Hypothesis Test for one variance

Section 11.5 Hypothesis Test for one variance

Proceeding in a similar manner, we can also perform hypothesis testing on variances using the

$\chi^2$ -distribution to determine probabilities.

$\begin{equation*} H_0 : \sigma^2 = \sigma_0^2 \\ H_a: \sigma^2 \neq \sigma_0^2 \end{equation*}$

$\begin{equation*} H_0 : \sigma^2 \ge \sigma_0^2 \\ H_a: \sigma^2 \lt \sigma_0^2 \end{equation*}$

$\begin{equation*} H_0 : \sigma^2 \le \sigma_0^2 \\ H_a: \sigma^2 \gt \sigma_0^2 \end{equation*}$

$\begin{equation*} \text{test statistic} = T = (n-1) \frac{s^2}{\sigma_0^2} \end{equation*}$

For two-tailed, reject if

$\begin{equation*} T \gt \chi_{1-\alpha/2,n-1}^2 \text{ or } T \lt \chi_{\alpha/2,n-1}^2 \end{equation*}$

and for one-tailed to the right if

$\begin{equation*} T \gt \chi_{1-\alpha,n-1}^2 \end{equation*}$

and for one-tailed to the left if

$\begin{equation*} T \lt \chi_{\alpha,n-1}^2. \end{equation*}$

Technically, for the two-tailed test you could pick T-values so that the total probability sums to

$\alpha$ in any fashion but generally this probability is split evenly between the two tails as noted above.