Concepts of Hypothesis Testing

Concepts of Hypothesis Testing

假说检验的目的是利用样本来测试一个或者多个群体的参数值

Steps for Testing a Hypothesis

1.设定虚无假说(null hypothesis,\(H_0\))和对立假说(alternative hypothesis,\(H_1/H_a\))

2.指定显著水准(level of significance,\(\alpha\))

3.决定适当的检定统计量(test statistic)

4.决定弃却域(rejection region)

5.下结论--推翻\(H_0\)/不推翻\(H_0\)(fail to reject \(H_0\) )

说明

1.对立假说(alternative hypothesis,\(H_1/H_a\))：研究者收集证据想要支持的假说

根据不同的\(H_1\),我们可以将检定分为：

1.单尾检定(One-sided Test)

对立假设里面有‘</>’出现，‘<'出现在\(\rightarrow\)左尾假设--

单个正态总体的均值与方差

\(\mu\)检验

1.单样本\(U\)检验法:
\(X_{1}, X_{2}, \cdots, X_{n}\) 是从正态总体 \(\quad N\left(\mu, \sigma_{0}^{2}\right) \quad\) 中抽取的简单随机样本
已知 \(\sigma_{0}^{2},\) 检验假设 \(\quad \boldsymbol{H}_{0}: \boldsymbol{\mu}=\boldsymbol{\mu}_{0}, \quad \boldsymbol{H}_{1}: \mu \neq \boldsymbol{\mu}_{0}\)
原假设成立时，检验统计量为: \(\quad U=\frac{\bar{X}-\mu_{0}}{\sigma_{0} / \sqrt{n}} \sim N(0,1)\)
拒绝域为: \(\quad|u|>u_{\frac{\alpha}{2}}\)
1.单样本\(t\)检验法:
\(X_{1}, X_{2}, \cdots, X_{n}\) 是从正态总体 \(N\left(\mu, \sigma^{2}\right)\) 中抽取的简单随机样本.
\(\sigma^{2}\) 未知，检验假设 \(\quad \boldsymbol{H}_{0}: \boldsymbol{\mu}=\boldsymbol{\mu}_{0}, \quad \boldsymbol{H}_{1}: \mu \neq \boldsymbol{\mu}_{0}\)
原假设成立时，检验统计量为: \(\quad T=\frac{\bar{X}-\mu_{0}}{S / \sqrt{n}} \sim t(n-1)\)
拒绝域为: \(\quad|t|>t_{\frac{\alpha}{2}}(n-1)\)

\(\delta^2\)检验

\(\chi^{2}\) 检验法
\(X_{1}, X_{2}, \cdots, X_{n}\) 是从正态总体 \(N\left(\mu, \sigma^{2}\right)\) 中抽取的简单随机样本.
检验假设: \(\quad H_{0}: \sigma^{2}=\sigma_{0}^{2}, \quad H_{1}: \sigma^{2} \neq \sigma_{0}^{2}\)

1. \(\mu\) 已知
   原假设成立时，检验统计量为: \(\quad \chi^{2}=\sum_{i=1}^{n}\left(\frac{X_{i}-\mu}{\sigma_{0}}\right)^{2} \sim \chi^{2}(n)\)
2. \(\mu\) 未知
   检验假设: \(\quad H_{0}: \sigma^{2}=\sigma_{0}^{2}, \quad H_{1}: \sigma^{2} \neq \sigma_{0}^{2}\)
   原假设成立时，检验统计量为:

\[\chi^{2}=(n-1) \frac{\mathrm{S}^{2}}{\sigma_{0}^{2}} \sim \chi^{2}(n-1) \]

标签：boldsymbol,frac,Testing,检验,mu,Hypothesis,Concepts,quad,sigma
来源： https://www.cnblogs.com/zonghanli/p/12886293.html