📚 Setting the Hypotheses | 假设的设定
In hypothesis testing, setting the hypotheses is the first and most important step. A clearly written null hypothesis H₀ and alternative hypothesis H₁ tell you exactly what you are testing, whether the test is one-tailed or two-tailed, and which parameter is under investigation. Get this step right and the rest of the test - the distribution, the critical region, the conclusion - falls into place almost automatically. Get it wrong and every later mark is at risk.
在假设检验中，设定假设是第一步，也是最重要的一步。清晰写出原假设 H₀ 和备择假设 H₁，能让你明确正在检验什么、检验是单尾还是双尾，以及所研究的参数是什么。这一步做对了，后面的内容——分布的选择、拒绝域的确定、结论的表述——几乎都会水到渠成；这一步做错了，后面每一步的分数都会受到牵连。
1. The Role of Hypotheses in a Test | 假设在检验中的作用
Every hypothesis test compares two competing statements about a population parameter. The null hypothesis represents the default or current belief, while the alternative hypothesis represents the claim we want to find evidence for. The data are then used to decide whether the evidence is strong enough to reject the default position.
每个假设检验都会对总体参数的两个对立陈述进行比较。原假设代表默认或当前的看法，而备择假设代表我们想要寻找证据支持的声明。随后我们利用数据来判断证据是否足够有力，从而拒绝默认的立场。
Edexcel questions often ask you to 'test whether...' or 'investigate whether...'. Your first line of working should always state H₀ and H₁ using correct notation. These two statements carry marks in their own right, usually B1 for a fully correct pair.
Edexcel 题目常要求你“检验是否……”或“调查是否……”。你的第一行解答应当始终使用正确记号写出 H₀ 和 H₁。这两个陈述本身就占分，通常一对完全正确的假设可以拿到 B1。
Notice that the hypotheses are always about the population, never about the particular sample you have collected. The sample provides evidence; the hypotheses describe the population.
请注意，假设永远是关于总体的，而不是关于你所收集的那个具体样本。样本提供证据，假设描述总体。
2. Null Hypothesis H₀ | 原假设 H₀
or H₀: λ = k, where k is the assumed value of the parameter before any data are collected.
原假设总是包含等号。在 Edexcel 评分方案中，通常写成 H₀: p = k、H₀: μ = k 或 H₀: λ = k，其中 k 是在收集数据之前所假定参数取的值。
eanest and safest form for exams is to write the equality version. Examiners accept the equality form in every one-tailed or two-tailed situation, so there is no reason to risk a stricter alternative.
有些教材也允许在单尾检验中写 H₀: p ≤ k 或 H₀: p ≥ k，但考试中最简洁、最安全的形式是写等式版本。无论在单尾还是双尾情形下，考官都接受等号形式，因此没有必要冒额外风险去写更严格的不等式。
tion proportion of students who pass the test on the first attempt. This extra clause costs you nothing and protects against ambiguity.
写得好的原假设还会用文字说明参数的含义。例如：H₀: p = 0.4，其中 p 是首次参加考试即通过的学生总体比例。这句补充说明不费什么力气，却能有效避免歧义。
The null hypothesis is never 'accepted'. If the evidence is not strong enough, we say there is insufficient evidence to reject H₀ - we do not claim that H₀ is true.
原假设永远不会被“接受”。如果证据不够有力，我们就说没有足够的证据拒绝 H₀——而不会声称 H₀ 是真实的。
3. Alternative Hypothesis H₁ | 备择假设 H₁
The alternative hypothesis is the statement you are testing for. It uses a strict inequality: <, > or ≠. It never contains an equals sign, because if it did, it would overlap with H₀ rather than compete with it.
备择假设是你正在寻找证据支持的陈述。它使用严格不等式：<、> 或 ≠。它绝不含等号，因为一旦含等号，它就会与原假设重叠，而不是与之对立。
The alternative hypothesis is what you conclude when the test result is significant. If you reject H₀, then H₁ is the statement you carry forward in your conclusion sentence.
