Answer to Question #122523 in Statistics and Probability for Saz

Question

The Head of the Mathematics Department is interested to know whether there is any variation in the way the two instructors in the deprtment grade mathematics quiz for the mid-term examination. They were given the same set of 30 exam papers selected randomaly with the same marking scheame. The first instructor's grades have a variance of 52.3. The second instructor's grades have a variance of 89.9. Test the claim that the first instructor's variance is smaller at the 10% level of significance.

Answer 1

The provided sample variances are $s_1^2=52.3$ and $s_2^2=89.9$ and the sample sizes are given by $n_1=30$ and $n_2=30.$

The following null and alternative hypotheses need to be tested:

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

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

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

Based on the information provided, the significance level is $\alpha=0.1,$ and the the rejection region for this left-tailed test is $R=\{F:F_L<0.617\}.$

The F-statistic is computed as follows:

$$F=\dfrac{s_1^2}{s_2^2}=\dfrac{52.2}{89.9}\approx0.582$$

Since from the sample information we get that  $F=0.582<0.617=F_L,$ it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the first instructor's variance of 52.3 is less than the second instructor's variance​ of 89.9, at the 10% level of significance.