### Comparison of Parametric and Nonparametric Tests for Differences in Distribution

#### Abstract

When the population is approximately normally distributed, the two-sample t-test is appropriate to conduct a hypothesis test for the difference between two means. However, when the population is not normally distributed, the two-sample t-test has low efficiency. The Wilcoxon-Mann-Whitney U two-sample test or the Kruskal-Wallis test can be considered. These nonparametric tests are used to test for a difference in two or more samples that are drawn from the same distribution. The Kruskal-Wallis test assumes homoscedasticity across samples. For the absence of homoscedasticity (heteroscedasticity), Mood’s median test is used to test if the medians of two or more populations are statistically identical or not. Using R, a well-known statistical software, simulations are conducted to explore the skewed distribution such as the Weibull and Chi-Squared distributions. The Weibull distribution has two parameters, the shape parameter (*k*) and the scale parameter (λ). The goal of this project to compare the efficiencies among the several different tests mentioned above. To measure the efficiency, the power – the probability of rejecting the null hypothesis when the null hypothesis is false – is calculated and compared for each test. Within this study, a real data analysis is conducted using income data which is known as typical skewed right data.

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