A manager is evaluating two baseball players based upon their batting averages (

Question

A manager is evaluating two baseball players based upon their batting averages (the proportion of the time that they get a hit when they come to bat). Russell has an average of 290 (gets a hit in 29% of at-bats), while Brown has an average of 285. On further inspection, however, the manager notices that Brown out hits Russell on both natural grass (265 to 260) and artificial surface (315 to 310). This is an example of what’s known as Simpson’s Paradox.

a. Determine the proportion of games that Brown played on natural grass.

b. Determine the proportion of games that Russell played on natural grass.

c. (Briefly) explain how Russell's statistics were better than Brown's overall despite the fact that Brown hit for higher averages on both surfaces.

Explanation / Answer

a. Let proportion of the games that Brown played on natural grass be, b.

=> 285 = b*265+(1-b)*315

Solving above equation for b yields, b = (315-285)/(315-265) = 0.6 or 60%

b. Let proportion of the games that Russell played on natural grass be, r.

=> 290 = r*260+(1-r)*310

Solving above equation for b yields, b = (310-290)/(310-260) = 0.4 or 40%

c. Russell has overall statistics better than Brown because he has played a higher proportion of matches on artificial surface where the hit rate higher than that on natural grass

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