📚 Rejection Region and Decision Rules in Two-Tailed Hypothesis Testing | 双尾检验的拒绝域与判断规则
In A-Level statistics, hypothesis testing is one of the most frequently examined topics, and the two-tailed test often confuses students because the rejection region is split into two tails. This article provides a complete and exam-focused explanation of the rejection region and decision rules for two-tailed tests, with worked examples and common traps to avoid.
在 A-Level 统计学中，假设检验是最常考的考点之一，而双尾检验由于拒绝域被分成左右两端，常常让同学们感到困惑。本文将从考试角度出发，系统地讲解双尾检验的拒绝域与判断规则，并通过典型例题和常见易错点帮助你彻底掌握这一内容。
1. What Is a Two-Tailed Test? | 什么是双尾检验？
: p = p₀ (or μ = μ₀), and the alternative hypothesis is written as H₁: p ≠ p₀ (or μ ≠ μ₀).
双尾检验适用于备择假设不指明方向的情况。我们不是检验参数是增大还是减小，而是检验参数是否发生了任何方向的变化。用数学符号表示，原假设写作 H₀: p = p₀（或 μ = μ₀），备择假设写作 H₁: p ≠ p₀（或 μ ≠ μ₀）。
The word "two-tailed" means that we are looking for evidence in both tails of the distribution. If the test statistic falls too far into either the left tail or the right tail, we reject the null hypothesis.
“双尾”意味着我们要在分布的两端寻找证据。如果检验统计量落在左尾或右尾过远的位置，我们就拒绝原假设。
2. Significance Level and Critical Values | 显著性水平与临界值
The significance level, denoted by α (alpha), is the probability of making a Type I error — rejecting H₀ when it is actually true. In a two-tailed test, this probability is divided equally between the two tails. Therefore, each tail contains α/2 of the total probability.
显著性水平用 α 表示，是指犯第一类错误的概率，即当原假设实际上为真时我们却拒绝了它。在双尾检验中，这个概率被平均分配到两个尾部，因此每个尾部各包含 α/2 的概率。
For example, if the significance level is 5% (α = 0.05), then each tail has 2.5% andard normal distribution with α = 0.05, the critical values are z = -1.96 and z = +1.96.
例如，若显著性水平为 5%（α = 0.05），则每个尾部各有 2.5%（0.025）的概率。临界值就是拒绝域的边界。在标准正态分布中，当 α = 0.05 时，临界值为 z = -1.96 和 z = +1.96。
Rejection region: z < -1.96 or z > +1.96
拒绝域：z < -1.96 或 z > +1.96
The total area of the rejection region equals α. In the normal distribution example above, the left tail area is 0.025 and the right tail area is 0.025, giving a total of 0.05.
拒绝域的总面积等于 α。在上述正态分布的例子中，左尾面积为 0.025，右尾面积为 0.025，合计为 0.05。
3. Critical Region for Discrete Distributions | 离散分布下的临界区域
For discrete distributions such as the binomial distribution, Poisson distribution, and geometric distribution, the rejection region cannot always be set to exactly α/2 in each tail. Instead, we choose the largest critical region such that the probability in each tail does not exceed α/2.
对于二项分布、泊松分布、几何分布等离散分布，我们无法总是让每个尾部的拒绝域面积恰好等于 α/2。此时，我们选择每个尾部概率不超过 α/2 的最大临界区域。
Suppose X ~ B(20, p) and we are testing at the 5% significance level. Then each tail has at most 2.5%. We find the critical values as follows:
设 X ~ B(20, p)，在 5% 显著性水平下检验。则每个尾部最多为 2.5%。临界值按如下方式确定：
t value a such that P(X ≤ a) < 0.025, and the smallest value b such that P(X ≥ b) < 0.025.
找到最大的 a 使得 P(X ≤ a) < 0.025，以及最小的 b 使得 P(X ≥ b) < 0.025。
For X ~ B(20, 0.5), we calculate:
对于 X ~ B(20, 0.5)，我们计算：
| k | P(X = k) | P(X ≤ k) |
| 0 | 0.0000 | 0.0000 |
| 5 | 0.0148 | 0.0207 |
| 6 | 0.0370 | 0.0577 |
| 14 | 0.0370 | 0.9423 |
| 15 | 0.0148 | 0.9793 |
| 20 | 0.0000 | 1.0000 |
critical value is 15 because P(X ≥ 15) = 0.0207 < 0.025, while P(X ≥ 14) = 0.0577 > 0.025.
下临界值为 5，因为 P(X ≤ 5) = 0.0207 < 0.025，而 P(X ≤ 6) = 0.0577 > 0.025。上临界值为 15，因为 P(X ≥ 15) = 0.0207 < 0.025，而 P(X ≥ 14) = 0.0577 > 0.025。
herefore X ≤ 5 or X ≥ 15. We would reject H₀ if the observed value of X is in this region.
