One of the assumptions of the two-sample t-test is that it should only be applied to pairs of samples if both samples were drawn from normal populations or if the samples are sufficiently large. In practice, many researchers check if this assumption is met by pre-testing. The pre-test allows them to determine whether to use a parametric or non-parametric test. This research explores the probability of the Type I error of a two-stage and a one-stage hypothesis test performed on two independent samples, both of which were drawn from different populations, such as the normal distribution, the uniform distribution, and the mixed-normal
distribution. The first step of the two-stage hypothesis test is to apply the Shapiro-Wilk test to pre-test for normality of the samples. The second step is to apply the Mann-Whitney test or the two-sample t-test, depending on the outcome of the first step. The one-stage hypothesis test with no preliminary testing performs all t-tests on the two independent samples. The probability of the Type I errors for different pairs of samples is calculated by running simulations in R. I also investigate the effect of different sample sizes and non-homogeneity of variance in both procedures. I conclude by comparing the robustness of the two-stage procedure to the robustness of the t-test.