I have 2 groups of sample sizes 10 and 11 respectively.
Can I apply a parametric or nonparametric test to find a significant difference in means between the groups?
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Since you're asking about a difference in means, if you're prepared to assume the shapes are the same under the null (that is, if the means are the same the shapes of the two population distributions are the same), and that population means exist, then you could certainly apply nonparametric tests -
i) you could apply a permutation test; the sample sizes are small enough to enumerate the full distribution.
ii) You could also apply a Wilcoxon-Mann-Whitney two sample test; this will have slightly less power than the permutation test at the normal, but may have better power if distributions are heavier-tailed. With the assumptions I mentioned, this will be a test for equality of means.
My advice would be that it's pretty pointless to test the assumption of identical shapes - firstly because the sample size is small; secondly because formal testing of assumptions answers the wrong question; and thirdly because you don't actually need to assume that they differ only by shape when H0 is false unless you want a CI for the difference in means, so you may not be able to test it in any case.
You could make pretty much any suitable parametric assumption (normal, exponential, gamma, Poisson, binomial, Weibull, ... etc etc) and test equality of means.
You can reasonably do a diagnostic check (such as a QQ plot or similar) of the assumption, but (again) I wouldn't advise formal testing of the assumption.
The answer depends whether you can assume any parametric distribution on the two groups. E.g., if you know that the two groups are normally distributed (either by statistical hypothesis testing - see Kolmogorov Smirnov, Anderson Darling or Shapiro Wilk tests - or theoretically) then you can apply a t test to check whether the means are statistically equal... In theory, if you can assume a specific distribution on the two groups you can perform any tests based on log-likelihood. You cannot perform a test based on sample's normality distribution if you cannot assume it on both samples, also taking into account that your sample size does not allow you to rely on asymptotic results. In such case you have to consider the use of non-parametric tests (e.g. median test, Wilcoxon or Mann Whitney test) that allows you to take inference on the parameters not relying on distributional assumptions.