Question

For the variable continues to be assigned by gender (men and women) it generates the
two-means hypothesis test to compare whether there is a difference between the mean of variable X of the Woman
and the mean of variable X for Men; and the hypothesis test of two variances
(standard deviation) to determine whether there is a difference between the standard deviation of the
variable X for Women and the standard deviation of variable X for Men

The claim that will use for both the mean and standard deviation parameter is "greater than"
Use a 95% confidence level (=0.05).

Two separate columns

LEG MAN   LEG WOMAN
42.5 41.6
40.2 42.8
44.4 39
42.8 40.2
40 36.2
47.3 43.2
43.4 38.7
40.1 41
42.1 43.8
36 37.3
44.2 42.3
36.7 39.1
48.4 40.3
41 48.6
39.8 33.2
45.2 43.4
40.2 41.5
46.2 40
39 38.2
44.8 38.2
40.9 38.2
43.1 41
38 38.1
41 38
46 36
41.4 32.1
42.7 31.1
40.5 39.4
44.2 40.2
41.8 39.2
47.2 39
48.2 36.6
42.9 27
42.8 38.5
40.8 39.9
42.6 37.5
44.9 39.7
41.1 39
44.5 41.6
44 33.8

Answer 1

we computed sample means from this data are shown below:

Also, we computed sample standard deviations as:

and the sample sizes are

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:

The critical values are, and since F=0.591, then the null hypothesis of equal variances is not rejected.

(2) Rejection Region

Based on the information provided, the significance level is , and the degrees of freedom are . In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this right-tailed test is for and df=78.

The rejection region for this right-tailed test is

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that , it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0, and since , it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean ​ is greater than ​, at the 0.05 significance level.

Confidence Interval

The 95% confidence interval is

Graphically