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There is no formal and generally accepted procedure for choosing an appropriate significance test for sample data when the assumption of normality is doubtful. Various tests of normality that have been proposed over the years have been found to have limited usefulness, and sometimes a preliminary test makes the situation worse. The present paper investigates a specific and easily applied rule for choosing between a parametric and non-parametric test, the Student t test and the Wilcoxon–Mann–Whitney test, that does not require a preliminary significance test of normality. Simulations reveal that the rule, which can be applied to sample data automatically by computer software, protects the Type I error rate and increases power for various sample sizes, significance levels, and non-normal distribution shapes. Limitations of the procedure in the case of heterogeneity of variance are discussed.