Chapter 3.1, Problem 9E

a.

To determine

Obtain the mean endowment for these universities.

Answer to Problem 9E

The mean endowment for these universities is 14.8.

Explanation of Solution

Calculation:

The data represent the 10 wealthiest universities for a sample of 435 colleges. The endowment is measured in billion.

Denote the endowment of 10 wealthiest universities as $x_1,x_2,\mathrm{...},x_{10}$.

The mean waiting time for 10 wealthiest universities is obtained from the calculation is given below:

$\begin{array}{c}\overline{x}=\frac{{\displaystyle \sum x_i}}{n}\\ =\frac{\left(7.2+36.6+10.1+7.6+7.2+16.4+17.2+16.1+6.7+22.9\right)}{10}\\ =\frac{148}{10}\\ =14.8\end{array}$

Thus, the mean waiting time for 10 wealthiest universities is 14.8.

b.

To determine

Obtain the median endowment.

Answer to Problem 9E

The median endowment is 13.1.

Explanation of Solution

Calculation:

The median endowment is obtained as follows:

Software Procedure:

Step-by-step procedure to obtain the descriptive statistics using EXCEL:

• In an EXCEL sheet enter 10 salaries and label it as Endowment.
• Go to Data > Data Analysis (in case it is not default, take the Analysis ToolPak from Excel Add Ins) > Descriptive statistics.
• Enter Input Range as $A$2:$A$11, select Columns in Grouped By, tick on Summary statistics.
• Click on OK.

Output using EXCEL is given as follows:

From the EXCEL output, the endowment is 13.1.

c.

To determine

Find the mode endowment.

Explanation of Solution

From Part (b) it is known that, the mode endowment is 7.2.

That is, the most repeated endowment amount for the 10 wealthiest universities is 7.2.

d.

To determine

Find the first and third quartiles.

Answer to Problem 9E

The first and third quartiles are $\underset{\_}{Q_1=7.2}$, and $\underset{\_}{Q_3=18.625}$, respectively.

Explanation of Solution

Calculation:

The first and third quartiles are obtained using EXCEL functions.

First quartile:

The EXCEL function to compute first quartile is $Q_1=\text{QUARTILE}\text{.EXC}\left(\text{Input range,1}\right)$.

Software Procedure:

Step-by-step procedure to obtain the first quartile using EXCEL:

• In an EXCEL sheet, enter the observations of first sample and label it as Endowment.
• In a cell enter the formula QUARTILE.EXC ($F$2:$F$11,1).
• Click on OK.

Output using EXCEL is given as follows:

From the EXCEL output, the first quartile is 7.2.

Third quartile:

The EXCEL function to compute third quartile is $Q_3=\text{QUARTILE}\text{.EXC}\left(\text{Input range,3}\right)$.

Software Procedure:

Step-by-step procedure to obtain the third quartile using EXCEL:

• In an EXCEL sheet, enter the observations of first sample and label it as Endowment.
• In a cell enter the formula QUARTILE.EXC ($F$2:$F$11,3).
• Click on OK.

Output using EXCEL is given as follows:

From the EXCEL output, the third quartile is 18.625.

Thus, the first and third quartiles in the previous year are $\underset{\_}{Q_1=7.2}$ and $\underset{\_}{Q_3=18.625}$, respectively.

e.

To determine

Find the total endowment at 10 universities.

Calculate the percentage of the total $413 billion in endowments that are held by the 10 universities.

Answer to Problem 9E

Total endowment at 10 universities is $148 billion.

The percentage of the total $413 billion in endowments that are held by the 10 universities is 35.8%.

Explanation of Solution

Calculation:

The total endowment at 10 universities is given below:

$\begin{array}{c}\text{Total}={\displaystyle \sum x_i}\\ =7.2+36.6+10.1+7.6+7.2+16.4+17.2+16.1+6.7+22.9\\ =148\end{array}$

Thus, the total endowment at 10 universities is $148 billion.

The required probability is as follows:

$\begin{array}{c}\text{Percentage}=\frac{148\times 100}{413}\\ =35.8\%\end{array}$

Thus, the percentage of total $413 billion in endowments is held by the 10 universities is 35.8%.

f.

To determine

Compute the estimate of the dollar amount of the decline in the total endowments held by the 10 universities.

Identify the steps that one would expect university administrators to consider.

Explanation of Solution

Calculation:

It is given that over a recent five-month period, a downturn in the economy has caused endowments to decline to 23%.

The amount of decline in endowment is as follows:

$0.23\times 148=34$

Thus, the estimate of the dollar amount of the decline in the total endowments held by the 10 universities is $34 billion.

The steps that one would expect university administrators to consider is as follows:

• Hiring faculty and staff should be controlled.
• Postponing construction projects.
• Increase tuition.
• Growing enrollments.