One-Tailed Tests: Rejection Regions and Decision Rules | 单尾检验的拒绝域与判断规则

📚 One-Tailed Tests: Rejection Regions and Decision Rules | 单尾检验的拒绝域与判断规则

In hypothesis testing, a one-tailed test checks whether a population parameter has moved in a particular direction: greater than or less than a stated value. The entire significance level is placed in one tail of the sampling distribution, which makes the test more powerful for detecting an effect in that direction.

在假设检验中，单尾检验用于判断总体参数是否朝某个特定方向变化：大于或小于某个指定值。整个显著性水平被放置在抽样分布的一侧尾部，因此对特定方向上的效应检测更敏感、功效更高。

1. Directional Hypotheses | 方向性假设

Every hypothesis test begins with two competing statements. The null hypothesis H₀ is a statement of no effect or no change, and the alternative hypothesis H₁ specifies what the researcher suspects is true. In a one-tailed test, H₁ is directional and uses > or <.

每个假设检验都从两个相互对立的陈述开始。原假设 H₀ 表示无效应或没有变化，备择假设 H₁ 说明研究者所怀疑的真实情况。在单尾检验中，H₁ 具有方向性，使用“>”或“<”。

For a right-tailed test, write H₀: μ ≤ μ₀ and H₁: μ > μ₀. For a left-tailed test, write H₀: μ ≥ μ₀ and H₁: μ < μ₀.

右尾检验写成 H₀: μ ≤ μ₀ 与 H₁: μ > μ₀；左尾检验写成 H₀: μ ≥ μ₀ 与 H₁: μ < μ₀。

2. Why Use a One-Tailed Test? | 为什么使用单尾检验？

A one-tailed test is appropriate when a theory or prior evidence predicts a specific direction. It is also useful when a deviation in one direction is scientifically meaningful while a deviation in the other direction is not.

当理论或先前证据预测了特定方向时，适合使用单尾检验。当只有某一方向的偏差具有科学意义，而另一方向的偏差没有意义时，单尾检验也更有针对性。

Because all of α is concentrated in one tail, the critical value is closer to the hypothesised parameter value than in a two-tailed test. As a result, the test has higher power in that direction for a fixed sample size.

由于全部 α 都集中在单侧尾部，临界值比双尾检验更靠近假设的参数值。因此，在固定样本量的条件下，单尾检验在该方向上的检验功效更高。

If there is no strong justification for a directional claim, a two-tailed test is usually safer because it protects against unexpected effects in both directions.

如果没有充分理由支持方向性主张，通常使用双尾检验更稳妥，因为它能同时防范两个方向上出现的意外效应。

3. Significance Level and Tail Probability | 显著性水平与尾部概率

The significance level α is the probability of rejecting a true null hypothesis. In a one-tailed test, α is placed entirely in one tail of the distribution. If α = 0.05, then the area of the rejection region on the chosen side is exactly 0.05.

显著性水平 α 是拒绝真实原假设的概率。在单尾检验中，α 完全放置在分布的一侧。若 α = 0.05，则所选一侧拒绝域的面积恰好为 0.05。

This tail probability determines the critical z or t value. For example, with α = 0.05 in a right-tailed z-test, the critical value is z = 1.645, meaning the area to the right of 1.645 equals 0.05.

该尾部概率决定了临界 z 值或 t 值。例如，右尾 z 检验中 α = 0.05 时，临界值为 z = 1.645，即 1.645 右侧的面积等于 0.05。

P(Z > z_α) = α for a right-tailed test

P(Z < -z_α) = α for a left-tailed test

4. Test Statistic and Null Distribution | 检验统计量与原假设分布

The test statistic measures the distance between the sample estimate and the hypothesised value in standard error units. For a population mean with known σ, a z-test uses:

检验统计量衡量样本估计值与假设值之间相差多少个标准误。当总体标准差 σ 已知时，z 检验使用：

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