A-Level Further Mathematics: How to Correctly Set Up Null and Alternative Hypotheses | A-Level进阶数学：如何正确建立原假设与备择假设

📚 A-Level Further Mathematics: How to Correctly Set Up Null and Alternative Hypotheses | A-Level进阶数学：如何正确建立原假设与备择假设

Hypothesis testing is one of the most examined topics in Edexcel A-Level Further Mathematics, appearing in both Paper 1 and Paper 2 of the Further Statistics modules. Many students lose marks not because they cannot perform the calculations, but because they set up the hypotheses incorrectly from the very beginning. This article will guide you through the correct logic, notation, and conventions for writing null and alternative hypotheses that match Edexcel marking schemes.

假设检验是Edexcel A-Level进阶数学中考查频率最高的内容之一，在Further Statistics模块的Paper 1和Paper 2中都会出现。许多学生失分并非因为不会计算，而是因为从一开始就错误地建立了假设。本文将系统地讲解如何按照Edexcel评分标准，正确书写原假设与备择假设的逻辑、符号和惯例。

1. What Are Null and Alternative Hypotheses? | 什么是原假设与备择假设？

The null hypothesis, denoted H₀, is a statement of no effect, no change, or no difference. It always contains an equality sign, which may be =, ≤, or ≥. The alternative hypothesis, denoted H₁, is the statement you are trying to find evidence for. It can be ≠, >, or <.

原假设（记作H₀）表示“无效应、无变化或无差异”的陈述，它必须包含等号，可以是 =、≤ 或 ≥。备择假设（记作H₁）是你试图寻找证据支持的陈述，可以是 ≠、> 或 <。

In essence, H₀ represents the status quo or the claim being tested, while H₁ represents the claim you suspect might be true based on the data.

从本质上讲，H₀代表现状或被检验的主张，而H₁代表你基于数据怀疑可能成立的另一种主张。

For example, if a coin manufacturer claims their coin is fair, you would set H₀: p = 0.5, where p is the probability of heads. If you suspect the coin is biased towards heads, you would set H₁: p > 0.5.

例如，若硬币制造商声称硬币是公平的，你应设 H₀: p = 0.5，其中p表示正面朝上的概率。若你怀疑硬币偏向正面，则应设 H₁: p > 0.5。

2. Why Must H₀ Always Contain ‘=’? | 为什么H₀必须始终包含“=”？

The reason H₀ must contain an equality is mathematical: the sampling distribution is calculated assuming H₀ is true. For example, if you are testing a binomial proportion, you substitute p = 0.5 into X ~ B(n, p) to compute probabilities. Without a fixed value, no probability can be computed.

H₀必须包含等号的原因在于数学计算：抽样分布是在假设H₀为真的前提下建立的。例如，检验二项分布的比例时，你需要将 p = 0.5 代入 X ~ B(n, p) 来计算概率。若无固定值，则无法计算任何概率。

This is why the exam board treats H₀ as the “working hypothesis” — it is the value you plug into the distribution. You never plug H₁ into the calculation; you only use it to decide the direction of the rejection region.

这也是为什么考试局将H₀视为“工作假设”——它是你代入分布进行计算的值。你从不将H₁代入计算，只使用它来决定拒绝域的方向。

Consequently, if the problem states “test whether the mean has decreased,” you write H₀: μ = 10 and H₁: μ < 10, not H₀: μ > 10. Equality always stays on the null side.

因此，若题目要求“检验均值是否下降”，你应写 H₀: μ = 10 和 H₁: μ < 10，而不是 H₀: μ > 10。等号始终保留在原假设一侧。

3. One-Tailed vs Two-Tailed Tests | 单尾检验与双尾检验

There are three possible forms for the alternative hypothesis:

备择假设有三种可能的形式：

H₁: p > p₀ (one-tailed, upper-tail test) — used when you suspect an increase.

H₁: p > p₀（单尾，上尾检验）——当怀疑比例增加时使用。
H₁: p < p₀ (one-tailed, lower-tail test) — used when you suspect a decrease.

H₁: p < p₀（单尾，下尾检验）——当怀疑比例减少时使用。
H₁: p ≠ p₀ (two-tailed test) — used when you suspect a change in either direction.