当检验结果显著时，你得出结论所依据的就是备择假设。如果你拒绝了 H₀，那么在结论句中你推进的就是 H₁ 这一陈述。
Three possible forms of H₁ are used at A Level:
A Level 中会用到 H₁ 的三种可能形式：
- H₁: p > k - a one-tailed test in the upper direction; the parameter is greater than k.
- H₁: p < k - a one-tailed test in the lower direction; the parameter is less than k.
- H₁: p ≠ k - a two-tailed test; the parameter is simply different from k.
- H₁: p > k ——均值为上方向的单尾检验，参数大于 k。
- H₁: p < k ——方向向下的单尾检验，参数小于 k。
- H₁: p ≠ k ——双尾检验，参数只是与 k 不同。
4. One-Tailed and Two-Tailed Tests | 单尾检验与双尾检验
The wording of the question determines the direction of the alternative hypothesis. Words such as 'increased', 'more than', 'greater than', 'has risen' or 'improved' point to an upper one-tailed test, so H₁ uses >. Words such as 'decreased', 'less than', 'lower than' or 'reduced' point to a lower one-tailed test, so H₁ uses <.
题目的措辞决定了备择假设的方向。像“增加”“多于”“大于”“上升”或“改善”这样的词，指向方向向上的单尾检验，因此 H₁ 用 >。像“减少”“少于”“低于”或“降低”这样的词，指向方向向下的单尾检验，因此 H₁ 用 <。
Words such as 'changed', 'different from', 'is there evidence that the value differs', or 'has altered' point to a two-tailed test, so H₁ uses ≠. Any question that mentions a change without a stated direction is two-tailed.
像“改变”“与……不同”“是否有证据表明该值有差异”或“发生变化”这样的词，指向双尾检验，因此 H₁ 用 ≠。任何提到变化但没有指明方向的问题都是双尾检验。
A useful habit is to read the final sentence of the question twice and underline the key verb. In most Edexcel questions, the direction is decided by one word, and that one word decides the whole tail structure of the test.
一个有用的习惯是把题目最后一句读两遍，并划出关键动词。在大多数 Edexcel 题目中，方向由一个词决定，而这一个词就决定了整个检验的尾部结构。
| Question wording | Hypothesis form | Test type |
| has increased / is greater / is more | H₁: θ > k | upper one-tailed |
| has decreased / is lower / is less | H₁: θ < k | lower one-tailed |
| has changed / is different / differs | H₁: θ ≠ k | two-tailed |
5. Choosing the Correct Parameter | 选择正确的参数
The parameter in the hypotheses must match the distribution being used. If the random variable is modelled by a binomial distribution B(n, p), the parameter is p, the population proportion of successes.
假设中的参数必须与所用的分布相匹配。如果随机变量用二项分布 B(n, p) 建模，参数就是 p，即总体中成功的比例。
If the random variable is modelled by a normal distribution N(μ, σ²), the parameter of interest is normally μ, the population mean. Occasionally a question tests the population standard deviation σ instead, in which case the hypotheses are written in terms of σ.
如果随机变量用正态分布 N(μ, σ²) 建模，通常关注的参数是 μ，即总体均值。偶尔题目会改为检验总体标准差 σ，这时假设要用 σ 来写。
If the random variable is modelled by a Poisson distribution Po(λ), the parameter is λ, the population mean rate of occurrence.
如果随机变量用泊松分布 Po(λ) 建模，参数就是 λ，即总体平均发生率。
For a geometric distribution Geo(p), the parameter is again p, the probability of success on each trial.
对于几何分布 Geo(p)，参数同样是 p，即每次试验成功的概率。
6. Parameters vs Sample Statistics | 总体参数与样本统计量
Do not put a sample statistic such as x̄ or p̂ hypotheses. Hypotheses are about population parameters, not sample results. Writing H₀: p̂ = 0.5 is a classic error that loses the notation mark immediately.
不要把样本统计量如 x̄ 或 p̂ 放进假设中。假设是关于总体参数的，而不是样本结果。写出 H₀: p̂ = 0.5 是典型错误，会立刻丢掉记号分。
The logic behind this is simple. The whole point of the test is to compare what the sample tells you with what the null hypothesis claims about the population. If the sample value appeared in the hypothesis, there would be nothing left to test.
