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I read this lecture note and found that it assumes normality of the data when we conduct z-test or t-test.

I can accept that when we have small samples we have to assume normality of the data, because there's not enough evidence to believe that sample mean was drawn from normal distribution.

My question is that if we have large enough samples (of course it is a bit subjective), do we need to assume normality of the data? Isn't it just okay to assume sample mean was drawn from normal distribution (by central limit theorem)?

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