Testing a Hypothesis | 假设检验

📚 Testing a Hypothesis | 假设检验

Hypothesis testing is a central statistical method in Edexcel A-Level Mathematics. It allows us to use sample data to decide whether there is enough evidence to reject an assumed value of a population parameter, such as a binomial probability p or a normal mean μ. A good hypothesis test combines clear hypotheses, a stated significance level, a correct probability calculation, and a conclusion written in the context of the original problem.

假设检验是爱德思 A-Level 数学中的核心统计方法。它允许我们利用样本数据，判断是否有足够证据拒绝总体参数的假定值，例如二项分布的概率 p 或正态分布的均值 μ。一个完整的假设检验应当包含清晰的假设、明确的显著性水平、正确的概率计算，以及结合原题情境写出的结论。

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

Every hypothesis test begins with two competing statements. The null hypothesis, H₀, is the assumption that the population parameter equals a given value. The alternative hypothesis, H₁, is the statement we are testing for; it says that the parameter is either greater than, less than, or not equal to that value.

每个假设检验都从两个相互竞争的命题开始。原假设 H₀ 是假定总体参数等于某个给定值的命题。备择假设 H₁ 是我们要检验的命题，它说明参数大于、小于或不等于该值。

For a binomial test, the hypotheses are usually written as H₀: p = 0.5 and H₁: p > 0.5, H₁: p < 0.5, or H₁: p ≠ 0.5. Notice that H₀ always contains the equality statement, while H₁ contains the inequality that matches the wording of the question.

对于二项分布检验，假设通常写成 H₀: p = 0.5 以及 H₁: p > 0.5、H₁: p < 0.5 或 H₁: p ≠ 0.5。注意 H₀ 始终包含等号，而 H₁ 包含与题目措辞相匹配的不等号。

H₀: p = 0.5, H₁: p > 0.5

2. One-Tailed and Two-Tailed Tests | 单尾与双尾检验

The alternative hypothesis determines whether a test is one-tailed or two-tailed. If we are testing for an increase or a decrease only, the test is one-tailed. If we are testing for any change, either up or down, the test is two-tailed.

备择假设决定了检验是单尾还是双尾。如果我们只检验增加或只检验减少，该检验就是单尾检验。如果我们检验任何方向的变化，即上升或下降，该检验就是双尾检验。

Upper one-tailed test: H₁: p > 0.5 — used when the claim involves “more than”, “greater than”, “increased”.
上单尾检验：H₁: p > 0.5 —— 当表述涉及“多于”“大于”“增加”时使用。
Lower one-tailed test: H₁: p < 0.5 — used when the claim involves “less than”, “smaller than”, “decreased”.
下单尾检验：H₁: p < 0.5 —— 当表述涉及“少于”“小于”“减少”时使用。
Two-tailed test: H₁: p ≠ 0.5 — used when the claim involves “changed”, “different from”, “not equal to”.
双尾检验：H₁: p ≠ 0.5 —— 当表述涉及“改变了”“不同于”“不等于”时使用。

In a two-tailed test, the significance level is split equally between the two tails. For example, at the 5% level, each tail receives 2.5%. This affects how we find critical values or calculate the rejection region.

在双尾检验中，显著性水平被平均分配到两侧尾部。例如，在 5% 水平下，每个尾部各占 2.5%。这会影响我们如何查找临界值或计算拒绝域。

3. Significance Levels and Critical Regions | 显著性水平与临界域

The significance level, denoted by α, is the maximum probability of rejecting H₀ when H₀ is actually true. In Edexcel A-Level questions, common significance levels are 5% and 1%, so α = 0.05 or α = 0.01.

显著性水平用 α 表示，是当 H₀ 实际上为真时拒绝 H₀ 的最大概率。在爱德思 A-Level 考题中，常见的显著性水平为 5% 和 1%，即 α = 0.05 或 α = 0.01。

The critical region, also called the rejection region, is the set of values of the test statistic for which we reject H₀. The boundary of this region is the critical value. In a binomial test, the critical region is a set of sample counts; in a normal test, it is a range of z-values or sample means.

临界域，也称为拒绝域，是检验统计量中导致我们拒绝 H₀ 的取值集合。该区域的边界就是临界值。在二项分布检验中，临界域是一组样本计数；在正态分布检验中，它是 z 值或样本均值的一个范围。

Reject H₀ if the test statistic falls in the critical region; otherwise fail to reject H₀.

如果检验统计量落在临界域内，则拒绝 H₀；否则不能拒绝 H₀。

Because a binomial distribution is discrete, the actual significance level is often smaller than the stated nominal level. For example, if we want a 5% test but the lowest tail probability below 5% is 0.0107, then the actual significance level is 1.07%.

由于二项分布是离散的，实际显著性水平通常小于名义显著性水平。例如，如果我们想要一个 5% 的检验，但低于 5% 的最小尾部概率是 0.0107，那么实际显著性水平就是 1.07%。

4. Binomial Hypothesis Tests | 二项分布假设检验

A binomial hypothesis test is used when we have a fixed number of trials n, a constant probability of success p under H₀, and a sample count x of successes. We define X ~ B(n, p) and compute the probability of obtaining the observed result or a more extreme one, assuming H₀ is true.

