# Multiple hypothesis tests, “comparing the expected values”

If I have two sets of random samples and I'd like to test whether there is some difference or not between them (or if one is better than the other) I'd test the null hypothesis $H_0: \mu_1 = \mu_2$ with a one-tailed test. So if it turns out that the null hypothesis is rejected on some significance level I could say the one has a significantly higher mean than the other (It was some time ago I studied this so please correct me if I'm wrong).

But what if I have more than two sets of random samples, is it possible to do a similar (I would assume: set of) test(s) so that I could draw the conclusion that one or some of has a significantly higher mean than the others.

If possible you someone please direct me to some keywords, papers and/or briefly explain how it's done.

• Mike, "whether there is some difference" is a two tailed alternative. If you're using a one tailed test with what (before seeing the sample) is a two tailed alternative, that's basically halving your p-values by cheating (choosing the direction of the alternative after the fact). If on the other hand your (a priori) alternatives are truly directional, then the F-tests that ANOVA gives you are always non-directional. If your $k$-sample alternative is actually directional then a non-directional test becomes increasingly weak against your directional alternative compared to a suitable test. – Glen_b -Reinstate Monica Sep 22 '13 at 7:15