Question

Miles Driven. The U.S Federal Highway Administration conducts studies on motor vehicle travel by type of vehicle. Results are published annually in highway statistics. A sample of 15 cars yield the following data on number of miles driven, in thousands, for last year.

a. obtain and interpret the quartiles.

b. determine and interpret the interquartile range.

c. find and interpret the five number summary.

d. identify potential outliers, if any

e. construct and interpret a boxplot.

Short answer

The quartiles are Q1= 11, Q2=12.3, Q3=14.2

And IQR= 3.2

The five-number summary of a data set is 3.3, 11, 12.2, 14.2, 16.7

and it is not a potential outlier.

Step-by-step solution

Given information

We are given that the H. William discussed the frequency of cremation burials found in 17 archaeological sites in eastern England and frequencies are listed in the table and we have to find out the quartile.

Step 2. Part a. Explanation

First, arrange all the frequencies in ascending order and after that we get

3.3, 8.7, 9.6, 10.7, 11.3, 11.6, 11.9, 12.2, 13.2, 13.3, 13.6, 14.8, 15.0, 15.7, 16.7 then find the median of entire data as no. of observations is 15, so the median will be at $\frac{(15+1)}{2}=8^{th}$

so we get 12.2 as the median which is the Q2.

Now divide it into two half and as observations are odd so we will include median in both part

we get bottom half is 3.3, 8.7, 9.6, 10.7, 11.3, 11.6, 11.9, 12.2

and top half is 12.2, 13.2, 13.3, 13.6, 14.8, 15.0, 15.7, 16.7

hen find the median of bottom half and median is at the position

$\frac{\left(8+1\right)}{2}=4.5^{th}$and this means median is between 4th and 5th position and that is $\frac{\left(10.7+11.3\right)}{2}$= 11 and this is Q1. and last find the median of the top half and median is between 4th and 5th position and that$\frac{\left(13.6+14.8\right)}{2}=14.2$that is Q3.

hence these are all quartiles.

part b. step c. Explanation

we have to find the interquartile range.

The interquartile range or IQR is the difference between the first and the third quartiles, that is IQR= Q3-Q1

IQR= 14.2-11 = 3.2

part c. step 4. Explanation

We need to find and interpret the five- number summary.

The five-number summary of a data set is min,Q1,Q2,Q3,max

that is, 3.3, 11, 12.2, 14.2, 16.7

where min. and max. observations are 3.3 and 16.7 respectively

part d. step 4. Explanation

We need to find the potential outliers

As we know, Q1=11, Q2=12.2 , Q3=14.2 and IQR=3.2

Therefore, lower limit= 11 - (1.5 .3.2) = 6.2

and upper limit = 14.2+(1.5. 3.2) =19

as one observation that is 16.7 lies inside both upper and lower limit,

so it is not a potential outlier.