Studdy AI Logo
Math  /  Data & Statistics

QuestionThe data set below on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set below on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds. Use the information in the table below to complete parts (a) through (d). Then complete part (e). \begin{tabular}{|c|c|c|c|} \hline \multicolumn{2}{|l|}{\begin{tabular}{l} Bond mutual \\ funds \end{tabular}} & \multicolumn{2}{|l|}{\begin{tabular}{l} Stock mutual \\ funds \end{tabular}} \\ \hline 3.2 & 1.8 & 9.4 & 7.6 \\ \hline 1.9 & 3.4 & 9.1 & 7.4 \\ \hline 2.4 & 2.7 & 8.4 & 7.2 \\ \hline 1.6 & 2.0 & 8.1 & 6.9 \\ \hline \end{tabular}
What is the CV of the data set for height in inches? □\square (Type an integer or decimal rounded to three decimal places as needed.)
What is the CV of the data set for height in centimeters? □\square (Type an integer or decimal rounded to three decimal places as needed.)
What is true of the coefficient of variation? A. The coefficient of variation is always more meaningful than the standard deviation. B. The coefficient of variation is best used when comparing two data sets that use the same units of measure. C. The coefficient of variation does not give as accurate a measurement as the standard deviation. D. When converting units of measure, the coefficient of variation is unchanged.

Studdy Solution

STEP 1

1. The CV (coefficient of variation) is defined as the ratio of the standard deviation to the mean, often expressed as a percentage.
2. We need to calculate the CV for both bond mutual funds and stock mutual funds.
3. The CV should be rounded to three decimal places.
4. We need to compare the CVs to understand if they change with different units of measurement.

STEP 2

1. Calculate the mean of the bond mutual funds.
2. Calculate the standard deviation of the bond mutual funds.
3. Compute the CV for the bond mutual funds.
4. Calculate the mean of the stock mutual funds.
5. Calculate the standard deviation of the stock mutual funds.
6. Compute the CV for the stock mutual funds.
7. Determine the impact of converting units on the CV.
8. Answer the given multiple-choice question regarding the coefficient of variation.

STEP 3

Calculate the mean of the bond mutual funds. Let the data set for bond mutual funds be: 3.2,1.8,1.9,3.4,2.4,2.7,1.6,2.03.2, 1.8, 1.9, 3.4, 2.4, 2.7, 1.6, 2.0. Mean=3.2+1.8+1.9+3.4+2.4+2.7+1.6+2.08 \text{Mean} = \frac{3.2 + 1.8 + 1.9 + 3.4 + 2.4 + 2.7 + 1.6 + 2.0}{8} Mean=19.08=2.375 \text{Mean} = \frac{19.0}{8} = 2.375

STEP 4

Calculate the standard deviation of the bond mutual funds. Standard Deviation=∑(xi−μ)2n−1 \text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}} Where μ\mu is the mean and nn is the number of data points. Standard Deviation=(3.2−2.375)2+(1.8−2.375)2+(1.9−2.375)2+(3.4−2.375)2+(2.4−2.375)2+(2.7−2.375)2+(1.6−2.375)2+(2.0−2.375)28−1 \text{Standard Deviation} = \sqrt{\frac{(3.2-2.375)^2 + (1.8-2.375)^2 + (1.9-2.375)^2 + (3.4-2.375)^2 + (2.4-2.375)^2 + (2.7-2.375)^2 + (1.6-2.375)^2 + (2.0-2.375)^2}{8-1}} Standard Deviation=0.6806+0.3306+0.2256+1.0506+0.0006+0.1056+0.6006+0.14067 \text{Standard Deviation} = \sqrt{\frac{0.6806 + 0.3306 + 0.2256 + 1.0506 + 0.0006 + 0.1056 + 0.6006 + 0.1406}{7}} Standard Deviation=3.13447=0.4478≈0.669 \text{Standard Deviation} = \sqrt{\frac{3.1344}{7}} = \sqrt{0.4478} \approx 0.669

STEP 5

Compute the CV for the bond mutual funds. CV=Standard DeviationMean×100% \text{CV} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\% CV=0.6692.375×100%≈28.184% \text{CV} = \frac{0.669}{2.375} \times 100\% \approx 28.184\%

STEP 6

Calculate the mean of the stock mutual funds. Let the data set for stock mutual funds be: 9.4,7.6,9.1,7.4,8.4,7.2,8.1,6.99.4, 7.6, 9.1, 7.4, 8.4, 7.2, 8.1, 6.9. Mean=9.4+7.6+9.1+7.4+8.4+7.2+8.1+6.98 \text{Mean} = \frac{9.4 + 7.6 + 9.1 + 7.4 + 8.4 + 7.2 + 8.1 + 6.9}{8} Mean=64.18=8.0125 \text{Mean} = \frac{64.1}{8} = 8.0125

STEP 7

Calculate the standard deviation of the stock mutual funds. Standard Deviation=∑(xi−μ)2n−1 \text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}} Where μ\mu is the mean and nn is the number of data points. Standard Deviation=(9.4−8.0125)2+(7.6−8.0125)2+(9.1−8.0125)2+(7.4−8.0125)2+(8.4−8.0125)2+(7.2−8.0125)2+(8.1−8.0125)2+(6.9−8.0125)28−1 \text{Standard Deviation} = \sqrt{\frac{(9.4-8.0125)^2 + (7.6-8.0125)^2 + (9.1-8.0125)^2 + (7.4-8.0125)^2 + (8.4-8.0125)^2 + (7.2-8.0125)^2 + (8.1-8.0125)^2 + (6.9-8.0125)^2}{8-1}} Standard Deviation=1.9166+0.1700+1.1796+0.3756+0.1506+0.6616+0.0076+1.23567 \text{Standard Deviation} = \sqrt{\frac{1.9166 + 0.1700 + 1.1796 + 0.3756 + 0.1506 + 0.6616 + 0.0076 + 1.2356}{7}} Standard Deviation=5.69727=0.8140≈0.902 \text{Standard Deviation} = \sqrt{\frac{5.6972}{7}} = \sqrt{0.8140} \approx 0.902

STEP 8

Compute the CV for the stock mutual funds. CV=Standard DeviationMean×100% \text{CV} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\% CV=0.9028.0125×100%≈11.256% \text{CV} = \frac{0.902}{8.0125} \times 100\% \approx 11.256\%

STEP 9

Determine the impact of converting units on the CV. The coefficient of variation is a dimensionless number (it is a ratio of standard deviation to mean) and hence is not affected by the units of measurement.

STEP 10

Answer the given multiple-choice question regarding the coefficient of variation. D. When converting units of measure, the coefficient of variation is unchanged.
Solution: The CV of the bond mutual funds is 28.184%28.184\%. The CV of the stock mutual funds is 11.256%11.256\%. The correct answer to the multiple-choice question is D.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord