Hypothesis Testing for Edexcel A-Level Maths | Edexcel A-Level 数学中的假设检验

📚 Hypothesis Testing for Edexcel A-Level Maths | Edexcel A-Level 数学中的假设检验

Hypothesis testing is a formal statistical method used to decide whether sample evidence is strong enough to reject a claim about a population parameter. In Edexcel A-Level Mathematics, the main tests involve the binomial distribution for proportions and the normal distribution for means.

假设检验是一种正式的统计方法，用来判断样本证据是否足够强，从而拒绝关于总体参数的某个主张。在 Edexcel A-Level 数学中，主要涉及二项分布的比例检验和正态分布的均值检验。

1. The Null and Alternative Hypotheses | 零假设与备择假设

The null hypothesis, written H₀, is the default claim that you assume to be true unless evidence says otherwise. The alternative hypothesis, written H₁, is the claim you are testing for.

零假设 H₀ 是默认成立的主张，除非有足够证据否则不拒绝它。备择假设 H₁ 是你要寻找证据支持的主张。

For example, if a coin is suspected of being biased towards heads, you could write H₀: p = 0.5 and H₁: p > 0.5, where p is the probability of heads. The hypotheses must always be about the population parameter, not the sample statistic.

例如，如果怀疑一枚硬币偏向正面，可以写出 H₀: p = 0.5 和 H₁: p > 0.5，其中 p 是出现正面的概率。假设必须针对总体参数，而不是样本统计量。

2. Significance Level and Critical Region | 显著性水平与临界域

The significance level, usually written as α, is the maximum probability of incorrectly rejecting H₀. Common Edexcel values are 1%, 5% and 10%.

显著性水平通常写作 α，是错误拒绝 H₀ 的最大概率。Edexcel 常见取值为 1%、5% 和 10%。

The critical region is the set of sample outcomes that would lead you to reject H₀. Its boundary values are called critical values. The acceptance region is the set of outcomes where H₀ is not rejected.

临界域是会导致你拒绝 H₀ 的样本结果集合。其边界值称为临界值。接受域是不会拒绝 H₀ 的结果集合。

If a test statistic falls inside the critical region, you reject H₀. If it falls outside, you do not reject H₀. Note that in A-Level wording, ‘accept H₀’ is often avoided because a lack of evidence is not the same as proving H₀ true.

如果检验统计量落入临界域，就拒绝 H₀。如果落在临界域之外，则不拒绝 H₀。注意在 A-Level 的表述中，通常避免使用“接受 H₀”，因为证据不足不等于证明 H₀ 为真。

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

A one-tailed test is used when the alternative hypothesis specifies a direction: H₁: p > 0.5 or H₁: μ < 50. The whole significance level goes into one tail of the distribution.

当备择假设指定方向时使用单尾检验：如 H₁: p > 0.5 或 H₁: μ < 50。整个显著性水平放在分布的一个尾部。

A two-tailed test is used when H₁ simply says the parameter is not equal to a value: H₁: p ≠ 0.5 or H₁: μ ≠ 50. The significance level is split equally between both tails.

当备择假设只说明参数不等于某个值时使用双尾检验：如 H₁: p ≠ 0.5 或 H₁: μ ≠ 50。显著性水平平均分配到两个尾部。

For a normal test at the 5% level, the one-tailed critical z-value is 1.645, while the two-tailed critical z-values are ±1.96. Using the wrong tail is a very common Edexcel error.

对 5% 显著性水平的正态检验，单尾临界 z 值为 1.645，而双尾临界 z 值为 ±1.96。用错尾部是 Edexcel 考试中非常常见的错误。

4. p-Values and Conclusions | p 值与结论

The p-value is the probability of obtaining a result at least as extreme as the observed sample result, assuming H₀ is true. It measures the strength of evidence against H₀.

p 值是在 H₀ 为真的条件下，得到至少与观测样本结果一样极端的结果的概率。它衡量反对 H₀ 的证据强度。

If the p-value is less than the significance level α, you reject H₀. If the p-value is greater than α, you do not reject H₀. For a two-tailed normal test, multiply the one-tail probability by 2 when calculating the p-value.

