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Student's one-sample t-test is a commonly used method when inference about the population mean is made. As advocated in textbooks and articles, the assumption of normality is often checked by a preliminary goodness-of-fit (GOF) test. In a paper recently published by Schucany and Ng it was shown that, for the uniform distribution, screening of samples by a pretest for normality leads to a more conservative conditional Type I error rate than application of the one-sample t-test without preliminary GOF test. In contrast, for the exponential distribution, the conditional level is even more elevated than the Type I error rate of the t-test without pretest. We examine the reasons behind these characteristics. In a simulation study, samples drawn from the exponential, lognormal, uniform, Student's t-distribution with 2 degrees of freedom (t2) and the standard normal distribution that had passed normality screening, as well as the ingredients of the test statistics calculated from these samples, are investigated. For non-normal distributions, we found that preliminary testing for normality may change the distribution of means and standard deviations of the selected samples as well as the correlation between them (if the underlying distribution is non-symmetric), thus leading to altered distributions of the resulting test statistics. It is shown that for skewed distributions the excess in Type I error rate may be even more pronounced when testing one-sided hypotheses.