A-Level Mathematics: Hypothesis Testing Steps and Logic | A-Level 数学：假设检验的步骤与逻辑

📚 A-Level Mathematics: Hypothesis Testing Steps and Logic | A-Level 数学：假设检验的步骤与逻辑

Hypothesis testing is a central topic in A-Level Mathematics and Statistics. It provides a formal method for deciding whether an observed result is genuinely significant or could easily have happened by chance. In this article, we will break down the logic and the standard decision-making steps so that you can apply them confidently in exams.

假设检验是 A-Level 数学与统计学的核心内容。它提供了一套规范的流程，用来判断观察到的结果究竟真的“显著”，还是仅仅可能由随机波动造成。本文将从逻辑与标准步骤两个角度，帮助你系统掌握假设检验，从而在考试中运用自如。

1. What is Hypothesis Testing? | 什么是假设检验

A statistical hypothesis is a claim about a population parameter such as p, μ, or σ. Because we rarely have data for the whole population, we use a sample to decide whether the claim is plausible. The reasoning is indirect: we first assume that a “null hypothesis” is true, then ask how likely the observed sample would be under that assumption.

统计假设是对总体参数（如 p、μ 或 σ）所作出的论断。我们很难获得完整总体数据，因此需要通过样本判断该论断是否合理。其推理是间接的：我们先假定“原假设”成立，然后计算在这样一个假设下，得到当前样本结果的可能性有多大。

If the observed data are very unlikely under the null hypothesis, we conclude that the data provide evidence against it. If the data are not unlikely, we say that there is insufficient evidence to reject the null hypothesis.

如果原假设成立时，当前样本出现的概率非常小，我们就认为数据提供了反对原假设的证据；如果概率并不小，我们就说“没有足够证据拒绝原假设”。

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

The null hypothesis, written H₀, usually represents “no change”, “no effect”, or “no difference”. It is assumed to be true when we perform the probability calculation. The alternative hypothesis, written H₁, is what we suspect might be true and what we wish to find evidence for.

原假设记作 H₀，通常表示“没有变化”“没有效应”或“没有差异”。在进行概率计算时，我们假定 H₀ 成立。备择假设记作 H₁，是我们怀疑可能成立的情况，也是我们希望通过样本寻找证据支持的方向。

• Two-tailed: H₀: p = 0.5, H₁: p ≠ 0.5. We are testing for a difference in either direction.
• 双尾检验：H₀: p = 0.5，H₁: p ≠ 0.5。我们检验的是任意方向上的差异。
• One-tailed: H₀: p = 0.5, H₁: p > 0.5. We are testing for an increase or a decrease in a specific direction.
• 单尾检验：H₀: p = 0.5，H₁: p > 0.5。我们检验的是某个特定方向的增大或减小。

Always write hypotheses in terms of the population parameter, not the sample statistic. For example, write H₀: p = 0.5, not H₀: X = 14. The sample statistic is the evidence; the hypothesis is about the population.

写假设时，必须使用总体参数，而不是样本统计量。例如应写 H₀: p = 0.5，而不是 H₀: X = 14。样本统计量是证据，而假设关心的是总体。

3. Significance Level and Test Statistic | 显著性水平与检验统计量

The significance level, usually denoted α, is the probability we are willing to accept of rejecting H₀ when H₀ is actually true. In A-Level questions, α is often 5% (0.05), but it can also be 1% or 10%. The significance level is chosen before looking at the data.

显著性水平通常记为 α，它表示我们愿意承担“原假设本身为真，却错误地拒绝了它”的最大概率。在 A-Level 考试中，α 通常取 5%（0.05），有时也取 1% 或 10%。显著性水平必须在看到样本数据之前预先设定。

A test statistic is a value calculated from the sample. It transforms the raw data into a number that can be compared with a known probability distribution. For example, in a binomial test, the test statistic may be the observed number of successes, X; in a normal test, it may be the z-score.

检验统计量是根据样本计算出来的数值。它把原始数据转化为一个可以与已知概率分布比较的量。例如，在二项分布检验中，检验统计量可以是观测到的成功次数 X；在正态分布检验中，检验统计量可以是 z 分数。

4. One-Tailed and Two-Tailed Tests | 单尾与双尾检验

Use a one-tailed test when H₁ is directional. For example, if we suspect a coin is biased towards heads, we use H₁: p > 0.5. If we suspect a new medicine reduces recovery time, we use H₁: μ < μ₀.

当 H₁ 具有方向性时，应使用单尾检验。例如，怀疑一枚硬币偏向正面时，用 H₁: p > 0.5；怀疑一种新药能缩短康复时间时，用 H₁: μ < μ₀。

Use a two-tailed test when H₁ is non-directional. For example, if we only want to know whether the coin is biased in either direction, we use H₁: p ≠ 0.5. For a two-tailed test, the total probability in the critical region is α, so it is split across both tails.

当 H₁ 没有方向性时，应使用双尾检验。例如，只想判断硬币是否有偏，无论偏正面还是偏反面，就用 H₁: p ≠ 0.5。在双尾检验中，拒绝域的总概率为 α，因此这个概率被分配到两个尾端。

Reject H₀ if the test statistic lies in the critical region, or if p-value ≤ α.

若检验统计量落在拒绝域内，或 p 值 ≤ α，则拒绝 H₀。

5. Critical Region and Critical Values | 拒绝域与临界值

The critical region is the set of values of the test statistic for which H₀ is rejected. The critical value is the boundary between the critical region and the rest of the distribution.

拒绝域是使原假设 H₀ 被拒绝的检验统计量取值集合。临界值则是拒绝域与其余取值范围之间的分界点。

For a continuous distribution, the critical region is chosen so that its total probability is exactly α. For a discrete distribution, such as the binomial distribution, we choose the largest critical region whose probability is less than or equal to α. Because of this, the actual significance level may be slightly smaller than the nominal α.

对于连续型分布，拒绝域的总概率恰好等于 α。对于离散型分布（例如二项分布），我们选择概率不超过 α 的最大拒绝域。因此，实际显著性水平可能略小于设定的名义 α。

Example: if X ∼ Bin(20, 0.5) and α = 0.05 for a one-tailed test towards heads, then P(X ≥ 15) = 0.0207, while P(X ≥ 14) = 0.0577. Since 0.0207 ≤ 0.05 but 0.0577 > 0.05, the critical region is X ≥ 15, not X ≥ 14.

例如：若 X ∼ Bin(20, 0.5)，对“偏向正面”做单尾检验，取 α = 0.05，则 P(X ≥ 15) = 0.0207，而 P(X ≥ 14) = 0.0577。

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