1. Business Statistics, 4e by Ken Black
Chapter 14
Multiple Regression Analysis
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

2. Learning Objectives Develop a multiple regression model.
Understand and apply significance tests of the regression model and its coefficients.
Compute and interpret residuals, the standard error of the estimate, and the coefficient of determination.
Interpret multiple regression computer output.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 2

3. Regression Models
Probabilistic Multiple Regression Model Y = 0 + 1X1 + 2X2 + 3X kXk+ 
Y = the value of the dependent (response) variable
0 = the regression constant
1 = the partial regression coefficient of independent variable 1
2 = the partial regression coefficient of independent variable 2
k = the partial regression coefficient of independent variable k
k = the number of independent variables
 = the error of prediction
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4

4. Estimated Regression Model
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 5

5. Multiple Regression Model with Two Independent Variables (First-Order)
Population
Model
Estimated
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 6

6. Response Plane for First-Order Two-Predictor Multiple Regression ModelY 
Vertical Intercept Y1
Response Plane X2 X1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 7

7. Least Squares Equations for k = 2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 8

8. Real Estate Data Observation Y X1 X2 1 63.0 65.1 1,605 35 13 79.7
2,121 14 2
2,489 45
84.5
2,485 9 3
69.9 7
1,553 20 15
96.0
2,300 19 4
76.8
2,404 32 16
109.5
2,714 5
73.9
1,884 25 17
102.5
2,463 6
77.9
1,558 18
121.0
3,076
74.9
1,748 8
104.9
3,048
78.0
3,105 10
128.0
3,267
79.0
1,682 28 21
129.0
3,069
63.4
2,470 30 22
117.9
4,765 11
79.5
1,820 23
140.0
4,540 12
83.9
2,143
Market
Price
($1,000)
Square
Feet
Age
(Years)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 9

9. MINITAB Output for the Real Estate Example
The regression equation is
Price = Sq.Feet Age
Predictor Coef StDev T P
Constant
Sq.Feet
Age
S = R-Sq = 74.1% R-Sq(adj) = 71.5%
Analysis of Variance
Source DF SS MS F P
Regression
Residual Error
Total
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 11

10. Predicting the Price of Home
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 12

11. Evaluating the Multiple Regression Model
Significance
Tests for
Individual
Regression
Coefficients
Testing
the
Overall
Model
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13

12. Testing the Overall Model for the Real Estate Example
ANOVA df SS MS F p
Regression 2
28.63
.000
Residual (Error) 20
143.1
Total 22
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 14

13. Significance Test of the Regression Coefficients for the Real Estate Example
tCal = 5.63 > 2.086, reject H0.
Coefficients
Std Dev
t Stat p
x1 (Sq.Feet)
0.0177
5.63
.000
x2 (Age)
-0.666
0.2280
-2.92
.008
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 15

14. Residuals and Sum of Squares Error for the Real Estate Example
SSE
Observation Y 1
43.0
42.466
0.534
0.285 13
59.7
65.602
-5.902
34.832 2
45.1
51.465
-6.365
40.517 14
64.5
75.383 3
49.9
51.540
-1.640
2.689 15
76.0
65.442
10.558 4
56.8
58.622
-1.822
3.319 16
89.5
82.772
6.728
45.265 5
53.9
54.073
-0.173
0.030 17
82.5
77.659
4.841
23.440 6
57.9
55.627
2.273
5.168 18
101.0
87.187
13.813 7
54.9
62.991
-8.091
65.466 19
84.9
89.356
-4.456
19.858 8
58.0
85.702 20
108.0
91.237
16.763 9
59.0
48.495
10.505 21
109.0
85.064
23.936 10
63.4
61.124
2.276
5.181 22
97.9 11
59.5
68.265
-8.765
76.823 23
120.0
7.540
56.854 12
63.9
71.322
-7.422
55.092
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 16

15. MINITAB Residual Diagnostics for the Real Estate Problem3 2 1 - 6 5 4 R e s i d u a l F r q n c y H t o g m f O b v N I C h X = 7 . E S L 9 8 P M D
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

16. SSE and Standard Error of the Estimate
ANOVA df SS MS F P
Regression 2
8189.7
4094.9
28.63
.000
Residual (Error) 20
2861.0
143.1
Total 22
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 20

17. Coefficient of Multiple Determination (R2)
SSE
ANOVA df SS MS F p
Regression 2
8189.7
28.63
.000
Residual (Error) 20
2861.0
143.1
Total 22
SSYY
SSR
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 21

18. Adjusted R2 SSYY SSE n-k-1 n-1 ANOVA df SS MS F p Regression 2 8189.7
4094.9
28.63
.000
Residual (Error) 20
2861.0
143.1
Total 22
SSYY
SSE
n-k-1
n-1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 22

19. Demonstration Problem 14.1: Freight Data
Country
Freight Cargo Shipped by Road (Million Short-Ton Miles)
Length 0f Roads (Miles)
Number of Commercial Vehicles
China
278,806
673,239
5,010,000
Brazil
178,359
1,031,693
1,371,127
India
144,000
1,342,000
1,980,000
Germany
138,975
395,367
2,923,000
Italy
125,171
188,597
2,745,500
Spain
105,824
206,271
2,859,438
Mexico
96,049
157,036
3,758,034
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.

20. Demonstration Problem 14.1: Excel Output
Regression Statistics
Multiple R
0.812
R Square
0.659
Adjusted R Square
0.488
Standard Error
Observations 7
ANOVA df SS MS F
Sig. F
Regression 2
7.57E+09
3.86
0.116
Residual 4
1.96E+09
Total 6
Coefficients
t Stat
P-value
Intercept
-0.39
0.716
Length 0f Roads
2.34
0.079
Commercial Vehicles
2.39
0.075
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.