Question

Penalty Worksheet – Due 10-19            Simpson’s Paradox     Name______________________________

Instructions:You must show your work and all work must be organized and easy to follow. You will have to make an appointment with me to discuss your work. You will not receive credit if you cannot adequately explain your work to me in person.

1) 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.

  

2) On the 70’s game show Let’s Make a Deal, a contestant was asked to select one of three doors. Behind one of the doors was a prize; while behind the other two doors were gag gifts. After the contestant made their selection, the game show host (Monty Hall) would open one of the two remaining doors to reveal one of the gag gifts. The contestant was then given an opportunity to switch doors. (For example, suppose the contestant initially selects door number 1, and then Monty Hall opens number 3 to reveal a gag gift. The contestant can then stay with door 1 or switch to door 2.) The question is: should the contestant switch doors? Justify your answer by calculating the probability of winning the prize under both strategies.

Answer 1

1)Let be the games played by Brown on natural grass. Let be the games played by Brown on artificial surface. Then,

The proportion of games that Brown played on natural grass is

b) Similarly, let be the games played by Brown on natural grass. Let be the games played by Russel on artificial surface. Then,

The proportion of games that Russel played on natural grass is .

c) The scores of both palyers on artificial surface is higher than that in natural grass. The proportion of games Russel played in artificial surface is greater than that of Brown. Hence Russell's statistics were better than Brown's overall despite the fact that Brown hit for higher averages on both surfaces.

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