H₁: p ≠ p₀（双尾检验）——当怀疑比例在任一方向发生变化时使用。

In a two-tailed test, the significance level is split equally between the two tails. For example, at the 5% significance level, each tail carries 2.5%.

在双尾检验中，显著性水平被平均分配在两个尾端。例如，在5%显著性水平下，每个尾端各占2.5%。

The key is to read the wording carefully. “Test whether the proportion has changed” implies two-tailed. “Test whether the proportion has increased” implies one-tailed upper. “Test whether the proportion has decreased” implies one-tailed lower.

关键在于仔细阅读题意。“检验比例是否发生变化”暗示双尾检验。“检验比例是否增加”暗示单尾上尾检验。“检验比例是否减少”暗示单尾下尾检验。

4. Common Symbols and Notation | 常用符号与记号

In Edexcel further mathematics, you will encounter the following notation:

在Edexcel进阶数学中，你会遇到以下记号：

p — population proportion in a binomial test | p — 二项检验中的总体比例
λ (lambda) — population mean in a Poisson test | λ（拉姆达）— 泊松检验中的总体均值
μ (mu) — population mean in a normal test | μ（缪）— 正态检验中的总体均值
σ² (sigma squared) — population variance | σ² — 总体方差
ρ (rho) — population correlation coefficient | ρ — 总体相关系数

When writing hypotheses, always define the parameter first. For example:

书写假设时，务必先定义参数。例如：

Let p be the probability that a randomly selected student passes the exam.
H₀: p = 0.7, H₁: p > 0.7

This definition of the parameter is worth marks in the exam and prevents ambiguity. A vague hypothesis without a parameter definition will not receive full credit.

参数的定义在考试中是得分点，且能避免歧义。未定义参数的模糊假设无法获得满分。

5. Step-by-Step: Setting Up Hypotheses for a Binomial Test | 分步指导：二项检验的假设建立

Let us work through a complete example. A factory claims that at most 10% of its components are defective. A quality inspector finds 15 defective components in a sample of 100.

让我们完整地分析一个例题。某工厂声称其产品中最多有10%有缺陷。质检员在100件样本中发现15件有缺陷。

Step 1: Define the parameter. Let p represent the true proportion of defective components.

步骤1：定义参数。设p表示有缺陷产品的真实比例。

Step 2: Identify the claim. The factory claims p ≤ 0.10. This claim contains an inequality, so it is tested against a null hypothesis that contains equality.

步骤2：识别主张。工厂声称 p ≤ 0.10。该主张包含不等式，因此需要与一个包含等号的原假设进行对照检验。

Step 3: Write H₀: p = 0.10 and H₁: p > 0.10, because the inspector suspects the true proportion is greater than claimed.

步骤3：写出 H₀: p = 0.10 和 H₁: p > 0.10，因为质检员怀疑真实比例高于工厂声称的值。

Step 4: State the distribution under H₀: X ~ B(100, 0.10), where X is the number of defective components.

步骤4：说明H₀下的分布：X ~ B(100, 0.10)，其中X为有缺陷产品数量。

Step 5: Calculate P(X ≥ 15) = 0.0726. At the 5% significance level, since 0.0726 > 0.05, we do not reject H₀.

步骤5：计算P(X ≥ 15) = 0.0726。在5%显著性水平下，由于0.0726 > 0.05，我们不拒绝H₀。

Conclusion: There is insufficient evidence to conclude that the defect rate exceeds 10%.

结论：没有足够证据表明缺陷率超过10%。

6. Setting Up Hypotheses for a Poisson Test | 泊松检验的假设建立

For Poisson tests, the parameter is λ, the mean number of occurrences in a fixed interval. Suppose a hospital claims the average number of emergency admissions per hour is 5. The hospital manager suspects the rate has increased.

对于泊松检验，参数是λ，即固定时间间隔内的平均发生次数。假设某医院声称每小时急诊收治人数平均为5人。医院管理者怀疑该比率已上升。

The hypotheses would be:

假设应写为：

Let λ be the mean number of emergency admissions per hour.
H₀: λ = 5, H₁: λ > 5

Under H₀, the distribution is X ~ Po(5), and you would calculate the probability of observing your sample result or something more extreme in the upper tail.

在H₀下，分布为X ~ Po(5)，你需要计算在样本中出现当前结果或更极端结果的概率，即上尾概率。

If instead the manager suspected the rate had decreased, you would write H₁: λ < 5 and calculate the lower-tail probability.

