📚 One-tailed tests | 单尾检验
In A-Level Statistics, a one-tailed test is a hypothesis test where the alternative hypothesis specifies a direction of the effect — either an increase or a decrease. It allows you to detect whether a parameter is significantly greater than or less than a hypothesised value, concentrating the entire significance level in one tail of the sampling distribution. Understanding one-tailed tests is essential for solving real-world problems, from quality control to clinical trials, where you have a clear expectation about the direction of change.
在A-Level统计学中，单尾检验是一种假设检验，其备择假设明确指出了效应的方向——要么增加，要么减少。它让你能够判断参数是显著大于还是小于某个假设值，将整个显著性水平集中在抽样分布的一个尾部。在质量控制、临床试验等实际问题中，当变化方向有明确预期时，理解单尾检验不可或缺。
1. What is a One-Tailed Test? | 什么是单尾检验？
A one-tailed test, also called a directional test, is used when the research question predicts that the population parameter will lie either above or below a certain value, but not both. Instead of splitting the significance level (α) between two tails, the entire α is placed in one tail of the distribution. This increases the test's power to detect an effect in the specified direction, but it cannot detect an opposite effect.
单尾检验，也称方向性检验，用于研究问题预测总体参数只会高于或低于某一特定值，而非两者都可能的情况。它不将显著性水平α分配到分布的两侧尾部，而是将整个α全部放在一个尾部。这增强了检验在指定方向上发现效应的能力，但无法检测相反方向的效应。
For example, if a manufacturer claims a new light bulb lasts longer than 1500 hours, a one-tailed upper-tail test is appropriate. If we want to prove the proportion of defective items is less than 5%, we use a lower-tail test.
例如，如果制造商声称新型灯泡的寿命超过1500小时，就适合采用上尾单尾检验。如果想证明缺陷品比例低于5%，则使用下尾检验。
Key point: one-tailed test → H₁ uses < or > ; two-tailed test → H₁ uses ≠
2. Stating the Hypotheses | 陈述假设
thesis H₁ expresses the strict directional claim (< or >). The pair must be complementary and cover all possibilities. For an upper-tail test, H₁: parameter > value; for a lower-tail test, H₁: parameter < value.
在单尾检验中，原假设H₀总包含等号（=、≤ 或 ≥），而备择假设H₁表达严格的方向性主张（< 或 >）。两者必须互补且涵盖所有可能性。对于上尾检验，H₁：参数 > 某个值；对于下尾检验，H₁：参数 < 某个值。
| Test Type | H₀ | H₁ |
| Upper-tail | μ = 50 (or μ ≤ 50) | μ > 50 |
| Lower-tail | p = 0.3 (or p ≥ 0.3) | p < 0.3 |
ime. The null hypothesis is assumed true unless there is sufficient evidence to reject it.
一定要用文字定义参数，例如 μ = 总体平均时间。除非有足够证据拒绝原假设，否则假定原假设成立。
3. Significance Level and Critical Region | 显著性水平与拒绝域
The significance level α is the probability of rejecting H₀ when it is actually true (Type I error). For a one-tailed test, the critical region consists of the most extreme α proportion of the sampling distribution in the direction specified by H₁. If the test statistic falls into this region, we reject H₀.
显著性水平α是当H₀为真时错误拒绝它的概率（第一类错误）。在单尾检验中，拒绝域由抽样分布在H₁指定方向上最极端的α比例构成。如果检验统计量落入该区域，就拒绝H₀。
ues are 0.05 and 0.01. In an upper-tail test at α = 0.05, the critical region is the top 5% of the distribution. The boundary is called the critical value.
常见的α值为0.05和0.01。对于α=0.05的上尾检验，拒绝域是分布顶部5%的区域。边界值称为临界值。
4. Finding Critical Values (Normal Distribution) | 查找临界值（正态分布）
or a one-tailed z-test at α = 0.05, the critical values are z = 1.645 (upper-tail) and z = –1.645 (lower-tail).
当检验统计量服从正态分布时，临界值是z值或t值，取决于总体标准差是否已知。对于α=0.05的单尾z检验，临界值为z=1.645（上尾）和z=–1.645（下尾）。
Upper-tail: reject H₀ if z > 1.645
Lower-tail: reject H₀ if z < –1.645
ribution (unknown σ), the critical value comes from t-tables with degrees of freedom ν = n – e critical t-value for an upper-tail test with α = 0.05 and ν = 20 is approximately 1.725.
