Question: Suppose that the sales manager of a large automotive parts distributor wants to estimate as early as April the total annual sales of a region. Based on regional sales, the total sales for the company can also be estimated. If, based on past experience, it is found that the April estimates of annual sales are reasonably accurate, then in future years the April forecast could be used to revise production schedules and maintain the correct inventory at the retail outlets.

Several factors appear to be related to sales, including the number of retail outlets in the region stocking the company's parts, the number of automobiles in the region registered as of April 1, and the total personal income for the first quarter of the year. A total of five independent variables were finally selected as being the most important (according to the sales manager). Then the data were gathered for a recent year. The total annual sales for that year for each region were also recorded. Note in the following table that for 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 sales for that year were $37,702,000.

a) Using the statistical package of your choice, run a multiple regression using all of the above variables and write the least squares multiple regression equation that relates y to the independent variables.

b) Interpret the slopes of X2 and X4.

c) Identify SSE, s², and s.

d) Find the total variation, the unexplained variation, and the explained variation.

e) Find and interpret R², the multiple coefficient of determination.

f) Conduct an overall test of this model at the 0.05 level.

g) Test whether each of the independent variables belongs in the model. Use a level of significance of .05. What are your recommendations for the model?

g) In a particular region, it is estimated that the number of retail outlets is 1,000, the number of automobiles registered is 8 million, personal income is 50 billion dollars, the average age of automobiles is 4 years, and the number of supervisors is 10. Report a point prediction of, and a 95 percent prediction interval, for annual sales for this region.

h) Use appropriate residual plots and comment on the appropriateness of the fit and whether or not the assumptions are satisfied.