二项分布假设检验适用于具有固定试验次数 n、在原假设下固定的成功概率 p，以及样本成功次数 x 的情形。我们定义 X ~ B(n, p)，并在假定 H₀ 为真的条件下，计算得到观测结果或更极端结果的概率。

For an upper-tail test H₁: p > p₀, we calculate P(X ≥ x). For a lower-tail test H₁: p < p₀, we calculate P(X ≤ x). If this probability is less than or equal to α, we reject H₀. Otherwise, there is insufficient evidence to reject H₀.

对于上尾检验 H₁: p > p₀，我们计算 P(X ≥ x)。对于下尾检验 H₁: p < p₀，我们计算 P(X ≤ x)。如果这个概率小于或等于 α，我们就拒绝 H₀。否则，没有足够证据拒绝 H₀。

For a two-tailed binomial test, we find both tails. We may compare the smaller tail probability with α/2, or check whether the observed count lies beyond the upper or lower critical value.

对于双尾二项分布检验，我们需要考虑两侧尾部。我们可以将较小的尾部概率与 α/2 比较，或者检查观测计数是否超出上侧或下侧临界值。

Example: X ~ B(10, 0.5), H₀: p = 0.5, H₁: p > 0.5, α = 0.05. P(X ≥ 8) = 0.0547, P(X ≥ 9) = 0.0107. Reject H₀ if X ≥ 9.

示例：X ~ B(10, 0.5)，H₀: p = 0.5，H₁: p > 0.5，α = 0.05。P(X ≥ 8) = 0.0547，P(X ≥ 9) = 0.0107。若 X ≥ 9，则拒绝 H₀。

5. Normal Hypothesis Tests | 正态分布假设检验

A normal hypothesis test is used when the population is normally distributed, or when the sample size is large enough for the Central Limit Theorem to apply. In Edexcel A-Level questions, the population variance σ² is often known, so we use the standard normal distribution Z ~ N(0, 1).

当总体服从正态分布，或者样本量足够大使得中心极限定理适用时，使用正态分布假设检验。在爱德思 A-Level 考题中，总体方差 σ² 通常是已知的，因此我们使用标准正态分布 Z ~ N(0, 1)。

The test statistic for a mean is calculated using the sample mean x̄, the hypothesised mean μ, the population standard deviation σ, and the sample size n:

均值检验的检验统计量使用样本均值 x̄、假设均值 μ、总体标准差 σ 和样本量 n 计算：

z = (x̄ − μ) ÷ (σ ÷ √n)

For a one-tailed test at the 5% level, the critical value is 1.645 for the upper tail or −1.645 for the lower tail. For a two-tailed test at the 5% level, the critical values are ±1.96. These come from the standard normal distribution.

对于 5% 水平的单尾检验，上尾的临界值为 1.645，下尾为 −1.645。对于 5% 水平的双尾检验，临界值为 ±1.96。这些值来自标准正态分布。

Reject H₀ if z > 1.645 for H₁: μ > μ₀ at α = 0.05.

若 H₁: μ > μ₀ 且 α = 0.05，则当 z > 1.645 时拒绝 H₀。

6. The p-Value Method | p 值法

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming H₀ is true. It measures the strength of the evidence against H₀. A small p-value means the observed result is unlikely if H₀ were true, so we have strong evidence against H₀.

p 值是在假定 H₀ 为真的条件下，得到至少与观测值同样极端的检验统计量的概率。它衡量了反对 H₀ 的证据强度。p 值很小意味着如果 H₀ 为真，观测结果不太可能发生，因此我们有较强的证据反对 H₀。

To make a decision, compare the p-value with α. If p ≤ α, reject H₀. If p > α, fail to reject H₀. In a two-tailed test, remember to double the one-tailed tail probability, or compare the smaller p-value with α/2, depending on the method used.

做决策时，将 p 值与 α 比较。如果 p ≤ α，拒绝 H₀。如果 p > α，不能拒绝 H₀。在双尾检验中，记得将单尾尾部概率乘以 2，或者根据所用方法将较小的 p 值与 α/2 比较。

If p ≤ α, reject H₀; if p > α, fail to reject H₀.

若 p ≤ α，拒绝 H₀；若 p > α，不能拒绝 H₀。

7. Critical Values and Rejection Rules | 临界值与拒绝规则

In a binomial test, critical values are found by looking at cumulative binomial probabilities. For an upper-tail test at α = 0.05, the critical value c is the smallest value such that P(X ≥ c) ≤ 0.05. For a lower-tail test, c is the largest value such that P(X ≤ c) ≤ 0.05.

在二项分布检验中，临界值通过查看累积二项概率来确定。对于 α = 0.05 的上尾检验，临界值 c 是满足 P(X ≥ c) ≤ 0.05 的最小值。对于下尾检验，c 是满足 P(X ≤ c) ≤ 0.05 的最大值。

In a normal test, the critical value is a z-value from the standard normal distribution. For a one-tailed test at the 5% level, use z = 1.645; at the 1% level, use z = 2.326. For a two-tailed test at the 5% level, use z = 1.96; at the 1% level, use z = 2.576.

