Question

Two soil samples consisting of 21 and 9 observations have variances of 18 and 8 respectively. Test the hypothesis that the first population variance is greater than the second at (a) 0.05 and (b) 0.01 levels of significance. (c) Repeat the same test If the two samples consist of 60 and 120 observations.

Answer 1

The provided sample variances are $s_1^2=18$ 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:

$$F={s_1^2\over s_2^2}={18\over 8}=2.25$$

Since from the sample information we get that $F=2.25\leq3.15=F_U,$ 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:

$$F={s_1^2\over s_2^2}={18\over 8}=2.25$$

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.

The provided sample variances are $s_1^2=18$ and $s_2^2=8$ and the sample sizes are given by $n_1=60$ and $n_2=120.$

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.

(c) Based on the information provided, the significance level is $\alpha=0.05,$ and the rejection region for this right-tailed test is $R=\{F:F>F_U=1.4323\}.$

The F-statistic is computed as follows:

$$F={s_1^2\over s_2^2}={18\over 8}=2.25$$

Since from the sample information we get that $F=2.25>1.4323=F_U,$ it is then concluded that we can reject the null hypothesis and claim that the population variance $\sigma_1^2$ is greater than the population variance $\sigma_2^2,$ at the  $\alpha=0.05$ significance level.

The significance level is $\alpha=0.01,$ and the rejection region for this right-tailed test is $R=\{F:F>F_U=1.661\}.$

The F-statistic is computed as follows:

$$F={s_1^2\over s_2^2}={18\over 8}=2.25$$

Since from the sample information we get that $F=2.25>1.661=F_U,$ it is then concluded that we can reject the null hypothesis and accept the alternative hypothesis. We can claim that the population variance $\sigma_1^2$ is greater than the population variance $\sigma_2^2,$ at the  $\alpha=0.01$ significance level.