因此拒绝域为 X ≤ 5 或 X ≥ 15。如果 X 的观测值落在这个区域内，我们就拒绝原假设。
4. The Decision Rule | 判断规则
Once the critical region is determined, the decision rule is straightforward. Calculate the test statistic from the sample data. If the test statistic falls inside the rejection region, reject H₀. If it falls outside the rejection region, do not reject H₀.
一旦确定了临界区域，判断规则就非常直接。根据样本数据计算检验统计量。如果检验统计量落在拒绝域内，则拒绝 H₀；如果落在拒绝域外，则不拒绝 H₀。
In a two-tailed test, the p-value for a symmetric distribution is calculated as 2 × P(Z ≥ |z|).
另一种等价的方法是比较 p 值与显著性水平。p 值是在原假设为真的前提下，得到至少与观测值一样极端的检验统计量的概率。在双尾检验中，对于对称分布，p 值计算为 2 × P(Z ≥ |z|)。
Decision rule: Reject H₀ if p-value < α. Do not reject H₀ if p-value ≥ α.
判断规则：若 p 值 < α，则拒绝 H₀；若 p 值 ≥ α，则不拒绝 H₀。
Both methods — critical region and p-value — always lead to the same conclusion. In the exam, you may use either, but you must clearly state your conclusion in the context of the question.
临界区域法和 p 值法两种方法总会得出相同的结论。在考试中，你可以任选其一，但必须结合题目背景清楚地写出结论。
5. Two-Tailed Test for Normal Distribution | 正态分布下的双尾检验
When testing a population mean μ with a known variance and a large sample, we use the standard normal distribution. The test statistic is:
当检验总体均值 μ 且方差已知、样本量较大时，我们使用标准正态分布。检验统计量为：
z = (x̄ - μ₀) / (σ / √n)
where x̄ is the sample mean, μ₀ is the hypothesised population mean, σ is the population standard deviation, and n is the sample size.
其中 x̄ 是样本均值，μ₀ 是假设的总体均值，σ 是总体标准差，n 是样本容量。
At the 5% es are z = ±1.96. If the calculated z-value lies below -1.96 or above +1.96, we reject H₀.
在 5% 显著性水平下，临界值为 z = ±1.96。如果计算出的 z 值小于 -1.96 或大于 +1.96，则拒绝 H₀。
At the 1% significance level, the critical values are z = ±2.5758, since each tail contains 0.5% of the probability.
在 1% 显著性水平下，临界值为 z = ±2.5758，因为每个尾部包含 0.5% 的概率。
6. Two-Tailed Test for Binomial Distribution | 二项分布下的双尾检验
d H₁: p ≠ p₀. The test statistic is simply the number of successes X observed in n trials.
对于二项检验，我们检验的是成功概率 p。假设为 H₀: p = p₀ 与 H₁: p ≠ p₀。检验统计量就是 n 次试验中观察到的成功次数 X。
Worked example: A coin is tossed 20 times and 4 heads are observed. Test at the 5% significance level whether the coin is fair.
例题：一枚硬币被抛掷 20 次，观察到 4 次正面。在 5% 显著性水平下检验这枚硬币是否均匀。
H₀: p = 0.5, H₁: p ≠ 0.5, where p is the probability of getting heads. Under H₀, X ~ B(20, 0.5).
H₀: p = 0.5，H₁: p ≠ 0.5，其中 p 是得到正面的概率。在 H₀ 下，X ~ B(20, 0.5)。
Since the observed value X = 4 falls into the lower critical region (X ≤ 5), we reject H₀.
根据前面的计算，拒绝域为 X ≤ 5 或 X ≥ 15。由于观测值 X = 4 落在下临界区域内（X ≤ 5），我们拒绝 H₀。
4) = 0.0059. Since this is less than 0.025, the result is significant at the two-tailed 5% level.
p 值为 H₀ 下的 P(X ≤ 4)。利用二项分布累积表，P(X ≤ 4) = 0.0059。由于该值小于 0.025，结果在 5% 双尾水平下显著。
Conclusion: There is sufficient evidence to suggest that the coin is biased.
结论：有充分证据表明这枚硬币是有偏的。
7. Two-Tailed Test for Poisson Distribution | 泊松分布下的双尾检验
istic is the observed count X, which under H₀ follows a Poisson distribution with mean λ₀.
对于泊松分布，我们检验平均发生率 λ。假设为 H₀: λ = λ₀ 与 H₁: λ ≠ λ₀。检验统计量是观测到的计数 X，在 H₀ 下服从均值为 λ₀ 的泊松分布。
Worked example: The number of accidents per day at a junction is known to follow a Poisson distribution with mean 2. After a new traffic light is installed, 6 accidents are recorded on one randomly chosen day. Test at the 5% significance level whether the accident rate has changed.
