  Comparisons of different treatments or production processes are the goals of
a significant fraction of applied research. Unsurprisingly, two-sample problems
play a main role in Statistics through natural questions such as `Is the the
new treatment significantly better than the old?'. However, this is only
partially answered by some of the usual statistical tools for this task. More
importantly, often practitioners are not aware of the real meaning behind these
statistical procedures. We analyze these troubles from the point of view of the
order between distributions, the stochastic order, showing evidence of the
limitations of the usual approaches, paying special attention to the classical
comparison of means under the normal model. We discuss the unfeasibility of
statistically proving stochastic dominance, but show that it is possible,
instead, to gather statistical evidence to conclude that slightly relaxed
versions of stochastic dominance hold.
