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So for mean inference, would there be a situation where you cannot use the T-test? Seems like you can always calculate a t-statistic, so it seems like you can always do the t-test for mean inference.

But what about for situations where the true population distribution either does not have a mean or a variance?

Richard Hardy
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confused
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you cannot use the t-test if you are not sure of the shape of the distribution you are testing, the classic example is when you have few observations, which therefore you have no convergence in a normal one and you have to use non-parametric methods

domiziano
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  • Thanks! I know rule of thumb says sample size needs to be 30 - is there a better rule of thumb as to when you need to use non-parametric methods? – confused Sep 24 '20 at 09:46
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    the rule of thumbs is fixed to 30 because at 30 we are sure that there is convergence to a normal distribution. If you have less than 30 observation and you have no assumption on the distribution of the data, therefore is it better choose non-parametric approach, like the Mann-Whitney test, that is the equivalent of the t-test but non-parametric. – domiziano Sep 24 '20 at 10:48
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    @confused, be aware that you don't necessarily need N=30 for the t-test to be valid (cf, [Is there a minimum sample size required for the t-test to be valid?](https://stats.stackexchange.com/q/37993/7290)), nor is the sampling distribution guaranteed to converge to a normal when N=30 (cf, [Regression when the OLS residuals are not normally distributed](https://stats.stackexchange.com/a/29748/)). – gung - Reinstate Monica Sep 24 '20 at 17:15
  • @gung-ReinstateMonica Thanks! Just to verify I understand that linked thread correctly. We can do t-test for mean inference (regardless of shape of true population distribution) with any sized sample as long as we can calculate the t-statistic. So basically 3 samples is enough. The fatter tails of the t-distribution will compensate for more uncertainty. The t-distribution is still the exact distribution of the sampling distribution of the sample mean. The only issue is the power of the test, as you mentioned. – confused Sep 25 '20 at 09:14
  • This would explain why I am having trouble finding a non-parametric version of the one-sample t-test. I have found two sample tests like Mann-Whitney U Test. Wilcoxon Signed-Rank Test. Kruskal-Wallis H Test. Friedman Test. – confused Sep 25 '20 at 09:15
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    The equivalent for one sample t-test should be the wilcoxon signed rank test https://sixsigmastudyguide.com/1-sample-wilcoxon-non-parametric-hypothesis-test/ – domiziano Sep 25 '20 at 10:06
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    @confused, no, the t-test isn't necessarily valid if the population isn't normal. There are assumptions for the t-test, but minimum N isn't one of them, nor is any 'high' N sufficient to make it valid. The validity of the t-test isn't primarily a function of N. – gung - Reinstate Monica Sep 25 '20 at 11:39