Chapter 2.2, Problem 21E

Largest University Endowments. University endowments are financial assets that are donated by supporters to be used to provide income to universities. There is a large discrepancy in the size of university endowments. The following table provides a listing of many of the universities that have the largest endowments as reported by the National Association of College and University Business Officers in 2017.

Summarize the data by constructing the following:

1. a. A frequency distribution (classes 0–1.9, 2.0–3.9, 4.0–5.9, 6.0–7.9, and so on).
2. b. A relative frequency distribution.
3. c. A cumulative frequency distribution.
4. d. A cumulative relative frequency distribution.
5. e. What do these distributions tell you about the endowments of universities?
6. f. Show a histogram. Comment on the shape of the distribution.
7. g. What is the largest university endowment and which university holds it?

To determine

Construct a frequency distribution for the data.

Answer to Problem 21E

The frequency distribution is as follows:

Endowment Amount ($ Billion)	Frequency
0-1.9	10
2-3.9	24
4-5.9	7
6-7.9	5
8-9.9	3
10-11.9	4
12-13.9	1
14-15.9	1
16-17.9	0
18-19.9	0
20-21.9	0
22-23.9	1
24-25.9	1
26-27.9	2
28-29.9	0
30-31.9	0
32-33.9	0
34-35.9	0
36-37.9	1
Total	60

Explanation of Solution

Calculation:

The data represents the Endowment amount of Universities in billions of dollars and the frequency distribution is constructed using the class 0-1.9, 2-3.9, 4-5.9 and so on.

Frequency:

The frequencies are calculated using the tally mark and the range of the data is from 0 to 37.9.

• Based on the given information, the class intervals are 0-1.9, 2-3.9, 4-5.9, …, 36-37.9.
• Make a tally mark for each value in the corresponding revenue class and continue for all values in the data.
• The number of tally marks in each class represents the frequency, f of that class.

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

Endowment Amount ($ Billion)	Tally	Frequency
0-1.9	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}$	10
2-3.9	$\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\bcancel{\left|\right|\left|\right|}\left|\right|\left|\right|$	24
4-5.9	$\bcancel{\left|\right|\left|\right|}\left|\right|$	7
6-7.9	$\bcancel{\left|\right|\left|\right|}$	5
8-9.9	$\left|\right||$	3
10-11.9	$\left|\right|\left|\right|$	4
12-13.9	$|$	1
14-15.9	$|$	1
16-17.9	-	0
18-19.9	-	0
20-21.9	-	0
22-23.9	$|$	1
24-25.9	$|$	1
26-27.9	$\left|\right|$	2
28-29.9	-	0
30-31.9	-	0
32-33.9	-	0
34-35.9	-	0
36-37.9	$|$	1
Total	60

To determine

Construct a relative frequency distribution for the data.

Answer to Problem 21E

The relative frequency distribution is as follows:

Endowment Amount ($ Billion)	Frequency	Relative frequency
0-1.9	10	0.17
2-3.9	24	0.40
4-5.9	7	0.12
6-7.9	5	0.08
8-9.9	3	0.05
10-11.9	4	0.07
12-13.9	1	0.02
14-15.9	1	0.02
16-17.9	0	0.00
18-19.9	0	0.00
20-21.9	0	0.00
22-23.9	1	0.02
24-25.9	1	0.02
26-27.9	2	0.03
28-29.9	0	0.00
30-31.9	0	0.00
32-33.9	0	0.00
34-35.9	0	0.00
36-37.9	1	0.02
Total	60	1

Explanation of Solution

Calculation:

Relative frequency:

The general formula for the relative frequency is given below:

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

For the class (0-1.9), substitute frequency as “10” and total frequency as “60”.

$\begin{array}{c}\text{Relative}\text{Frequency}\text{for class(0-1}\text{.9)}=\frac{10}{60}\\ =0.17\end{array}$

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

Endowment Amount ($ Billion)	Frequency	Relative frequency
0-1.9	10	0.17
2-3.9	24	0.40
4-5.9	7	0.12
6-7.9	5	0.08
8-9.9	3	0.05
10-11.9	4	0.07
12-13.9	1	0.02
14-15.9	1	0.02
16-17.9	0	0.00
18-19.9	0	0.00
20-21.9	0	0.00
22-23.9	1	0.02
24-25.9	1	0.02
26-27.9	2	0.03
28-29.9	0	0.00
30-31.9	0	0.00
32-33.9	0	0.00
34-35.9	0	0.00
36-37.9	1	0.02
Total	60	1

To determine

Construct a cumulative frequency distribution for the data.

