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. approximately normal.

b. highly skewed; the sample size is $20$.

c. symmetric bimodal.

Given in the question that, you want to perform a hypothesis test to compare the means of two populations, using a paired sample.

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

The distribution of the paired difference variable appears to be approximately normal, according on 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 compare the means of two populations.

Given in the question that, the sample size is $20$and highly skewed

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

Preliminary data assessments of the sample of pair distinctions suggest that the paireddifference variable's distribution is highly skewed, and sample size 20 is used for neither test because the data is highly skewed, and neither paired t-test nor Wilcoxon signed-rank test is appropriate, and the sample size is small (20 instead of large).

To explain Whether paired t-test, paired Wilcoxon signed-rank test or neither of the tests is used.

Wilcoxon signed-rank test is employed because the sample is collected from simple random paired sampling to evaluate the means of the two populations and the paired differential variable follows symmetric distribution, according to preliminary data assessments of the sample of pair differences.