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For example,

\begin{array} {|r|r|}\hline & \text{Sample}\enspace X & \text{Sample}\enspace Y \\ \hline mean & 14 & 20 \\ \hline median & 5 & 5 \\ \hline \end{array}

How should we approach in order to test if $mean_Y > mean_X$ is statistical significant?

My thought is: It seems that the variables in the 2 groups don't follow a normal distribution (mean != median). But if the sample size is large enough, we can use the two-sample t-test (parametric). However, if our sample size is too small, we should use Wilcoxon-Mann-Whitney (or rank sum) test (non-parametric).

Is that a good approach? Not sure if I'm missing anything here.

Thank you!

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  • $\begingroup$ Note that hypotheses are about populations, not samples. If (as it seems) your hypothesis of interest involves the mean, you could directly do a nonparametric test of means (e.g a permutation test). How can you decide if sample size is 'large enough'? $\endgroup$
    – Glen_b
    Sep 15, 2020 at 23:36
  • $\begingroup$ Thanks for the suggestion. I'm not sure but I think used on the Central Limit Theorem definition, usually, sample size >= 30 should be large enough? Most importantly, why don't we use two-sample t-test or Wilcoxon-Mann-Whitney test here? My ultimate goal is to test the statistical significance of the mean difference. $\endgroup$
    – KatieN
    Sep 16, 2020 at 0:00
  • $\begingroup$ 1. Note that the actual central limit theorem makes no reference to any specific sample size; it's about the behaviour of the standardized sample mean as n increases beyond any finite value. e.g. see en.wikipedia.org/wiki/Central_limit_theorem ... 2. A t-statistic is not just a mean 3. 30 is not always sufficient. See (i) stats.stackexchange.com/questions/81074/… (ii) stats.stackexchange.com/questions/412606/… ... ctd $\endgroup$
    – Glen_b
    Sep 16, 2020 at 0:14
  • $\begingroup$ (iii) stats.stackexchange.com/questions/437372/… (and many more on site) 4. when this works it just implies that the test will have about the right significance level; but people usually also care about power. ... So far I've basically been addressing the premises of the question, but that last question in your comment there would be best addressed in an answer. $\endgroup$
    – Glen_b
    Sep 16, 2020 at 0:15
  • $\begingroup$ You might find a few relevant questions and answers for your issues are on site already -- such as stats.stackexchange.com/questions/121852/… ... it would be good to search for existing questions and identify what is not already duplicating questions already here $\endgroup$
    – Glen_b
    Sep 16, 2020 at 0:18

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