Answer to Problem 21E

The cumulative frequency distribution is as follows:

Endowment Amount ($ Billion)	Frequency	Cumulative frequency
0-1.9	10	10
2-3.9	24	34
4-5.9	7	41
6-7.9	5	46
8-9.9	3	49
10-11.9	4	53
12-13.9	1	54
14-15.9	1	55
16-17.9	0	55
18-19.9	0	55
20-21.9	0	55
22-23.9	1	56
24-25.9	1	57
26-27.9	2	59
28-29.9	0	59
30-31.9	0	59
32-33.9	0	59
34-35.9	0	59
36-37.9	1	60

Explanation of Solution

Calculation:

Cumulative frequency:

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

Thus, the cumulative frequencies for the endowment classes are obtained below:

Endowment Amount ($ Billion)	Frequency	Cumulative frequency
0-1.9	10	10
2-3.9	24	$10+24=34$
4-5.9	7	$34+7=41$
6-7.9	5	$41+5=46$
8-9.9	3	$46+3=49$
10-11.9	4	$49+4=53$
12-13.9	1	$53+1=54$
14-15.9	1	$54+1=55$
16-17.9	0	$55+0=55$
18-19.9	0	$55+0=55$
20-21.9	0	$55+0=55$
22-23.9	1	$55+1=56$
24-25.9	1	$56+1=57$
26-27.9	2	$57+2=59$
28-29.9	0	$59+0=59$
30-31.9	0	$59+0=59$
32-33.9	0	$59+0=59$
34-35.9	0	$59+0=59$
36-37.9	1	$59+1=60$

To determine

Construct a cumulative relative frequency distribution for the data.

Answer to Problem 21E

The cumulative relative frequency distribution is given below:

Endowment Amount ($ Billion)	Frequency	Cumulative Relative Frequency
0-1.9	10	0.17
2-3.9	24	0.57
4-5.9	7	0.69
6-7.9	5	0.77
8-9.9	3	0.82
10-11.9	4	0.89
12-13.9	1	0.90
14-15.9	1	0.92
16-17.9	0	0.92
18-19.9	0	0.92
20-21.9	0	0.92
22-23.9	1	0.94
24-25.9	1	0.95
26-27.9	2	0.99
28-29.9	0	0.99
30-31.9	0	0.99
32-33.9	0	0.99
34-35.9	0	0.99
36-37.9	1	1.00

Explanation of Solution

Calculation:

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

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

Endowment Amount ($ Billion)	Frequency	Relative frequency
0-1.9	10	0.17
2-3.9	24	0.40
4-5.9	7	0.12
6-7.9	5	0.08
8-9.9	3	0.05
10-11.9	4	0.07
12-13.9	1	0.02
14-15.9	1	0.02
16-17.9	0	0.00
18-19.9	0	0.00
20-21.9	0	0.00
22-23.9	1	0.02
24-25.9	1	0.02
26-27.9	2	0.03
28-29.9	0	0.00
30-31.9	0	0.00
32-33.9	0	0.00
34-35.9	0	0.00
36-37.9	1	0.02
Total	60	1

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

Endowment Amount ($ Billion)	Relative frequency	Cumulative Relative Frequency
0-1.9	0.17	0.17
2-3.9	0.40	$0.17+0.40=0.57$
4-5.9	0.12	$0.57+0.12=0.69$
6-7.9	0.08	$0.69+0.08=0.77$
8-9.9	0.05	$0.77+0.05=0.82$
10-11.9	0.07	$0.82+0.07=0.89$
12-13.9	0.02	$0.89+0.02=0.91$
14-15.9	0.02	$0.91+0.02=0.93$
16-17.9	0.00	$0.93+0=0.93$
18-19.9	0.00	$0.93+0=0.93$
20-21.9	0.00	$0.93+0=0.93$
22-23.9	0.02	$0.93+0.02=0.95$
24-25.9	0.02	$0.93+0.02=0.97$
26-27.9	0.03	$0.97+0.03=1$
28-29.9	0.00	$1+0=1$
30-31.9	0.00	$1+0=1$
32-33.9	0.00	$1+0=1$
34-35.9	0.00	$1+0=1$
36-37.9	0.02	1.00

To determine

Explain about the endowment of universities using the distributions.

Explanation of Solution

From the given data set and obtained distributions , it is observed that the frequency for the endowment of universities in the range of 0 billion dollars to less than $16 billion dollars is obtained by majority of 55 Universities over 60 Universities.

Further, the endowment of universities greater $16 billion is obtained by only 5 Universities.

Moreover, 92% of universities have endowment amount under 16 billion dollars. Only 8% of universities have endowment amount over 16 billion dollars.

To determine

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

Answer to Problem 21E

• Output using Excel is given below:

The histogram is skewed to the right.

Explanation of Solution

Calculation:

Step-by-step procedure to draw the frequency histogram chart using Excel is as follows:

• In Excel sheet, enter Endowment Amount ($ Billion) in one column and Frequency in another column.
• Select the data and then choose Insert > Insert Column Bar Charts.
• Select Clustered Column Under More Column Charts.
• Double click the bars
• In Format Data Series, enter 0 in Gap Width under Series Options.

Skewness:

The data is said to be skewed if there is lack of symmetry and 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, more over it attains its peak rapidly than its horizontal axis and then it is a right skewed distribution. It is also called positively skewed.

The distribution of Endowment in the histogram has elongated tail toward right side. There are five universities in the range 20 billion dollars to 37.9 billion dollars.

Therefore, the distribution of the histogram with Endowment is skewed right.

To determine

Find the largest university endowment and find the university that holds the largest endowment.

Answer to Problem 21E

The largest university endowment is $36 billion.

The University H holds the largest endowment.

Explanation of Solution

From the data set of 60 universities, it is observed that the largest university endowment is $36 billion and the university is University H.

Moreover, the endowment of remaining universities has less than 28 billion dollars and most of the universities have less than 16 billion dollars and is approximately 92 percent.