Wilcoxon-Mann-Whitney as an alternative to the t-test

The two sample t-test is one of the most used statistical procedures. Its purpose is to test the hypothesis that the means of two groups are the same. The test assumes that the variable in question is normally distributed in the two groups. When this assumption is in doubt, the non-parametric Wilcoxon-Mann-Whitney (or rank sum ) test is sometimes suggested as an alternative to the t-test (e.g. the Wikipedia page on the t-test), which doesn't rely on distributional assumptions. But is this necessarily a good 'replacement'?

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The t-test and robustness to non-normality

The t-test is one of the most commonly used tests in statistics. The two-sample t-test allows us to test the null hypothesis that the population means of two groups are equal, based on samples from each of the two groups. In its simplest form, it assumes that in the population, the variable/quantity of interest X follows a normal distribution N(\mu_{1},\sigma^{2}) in the first group and isĀ N(\mu_{2},\sigma^{2}) in the second group. That is, the variance is assumed to be the same in both groups, and the variable is normally distributed around the group mean. The null hypothesis is then that \mu_{1}=\mu_{2}.

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