Do you ever test the pre-asymptotic behaviour of a variable before performing a comparison of means t-test? I understand that it is not the normality of a random variable that matters in a t-test, but rather the fact that the distribution of the mean follows a normal distribution for large samples. However, is it sometimes useful to test how large a sample must be before the assumption of a normal distribution is warranted? For example, if the number of samples required were very large (much larger than n=30), would assuming a normal distribution be inappropriate? If so, how would you go about checking this?
The inspiration for this question comes from reading about Nassim Nicholas Taleb's Kappa metric.

I think the normality on the population does matter. It is a necessity for the t-distribution to hold. See my own question body for why.
So I think it is a good idea to check the normality of the population.
Note: when normality is not there, t-test has certain robustness against it, but that is another topic.
