Question

A police officer exam have an average score of 79 with a standard deviation of 4.5. Assume police officer exam scores follow a normal distribution.

1. 1. What is the minimum score a recruit needs to be in the top 10% on the exam?
2. 2. A minimum score of 70 is needed to pass. If 75 recruits take the exam, how many are expected to pass?

Answer 1

a)

X ~ N ( µ = 79 , σ = 4.5 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.1 = 0.9
To find the value of x
Looking for the probability 0.9 in standard normal table to calculate critical value Z = 1.2816
Z = ( X - µ ) / σ
1.2816 = ( X - 79 ) / 4.5
X = 84.767

b)

X ~ N ( µ = 79 , σ = 4.5 )
We covert this to standard normal as
P ( X < x) = P ( (Z < X - µ ) / σ )
P ( X > 70 ) = P(Z > (70 - 79 ) / 4.5 )
= P ( Z > -2 )
= 1 - P ( Z < -2 )
= 1 - 0.0228
= 0.9772

So, of the 75 students we expect 75 * 0.9772 = 73 to pass