Question

A sample of final exam scores is normally distributed with a mean equal to 29 and a variance equal to 25.

• Part (a) What percentage of scores are between 24 and 34? (Round your answer to two decimal places.)

• Part (b) What raw score is the cutoff for the top 10% of scores? (Round your answer to one decimal place.)

• Part (c) What is the proportion below 23? (Round your answer to four decimal places.)

• Part (d) What is the probability of a score less than 35? (Round your answer to four decimal places.)

Answer 1

Part a)

P ( 24 < X < 34 )
Standardizing the value

Z = ( 24 - 29 ) / 5
Z = -1
Z = ( 34 - 29 ) / 5
Z = 1
P ( -1 < Z < 1 )
P ( 24 < X < 34 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 24 < X < 34 ) = 0.8413 - 0.1587
P ( 24 < X < 34 ) = 0.6827   0.68

Part b)

P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.1 = 0.9
Looking for the probability 0.9 in standard normal table to calculate critical value Z = 1.28

1.28 = ( X - 29 ) / 5
X = 35.4
P ( X > 35.4 ) = 0.1

Part c)

P ( X < 23 )
Standardizing the value

Z = ( 23 - 29 ) / 5
Z = -1.2

P ( X < 23 ) = P ( Z < -1.2 )
P ( X < 23 ) = 0.1151

Part d)

P ( X < 35 )
Standardizing the value

Z = ( 35 - 29 ) / 5
Z = 1.2

P ( X < 35 ) = P ( Z < 1.2 )
P ( X < 35 ) = 0.8849