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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 calculated using the formula \( \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} \), where \( w_i \) is the weight (or value) and \( x_i \) is the corresponding value for that weight.
02

Identify Values and Weights

Here, the 'Days to Maturity' are the \( x_i \) values and the 'Dollar Values' in millions are the \( w_i \) values. \( x_1 = 20, x_2 = 12, x_3 = 7, x_4 = 5, x_5 = 6 \) with corresponding weights \( w_1 = 20, w_2 = 30, w_3 = 10, w_4 = 15, w_5 = 10 \).
03

Calculate Weighted Sum

Multiply each day's maturity by its respective dollar value: - \( 20 \times 20 = 400 \)- \( 12 \times 30 = 360 \)- \( 7 \times 10 = 70 \)- \( 5 \times 15 = 75 \)- \( 6 \times 10 = 60 \)Sum these results: \( 400 + 360 + 70 + 75 + 60 = 965 \).
04

Calculate Sum of Weights

Sum the dollar values to find the total weight: \( 20 + 30 + 10 + 15 + 10 = 85 \).
05

Compute the Weighted Mean

Use the weighted mean formula: \( \bar{x}_w = \frac{965}{85} \). Divide the weighted sum by the total weights: \( 11.35 \).

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Key Concepts

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

Statistics
Statistics is a powerful branch of mathematics that deals with collecting, analyzing, and interpreting data. One essential concept in statistics is the 'weighted mean'.
The weighted mean is similar to the arithmetic mean, which is the standard average. However, it takes into account the different degrees of importance of various data points.
  • This approach is particularly useful when some values are more significant than others in a dataset.
  • For example, in finance, the concept helps determine metrics such as average days to maturity, factoring in different investment amounts.
Understanding statistics and using tools like the weighted mean allows analysts to draw meaningful conclusions from data. Methods like these help in making informed decisions based on complex datasets.
Data Analysis
Data analysis involves inspecting and modeling data to discover useful information. As seen in the exercise, data analysis starts with understanding the dataset.
In our example, the dataset comprised 'Days to Maturity' and 'Dollar Values'. Each entry or observation has a significant impact on the analysis outcome.
  • The initial step is to identify the variables: 'Days to Maturity' as the values ( \( x_i \)) and 'Dollar Values' as the weights ( \( w_i \)).
  • It is crucial to correctly pair each value with its respective weight to ensure accurate analysis.
Each value and weight relationship helps shape the final calculated weighted mean. Systematic data analysis guarantees that each step builds towards an accurate interpretation of the weighted mean.
Mathematical Calculation
Mathematical calculations provide the crux of data-driven insights. The formula for calculating the weighted mean is a simple yet powerful tool.
The formula \( \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} \) might initially seem complex, but breaking it down makes it manageable.
  • The numerator represents the summation of all weighted values, \( \sum w_i x_i \).
  • Each value of 'Days to Maturity' must be multiplied by its corresponding 'Dollar Value'.
  • These products are then summed up to get the total weighted value.
  • Lastly, the denominator is the total of the weights, \( \sum w_i \).
By dividing the total weighted value by the sum of the weights, we derive the weighted mean. This process highlights the significance of methodical calculations in achieving meaningful statistical results.

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Most popular questions from this chapter

The following data show the trailing 52 -week primary share earnings and book values as reported by 10 companies (The Wall Street Journal, March 13,2000 ). $$\begin{array}{lrr} & \text { Book } & \\ \text { Company } & \text { Value } & \text { Earnings } \\ \text { Am Elec } & 25.21 & 2.69 \\ \text { Columbia En } & 23.20 & 3.01 \\ \text { Con Ed } & 25.19 & 3.13 \\ \text { Duke Energy } & 20.17 & 2.25 \\ \text { Edison Int'l } & 13.55 & 1.79 \\ \text { Enron Cp. } & 7.44 & 1.27 \\ \text { Peco } & 13.61 & 3.15 \\ \text { Pub Sv Ent } & 21.86 & 3.29 \\ \text { Southn Co. } & 8.77 & 1.86 \\ \text { Unicom } & 23.22 & 2.74 \end{array}$$ a. Develop a scatter diagram for the data with book value on the \(x\) -axis. b. What is the sample correlation coefficient, and what does it tell you about the relationship between the earnings per share and the book value?

Consider a sample with data values of \(10,20,12,17,\) and \(16 .\) Compute the variance and standard deviation.

The results of a national survey showed that on average, adults sleep 6.9 hours per night. Suppose that the standard deviation is 1.2 hours. a. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours. b. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 hours. c. Assume that the number of hours of sleep follows a bell-shaped distribution. Use the empirical rule to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours per day. How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)?

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 (http://www.census.gov, 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?

Consider a sample with data values of \(10,20,12,17,\) and \(16 .\) Compute the range and interquartile range.
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