例题：某路口每天的事故数服从均值为 2 的泊松分布。安装新的交通灯后，随机选择的一天记录了 6 起事故。在 5% 显著性水平下检验事故率是否发生了变化。
H₀: λ = 2, H₁: λ ≠ 2. Under H₀, X ~ Po(2).
H₀: λ = 2，H₁: λ ≠ 2。在 H₀ 下，X ~ Po(2)。
itical value a and upper critical value b such that P(X ≤ a) < 0.025 and P(X ≥ b) < 0.025.
我们需要找到下临界值 a 和上临界值 b，使得 P(X ≤ a) < 0.025 且 P(X ≥ b) < 0.025。
, which is not less than 0.025, so the lower tail has no critical region. This means we can never reject H₀ on the lower side, no matter how small the observed value is.
从泊松分布表中查得：P(X ≤ 0) = 0.1353，不小于 0.025，因此下尾没有临界区域。这意味着无论观测值多么小，我们都不能在低侧拒绝 H₀。
.025. But P(X ≥ 5) = 1 - 0.9473 = 0.0527 > 0.025. Therefore the upper critical value is 6.
对于上尾，P(X ≥ 6) = 1 - P(X ≤ 5) = 1 - 0.9834 = 0.0166 < 0.025。但 P(X ≥ 5) = 1 - 0.9473 = 0.0527 > 0.025。因此上临界值为 6。
X = 6 falls in this region, we reject H₀. We conclude that the accident rate has changed.
拒绝域为 X ≥ 6。由于观测值 X = 6 落在此区域内，我们拒绝 H₀。结论是事故率发生了变化。
8. One-Tailed vs Two-Tailed Tests | 单尾检验与双尾检验的区别
The key difference between one-tailed and two-tailed tests lies in the alternative hypothesis. A one-tailed test uses either H₁: p > p₀ or H₁: p < p₀, while a two-tailed test uses H₁: p ≠ p₀.
单尾检验与双尾检验的关键区别在于备择假设。单尾检验使用 H₁: p > p₀ 或 H₁: p < p₀，而双尾检验使用 H₁: p ≠ p₀。
critical value for a one-tailed test at significance level α/2. For example, at α = 0.05:
双尾检验在显著性水平 α 下的临界值与单尾检验在显著性水平 α/2 下的临界值相同。例如，当 α = 0.05 时：
| Test type | H₁ | Critical value (z) | Reject H₀ if |
| Two-tailed | p ≠ p₀ | ±1.96 | |z| > 1.96 |
| One-tailed (right) | p > p₀ | +1.645 | z > 1.645 |
| One-tailed (left) | p < p₀ | -1.645 | z < -1.645 |
Note that a two-tailed test requires stronger evidence to reject H₀ in any given direction. This is why a result that is significant in a one-tailed test may not be significant in a two-tailed test at the same significance level.
注意，双尾检验在任一方向上都需要更强的证据才能拒绝 H₀。这就是为什么在相同显著性水平下，单尾检验显著的结果在双尾检验中可能不显著。
9. Common Mistakes in Two-Tailed Tests | 双尾检验的常见错误
Students often make the following mistakes when performing two-tailed tests:
同学们在做双尾检验时经常会犯以下错误：
Dividing the significance level by 2 when comparing the p-value: This is incorrect. The p-value in a two-tailed test is already doubled. Compare the p-value directly to α, not to α/2.
将显著性水平除以 2 再与 p 值比较：这是错误的。双尾检验中的 p 值已经乘以 2 了。应直接将 p 值与 α 比较，而不是与 α/2 比较。
ne-tailed p-value: When the distribution is symmetric, the two-tailed p-value is 2 × P(Z ≥ |z| alculate only one side and then compare it with α/2, which leads to incorrect conclusions.
忘记将单尾 p 值乘以 2：当分布对称时，双尾 p 值为 2 × P(Z ≥ |z|)。许多同学只计算了一侧的概率，然后与 α/2 比较，导致结论错误。
Using the wrong critical value: At the 5% level, some students use 1.645 for a two-tailed test, which is the critical value for a one-tailed test. The correct value is 1.96.
使用错误的临界值：在 5% 水平下，有些同学使用 1.645，而这其实是单尾检验的临界值。正确的值是 1.96。
Writing the conclusion without referencing the context: A conclusion should always mention the actual parameter and the real-world situation described in the question.
