This problem is taken from the 2011 AP Statistics Exam, which we reviewed in class. The exact problem can be found here, and solution after:
Its Question 1, part c.
However, my problem is more general.
Suppose I have two players, each with measured running times and weight lifting amounts. It has already been concluded that at least one of these distributions is not normal. The z score table is reproduced(approximately) below:
Amount Held 2.4 2.6
Running Time -1.2 -.2
According to the solution, A is slightly worse at weight lifting, but much greater at running, thus A is the overall better player. However, in concluding that, we saw that the B led A in amount held by .2, and A led B in running time by 1.0. We compared the two z scores, and determined A's ability in running overshadows B's ability in amount held. But by doing this, aren't we comparing z scores(or differences in them) across distributions? Specifically, how do we know that a 1.0 lead in running is more impressive than a .2 lead in Amount held, if we dont know the skewness of the distributions?
This was a long question, and I wasn't quite sure how to phrase it. Any help in understanding this would be great.