这背后的逻辑很简单。检验的全部意义就在于把样本告诉你的信息与原假设关于总体的断言作比较。如果假设里出现了样本值，那就没有什么可检验的了。
A clean set of hypotheses therefore looks like this:
因此，一组干净的假设应当写成这样：
H₀: p = 0.5 H₁: p ≠ 0.5
where p is the population proportion, not the observed proportion p̂.
7. Notation for Different Distributions | 不同分布的记号
Being fluent in the notation saves time and prevents errors. The table below summarises the standard forms used across Edexcel and AQA papers.
熟练使用记号既能节省时间，也能避免出错。下表总结了 Edexcel 与 AQA 试卷中使用的标准形式。
| Model | Parameter | Typical H₀ / H₁ pair |
| X ~ B(n, p) | p | H₀: p = 0.3, H₁: p > 0.3 |
| X ~ N(μ, σ²) | μ | H₀: μ = 25, H₁: μ ≠ 25 |
| X ~ Po(λ) | λ | H₀: λ = 4, H₁: λ < 4 |
| X ~ Geo(p) | p | H₀: p = 1/6, H₁: p ≠ 1/6 |
The Greek letters μ and λ are reserved for population parameters. Never use them for sample values; use x̄ and the sample mean or sample count instead.
希腊字母 μ 和 λ 专门用于总体参数。绝不要用它们表示样本值；样本均值或样本计数应使用 x̄ 等记号。
8. Worked Example 1: Binomial Proportion | 例题一：二项分布比例
A six-sided die is rolled 60 times and a 2 appears 15 times. A student claims the die is biased in favour of 2. Test, at the 5% level, whether there is evidence that the probability of rolling a 2 is greater than 1/6.
一枚六面骰子被掷 60 次，其中出现 2 的次数为 15 次。一名学生声称这枚骰子偏向于掷出 2。在 5% 显著性水平下，检验是否有证据表明掷出 2 的概率大于 1/6。
Step 1: define the random variable and its distribution. Let X be the number of 2s in 60 rolls, so X ~ B(60, p).
第一步：定义随机变量及其分布。设 X 为 60 次投掷中出现 2 的次数，则 X ~ B(60, p)。
Step 2: write the hypotheses. The word 'greater' signals an upper one-tailed test:
第二步：写出假设。“greater”（大于）一词提示这是方向向上的单尾检验：
H₀: p = 1/6 H₁: p > 1/6
Note that the observed proportion 15 ÷ 60 = 0.25 does not appear in the hypotheses at all.
其中 p 是单次投掷出现 2 的总体概率。注意，观测到的比例 15 ÷ 60 = 0.25 完全没有出现在假设中。
9. Worked Example 2: Normal Mean | 例题二：正态分布均值
A factory produces batteries whose lifetimes are normally distributed with mean μ hours and standard deviation 12 hours. A new process is introduced and the manufacturer wants to know whether the mean lifetime has changed. A sample of 25 batteries gives a sample mean of 96.4 hours.
某工厂生产的电池寿命服从正态分布，均值为 μ 小时，标准差为 12 小时。工厂引入了一种新工艺，想要知道平均寿命是否发生了变化。抽取 25 节电池的样本，样本均值为 96.4 小时。
Step 1: the sample mean follows X̄ ~ N(μ, 12² ÷ 25), so the standard error is 12 ÷ √25 = 2.4.
第一步：样本均值服从 X̄ ~ N(μ, 12² ÷ 25)，因此标准误为 12 ÷ √25 = 2.4。
Step 2: the word 'changed' has no direction, so the test is two-tailed:
第二步：“changed”（变化）没有指明方向，因此为双尾检验：
H₀: μ = 100 H₁: μ ≠ 100
question stem. Both tails must be considered, and the critical values are found at the 2.5% points of the standard normal distribution, giving approximately ±1.96 for a 5% test.
