Question

Arbitron Media Research Inc. conducted a study of the iPod listening habits of men and women. One facet of the study involved the mean listening time. It was discovered that the mean listening time for a sample of 10 men was 32 minutes per day. The standard deviation was 19 minutes per day. The mean listening time for a sample of 16 women was also 32 minutes, but the standard deviation of the sample was 9 minutes. At the 0.02 significance level, can we conclude that there is a difference in the variation in the listening times for men and women?

Answer 1

Answer:

The null and alternative hypotheses are:

$H_{0}:\sigma_{men} = \sigma_{women}$

$H_{a}:\sigma_{men} \neq \sigma_{women}$

Under the null hypothesis, the test statistic is:

$F=\frac{\sigma^{2}_{men}}{\sigma^{2}_{women}}$

       $=\frac{19^{2}}{9^{2}}$

       $=\frac{361}{81}$

       $=4.46$

Therefore, the test-statistic is $F=4.46$

 Now the F critical value at 0.02 significance level for df1 = 10- 1 =9 and df2 = 16 - 1 =15 is:

$F_{critical} = 3.303$

Since F statistic is greater than the F critical value, we therefore, reject the null hypothesis and conclude that there is sufficient evidence to support the claim that there is a difference in the variation in the listening times for men and women.