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Chapter 3: Problem 69

The days to maturity for a sample of five money market funds are shown here. The dollar amounts invested in the funds are provided. Use the weighted mean to determine the mean number of days to maturity for dollars invested in these five money market funds. $$\begin{array}{cc} \text { Days to } & \text { Dollar Value } \\ \text { Maturity } & \text { (\$millions) } \\ 20 & 20 \\ 12 & 30 \\ 7 & 10 \\ 5 & 15 \\ 6 & 10 \end{array}$$

Short Answer

Expert verified
The weighted mean number of days to maturity is approximately 11.35 days.

Step by step solution

01

Understand the Weighted Mean Formula

The weighted mean is a type of average where each quantity to be averaged is assigned a weight. The formula for the weighted mean is \( \bar{x} = \frac{\sum (x_i \cdot w_i)}{\sum w_i} \), where \( x_i \) is each data point, and \( w_i \) is the corresponding weight.
02

Identify Data Points and Weights

Identify each data point \( x_i \) and its corresponding weight \( w_i \). Here, the days to maturity are the data points \( x_i \) and the dollar amounts in millions are the weights \( w_i \).
03

Calculate Weighted Products

For each pair \((x_i, w_i)\), multiply the number of days \( x_i \) by the corresponding dollars \( w_i \). The products are:\( 20 \times 20 = 400 \), \( 12 \times 30 = 360 \), \( 7 \times 10 = 70 \), \( 5 \times 15 = 75 \), \( 6 \times 10 = 60 \).
04

Sum the Weighted Products

Add all the products calculated in Step 3: \( 400 + 360 + 70 + 75 + 60 = 965 \).
05

Sum the Weights

Add all the weights (dollar amounts in millions): \( 20 + 30 + 10 + 15 + 10 = 85 \).
06

Calculate the Weighted Mean

Now use the formula \( \bar{x} = \frac{\sum (x_i \cdot w_i)}{\sum w_i} \) with the values from Step 4 and Step 5: \( \bar{x} = \frac{965}{85} \approx 11.35 \).
07

Interpret the Result

The weighted mean of the days to maturity for invested dollars is approximately 11.35 days.

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Key Concepts

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

Days to Maturity
Understanding the concept of "Days to Maturity" is crucial when dealing with investments like money market funds. This term refers to the number of days left until the investment reaches its full value and can be cashed in. It is an important metric because it signifies the duration until those funds are liquid and available for use.

For instance, if a money market fund has a maturity of 20 days, that means you have to wait 20 days before you can access the full return on that investment. This metric is vital for making strategic financial decisions, especially when aligning investments with future financial needs.

In our exercise, each money market fund has a different number of days to maturity, i.e., 20, 12, 7, 5, and 6, which means they each reach maturity at different timings. When calculating the mean days to maturity, it's important to consider how each of these times contributes to the overall average, especially when amounts invested are unequal.
Data Points
Data points in any statistical calculation are the individual values or measurements gathered from your subject of study. Here, the data points are the various "Days to Maturity" for each of the five money market funds, which are recorded as 20, 12, 7, 5, and 6 days.

These data points alone do not give us complete insight, but when paired with their corresponding dollar values, they help us determine the weighted influence of each point.

The weight of each data point in our exercise is determined by the amount of dollars invested in the respective fund. This concept underpins calculating a weighted mean, where not every day to maturity is of equal significance without considering the investment it represents.
  • The more money tied to a particular maturity date, the greater its impact on the overall weighted mean.
  • Having a list of data points helps us organize our calculation and apply the correct statistical methods to derive meaningful insights from what might otherwise seem like a jumble of numbers.
Statistical Calculation
Statistical calculation begins with understanding the process of achieving a weighted mean in this context. The weighted mean is useful when different data points contribute unequally towards a general average.

In the exercise, we calculated the weighted mean of days to maturity by multiplying each day by its corresponding investment value (in millions) to get a 'weighted product.' Here are the steps:
  • Multiply the days each fund takes to mature (data points) by the dollar value invested (weights).
  • Add all these "weighted" results to get a total sum.
  • Finally, divide that sum by the total sum of the weights (dollar amounts).

This detailed process allows us to compute the weighted mean accurately by focusing on not just average maturity but how each investment affects this average.

The formulas and computation steps involve little more than basic multiplication and division but yield robust insights. By carefully understanding each part, one can grasp how statistical calculations like these make decision-making more data-driven.

