A manager is evaluating two baseball players based upon their batting averages (
ID: 3203528 • Letter: A
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 out of 1000, Brown plays x games on natural grass and (1000-x) games on artificial
from weighted average method
his average 285 =(x*265+(1000-x)*315)/1000
50x =315000-285000
x =30000/50 =600
hence he played 600 on natural grass and 400 on artificial grass.
therefore proportion of games that Brown played on natural grass =600/1000 =0.6
b) let out of 1000, russel plays x games on natural grass and (1000-x) games on artificial
from weighted average method
his average 290 =(x*260+(1000-x)*310)/1000
50x =310000-290000
x =20000/50 =400
hence he played 400 on natural grass and 600 on artificial grass.
therefore proportion of games that Russell played on natural grass =400/600 =0.40
c) as on artificial grass both player has higher score then natural grass. But as BRown has played less games on Artificial grass with respect to Russel, his weighted average batting score is less then Russel.
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