A media firm 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 men was 35 minutes a day. The standard deviation of the sample of the 10 men studied was 10 minutes per day. The mean listening time for the 12 women studied was also 35 minutes, but the standard deviation of the sample was 12 minutes. At the .10 significance level, can we conclude that there is a difference in the variation in the listening habits for men and women?
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To determine if there is a difference in the variation in the listening habits for men and women, we can conduct an F-test for the equality of two variances. The F-test is used to test if the variances of two populations are equal.
The null hypothesis (H0) is that the variances are equal, and the alternative hypothesis (H1) is that the variances are not equal.
The F statistic is calculated as the ratio of the two sample variances, which in this case is (12^2) / (10^2) = 1.44.
The degrees of freedom for the numerator and denominator are the sample sizes minus 1, which are 12-1 = 11 and 10-1 = 9 respectively.
Using an F-distribution table or an online F-distribution calculator, we can find the critical values for a two-tailed test at the 0.10 significance level. The critical values are approximately 0.39 and 2.73.
Since the calculated F statistic (1.44) falls within the critical region (0.39 to 2.73), we fail to reject the null hypothesis.
Therefore, at the 0.10 significance level, we cannot conclude that there is a difference in the variation in the listening habits for men and women.
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