Here's a fun bit in the NYT looking into the allegation that Alex Rodriguez tipped pitches to opponents. In a recent book, author Selena Roberts alleges that Rodriguez tipped (i.e., signaled to the opposing batter) what pitch was on its way. And while statistical analysis can't say whether Rodriguez did this or not, it can determine that if it was happening, was it benefiting Rodriguez. The analysis says Rodriguez did not benefit from the alleged activity.
Interesting, though, is the assumption that all performance should be the same across different game scenarios. Spring training sees high offensive output not only for favorable weather/atmospheric conditions but, as is speculated, due to the lack of pressure to perform. Pressure within a game changes as well, and if hitting is a function of pressure, then it would be necessary to adjust the assumptions accordingly. Of course, as more important at bats have more pressure, at bats with more runners on base should (in general) lead to the batter receiving better pitches. Whether you're facing a starter or reliever matters, too. It's not possible to extricate a one-way influence from all of this, but I'm not convinced that the null hypothesis should be that all of the averages should be equal.