Question

To qualify for a police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score to qualify to be accepted into the academy. Assume the test scores are normally distributed.

Answer 1

X ~ $N(200,20^2)$

Let a be the lowe score that allows to qualify for a police academy, then

$P(X<a)=0.9\implies P(N(200,20^2)<a)=0.9\implies P(200+20N(0,1)<a)=0.9\implies P(N(0,1)<{\frac {a-200} {20}})=0.9\implies {\frac {a-200} {20}}=1.29\implies a=225.8$