Test difference between samples with very small sample size Suppose I have observed 3 realizations of two non-negative, integer random variables $X$ and $Y$.  Nothing is known about their underlying distribution.  The results were $x = \{4,~8,~2\}$, and $y = \{22,~11,~8\}$.  Hence, $\overline{x} = 14/3$ and $\overline{y} = 41/3$, for a sample mean difference of $9$.
My question is: is there any way to conduct a meaningful statistical hypothesis test in order to decide whether $H_0: E(X) = E(Y)$, can be rejected? 
My first idea was to use a two-sample t-test or Welch-test, but I guess the sample size is way too low for that. Is there anything one can reasonably test?
 A: There are potentially a number of ways of testing if these two samples differ, but all will probably have low power.  You could use a t-test, but its validity will depend on whether the underlying populations are normally distributed and have equal variances.  With so few data, you really can't check that very well so you have to rely entirely on prior knowledge (of which you say you have none) and the assumptions you are willing to make.  Given that your variances are $9.3$ and $54.3$, I would not want to make the assumption of equal variances (although, again, with so few data they actually could be), so the Satterthwaite-Welch correction seems appropriate.  If you weren't willing to assume the populations were exactly normal (since the central limit theorem cannot cover you with samples this small), you could use the Mann-Whitney U test.  As it happens, that test gives a lower p-value ($.12$) than the corrected t-test ($.16$).  The question then, is what you want to conclude from these results.  My opinion is that using a rigid $.05$ cutoff is typically not appropriate (see my answer here: When to use Fisher and Neyman-Pearson framework?); so I would say this result is somewhat ambiguous, but you might find it does provide some evidence against the null, depending on how plausible the null is a-priori.  
A: You say that nothing is known about the distribution, but you also say that the values are non-negative integers, so that tells you something, can you learn more about the theory?
An exact permutation test on your data gives a p-value of 0.10 and does not require assumptions about the distribution (just testing that it is identical for the 2 groups).
If your data represent counts (non-negative integers) then a Poisson model may be appropriate (which gives a p-value less than 0.001), but make sure that the assumptions are reasonable.  
As @gung mentions, there are many ways to look at this and which is best is going to depend on the source of the data and the science that underlies it.  learning about the data, talking to the person that gave you the data, talking to other experts, etc. is going to be of the most benefit.
If you really cannot learn anything more about the underlying distribution or approximations to it then you may have to resort to SnowsCorrectlySizedButOtherwiseUselessTestOfAnything (a function in the TeachingDemos package for R) which will give a p-value without requiring any assumptions about your data.  But note that that function is considered less useful than its documentation.
