Suppose that the sales manager of a large automotive parts distributor wants to
ID: 3232369 • Letter: S
Question
Suppose that the sales manager of a large automotive parts distributor wants to estimate as early as April the total annual sales.
According to the manager of the distribution warehouse, several factors are related to annual sales (measured in millions of dollars) (sales), including the number of retail outlets in the region stocking the company’s parts (outlets), the number of automobiles in the region registered as of April 1 (measured in millions) (cars), the total personal income for the first quarter of the year (measured in billions of dollars) (income), the average age of automobiles in years (age), and the number of supervisors at the distribution warehouse (bosses). The data for all these variables were gathered for a recent year.
Consider the following correlation matrix.
sales
outlets
cars
income
age
outlets
0.899
cars
0.605
0.775
income
0.964
0.825
0.409
age
-0.323
-0.489
-0.447
-0.349
bosses
0.286
0.183
0.395
0.155
0.291
A. Which single variable has the strongest correlation with the dependent variable? Is there evidence of multicollinearity? If so, between what variables?
Using the data, the following multivariate regression equation was estimated:
sales = -19.7 – 0.00063 outlets – 1.74 cars + 0.410 income + 2.04 age – 0.034 bosses
The output for all five variables is shown below.
Predictor
Coef
SE Coef
T
P
Constant
-19.672
5.422
-3.63
0.022
Outlets
-0.000629
0.002638
-0.24
0.823
Cars
-1.7399
0.5530
3.15
0.035
Income
0.40994
0.04385
9.35
0.001
Age
2.0357
0.8779
2.32
0.081
bosses
-0.0344
0.1880
-0.18
0.864
Analysis of Variance
SOURCE
DF
SS
MS
F
P
Regression
5
1593.81
318.76
140.36
0.000
Residual Error
4
9.08
2.27
Total
9
1602.89
B. State the null hypothesis concerning the statistical significance of the overall regression, test this hypothesis, and interpret the results. (use a .05 level of significance)
C. What percent of the variation is explained by the regression equation?
D. Interpret the results (both statistical significance and magnitude of effect) for each of the independent variables in the model. (Use a .05 level of significance)
E. What would be the projected value in annual sales if the following were true?
outlets = 1739, cars = 9.27, income = 85.4, age = 3.5, and bosses = 9.0
If these values are outside the range of values used for the regression, would this be a reliable forecast? Why or why not?
sales
outlets
cars
income
age
outlets
0.899
cars
0.605
0.775
income
0.964
0.825
0.409
age
-0.323
-0.489
-0.447
-0.349
bosses
0.286
0.183
0.395
0.155
0.291
Explanation / Answer
A.
Income has the strongest correlation with the dependent variable i.e. sales. Cars and outlets and income and outlets are highly correlated amongst themselves. Hence, these variables suggest multicollinearity.
B.
H0: 1 = 2 = 3 = 4 = 5 = 0 (no linear relationship)
H1: at least one i 0 (at least one independent variable affects Y)
FSTAT is 140.36 and FCRIT is 6.256 and Hence FSTAT > FCRIT and lies in the rejection region and hence reject null hypothesis. This proves that at least one independent variable affects Y.
C.
r^2=SSR/SST=1593.81/1602.89=0.9943 or 99.43%
D.
Outlets:- p-value= 0.823, not significant, For every increase of 1 in outlets, sales would go down by 0.000629.
Cars:- p-value= 0.035, significant, For every increase of 1 in cars, sales would go down by 1.7399.
Income:- p-value= 0.001, significant, For every increase of 1 in income, sales would go up by 0.40994.
Age:- p-value= 0.081, not significant, For every increase of 1 in age, sales would go up by 2.0357.
Bosses:- p-value= 0.864, not significant, For every increase of 1 in bosses, sales would go down by 0.0344.
E.
Sales = -19.7 – 0.00063 outlets – 1.74 cars + 0.410 income + 2.04 age – 0.034 bosses
= -19.7-0.00063(1739)-17.4(9.27)+0.0410(85.4)+2.04(3.5)-0.034(9)
= -19.7-1.09557-161.28+3.5014+7.14-0.306
= -171.75817
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