Question

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			% (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 (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 6 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 4 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 (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 $

Answer 1

a)

Variable	Mean	S	CV%
X1	286.57	192.06	67.02027
X2	3.3259	2.011	60.46484
X3	387.48	191.17	49.33674
X4	8.1	3.7745	46.59877
X5	9.6929	5.14	53.02851
X6	7.7407	4.8958	63.24751

Mean:

b)

	r
X1,X2	0.894
X1,X3	0.946
X1,X4	0.914
X1,X5	0.954
X1,X6	-0.912
X2,X3	0.844
X3,X4	0.749
X2,X5	0.838
X2,X6	-0.766
X3,X4	0.906
X3,X5	0.864
X3,X6	-0.807
X4,X5	0.795
X4,X6	-0.841
X5,X6	-0.87

c)

Coefficient of determination or R-squared: 99.32%

d) Regression equation:

X1' =-18.9+16.2* X2+ 0.1746* X3 + 11.53 * X4 + 13.58 * X5 -5.31 * X6

X6= 6 and other variables are same:

X1' =-18.9+16.2* 1+ 0.1746* 1 + 11.53 * 1 + 13.58 * 1 -5.31 * 6 = -9.2754

X4 = 4 and other variables are same:

X1'= -18.9 +16.2* 1+ 0.1746* 1 + 11.53 * 4 + 13.58 * 1 - 5.31 * 1 =51.8646

e)

f)

Prediction for x1

Regression Equation

x1 = -18.9 + 16.20 x2 + 0.1746 x3 + 11.53 x4 + 13.58 x5 - 5.31 x6

Variable Setting
x2 2.8
x3 250
x4 3.1
x5 7.3
x6 2

Fit SE Fit 80% CI 80% PI
194.410 18.3055 (170.188, 218.631) (160.763, 228.056) X

g)

h)

Prediction for x4

Regression Equation

x4 = 4.14 + 0.04309 x1 - 0.800 x2 + 0.00059 x3 - 0.661 x5 + 0.057 x6

Variable Setting
x1 163
x2 2.4
x3 188
x5 6.6
x6 10

Fit SE Fit 80% CI 80% PI
5.57065 0.400115 (5.04122, 6.10007) (4.04779, 7.09350)