Tests around Variance of Normal Population¶

For a \(n\) sized sample of independent observations from a normal population, we are interested in checking

\[ \begin{aligned} H_{0}: \sigma^{2} = \sigma_{0}^{2} \quad \text{versus} \quad \sigma^{2} \neq \sigma_{0}^{2} \end{aligned} \]

Recall from section

\[ \begin{aligned} \frac{(n-1)S^{2}}{\sigma^{2}} \sim \chi_{n-1}^{2} \end{aligned} \]

Then if \(H_{0}\) is true, our test statistic

\[ \begin{aligned} TS = \frac{(n-1)S^{2}}{\sigma_{0}^{2}} \sim \chi_{n-1}^{2} \end{aligned} \]

and from the test simply becomes

\[ \begin{alignat}{2} \text{Accept} \quad &H_{0} \quad &&\text{if} \quad \chi_{1-\alpha/2, n-1}^{2} \leq TS \leq \chi_{\alpha/2, n-1}^{2}\newline \text{Reject} \quad &H_{0} \quad &&\text{otherwise} \end{alignat} \]

One sided test can be done in a similar manner, comparing with \(\chi_{1-\alpha, n-1}^{2}\) or \(\chi_{\alpha, n-1}^{2}\) based on which side we want to reject \(H_{0}\).

Comparing Variance of Two Normal Populations¶

We are interested in comparing

\[ \begin{aligned} H_{0}: \sigma_{x}^{2} = \sigma_{y}^{2} \quad \text{versus} \quad \sigma_{x}^{2} \neq \sigma_{y}^{2} \end{aligned} \]

Recall that the ratio of sample variance with population variance is \(\chi^{2}\)-distributed, and the ratio of two \(\chi^{2}\)-distributed variables has an F-distribution. Hence, when \(H_{0}\) is true,

\[ \begin{aligned} TS = \frac{S_{x}^{2}}{S_{y}^{2}} \sim F_{n-1, m-1} \end{aligned} \]

Noting that F-distribution is always positive, the region for accepting \(H_{0}\) simply become

\[ \begin{alignat}{2} \text{Accept}\quad &H_{0} \quad &&\text{if} \quad F_{1-\alpha/2, n-1, m-1} \leq TS \leq F_{\alpha/2, n-1, m-1}\newline \text{Reject}\quad &H_{0} \quad &&\text{otherwise} \end{alignat} \]