One-tailed Tests | 单尾检验

📚 One-tailed Tests | 单尾检验

In Edexcel A-Level Mathematics, hypothesis testing with the binomial distribution often asks you to choose between a one-tailed test and a two-tailed test. A one-tailed test is used when the alternative hypothesis predicts a specific direction of change, such as ‘more than’ or ‘less than’. This article explains the key ideas, critical regions, p-values and common exam techniques for one-tailed tests.

在爱德思 A-Level 数学中，二项分布的假设检验经常要求你选择单尾检验或双尾检验。当备择假设预测了明确的变化方向时，例如“大于”或“小于”，就使用单尾检验。本文解释单尾检验的核心概念、临界域、p 值以及常见考试技巧。

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1. What Is a One-tailed Test? | 什么是单尾检验？

A one-tailed test, also called a one-sided test, is a hypothesis test where the alternative hypothesis H₁ specifies that the population parameter is either greater than or less than the null value, but not both. It focuses all of the significance level in one tail of the sampling distribution.

单尾检验，也称为单侧检验，是一种假设检验，其中备择假设 H₁ 指定总体参数要么大于要么小于原假设值，但不会同时包含两个方向。它将全部显著性水平集中在抽样分布的一个尾部。

For example, if we test a coin to see whether it is biased in favour of heads, we only care about unusually high numbers of heads. That is a right-tailed test. If we test whether the proportion of defective items has decreased, we only care about unusually low numbers of defective items. That is a left-tailed test.

例如，如果我们检验一枚硬币是否偏向正面，我们只关心正面次数是否异常多。这就是右尾检验。如果我们检验次品率是否下降，我们只关心次品数是否异常少。这就是左尾检验。

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

In a binomial hypothesis test, the null hypothesis H₀ states that the population proportion p equals a given value p₀. The alternative hypothesis H₁ states the direction we are testing for. For a one-tailed test, H₁ must be either p > p₀ or p < p₀.

在二项分布假设检验中，原假设 H₀ 表明总体比例 p 等于给定值 p₀。备择假设 H₁ 说明我们要检验的方向。对于单尾检验，H₁ 必须是 p > p₀ 或 p < p₀。

H₀: p = p₀

H₁: p > p₀ or H₁: p < p₀

You should never write H₁: p ≠ p₀ for a one-tailed test; that would be a two-tailed test.

对于单尾检验，你绝不能写成 H₁: p ≠ p₀；那是双尾检验。

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3. Left-tailed vs Right-tailed Tests | 左尾检验与右尾检验

The direction of the alternative hypothesis determines which tail is used. A right-tailed test has H₁: p > p₀ and rejects H₀ for unusually large values of the test statistic. A left-tailed test has H₁: p < p₀ and rejects H₀ for unusually small values.

备择假设的方向决定使用哪一个尾部。右尾检验的 H₁: p > p₀，当检验统计量异常大时拒绝 H₀。左尾检验的 H₁: p < p₀，当检验统计量异常小时拒绝 H₀。

Test type | 检验类型	Alternative hypothesis | 备择假设	Critical region | 临界域
Right-tailed | 右尾	H₁: p > p₀	X ≥ c
Left-tailed | 左尾	H₁: p < p₀	X ≤ c

In both cases, the critical value c is chosen so that the probability of being in the critical region when H₀ is true is as close as possible to the significance level α without exceeding it.

在两种情况下，临界值 c 的选择要使在 H₀ 为真时落入临界域的概率尽可能接近显著性水平 α，但不超过 α。

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

The significance level α is the maximum probability of rejecting H₀ when H₀ is actually true. This is the probability of a Type I error. In Edexcel questions, α is usually 5%, 2% or 1%, written as 0.05, 0.02 or 0.01.

显著性水平 α 是当 H₀ 实际为真时拒绝 H₀ 的最大概率。这是第一类错误的概率。在爱德思考试题中，α 通常为 5%、2% 或 1%，写作 0.05、0.02 或 0.01。

The critical region is the set of values of the test statistic that lead to rejecting H₀. For a right-tailed test with test statistic X, the critical region has the form X ≥ c. For a left-tailed test, it has the form X ≤ c.

