Chapter 2.2, Problem 21E

a.

To determine

Construct a frequency distribution for the data.

Answer to Problem 21E

The frequency distribution is obtained as shown below:

Revenue	Frequency
0-49	6
50-99	29
100-149	11
150-199	0
200-249	1
250-299	1
300-349	0
350-399	0
400-449	2
Total	50

Explanation of Solution

Calculation:

The data represent the list of 50 Country A’s largest corporations with annual revenue expressed in billions of dollars and the frequency distribution is constructed using the classes 0-49, 50-99,100-149, and so on.

Frequency:

The frequencies are calculated using the tally mark and the range of the data is from 0 to 449.

• Based on the given information, the class intervals are 0-49, 50-99, 100-149, …,400-449.
• Make a tally mark for each value in the corresponding revenue class and continue for all the values in the data.
• The number of tally marks in each class represents the frequency, f, of that class.

Similarly, the frequency of the remaining classes for the revenue is given below:

Revenue	Tally	Frequency
0-49	$\bcancel{\left|\right|\left|\right|}|$	6
50-99	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right|\left|\right|$	29
100-149	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}|$	11
150-199		0
200-249	$|$	1
250-299	$|$	1
300-349		0
350-399		0
400-449	$|$|	2
Total		50

b.

To determine

Construct a relative frequency distribution for the data.

Answer to Problem 21E

The relative frequency distribution is tabulated below:

Revenue	Frequency	Relative frequency
0-49	6	0.12
50-99	29	0.58
100-149	11	0.22
150-199	0	0.00
200-249	1	0.02
250-299	1	0.02
300-349	0	0.00
350-399	0	0.00
400-449	2	0.04
Total	50	1.00

Explanation of Solution

Calculation:

Relative frequency:

The general formula for the relative frequency is as follows:

$\text{Relative}\text{Frequency = }\frac{\text{Frequency}}{\text{Total frequency}}$

For the revenue class (0-49), substitute frequency as “6” and total frequency as “50”.

$\begin{array}{c}\text{Relative}\text{Frequency}\text{for revenue class(0-49)}=\frac{6}{50}\\ =0.12\end{array}$

Similarly, the relative frequencies for the remaining revenue classes are obtained below:

Revenue	Frequency	Relative frequency
0-49	6	0.12
50-99	29	0.58
100-149	11	0.22
150-199	0	0.00
200-249	1	0.02
250-299	1	0.02
300-349	0	0.00
350-399	0	0.00
400-449	2	0.04
Total	50	1.00

c.

To determine

Construct a cumulative frequency distribution for the data.

Answer to Problem 21E

The cumulative frequency distribution is as follows:

Revenue	Frequency	Cumulative frequency
0-49	6	6
50-99	29	35
100-149	11	46
150-199	0	46
200-249	1	47
250-299	1	48
300-349	0	48
350-399	0	48
400-449	2	50

Explanation of Solution

Calculation:

Cumulative frequency:

Cumulative frequency of a particular class is the sum of all frequencies up to a class. The last class’s cumulative frequency is equal to the sample size $n$.

Thus, the cumulative frequency for each class is tabulated below:

The relative frequencies for the five types of classes are obtained below:

Revenue	Frequency	Cumulative frequency
0-49	6	6
50-99	29	$29+6=35$
100-149	11	$35+1=46$
150-199	0	$46+0=46$
200-249	1	$46+1=47$
250-299	1	$47+1=48$
300-349	0	$48+0=48$
350-399	0	$48+0=48$
400-449	2	$48+2=50$

d.

To determine

Construct a cumulative relative frequency distribution for the data.

Answer to Problem 21E

The cumulative relative frequency distribution is given below:

Revenue	Cumulative relative frequency
0-49	0.12
50-99	0.70
100-149	0.92
150-199	0.92
200-249	0.94
250-299	0.96
300-349	0.96
350-399	0.96
400-449	1.00

Explanation of Solution

Calculation:

Cumulative relative frequency of a particular class is the sum of all relative frequencies up to a class. The last class’s cumulative relative frequency is equal to the approximate value 1.00.

The relative frequencies for the revenue classes from part (b) is given below:

Revenue	Frequency	Relative frequency
0-49	6	0.12
50-99	29	0.58
100-149	11	0.22
150-199	0	0.00
200-249	1	0.02
250-299	1	0.02
300-349	0	0.00
350-399	0	0.00
400-449	2	0.04
Total	50	1.00

Thus, the cumulative relative frequencies for the revenue classes are obtained below:

Revenue	Relative frequency	Cumulative relative frequency
0-49	0.12	0.12
50-99	0.58	$0.12+0.58=0.70$
100-149	0.22	$0.70+0.22=0.92$
150-199	0.00	$0.92+0.00=0.92$
200-249	0.02	$0.92+0.02=0.94$
250-299	0.02	$0.94+0.02=0.96$
300-349	0.00	$0.96+0.00=0.96$
350-399	0.00	$0.96+0.00=0.96$
400-449	0.04	$0.96+0.04=1.00$

e.

To determine

Explain about the annual revenue of the largest corporations in Country A using the distributions.

Explanation of Solution

From the given data set and obtained distributions, it is observed that the frequency for the revenues in the range of 50 billion dollars to 149 billion dollars is obtained by the majority of 40 corporations over 50 largest corporations.

Further, the revenue over $200 billion is obtained by only 4 corporations and the revenue over 400 billion dollars is obtained by only 2 corporations.

Moreover, 70 percent of the corporations have revenues under 100 billion dollars. Only 30 percent of corporations have revenues over 100 billion dollars.

f.

To determine

Construct the histogram and comment on the shape of the distribution.

Answer to Problem 21E

• The histogram of the data is given below:

The histogram is skewed to the right.

Explanation of Solution

Calculation:

Software procedure:

Step-by-step procedure to draw the frequency histogram chart using MINITAB software:

• Choose Graph > Histogram.
• Choose Simple, and then click OK.
• In Graph variables, enter the Revenue column of data.
• In scale on y-axis, make click on frequency.
• Click on ok.

Skewness:

The data are said to be skewed if there is a lack of symmetry and the values fall on one side, that is, either left or right of the distribution.

Right skewed:

If the tail on the distribution is elongated toward the right and it attains its peak rapidly than its horizontal axis, then it is a right-skewed distribution. It is also called positively skewed.

The distribution of revenue in the histogram has elongated tail toward the right side. There are four corporations in the range of 200 billion dollars to 449 billion dollars.

Therefore, the distribution of the histogram with revenue is skewed to the right.

g.

To determine

Find the largest corporation in Country A and find its annual revenue.

Answer to Problem 21E

Exxon-Mobil is the largest corporation in Country A and its annual revenue is 443 billion dollars.

Explanation of Solution

From the data set of 50 largest corporations, it is observed that the largest corporation in Country A is Exxon-Mobil and its annual revenue is 443 billion dollars.

The second largest corporation in Country A is Walmart and its annual revenue is 406 billion dollars.

Moreover, the possibility of corporation that has annual revenues less than 150 billion dollars is approximately 92 percent, and the remaining other corporations have annual revenues less than 300 billion dollars.