From here I read that
The t-test assumes that the means of the different samples are normally distributed; it does not assume that the population is normally distributed. By the central limit theorem, means of samples from a population with finite variance approach a normal distribution regardless of the distribution of the population ...... t-test is invalid for small samples from non-normal distributions, but it is valid for large samples from non-normal distributions.
So my follow up question is, how do you know if the means are normally distributed when population is believed to be non-normal?
It basically says that if sample size is large enough, you can assume normality of the statistic of your interest, regardless of the shape of the actual population. But how do you know what the threshold of "large enough" is?
I'm interested in checking normality of both means and variances.