Suppose that you want to perform a hypothesis test based on...

Suppose that you want to perform a hypothesis test based on a simple random paired sample to compare the means of two populations. For each part, decide whether you would use the paired $t$ -test, the paired Wilcoxon signed-rank test, or neither of these tests. Preliminary data analyses of the sample of paired differences suggest that the distribution of the paired-difference variable is a. uniform. b. not symmetric; the sample size is $132 .$ c. moderately skewed but otherwise approximately bell shaped.

Suppose that you want to perform a hypothesis test based on a simple random paired sample to compare the means of two populations. For each part, decide whether you would use the paired $t$ -test, the paired Wilcoxon signed-rank test, or neither of these tests. Preliminary data analyses of the sample of paired differences suggest that the distribution of the paired-difference variable is
a. uniform.
b. not symmetric; the sample size is $132 .$
c. moderately skewed but otherwise approximately bell shaped.

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Key Concepts

Paired Wilcoxon signed-rank test

The paired Wilcoxon signed-rank test is a nonparametric alternative to the paired t-test. It is used when the normality assumption is violated, though it does require that the distribution of the paired differences be symmetric. This test ranks the absolute differences while taking into account the sign of the differences, allowing for analysis without relying on the normality assumption.

Paired t-test

The paired t-test is a parametric method used to compare the means of two related populations. It requires that the differences between paired observations are approximately normally distributed. When the sample size is small, this condition is crucial, though with larger samples the test becomes more robust to mild deviations from normality due to the central limit theorem.

Distributional Assumptions

Distributional assumptions refer to the expectations about the shape and characteristics of the data’s distribution that underpin various statistical tests. For example, the paired t-test assumes normally distributed differences, while the Wilcoxon signed-rank test requires symmetry (even if not perfect normality). Understanding these assumptions is key to selecting the appropriate test based on the data's characteristics.

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