📚 A-Level Maths: How to Determine a One-Tailed Test | A-Level 数学：单尾检验的判定方法
In A-Level Mathematics, one of the most common mistakes in hypothesis testing is failing to identify whether a test is one-tailed or two-tailed. This decision determines the critical region, the p-value calculation, and ultimately whether you reject the null hypothesis. In this article, we will explore how to determine a one-tailed test quickly and accurately.
在 A-Level 数学中，假设检验最常见的错误之一就是未能判断检验是单尾还是双尾。这一判断决定了临界区域、p 值计算，并最终影响你是否拒绝原假设。本文将帮助你快速而准确地判定单尾检验。
1. Setting Up a Hypothesis Test | 建立假设检验
f a coin is claimed to be fair, we write H₀: p = 0.5, where p is the probability of heads.
每一个假设检验都从两个陈述开始。原假设记作 H₀，代表当前的观点或无变化的说法；备择假设记作 H₁，代表我们试图寻找证据支持的说法。例如，若一枚硬币被宣称是均匀的，我们写 H₀: p = 0.5，其中 p 是出现正面的概率。
The alternative hypothesis is the most important part when deciding the tail of the test. It tells us the direction of the suspected deviation from H₀.
在判断检验尾部时，备择假设是最关键的部分。它告诉我们怀疑偏离 H₀ 的方向。
2. One-Tailed vs Two-Tailed Tests | 单尾检验与双尾检验
A one-tailed test is directional. It tests whether a parameter has increased or decreased in a specific direction. For a binomial proportion p, a one-tailed test has an alternative hypothesis of the form H₁: p > p₀ (upper-tailed) or H₁: p < p₀ (lower-tailed).
单尾检验是方向性的。它检验参数是否朝某个特定方向增大或减小。对于二项分布比例 p 而言，单尾检验的备择假设形如 H₁: p > p₀（上尾）或 H₁: p < p₀（下尾）。
A two-tailed test is non-directional. It checks whether the parameter has changed without specifying a direction, so the alternative hypothesis is H₁: p ≠ p₀. This means both extremes of the distribution contribute to the critical region.
双尾检验是非方向性的。它检查参数是否发生变化但不指定方向，因此备择假设是 H₁: p ≠ p₀。这意味着分布两端极端值共同构成临界区域。
3. The Alternative Hypothesis Tells You the Tail | 备择假设决定尾部
To determine whether a test is one-tailed, look only at the inequality sign in H₁. If H₁ contains the symbol ">" then it is an upper-tailed (right-tailed) test. If H₁ contains the symbol "<" then it is a lower-tailed (left-tailed) test. If H₁ contains "≠" then the test is two-tailed.
要判断是否为单尾检验，只需看 H₁ 中的不等号。若 H₁ 包含 “>” 则为上尾（右尾）检验；若 H₁ 包含 “<” 则为下尾（左尾）检验；若 H₁ 包含 “≠” 则为双尾检验。
H₁: p > p₀ → upper-tailed test
H₁: p < p₀ → lower-tailed test
H₁: p ≠ p₀ → two-tailed test
orm H₀: p = p₀ (or sometimes H₀: p ≤ p₀ or H₀: p ≥ p₀, but you will usually use equality).
注意原假设通常为 H₀: p = p₀（有时也可写 H₀: p ≤ p₀ 或 H₀: p ≥ p₀，但通常你使用等号形式）。
4. Keywords in the Question | 题目中的关键词
Exam questions often use words that indicate the direction. The table below lists common phrases and the corresponding tail.
考试题目经常使用一些表明方向的词。下表列出常见短语及其对应的尾部。
| Direction | English phrases | 中文短语 | Tail |
| Increase | "greater than", "more than", "improve", "increase", "higher", "exceeds" | “大于”“超过”“增加”“提高”“优于” | upper-tailed |
| Decrease | "less than", "fewer than", "decrease", "reduce", "lower", "worse" | “小于”“少于”“减少”“降低”“不如” | lower-tailed |
| Change either way | "changed", "different from", "not equal to", "has not remained" | “改变”“与…不同”“不等于” | two-tailed |
Be careful: the phrase "no more than" a lower-tailed test for p (testing whether p is too high). Always translate the wording into an inequality before deciding.
