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Multiple Standard Error Estimates

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The district manager of Jasons, a large discount electronics chain, is investigating why certain stores in her region are performing better than others. She believes that three factors are related to total sales: the number of competitors in the region, the population in the surrounding area, and the amount spent on advertising. From her district, consisting of several hundred stores, she selects a random sample of 30 stores. For each store she gathered the following information.
Y = total sales last year (in $ thousands). The sample data were run on MINITAB, with the following results.
X1 = number of competitiors in the region. Analysis of variance
X2 =population of the region (in millions). SOURCE DF SS MS
X3 = advertising expense (in $ thousands). Regression 3 3050.00 1016.67
Error 26 2200.00 84.62
Total 29 5250.00

Predictor Coef StDev t-ratio
Constant 14.00 7.00 2.00
X1 -1.00 0.70 -1.43
X2 30.00 5.20 5.77
X3 0.20 0.08 2.50

a. What are the estimated sales for the Bryne store, which has four competitors, a regional population of 0.4 (400,000), and advertising expense of 30 ($30,000)?

b. Compute the R2 value.

c. Compute the multiple standard error of estimate.

d. Conduct a global test of hypothesis to determine whether any of the regression coefficients are not equal to zero. Use the .05 level of significance.

e. Conduct tests of hypotheses to determine which of the independent variables have significant regression coefficients. Which variables would you consider eliminating? Use the .05 significance level.

Suppose that the sales manager of a large automotive parts distributor wants to estimate as early as April the total annual sales of a region. On the basis of regional sales, the total sales for the company can also be estimated. If, based on past experience, it is found that the April estimates of annual sales are reasonably accurate, then in future years the April forecast could be used to revise production schedules and maintain the correct inventory at the retail outlets.
Several factors appear to be related to sales, including the number of retail outlets in the region stocking the company's parts, the number of automobiles in the region registered as of April 1, and the total personal income for the first quarter of the year. Five independent variables were finally selected as being the most important (according to the sales manager). Then the data were gathered for a recent year. The total annual sales for that year for each region were also recorded. Note in the following table that for region 1 there were 1,739 retail outlets stocking the company's automotive parts, there were 9,270,000 registered automobiles in the region as of April 1 and so on. The sales for that year were $37,702,000.
Number of Average
Annual Number of Automobiles Personal Age of
Sales Retail Registered Income Automobiles Number of
($ millions), Outlets, (millions), ($ billions), (years) Supervisors
Y X1 X2 X3 X4 X5
37,702 1,739 9.27 85.4 3.5 9
24,196 1,221 5.86 60.7 5 5
32,055 1,846 8.81 68.1 4.4 7
3,611 120 3.81 20.2 4 5
17,625 1,096 10.31 33.8 3.5 7
45,919 2,290 11.62 95.1 4.1 13
29,600 1,687 8.96 69.3 4.1 15
8,114 241 6.28 16.3 5.9 11
20,116 649 7.77 34.9 5.5 16
12,994 1,427 10.92 15.1 4.1 10

a. Consider the following correlation matrix. Which single variable has the strongest correlation with the dependent variable? The correlations between the independent variables outlets and income and between cars and outlets are fairly strong. Could this be a problem? What is this condition called?
sales outlets cars inome 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

b. The output for all five variables is on the following page. What percent of the variation is explained by the regression equation?
The regression equation is
sales = -19.7 - 0.00063 outlets + 1.74 cars + 0.410 income + 2.04 age - 0.034 bosses
Predictor Coef StDev t-ratio
Constant -19.672 5.422 -3.63
outlets -0.000629 0.002638 -0.24
cars 1.7399 0.553 3.15
income 0.40994 0.04385 9.35
age 2.0357 0.8779 2.32
bosses -0.0344 0.188 -0.18

Analysis of variance
SOURCE DF SS MS
Regression 5 1593.81 318.76
Error 4 9.08 2.27
Total 9 1602.89

c. Conduct a global test of hypothesis to determine whether any of the regression coefficients are not zero. Use the .05 significance level.

d. Conduct a test of hypothesis on each of the independent variables. Would you consider eliminating "outlets" and "bosses"? Use the .05 significance level.
e. The regression has been rerun below with "outlets" and "bosses" eliminated. Compute the coefficient of determination. How much has R2 changed from the previous analysis?
The regression equation is
sales = -18.9 + 1.61 cars + 0.400 income + 1.96 age

Predictor Coef StDev t-ratio
Constant -18.924 3.636 -5.20
cars 1.6129 0.1979 8.15
income 0.40031 0.01569 25.52
age 1.9637 0.5846 3.36

Analysis of variance
SOURCE DF SS MS
Regression 3 1593.66 531.22
Error 6 9.23 1.54
Total 9 1602.89

f. Following is a histogram and a stem-and-leaf chart of the residuals. Does the normality assumption appear reasonable?
Histogram of residual N = 10 Stem-and-leaf of residual N = 10
Leaf Unit = 0.10
Midpoint Count
-1.5 1 * 1 -1 7
-1.0 1 * 2 -1 2
-0.5 2 ** 2 -0
0.0 2 ** 5 -0 440
0.5 2 ** 5 0 24
1.0 1 * 3 0 68
1.5 1 * 1 1

g. Following is a plot of the fitted values of Y (i.e. Y, and the residuals. Do you see any violations of the assumptions?
Multiple Regression and Correlation Analysis

Banner Mattress and Furniture Company wishes to study the number of credit applications received per day for the last 300 days.
Number of Credit Frequency
Applications (Number of Days)
0 50
1 77
2 81
3 48
4 31
5 or more 13

To interpret, there were 50 days on which no credit applications were received, 77 days on which only one application was received, and so on. Would it be reasonable to conclude that the population distribution is Poisson with a mean of 2.0? Use the .05 significance level. Hint: To find the expected frequencies use the Poisson distribution with a mean of 2.0. Find the probability of exactly one success given a Poisson distribution with a mean of 2.0. Multiply this probability by 300 to find the expected frequency for the number of days in which there was exactly one application. Determine the expected frequency for the other days in a similar manner.

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Solution Summary

The solution examines multiple standard error estimates. Minitab is used to run the data to determine the ANOVA.

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