1. CHAPTER 17 Model Building
to accompany
Introduction to Business Statistics
fourth edition, by Ronald M. Weiers
Presentation by Priscilla Chaffe-Stengel
Donald N. Stengel
© 2002 The Wadsworth Group

2. Chapter 17 - Learning Objectives
Build polynomial regression models to describe curvilinear relationships
Apply qualitative variables representing two or three categories.
Use logarithmic transforms in constructing exponential and multiplicative models.
Identify and compensate for multicollinearity
Apply stepwise regression
Select the most suitable among competing models
© 2002 The Wadsworth Group

3. Polynomial Models with One Quantitative Predictor Variable
Simple linear regression equation:
Equation for second-order polynomial model:
Equation for third-order polynomial model:
Equation for general polynomial model:
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4. Polynomial Models with Two Quantitative Predictor Variables
First-order model with no interaction:
First-order model with interaction:
Second-order model with no interaction:
Second-order model with interaction:
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5. Models with Qualitative Variables
Equation for a model with a categorical independent variable with two possible states:
where state 1 is shown x = 1
where state 2 is shown x = 0
Equation for a model with a categorical independent variable with three possible states:
where state 1 is shown x1 = 1, x2 = 0
where state 2 is shown x1 = 0, x2 = 1
Where state 3 is shown x1 = 0, x2 = 0
© 2002 The Wadsworth Group

6. Models with Data Transformations
Exponential Model:
General equation for an exponential model:
Corresponding linear regression equation for an exponential model:
Multiplicative Model:
General equation for a multiplicative model:
Corresponding linear regression equation for a multiplicative model:
© 2002 The Wadsworth Group

7. Example, Problem 17.8
International Data Corporation has reported the following costs per gigabyte of hard drive storage space for years 1995 through Using x = 1 through 6 to represent years 1995 through 2000, fit a second-order polynomial model to the data and estimate the cost per gigabyte for the year 2008.
The regression equation
will have the form:
Year
x = Yr
y = Cost
1995 1
$261.84
1996 2
137.94
1997 3
69.68
1998 4
29.30
1999 5
13.09
2000 6
6.46
© 2002 The Wadsworth Group

8. Example, Problem 17.8, cont. Microsoft Excel Output SUMMARY OUTPUT
Regression
Statistics
Multiple R
R Square
Adj R Square
Standard Error
Observations 6
© 2002 The Wadsworth Group

9. Example, Problem 17.8, cont. Microsoft Excel Output
The regression equation is:
Coefficients
Standard
Error
t Stat
P-value
Intercept x
x^2
© 2002 The Wadsworth Group

10. Example, Problem 17.8, cont.
To estimate the cost per gigabyte for the year 2008, evaluate when x = 14.
So the cost per gigabyte in 2008 is estimated to be $
Does this make sense? Of course not.
Explanation: Although the polynomial equation provides a good fit for the data during the period , this form is not appropriate to extrapolate the data out to 2008.
© 2002 The Wadsworth Group

11. Example, Problem 17.32
An exponential model will probably be more appropriate to the data used in Problem 17.8. y
Log y x
$261.84 1
137.94 2
69.68 3
29.30 4
13.09 5
6.46 6
© 2002 The Wadsworth Group

12. Example, Problem 17.32, cont. Microsoft Excel Output SUMMARY OUTPUT
Regression
Statistics
Multiple R
R Square
Adj R Square
Standard Error
Observations 6
© 2002 The Wadsworth Group

13. Example, Problem 17.32, cont. Microsoft Excel Output
The regression equation is:
Coefficients
Standard
Error
t Stat
P-value
Intercept
8.12E-08 x
1.82E-06
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14. Example, Problem 17.32, cont. For x = 14,
Based on the exponential model, the cost per gigabyte in 2008 will be $0.0154, or just under 2 cents.
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15. Example, Problem 17.27
An efficiency expert has studied 12 employees who perform similar assembly tasks, recording productivity (units per hour), number of years of experience, and which one of three popular assembly methods the individual has chosen to use in performing the task. Given the data, shown on the next slide, determine the linear regression equation for estimating productivity based on the other variables. For any qualitative variables that are used, be sure to specify the coding strategy each will employ.
© 2002 The Wadsworth Group

16. Example, Problem 17.27, cont. 1 75 7 A 97 12 B 2 88 10 C 8 85 3 91 4 9
Worker
Prod.
Yrs.Exp
Method 1 75 7 A 97 12 B 2 88 10 C 8 85 3 91 4 9
102 93 5 13 95 11
112 6 77 86 14
© 2002 The Wadsworth Group

17. Example, Problem 17.27, cont.
The equation for a model with one quantitative variable and a categorical independent variable with three possible states is:
where x1 represents the years of experience
where state 1 is shown x2 = 1 if method A is used, 0 if otherwise
where state 2 is shown x3 = 1 if method B is used, 0 if otherwise
where state 3 is shown x2 = 0 and x3 = 0 if method C is used.
© 2002 The Wadsworth Group

18. Example, Problem 17.27, cont. So the data to be analyzed are: Worker y75 7 2 88 10 3 91 4 93 5 95 11 6 77
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19. Example, Problem 17.27, cont. Worker y x1 x2 x3 7 97 12 1 8 85 10 91 8 85 10 9
102 93 13 11
112 86 14
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20. Example, Problem 17.27, cont. Microsoft Excel Output SUMMARY OUTPUT
Regression
Statistics
Multiple R
R Square
Adj R Square
Standard Error
Observations 12
© 2002 The Wadsworth Group

21. Example, Problem 17.27, cont. Microsoft Excel Output
The regression equation is:
Coefficients
Standard
Error
t Stat
P-value
Intercept
2.214E-06 x1 x2 x3
© 2002 The Wadsworth Group

22. Example, Problem 17.27, cont.
The regression equation has an adjusted R-square of This indicates that the regression model provides a reasonable explanation for the variation in the data set.
Only the coefficient for x1 is significant at the 0.05 level. One might consider removing the assembly method from the model.
© 2002 The Wadsworth Group