Question

A sample of final exam scores is normally distributed with a mean equal to 20 and a variance equal to 25. a) what percentage of scores is between 15 and 25. b) what raw score is the cutoff for the top 10% of scores. c) what is the probability of a score less than 27. d) what is the proportion below 13.

Answer 1

Part a)

X ~ N ( µ = 20 , σ = 5 )
P ( 15 < X < 25 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 15 - 20 ) / 5
Z = -1
Z = ( 25 - 20 ) / 5
Z = 1
P ( -1 < Z < 1 )
P ( 15 < X < 25 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 15 < X < 25 ) = 0.8413 - 0.1587
P ( 15 < X < 25 ) = 0.6827
Percentage = 0.6827 * 100 = 68.27%

Part b)

X ~ N ( µ = 20 , σ = 5 )
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
Z = ( X - µ ) / σ
1.28 = ( X - 20 ) / 5
X = 26.4
P ( X > 26.4 ) = 0.1

Part c)

X ~ N ( µ = 20 , σ = 5 )
P ( X < 27 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 27 - 20 ) / 5
Z = 1.4
P ( ( X - µ ) / σ ) < ( 27 - 20 ) / 5 )
P ( X < 27 ) = P ( Z < 1.4 )
P ( X < 27 ) = 0.9192

Part d)

X ~ N ( µ = 20 , σ = 5 )
P ( X < 13 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 13 - 20 ) / 5
Z = -1.4
P ( ( X - µ ) / σ ) < ( 13 - 20 ) / 5 )
P ( X < 13 ) = P ( Z < -1.4 )
P ( X < 13 ) = 0.0808