Q 3.170 - Elementary Statistics

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 Stutisties. A sample of 15 cars yields 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 am:

e. construct and interpret a boxplot.

Step-by-Step Solution

Answer

a) The first quartile of the data set is 11

 The second quartile of the data set is 12.2

 The third quartile of the data set is 14.2

b) The inter quartile range is 3.2

c)Interpretation: It can be seen from the given statistics that the third and fourth quartiles have less fluctuation. However, there is a lot of variety in the first quarter.

d) The potential outliers total 3.3

e)The left whisker and asterisks in the above box plot reflect the spread of the first quarter of the data. The asterisks indicate a possible outlier in the data.

Part a Step 1 Given Information

Given in the question that, A sample of 15 cars yields the following data on number of miles driven, in thousands, for last year. 

Part (a) Step 2: Explanation

The quartiles should be calculated as follows:

To begin, arrange the data in ascending order:

$\begin{array}{|lllllllllllllll|}\hline 3.3& 8.7& 9.6& 10.7& 11.3& 11.6& 11.9& 12.2& 13.2& 13.3& 13.6& 14.8& 15.0& 15.7& 16.7\\ \hline\end{array}$

There are 15 observations in total. As a result, the median is the data's middle term.

The given data set's second quartile is $Q_2=12.2$

As a result, the data set's second quartile is 12.2

Consider the first section of the complete data set that is equal to or less than the data set's median.

$\begin{array}{|llllllll|}\hline 3.3& 8.7& 9.6& 10.7& 11.3& 11.6& 11.9& 12.2\\ \hline\end{array}$

There are 8 observations in all.

So the median is at position$=\frac{(n+1)}{2}$

$=\frac{9}{2}$

= 4.5

The median is average of $4^{th}$ and$5^{th}$ position values, it can be shown in boldface in ordered data set. Thus, the first quartile of the data set is

$Q_2=\frac{10.7+11.3}{2}$

= 11

As a result, the data set's first quartile is 11.

Consider the second section of the complete data set that is equal to or less than the data set's median.

$\begin{array}{|llllllll|}\hline 12.2& 13.2& 13.3& 13.6& 14.8& 15.0& 15.7& 16.7\\ \hline\end{array}$

The Number of observations is 8 .

So the median is at position

$=\frac{(n+1)}{2}$

$=\frac{9}{2}$

= 4.5

The median is average of$4^{\text{th }}$ and$5^{\text{th }}$ position values, it can be shown in boldface in ordered data set. Thus, the first quartile of the data set is

$Q_3=\frac{13.6+14.8}{2}$

= 14.2

As a result, the data set's third quartile is 14.2

Part (b) step 1: Given Information

Given in the question that, A sample of 15 cars yields the following data on number of miles driven, in thousands, for last year. 

Part (b) Step 2 Explanation

We have to calculate the inter quartile range

The difference between the first and third quartiles is the inter quartile range (IQR).

$IQR=Q_3-Q_1$

= 14.2 - 11

= 3.2

The interquartile range is$3.2$

Part (c) step 1 Given Information

Let's consider the table given in the question:

We have to find and interpret the five-number summary: 

Part c Step 2 Explanation

Calculate the five numbers summery

The data set's minimum value is$3.3.$

The data set's lower quartile is $Q 1=11.$

The data set's median is$Q 2=12.2.$

The data set's top quartile is$Q 3=14.2.$

The given data set's highest value is $16.7.$

The middle quarter's variance is measured in

$=Q_2-Q_1$

= 12.2 - 11

= 1.2

The measure of variation of the third quarter is

$=Q_3-Q_2$

= 14.2 -12.2

=    2

The Variation of the first quarter is

$=Q_1-\min$

= 11 - 3.3

= 7.7

The Variation of the fourth quarter is

$=\max -Q_3$

= 16.7 - 14.2

= 2.5

Interpretation: It can be seen from the given statistics that the third and fourth quartiles have less fluctuation. However, there is a lot of variety in the first quarter.

Part (d) Step 1: Given Information

A sample of 15 cars yields the following data on number of miles driven, in thousands, for last year. 

Part (d) Step 2: Explanation

We have to calculate the lower and upper limits of the data set.

Lower limit

$=Q_1-1.5\left(IQR\right)$

= 11 -1.5(3.2)

= 6.2

Upper limit

$=Q_3+1.5\left(IQR\right)$

= 14.2 + 1.5(3.2)

= 19

Potential outliers are observations that fall below or over the lower or higher limits.

$3.3$ is below the bottom limit based on the info provided.

As a result, the potential outliers total$3.3$.

Part (e) Step 1: Given Information

We have to construct and interpret a boxplot

Part (e) Step 2: Explanation

We can use MINITAB for the box plot:

The left whisker and asterisks in the above box plot reflect the spread of the first quarter of the data. The asterisks indicate a possible outlier in the data.