在正态检验中，临界值是来自标准正态分布的 z 值。对于 5% 水平的单尾检验，使用 z = 1.645；对于 1% 水平，使用 z = 2.326。对于 5% 水平的双尾检验，使用 z = 1.96；对于 1% 水平，使用 z = 2.576。

Test type	Significance level	Critical z-value
One-tailed	5%	1.645
Two-tailed	5%	1.96
One-tailed	1%	2.326
Two-tailed	1%	2.576

The rejection rule is simple: reject H₀ if the test statistic falls in the critical region, or equivalently if the p-value is less than or equal to α. Otherwise, we do not reject H₀.

拒绝规则很简单：如果检验统计量落在临界域内，或等价地，如果 p 值小于或等于 α，则拒绝 H₀。否则，不拒绝 H₀。

8. Writing Conclusions in Context | 结合情境写结论

A hypothesis test conclusion must always be written in the context of the question. It is not enough to say “reject H₀” or “fail to reject H₀”. You must state what this means for the original claim, and mention the significance level used.

假设检验的结论必须始终结合题目情境写出。仅说“拒绝 H₀”或“不能拒绝 H₀”是不够的。你必须说明这对原始命题意味着什么，并提及所使用的显著性水平。

For example, if H₀ is rejected, a suitable conclusion might be: “There is sufficient evidence at the 5% significance level to reject H₀ and conclude that the proportion of successes is greater than 0.5.”

例如，如果拒绝 H₀，合适的结论可以是：“在 5% 显著性水平下，有充分证据拒绝 H₀，并得出结论：成功比例大于 0.5。”

If H₀ is not rejected, write: “There is insufficient evidence at the 5% significance level to reject H₀, so we cannot conclude that the proportion has changed.” Never say that H₀ is true or that the parameter equals the null value.

如果没有拒绝 H₀，应写：“在 5% 显著性水平下，没有足够证据拒绝 H₀，因此我们无法得出比例已改变的结论。”永远不要说 H₀ 为真，或参数等于原假设值。

9. Type I and Type II Errors | 第一类与第二类错误

A Type I error occurs when we reject H₀ when H₀ is actually true. The probability of a Type I error is exactly the significance level α. A Type II error occurs when we fail to reject H₀ when H₁ is actually true. Its probability is denoted by β.

第一类错误发生在 H₀ 实际上为真时却拒绝了 H₀。第一类错误的概率恰好是显著性水平 α。第二类错误发生在 H₁ 实际上为真时却没有拒绝 H₀。其概率用 β 表示。

The power of a test is 1 − β, the probability of correctly rejecting H₀ when H₁ is true. In A-Level questions, you may be asked to identify a Type I or Type II error in context, or to explain what a change in significance level does to these error probabilities.

检验的功效为 1 − β，即当 H₁ 为真时正确拒绝 H₀ 的概率。在 A-Level 考题中，可能会要求你在具体情境中识别第一类或第二类错误，或者解释显著性水平的变化对这些错误概率的影响。

P(Type I error) = α; P(Type II error) = β; power = 1 − β

P(第一类错误) = α；P(第二类错误) = β；功效 = 1 − β

Increasing α makes it easier to reject H₀, so it reduces the chance of a Type II error but increases the chance of a Type I error. Decreasing α has the opposite effect.

增大 α 会使拒绝 H₀ 更容易，因此会降低第二类错误的概率，但会增加第一类错误的概率。减小 α 则产生相反的效果。

10. Common Mistakes and Exam Tips | 常见错误与应试技巧

One common mistake is stating the hypotheses the wrong way round or forgetting to include the equality in H₀. Always check the wording carefully: “test whether p has increased” means H₁: p > p₀, not H₁: p < p₀.

一个常见错误是把假设写反，或者忘记在 H₀ 中包含等号。务必仔细审题：“检验 p 是否增加”意味着 H₁: p > p₀，而不是 H₁: p < p₀。

Another mistake is using the normal critical value for a binomial test or vice versa. Use cumulative binomial probabilities for discrete counts, and use the standard normal distribution only when the test statistic is z.

另一个错误是在二项分布检验中使用正态临界值，或反过来。对于离散计数，应使用累积二项概率；只有检验统计量为 z 时，才使用标准正态分布。

Many students also compare the p-value in the wrong direction. Remember: reject H₀ when the probability is small, not large. If P(X ≥ x) = 0.23 at α = 0.05, do not reject H₀.

许多学生还会把 p 值的比较方向弄反。记住：当概率很小时拒绝 H₀，而不是很大时。如果 P(X ≥ x) = 0.23 且 α = 0.05，则不要拒绝 H₀。

Finally, always write a conclusion in context. A bare statement such as “reject H₀” gains no context mark. State the evidence, the significance level, and what the result means for the original problem.

最后，一定要结合情境写结论。像“拒绝 H₀”这样孤立的陈述得不到情境分。请说明证据、显著性水平，以及该结果对原题意味着什么。

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Testing a Hypothesis | 假设检验