Question

In most situations the t-test is a robust test regarding normality requirements. At what point does “robustness” of normality fail and become a problem in t-tests?

Answer 1

let us consider a large population from where we can take many different samples of a particular size.

The t-test assumes that the means of the different samples are normally distributed, it does not assume that the population is normally distributed.It is invalid for small samples from non-normal distributions, but it is valid for large samples from non-normal distributions.

When the two samples are mildly skew in the same direction, the one-tailed t-test is no longer unbiased. If they are skew in opposite directions, the type I error rate can be heavily affected.

Heavy skewness could have bigger impacts, but generally , moderate skewness with a two-tailed test isn't too bad if you don't mind your test in essence allocating more of its power to one direction that the other.

ISo basically the two-tailed, two-sample t-test is reasonably robust to those kinds of things if we can tolerate some impact on the significance level and limited bias.