请注意：“no more than” 表示 ≤，因此对应的是对 p 的下尾检验（检验 p 是否过高）。在判断之前，一定先把文字翻译成不等式。
5. Lower-Tailed (Left-Tailed) Test | 下尾（左尾）检验
In a lower-tailed test, the critical region lies entirely in the lower end of the sampling distribution. We suspect the true proportion is smaller than the claimed value.
在下尾检验中，临界区域完全位于抽样分布的低端。我们怀疑真实比例小于所声称的值。
uch that P(X ≤ c) ≤ α, where α is the significance level. The critical region is {0, 1, 2, …, c}.
假设我们抽取大小为 n 的随机样本并统计成功次数 X。对于下尾检验，我们寻找临界值 c，使得 P(X ≤ c) ≤ α，其中 α 是显著性水平。临界区域为 {0, 1, 2, …, c}。
P(X ≤ c) ≤ α
We then reject H₀ if the observed number of successes x is less than or equal to c.
然后若观测到的成功次数 x 小于或等于 c，我们就拒绝 H₀。
6. Upper-Tailed (Right-Tailed) Test | 上尾（右尾）检验
In an upper-tailed test, the critical region lies entirely in the upper end of the distribution. We suspect the true proportion is larger than the claimed value.
在上尾检验中，临界区域完全位于分布的高端。我们怀疑真实比例大于所声称的值。
tion, we find the critical value d such that P(X ≥ d) ≤ α. The critical region is {d, d+1, …, n}.
对于二项分布的上尾检验，我们寻找临界值 d，使得 P(X ≥ d) ≤ α。临界区域为 {d, d+1, …, n}。
P(X ≥ d) ≤ α
We reject H₀ if the observed number of successes x is greater than or equal to d.
若观测到的成功次数 x 大于或等于 d，我们就拒绝 H₀。
7. Using p-values | 使用 p 值
Many A-Level questions ask you to use the p-value method. For a one-tailed test, the p-value is computed on the side indicated by H₁:
许多 A-Level 题目要求使用 p 值法。对于单尾检验，p 值在 H₁ 所指示的那一侧计算：
For H₁: p > p₀, the p-value is P(X ≥ x) under H₀.
对于 H₁: p > p₀，p 值为在原假设下 P(X ≥ x)。
For H₁: p < p₀, the p-value is P(X ≤ x) under H₀.
对于 H₁: p < p₀，p 值为在原假设下 P(X ≤ x)。
If the p-value is less than the significance level α, reject H₀. If it is greater than α, do not reject H₀.
若 p 值小于显著性水平 α，则拒绝 H₀；若 p 值大于 α，则不拒绝 H₀。
8. Worked Example: Lower-Tailed Test | 完整示例：下尾检验
A pesticide company claims that 40% of a certain pest population is killed by its new product. A researcher suspects the product is actually less effective. In a sample of 20 pests, only 5 were killed. Test at the 5% significance level.
某农药公司声称其新产品能杀死 40% 的某种害虫。研究人员怀疑该产品实际效果较低。在 20 只害虫的样本中，只有 5 只被杀死。在 5% 显著性水平下进行检验。
Let p be the probability that a pest is killed. Since the researcher suspects a lower effectiveness, we write:
设 p 为害虫被杀死概率。由于研究人员怀疑效果更低，我们写：
H₀: p = 0.4, H₁: p < 0.4
This is a lower-tailed test. Under H₀, X ~ B(20, 0.4). We need the probability of seeing 5 or fewer successes:
这是下尾检验。在原假设下，X ~ B(20, 0.4)。我们需要计算出现 5 次或更少成功的概率：
P(X ≤ 5) = 0.0210 (from tables or calculator)
Since 0.0210 < 0.05, we reject H₀. There is sufficient evidence that the product is less effective than claimed.
因为 0.0210 < 0.05，我们拒绝 H₀。有充分证据表明该产品不如声称的那么有效。
9. Worked Example: Upper-Tailed Test | 完整示例：上尾检验
A machine is supposed to produce screws with a defect rate of 10%. After a repair, an engineer believes the defect rate has not increased, but a quality inspector claims it has increased. The inspector tests 15 screws and finds 5 defective. Test at the 1% significance level.
