Chapter 14, Problem 18CE

Suppose that the sales manager of a large automotive parts distributor wants to estimate the total annual sales for each of the company’s regions. Five factors appear to be related to regional sales: the number of retail outlets in the region, the number of automobiles in the region registered as of April 1, the total personal income recorded in the first quarter of the year, the average age of the automobiles (years), and the number of sales supervisors in the region. The data for each region were gathered for last year. For example, see the following table. In region 1 there were 1,739 retail outlets stocking the company’s automotive parts, there were 9,270,000 registered automobiles in the region as of April 1, and so on. The region’s sales for that year were $37,702,000.

1. a. Consider the following correlation matrix. Which single variable has the strongest correlation with the dependent variable? The correlations between the independent variables outlets and income and between outlets and number of automobiles are fairly strong. Could this be a problem? What is this condition called?

1. b. The output for all five variables is shown below. What percent of the variation is explained by the regression equation?

1. c. Conduct a global test of hypothesis to determine whether any of the regression coefficients are not zero. Use the .05 significance level.
2. d. Conduct a test of hypothesis on each of the independent variables. Would you consider eliminating “outlets” and “bosses”? Use the .05 significance level.
3. e. The regression has been rerun below with “outlets” and “bosses” eliminated. Compute the coefficient of determination. How much has R² changed from the previous analysis?

1. f. Following is a histogram of the residuals. Does the normality assumption appear reasonable? Why?

1. g. Following is a plot of the fitted values of y (i.e., $\hat{y}$) and the residuals. What do you observe? Do you see any violations of the assumptions?