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 12 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 5 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)

Descriptive Statistics          
   N   Mean   Std. Deviation
X1   27   286.5741   192.06172
X2   27   3.3259   2.01105
X3   27   387.4815   191.16774
X4   27   8.1000   3.77451
X5   27   9.6926   5.14003
X6   27   7.7407   4.89578

CV

.67
.60
.49
.47
.53
.63

b)

r R2

.894   .799
.946   .895
.914   .835
.954   .910
-.912   .832
.844   .712
.749   .561
.838   .702
-.766   .587
.906   .821
.864   .746
-.807   .651
.795   .632
-.841   .707
-.870   .757

c)

Model Summary                                  
Model   R   R Square   Adjusted R Square   Std. Error of the Estimate   Change Statistics              
                   R Square Change   F Change   df1   df2   Sig. F Change
1   .997a   .993   .992   17.64924   .993   611.590   5   21   .000
a Predictors: (Constant), X6, X2, X4, X5, X3                                  

Coefficientsa                      
Model       Unstandardized Coefficients       Standardized Coefficients   t   Sig.
       B   Std. Error   Beta      
1   (Constant)   -18.859   30.150       -.626   .538
   X2   16.202   3.544   .170   4.571   .000
   X3   .175   .058   .174   3.032   .006
   X4   11.526   2.532   .227   4.552   .000
   X5   13.580   1.770   .363   7.671   .000
   X6   -5.311   1.705   -.135   -3.114   .005
a Dependent Variable: X1                      

99.3% of variation can be explained by the corresponding variations in x₂, x₃, x₄, x₅, and x₆ taken together.

d)

X1 = -18.9 + 16.2*X2 + 0.17*X3 + 11.5*X4 + 13.6*X5 - 5.31*X6

If 12 new competing stores moved into the sales district but the other explanatory variables did not change:

X1 = -82.62 + 16.2*X2 + 0.17*X3 + 11.5*X4 + 13.6*X5

If you increased the local advertising by 5 thousand dollars but the other explanatory variables did not change:

X1 = 38.6 + 16.2*X2 + 0.17*X3 + 13.6*X5 - 5.31*X6