Question

Required: Show your complete solution.  Write your formula (explicitly) first before actually solving the problem.Clearly define your random variables or events

The amount that airlines spend on food per passenger is normally distributed with mean $8.00 and a standard deviation $2.00. a. What percent spend less than $5.00 per passenger? b. What percent spend between $6.00 and $10.00? c. What percent spend more than $12.50?

Answer 1

Part a)

X ~ N ( µ = 8 , σ = 2 )
P ( X < 5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 5 - 8 ) / 2
Z = -1.5
P ( ( X - µ ) / σ ) < ( 5 - 8 ) / 2 )
P ( X < 5 ) = P ( Z < -1.5 )
P ( X < 5 ) = 0.0668
Percentage = 0.0668 * 100 = 6.68%

Part b)

X ~ N ( µ = 8 , σ = 2 )
P ( 6 < X < 10 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 6 - 8 ) / 2
Z = -1
Z = ( 10 - 8 ) / 2
Z = 1
P ( -1 < Z < 1 )
P ( 6 < X < 10 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 6 < X < 10 ) = 0.8413 - 0.1587
P ( 6 < X < 10 ) = 0.6827
Percentage = 0.6827 * 100 = 68.27%

Part c)

X ~ N ( µ = 8 , σ = 2 )
P ( X > 12.5 ) = 1 - P ( X < 12.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 12.5 - 8 ) / 2
Z = 2.25
P ( ( X - µ ) / σ ) > ( 12.5 - 8 ) / 2 )
P ( Z > 2.25 )
P ( X > 12.5 ) = 1 - P ( Z < 2.25 )
P ( X > 12.5 ) = 1 - 0.9878
P ( X > 12.5 ) = 0.0122
Percentage = 0.0122 * 100 = 1.22%