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Central Limit Theorem and the assumption of Normality

Hi all,

Parametric tests assumes that the distributions of means should be normally distributed, in order for its results to be valid (along other assumptions, of course).

CLT says that the sampling distribution of means of most distributions follows a Normal distribution (for N > 30 or so).

People often check the Normality of the sample (not the means), and if it's not Normal, they go with non-parametric tests. Why's that? Why not go with parametric tests even if the sample is not normally distributed given the CLT?

What would be a good formal test for the normality of the means for a given sample? Bootstrapping the sample would be valid?
gpt
First, ANOVA’s assumption of normality does not pertain only to the sampling distribution of the mean (although that is a frequent misconception). Remember that the F statistic also has a denominator. Second, significance tests of normality are not informative because they test the null hypothesis that the distribution is exactly normal, which it never is. You can consider the robustness of ANOVA or whatever analysis you are using to the kind and degree of normality you have.