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Given the central limit theorem, can you always use a t-test to test a difference between two groups even if the data are not normally distributed but the sample size is large enough? Stated in another way, if data are not normally distributed, would you only use Mann-Whitney in cases where the sample size is not large enough?

Is there a general rule of thumb for sample size minimum to be able to use the central limit theorem to state that the sampling distribution of the mean is approximately normal (despite the underlying data not being normally distributed)?

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    $\begingroup$ A highly relevant example is discussed at stats.stackexchange.com/questions/69898. $\endgroup$ – whuber Nov 6 '18 at 17:41
  • $\begingroup$ @puckT 1. how does asymptotic normality for the numerator of a t-statistic imply that the t-statistic itself has a t-distribution? I presume you're actually trying to get to an argument that the t-statistic will be asymptotically normal, not t (in which case, you'll need an additional theorem and the argument will justify doing a Z-test rather a t-test). 2. Note that people tend to care about power, not just significance level, and asymptotic normality of the numerator doesn't get around the fact that if tails are very heavy the t-test can have relatively low power (ARE -> 0 in some cases). $\endgroup$ – Glen_b -Reinstate Monica Nov 6 '18 at 23:48
  • $\begingroup$ Besides the indicated duplicates, many other answers to questions on site address various aspects of the issues in your question. Searching on keywords in your question will turn many of them up. One question on site discusses a real-data example where thousands of observations were not sufficient to get the numerator of the t-statistic to be reasonably close to normal (and much larger counterexamples are trivial to generate). $\endgroup$ – Glen_b -Reinstate Monica Nov 7 '18 at 0:09
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The issue of normal distribution assumption for Student t-test was already well explained and disused here: T-test for non normal when N>50?

For both test, the data needs to fulfil certain assumptions (discussed here) and the choice is largely determinated by the research question and features of your data. Sample size is important but it is only one of them.

Another good explanation is provided here.

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