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The title is a bit misleading I suppose, so let me explain in detail what I mean.

For the sake of argument, let's say that we have an experiment, where population is divided into two groups and both perform same activities, but group A uses our new drug and group B uses placebo. Now, when analysing the data, we notice that group A has a "normal" distribution of the data and the group B appears to be "non-normally" distributed (please forgive my unprofessional terminology, I'm using quotation marks where I feel that the terminology is lacking).

Which tests should we use now? Is t-test still appropriate, even though we're "violating the normality of the data"?

I'd really appreciate some sources (e.g. books, articles) that clearly define what happens in this case or how to approach it.

Thank you in advance.

uglycode
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  • An ordinary t-test is appropriate to investigate the effect of a drug that may change the mean parameter of a normally distributed response, leaving scale & shape unchanged. When the response follows some other distribution in either or both groups, the difference in means will (usually) approximate a normal distribution better as the sample sizes increase - see [What normality assumptions are required for an unpaired t-test? And when are they met?](https://stats.stackexchange.com/q/19675/17230) or ... – Scortchi - Reinstate Monica Sep 09 '17 at 15:05
  • ... [How robust is the independent samples t-test when the distributions of the samples are non-normal?](https://stats.stackexchange.com/q/38967/17230) - ; for small samples, when there's no obvious reason to choose between parametric models, non-parametric models are often used - see [How to choose between t-test or non-parametric test e.g. Wilcoxon in small samples](https://stats.stackexchange.com/q/121852/17230) - : but if administration of the drug is evidently changing the scale or shape of the response distribution you need to think more about what exactly you want to test for. – Scortchi - Reinstate Monica Sep 09 '17 at 15:10
  • Sample size matters, with small sample sizes, it may be difficult to verify normality. If you provide some information on sample size, I might attempt an answer. – Heteroskedastic Jim Sep 10 '17 at 15:40

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