If there's no answer, is it at least true that for a very large sample size the normality assumption is relaxed for the t-test? My question lies in the fact that I ve seen there's a two sample distribution t-test which assumes unequal variances and can perform a tail test to reject the null hypothesis (μ1 = μ2) over one of the pre-mentioned alternative hypotheses, only that it assumes the two sample distributions follow the Normal distribution.

If the normality assumption can be ommited, for large samples I am done. There's one point however. I've read somewhere that this is true only if sample sizes do not differ significantly. Well my samples differ: I got two such pairs. The first wo samples have sizes 209826 and 11588 (which unfortunately differ by an order of magnitude) and I also have two samples made of averages of previous samples, which, sadly and contrary to the central limit theorem, fail to pass normality tests and have sizes 14958 and 1070.

Finally if there's nothing concerning the above, is there a test that proves that one distribution has more lower values than another distribution, which has more higher values?

Thank you all in advance! -If there's anyone out there...-

Michael