若管理者怀疑比率下降，则应写 H₁: λ < 5 并计算下尾概率。

7. Normal Distribution Tests: When σ² Is Known | 正态检验：当σ²已知时

When testing a population mean μ using a normal distribution with known variance, the hypotheses follow the same pattern. The test statistic is calculated using the sample mean and the standard error.

当使用方差已知的正态分布检验总体均值μ时，假设的写法遵循相同模式。检验统计量使用样本均值和标准误计算。

For example, a machine fills bottles with 500 ml of juice. The standard deviation is known to be 4 ml. A consumer group claims the machine is under-filling.

例如，一台机器灌装500ml果汁。已知总体标准差为4ml。一个消费者团体声称该机器灌装量不足。

Let μ be the true mean fill volume.
H₀: μ = 500, H₁: μ < 500

The test statistic is z = (x̄ − 500) / (4/√n). If this z-value is less than the critical value at the chosen significance level, we reject H₀.

检验统计量为 z = (x̄ − 500) / (4/√n)。若该z值小于所选显著性水平下的临界值，则拒绝H₀。

Note that the direction of H₁ determines the critical region. In a lower-tail test, the critical region is z < −zₐ. In an upper-tail test, it is z > zₐ. In a two-tailed test, both tails are used.

注意，H₁的方向决定了临界区域的方向。在下尾检验中，临界区域为z < −zₐ。在上尾检验中，临界区域为z > zₐ。在双尾检验中，两个尾端均被使用。

8. Classic Mistakes and How to Avoid Them | 常见错误及避免方法

Even strong students frequently make the same errors. Here are the most common ones:

即使优秀的学生也常犯同样的错误。以下是最常见的几种：

Mistake 1: Putting the claim in H₀. If the question says “test whether the mean has increased,” some students write H₀: μ > 10 and H₁: μ = 10. This is completely wrong. H₀ must always contain “=”.

错误1：将主张写入H₀。若题目说“检验均值是否增加”，有些学生写H₀: μ > 10 和 H₁: μ = 10。这完全错误。H₀必须始终包含“=”。
Mistake 2: Wrong direction in H₁. The inequality sign in H₁ must match the wording: increase means >, decrease means <, change means ≠.

错误2：H₁方向错误。H₁中的不等号必须与题意匹配：增加用>，减少用<，变化用≠。
Mistake 3: Not defining the parameter. Writing “H₀: p = 0.2” without stating what p represents loses method marks.

错误3：未定义参数。不说明p代表什么就直接写“H₀: p = 0.2”会丢失方法分。
Mistake 4: Confusing p-value with significance level. Remember: the p-value is a probability computed from data; the significance level is chosen in advance.

错误4：混淆p值与显著性水平。记住：p值是根据数据计算出的概率；显著性水平是事先选定的。
Mistake 5: Forgetting to state the distribution under H₀. Always write X ~ B(n, p₀) or X ~ Po(λ₀) explicitly — this is a mark in itself.

错误5：忘记说明H₀下的分布。务必明确写出X ~ B(n, p₀) 或 X ~ Po(λ₀)——这本身就是得分点。

9. Worked Example: Full Mark Answer | 完整示例：满分答案

The following is a full exam-style question with a model answer:

下面是一道完整的考试风格题目及模范答案：

Question: A pharmaceutical company claims that a new drug cures 70% of patients. A doctor believes the cure rate is lower. In a trial of 20 patients, 11 are cured. Test at the 5% significance level.

题目：某制药公司声称一种新药能治愈70%的患者。一位医生认为治愈率更低。在一项20名患者的试验中，11人被治愈。在5%显著性水平下进行检验。

Solution:

解答：

Let p be the true proportion of patients cured by the drug.

设p为该药治愈患者的真实比例。

H₀: p = 0.70, H₁: p < 0.70

Under H₀, X ~ B(20, 0.70), where X is the number of patients cured.

在H₀下，X ~ B(20, 0.70)，其中X为被治愈的患者人数。

P(X ≤ 11) = P(X = 0) + P(X = 1) + … + P(X = 11)

Using the binomial CDF: P(X ≤ 11) = 0.1133.

