Problem 1: In a study of the demand for life insurance, Executive Insurers, Inc. is examining the factors that affect the amount of life insurance held by executives. The following data on the amount of insurance and annual incomes of a random sample of 12 executives were collected.
Observation Amount of Life Insurance Annual Income
(X $1,000) (X $1,000)
1 90 50
2 180 84
3 225 74
4 210 115
5 150 104
6 150 96
7 60 56
8 135 102
9 150 104
10 150 108
11 60 65
12 90 58
a. Given the nature of the problem, which would be the dependent variable and which would be the independent variable?
b. Plot the data
c. Determine the estimated regression lnc. Give an economic interpretation of the slope (b) coefficient
d. Test the hypothesis that there is no relationship between the variables. (i.e., B= 0)
e. Calculate the coefficient of determination
f. Perform an analysis of variance on the regression, including an F test on the overall significance of the results
g. Determine the best estimate, based on the regression model, of the amount of life insurance held by an executive whose annual income is $80,000. Construct an approximate 95 percent prediction interval.
Problem 2: 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?
Problem 3: 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?