In its Fuel Economy Guide for 2014 model vehicles, the Environmental Protection
ID: 3273264 • Letter: I
Question
In its Fuel Economy Guide for 2014 model vehicles, the Environmental Protection Agency gives data on 1160 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 1134 vehicles is approximately Normal with mean 22.7 miles per gallon (mpg) and standard deviation 5.2 mpg. The quintiles of any distribution are the values with cumulative proportions 0.20, 0.40, 0.60, and 0.80. What are the quintiles of the distribution of gas mileage? (Use software what are the first, second, third, and fourth quintiles?
Explanation / Answer
here we use standard normal variate Z=(X-mean)/sd
for the cumulative proportion we use cdf of z
so for cumulative proportion 0.2 we find the corresponding z-value P(Z<z)=0.2
and corresponding z=-0.8416 ( using ms-excel command=normsinv(0.2)
and corresponding X=22.7-0.8416*5.2=18.32 ( first quintiles)
similarly
for cumulative proportion 0.4 the z-value=-0.2533 and corresponding X=22.7-0.2533*5.2=21.38 (second quintiles)
for cumulative proportion 0.6 the z-value=0.2533 and corresponding X=22.7+0.2533*5.2=24.02 (third quintiles)
for cumulative proportion 0.8 the z-value=0.8416 and corresponding X=22.7+0.8416*5.2=27.02 (foruth quintiles)