对于t分布（σ未知），临界值来自自由度为ν=n–1的t表。α=0.05、ν=20的上尾检验临界t值约为1.725。
5. One-Tailed Test with a Binomial Distribution | 二项分布的单尾检验
e test involves a proportion p, we often use the binomial distribution directly. Suppose X ~ per-tail test H₁: p > p₀, we calculate P(X ≥ x). This probability is then compared with α.
当检验涉及比例p时，通常直接使用二项分布。假设在H₀下X~B(n, p₀)，观察到的成功次数为x。对于下尾检验H₁: p < p₀，计算P(X ≤ x)；对于上尾检验H₁: p > p₀，计算P(X ≥ x)。然后将该概率与α比较。
If the calculated probability is less than α, the result is significant and we reject H₀. Note that the significance level often cannot be achieved exactly due to the discrete nature of the binomial; the actual size of the test may be slightly less than α.
如果计算出的概率小于α，则结果显著，拒绝H₀。注意，由于二项分布的离散性，往往无法精确达到显著性水平；检验的实际规模可能会略小于α。
6. Calculating the p-Value | 计算p值
value = P(statistic ≥ observed). For a lower-tail test, p-value = P(statistic ≤ observed).
p值是在原假设下，得到与观察结果同样极端或更极端（沿H₁方向）的概率。对于上尾检验，p值 = P(统计量 ≥ 观察值)；对于下尾检验，p值 = P(统计量 ≤ 观察值)。
We reject H₀ if p-value < α. The p-value approach provides more information than the critical region method, as it tells us how significant the result is.
如果p值 < α，则拒绝H₀。p值法比拒绝域法提供更多信息，因为它能告诉我们结果有多么显著。
7. Decision Rules: Reject or Do Not Reject H₀ | 决策规则：拒绝或不拒绝原假设
A one-tailed test gives a clear decision rule. If the test statistic falls in the critical region (e.g. z > zcrit for upper-tail) or if p-value < α, we reject H₀. Otherwise, we do not reject H₀. It is important to phrase conclusions in context: "There is sufficient evidence at the α level to suggest the mean is greater than 50."
单尾检验给出明确的决策规则。如果检验统计量落入拒绝域（如上尾检验中z > zcrit），或者p值 < α，则拒绝H₀。否则，不拒绝H₀。重要的是结合上下文表述结论：“在α水平下，有足够证据表明均值大于50。”
Never say "accept H₀" — failing to reject H₀ simply means the evidence is not strong enough to support H₁.
永远不要说“接受H₀”——未能拒绝H₀仅仅意味着证据不足以支持H₁。
8. Worked Example 1: Upper-Tail Test for a Mean | 实例1：均值的上尾检验
A machine fills bags of sugar. The stated weight is 500 g, and past records show the standard deviation is 4 g. A new technician suspects overfilling. A random sample of 25 bags gives a mean weight of 502.1 g. Test at the 5% significance level whether there is evidence of overfilling.
一台机器装袋糖。标称重量为500 g，历史记录显示标准差为4 g。一位新技术员怀疑装填过量。随机抽取25袋，得出平均重量为502.1 g。在5%显著性水平下检验是否存在装填过量的证据。
Solution = 500, H₁: μ > 500 (upper-tail). Population σ known, so use z-test. Test statistic: z = (x̄ – μ) / (σ/√n) = (502.1 – 500) / (4/√25) = 2.1 / 0.8 = 2.625. Critical value for α=0.05 upper-tail: zcrit p-value = P(Z > 2.625) ≈ 0.0043 < 0.05. Conclusion: There is significant evidence at the 5% level that the mean fill weight exceeds 500 g.
解：H₀: μ = 500, H₁: μ > 500（上尾）。总体σ已知，用z检验。检验统计量：z = (x̄ – μ) / (σ/√n) = (502.1 – 500) / (4/√25) = 2.1 / 0.8 = 2.625。α=0.05上尾临界值：zcrit = 1.645。由于2.625 > 1.645，拒绝H₀。或p值 = P(Z > 2.625) ≈ 0.0043 < 0.05。结论：在5%水平下，有显著证据表明平均填充重量超过500 g。
9. Worked Example 2: Lower-Tail Test for a Proportion | 实例2：比例的下尾检验
A school claims that at least 70% of its students achieve grade B or above in Mathematics. A parent survey of 50 students finds 30 with B or above. Test at the 1% significance level whether the proportion is less than 70%.
