Hypothesis Tests for One Variance

When the data are sampled from a normal population, we can test claims about the population variance $\sigma^2$ by comparing the sample variance $s^2$ to a hypothesized value $\sigma_0^2.$

The null hypothesis takes the form

$$H_0: \sigma^2 = \sigma_0^2,$$

and the alternative can be right-tailed ($H_a: \sigma^2 > \sigma_0^2$), left-tailed ($H_a: \sigma^2 < \sigma_0^2$), or two-tailed ($H_a: \sigma^2 \ne \sigma_0^2$).

The test statistic is

$$\chi^2 = \dfrac{(n-1)\,s^2}{\sigma_0^2}.$$

When $H_0$ is true, this statistic follows a chi-square distribution with $n-1$ degrees of freedom.

For a right-tailed test at significance level $\alpha,$ we reject $H_0$ when

$$\chi^2 > \chi^2_{\alpha,\,n-1},$$

where $\chi^2_{\alpha,\,n-1}$ is the critical value with area $\alpha$ to its right.

Illustration. Suppose $n=10,$ $s^2=5,$ $\sigma_0^2=3,$ and $\alpha=0.05.$ The test statistic is

$$\chi^2 = \dfrac{9\cdot 5}{3} = 15.$$

Given $\chi^2_{0.05,\,9} = 16.919,$ we have $15 < 16.919,$ so we fail to reject $H_0.$