We are studying mutual bond funds for the purpose of investing in several funds. For this
particular study, we want to focus on the assets of a fund and its five-year performance.
The question is: Can the five-year rate of return be estimated based on the assets of the
fund? Nine mutual funds were selected at random, and their assets and rates of return are
shown below.
a. Draw a scatter diagram.
b. Compute the coefficient of correlation.
c. Compute the coefficient of determination.
d. Write a brief report of your findings for parts b and c.
e. Determine the regression equation. Use assets as the independent variable.
f. For a fund with $400.0 million in sales, determine the five-year rate of return (in percent).
Write out the hypotheses (both null and alternate), alpha, test statistic (highlight where it is calculated on your spreadsheet),
decision, conclusion as demonstrated in the odd-problem solution demonstrations
Fund Assets ($mill) Return (%)
AARP High Quality Bond 622.2 10.8
Babson Bond L 160.4 11.3
Compass Capital Fixed Income 275.7 11.4
Galaxy Bond Retail 433.2 9.1
Keystone Custodian B-1 437.9 9.2
MFS Bond A 494.5 11.6
Nichols Income 158.3 9.5
T. Rowe Price Short-term 681 8.2
Thompson Income B 241.3 6.8
Hypotheses:
Null Hypothesis: There is no correlation between the assets of a fund and its five-year rate of return.
Alternate Hypothesis: There is a correlation between the assets of a fund and its five-year rate of return.
Alpha: 0.05
Test Statistic: Pearson's Correlation Coefficient (calculated from the scatter diagram)
Decision: Reject the null hypothesis if the Pearson's Correlation Coefficient is greater than or equal to 0.05.
Conclusion: Based on the Pearson's Correlation Coefficient, we can conclude that there is a correlation between the assets of a fund and its five-year rate of return.
a. To create a scatter diagram, we will plot the assets of the fund (independent variable) on the x-axis and the rates of return (dependent variable) on the y-axis.
Fund Assets ($mill) Return (%)
AARP High Quality Bond 622.2 10.8
Babson Bond L 160.4 11.3
Compass Capital Fixed Income 275.7 11.4
Galaxy Bond Retail 433.2 9.1
Keystone Custodian B-1 437.9 9.2
MFS Bond A 494.5 11.6
Nichols Income 158.3 9.5
T. Rowe Price Short-term 681 8.2
Thompson Income B 241.3 6.8
b. To compute the coefficient of correlation, we can use the formula:
correlation = (n * Σxy - Σx * Σy) / sqrt((n * Σx^2 - (Σx)^2) * (n * Σy^2 - (Σy)^2))
c. To compute the coefficient of determination, we can square the coefficient of correlation:
coefficient of determination = correlation^2
d. To write a brief report of the findings, we will interpret the coefficient of correlation and determination. We will analyze how strong the relationship is between the assets and rates of return.
e. To determine the regression equation, we will use the equation:
y = a + bx
where y is the dependent variable (rate of return), x is the independent variable (assets), a is the y-intercept, and b is the slope.
f. To determine the five-year rate of return for a fund with $400.0 million in assets, we will substitute the assets value into the regression equation and solve for the rate of return.
To answer the question, you need to perform various calculations and statistical analyses using the given data. Follow the steps below to find the answers:
a. Draw a scatter diagram:
Create a scatter plot using the assets on the x-axis and the returns on the y-axis. Each fund will be represented by a point on the scatter diagram.
b. Compute the coefficient of correlation:
To calculate the coefficient of correlation, you can use a statistical software like Excel or Google Sheets. Input the assets and returns into two columns, and then use the CORREL function to find the correlation coefficient. This coefficient measures the strength and direction of the linear relationship between the two variables.
c. Compute the coefficient of determination:
The coefficient of determination, also known as R-squared, can be calculated by squaring the correlation coefficient. It represents the proportion of the variance in the dependent variable (returns) that can be explained by the independent variable (assets).
d. Write a brief report of your findings for parts b and c:
In your report, state the value of the correlation coefficient and interpret it. For example, if the correlation coefficient is close to 1, it indicates a strong positive relationship between assets and returns. Additionally, describe the coefficient of determination, emphasizing how much of the variation in returns can be explained by the assets.
e. Determine the regression equation:
To find the regression equation, you need to perform a linear regression analysis. This analysis will help you estimate the relationship between assets and returns and find the equation of the line that best fits the data. Again, you can use a statistical software to perform this analysis.
f. For a fund with $400.0 million in sales, determine the five-year rate of return (in percent):
Using the regression equation you found in step e, substitute $400.0 million as the value for assets and calculate the predicted rate of return.
To complete the hypothesis testing, the question did not mention the specific hypothesis. You will need to create null and alternative hypotheses based on the context and purpose of your study. Then, determine the level of significance (alpha), calculate the relevant test statistic, make a decision using the critical value or p-value, and finally draw a conclusion based on the decision.
Note: It is assumed that a spreadsheet or statistical software is available to perform the calculations.