91Ó°ÊÓ

Endowment income is a critical part of the annual budgets at colleges and universities. A study by the National Association of College and University Business Officers reported that the 435 colleges and universities surveyed held a total of \(\$ 413\) billion in endowments. The 10 wealthiest universities are shown in the following table (The Wall Street Journal, January 27,2009 ). Amounts are in billions of dollars. $$\begin{array}{lccc} \text { University } & \text { Endowment (\$billion) } & \text { University } & \text { Endowment (Sbillion) } \\ \text { Columbia } & 7.2 & \text { Princeton } & 16.4 \\ \text { Harvard } & 36.6 & \text { Stanford } & 17.2 \\ \text { M.I.T. } & 10.1 & \text { Texas } & 16.1 \\ \text { Michigan } & 7.6 & \text { Texas A\&M } & 6.7 \\ \text { Northwestern } & 7.2 & \text { Yale } & 22.9 \end{array}$$ a. What is the mean endowment for these universities? b. What is the median endowment? c. What is the mode endowment? d. Compute the first and third quartiles. e. What is the total endowment at these 10 universities? These universities represent \(2.3 \%\) of the 435 colleges and universities surveyed. What percentage of the total \(\$ 413\) billion in endowments is held by these 10 universities? f. The Wall Street Journal reported that over a recent five-month period, a downturn in the economy has caused endowments to decline \(23 \%\). What is the estimate of the dollar amount of the decline in the total endowments held by these 10 universities? Given this situation, what are some of the steps you would expect university administrators to be considering?

Step by step solution

01

Calculate the Mean Endowment

The mean (average) is calculated by summing all endowments and dividing by their count. Sum up: \[7.2 + 36.6 + 10.1 + 7.6 + 7.2 + 16.4 + 17.2 + 16.1 + 6.7 + 22.9 = 148\] Divide the total by 10 (number of universities): \[\text{Mean} = \frac{148}{10} = 14.8\] billion dollars.

02

Determine the Median Endowment

First, sort the endowments in ascending order: \[6.7, 7.2, 7.2, 7.6, 10.1, 16.1, 16.4, 17.2, 22.9, 36.6\] Since there are 10 numbers, the median is the average of the 5th and 6th values: \[\text{Median} = \frac{10.1 + 16.1}{2} = 13.1\] billion dollars.

03

Determine the Mode Endowment

The mode is the number that appears most frequently. Here, 7.2 billion appears twice, and no other endowments repeat. Hence, the mode is 7.2 billion dollars.

04

Calculate the First and Third Quartiles

The first quartile (Q1) is the median of the first half, and the third quartile (Q3) is the median of the second half of the sorted data. For Q1, median of \[6.7, 7.2, 7.2, 7.6, 10.1\] is 7.2 (3rd number).For Q3, median of \[16.1, 16.4, 17.2, 22.9, 36.6\] is 17.2 (3rd number).Thus, Q1 = 7.2 billion and Q3 = 17.2 billion.

05

Calculate Total Endowment and Percentage of Total

Sum the endowments for these universities: \[7.2 + 36.6 + 10.1 + 7.6 + 7.2 + 16.4 + 17.2 + 16.1 + 6.7 + 22.9 = 148\] billion. To find the percentage of the total \(\$413\) billion: \[\text{Percentage} = \left(\frac{148}{413}\right) \times 100 \approx 35.84\%\] of the total endowments.

06

Estimate the Decline in Endowments due to Economic Downturn

The decline is given as \(23\%\). To find the dollar amount, calculate 23% of the total endowment for these universities: \[\text{Decline} = 0.23 \times 148 = 34.04\] billion dollars. University administrators might consider cost-cutting, fundraising, or leveraging other revenue sources to manage this decline.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Endowment

When we look at the term *mean*, we are referring to the average value. In the context of endowments, which are significant funds maintained by universities, the mean endowment provides a sense of the typical fund amount universities hold. To calculate the mean, sum up all the endowment values provided and divide by the total number of universities.
The mathematical formula for mean is:\[\text{Mean} = \frac{\text{Sum of all endowments}}{\text{Number of universities}}\]

• For the given data: \[7.2, 36.6, 10.1, 7.6, 7.2, 16.4, 17.2, 16.1, 6.7, 22.9\] billion dollars
• The sum of the endowments is 148 billion dollars.
• With 10 universities in total, the mean endowment is calculated as \(\frac{148}{10} = 14.8\) billion dollars.

Knowing the mean helps universities benchmark their endowment fund sizes against others, aiding in financial planning and educational investments.

Median Endowment

The *median* is often considered a better measure than the mean when analyzing skewed data because it provides the mid-point value. It is especially significant in finance as it indicates the typical value without the distortion of extremely high or low values. To find the median, arrange the endowments in increasing order and locate the middle number.
With an even number of observations, you take the average of the two middle numbers.

• Sorted endowments: \[6.7, 7.2, 7.2, 7.6, 10.1, 16.1, 16.4, 17.2, 22.9, 36.6\] billion dollars
• The median therefore is the average of the 5th and 6th values: \(\frac{10.1 + 16.1}{2} = 13.1\) billion dollars.

Understanding the median gives insight into how evenly endowments are distributed among various institutions, which is crucial for stakeholders assessing economic parity and decision-making.

Quartiles

Quartiles divide a ranked data set into four equal parts, making it easy to understand the spread and the central tendency of the data. The first quartile (\(Q_1\)) represents the 25th percentile, meaning 25% of the data falls below this value, whereas the third quartile (\(Q_3\)) shows the 75th percentile.In the context of endowments:

• The first quartile (\(Q_1\)) is the median of the first half of the data set. For the sorted endowments: \[6.7, 7.2, 7.2, 7.6, 10.1\], \(Q_1\) is 7.2 billion dollars.
• The third quartile (\(Q_3\)) is the median of the second half of the dataset \[16.1, 16.4, 17.2, 22.9, 36.6\], and \(Q_3\) is 17.2 billion dollars.

This distribution knowledge helps universities understand variability in endowments. It informs strategic decisions, especially in resource allocation and identifying outlier institutions.