Question

Mandy has an IQ of 105. We know that the mean IQ is 100 with a standard deviation of 15. There are 100 people in Mandy’s Alcoholics Anonymous meeting. Taken at random, how many members are smarter than Mandy? How many people are not as smart as Mandy? How many members have an IQ between 120 and 135?

Answer 1

Part a)

X ~ N ( µ = 100 , σ = 15 )
P ( X > 105 ) = 1 - P ( X < 105 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 105 - 100 ) / 15
Z = 0.3333
P ( ( X - µ ) / σ ) > ( 105 - 100 ) / 15 )
P ( Z > 0.3333 )
P ( X > 105 ) = 1 - P ( Z < 0.3333 )
P ( X > 105 ) = 1 - 0.6305
P ( X > 105 ) = 0.3695

Number of members would be 0.3695 * 100 = 36.9454 ≈ 37

Part b)
X ~ N ( µ = 100 , σ = 15 )
P ( X < 105 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 105 - 100 ) / 15
Z = 0.3333
P ( ( X - µ ) / σ ) < ( 105 - 100 ) / 15 )
P ( X < 105 ) = P ( Z < 0.3333 )
P ( X < 105 ) = 0.6305

Number of members would be 0.6305 * 100 = 63.0546 ≈ 63

Part c)
X ~ N ( µ = 100 , σ = 15 )
P ( 120 < X < 135 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 120 - 100 ) / 15
Z = 1.3333
Z = ( 135 - 100 ) / 15
Z = 2.3333
P ( 1.33 < Z < 2.33 )
P ( 120 < X < 135 ) = P ( Z < 2.33 ) - P ( Z < 1.33 )
P ( 120 < X < 135 ) = 0.9902 - 0.9088
P ( 120 < X < 135 ) = 0.0814

Number of members would be 0.0814 * 100 = 8.1400 ≈ 8