In the book "Causal Inference In Statistics" by Pearl et al., there is the following problem (study question 1.2.2.)
A baseball batter Tim has a better batting average than his teammate Frank. However, someone notices that Frank has a better batting average than Tim against both right-handed and left-handed pitchers. How can this happen? (Present your answer in a table.)
This is basically an example of the Simpson's paradox. The following table of proportions is used in the solution
And, apparently, the "casual story" would be
- left-handed batters, on average, hit better than their right-handed counterparts
- Frank met significantly more left-handed than Tim
And the solution is then "left-handed batters is a common cause of meeting the player and failure", thus we should look at the segregated data, that is, we should compare Frank and Time with left-hand and right-hand batters separately.
I am not fully understanding this "causal story". It is clear that Frank met more left-handed than Tim, according to the table. But how can we solve this problem if we do not know that "left-handed batters, on average, hit better than their right-handed counterparts". Where does this information come from? Or is this just a supposition in order to draw a conclusion? For example, could we have supposed that right-handed batters are better than left-handed? Then what? Furthermore, how can we come up with these tables?
In general, how do we know whether to draw some conclusions based on the segregated or aggregated data?