临界域是导致拒绝 H₀ 的检验统计量取值的集合。对于检验统计量为 X 的右尾检验，临界域形式为 X ≥ c。对于左尾检验，临界域形式为 X ≤ c。

We require:

我们要求：

P(X ≥ c | p = p₀) ≤ α   or   P(X ≤ c | p = p₀) ≤ α

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5. The Binomial Test Statistic | 二项分布检验统计量

In A-Level hypothesis tests, the test statistic is usually X, the number of successes in n independent trials. Under H₀, X follows a binomial distribution:

在 A-Level 假设检验中，检验统计量通常是 X，即 n 次独立试验中成功的次数。在 H₀ 下，X 服从二项分布：

X ~ B(n, p₀)

Here n is the sample size and p₀ is the assumed probability of success under the null hypothesis. You can calculate probabilities using the binomial formula:

这里 n 是样本容量，p₀ 是原假设下假定的成功概率。你可以使用二项分布公式计算概率：

P(X = x) = ⁿCₓ p₀ˣ (1 – p₀)ⁿ⁻ˣ

In the exam you may also use binomial cumulative distribution tables or a calculator to find P(X ≤ x) and P(X ≥ x).

在考试中，你也可以使用二项分布累积表或计算器求出 P(X ≤ x) 和 P(X ≥ x)。

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6. Finding Critical Values | 求临界值

To find the critical value for a right-tailed test, you look for the smallest integer c such that P(X ≥ c | H₀) ≤ α. This is equivalent to finding the largest integer c – 1 for which P(X ≤ c – 1 | H₀) ≥ 1 – α.

对于右尾检验，求临界值时要寻找满足 P(X ≥ c | H₀) ≤ α 的最小整数 c。这等价于寻找满足 P(X ≤ c – 1 | H₀) ≥ 1 – α 的最大整数 c – 1。

For a left-tailed test, you look for the largest integer c such that P(X ≤ c | H₀) ≤ α.

对于左尾检验，要寻找满足 P(X ≤ c | H₀) ≤ α 的最大整数 c。

Because the binomial distribution is discrete, the actual probability of being in the critical region is often less than α. This actual probability is called the actual significance level.

由于二项分布是离散的，落入临界域的实际概率通常小于 α。这个实际概率称为实际显著性水平。

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7. Worked Example: Right-tailed Test | 例题：右尾检验

A coin is tossed 20 times and it is suspected that the coin is biased in favour of heads. Let p be the probability of getting a head. Test at the 5% significance level whether the proportion of heads is greater than 0.5. It is observed that 15 heads occur.

一枚硬币抛掷 20 次，怀疑它偏向正面。设 p 为得到正面的概率。在 5% 显著性水平下检验正面比例是否大于 0.5。实际观察到 15 次正面。

State the hypotheses:

写出假设：

H₀: p = 0.5

H₁: p > 0.5

Under H₀, X ~ B(20, 0.5), where X is the number of heads.

在 H₀ 下，X ~ B(20, 0.5)，其中 X 是正面次数。

Find the critical region. We need the smallest c with P(X ≥ c | p = 0.5) ≤ 0.05.

求临界域。我们需要满足 P(X ≥ c | p = 0.5) ≤ 0.05 的最小 c。

Using a calculator or tables:

使用计算器或查表得：

P(X ≥ 14 | p = 0.5) = 0.0577 > 0.05

P(X ≥ 15 | p = 0.5) = 0.0207 ≤ 0.05

Therefore the critical region is X ≥ 15. The actual significance level is 0.0207, or 2.07%.

因此临界域为 X ≥ 15。实际显著性水平为 0.0207，即 2.07%。

Since the observed value 15 lies in the critical region, we reject H₀. There is sufficient evidence at the 5% significance level to suggest that the coin is biased in favour of heads.

由于观察值 15 落在临界域内，我们拒绝 H₀。在 5% 显著性水平下，有充分证据表明这枚硬币偏向正面。

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8. Worked Example: Left-tailed Test | 例题：左尾检验

A factory claims that the proportion of defective components is 0.2. A new production process is expected to reduce this proportion. A random sample of 25 components is tested, and 1 defective component is found. Test at the 5% significance level whether the new process has a lower defective rate.

一家工厂声称次品率为 0.2。新的生产工艺有望降低这一比例。随机抽取 25 个元件进行检测，发现 1 个次品。在 5% 显著性水平下检验新工艺的次品率是否更低。

Let p be the proportion of defective components. State the hypotheses:

设 p 为次品率。写出假设：

H₀: p = 0.2

H₁: p < 0.2

Under H₀, X ~ B(25, 0.2), where X is the number of defective components.

在 H₀ 下，X ~ B(25, 0.2)，其中 X 为次品数。

For a left-tailed test at α = 0.05, we need the largest c with P(X ≤ c | p = 0.2) ≤ 0.05.

