林深见鹿（十）：概率论与数理统计（9）

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假设检验：统计推断的科学决策方法

Hypothesis Testing: A Scientific Decision-Making Method in Statistical Inference

基本概念

Basic Concepts

假设检验是统计推断的核心内容，其基本思想是建立关于总体参数的两个对立假设——原假设H₀和备择假设H₁。检验过程中，我们基于"小概率事件实际不可能性原理"，在给定的显著性水平α下作出统计决策。值得注意的是，假设检验可能犯两类错误：第一类错误是拒绝真实原假设的"弃真"错误，其概率为α；第二类错误是接受错误原假设的"取伪"错误，其概率为β。检验功效1-β反映了检验正确拒绝错误原假设的能力，是评价检验方法优劣的重要指标。p值是反对原假设的证据强度度量，其定义为在原假设成立时，获得当前样本结果或更极端结果的概率。

Hypothesis testing is a core component of statistical inference. Its fundamental idea involves establishing two opposing hypotheses about a population parameter—the null hypothesis H₀ and the alternative hypothesis H₁. During the testing process, statistical decisions are made based on the "principle of the practical impossibility of small probability events" at a given significance level α. It is important to note that hypothesis testing may involve two types of errors: a Type I error, which is the "rejection of a true null hypothesis" with probability α, and a Type II error, which is the "acceptance of a false null hypothesis" with probability β. The test power, 1-β, reflects the ability of the test to correctly reject a false null hypothesis and is a key indicator for evaluating the effectiveness of a testing method. The p-value measures the strength of evidence against the null hypothesis and is defined as the probability of obtaining the current sample result or more extreme results under the assumption that the null hypothesis is true.

单正态总体参数的假设检验

Hypothesis Testing for Parameters of a Single Normal Population

对于单正态总体N(μ,σ²)，我们常需要检验均值μ和方差σ²的假设。当检验均值μ时，若总体方差σ²已知，采用Z检验法，构造检验统计量Z=(X̄-μ₀)/(σ/√n)～N(0,1)；若σ²未知，则使用t检验法，统计量t=(X̄-μ₀)/(S/√n)～t(n-1)。对于方差σ²的检验，采用χ²检验法，检验统计量χ²=(n-1)S²/σ₀²～χ²(n-1)。每种检验都需要根据备择假设的形式确定拒绝域，并严格满足检验方法的前提条件——总体服从正态分布。

For a single normal population N(μ, σ²), it is often necessary to test hypotheses about the mean μ and the variance σ². When testing the mean μ, if the population variance σ² is known, the Z-test is used, with the test statistic Z = (X̄ - μ₀) / (σ / √n) ~ N(0,1). If σ² is unknown, the t-test is employed, with the statistic t = (X̄ - μ₀) / (S / √n) ~ t(n-1). For testing the variance σ², the χ²-test is used, with the test statistic χ² = (n-1)S² / σ₀² ~ χ²(n-1). Each test requires determining the rejection region based on the form of the alternative hypothesis and strictly satisfying the prerequisite condition of the testing method—that the population follows a normal distribution.

双正态总体参数的假设检验

Hypothesis Testing for Parameters of Two Normal Populations

在实际研究中，我们常需要比较两个正态总体的参数。对于均值差的检验，当两总体方差已知时使用Z检验，未知但相等时使用t检验，此时需要计算合并方差。对于方差比的检验，采用F检验法，构造检验统计量F=S₁²/S₂²～F(n₁-1,n₂-1)。双总体检验在医学研究、质量控制和社会科学中应用广泛，如比较两种药物的疗效、两种生产方法的精度等。进行双总体检验时，需要特别注意方差齐性假设的检验，这是选择适当检验方法的前提。

In practical research, it is often necessary to compare the parameters of two normal populations. For testing the difference in means, the Z-test is used when the variances of the two populations are known, and the t-test is used when the variances are unknown but assumed equal, in which case the pooled variance must be calculated. For testing the ratio of variances, the F-test is employed, with the test statistic F = S₁² / S₂² ~ F(n₁-1, n₂-1). Two-population tests are widely applied in medical research, quality control, and social sciences, such as comparing the efficacy of two drugs or the precision of two production methods. When conducting two-population tests, special attention must be paid to testing the assumption of homogeneity of variances, as this is a prerequisite for selecting the appropriate testing method.

独立性检验与拟合优度检验

Independence Tests and Goodness-of-Fit Tests

χ²检验在分类数据分析中具有重要地位。独立性检验用于判断两个分类变量是否相互独立，其检验统计量基于观测频数与期望频数的差异构造。拟合优度检验则用于判断样本数据是否来自某一特定分布，通过比较经验分布与理论分布的吻合程度作出推断。这两种检验在社会科学、市场研究和医学领域应用广泛，但需要注意，χ²检验要求每个类别的期望频数不能过小，通常要求不少于5，否则需要合并类别或使用精确检验方法。

The χ² test plays a significant role in the analysis of categorical data. Independence tests are used to determine whether two categorical variables are independent of each other, with the test statistic constructed based on the differences between observed frequencies and expected frequencies. Goodness-of-fit tests are used to determine whether sample data come from a specific distribution, making inferences by comparing the agreement between the empirical distribution and the theoretical distribution. These two types of tests are widely used in social sciences, market research, and medical fields. However, it is important to note that the χ² test requires that the expected frequency for each category should not be too small, typically not less than 5. Otherwise, categories need to be merged or exact testing methods should be used.

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翻译：文心一言

参考资料：百度百科