Q 3.173 - Elementary Statistics

Question

 Capital Spending. An issue of the Brokerage Report discussed the capital spending of telecommunications companies in the United States and Canada. The capital spending, in thousands of dollars, for each of the 27 telecommunications companies is shown in the following table.

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 quartiles is $660,1800,4749.5$
(b) The interquartile range is $4643$
(c) The five-number summary is $21,660,1800,4749.5,17341$
(d) The potential outliers is $17,341$
(e) Obtained box-and-whisker plot as:

Part a Step 1 Given information

To obtain and interpret the quartiles.

Part a Step 2 Explanation

The sort of given data and arrange the data in ascending order to begin:

$21,70,125,195,389,649,656,664,682,1006,1300,1403,1433,1800,1982$$2205,2515,3027,3634,4200,5299,5947,7886,8543,9310,11189,14341$

The middle value of the sorted data set is the second quartile or median:
$Q_2=1800$
The median of the numbers below or equal to the median is the first quartile:
$Q_1=\frac{656+664}{2}$

$=660$

The median of the values above or equal to the median is the third quartile:

$Q_3=\frac{4200+5299}{2}$

$=4749.5$

As a result, the quartiles is $660,1800,4749.5$.

Part (b) Step 1: Given information

To determine and interpret the interquartile range.

Part (b) Step 2: Explanation

The difference between the third and first quartiles is the interquartile range:
$IQR=Q_3-Q_1$

$=4749.5-660$

$=4089.5$

As a result, the interquartile range is $4643$.

Part (c) Step 1: Given information

To find and interpret the five-number summary.

Part (c) Step 2: Explanation

The minimum, first quartile, second quartile, third quartile, and maximum are the five numbers in the summary:
Minimum: $21$
$Q_1:660$
$Q_2:1800$
$Q_3:4749.5$
Maximum: $17341$

As a result, the five-number summary is $21,660,1800,4749.5,17341$.

Part (d) Step 1: Given information

To identify potential outliers for given data set.

Part (d) Step 2: Explanation

More than $1.5 IQR\text{'}s$  larger than $Q_3$ or less than $Q_1$ is considered an unusual.
$Q_3+1.5IQR=4749.5+1.5(4089.5)$

$=10,883.75$

$Q_1-1.5IQR=660-1.5(4089.5)$

$=-6308.5$

As a result, $11,189$ and$17,341$ are outliers.

Part (e) Step 1: Given information

To construct and interpret a boxplot.

Part (e) Step 2: Explanation

The boxplot's whiskers are at their smallest and largest values (that are not outliers).
The box has a vertical line at the median and starts at the first quartile and finishes at the third quartile.
An "$x$" is used to represent outliers.