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# Coefficients of correlation and determination, regression equations, and weighted moving averages

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This posting has 5 examples of how to calculate a coefficient of correlation (r), coefficient of determination (r^2), regression equation, and weighted moving average.

1. The following sample observations were randomly selected.

[table of data] - SEE ATTACHMENT

Determine the coefficient of correlation and the coefficient of determination. Interpret.

2. Bi-lo Appliance Stores has outlets in several large metropolitan areas. The general sales manager
plans to air a commercial for a digital camera on selected local TV stations prior to a sale starting
on Saturday and ending Sunday. She plans to get the information for Saturday-Sunday digital
camera sales at the various outlets and pair them with the number of times the advertisement was
shown on the local TV stations. The purpose is to find whether there is any relationship between
the number of times the advertisement was aired and digital camera sales. The pairings are:

[table of data]

a. What is the dependent variable?
b. Draw a scatter diagram.
c. Determine the coefficient of correlation.
d. Determine the coefficient of determination.
e. Interpret these statistical measures.

3. Mr. James McWhinney, president of Daniel-James Financial Services, believes there is a
relationship between the number of client contacts and the dollar amount of sales. To document
this assertion, Mr. McWhinney gathered the following sample information. The X column indicates
the number of client contacts last month, and the Y column shows the value of sales (\$ thousands)
last month for each client sampled.

[table of data]

a. Determine the regression equation.
b. Determine the estimated sales if 40 contacts are made.

4. 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.

[table of data]

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).

5. The yield on a 30-year treasury note at the end of each year since 1990 is recorded below. Compute
a five-year weighted moving average using weights of .1, .1, .2, .3 and .3, respectively.
Describe the trend in yield.

[table of data]

##### Solution Preview

1. The following sample observations were randomly selected.

[table of data]

Determine the coefficient of correlation and the coefficient of determination. Interpret.

The coefficient of correlation (r) is a number between -1 and +1 that measures the linear dependence between two random variables. The limiting values of -1 and +1 indicate perfect negative and perfect positive correlation, respectively; a correlation of zero suggests a complete lack of association between the two variables.

[equations]

See attached Excel spreadsheet for calculations of Sxx, Sxy, and Syy.

r = -.89

The variables x and y have a fairly strong negative relationship. That is, higher values of x are typically associated with lower values of y.

The coefficient of determination (r2) is a measure of the goodness of fit of a regression model that is equal to the square of the coefficient of correlation.

r2 = (-.89) 2 = .79

This means that 79% of the variation in y is explained by the linear relationship with x.

2. Bi-lo Appliance Stores has outlets in several large metropolitan areas. The general sales manager
plans to air a commercial for a digital camera on selected local TV stations prior to a sale starting
on Saturday and ending Sunday. She plans to get the information for Saturday-Sunday digital
camera sales at the various outlets and pair them with the number of times the advertisement was
shown on the local TV stations. The purpose is to find whether there is any relationship between

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