写结论时不结合题目背景：结论应始终提及实际参数以及题目中所描述的现实情境。
10. Worked Example: Full Exam-Style Question | 完整例题：考试风格题目
A manufacturer claims that the mean weight of a bag of flour is 500 g. A random sample of 50 bags is taken and the sample mean is found to be 496 g. The population standard deviation is known to be 12 g. Test at the 5% significance level whether the mean weight differs from 500 g.
某制造商声称一袋面粉的平均重量为 500 g。随机抽取 50 袋，样本均值为 496 g。已知总体标准差为 12 g。在 5% 显著性水平下检验平均重量是否与 500 g 不同。
Step 1: State the hypotheses.
第一步：写出假设。
H₀: μ = 500, H₁: μ ≠ 500
Step 2: Determine the significance level and critical values.
第二步：确定显著性水平和临界值。
α = 0.05. For a two-tailed normal test, the critical values are z = ±1.96.
α = 0.05。对于正态分布双尾检验，临界值为 z = ±1.96。
Step 3: Calculate the test statistic.
第三步：计算检验统计量。
z = (496 - 500) / (12 / √50) = -4 / 1.6971 = -2.357
Step 4: Compare with the critical value and make a decision.
第四步：与临界值比较并作出判断。
Since -2.357 < -1.96, the test statistic falls in the rejection region. We reject H₀.
因为 -2.357 < -1.96，检验统计量落在拒绝域内。我们拒绝 H₀。
the p-value is 2 × P(Z ≥ 2.357) = 2 × 0.0092 = 0.0184. Since 0.0184 < 0.05, we reject H₀.
或者，p 值为 2 × P(Z ≥ 2.357) = 2 × 0.0092 = 0.0184。因为 0.0184 < 0.05，我们拒绝 H₀。
Step 5: State the conclusion in context.
第五步：结合背景写出结论。
There is sufficient evidence at the 5% significance level to conclude that the mean weight of a bag of flour differs from 500 g.
在 5% 显著性水平下有充分证据表明，一袋面粉的平均重量与 500 g 不同。
11. Interpreting the Results Correctly | 正确解读结果
When we say "do not reject H₀", we do not mean that H₀ is proven to be true. It simply means that there is not enough evidence to conclude that the parameter is different from the hypothesised value. This is a subtle but important distinction in statistics.
当我们说“不拒绝 H₀”时，并不意味着 H₀ 被证明为真。它只是意味着没有足够的证据表明参数与假设值不同。这是统计学中一个微妙但重要的区别。
Similarly, "reject H₀" means that the sample data provide sufficient evidence against H₀, but it does not mean the alternative hypothesis is definitely true. There is always a possibility of a Type I error.
同样，“拒绝 H₀”意味着样本数据提供了充分的证据反对 H₀，但这并不意味着备择假设一定为真。总是存在犯第一类错误的可能性。
In a two-tailed test, rejecting H₀ tells us only that the parameter has changed, but it does not tell us whether it has increased or decreased. To determine the direction, we look at the sign of the test statistic or the sample mean.
在双尾检验中，拒绝 H₀ 只能告诉我们参数发生了变化，但不能告诉我们它是增大还是减小。要确定方向，我们需要看检验统计量或样本均值的符号。
For example, in the previous question, the test statistic was negative, so the evidence suggests the mean weight is likely below 500 g. However, the formal conclusion of a two-tailed test does not make this claim.
例如，在前一题中，检验统计量为负，因此证据表明平均重量可能低于 500 g。然而，双尾检验的正规结论并不会作出这一断言。
12. Summary of Key Points | 关键要点总结
The following table summarises the most important ideas to remember for two-tailed hypothesis testing in A-Level exams:
下表总结了 A-Level 考试中双尾假设检验最需要记住的要点：
| Key idea | Important detail | 关键要点 |
| Alternative hypothesis | H₁: p ≠ p₀ or H₁: μ ≠ μ₀ | 备择假设为不等于 |
| Significance level split | α/2 in each tail | 每个尾部各占 α/2 |
| Normal critical values | 5% → ±1.96; 1% → ±2.5758 | 正态临界值 |
| Decision rule | Reject H₀ if statistic is in critical region | 统计量落入拒绝域则拒绝 |
| p-value method | Reject H₀ if p-value < α | p 值小于 α 则拒绝 |
| Discrete distributions | Use largest region with probability < α/2 per tail | 离散分布取每尾不超过 α/2 的最大区域 |
Mastering the rejection region and decision rules for two-tailed tests requires consistent practice. Always write out the hypotheses, check whether you are using the correct critical value, and state your conclusion in the context of the problem. With careful attention to these details, you can confidently earn full marks on hypothesis testing questions.
掌握双尾检验的拒绝域与判断规则需要持续的练习。始终写出假设，检查使用的是否是正确的临界值，并结合作答背景写出结论。只要注意这些细节，你就能自信地在假设检验题目中获得满分。
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