Question

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.What is the first quartile of the distribution of gas mileage?

Answer 1

The first quartile of the distribution of gas mileage is 19.6925 mpg.

Explanation:

Problems of normally distributed samples can be 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 = 23, \sigma = 4.9`

What is the first quartile of the distribution of gas mileage?

The first quartile has a proportion of 0.25. So this is the value of X when Z has a pvalue of 0.25. It happens between Z = -0.67 and Z = -0.68, so i am going to use
`Z = -0.675`

So

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

`-0.675 = (X - 23)/(4.9)`

`X - 23 = -0.675*4.9`

`X = 19.6925`

The first quartile of the distribution of gas mileage is 19.6925 mpg.