Question

The mean cost of domestic airfares in the United States rose to an all-time high of $385 per ticket (Bureau of Transportation Statistics website, November 2, 2012). 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 $110.a. What is the probability that a domestic airfare is $550 or more (to 4 decimals)?b. What is the probability than a domestic airfare is $250 or less (to 4 decimals)?c. What if the probability that a domestic airfare is between $300 and $500 (to 4 decimals)?d. What is the cost for the 3% highest domestic airfares?

Answer 1

a) 0.0668 = 6.68% probability that a domestic airfare is $550 or more

b) 0.1093 = 10.93% probability than a domestic airfare is $250 or less

c) 0.6313 = 63.13% probability that a domestic airfare is between $300 and $500

d) The cost for the 3% highest domestic airfares are $592 and higher.

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 385, \sigma = 110`

a. What is the probability that a domestic airfare is $550 or more (to 4 decimals)?

This is 1 subtracted by the pvalue of Z when X = 550. So

`Z = (X - \mu)/(\sigma)`

`Z = (550 - 385)/(110)`

`Z = 1.5`

`Z = 1.5` has a pvalue of 0.9332

1 - 0.9332 = 0.0668

0.0668 = 6.68% probability that a domestic airfare is $550 or more

b. What is the probability than a domestic airfare is $250 or less (to 4 decimals)?

This is the pvalue of Z when X = 250. So

`Z = (X - \mu)/(\sigma)`

`Z = (250 - 385)/(110)`

`Z = -1.23`

`Z = -1.23` has a pvalue of 0.1093

0.1093 = 10.93% probability than a domestic airfare is $250 or less

c. What if the probability that a domestic airfare is between $300 and $500 (to 4 decimals)?

This is the pvalue of Z when X = 500 subtracted by the pvalue of Z when X = 300. So

X = 500

`Z = (X - \mu)/(\sigma)`

`Z = (500 - 385)/(110)`

`Z = 1.045`

`Z = 1.045` has a pvalue of 0.8519

X = 300

`Z = (X - \mu)/(\sigma)`

`Z = (300 - 385)/(110)`

`Z = -0.77`

`Z = -0.77` has a pvalue of 0.2206

0.8519 - 0.2206 = 0.6313

0.6313 = 63.13% probability that a domestic airfare is between $300 and $500

d. What is the cost for the 3% highest domestic airfares?

At least the 97th percentile, so at least X when Z has a pvalue of 0.97. So X when Z = 1.88.

`Z = (X - \mu)/(\sigma)`

`1.88 = (X - 385)/(110)`

`X - 385 = 1.88*110`

`X = 592`

The cost for the 3% highest domestic airfares are $592 and higher.