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In its Fuel Economy Guide for 2016 model vehicles, the Environmental Protection Agency gives data on 1170 vehicles. There are a number of high outliers, mainly hybrid gas‑electric vehicles. If we ignore the vehicles identified as outliers, however, the combined city and highway gas mileage of the other 1146 vehicles is approximately Normal with mean 23.0 miles per gallon (mpg) and standard deviation 4.9 mpg.
The quartiles of any distribution are the values with cumulative proportions 0.25 and 0.75. They span the middle half of the distribution.
a. What is the first quartile of the distribution of gas mileage?
b. What is the third quartile of the distribution of gas mileage?

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Answer:

a. 19.68 miles per gallon.

b. 26.32 miles per gallon.

Step-by-step explanation:

Mean gas mileage (μ) = 23.0 mpg

Standard deviation (σ) = 4.9 mpg

In a normal distribution, for any length X, the z-score is determined by the following expression:

 [tex]z=\frac{X-\mu}{\sigma}[/tex]

In a normal distribution, the 25th percentile  (first quartile) of a normal distribution has a corresponding z-score of z = -0.677 and the 75th percentile  has a corresponding z-score of z = 0.677

a. The first quartile of the distribution of gas mileage is

[tex]-0.677=\frac{X_{25}-23}{4.9}\\X_{25}=19.68\ mpg[/tex]

19.68 miles per gallon.

b. The third quartile of the distribution of gas mileage is

[tex]0.677=\frac{X_{25}-23}{4.9}\\X_{25}=26.32\ mpg[/tex]

26.32 miles per gallon.

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