Question

The Environmental Protection Agency records data on the fuel economy of many different makes of cars. Data on the mileage of 20 randomly selected cars are listed below. The values are ordered for convenience.

12 13 15 16 16 17 18 18 19 19
20 20 22 23 24 26 26 27 27 29
What is the interquartile range for the mileage data?
A) 8.5 miles per gallon
B) 16.5 miles per gallon
C) 17 miles per gallon
D) 25 miles per gallon

Answer 1

A) 8.5 miles per gallon

Explanation:

Given

`x = \{12\ 13\ 15\ 16\ 16\ 17\ 18\ 18\ 19\ 19\ 20\ 20\ 22\ 23\ 24\ 26\ 26\ 27\ 27\ 29\}`

`n = 20`

Required

The IQR

This is calculated as:

`IQR = Q_3 - Q_1`

Calculate
`Q_1`

`Q_1 = (1)/(4)(n + 1)th`

`Q_1 = (1)/(4)(20 + 1)th`

`Q_1 = (1)/(4)(21)th`

Remove bracket

`Q_1 = 5.25th`

This means that:

`Q_1` is the mean of 5th and 6th item

From the given data:

`5th \to 16`

`6th \to 17`

So, we have:

`Q_1 = (1)/(2)(16 + 17)`

`Q_1 = (1)/(2)(33)`

`Q_1 = 16.5`

`Q_1 = 16.25`

Calculate
`Q_3`

`Q_3 = (3)/(4)(n + 1)th`

`Q_3 = (3)/(4)(20 + 1)th`

`Q_3 = (3)/(4)(21)th`

Remove bracket

`Q_3 = 15.75th`

This means that:

`Q_3` is the mean of 15th and 16th item

From the data:

`15th \to 24`

`16th \to 26`

So:

`Q_3 = (1)/(2)(24 + 26)`

`Q_3 = (1)/(2)(50)`

`Q_3 = 25`

Recall that:

`IQR = Q_3 - Q_1`

`IQR = 25 -16.5`

`IQR = 8.5`