📚 Hypothesis Testing for OCR GCSE Maths | OCR GCSE数学假设检验考点精讲
Hypothesis testing is a key topic in the OCR GCSE Maths Statistics component. It allows you to use probability to decide whether a claim about a population is likely to be true, based on evidence from a sample. This revision guide will take you through the essential concepts, from setting up hypotheses to interpreting p‑values and critical regions, using clear binomial examples that mirror exam‑style questions.
假设检验是 OCR GCSE 数学统计部分的核心考点。它让你能够利用概率，根据样本证据来判断关于总体的某个声明是否很可能成立。本考点精讲将带你梳理从建立假设到解读 p 值和拒绝域的所有核心概念，并通过清晰的二项分布示例，还原考试风格的题目。
1. What Is Hypothesis Testing? | 什么是假设检验？
Hypothesis testing is a method for testing a claim or theory about a population parameter. You begin with a belief (the null hypothesis) and look for evidence in the sample data that contradicts it. If the sample result is highly unlikely under that belief, you conclude that the belief is probably wrong.
假设检验是检验关于总体参数的说法或理论的方法。你从一个信念（原假设）出发，然后在样本数据中寻找与之矛盾的证据。如果在该信念下样本结果出现的概率极低，你就可以推断该信念很可能是错误的。
In GCSE Maths, hypothesis tests are nearly always about a binomial probability p, such as whether a coin is fair or whether a new treatment improves recovery rates. You assume a specific value for p under the null hypothesis, then calculate probabilities based on that assumption.
在 GCSE 数学中，假设检验几乎总是围绕二项分布的概率 p 展开，例如硬币是否公平，或者新疗法是否能提升康复率。你在原假设下假定 p 取某个特定值，然后基于该假定计算概率。
2. Null and Alternative Hypotheses | 原假设与备择假设
The null hypothesis, written H₀, is the statement you assume to be true at the start. It usually represents ‘no change’ or ‘no effect’. In binomial contexts, it is typically H₀: p = [a given value], for example H₀: p = 0.5.
原假设记为 H₀，是你在开始时假定为真的陈述。它通常代表“没有变化”或“没有效果”。在二项分布背景下，原假设一般写作 H₀: p = [给定值]，例如 H₀: p = 0.5。
The alternative hypothesis, H₁, is what you might conclude if the evidence is strong enough. It can be two‑tailed: H₁: p ≠ 0.5, or one‑tailed: H₁: p < 0.5 or H₁: p > 0.5. The type of H₁ affects where you look for extreme results.
备择假设记为 H₁，如果证据足够充分，这便是你可能会得出的结论。它可以是双尾的：H₁: p ≠ 0.5，也可以是单尾的：H₁: p < 0.5 或 H₁: p > 0.5。H₁ 的类型会影响你在哪里寻找极端结果。
3. Significance Level α and Critical Region | 显著性水平 α 与拒绝域
The significance level, denoted by α, is the threshold probability that determines when we reject H₀. It is usually chosen before the test, with common values being 5% (0.05) or 1% (0.01). In OCR GCSE exams, α is often given in the question.
显著性水平用 α 表示，是决定我们何时拒绝原假设的概率门槛。通常在检验前选定，常用值为 5% (0.05) 或 1% (0.01)。在 OCR GCSE 考试中，α 通常会直接给出。
The critical region is the set of outcomes that are so extreme that, if the test statistic falls into it, we reject H₀. The boundary value that separates the critical region from the rest is called the critical value. You find it using binomial probability tables or calculations.
拒绝域是指那些极端到如果检验统计量落入其中我们就拒绝原假设的结果集合。将拒绝域与其余部分分开的边界值称为临界值。你需要利用二项分布概率表或通过计算来找到它。
4. Test Statistic and p‑value | 检验统计量与 p 值
The test statistic is the number of successes observed in your sample. For a binomial test with n trials, the random variable X ~ B(n, p) is used, and the test statistic is the observed value x.