某机器生产的螺丝缺陷率应为 10%。维修后，工程师认为缺陷率没有增加，但质量检查员声称缺陷率增加了。检查员测试 15 颗螺丝，发现 5 颗有缺陷。在 1% 显著性水平下检验。
Let p be the defect rate. Since the claim is an increase, we write:
设 p 为缺陷率。由于声称的是增加，我们写：
H₀: p = 0.1, H₁: p > 0.1
This is an upper-tailed test. Under H₀, X ~ B(15, 0.1). We need P(X ≥ 5):
这是上尾检验。在原假设下，X ~ B(15, 0.1)。我们需要 P(X ≥ 5)：
P(X ≥ 5) = 1 − P(X ≤ 4) = 1 − 0.9873 = 0.0127
Since 0.0127 > 0.01, we do not reject H₀. There is not enough evidence to support the inspector's claim.
因为 0.0127 > 0.01，我们不拒绝 H₀。没有足够证据支持检查员的说法。
10. Finding Critical Values for a One-Tailed Test | 求单尾检验的临界值
h that P(X ≤ c) ≤ α. For an upper-tailed test, find the smallest d such that P(X ≥ d) ≤ α.
有时题目要求求临界区域而不是 p 值。对于显著性水平 α 的下尾检验，求最大的 c 使得 P(X ≤ c) ≤ α。对于上尾检验，求最小的 d 使得 P(X ≥ d) ≤ α。
For an upper-tailed test, you may need to use the complement rule:
对于上尾检验，你可能需要使用补集法则：
P(X ≥ d) = 1 − P(X ≤ d − 1) ≤ α
Therefore, find the smallest d such that P(X ≤ d − 1) ≥ 1 − α.
因此，求最小的 d 使得 P(X ≤ d − 1) ≥ 1 − α。
11. Common Mistakes and Exam Tips | 常见错误与考试建议
Confusing the tail direction: Always write H₁ first and check the inequality sign. If H₁ uses ">", shade the right tail; if it uses "<", shade the left tail.
混淆尾部方向：先写出 H₁，再检查不等号。若 H₁ 用 “>”，涂右侧尾部；若用 “<”，涂左侧尾部。
Using the wrong p-value: For one-tailed tests, never double the p-value. Doubling is only for two-tailed tests.
用错 p 值：单尾检验绝不可以把 p 值翻倍。翻倍只适用于双尾检验。
Writing H₁ incorrectly: H₁ must be expressed in terms of the population parameter, not the sample statistic. Do not write H₁: x > 0.4; write H₁: p > 0.4.
写错 H₁：H₁ 必须用总体参数表达，而不是样本统计量。不要写 H₁: x > 0.4，要写 H₁: p > 0.4。
Forgetting the actual observed value: State whether x falls in the critical region, and always conclude in the context of the question.
忘记实际观测值：说明 x 是否落入临界区域，并始终用题目的情境下结论。
Saying "accept H₀": When the test statistic is not in the critical region, say "there is insufficient evidence to reject H₀" rather than "accept H₀".
说“接受 H₀”：当检验统计量不在临界区域内时，应说“没有足够证据拒绝 H₀”，而不是“接受 H₀”。
12. Quick Checklist for One-Tailed Tests | 单尾检验快速检查清单
Use this checklist in the exam to avoid losing marks.
在考试中使用这个检查清单，以避免失分。
Read the question: find the key phrase that indicates direction (increase / decrease / different).
阅读题目：找出表明方向的关键短语（增加 / 减少 / 不同）。
Define the parameter p and write H₀ and H₁ with the correct inequality.
定义参数 p，并写出带有正确不等号的 H₀ 和 H₁。
State the distribution of X under H₀, e.g. X ~ B(20, 0.4).
写出在原假设下 X 的分布，例如 X ~ B(20, 0.4)。
Calculate the p-value or find the critical region using the appropriate tail.
利用相应的尾部计算 p 值或求出临界区域。
Compare the p-value with α, or check whether the observed value lies in the critical region.
将 p 值与 α 比较，或检查观测值是否落在临界区域内。
Write a conclusion in context, using "reject H₀" or "insufficient evidence to reject H₀".
结合情境写出结论，使用“拒绝 H₀”或“没有足够证据拒绝 H₀”。
Remember: the single most important step is to identify the direction of H₁. Once you have the correct inequality, the rest of the test follows mechanically.
请记住：最关键的一步是判断 H₁ 的方向。一旦你得到了正确的不等号，其余检验步骤就会按部就班地完成。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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