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Most popular questions from this chapter

The U.S. Census Bureau provides statistics on family life in the United States, including the age at the time of first marriage, current marital status, and size of household (U.S. Census Bureau website, March 20,2006 ). The following data show the age at the time of first marriage for a sample of men and a sample of women. $$\begin{array}{lcccccccc} \text { Men } & 26 & 23 & 28 & 25 & 27 & 30 & 26 & 35 & 28 \\ & 21 & 24 & 27 & 29 & 30 & 27 & 32 & 27 & 25 \\ \text { Women } & 20 & 28 & 23 & 30 & 24 & 29 & 26 & 25 & \\ & 22 & 22 & 25 & 23 & 27 & 26 & 19 & & \end{array}$$ a. Determine the median age at the time of first marriage for men and women. b. Compute the first and third quartiles for both men and women. c. Twenty-five years ago the median age at the time of first marriage was 25 for men and 22 for women. What insight does this information provide about the decision of when to marry among young people today?

Dividend yield is the annual dividend per share a company pays divided by the current market price per share expressed as a percentage. A sample of 10 large companies provided the following dividend yield data (The Wall Street Journal, January 16,2004 ). $$\begin{array}{lclc} \text { Company } & \text { Yield % } & \text { Company } & \text { Yield % } \\\ \text { Altria Group } & 5.0 & \text { General Motors } & 3.7 \\ \text { American Express } & 0.8 & \text { JPMorgan Chase } & 3.5 \\ \text { Caterpillar } & 1.8 & \text { McDonald's } & 1.6 \\ \text { Eastman Kodak } & 1.9 & \text { United Technology } & 1.5 \\ \text { ExxonMobil } & 2.5 & \text { Wal-Mart Stores } & 0.7 \end{array}$$ a. What are the mean and median dividend yields? b. What are the variance and standard deviation? c. Which company provides the highest dividend yield? d. What is the \(z\) -score for McDonald's? Interpret this z-score. e. What is the \(z\) -score for General Motors? Interpret this z-score. f. Based on z-scores, do the data contain any outliers?

Small business owners often look to payroll service companies to handle their employee payroll. Reasons are that small business owners face complicated tax regulations, and penalties for employment tax errors are costly. According to the Internal Revenue Service, \(26 \%\) of all small business employment tax returns contained errors that resulted in a tax penalty to the owner (The Wall Street Journal, January 30,2006 ). The tax penalty for a sample of 20 small business owners follows: $$\begin{array}{rrrrrrrrr} 820 & 270 & 450 & 1010 & 890 & 700 & 1350 & 350 & 300 & 1200 \\ 390 & 730 & 2040 & 230 & 640 & 350 & 420 & 270 & 370 & 620 \end{array}$$ a. What is the mean tax penalty for improperly filed employment tax returns? b. What is the standard deviation? c. Is the highest penalty, \(\$ 2040,\) an outlier? d. What are some of the advantages of a small business owner hiring a payroll service company to handle employee payroll services, including the employment tax returns?

How do grocery costs compare across the country? Using a market basket of 10 items including meat, milk, bread, eggs, coffee, potatoes, cereal, and orange juice, Where to Retire magazine calculated the cost of the market basket in six cities and in six retirement areas across the country (Where to Retire, November/December 2003 ). The data with market basket cost to the nearest dollar are as follows: $$\begin{array}{lclr} \text { City } & \text { cost } & \text { Retirement Area } & \text { cost } \\\ \text { Buffalo, NY } & \$ 33 & \text { Biloxi-Gulfport, MS } & \$ 29 \\ \text { Des Moines, IA } & 27 & \text { Asheville, NC } & 32 \\ \text { Hartford, CT } & 32 & \text { Flagstaff, AZ } & 32 \\ \text { Los Angeles, CA } & 38 & \text { Hilton Head, SC } & 34 \\ \text { Miami, FL } & 36 & \text { Fort Myers, FL } & 34 \\ \text { Pittsburgh, PA } & 32 & \text { Santa Fe, NM } & 31 \end{array}$$ a. Compute the mean, variance, and standard deviation for the sample of cities and the sample of retirement areas. b. What observations can be made based on the two samples?

A home theater in a box is the easiest and cheapest way to provide surround sound for a home entertainment center. A sample of prices is shown here (Consumer Reports Buying Guide, 2004 ). The prices are for models with a DVD player and for models without a DVD player. $$\begin{array}{lclr} \text { Models with DVD Player } & \text { Price } & \text { Models without DVD Player } & \text { Price } \\ \text { Sony HT-1800DP } & \$ 450 & \text { Pioneer HTP-230 } & \$ 300 \\ \text { Pioneer HTD-330DV } & 300 & \text { Sony HT-DDW750 } & 300 \\ \text { Sony HT-C800DP } & 400 & \text { Kenwood HTB-306 } & 360 \\ \text { Panasonic SC-HT900 } & 500 & \text { RCA RT-2600 } & 290 \\ \text { Panasonic SC-MTI } & 400 & \text { Kenwood HTB-206 } & 300 \end{array}$$ a. Compute the mean price for models with a DVD player and the mean price for models without a DVD player. What is the additional price paid to have a DVD player included in a home theater unit? b. Compute the range, variance, and standard deviation for the two samples. What does this information tell you about the prices for models with and without a DVD player?
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