假定题干中说明原来的平均寿命为 100 小时。此时必须考虑两侧尾部，关键值取自标准正态分布的 2.5% 分位点，在 5% 检验下约为 ±1.96。
10. Worked Example 3: Poisson Mean | 例题三：泊松分布均值
3.5. After a new advertising campaign, the manager believes the rate has increased. In a 10-minute period, 47 calls are recorded.
某呼叫中心每分钟接到电话的平均速率为 λ。历史上 λ = 3.5。在新的广告宣传活动之后，经理认为来电速率有所上升。在 10 分钟的时间段内记录到 47 个来电。
Step 1: let X be the number of calls in 10 minutes, so X ~ Po(10λ). Taking the historical rate as the null value gives X ~ Po(35) under H₀.
第一步：设 X 为 10 分钟内的来电数，则 X ~ Po(10λ)。把历史速率作为原假设取值，则在 H₀ 下 X ~ Po(35)。
Step 2: 'increased' signals an upper one-tailed test:
第二步：“increased”（上升）提示方向向上的单尾检验：
H₀: λ = 3.5 H₁: λ > 3.5
he parameter used in the calculation is 10λ = 35. Mixing these two scales is a frequent source of lost marks.
注意这里如何处理时间区间。假设以每分钟的速率书写，但分布针对的是 10 分钟的总数，因此计算中所用的参数为 10λ = 35。混淆这两个尺度是失分的常见原因。
11. How to Write the Hypotheses in the Exam | 考试中如何书写假设
Follow a fixed routine so that the first line of every hypothesis test answer is automatic.
遵循固定的流程，让每一道假设检验题的第一行都成为自动反应。
- Write 'Let X be ...' and give the distribution with the parameter symbol still in place.
- State H₀ and H₁ on one line, in that order, with H₁ on the right.
- Add a short clause defining the parameter in the context of the question.
- Check that H₀ contains = and H₁ contains one of <, >, ≠.
- Check that no sample value appears in either statement.
- 写出“设 X 为……”，并给出分布，参数仍以符号表示。
- 在同一行写出 H₀ 与 H₁，顺序如此，H₁ 在后。
- 用一句简短的话结合题目情境说明参数的含义。
- 检查 H₀ 中含有 =，H₁ 中含有 <、>、≠ 之一。
- 检查两个陈述中都没有出现样本值。
0.25, H₁: p > 0.25, where p is the population proportion of customers who renew their subscription.
标准答案的一行写法如下：H₀: p = 0.25，H₁: p > 0.25，其中 p 是续订客户的总体比例。
12. Common Mistakes and Examiner Tips | 常见错误与考官提示
Most lost marks in this topic come from a small number of repeated errors. Check your work against this list before moving on to the calculation.
本主题的大部分失分都来自少数几个反复出现的错误。在进入计算之前，先对照下面这份清单检查你的作答。
- Using p̂ or x̄ in the hypotheses instead of p or μ.
- Putting an equals sign in H₁, for example H₁: p = 0.3.
- Putting ≠ or < or > in H₀ when the mark scheme expects a single value.
- Choosing the wrong tail direction for a one-tailed test.
- Failing to define what the parameter represents in context.
- Testing the wrong parameter, for example using p when the distribution is Poisson.
- Forgetting that two-tailed tests split the significance level across both tails.
- 在假设中使用 p̂ 或 x̄，而不是 p 或 μ。
- 在 H₁ 中写了等号，例如 H₁: p = 0.3。
- 在评分方案期望单一取值时，却在 H₀ 中写了 ≠、< 或 >。
- 单尾检验的尾部方向选错。
- 没有结合情境说明参数代表什么。
- 检验了错误的参数，例如分布是泊松分布却使用 p。
- 忘记双尾检验要把显著性水平分配到两侧尾部。
The single most valuable habit is to write the hypotheses before doing any arithmetic. Once H₀ and H₁ sit correctly at the top of your page, you always know which tail you are working in, and the conclusion sentence writes itself from the alternative hypothesis.
最有价值的习惯就是在做任何算术之前先写出假设。只要 H₀ 与 H₁ 正确地写在你答题纸的最上方，你就始终清楚自己在哪一侧尾部工作，结论句也会依据备择假设自然写出来。
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