Choosing distribution for hypotheses testing I am testing hypotheses, that there is a significant difference between skin temp in the morning and in the evening. 
My null hypotheses  is that $$ H_0: |t_m - t_e| = 0 $$
My alte hypotheses  is that $$ H_a: |t_m - t_e| > 0 $$
There is a hist and normplot of morning data:


And there is a hist and normplot of evening data


What distribution should I use for statistical testing of my hypotheses? 
 A: You're probably going to want to use a non-parametric test. One user in a comment suggested to use a $t$-test, but I think this would be a bad idea. The normality assumptions do not seem to be satisfied. In fact it appears that the two populations do not even have the same distribution. A $t$-test is somewhat robust against non-normality, but I think I better fit in this case would be a randomization test. This involves repeatedly randomly shuffling the data points into two classes and comparing the means of the two classes. You can't try all combinations so you perform a random sample of them basically. The idea is that if the means of the two populations are truly the same, then the observed partition of the sample points into classes from your experiment shouldn't be too extreme. The absolute value of the difference of means shouldn't be too unlikely to have occurred as a result of random shuffling. The linked article explains this much better than I can. 
You can perform such a test easily using the coin package of R. This answer
should help you get started if you decide to go this route.
