Question

The mean cost of domestic airfares in the United States rose to an all-time high of $375 per ticket. Airfares were based on the total ticket value, which consisted of the price charged by the airlines plus any additional taxes and fees. Assume domestic airfares are normally distributed with a standard deviation of $100. Use Table 1 in Appendix B.

a. What is the probability that a domestic airfare is $540 or more (to 4 decimals)?

b. What is the probability that a domestic airfare is $265 or less (to 4 decimals)?

c. What if the probability that a domestic airfare is between $320 and $480 (to 4 decimals)?

d. What is the cost for the 5% highest domestic airfares? (rounded to nearest dollar)

Answer 1

Part a)

X ~ N ( µ = 375 , σ = 100 )
P ( X >= 540 ) = 1 - P ( X < 540 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 540 - 375 ) / 100
Z = 1.65
P ( ( X - µ ) / σ ) > ( 540 - 375 ) / 100 )
P ( Z > 1.65 )
P ( X >= 540 ) = 1 - P ( Z < 1.65 )
P ( X >= 540 ) = 1 - 0.9505
P ( X >= 540 ) = 0.0495

Part b)

X ~ N ( µ = 375 , σ = 100 )
P ( X <= 265 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 265 - 375 ) / 100
Z = -1.1
P ( ( X - µ ) / σ ) < ( 265 - 375 ) / 100 )
P ( X <= 265 ) = P ( Z < -1.1 )
P ( X <= 265 ) = 0.1357

Part c)

X ~ N ( µ = 375 , σ = 100 )
P ( 320 < X < 480 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 320 - 375 ) / 100
Z = -0.55
Z = ( 480 - 375 ) / 100
Z = 1.05
P ( -0.55 < Z < 1.05 )
P ( 320 < X < 480 ) = P ( Z < 1.05 ) - P ( Z < -0.55 )
P ( 320 < X < 480 ) = 0.8531 - 0.2912
P ( 320 < X < 480 ) = 0.562

Part d)

X ~ N ( µ = 375 , σ = 100 )
P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.05 = 0.95
Looking for the probability 0.95 in standard normal table to calculate critical value Z = 1.64
Z = ( X - µ ) / σ
1.64 = ( X - 375 ) / 100
X = 539
P ( X > 539 ) = 0.05