Question

A normal population has mean u = 9 and standard deviation = 5. (a) What proportion of the population is less than 20? (b) What is the probability that a randomly chosen value will be greater than 6? Round the answers to four decimal places. Part 1 of 2 The proportion of the population less than 20 is . Part 2 of 2 The probability that a randomly chosen value will be greater than 6 is .

Answer 1

Part a)
X ~ N ( µ = 9 , σ = 5 )
P ( X < 20 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 20 - 9 ) / 5
Z = 2.2
P ( ( X - µ ) / σ ) < ( 20 - 9 ) / 5 )
P ( X < 20 ) = P ( Z < 2.2 )
P ( X < 20 ) = 0.9861

Part b)
X ~ N ( µ = 9 , σ = 5 )
P ( X > 6 ) = 1 - P ( X < 6 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 6 - 9 ) / 5
Z = -0.6
P ( ( X - µ ) / σ ) > ( 6 - 9 ) / 5 )
P ( Z > -0.6 )
P ( X > 6 ) = 1 - P ( Z < -0.6 )
P ( X > 6 ) = 1 - 0.2743
P ( X > 6 ) = 0.7257