Question

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.

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) Let the population variance for Business major is and the population variance for Non-business major is .

The null and alternative hypothesis for testing the two-variance are:

And the significance level is given as

The test-statistic is given as:

The data for Business major and Non-business major is given as-

	Business Major	Non-business Major
sample mean
sample standard deviation
sample size

Calculation for test-statistic:

So, the test-statistic is calculated as

Critical value: Since it is a two-tailed test so, the critical values are-

So, we Reject the null hypothesis:

If the test-statistic F:

Or if the test-statistic F:

Since, the test-statistic is neither greater than 9.0741 nor less than 0.1811, hence "We fail to reject the null hypothesis H₀"

In symbols since, and also ,

So, at we have insufficient evidence to reject null hypothesis H₀

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(b) The null and alternative hypothesis for the population average score of Business Major and Non-business Major is given as -

; i.e., the true average score of Business Major and Non-business Major are not different.

; i.e., the true average GMAT score of Business Major is greater than the true average score of Non-business Major.

We need to test this hypothesis at a given significance level of and also it is given that population variance is UNEQUAL.

Test-statistic:

Degrees of freedom:

The data for GMAT score for Business Major and Non-business Major is given as-

	Business Major	Non-business Major
sample mean
sample standard deviation
sample size

Calculation for test-statistic:

The test-statistic is calculated as

Calculation for degrees of freedom(df):

So, the degrees of freed is calculated as

P-value: We find the p-value for the test-statistic we have calculated i.e., in order to make a decision that whether to reject the null hypothesis or do not reject the null hypothesis.

Since it is a Right-tailed test, so the p-value is calculated as-

So, the p-value is calculated as

Decision:

Since,

At the sample data does not provide sufficient evidence to reject null hypothesis, hence we "Fail to reject the null hypothesis".

So, we conclude that we did not find enough evidence to believe that the true average GMAT scores for Business Major is greater than the Non-business Major.