# Multiple Regression

Thompson Machine Works purchased several new, highly sophisticated machines. The production department needed some guidance with respect to qualifications needed by an operator. Four variables were listed.

X1 = Length of time employee was a machinist

X2 = Mechanical aptitude test score

X3 = Prior on -the-job rating

X4 = Age

Performance on the machine is designated Y

The following are the results of the regression analysis;

Independent

Variable Coefficient t-stat

Intercept 11.600 2.95

X1 0.400 1.23

X2 0.286 3.71

X3 0.112 4.10

X4 0.002 1.86

a. Write out the multiple regression equation.

b. How many independent variables are there?

c. Would you consider eliminating any of the independent variables?

Assume a very large sample size with the degrees of freedom being greater than 60.

2. Suppose the regression equation that has been used to estimate the value of existing homes is as follows;

Value = 10,000 + 50X1 + 5X2 + 10,000X3

Where;

X1 is square feet of livable area

X2 is the size of the lot measured in square feet

X3 is an indicator variable for the presence of a pool (1 if yes, 0 if no)

a. Interpret the meaning of the slopes of this equation.

b. Estimate the value of a home with 2,000 square feet of livable area, 10,000 square foot lot, and a pool.

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#### Solution Preview

1.a. Write out the multiple regression equation.

The multiple regression equation is

Y = 11.6 + 0.4 X1 + 0.286 X2 + 0.112 X3 + 0.002 X4

b. How many independent variables are there?

There are four independent variables and they are X1, X2, X3 and X4.

c. Would you consider eliminating any of the independent variables?

Assume a very large sample size with the degrees of freedom being greater than 60.

Since the sample size is large we can approximate the distribution of the test statistic for testing the significance of the coefficient of the independent variable to a Standard Normal distribution. ...

#### Solution Summary

The explanation of the output of a multiple regression analysis is discussed in the solution.

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