The authors performed a Monte Carlo simulation to empirically investigate the robustness and power of 4 methods in testing mean differences for 2 independent groups under conditions in which 2 populations may not demonstrate the same pattern of nonnormality. The approaches considered were the t test, Wilcoxon rank-sum test, Welch-James test with trimmed means and Winsorized variances, and a nonparametric bootstrap test. Results showed that the Wilcoxon rank-sum test and Welch-James test with trimmed means and Winsorized variances were not robust in terms of type I error control when the 2 populations showed different patterns of nonnormality. The nonparametric bootstrap test provided power advantages over the t test. The authors discuss other results from the simulation study and provide recommendations.