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Just a quick sanity check, if a want to do an hypothesis testing about the mean, following the Central Limit theorem the sampling distribution of the mean is asymptotically normal, then the only tests I care about are the Wald test and the t-test.

Even the distribution of the population is very weird, I don't need to too much think, only use one of these two tests. Am I right?

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    $\begingroup$ As a general proposition this cannot be right, because (a) you don't know how large the sample size might need to be for the CLT to be a decent approximation and (b) there's always the possibility of a more powerful test (adapted to your assumptions about the population distribution). See stats.stackexchange.com/a/69967/919 for a real example. $\endgroup$
    – whuber
    Feb 18 at 14:12
  • $\begingroup$ @whuber Thank you very much, everything is clear now. So what to do if I have a very skew distribution and I need to test the mean? It wasn't so clear for me in the link you showed me. $\endgroup$
    – user45523
    Feb 18 at 14:29
  • $\begingroup$ Bootstrapping immediately comes to mind. Or, following my answer in the link I give, a quick simulation will indicate whether you can rely on the CLT. There are also analytical approaches equivalent to such a simulation (based on inspecting higher moments of the data, which when appropriate scaled will estimate cumulants of the sampling distribution). $\endgroup$
    – whuber
    Feb 18 at 14:30

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