Question

3. It is increasingly true that government agencies, on all levels, use examinations to screen applicants and remove bias from the hiring process. Consider a police department in a large northeastern city that uses such an examination to hire new officers. The mean score for all applicants this year on the exam is 120, with a standard deviation of 20. The distribution of scores for the applicants is approximately normal.

a) Assume that only 10 percent of all applicants can be accepted this year. Will an applicant be accepted if his or her score on the exam is 20 points above the mean of the exam?

b) What is the cutoff score of this year’s test? In other words, what score is above 90 percent of all scores in the distribution?

c) What is the Z value for this score?

Answer 1

Mean = 120

Sd = 20

a)

He will pass the if the P[ X > mean + 20 ] = P[ X > 120+20 ] = P[ X > 140 ] > 10%

P[ X > 140 ] = P[ ( X - mean )/sd > ( 140 - mean )/sd ] = P[ ( X - 120 )/20 > ( 140 - 120 )/20 ] = P[ Z > 1 ] = 0.1587

Since it is more than 10% (0.1 )

hence, he will not clear

b)

Cut off will be x if

P[ X > x ] = 90% = 0.9

P[ X > x ] = P[ ( X - mean )/sd > ( x - mean )/sd ] = P[ ( X - 120 )/20 > ( x - 120 )/20 ] = P[ Z > ( x - 120 )/20 ] = 0.9

We know that P[ Z > 1.28 ] = 0.9

( x - 120 )/20 = 1.28

x - 120 = 1.28*20

x - 120 = 25.6

x = 120 + 25.6

x = 145.6

cut off = 145.6

The corresponding Z score is 1.28