Chapter 2, Problem 2.52E

a.

To determine

To find: The five-number summary for the distribution of revenues for the top global 500

To explain: Why the five-number summary suggests that the distribution is right skewed.

Answer to Problem 2.52E

The five-number summary for the distribution of revenues for the top global 500 is $139.4149.1182.0286.3485.7$.

The difference between the median and minimum is less than the difference between maximum and median in the five-number summary suggests that the distribution is right skewed.

Explanation of Solution

Given info: Table 2.3gives the highest revenues for 30 companies from global 500 companies in 2014.

Calculation:

Five-number summary:

The five-number summary of a distribution consists of minimum, the first quartile, the median, the third quartile, and the maximum. The order of five-number summary is as follows:

$Minimum{Q}_{1}M{Q}_{3}Maximum$.

Software procedure:

Step-by-step software procedure for five-number summary using MINITAB software is as follows:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns GLOBE500.
• Choose option statistics, and select minimum, first quartile, median, third quartile, maximum.
• Click OK.

Output using MINITAB software is as follows:

The five-number summary for the distribution of the highest revenues of 30 companies for the top global 500 is $139.4149.1182.0286.3485.7$.

Justification:

The distance from median and minimum is,

$\begin{array}{c}\text{Median}-\text{Minimum}=182.0-139.4\\ =42.6\end{array}$

The distance from the maximum to the median is,

$\begin{array}{c}\text{Maximum}-\text{Median}=485.7-182.0\\ =303.7\end{array}$

From the five-number summary, the distribution is right skewed because the difference between the median and minimum is less than the difference between the median and maximum.

b.

To determine

To find: The companies which are outliers according to the $1.5\times IQR$ rule.

To check: Whether the $1.5\times IQR$ rule suggest the same about the companies that are outliers or not.

Answer to Problem 2.52E

According to $1.5\times IQR$ rule, there are no companies that are considered as outliers.

Yes, the $1.5\times IQR$ rule’s suggestion is agreed with stemplot.

Explanation of Solution

Calculation:

From the output in part (a) $Q_1=149.1$ and $Q_3=286.3$

Interquartile range (IQR):

The difference between the first quartile and the third quartile is considered as interquartile range.

Thus,

$\begin{array}{c}IQR=Q_3-Q_1\\ =286.3-149.1\\ =137.2\end{array}$

For revenues of 30 companies, $IQR=137.2$.

The $1.5\times IQR$ rule for outliers:

A observation is a suspected as outlier if it is more than $Q_3+\left(1.5\times IQR\right)$ or less than $Q_1-\left(1.5\times IQR\right)$.

$\begin{array}{c}{Q}_3+\left(1.5\times IQR\right)=286.3+\left(1.5\times 137.2\right)\\ =286.3+205.8\\ =492.1\end{array}$

$\begin{array}{c}{Q}_1-\left(1.5\times IQR\right)=149.1-\left(1.5\times 137.2\right)\\ =149.1-205.8\\ =-56.7\end{array}$

The $1.5\times IQR$ rule suspects no outliers in the distribution of revenues for the companies. Because there are no values that are out of the limit $Q_3+\left(1.5\times IQR\right)=492.1$ and $Q_1-\left(1.5\times IQR\right)=-56.7$.

Stemplot:

Software procedure:

Step-by-step software procedure to draw stemplot using MINITAB software is as follows:

• Select Graph>Stem and leaf.
• Select the column of GLOBE500 in Graph variables.
• Select OK.

Output of stemplot is shown below:

Justification:

The stemplot does not show any outliers which indicates that the $1.5\times IQR$ rule’s suggestion that there are no outliers is agreed.

c.

To determine

To explain: The common feature that the 30 companies in the Table 2.3 share.

To find: The proportion of the total of the global 500 revenues accounted by the 30 companies.

Answer to Problem 2.52E

The common feature that the 30 companies share is that they are the high outliers and have high revenues out of global 500 companies.

The proportion of the total global 500 revenues by the 30 companies is 0.06.

Explanation of Solution

Justification:

From the global 500 companies, 30 companies in Table2.3 have high revenues than the remaining companies, and these 30 companies are the high outliers that has a common feature of sharing.

The proportion of the total global 500 companies accounted by the 30 companies is:

$\begin{array}{c}\text{Proportion}=\frac{30}{500}\\ =0.06\end{array}$

Thus, the proportion of the total global 500 revenues by the 30 companies is 0.06.