# Statistics Problems

1. The owner of Maumee Motors wants to study the relationship between the age of a car ad its selling price. Listed below is a random sample of 12 used cars sold at Maumee Motors during the last year.

Car Age (years) Selling Price ($000) Car Age (years) Selling Price ($000)

1 9 8.1 7 8 7.6

2 7 6.0 8 11 8.0

3 11 3.6 9 10 8.0

4 12 4.0 10 12 6.0

5 8 5.0 11 6 8.6

6 7 10.0 12 6 8.0

a.Determine the coefficient of correlation

b.Determine the coefficient of determination

2. The Bradford Electric Illuminating Company is studying the relationship between kilowatt-hours (thousands) used and the number of rooms in a private single-family residence. A random sample of 10 homes yielded the following.

Number of Rooms Kilowatt-Hours (thousands) Number of Rooms Kilowatt-Hours (thousands)

12 9 8 6

9 7 10 8

14 10 10 10

6 5 5 4

10 8 7 7

a.Determine the regression equation.

b.Determine the number of kilowatt-hours, in thousands, for a six-room house.

3. Mr. James McWinney, President of Daniel-Jones Financial Services, believes there is a relationship between the number of client contacts and the dollar amount of sales. To document this assertion, Mr. McWinney 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 samples.

Number of Contacts, X Sales ($ thousands), Y Number of Contacts, X Sales ($ thousands), Y

14 24 23 30

12 14 48 90

20 28 50 85

16 30 55 120

46 80 50 110

a.Determine the regression equation.

b.Determine the estimated sales if 40 contacts are made.

4. A recent article in Business Week listed the "Best Small Companies." We are interested in the current results of the companies' sales and earnings. A random sample of 12 companies was selected and the sales and earnings, in millions of dollars, are reported below.

Company Sales ($ millions) Earnings (S millions) Company Sales ($ millions) Earnings ($ millions)

Papa John's International 89.2 4.9 Checkmate Electronics 17.5 2.6

Applied Innovation 18.6 4.4 Royal Grip 11.9 1.7

Integracare 18.2 1.3 M-Wave 19.6 3.5

Wall Data 71.7 8.0 Serving-N-Slide 51.2 8.2

Davidson Associates 58.6 6.6 Daig 2 8.6 6.0

Chico's Fas 46.8 4.1 Cobra Golf 69.2 12.8

a.Compute the correlation coefficient.

b.Compute the coefficient of determination.

c.Determine the regression equation.

d.For a small company with $50.0 million in sales, estimate the earnings.

5. A random sample of five observations from the first population resulted in a standard deviation of 12.1

A random sample of seven observations from the second population showed a standard deviation of 7.0

At the .01 significance level, is there more variation in the first population?

6. Stargell Research Associates conducted a study of the radio listening habits of men and women. One facet of the study involved the mean listening time. It was discovered that the mean listening time for men was 35 minutes per day. The standard deviation of the sample of the 10 men studied was 10 minutes per day. The mean listening time for the 12 women studied was also 35 minutes, but the standard deviation of the sample was 12 minutes. At the .10 significance level, can we conclude that there is a difference in the variation in the listening times for men and women?

7. The mean rate of return on a sample of 12 utility stocks was 12.6 percent with a standard deviation of 3.9 percent. The mean rate of return on a sample of 8 utility stocks was 10.9 percent with a standard deviation of 3.5 percent. At the .05 significance level, can we conclude that there is more variation in the oil stocks?

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#### Solution Summary

This solution is comprised of detailed step-by-step calculations and analysis of the given problems related to Statistics and provides students with a clear perspective of the underlying concepts.

Statistics Problems - Regression Analysis, Autocorrelation, Multicollinearity

1. Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present.

a. What are some of the possible causes of this autocorrelation?

b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model?

c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts?

d. What techniques might be used to remove this autocorrelation from the model?

2. Suppose the appliance manufacturer discussed in Exercise 1 also developed another model, again using time-series data, where appliance sales was the dependent variable and disposable personal income and retail sales of durable goods were the independent variables. Although the r2 statistic is high, the manufacturer also suspects that serious multicollinearity exists between the two independent variables.

a. In what ways does the presence of this multicollinearity affect the results of the regression analysis?

b. Under what conditions might the presence of multicollinearity cause problems in the use of this regression equation in designing a marketing plan for appliance sales?

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