I have what is probably a silly question regarding normality assumptions for t/Z tests. As I understand, t/z tests require that sample data was obtained from populations following a normal distribution.
...So, what does this actually mean in practice?
Does it imply that the distributions of each variable we collect should be approximately normal as well? Or does it just mean that, even if they are not normal, it's fine as long as the population they were sampled from is arguably normal? The logic in the last sentence seems tenuous.
Additionally, what does this imply for regression analysis? I understand that regression analysis 'only' requires that the error terms are normally distributed. But, I am wondering if the above implies anything for t tests on the OLS parameters.