检验统计量就是在样本中观察到的成功次数。对于 n 次试验的二项检验，使用随机变量 X ~ B(n, p)，检验统计量就是观察值 x。
The p‑value is the probability of obtaining a result at least as extreme as the observed test statistic, given that H₀ is true. If the p‑value is less than or equal to α, we reject H₀. If it is greater than α, we do not have enough evidence to reject H₀.
p 值是在原假设为真的条件下，得到至少与观察到的检验统计量同样极端的结果的概率。如果 p 值小于或等于 α，我们就拒绝原假设；如果大于 α，就没有足够的证据拒绝原假设。
5. One‑tailed vs Two‑tailed Tests | 单尾检验与双尾检验
A one‑tailed test is used when the alternative hypothesis predicts a direction, e.g. H₁: p > 0.3. The whole significance level α is placed in one tail of the distribution. This increases the chance of rejecting H₀ if the true p is indeed larger.
当备择假设指明了方向，比如 H₁: p > 0.3，就使用单尾检验。整个显著性水平 α 被放在分布的某一个尾部，这样在真实的 p 确实更大时，增加了拒绝原假设的机会。
A two‑tailed test is used for H₁: p ≠ 0.3. In this case, α is split equally between the two tails, e.g. 2.5% in each tail for a 5% significance level. The p‑value is then the probability of an extreme result in either direction.
当备择假设为 H₁: p ≠ 0.3 时，使用双尾检验。此时 α 会被均分到两个尾部，比如在 5% 显著性水平下每个尾部各 2.5%。p 值就是任一方向上极端结果的概率。
It is crucial to identify the correct type of test before calculating probabilities, as an error will lead to the wrong conclusion.
在计算概率之前，必须正确识别检验类型，否则会导致错误的结论。
6. Steps in a Binomial Hypothesis Test | 二项假设检验的步骤
Step 1: Define the test statistic X ~ B(n, p) and state the hypotheses H₀ and H₁.
第一步：定义检验统计量 X ~ B(n, p) 并陈述原假设 H₀ 和备择假设 H₁。
Step 2: Choose the significance level α (if not already given).
第二步：选择显著性水平 α（如果题目未给出）。
Step 3: Calculate the probability of the observed result, or of results more extreme in the direction of H₁, assuming H₀ is true.
第三步：在原假设为真的前提下，计算观察到结果或比 H₁ 方向更极端的结果的概率。
Step 4: Compare this probability with α. If p‑value ≤ α, reject H₀; otherwise, do not reject H₀.
第四步：将该概率与 α 比较。如果 p 值 ≤ α，拒绝原假设；否则不拒绝原假设。
Step 5: Write a conclusion in the context of the question, making sure to reference the evidence.
第五步：在题目背景下写出结论，确保引用了证据。
7. Example 1: Testing a Coin for Fairness | 示例 1：检验硬币是否公平
A student suspects a coin is biased in favour of heads. She tosses the coin 10 times and gets 8 heads. Test, at the 5% significance level, whether the coin is biased towards heads. Clearly state your hypotheses.
某学生怀疑一枚硬币偏向正面。她抛掷硬币 10 次，得到 8 次正面。在 5% 显著性水平下，检验该硬币是否偏向正面。明确写出你的假设。
Let X be the number of heads in 10 tosses, so X ~ B(10, p). H₀: p = 0.5, H₁: p > 0.5 (one‑tailed). The observed test statistic is 8. We need P(X ≥ 8) under H₀.
令 X 为 10 次抛掷中出现正面的次数，则 X ~ B(10, p)。H₀: p = 0.5，H₁: p > 0.5（单尾）。观察到的检验统计量为 8。需要计算 H₀ 下 P(X ≥ 8)。
Using binomial probabilities: P(X = 8) = ₁₀C₈ × 0.5¹⁰ = 45 × 0.0009766 ≈ 0.0439; P(X = 9) = ₁₀C₉ × 0.5¹⁰ = 10 × 0.0009766 ≈ 0.0098; P(X = 10) = ₁₀C₁₀ × 0.5¹⁰ = 1 × 0.0009766 ≈ 0.0010.
