Finding Type I and Type II Errors Using the Normal Distribution | 利用正态分布求第一类与第二类错误

📚 Finding Type I and Type II Errors Using the Normal Distribution | 利用正态分布求第一类与第二类错误

In Edexcel A-Level Statistics, hypothesis testing often assumes a normal distribution for the sample mean. This topic focuses on calculating the probabilities of two possible decision errors: rejecting a true null hypothesis and failing to reject a false null hypothesis.

在 Edexcel A-Level 统计学中，假设检验通常假定样本均值服从正态分布。本主题重点计算两种可能决策错误的概率：拒绝正确原假设的概率，以及未拒绝错误原假设的概率。

1. Hypothesis Testing Recap | 假设检验回顾

A significance test starts with a null hypothesis H₀, such as μ = μ₀, and an alternative hypothesis H₁, which can be one-tailed or two-tailed. The test statistic for a normal mean with known σ is Z = (X̄ − μ₀) / (σ / √n).

显著性检验从原假设 H₀ 开始，例如 μ = μ₀，以及备择假设 H₁，可以是单尾或双尾。已知 σ 时正态均值的检验统计量为 Z = (X̄ − μ₀) / (σ / √n)。

The decision rule compares the observed Z value with critical values such as 1.645 for a 5% one-tailed test or 1.960 for a 5% two-tailed test. If the statistic falls in the critical region, we reject H₀.

决策规则将观测到的 Z 值与临界值比较，例如 5% 单尾检验的 1.645，或 5% 双尾检验的 1.960。若统计量落入拒绝域，就拒绝 H₀。

2. The Two Error Types | 两类错误

A Type I error occurs when H₀ is true but the test rejects it. A Type II error occurs when H₀ is false but the test does not reject it.

第一类错误发生在 H₀ 为真但检验拒绝了它。第二类错误发生在 H₀ 为假但检验没有拒绝它。

These errors are mutually exclusive in a single decision, but their probabilities trade off against each other when the sample size is fixed.

两类错误在一次决策中互斥，但在样本量固定时，它们的概率相互制约。

3. Type I Error and Significance Level |

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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