Test of Variances

Given a random sample $X_{1}, \dots, X_{n} \sim N (μ, σ^{2})$ , we want to test if $σ^{2} = σ_{0}^{2}$ or not at an $α$ significance level

\begin{aligned} \sum_{i = 1}^{n} (X_{i} - μ)^{2} / σ^{2} \sim χ_{n}^{2}, \\ \sum_{i = 1}^{n} (X_{i} - \overset{―}{X})^{2} / σ^{2} \sim χ_{n - 1}^{2}, \\ (n - 1) S^{2} / σ^{2} \sim χ_{n - 1}^{2}, \end{aligned}

Tests for one variance

Case 1: $μ$ is known

\begin{aligned} {\hat{σ}}^{2} & = \sum \frac{(x_{i} - μ)^{2}}{n} \\ χ^{2} & = \frac{n {\hat{σ}}^{2}}{σ_{0}^{2}} = \frac{\sum (x_{i} - μ)^{2}}{σ_{0}^{2}} \sim χ_{n}^{2} | H_{0} \\ α & = P (χ^{2} \leq χ_{\frac{α}{2}, n}^{2} or χ^{2} \geq χ_{1 - \frac{α}{2}, n}^{2}) \end{aligned}

Compute $χ_{o b s}^{2}$ and reject $H_{0}$ if the value is too large or too small

Case 2: $μ$ is unknown
$\hat{μ} = \bar{x}, {\bar{σ}}^{2} = s^{2} = \frac{1}{n - 1} \sum (x_{i} - \bar{x})^{2}$

\begin{aligned} pivot statistic \to χ^{2} & = \frac{(n - 1) {\hat{σ}}^{2}}{σ_{0}^{2}} = \frac{(n - 1) S^{2}}{σ_{0}^{2}} = \frac{\sum (x_{i} - \bar{x})^{2}}{σ_{0}^{2}} \sim χ_{n - 1}^{2} \\ α & = P (χ^{2} \leq χ_{\frac{α}{2}, n - 1}^{2} or χ^{2} \geq χ_{1 - \frac{α}{2}, n - 1}^{2}) \\ compute χ_{o b s}^{2} & , reject H_{0} if χ_{o b s}^{2} \leq χ_{\frac{α}{2}, n - 1}^{2} or χ_{o b s}^{2} \geq χ_{1 - \frac{α}{2}, n - 1}^{2} \end{aligned}

$X_{1}, \dots, X_{n} \sim i i d N (μ_{1}, σ_{1}^{2}), Y_{1}, \dots, Y_{n} \sim i d d N (μ_{2}, σ_{2}^{2})$

F = \frac{S_{1}^{2} / σ_{1}^{2}}{S_{2}^{2} / σ_{2}^{2}} \sim F_{n_{1} - 1, n_{2} - 1}

\begin{aligned} H_{0} & : σ_{1}^{2} = σ_{2}^{2} & v s & H_{a} : & σ_{1} \neq σ_{2} \\ H_{0} & : \frac{σ_{1}^{2}}{σ_{2}^{2}} = 1 & v s & H_{a} : & \frac{σ_{1}^{2}}{σ_{2}^{2}} \neq 1 \\ {\hat{σ}}_{1} & = S_{1}^{2} = \frac{\sum (x_{i} - \bar{x})}{n - 1}, & \hat{σ} 2 = & S_{2}^{2} = \frac{\sum (y_{i} - \bar{y})^{2}}{n_{2} - 1} \\ F & = \frac{S_{1}^{2} / σ_{1}^{2}}{S_{2}^{2} / σ_{2}^{2}} = \frac{S_{1}^{2}}{S_{2}^{2}} \sim F_{n_{1} - 1, n_{2} - 1} & under H_{0} : & σ_{1}^{2} = σ_{2}^{2} \\ reject & H_{0} if F_{o b s} > F_{n_{1} - 1, n_{2} - 1, 1 - \frac{α}{2}} & or the other & way i guess \end{aligned}