使用二项分布累积函数：P(X ≤ 11) = 0.1133。

Since 0.1133 > 0.05, we do not reject H₀.

由于0.1133 > 0.05，我们不拒绝H₀。

There is insufficient evidence at the 5% significance level to support the doctor’s claim that the cure rate is lower than 70%.

在5%显著性水平下，没有足够证据支持医生关于治愈率低于70%的主张。

10. Critical Values vs P-Values: Which Method to Use? | 临界值法与p值法：应该用哪种？

Edexcel accepts both methods, but they require different written conclusions.

Edexcel对两种方法都接受，但写结论的方式不同。

Method \| 方法	Decision Rule \| 决策规则	Conclusion \| 结论
Critical value \| 临界值	If test statistic falls in critical region → reject H₀. 若检验统计量落入临界区域 → 拒绝H₀。	State “reject/do not reject H₀” and interpret in context. 写出“拒绝/不拒绝H₀”并结合上下文解释。
P-value \| p值	If p-value < significance level → reject H₀. 若p值 < 显著性水平 → 拒绝H₀。	Same as above. 同上。

Whichever method you use, always state the conclusion in the context of the question, not just as “reject H₀.” For example, “There is sufficient evidence that the mean waiting time has decreased.”

无论使用哪种方法，务必结合题目情境给出结论，而不仅仅是“拒绝H₀”。例如：“有充分证据表明平均等待时间已减少。”

11. Exam Tips Specific to Edexcel Further Statistics | 针对Edexcel进阶统计的考试技巧

Based on examiner reports, the following tips will help you avoid losing easy marks:

根据考官报告，以下技巧将帮助你避免丢失容易获得的分数：

Always define parameters before writing hypotheses. This is usually worth one mark in part (a).

写假设前务必定义参数。这通常在(a)部分占一分。
Match the significance level to the context. If the question says “at the 1% level,” you must use 0.01, not 0.05.

将显著性水平与题目匹配。若题目说“在1%水平下”，你必须使用0.01而非0.05。
In two-tailed tests, remember to halve the significance level. For a 5% two-tailed test, the critical region is split as 2.5% in each tail.

在双尾检验中，记得将显著性水平对半分。对于5%双尾检验，每个尾端的临界区域各占2.5%。
Use “insufficient evidence” instead of “accept H₀.” The exam board prefers “do not reject H₀” because we cannot prove H₀ to be true.

用“证据不足”代替“接受H₀”。考试局更偏好“不拒绝H₀”，因为我们无法证明H₀为真。
For continuous distributions, write the inequality carefully. In normal tests, P(X ≥ c) = P(X > c) because the distribution is continuous, but in binomial tests, this equality does not hold.

对于连续分布，注意不等号的写法。在正态检验中P(X ≥ c) = P(X > c)，因为分布是连续的；但在二项检验中，此等式不成立。

12. Final Summary | 最终总结

To write hypotheses correctly in every situation, follow this checklist:

要确保在任何情况下正确书写假设，请遵循以下清单：

By following these five steps, you eliminate the most common source of lost marks in hypothesis testing questions. The logic is simple: H₀ is the conservative default, H₁ is the claim you seek evidence for, and every calculation flows from H₀.

遵循这五步，你将消除假设检验题中最大的失分来源。逻辑很简单：H₀是保守的默认假设，H₁是你寻求证据支持的主张，所有计算都从H₀出发。

Practice writing hypotheses for every past paper question you attempt. Within a few hours, you will find it becomes automatic — and this skill alone could be worth 5 to 7 marks on your final exam.

在练习每道历年真题时都有意识地书写假设。几小时后，你会发现这已形成本能——而这一技能本身就值最终考试中的5到7分。

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Method \| 方法	Decision Rule \| 决策规则	Conclusion \| 结论
Critical value \| 临界值	If test statistic falls in critical region → reject H₀. 若检验统计量落入临界区域 → 拒绝H₀。	State “reject/do not reject H₀” and interpret in context. 写出“拒绝/不拒绝H₀”并结合上下文解释。
P-value \| p值	If p-value < significance level → reject H₀. 若p值 < 显著性水平 → 拒绝H₀。	Same as above. 同上。

A-Level Further Mathematics: How to Correctly Set Up Null and Alternative Hypotheses | A-Level进阶数学：如何正确建立原假设与备择假设