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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.

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This post may be helpful: https://www.johndcook.com/blog/2018/05/11/two-sample-t-test/

It shows that when you sample from populations that are non-normal, but symmetric and with tails that aren't too heavy, the t-test should work fine.

Here's the final conclusion:

These two examples show that you can replace the normal distribution with a moderately heavy tailed symmetric distribution, but don’t overdo it. When the data some from a heavy tailed distribution, even one that is symmetric, the two-sample t-test may not have the operating characteristics you’d expect.

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  • $\begingroup$ While I really like the simulation-based approach to illustrate something, I think that a very crucial point is missing: It depends on sample size! The t/z-Test will work as sample sizes approaches infinity (central limit theorem, the sampling distribution of the mean is normal independent of the distribution of the original variable) - asymmetry simply changes the point at which non-normality becomes negligible not the fact that at some point it does become negligible. $\endgroup$ – StoryTeller0815 Aug 17 '19 at 8:03

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