Question

The students of same age group from two different schools were compared for variability in their mathematical skill. A random sample of 25 pupils from one school had a variance of 16 marks while a random sample of 22 pupils from the other school had a variance of 8 marks. Examine if the difference in variability is significant. [F at (24,21) at 5% level of significance is 2.05] 

Answer 1

The following null and alternative hypotheses need to be tested:

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

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

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

Numerator degrees of freedom: $d_1=25-1=24.$

Denomirator degrees of freedom: $d_2=22-1=21.$

Based on the information provided, the significance level is $\alpha = 0.05,$ and the the rejection region for this right-tailed test test is

$$R = \{F: F > 2.054\}.$$

The F-statistic is computed as follows:

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

Since from the sample information we get that $F = 2 \le F_c = 2.054,$

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.