Question

In a certain city within Florida, the IQ scores of its citizens follow a normal distribution with a mean of 85 and a standard deviation of 10.

a) What percentage of citizens have an IQ score that falls between 80 and 100?

b) 20% of the city's residents are above what IQ score?

Answer 1

Part a)

X ~ N ( µ = 85 , σ = 10 )
P ( 80 < X < 100 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 80 - 85 ) / 10
Z = -0.5
Z = ( 100 - 85 ) / 10
Z = 1.5
P ( -0.5 < Z < 1.5 )
P ( 80 < X < 100 ) = P ( Z < 1.5 ) - P ( Z < -0.5 )
P ( 80 < X < 100 ) = 0.9332 - 0.3085
P ( 80 < X < 100 ) = 0.6247

Part b)

X ~ N ( µ = 85 , σ = 10 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.2 = 0.8
To find the value of x
Looking for the probability 0.8 in standard normal table to calculate Z score = 0.8416
Z = ( X - µ ) / σ
0.8416 = ( X - 85 ) / 10
X = 93.416
P ( X > 93.416 ) = 0.2