如果 p 值小于显著性水平 α，则拒绝 H₀。如果 p 值大于 α，则不拒绝 H₀。对双尾正态检验，计算 p 值时要把单尾概率乘以 2。

p-value	Conclusion \| 结论
p < 0.05	Reject H₀ at 5% level \| 在 5% 水平拒绝 H₀
p > 0.05	Do not reject H₀ \| 不拒绝 H₀

5. Binomial Hypothesis Test Setup | 二项假设检验的建立

For a binomial test, define the test statistic X as the number of successes in n trials. Under H₀, X ~ B(n, p₀), where p₀ is the claimed probability of success.

对于二项检验，将检验统计量 X 定义为 n 次试验中成功的次数。在 H₀ 下，X ~ B(n, p₀)，其中 p₀ 是声称的成功概率。

Write the hypotheses using p, not X or p̂. For example: H₀: p = 0.25, H₁: p < 0.25. The sample proportion is only used to calculate P(X ≤ x) or P(X ≥ x).

使用 p 而不是 X 或 p̂ 来书写假设。例如：H₀: p = 0.25，H₁: p < 0.25。样本比例只用于计算 P(X ≤ x) 或 P(X ≥ x)。

The direction of the alternative hypothesis tells you which tail to examine. H₁: p < p₀ uses the lower tail; H₁: p > p₀ uses the upper tail; H₁: p ≠ p₀ considers both tails.

备择假设的方向告诉你应考察哪个尾部。H₁: p < p₀ 使用下尾；H₁: p > p₀ 使用上尾；H₁: p ≠ p₀ 则考虑双尾。

6. Binomial Critical Region and Actual Significance Level | 二项临界域与实际显著性水平

Because the binomial distribution is discrete, you often cannot hit the nominal significance level exactly. The actual significance level is the true probability of falling in the chosen critical region.

由于二项分布是离散的，通常无法恰好达到名义显著性水平。实际显著性水平是落入所选临界域的真实概率。

Suppose X ~ B(20, 0.35) and you test H₀: p = 0.35 against H₁: p < 0.35 at the 5% level. Using a calculator, P(X ≤ 3) ≈ 0.0444 and P(X ≤ 4) ≈ 0.1182. The critical region is therefore X ≤ 3, with actual significance level 4.44%.

假设 X ~ B(20, 0.35)，在 5% 水平下检验 H₀: p = 0.35 对 H₁: p < 0.35。使用计算器可得 P(X ≤ 3) ≈ 0.0444，P(X ≤ 4) ≈ 0.1182。因此临界域为 X ≤ 3，实际显著性水平为 4.44%。

You should state the actual significance level when the question says ‘find the critical region’ or ‘state the actual significance level’. Do not just say 5% if the true probability is not 5%.

当题目要求“找出临界域”或“写出实际显著性水平”时，应说明实际显著性水平。不要直接写 5%，因为真实概率可能不是 5%。

7. Normal Hypothesis Test for the Mean | 正态均值假设检验

For a normal mean test, you assume the sample mean x̄ comes from a normal distribution. Under H₀: μ = μ₀, the test statistic is:

对于正态均值检验，假设样本均值 x̄ 来自正态分布。在 H₀: μ = μ₀ 下，检验统计量为：

z = (x̄ – μ₀) / (σ / √n)

This z-value is compared with critical z-values from the standard normal distribution. You must know the population standard deviation σ, or be given it in the question.

该 z 值与标准正态分布的临界 z 值进行比较。你必须知道总体标准差 σ，或由题目给出。

If σ is unknown, Edexcel A-Level Maths does not usually require the t-distribution; the question will either give σ or say that the population variance is known. Always check the wording carefully.