利用二项分布概率：P(X = 8) = ₁₀C₈ × 0.5¹⁰ = 45 × 0.0009766 ≈ 0.0439；P(X = 9) ≈ 0.0098；P(X = 10) ≈ 0.0010。
Summing gives p‑value = P(X ≥ 8) ≈ 0.0439 + 0.0098 + 0.0010 = 0.0547. Since 0.0547 > 0.05, the p‑value is greater than the significance level.
求和得 p 值 = P(X ≥ 8) ≈ 0.0547。由于 0.0547 > 0.05，p 值大于显著性水平。
We do not reject H₀. There is insufficient evidence, at the 5% level, to conclude that the coin is biased towards heads.
我们不拒绝原假设。在 5% 显著性水平下，没有足够证据表明这枚硬币偏向正面。
8. Example 2: Testing a New Treatment | 示例 2：检验新疗法的有效性
An old medicine cures 60% of patients. A new drug is tested on 20 volunteer patients, and 15 are cured. Test, at the 5% significance level, whether the new drug has a higher cure rate. State the conclusion clearly.
一种旧药物的治愈率为 60%。新药在 20 名志愿患者上进行试验，其中 15 人被治愈。在 5% 显著性水平下，检验新药是否有更高的治愈率。清晰陈述结论。
Define X as the number of patients cured out of 20. Under H₀ the cure rate is unchanged, so X ~ B(20, 0.6). H₀: p = 0.6; H₁: p > 0.6. The observed value is 15.
定义 X 为 20 名患者中被治愈的人数。在原假设下治愈率不变，因此 X ~ B(20, 0.6)。H₀: p = 0.6，H₁: p > 0.6。观察值为 15。
We need P(X ≥ 15) assuming p = 0.6. From binomial tables or calculation: P(X = 15) ≈ 0.0746, P(X = 16) ≈ 0.0311, P(X = 17) ≈ 0.0099, P(X = 18) ≈ 0.0024, P(X = 19) ≈ 0.0005, P(X = 20) ≈ 0.00005. Summing yields p‑value ≈ 0.118.
我们需要计算 p = 0.6 下 P(X ≥ 15)。查表或计算得：P(X = 15) ≈ 0.0746, P(X = 16) ≈ 0.0311, P(X = 17) ≈ 0.0099, P(X = 18) ≈ 0.0024, P(X = 19) ≈ 0.0005, P(X = 20) ≈ 0.00005。累加得 p 值 ≈ 0.118。
Since 0.118 > 0.05, the p‑value is larger than α. We do not reject H₀. There is not enough evidence to claim that the new drug has a higher cure rate than 60%.
因为 0.118 > 0.05，p 值大于显著性水平，我们不拒绝原假设。没有足够的证据表明新药的治愈率超过 60%。
9. What Does ‘Do Not Reject’ Actually Mean? | “不拒绝”究竟意味着什么？
In hypothesis testing, you never formally ‘accept’ H₀. Failing to reject H₀ simply means the data are not strong enough to disprove it. The test starts by assuming H₀ is true, and you only change your mind if the evidence is overwhelmingly against it.
在假设检验中，你永远不能正式“接受”原假设。未能拒绝原假设只意味着数据尚不足以推翻它。检验开始时假定原假设为真，只有当证据强到足以压倒它时，你才改变看法。
This careful language is important in exam marking. Always use phrasing like ‘do not reject H₀’ or ‘there is insufficient evidence to support H₁’. Avoid saying ‘H₀ is proved’.
这种谨慎的措辞在考试评分中非常重要。一定要使用“不拒绝原假设”或“没有足够证据支持备择假设”等表述。避免说“证明了原假设”。
10. Common Mistakes Students Make | 学生常犯的错误
Mistake 1: Using the wrong tail. If H₁: p < 0.4, you must calculate P(X ≤ observed), not P(X ≥ observed).
