A certain NBA player had a long-run field goal percentage of 50% before needing

Question

A certain NBA player had a long-run field goal percentage of 50% before needing to take a season off to recover from an injury. (a) Suppose that since returning to the game from injury, the NBA player has had a field goal percentage of 60% from 100 shots. Has the player increased his field goal percentage? Perform a suitable hypothesis test and state your conclusion, taking -0.1. (b) Povide an approximate formula for the number of shots needed since returning from injury in order to have at least a 95% chance of concluding that the NBA player has increased his field goal percentage, as a function of the true new field goal percentage 100p, for pe (0.5,1). (Notice that if the new field goal percentage is very close to 50 then you will need a very large number of shots.) Compute the values of n for each of p = 0.7, 0.6, 0.51

Explanation / Answer

a) H0 : P = 0.5

H1 : P = 0.5

Decision rule : if Z statistic > 1.28 at 10% rejection level, then Reject H0

Z statistic = p - P / sqrt(PQ/n)

= 0.6 - 0.5 / sqrt(0.5*0.5/100)

= 2

Here Z statistic = 2 > 128, Reject H0

we conclude that player has increased his field goal percentage

b) sample size = [Z critical / margin error]2 PQ

margin error = [1 - 0.5]/2 = 0.25

if p = 0.7

n = [1.96/0.25]2 0.7*0.3 = 13

if p = 0.6

n = [1.96/0.25]2 0.6*0.4 = 15

if p = 0.51

n = [1.96/0.25]2 0.51*0.49 = 16

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