Many scientists who are told for the first time to use a -test, are wondering how they are supposed to know that the data are distributed according to the normal or Gaussian distribution, which according to mathematical statistics is one of the assumptions for the validity of the -test. Statistical goodness-of-fit tests can never (statistically) prove, only disprove normality, and when there are enough data, they usually do disprove it. However, such tests can often prove ``approximate'' normality (to be defined in a suitable sense), in accordance with frequent informal practical experience. Now, the scientists are often told (or hope) that small deviations from normality do not matter, that the -test is ``robust'' against small deviations from the mathematical assumption of normality. But this is true, roughly speaking, only for the level of the -test; the power (and the corresponding length of confidence intervals, as well as the efficiency of the arithmetic mean) is very sensitive even to small deviations from normality. For not too small samples, there are other tests, such as the Wilcoxon- (Mann-Whitney -) test, with a much better behavior.
If the problem considered by the scientist is a two-sample problem, there is also the assumption of equality of variances for the validity of the -test. It can be shown that this assumption is of limited importance unless the sample sizes differ considerably. But even in recent years there have been textbooks written which suggest or require first an -test for the equality of the variances. Only when this test comes out nonsignificant, is the application of the two-sample -test permitted. This pseudologic ignores not only the fact that exact equality of variances can never be statistically proven, it ignores the much more important fact, known since around 1930, that already the level of chisquare- and -tests for variances is so sensitive to tiny deviations from normality that J.W. Tukey later suggested these tests might better be used as tests of normality.
There is another assumption for the -test, which is hardly ever discussed in the literature, but for example according to the highly experienced practical statistician Cuthbert Daniel it is the most important one: the assumption of independence. Great data analysts, such as K. Pearson, Gosset (``Student'') and Jeffreys, have for a long time been aware of the nonvalidity of this assumption and the dangers arising from this fact, but only fairly recently has there been some systematic work trying to bring more light into these issues.