Question

Suppose that you want to perform a hypothesis test to compare the means of two populations, using a paired sample. For each part, decide whether you would use the paired $t-$test, the paired Wilcoxon signed-rank test, or neither of these tests if preliminary data analyses of the sample of paired differences suggest that the distribution of the paired-difference variable is

a. uniform.

b. neither symmetric nor normal; the sample size is $132.$

c. moderately skewed but otherwise roughly bell-shaped.

Short answer

a) The Wilcoxon signed-rank test is utilised.

b) Neither the paired t-test nor the Wilcoxon signed-rank test is appropriate.

c) The paired $t$test is performed

Step-by-step solution

Part (a) Step 1: Given Information

Given in the question that, the sample is uniform.

We have to decide Whether paired t-test, paired Wilcoxon signed-rank test or neither of the tests is used.

Part(a) Explanation: Step 2

Because the sample is drawn from a simple random paired sample to compare the means of two populations, and the paired-difference variable follows a uniform distribution rather than a normal distribution.

Part(b) Step 1: Given Information : Step 1

Given in the question that, the sample is neither symmetric nor normal; the sample size is $132$.

We have to decide whether paired t-test, paired Wilcoxon signed-rank test, or neither of the tests is used.

Part(b) Explanation : Step 2

The distribution of the paired difference variable is not symmetric, according to preliminary data studies, and sample size 132 is utilised for neither test since the data are not symmetric.

Part(c) Step 1: Given Information

Given in the question that, the sample is moderately skewed but otherwise roughly bell-shaped.

We have to decide whether paired t-test paired Wilcoxon signed-rank test or neither of the test is used.

Part (c) Step 2: Explanation

The distribution of the paireddifference variable appears to be highly skewed but generally bell-shaped, according to preliminary data analysis of the sample of pair differences. The paired t-test is performed because the data follows an almost normal distribution and the sample is taken from a simple random paired sample to evaluate the means of two populations.