It is sometimes said that confidence interval methods for the mean are robust against departures of normality. But does this refer to the population distribution, or the sampling distribution (of the sample mean)?
For instance, the criterion $n\geqslant 30$ is often given as a kind of rule of thumb for using the $t$-distribution. So in that case it doesn't matter very much what the population distribution is, as long as it's not too terribly wonky. Is that what is meant by "robust against departures from normality"? Maybe, but I suspect not.
My guess is that it's referring instead to the sampling distribution. I am told, for instance, that even for "nice" distributions (e.g. the uniform of chi-square) this $n\geqslant 30$ condition sometimes only gives a very rough normal approximation in the sampling distribution. But apparently it's still okay to use run the $t$-interval. Is that what is meant when we are told that these methods are "robust against departures from normality"?
That's what I'm guessing, but I don't want to guess! So, any help would be appreciated.
Thanks guys!