Seth, very nice work and I agree completely! However, Mathematica does test the null hypothesis, provided that the data is distributed according to the Normal Distribution. We know that data based on samples are problematic and various kinds of errors are possible. Many researchers in social sciences do not examine their data carefully and simply assume that some kind of normality is the rule...and often they make mistakes. Personally, I often ask Mathematica to "FindDistribution", before I run statistics and if it rejects it I am more careful.
Still, normality might not be required to run some kind of regressions. But it is always a good idea to not rely on simple regressions only, but try with different alternatives, like Quantile or Kernel estimates.
