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# Statistics Problem

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5. The city council of Pine Bluffs is considering increasing the number of police in an effort
to reduce crime. Before making a final decision, the council asks the chief of police to survey
other cities of similar size to determine the relationship between the number of police
and the number of crimes reported. The chief gathered the following sample information.

City Police Number of Crimes City Police Number of Crimes
Oxford 15 17 Holgate 17 7
Starksville 17 13 Carey 12 21
Danville 25 5 Whistler 11 19
Athens 27 7 Woodville 22 6

a. If we want to estimate crimes on the basis of the number of police, which variable is
the dependent variable and which is the independent variable?
b. Draw a scatter diagram.
c. Determine the coefficient of correlation.
d. Determine the coefficient of determination.
e. Interpret these statistical measures. Does it surprise you that the relationship is inverse?

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

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

21. The following sample observations were randomly selected.
X: 4 5 3 6 10
Y: 4 6 5 7 7
a. Determine the regression equation.
b. Determine the value of when X is 7.

c. Determine the standard error of estimate.
d. Suppose a large sample is selected (instead of just five). About 68 percent of the predictions
would be between what two values?

1. The director of marketing at Reeves Wholesale Products is studying monthly sales. Three
independent variables were selected as estimators of sales: regional population, per capita income, and regional unemployment rate. The regression equation was computed
to be (in dollars):

Y= 64,100 + .394X1 + 9.6X2 - 11,600X3

a. What is the full name of the equation?
b. Interpret the number 64,100.
c. What are the estimated monthly sales for a particular region with a population of 796,000, per capita income of \$6,940, and an unemployment rate of 6.0 percent?

3. A sample of General Mills employees was studied to determine their degree of satisfaction with their present life. A special index, called the index of satisfaction, was used to measure satisfaction. Six factors were studied, namely, age at the time of first marriage (X1, annual income (X2), number of children living (X3), value of all assets (X4), status of health in the form of an index (X5) and the average number of social activities
per week-such as bowling and dancing (X6) Suppose the multiple regression
equation is:

Y= 16.24 + .017X1 + .0028X2 + 42X3 + .0012X4 + .19X5 + 26.8X6

a. What is the estimated index of satisfaction for a person who first married at 18,
has an annual income of \$26,500, has three children living, has assets of \$156,000,
has an index of health status of 141, and has 2.5 social activities a week on the
average?
b. Which would add more to satisfaction, an additional income of \$10,000 a year or two
more social activities a week?

5. The following table lists the annual amounts of glass cullet produced by Kimble Glass
Works, Inc.
Scrap
Year Code (tons)
2002 1 2.0
2003 2 4.0
2004 3 3.0
2005 4 5.0
2006 5 6.0
Determine the least squares trend equation. Estimate the amount of scrap for the year 2008.

#### Solution Summary

Step by step solutions to all the problems are provided.

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## 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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