错误一：用错尾部。如果 H₁: p < 0.4，必须计算 P(X ≤ 观察值)，而不是 P(X ≥ 观察值)。
Mistake 2: Forgetting to include the observed value when computing the p‑value, especially in two‑tailed tests where you need to consider both sides.
错误二：计算 p 值时忘记将观察值本身纳入，尤其是在需要考虑两侧的双尾检验中。
Mistake 3: Confusing the p‑value with the probability of H₀ being true. The p‑value is the probability of the data given that H₀ is true, not the reverse.
错误三：将 p 值与“原假设为真的概率”混淆。p 值是假定原假设为真时得到该数据的概率，而不是反过来。
Mistake 4: Writing a conclusion that ignores the context of the problem. Always link your decision back to the real‑world question.
错误四：写结论时脱离问题背景。务必将你的判定与题目中的现实问题联系起来。
11. Using Binomial Probability Tables Efficiently | 高效使用二项分布概率表
OCR often provides extracted binomial tables in the exam. For a given n and p, you will see cumulative probabilities P(X ≤ k). To find P(X ≥ k), use the formula P(X ≥ k) = 1 – P(X ≤ k–1).
OCR 考试通常会提供节选的二项分布表。对于给定的 n 和 p，你会看到累积概率 P(X ≤ k)。要计算 P(X ≥ k)，可使用公式 P(X ≥ k) = 1 – P(X ≤ k–1)。
For a two‑tailed test with α = 0.05, you need to check both tails. Often it is easier to double the smaller tail probability if the distribution is symmetric, but for binomial distributions you must calculate each tail carefully.
对于 α = 0.05 的双尾检验，需要检查两个尾部。若分布对称，常常可以直接将较小尾部的概率翻倍，但在二项分布中必须仔细计算每个尾部。
Here is an example cumulative table for n = 10, p = 0.5:
| k | P(X ≤ k) |
|---|---|
| 0 | 0.0010 |
| 1 | 0.0107 |
| 2 | 0.0547 |
| 3 | 0.1719 |
| 4 | 0.3770 |
| 5 | 0.6230 |
| 6 | 0.8281 |
| 7 | 0.9453 |
| 8 | 0.9893 |
| 9 | 0.9990 |
| 10 | 1.0000 |
Notice that P(X ≥ 8) = 1 – P(X ≤ 7) = 1 – 0.9453 = 0.0547, which matches our earlier calculation.
注意 P(X ≥ 8) = 1 – P(X ≤ 7) = 1 – 0.9453 = 0.0547，与我们之前的计算一致。
12. Summary and Top Exam Tips | 总结与高分技巧
Hypothesis testing questions in OCR GCSE Maths are straightforward if you follow the structured steps. Always define the random variable and explicitly write H₀ and H₁. Show your probability calculations clearly, and make sure your conclusion is phrased in context.
OCR GCSE 数学中的假设检验题只要按部就班完成，其实非常简单。务必定义随机变量并明确写出 H₀ 和 H₁。清楚地展示概率计算过程，并确保结论紧密结合题意。
Double‑check whether the test is one‑tailed or two‑tailed from the wording of the question. If it says ‘biased’ without direction, it usually implies a two‑tailed test. If the question explicitly asks for ‘greater than’ or ‘improved’, a one‑tailed test is required.
根据题目的措辞仔细确认是单尾还是双尾检验。如果题目只说“有偏”而没有指明方向，通常意味着双尾检验。若题目明确要求检验“大于”或“改善”，则需使用单尾检验。
Remember that the p‑value is compared against the significance level, and you should never accept H₀ as proven. Practice with past papers to become confident in reading binomial tables and carrying out quick probability checks.
记住，p 值是与显著性水平比较，你永远不能宣称已证明原假设为真。通过练习历年真题，熟练掌握二项分布表的查阅和快速概率校验。
These skills will not only secure marks in the statistics section but also strengthen your overall data handling abilities for the GCSE.
这些技能不仅将帮助你在统计部分稳拿分数，还将强化你 GCSE 整体数据处理的能力。
Published by TutorHao | Maths Revision Series | aleveler.com
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