Hypothesis Testing: Basic Steps and Logic | 假设检验的基本步骤与逻辑

📚 Hypothesis Testing: Basic Steps and Logic | 假设检验的基本步骤与逻辑

Hypothesis testing is a formal statistical procedure used to decide whether a claim about a population is supported by sample data. It starts from a default assumption, called the null hypothesis, and uses probability to measure how surprising the observed sample would be if that assumption were true.

假设检验是一种正式的统计推断方法，用于判断样本数据是否支持关于总体的某个主张。它从一个默认假设——原假设——出发，并利用概率来衡量：如果这个假设为真，观察到的样本会有多么“反常”。

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

The null hypothesis, written H₀, is the statement being tested. It always contains an equality, for example μ = 120 or p = 0.4. It is the “default” position that we try to disprove.

原假设记作 H₀，是被检验的陈述。它总是包含等号，例如 μ = 120 或 p = 0.4。它是我们试图反驳的“默认”状态。

The alternative hypothesis, written H₁ or Hₐ, describes the situation we suspect is true. It can take three common forms:

备择假设记作 H₁ 或 Hₐ，描述我们怀疑为真的情况。它有三种常见形式：

• One-tailed upper: H₁: μ > μ₀ or H₁: p > p₀. This is used when the parameter is suspected to have increased.

  上尾单侧检验：H₁: μ > μ₀ 或 H₁: p > p₀，用于怀疑参数增大时。

• One-tailed lower: H₁: μ < μ₀ or H₁: p < p₀. This is used when the parameter is suspected to have decreased.

  下尾单侧检验：H₁: μ < μ₀ 或 H₁: p < p₀，用于怀疑参数减小时。

• Two-tailed: H₁: μ ≠ μ₀ or H₁: p ≠ p₀. This is used when we are asking whether the parameter has changed in either direction.

  双侧检验：H₁: μ ≠ μ₀ 或 H₁: p ≠ p₀，用于怀疑参数发生了方向不明的变化时。

In exam questions, the wording decides which alternative to use. Words like “greater than”, “less than”, “improved”, “reduced” suggest a one-tailed test, while “changed”, “different”, or “not equal” suggest a two-tailed test.

在考试题目中，措辞决定应使用哪一种备择假设。诸如“大于”“小于”“提高”“降低”等词指向单侧检验；而“改变”“不同”“不等于”等词则指向双侧检验。

3. Significance Level and Type I Error | 显著性水平与第一类错误

The significance level, written α, is the probability of rejecting H₀ when H₀ is actually true. This mistake is called a Type I error. Common values are α = 0.10, α = 0.05 and α = 0.01.

显著性水平记作 α，是“原假设实际为真时却拒绝原假设”的概率。这种错误称为第一类错误。常用取值为 α = 0.10、α = 0.05 和 α = 0.01。

The significance level is chosen before the data are collected. In A-Level questions it is usually given, but you must still state its value. For example, “test at the 5% significance level” means α = 0.05.

显著性水平应在收集数据之前选定。在 A-Level 题目中通常会直接给出，但你仍需要明确写出它的数值。例如，“在 5% 显著性水平下检验”即表示 α = 0.05。

A smaller α makes the test more demanding, because we need stronger evidence to reject H₀. However, this also makes it harder to detect a real effect.

α 越小，检验就越严格，因为我们需要更强的证据才能拒绝 H₀。但这也会使真实效应更难被发现。

4. Test Statistic and Sampling Distribution | 检验统计量与抽样分布

A test statistic is a value calculated from the sample data. Under the null hypothesis, it has a known distribution, called its sampling distribution. We use this distribution to decide whether the observed value is unusual.

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）