 Correct	第二类错误（概率 β）Type II Error

AQA 对两类错误的考查方式通常是概念辨析题：给出一个情境，问”如果 H0 实际上为真而我们拒绝了她，这叫什么错误？概率是多少？”答案就是”第一类错误，概率等于显著性水平 5%”；如果问”如何减少第二类错误”，标准答案是”增大样本容量”。

AQA usually examines the two types of error through concept-discrimination questions: given a scenario, they ask “if H0 is actually true and we reject it, what is this error called and what is its probability?” The answer is “a Type I error, with probability equal to the significance level, 5%”. If they ask “how can the Type II error be reduced”, the standard answer is “increase the sample size”.

十、AQA 考试解题模板：五步拿到满分 | AQA Exam Answer Framework: Five Steps to Full Marks

把前面的内容整合起来，AQA 假设检验大题的完整解题流程可以总结为五步模板。第一步：定义随机变量并说明分布，如”设 X 为 20 次抛掷中的正面次数，X ~ B(20, p)”。第二步：写出 H0 与 H1，参数用总体参数，方向与题目关键词一致。第三步：计算检验统计量或概率，二项分布用累积概率表，正态分布用 z 统计量。

Putting everything together, the complete procedure for an AQA hypothesis testing question can be summarised as a five-step template. Step one: define the random variable and state its distribution, for example “let X be the number of heads in 20 tosses, X ~ B(20, p)”. Step two: write out H0 and H1, using population parameters, with the direction matching the keywords of the question. Step three: calculate the test statistic or probability, using the cumulative binomial table for binomial tests and the z statistic for normal tests.

第四步：比较。把 p 值与显著性水平比较，或把检验统计量与临界值比较，明确写出不等号方向。第五步：下结论。先说统计结论（拒绝或不拒绝 H0），再用题目背景语言复述一遍（”有证据表明……”），最后可补充”在 5% 显著性水平下”字样。这五步全部写清楚，一道 5 分的题基本可以拿满。

Step four: compare. Compare the p-value with the significance level, or the test statistic with the critical value, explicitly writing the direction of the inequality. Step five: conclude. First state the statistical conclusion (reject or do not reject H0), then restate it in the language of the question’s context (“there is evidence that…”), and finally add “at the 5% significance level”. If you write all five steps clearly, you can basically secure full marks on a 5-mark question.

历年 AQA 学生最常见的失分点有三个：第一，结论没有结合题目背景，只写”拒绝 H0″；第二，把”不拒绝 H0″误写成”接受 H0 为真”；第三，二项分布检验中把 P(X ≥ 15) 错算成 P(X = 15)（漏掉”或更极端”）。此外，检查答案时务必确认 H1 的方向与结论一致：如果 H1 是 p > 0.5，结论必须是”正面偏多”，不能写成”硬币不公平”这种含糊说法。

There are three most common mark-losing mistakes among AQA students over the years. First, the conclusion is not written in context, only “reject H0”. Second, “do not reject H0” is wrongly written as “accept H0 as true”. Third, in binomial tests, P(X ≥ 15) is miscomputed as P(X = 15), omitting the “or more extreme” part. In addition, when checking your answer, make sure the direction of H1 agrees with the conclusion: if H1 is p > 0.5, the conclusion must be “the coin is biased towards heads”, not a vague statement like “the coin is unfair”.

Summary | 总结

假设检验是 A-Level 数学统计学部分的核心考点，也是 AQA Paper 3 中性价比最高的题型之一。它的本质是用样本数据判断关于总体的说法是否可信，完整流程包括：设定 H0 与 H1（H0 含等号、H1 定方向）、确定显著性水平 α、计算 p 值或检验统计量、与临界值比较、写出结合背景的结论。

Hypothesis testing is a core topic in the statistics section of A-Level Mathematics and one of the highest-value question types in AQA Paper 3. Its essence is using sample data to judge whether a claim about a population is credible. The complete procedure includes: stating H0 and H1 (H0 contains the equals sign, H1 fixes the direction), setting the significance level α, calculating the p-value or test statistic, comparing with the critical value, and writing a conclusion linked to the context.

关键记忆点：单尾检验把 α 集中在一侧，双尾检验把 α 平分到两侧；二项分布检验 X ~ B(n, p) 用累积概率表，正态检验 z = (x̄ – μ0) / (σ / √n)；临界值是满足概率条件的最小整数；第一类错误概率恰为 α，第二类错误只能通过增大样本量来同时压低。掌握五步模板并配合真题练习，假设检验分数可以稳定拿到。

Key points to remember: a one-tailed test concentrates α on one side while a two-tailed test splits α evenly across both sides; binomial tests X ~ B(n, p) use the cumulative probability table while normal tests use z = (x̄ – μ0) / (σ / √n); the critical value is the smallest integer satisfying the probability condition; the Type I error probability is exactly α, and both errors can only be reduced together by increasing the sample size. Master the five-step template, practise with past papers, and you can secure the hypothesis testing marks consistently.

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