Promotion of supermarket vegetables. A supermarket chain is interested in exploring the relationship between the sales of its store-brand canned vegetables 1y2, the amount spent on promotion of the vegetables in local newspapers 1x12, and the amount of shelf space allocated to the brand 1x22. One of the chain’s supermarkets was randomly selected, and over a 20-week period x1 and x2 were varied, as reported in the table. Week Sales ($) Advertising Expenditures ($) Shelf Space (sq. ft.) 1 2,010 201 75 2 1,850 205 50 3 2,400 355 75 4 1,575 208 30 5 3,550 590 75 6 2,015 397 50 7 3,908 820 75 8 1,870 400 30 9 4,877 997 75 10 2,190 515 30 11 5,005 996 75 12 2,500 625 50 13 3,005 860 50 14 3,480 1,012 50 15 5,500 1,135 75 16 1,995 635 30 17 2,390 837 30 18 4,390 1,200 50 19 2,785 990 30 20 2,989 1,205 30

a. Fit the following model to the data: y = b0 + b1x1 + b2x2 + b3x1x2 + e

b. Conduct an F-test to investigate the overall usefulness of this model. Use a = .05.

c. Test for the presence of interaction between advertising expenditures and shelf space. Use a = .05.

d. Explain what it means to say that advertising expenditures and shelf space interact.

e. Explain how you could be misled by using a first-order model instead of an interaction model to explain how advertising expenditures and shelf space influence sales.

f. Based on the type of data collected, comment on the assumption of independent errors.