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multiple regression

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Qty. Price Advertise Distance
180 475 1100 120
590 400 2400 65
430 450 1500 50
250 550 3100 75
275 575 3400 45
720 375 2200 20
660 375 1200 50
490 450 2400 75
700 400 2200 45
210 500 1000 55

A firm's marketing dept. obtained data from 10 of the firm's outlets. The data consists of the qty. and price of the products sold at each outlet from the distribution center. Using the data assume the underlying demand relationship is linear function of price. Run a regression to obtain the least squares estimate of the demand (make sure you include the constant term). Provide a detailed economic interpretation of the regression outputs, including an estimate of the optimal price that each regression generates. Which regression is recommended for forecasting future sales and price?

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https://brainmass.com/economics/regression/multiple-regression-135089

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Please see the attached files.

The first thing we should always do is to set up scatter diagrams and investigate the associated correlation. The Scatterplot function can be easily generated using the Chart procedure in Excel. We can do this under Chart menu.

After copying the date, paste them into the Excel file, which I have created. You then select the whole area. Go to the Chart Wizard icon, select Scatter diagram, and the end result is presented in the 4 charts. In each of these charts, I've used the function fit trendline to estimate the ...

Solution Summary

Insight is given about multiple regression.

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Multiple Regression Models and Simple Linear Regression Models (21 Problems) : Least Squares, Durbin-Watson, Correlation Coefficient, Standard Error and p-Values

1. The y-intercept (b0) represents the
a. predicted value of Y when X = 0.
b. change in estimated average Y per unit change in X.
c. predicted value of Y.
d. variation around the sample regression line.

2. The least squares method minimizes which of the following?
a. SSR
b. SSE
c. SST
d. All of the above

TABLE 1
A candy bar manufacturer is interested in trying to estimate how sales are
influenced by the price of their product. To do this, the company randomly
chooses 6 small cities and offers the candy bar at different prices. Using
candy bar sales as the dependent variable, the company will conduct a
simple linear regression on the data below:

City Price ($) Sales
River Falls 1.30 100
Hudson 1.60 90
Ellsworth 1.80 90
Prescott 2.00 40
Rock Elm 2.40 38
Stillwater 2.90 32

Referring to Table 1, what is the estimated slope parameter for the
candy bar price and sales data?
a. 161.386
b. 0.784
c. -3.810
d. -48.193

4. Referring to Table 1, what is the percentage of the total variation in
candy bar sales explained by the regression model?
a. 100%
b. 88.54%
c. 78.39%
d. 48.19%

5. Referring to Table 1, what is the standard error of the estimate, SYX,
for the data?
a. 0.784
b. 0.885
c. 12.650
d. 16.299

6. Referring to Table 1, if the price of the candy bar is set at $2, the
predicted sales will be
a. 30
b. 65
c. 90
d. 100

7. If the Durbin-Watson statistic has a value close to 0, which
assumption is violated?
a. Normality of the errors.
b. Independence of errors.
c. Homoscedasticity.
d. None of the above.

8. If the Durbin-Watson statistic has a value close to 4, which
assumption is violated?
a. Normality of the errors.
b. Independence of errors.
c. Homoscedasticity.
d. None of the above.

9. If the correlation coefficient (r) = 1.00, then
a. the y-intercept (b0) must equal 0.
b. the explained variation equals the unexplained variation.
c. there is no unexplained variation.
d. there is no explained variation.

10. In a simple linear regression problem, r and b1
a. may have opposite signs.
b. must have the same sign.
c. must have opposite signs.
d. are equal.

11. The strength of the linear relationship between two numerical
variables may be measured by the
a. scatter diagram.
b. y-intercept.
c. slope.
d. coefficient of correlation.

12. The width of the prediction interval estimate for the predicted value
of Y is dependent on
a. the standard error of the estimate.
b. the value of X for which the prediction is being made.
c. the sample size.
d. All of the above.

TABLE 2
The following Excel tables are obtained when "Score received on an
exam (measured in percentage points)" (Y) is regressed on
"percentage attendance" (X) for 22 students in a Statistics for
Business and Economics course.

Regression Statistics
Multiple R 0.142620229
R Square 0.02034053
Adjusted R Square -0.028642444
Standard Error 20.25979924
Observations 22

Coefficients Standard Error t Stat p-value
Intercept 39.39027309 37.24347659 1.057642216 0.302826622
Attendance 0.340583573 0.52852452 0.644404489 0.526635689

13. Referring to Table 2, which of the following statements is true?
a. -2.86% of the total variability in score received can be
explained by percentage attendance.
b. -2.86% of the total variability in percentage attendance can
be explained by score received.
c. 2% of the total variability in score received can be explained
by percentage attendance.
d. 2% of the total variability in percentage attendance can be
explained by score received.

14. In a multiple regression problem involving two independent
variables, if b1 is computed to be +2.0, it means that
a. the relationship between X1 and Y is significant.
b. the estimated average of Y increases by 2 units for each
increase of 1 unit of X1, holding X2 constant.
c. the estimated average of Y increases by 2 units for each
increase of 1 unit of X1, without regard to X2.
d. the estimated average of Y is 2 when X1 equals zero.

15. In a multiple regression model, which of the following is correct
regarding the value of the adjusted r2?
a. It can be negative.
b. It has to be positive.
c. It has to be larger than the coefficient of multiple
determination.
d. It can be larger than 1.

16. A manager of a product sales group believes the number of sales
made by an employee (Y) depends on how many years that employee
has been with the company (X1) and how he/she scored on a business
aptitude test (X2). A random sample of 8 employees provides the
following:

TABLE 3
Employee Y X1 X2
1 100 10 7
2 90 3 10
3 80 8 9
4 70 5 4
5 60 5 8
6 50 7 5
7 40 1 4
8 30 1 1
Referring to Table 3, for these data, what is the value for the
regression constant, b0?
a. 0.998
b. 3.103
c. 4.698
d. 21.293

17. Referring to Table 3, if an employee who had been with the company
5 years scored a 9 on the aptitude test, what would his estimated
expected sales be?
a. 79.09
b. 60.88
c. 55.62
d. 17.98

TABLE 4
An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross
domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below.

SUMMARY OUTPUT

Regression Statistics
Multiple R 0.991
R Square 0.982
Adjusted R Square 0.976
Standard Error 0.299
Observations 10

ANOVA
Df SS MS F Signif F
Regression 2 33.4163 16.7082 186.325 0.0001
Residual 7 0.6277 0.0897
Total 9 34.0440

Coeff StdError t Stat P-value
Intercept -0.0861 0.5674 -0.152 0.8837
GDP 0.7654 0.0574 13.340 0.0001
Price -0.0006 0.0028 -0.219 0.8330
18. Referring to Table 4, when the economist used a simple linear
regression model with consumption as the dependent variable and
GDP as the independent variable, he obtained an r2 value of 0.971.
What additional percentage of the total variation of consumption
has been explained by including aggregate prices in the multiple
regression?
a. 98.2
b. 11.1
c. 2.8
d. 1.1

19. Referring to Table 4, what is the predicted consumption level for an
economy with GDP equal to $4 billion and an aggregate price index
of 150?
a. $1.39 billion
b. $2.89 billion
c. $4.75 billion
d. $9.45 billion

20. Referring to Table 4, to test for the significance of the coefficient on
aggregate price index, the value of the relevant t-statistic is
a. 2.365
b. 0.143
c. -0.219
d. -1.960

21. Referring to Table 4, to test whether gross domestic product has a
positive impact on consumption, the p-value is
a. 0.00005
b. 0.0001
c. 0.9999
d. 0.99995

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