Question

describe the F-tests used for testing equality of the variation of the two different populations. if ten ring bobbins are selected from the day shift and night shift productions have shown standard deviations of the count as 1.2 and 2.2 units respectively. can we say that variation in count is more for the night shift production than that of day shift production?

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\sigma^2_1=\sigma_2^2$

$H_0:\sigma^2_1<\sigma_2^2$

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 $\alpha=0.05,$ the F-critical value for a left-tailed test, for $df_1=df_2=10-1=9$ degrees of freedom is $F_c=0.3146.$

The rejection region for this left-tailed test test is $R=\{F:F<0.3146\}.$

The F-statistic is computed as follows:

$$F=\dfrac{s^2_1}{s^2_1}=\dfrac{1.2^2}{2.2^2}\approx0.2975$$

Since from the sample information we get that $F=0.2975<F_c=0.3146,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population $\sigma^2_1$ is less than the population variance $\sigma^2_2,$ at the $\alpha=0.05$ significance level.

We can say that variation in count is more for the night shift production than that of day shift production at the $\alpha=0.05$ significance level.