Learning Objectives

1. Understand how multiple regression analysis can be used to develop relationships involving one dependent variable and several independent variables.

2. Be able to interpret the coefficients in a multiple regression analysis.

3. Know the assumptions necessary to conduct statistical tests involving the hypothesized regression model.

4. Understand the role of computer packages in performing multiple regression analysis.

5. Be able to interpret and use computer output to develop the estimated regression equation.

6. Be able to determine how good a fit is provided by the estimated regression equation.

7. Be able to test for the significance of the regression equation.

8. Understand how multicollinearity affects multiple regression analysis.

9. Know how residual analysis can be used to make a judgement as to the appropriateness of the model, identify outliers, and determine which observations are influential.

10. Understand how logistic regression is used for regression analyses involving a binary dependent variable.

Solutions:

1. a. b1 = .5906 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2 is held constant.

b2 = .4980 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1 is held constant.

b. [pic]= 29.1270 + .5906(180) + .4980(310) = 289.82

2. a. The estimated regression equation is

[pic]= 45.06 + 1.94x1

An estimate of y when x1 = 45 is

[pic]= 45.06 + 1.94(45) = 132.36

b. The estimated regression equation is

[pic]= 85.22 + 4.32x2

An estimate of y when x2 = 15 is

[pic]= 85.22 + 4.32(15) = 150.02

c. The estimated regression equation is

[pic]= -18.37 + 2.01x1 + 4.74x2

An estimate of y when x1 = 45 and x2 = 15 is

[pic]= -18.37 + 2.01(45) + 4.74(15) = 143.18

3. a. b1 = 3.8 is an estimate of the change in y corresponding to a 1 unit change in x1 when x2, x3, and x4 are held constant.

b2 = -2.3 is an estimate of the change in y corresponding to a 1 unit change in x2 when x1, x3, and x4 are held constant.

b3 = 7.6 is an estimate of the change in y corresponding to a 1 unit change in x3 when x1, x2, and x4 are held constant.

b4 = 2.7 is an estimate of the change in y corresponding to a 1 unit change in x4 when x1, x2, and x3 are held constant.

b. [pic]= 17.6 + 3.8(10) – 2.3(5) + 7.6(1) + 2.7(2) = 57.1

4. a. [pic]= 25 + 10(15) + 8(10) = 255; sales estimate: $255,000

b. Sales can be expected to increase by $10 for every dollar increase in inventory investment when advertising expenditure is held constant. Sales can be expected to increase by $8 for every dollar increase in advertising expenditure when inventory investment is held constant.

5. a. The Minitab output is shown below:

The regression equation is Revenue = 88.6 + 1.60 TVAdv

Predictor Coef SE Coef T P Constant 88.638 1.582 56.02 0.000 TVAdv 1.6039 0.4778 3.36 0.015

S = 1.215 R-Sq = 65.3% R-Sq(adj) = 59.5%

Analysis of Variance

Source DF SS MS F P Regression 1 16.640 16.640 11.27 0.015 Residual Error 6 8.860 1.477 Total 7 25.500

b. The Minitab output is shown below:

The regression equation is Revenue = 83.2 + 2.29 TVAdv + 1.30 NewsAdv

Predictor