Franchise Business: Market Analysis All Greens is a franchise store that sells house plants and lawn and garden supplies. Although All Greens is a franchise, each store is owned and managed by private individuals. Some friends have asked you to go into business with them to open a new All Greens store in the suburbs of San Diego. The national franchise headquarters sent you the following information at your request. These data are about 27 All Greens stores in California. Each of the 27 stores has been doing very well, and you would like to use the information to help set up your own new store. The variables for which we have data are annual net sales, in thousands of dollars number of square feet of floor display in store, in thousands of square feet value of store inventory, in thousands of dollars amount spent on local advertising, in thousands of dollars size of sales district, in thousands of families number of competing or similar stores in sales district 1

2 3

4 5

6 x

x x

x x

x

=

=

=

=

=

=

A sales district was defined to be the region within a 5-mile radius of an All Greens store.

x1 x2 x3 x4 x5 x6 x1 x2 x3 x4 x5 x6 231 3 294 8.2 8.2 11 65 1.2 168 4.7 3.3 11 156 2.2 232 6.9 4.1 12 98 1.6 151 4.6 2.7 10 10 0.5 149 3 4.3 15 398 4.3 342 5.5 16.0 4 519 5.5 600 12 16.1 1 161 2.6 196 7.2 6.3 13 437 4.4 567 10.6 14.1 5 397 3.8 453 10.4 13.9 7 487 4.8 571 11.8 12.7 4 497 5.3 518 11.5 16.3 1 299 3.1 512 8.1 10.1 10 528 5.6 615 12.3 16.0 0 195 2.5 347 7.7 8.4 12 99 0.8 278 2.8 6.5 14 20 1.2 212 3.3 2.1 15 0.5 1.1 142 3.1 1.6 12 68 0.6 102 4.9 4.7 8 347 3.6 461 9.6 11.3 6 570 5.4 788 17.4 12.3 1 341 3.5 382 9.8 11.5 5 428 4.2 577 10.5 14.0 7 507 5.1 590 12.0 15.7 0 464 4.7 535 11.3 15.0 3 400 8.6 517 7.0 12.0 8 15 0.6 163 2.5 2.5 14

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation (see Section 3.2) for each variable. Relative to its mean, which variable has the largest spread of data values? Which variable has the least spread of data values relative to its mean?

(b) For each pair of variables, generate the sample correlation coefficient r. For all pairs involving x1, compute the corresponding coefficient of determination r2. Which variable has the greatest influence on annual net sales? Which variable has the least influence on annual net sales?

(c) Perform a regression analysis with x1 as the response variable. Use x2, x3, x4, x5, and x6 as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x1 can be explained by the corresponding variations in x2, x3, x4, x5, and x6 taken together?

(d) Write out the regression equation. If two new competing stores moved into the sales district but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? Explain your answer. If you increased the local advertising by a thousand dollars but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales?
Explain.

(e) Test each coefficient to determine if it is or is not zero. Use level of significance 5%.

(f) Suppose you and your business associates rent a store, get a bank loan to start up your business, and do a little research on the size of your sales district and the number of competing stores in the district. If x2 5 2.8, x3 5 250, x4 5 3.1, x5 5 7.3, and x6 5 2, use a computer to forecast x1 5annual net sales and find an 80% confidence interval for your forecast (if your software produces prediction intervals).
(g) Construct a new regression model with x4 as the response variable and x1, x2, x3, x5, and x6 as explanatory variables. Suppose an All Greens store in Sonoma, California, wants to estimate a range of advertising costs appropriate to its store. If it spends too little on advertising, it will not reach enough customers. However, it does not want to overspend on advertising for this type and size of store. At this store, x1 5163, x2 5 2.4, x3 5188, x5 5 6.6, and x6 510. Use these data to predict x4 (advertising costs) and find an 80% confidence interval for your prediction. At the 80% confidence level, what range of advertising costs do you think is appropriate for this store?

Problems 3–6, use appropriate multiple regression software of your choice and enter the data. Note that the data are also available for download at the Companion Sites for this text.AppendixLO1