一所学校声称至少70%的学生数学获得B级或以上。一位家长调查了50名学生，发现30名获得B或以上。在1%显著性水平下检验该比例是否低于70%。
H₀: p = 0.7 (or p ≥ 0.7), H₁: p < 0.70. X ~ = 0.01. Since 0.0319 > 0.01, we do not reject H₀. There is insufficient evidence at the 1% level to conclude the true proportion is less than 0.70.
H₀: p = 0.7（或p ≥ 0.7），H₁: p < 0.70。X ~ B(50, 0.7)。观察成功次数 = 30。p值 = P(X ≤ 30)。查二项分布表：P(X ≤ 30) ≈ 0.0319。α = 0.01。由于0.0319 > 0.01，不拒绝H₀。在1%水平下，没有足够证据得出结论真实比例低于0.70。
Note: The binomial test often requires careful continuity correction if approximated by normal.
注意：如果用正态近似，二项检验通常需要谨慎进行连续性修正。
10. Common Mistakes and Tips | 常见错误与提示
One common error is using a one-tailed test when the direction of effect is not specified in advance. Choosing a one-tailed test after seeing the data inflates the Type I error rate. Always decide the test type based on the research question before collecting data.
一个常见错误是在事先未指定效应方向时使用单尾检验。看到数据之后再选择单尾检验会扩大第一类错误率。务必在收集数据之前根据研究问题确定检验类型。
Another pitfall is mixing up the inequality sign: make sure the alternative hypothesis matches the suspicion. Double-check that critical values match the tail—negative for lower-tail, positive for upper-tail in symmetric distributions.
另一个陷阱是搞乱不等号方向：确保备择假设与怀疑一致。仔细检查临界值与尾部匹配——对称分布中下尾为负值，上尾为正值。
When using tables, always state the actual significance level if it falls between tabled values, especially for discrete distributions.
使用表格时，如果实际显著性水平介于表列值之间，务必说明，特别是离散分布。
11. One-Tailed vs Two-Tailed Tests | 单尾与双尾检验对比
A two-tailed test splits α equally into both tails and uses H₁: ≠. It is safer when the direction of effect is unknown, but requires a more extreme test statistic to reject H₀ for the same α. The one-tailed test is more powerful for the specified direction, but completely misses an effect in the opposite direction.
双尾检验将α平分到两个尾部，并使用H₁: ≠。当效应方向未知时更为稳妥，但在相同α下需要更极端的检验统计量才能拒绝H₀。单尾检验对指定方向功效更高，但会完全错过相反方向的效应。
| Feature | One-tailed | Two-tailed |
| H₁ | > or < | ≠ |
| Critical region | One tail (α) | Both tails (α/2 each) |
| Power for direction | Higher | Lower |
| Critical z (α=0.05) | ±1.645 | ±1.96 |
In Edexcel exam questions, carefully read the wording: "test whether the mean has increased" signals a one-tailed upper test; "test whether the proportion has changed" signals two-tailed.
在Edexcel考试题目中，仔细阅读措辞：“检验均值是否增加”表示单尾上侧检验；“检验比例是否发生变化”表示双尾检验。
12. Practice and Summary | 练习与总结
Mastering one-tailed tests involves being able to formulate hypotheses, select the correct distribution, calculate test statistics, find critical values or p-values, and draw conclusions in context. Practice with a variety of scenarios — means, proportions, differences between means — using both normal and t distributions as well as the binomial. Always sketch a graph of the distribution, mark the rejection region, and write your conclusion addressing H₁.
掌握单尾检验需要能建立假设、选择正确分布、计算检验统计量、找出临界值或p值，并结合上下文得出结论。通过多种场景进行练习——均值、比例、均值差——使用正态分布、t分布以及二项分布。始终画出分布示意图，标出拒绝域，并针对H₁给出结论。
Remember: one-tailed tests are a powerful tool when you have a clear directional research hypothesis. Used correctly, they allow you to draw meaningful conclusions with less data than their two-tailed counterparts. Keep reviewing the Edexcel formula booklet and tables to speed up your work in the exam.
记住：当你拥有明确的方向性研究假设时，单尾检验是一个有力的工具。正确使用，它能让你用更少的数据得出有意义的结论，相比双尾检验更高效。持续复习Edexcel公式手册和表格，以加快考试中的解题速度。
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