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?
x4; x1x3; x4 x1; x4x1; x3
(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?
x2; x5x4; x2 x5; x4x5; 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 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 2 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 sufficient evidence that all βi differ from 0.We reject all null hypotheses, there is insufficient 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 | $ |
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 aske...
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...
Al Greens is a franchise store that sells house plants and lawn and garden supplies. Although Al 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 Al Greens stores in California. Each of the 27...
3. Description of each X and data for 27 franchise stores are given below The data (X1, X2, X3, X4, X5, X6) are for each franchise store. X1 annual net sales/$1000 X2 number sq. ft/1000 X3 - inventory I$1000 X4- amount spent on advertising /$1000 X5 size of sales district/1000 families X6 number of competing stores in distric 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...
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