如果 σ 未知，Edexcel A-Level 数学通常不要求使用 t 分布；题目要么给出 σ，要么说明总体方差已知。务必仔细审题。

8. Normal Test Example and Critical Values | 正态检验例题与临界值

A machine fills bottles with mean volume μ ml. It is known that σ = 8 ml. A sample of 36 bottles has mean x̄ = 52.5 ml. Test at the 5% level whether the mean has increased from 50 ml.

一台机器装瓶的容量均值为 μ ml。已知 σ = 8 ml。随机抽取 36 瓶，样本均值 x̄ = 52.5 ml。在 5% 水平下检验均值是否从 50 ml 增加。

Write H₀: μ = 50 and H₁: μ > 50. This is an upper-tail test. The critical z-value at 5% is 1.645. The test statistic is:

写出 H₀: μ = 50 和 H₁: μ > 50。这是一个上尾检验。5% 对应的临界 z 值为 1.645。检验统计量为：

z = (52.5 – 50) / (8 / √36) = 2.5 / 1.333 = 1.875

Since 1.875 > 1.645, the test statistic lies in the critical region, so reject H₀. There is sufficient evidence at the 5% level that the mean volume has increased.

因为 1.875 > 1.645，检验统计量落在临界域内，所以拒绝 H₀。在 5% 水平下有足够证据表明平均容量增加了。

The p-value is P(Z > 1.875) ≈ 0.0304. Since 0.0304 < 0.05, the same conclusion follows. If the test had been two-tailed, the critical values would be ±1.96 and the conclusion would change.

p 值为 P(Z > 1.875) ≈ 0.0304。由于 0.0304 < 0.05，结论相同。如果这是双尾检验，临界值将为 ±1.96，结论也会改变。

9. Common Errors in Hypothesis Testing | 假设检验中的常见错误

One common mistake is writing the hypotheses using the sample statistic, such as H₀: x̄ = 50 or H₁: p̂ > 0.5. Hypotheses must be written in terms of population parameters like μ and p.

一个常见错误是用样本统计量书写假设，例如 H₀: x̄ = 50 或 H₁: p̂ > 0.5。假设必须使用总体参数，如 μ 和 p。

Another error is confusing the direction of the test. For H₁: p < 0.4, you must calculate P(X ≤ observed), not P(X ≥ observed). For a two-tailed test, do not forget to double the tail probability when finding the p-value.

另一个错误是混淆检验方向。对于 H₁: p < 0.4，应计算 P(X ≤ 观测值)，而不是 P(X ≥ 观测值)。对于双尾检验，计算 p 值时不要忘记将单尾概率乘以 2。

In binomial tests, do not claim the actual significance level is exactly 5% unless P(rejecting H₀) equals 0.05. State the exact probability from the calculator, often using binomCdf or binomial tables.

在二项检验中，不要声称实际显著性水平恰好为 5%，除非 P(拒绝 H₀) 确实等于 0.05。应写出计算器给出的精确概率，通常使用 binomCdf 或二项分布表。

10. Exam Strategy and Quick Checks | 考试策略与快速检查

Always state H₀ and H₁ clearly, define the test statistic and its distribution, calculate the probability or test statistic, compare with α or the critical value, and write a conclusion in context.

始终清晰写出 H₀ 和 H₁，定义检验统计量及其分布，计算概率或检验统计量，与 α 或临界值比较，最后结合题意写出结论。

Use the context of the question in your conclusion. Instead of only writing ‘reject H₀’, write ‘there is sufficient evidence to suggest that the proportion of defective items has decreased’.

在结论中要结合题目背景。不要只写“拒绝 H₀”，而要写“有足够证据表明次品比例有所下降”。

Check your answer by asking: Were the hypotheses about parameters? Did I use the correct one-tailed or two-tailed critical value? Did I write the final conclusion in words?

通过以下问题检查答案：假设是否针对参数？是否使用了正确的单尾或双尾临界值？最终结论是否用文字表述？

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Hypothesis Testing for Edexcel A-Level Maths | Edexcel A-Level 数学中的假设检验