Question

1. Suppose we have the following information on GMAT scores for business and non-business majors:

Business Majors                      Non-Business Majors

n₁ = 8                                       n₂ = 5

_                                              _

X₁ = 545                                  X₂ = 525

s₁ = 120                                   s₂ = 60

a. Using a 0.05 level of significance, test to see whether the population variances are equal. (4 points)

b. Using a 0.05 level of significance, test the clam that average GMAT scores for business majors is above the average GMAT scores for non-business majors in the population. Assume unequal population variances.

Answer 1

a.

The provided sample variances are and and the sample sizes are given by and

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

​

​

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

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the the rejection region for this two-tailed test is R={F:F<0.181 or F>9.074}.

(3) Test Statistics

The F-statistic is computed as follows:

(4) Decision about the null hypothesis

Since from the sample information we get that FL​=0.181<F=4<FU​=9.074, it is then concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population variance ​ is different than the population variance ​, at the α=0.05 significance level.

b.

(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.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=10.719. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal:

Hence, it is found that the critical value for this right-tailed test is tc​=1.8, for α=0.05 and df=10.719.

The rejection region for this right-tailed test is R={t:t>1.8}.

(3) Test Statistics

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

(4) Decision about the null hypothesis

Since it is observed that t=0.398≤tc​=1.8, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.3491, and since p p=0.3491≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is greater than μ2​, at the 0.05 significance level.

Graphically