Question

Two samples consisting of 21 and 9 observations have variances given by s 1 2 =16 and s 2 2

=8 respectively. Test the hypothesis that the first population variance is greater than the

second at a (a) 0.05, (b) 0.01 level of significance.

Answer 1

The provided sample variances are $s_1^2=16$ and $s_2^2=8$  and the sample sizes are given by $n_1=21$ and $n_2=9.$

The following null and alternative hypotheses need to be tested:

$H_0:\sigma_1^2=\sigma_2^2$

$H_1:\sigma_1^2>\sigma_2^2$

This corresponds to a right-tailed test, for which a F-test for two population variances needs to be used.

(a) Based on the information provided, the significance level is $\alpha=0.05,$ and the rejection region for this right-tailed test is $R=\{F:F>F_U=3.15\}.$

The F-statistic is computed as follows:

$$F={s_1^2\over s_2^2}={16\over 8}=2$$

Since from the sample information we get that $F=2\leq3.15=F_U,$ it is then concluded that the null hypothesis is not rejected. Therefore, there is not enough evidence to claim that the population variance $\sigma_1^2$ is greater than the population variance $\sigma_2^2,$ at the $\alpha=0.05$ significance level.

(b) The significance level is $\alpha=0.01,$ and the rejection region for this right-tailed test is $R=\{F:F>F_U=5.36\}.$

The F-statistic is computed as follows:

$$F={s_1^2\over s_2^2}={16\over 8}=2$$

Since from the sample information we get that $F=2.25\leq5.36=F_U,$ it is then concluded that the null hypothesis is not rejected. Therefore, there is not enough evidence to claim that the population variance $\sigma_1^2$ is greater than the population variance $\sigma_2^2,$ at the $\alpha=0.01$ significance level.