Question

In: Statistics and Probability

All Greens is a franchise store that sells house plants and lawn and garden supplies. Although...

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 detailed below.

x1 = annual net sales, in thousands of dollars
x2 = number of square feet of floor display in store, in thousands of square feet
x3 = value of store inventory, in thousands of dollars
x4 = amount spent on local advertising, in thousands of dollars
x5 = size of sales district, in thousands of families
x6 = number of competing or similar stores in sales district

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

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

x s CV
x1 %
x2 %
x3 %
x4 %
x5 %
x6 %

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?

x1; x3

x4; x1     

x3; x4

x1; x4



(b) For each pair of variables, generate the correlation coefficient r. For all pairs involving x1, compute the corresponding coefficient of determination r2. (Use 3 decimal places.)

r r2
x1, x2
x1, x3
x1, x4
x1, x5
x1, x6
x2, x3
x2, x4
x2, x5
x2, x6
x3, x4
x3, x5
x3, x6
x4, x5
x4, x6
x5, x6

Which variable has the greatest influence on annual net sales? Which variable has the least influence on annual net sales?

x5; x2

x2; x5     

x5; x4

x4; x2



(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? (Use 1 decimal place.)
%

(d) Write out the regression equation. (Use 2 decimal places for x3 and x6. Use 1 decimal place otherwise.)

x1 = +  x2 +  x3 +  x4 +  x5 +  x6


If 5 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? (Use 2 decimal places.)


If you increased the local advertising by 10 thousand dollars but the other explanatory variables did not change, what would you expect for the corresponding change in annual net sales? (Use 2 decimal places.)


(e) Test each coefficient to determine if it is or is not zero. Use a 5% level of significance. (Use 2 decimal places for t and 3 decimal places for the P-value.)

t P-value
β2
β3
β4
β5
β6

Conclusion

We fail to reject all null hypotheses, there is insufficient evidence that all βi differ from 0.

We fail to reject all null hypotheses, there is sufficient evidence that all βi differ from 0.     

We reject all null hypotheses, there is insufficient evidence that all βi differ from 0.

We reject all null hypotheses, there is sufficient evidence that all βi differ from 0.



(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 = 2.8, x3 = 250, x4 = 3.1, x5 = 7.3, and x6 = 2, use a computer to forecast x1 = annual net sales and find an 80% confidence interval for your forecast (if your software produces prediction intervals). (Use 2 decimal places.)

forecast
lower limit
upper limit


(g) Construct a new regression model with x4 as the response variable and x1, x2, x3, x5, and x6 as explanatory variables. (Use 2 decimal places for the intercept, 4 for x1, 5 for x3, and 3 otherwise.)

x4 = +  x1 +  x2 +  x3 +  x5 +  x6


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 = 163, x2 = 2.4, x3 = 188, x5 = 6.6, and x6 = 10. Use these data to predict x4 (advertising costs) and find an 80% confidence interval for your prediction. (Use 2 decimal places.)

prediction
lower limit
upper limit


At the 80% confidence level, what range of advertising costs do you think is appropriate for this store? (Round to nearest integer.)

lower limit $
upper limit $

Solutions

Expert Solution


Related Solutions

All Greens is a franchise store that sells house plants and lawn and garden supplies. Although...
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...
All Greens is a franchise store that sells house plants and lawn and garden supplies. Although...
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...
Everyday Supplies Pty Ltd is a single-store retailer that sells a variety of tools, garden supplies,...
Everyday Supplies Pty Ltd is a single-store retailer that sells a variety of tools, garden supplies, timber, small appliances, and electrical fixtures to the public, although about half of Everyday Supplies’ sales are to construction contractors on account. Retail customers pay for merchandise by cash or credit card at cash registers when merchandise is purchased. A contractor may purchase merchandise on account, if approved by the credit manager based only on the manager’s familiarity with the contractor’s reputation. After credit...
Everyday Supplies Pty Ltd is a single-store retailer that sells a variety of tools, garden supplies,...
Everyday Supplies Pty Ltd is a single-store retailer that sells a variety of tools, garden supplies, timber, small appliances, and electrical fixtures to the public, although about half of Everyday Supplies’ sales are to construction contractors on account. Retail customers pay for merchandise by cash or credit card at cash registers when merchandise is purchased. A contractor may purchase merchandise on account, if approved by the credit manager based only on the manager’s familiarity with the contractor’s reputation. After credit...
Everyday Supplies Pty Ltd is a single-store retailer that sells a variety of tools, garden supplies,...
Everyday Supplies Pty Ltd is a single-store retailer that sells a variety of tools, garden supplies, timber, small appliances, and electrical fixtures to the public, although about half of EverydaySupplies’ sales are to construction contractors on account. Retail customers pay for merchandise by cash or credit card at cash registers when merchandise is purchased. A contractor may purchase merchandise on account, if approved by the credit manager based only on the manager’s familiarity with the contractor’s reputation. After credit is...
A typical lawn averages 21 blades of grass per square inch. A garden store sells an...
A typical lawn averages 21 blades of grass per square inch. A garden store sells an expensive fertilizer designed to transform an average lawn into a thick carpet within three weeks. To test the claim, a random sample of 32 average lawn plots were selected and treated with the fertilizer. Three weeks later the density of each plot was measured by counting the blades of grass per square inch. n = 32 and X = 22.4 Based on data from...
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter.
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter. The company usually has to borrow money during this quarter to support peak sales of lawn care equip­ment, which occur during May. The following information has been assembled to assist in preparing a cash budget for the quarter: a. Budgeted monthly absorption costing income statements for April-July are:   April May June July Sales ... $600,000 $900,000 $500,000 $400,000 Cost of goods sold ... 420,000...
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter....
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter. The company usually has to borrow money during this quarter to support peak sales of lawn care equipment, which occur during May. The following information has been assembled to assist in preparing a cash budget for the quarter: Budgeted monthly absorption costing income statements for April–July are: April May June July Sales $ 730,000 $ 1,190,000 $ 650,000 $ 560,000 Cost of goods sold...
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter....
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter. The company usually has to borrow money during this quarter to support peak sales of lawn care equipment, which occur during May. The following information has been assembled to assist in preparing a cash budget for the quarter: Budgeted monthly absorption costing income statements for April–July are: April May June July Sales $ 640,000 $ 1,140,000 $ 600,000 $ 500,000 Cost of goods sold...
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter....
Garden Sales, Inc., sells garden supplies. Management is planning its cash needs for the second quarter. The company usually has to borrow money during this quarter to support peak sales of lawn care equipment, which occur during May. The following information has been assembled to assist in preparing a cash budget for the quarter: Budgeted monthly absorption costing income statements for April–July are: April May June July Sales $ 690,000 $ 860,000 $ 570,000 $ 470,000 Cost of goods sold...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT