Question

A national chain of women’s clothing stores with locations in the large shopping malls thinks that it can do a better job of planning more renovations and expansions if it understands what variables impact sales. It plans a small pilot study on stores in 25 different mall locations. The data it collects consist of monthly sales, store size (sq. ft), number of linear feet of window display, number of competitors located in mall, size of the mall (sq. ft),and distance to nearest competitor (ft).

1. Test the individual regression coefficients. At the 0.05 level of significance, what are your conclusions?

2. f you were going to drop just one variable from the model, which one would you choose? Why?

The store planners for the women’s clothing chain want to find the best model that they can for understanding what store characteristics impact monthly sales.

3. Use stepwise regression to find the best model for the data.

4. Analyze the model you have identified to determine whether it has any problem

5. Write a memo reporting your findings to your boss. Identify the strengths and weaknesses of the model you have chosen.

Sales	Size	Windows	Competitors	Mall Size	Nearest Competitor
4453	3860	39	12	943700	227
4770	4150	41	15	532500	142
4821	3880	39	15	390500	263
4912	4000	39	13	545500	219
4774	4140	40	10	329600	232
4638	4370	48	14	802600	257
4076	3570	37	16	463300	241
3967	3870	39	16	855200	220
4000	4020	44	21	443000	188
4379	3990	38	16	613400	209
5761	4930	50	15	420300	220
3561	3540	34	15	626700	167
4145	3950	36	14	601500	187
4406	3770	36	12	593000	199
4972	3940	38	11	347100	204
4414	3590	35	10	355900	146
4363	4090	38	13	490100	206
4499	4580	45	16	649200	144
3573	3580	35	18	685900	178
5287	4380	42	15	106200	149
5339	4330	40	10	354900	231
4656	4060	37	11	598700	225
3943	3380	34	16	381800	163
5121	4760	44	17	597900	224
4557	3800	36	14	745300	195

Answer 1

1)to check the individual regression coefficient the test statistic is:

H0: Beta is equal to zero.

vs

H1: Beta not equal to zero.

The minitab output is as follows:

Regression Analysis: sales versus size, window, compititors, mall size, nearest compititors

Analysis of Variance

Source DF Adj SS Adj MS F-Value   P-Value

Regression 5 5761406   1152281 19.21 0.000

size 1 560848    560848 9.35 0.006

window 1 5946 5946 0.10 0.756

compititors 1 570069    570069 9.51 0.006

mall size 1 620772    620772 10.35 0.005

nearest compititors   1 103620    103620 1.73 0.204

Error 19 1139390    59968

Total    24   6900796

Model Summary

      S    R-sq R-sq(adj) R-sq(pred)

244.883 83.49%     79.14%      72.06%

Coefficients

Term                      Coef SE Coef T-Value P-Value VIF

Constant 1507 672 2.24    0.037

size 0.919 0.301 3.06    0.006 5.26

window 9.1 28.8 0.31    0.756 5.82

compititors -67.7 22.0 -3.08    0.006   1.40

mall size -0.000903 0.000281    -3.22    0.005 1.12

nearest compititors      2.10 1.59 1.31 0.204 1.24

Regression Equation

sales = 1507 + 0.919 size + 9.1 window - 67.7 compititors - 0.000903 mall size

        + 2.10 nearest compititors

If p-value > level of significance, then we accept H0.

Here, the regression coefficients for window and nearest compititors is greater than 0.05. Therefore, the two variables window and nearest neigbour must be deleted from model.

2) If I were to drop only one variable from the model then i would drop the window variable since its p-value is greater than 0.05 and also the VIF is 5.82 which indicates that the variable is correlated to other variable causing problem of multicollinearity.

3)

Regression Analysis: sales versus size, window, compititors, mall size, nearest compititors

Stepwise Selection of Terms

α to enter = 0.15, α to remove = 0.15

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Regression 3 5627674 1875891 30.94 0.000

size 1 3755185 3755185 61.94 0.000

compititors 1 856069 856069 14.12    0.001

mall size 1 514713 514713 8.49 0.008

Error 21 1273122 60625

Total 24 6900796

Model Summary

S R-sq R-sq(adj) R-sq(pred)
246.221 81.55% 78.92% 75.09%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 1770 611 2.90 0.009
size 1.045 0.133 7.87   0.000 1.01
compititors -71.0 18.9 -3.76 0.001 1.03
mall size -0.000792 0.000272 -2.91 0.008 1.04

Regression Equation

sales = 1770 + 1.045 size - 71.0 compititors - 0.000792 mall size

4)

a) Normal probability plot implies residuals are normally distribued.

b)Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Here, the points fall randomly on both sides of 0, with no recognizable patterns in the points. Therefore residual are independently distributed.

c)Use the histogram of the residuals to determine whether the data are skewed or include outliers. The bar that is far from other bars may indicates presence of an  outlier.

d)similary as in (b) it indicates no problem as residuals indicate no pattern.