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 22.2 miles per gallon (mpg) and standard deviation 5.2 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 are the quartiles of the distribution of gas mileage? (Round your answers to two decimal places.)

Answer 1

Answer:

$Q_1=18.69\\Q_2=22.2\\Q_3=25.7$

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 22.2 miles per gallon

Standard Deviation, σ = 5.2 mpg

We are given that the distribution of gas mileage is a bell shaped distribution that is a normal distribution.

Formula:

$z_{score} = \displaystyle\frac{x-\mu}{\sigma}$

First Quartile(Cumulative proportion 0.25)

We have to find the value of x such that the probability is 0.25

P(X < x)

$=P( z \leq \displaystyle\frac{x - 22.}{5.2})=0.25$  

Calculation the value from standard normal z table, we have,  

$P(z<-0.674) = 0.25$

$\displaystyle\frac{x - 22.2}{5.2} = -0.674\\\\x = 18.69$

$Q_1 = 18.69$

Second Quartile(Cumulative proportion 0.5)

We have to find the value of x such that the probability is 0.5

P(X < x)

$=P( z \leq \displaystyle\frac{x - 22.}{5.2})=0.5$  

Calculation the value from standard normal z table, we have,  

$P(z<0.00) = 0.5$

$\displaystyle\frac{x - 22.2}{5.2} = 0.0\\\\x = 22.2$

$Q_2 = 22.2$

Third Quartile(Cumulative proportion 0.75)

We have to find the value of x such that the probability is 0.75

P(X < x)

$=P( z \leq \displaystyle\frac{x - 22.}{5.2})=0.75$  

Calculation the value from standard normal z table, we have,  

$P(z<0.674) = 0.75$

$\displaystyle\frac{x - 22.2}{5.2} = 0.674\\\\x = 25.70$

$Q_3 = 25.70$

Inter-quartile range:

$Q_3-Q_1=25.70-22.2 = 3.5$