Question

Suppose that you want to perform a hypothesis test to compare the means of two populations, using independent simple random samples. Assume that the two distributions of the variable under consideration have the same shape, but are not normal, and both sample sizes are large. Answer the following questions and explain your answers.

a. Is it permissible to use the pooled $t$-test to perform the hypothesis test?

b. Is it permissible to use the Mann-Whitney test to perform the hypothesis test?

c. Which procedure is preferable, the pooled$t$-test or the Mann-Whitney test?

Short answer

Part $\left(a\right)$: Yes, it is permissible to use the pooled $t$-test to perform the hypothesis test.

Part $\left(b\right)$: Yes, it is permissible to use the Mann-Whitney test to perform the hypothesis test.

Part $\left(c\right)$: The pooled $t$-test should be preferable.

Step-by-step solution

Part a: Step 1. Given information

Assume that the two distributions of the variable under consideration have the same shape, but are not normal, and both sample sizes are large.

Part a Step 2. Assumption for the pooled t-test is given below:

• The selected sample should be a simple random sample from two populations.
• The samples are independent of one another.
• Population is approximately normal.
• Equal population standard deviations.

So, it is permissible to use the pooled$t$-test to perform the hypothesis test because the given information satisfied the assumptions of the pooled$t$-test.

Part b Step 1. Explanation for the answer:

Yes, it is permissible to use the Mann-Whitney test to perform the hypothesis test because the two distributions of the variable under the consideration have the same shape but it is not normal.

Part c Step 1. Explanation for the answer:

Here, the pooled $t$-test procedure should be preferable because the pooled$t$-test provide more accurate results than the Mann-Whitney test.