对于 α = 0.05 的左尾检验，我们需要满足 P(X ≤ c | p = 0.2) ≤ 0.05 的最大 c。

Using a calculator or tables:

使用计算器或查表得：

P(X ≤ 1 | p = 0.2) = 0.0274 ≤ 0.05

P(X ≤ 2 | p = 0.2) = 0.0982 > 0.05

Therefore the critical region is X ≤ 1. The actual significance level is 0.0274, or 2.74%.

因此临界域为 X ≤ 1。实际显著性水平为 0.0274，即 2.74%。

Since the observed value 1 lies in the critical region, we reject H₀. There is sufficient evidence at the 5% significance level to suggest that the new process has reduced the defective rate.

由于观察值 1 落在临界域内，我们拒绝 H₀。在 5% 显著性水平下，有充分证据表明新工艺降低了次品率。

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9. p-value Approach | p 值方法

Instead of comparing the observed value with a critical value, you can calculate the p-value. The p-value is the probability of obtaining a result at least as extreme as the observed value, assuming H₀ is true.

除了将观察值与临界值进行比较，你还可以计算 p 值。p 值是在 H₀ 为真的条件下，得到至少与观察值一样极端的结果的概率。

For a right-tailed test, the p-value is:

对于右尾检验，p 值为：

p-value = P(X ≥ x | H₀)

For a left-tailed test, the p-value is:

对于左尾检验，p 值为：

p-value = P(X ≤ x | H₀)

If p-value < α, reject H₀. If p-value ≥ α, do not reject H₀. In the right-tailed example above, the observed value was 15, so the p-value was 0.0207 < 0.05, leading to rejection of H₀.

如果 p 值 < α，则拒绝 H₀。如果 p 值 ≥ α，则不拒绝 H₀。在上面的右尾例题中，观察值为 15，因此 p 值为 0.0207 < 0.05，从而拒绝 H₀。

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10. One-tailed vs Two-tailed Tests | 单尾检验与双尾检验

A two-tailed test is used when the alternative hypothesis is H₁: p ≠ p₀. In that case, the significance level is split equally between the two tails, so each tail has probability α/2. A one-tailed test puts the entire α in one tail, making it easier to reject H₀ in that direction but useless for detecting the opposite direction.

当备择假设为 H₁: p ≠ p₀ 时，使用双尾检验。此时显著性水平被平均分配到两个尾部，每个尾部概率为 α/2。单尾检验将整个 α 放在一个尾部，因此在该方向上更容易拒绝 H₀，但无法检测相反方向。

In Edexcel questions, the wording of the claim decides the type of test. Words such as ‘greater than’, ‘increased’, ‘more than’ indicate a right-tailed test. Words such as ‘less than’, ‘decreased’, ‘reduced’ indicate a left-tailed test. Words such as ‘different’, ‘changed’, ‘not equal’ indicate a two-tailed test.

在爱德思考题中，题目中的措辞决定检验类型。诸如“大于”“增加”“多于”等词语表示右尾检验。诸如“小于”“减少”“降低”等词语表示左尾检验。诸如“不同”“改变”“不等于”等词语表示双尾检验。

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11. Common Mistakes and Exam Tips | 常见错误与考试提示

Students often lose marks by using the wrong tail, writing H₁: p ≠ p₀ when a one-tailed test is required, or using the wrong inequality in the critical region. Always define p clearly and check the wording of the question before choosing the test direction.

学生常因使用错误尾部、在需要单尾检验时写成 H₁: p ≠ p₀、或在临界域中使用错误不等式而失分。务必清晰定义 p，并在选择检验方向前仔细审题。

• Define p before stating H₀ and H₁. | 在写出 H₀ 和 H₁ 之前先定义 p。

• Use the exact significance level α given in the question. | 使用题目给出的确切显著性水平 α。

• For a right-tailed test, the critical region is X ≥ c; for a left-tailed test, it is X ≤ c. | 右尾检验的临界域为 X ≥ c；左尾检验的临界域为 X ≤ c。

• Always write a conclusion in context, using ‘sufficient evidence’ or ‘insufficient evidence’. | 始终在结论中结合背景，使用“充分证据”或“证据不足”。

• Do not say ‘accept H₀’; say ‘do not reject H₀’. | 不要说“接受 H₀”；要说“不拒绝 H₀”。

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12. Summary | 小结

A one-tailed test is appropriate when the alternative hypothesis is directional: H₁: p > p₀ or H₁: p < p₀. The entire significance level lies in one tail of the binomial distribution. You can use a critical region or a p-value to make the decision. The critical value must be chosen so that the probability of the critical region under H₀ does not exceed α. Always interpret your conclusion in the context of the question.

当备择假设具有方向性时，即 H₁: p > p₀ 或 H₁: p < p₀，应使用单尾检验

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