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Copyright  2007 by Oxford University Press, Inc.
Slide 1
1
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 2
Regression Analysis
Year
X
Y
1
10
44
2
9
40
3
11
42
4
12
46
5
11
48
6
12
52
7
13
54
8
13
58
9
14
56
10
15
60
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Scatter Diagram
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Slide 3
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Slide 4
Regression Analysis
• Regression Line: Line of Best Fit
• Regression Line: Minimizes the sum of
the squared vertical deviations (et) of
each point from the regression line.
• Ordinary Least Squares (OLS) Method
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Copyright  2007 by Oxford University Press, Inc.
Slide 5
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 6
Ordinary Least Squares (OLS)
Model:
Yt  a  bX t  et
€
Y€t  a€ bX
t
et  Yt  Y€t
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Slide 7
Ordinary Least Squares (OLS)
Objective: Determine the slope and
intercept that minimize the sum of
the squared errors.
n
n
n
t 1
t 1
t 1
2
2
€ )2
€
€
e

(
Y

Y
)

(
Y

a

bX
t  t t  t
t
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 8
Ordinary Least Squares (OLS)
Estimation Procedure
n
b€ 
(X
t 1
t
 X )(Yt  Y )
n
2
(
X

X
)
 t
€
a€  Y  bX
t 1
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 9
Ordinary Least Squares (OLS)
Estimation Example
Time
Xt
1
2
3
4
5
6
7
8
9
10
10
9
11
12
11
12
13
13
14
15
120
n  10
Yt
44
40
42
46
48
52
54
58
56
60
500
n
n
 X t  120
 Yt  500
t 1
n
X 
t 1
X t 120

 12
n
10
t 1
n
Yt 500

 50
10
t 1 n
Y 
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Xt  X
Yt  Y
-2
-3
-1
0
-1
0
1
1
2
3
-6
-10
-8
-4
-2
2
4
8
6
10
n
(X
t 1
t 1
( X t  X )2
12
30
8
0
2
0
4
8
12
30
106
4
9
1
0
1
0
1
1
4
9
30
t
 X ) 2  30
106
b€ 
 3.533
30
t
 X )(Yt  Y )  106
a€  50  (3.533)(12)  7.60
n
(X
( X t  X )(Yt  Y )
Copyright  2007 by Oxford University Press, Inc.
Slide 10
Ordinary Least Squares (OLS)
Estimation Example
n
X 
n  10
n
X
t 1
t
n
(X
t 1
t 1
 120
n
Y
t 1
t
n
 500
Yt 500

 50
10
t 1 n
Y 
t
 X )  30
106
€
b
 3.533
30
t
 X )(Yt  Y )  106
a€  50  (3.533)(12)  7.60
2
n
(X
t 1
X t 120

 12
n
10
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Copyright  2007 by Oxford University Press, Inc.
Slide 11
Tests of Significance
Standard Error of the Slope Estimate
sbˆ 
2
ˆ
 (Yt  Y )
(n  k ) ( X t  X )
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2

e
(n  k ) ( X
Copyright  2007 by Oxford University Press, Inc.
2
t
t
 X )2
Slide 12
Tests of Significance
Example Calculation
Y€t
et  Yt  Y€t
et2  (Yt  Y€t )2
( X t  X )2
44
42.90
1.10
1.2100
4
9
40
39.37
0.63
0.3969
9
3
11
42
46.43
-4.43
19.6249
1
4
12
46
49.96
-3.96
15.6816
0
5
11
48
46.43
1.57
2.4649
1
6
12
52
49.96
2.04
4.1616
0
7
13
54
53.49
0.51
0.2601
1
8
13
58
53.49
4.51
20.3401
1
9
14
56
57.02
-1.02
1.0404
4
10
15
60
60.55
-0.55
0.3025
9
65.4830
30
Time
Xt
Yt
1
10
2
n
n
 e   (Yt  Y€t )2  65.4830
t 1
2
t
t 1
(X
t 1
 (Y  Y€)
( n  k ) ( X  X )
2
n
 X )  30
2
t
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
sb€ 
t
2
t
Copyright  2007 by Oxford University Press, Inc.

65.4830
 0.52
(10  2)(30)
Slide 13
Tests of Significance
Example Calculation
n
n
t 1
t 1
2
2
€
e

(
Y

Y
)
 t  t t  65.4830
n
2
(
X

X
)
 30
 t
t 1
2
€
 (Yt  Y )
65.4830
sb€ 

 0.52
2
( n  k ) ( X t  X )
(10  2)(30)
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 14
Tests of Significance
Calculation of the t Statistic
b€ 3.53
t 
 6.79
sb€ 0.52
Degrees of Freedom = (n-k) = (10-2) = 8
Critical Value at 5% level =2.306
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 15
Tests of Significance
Decomposition of Sum of Squares
Total Variation = Explained Variation + Unexplained Variation
2
2
€
€
 (Yt  Y )   (Y  Y )   (Yt  Yt )
2
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 16
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 17
Tests of Significance
Coefficient of Determination
R2 
2
€
(
Y

Y
)

Explained Variation

2
TotalVariation
(
Y

Y
)
 t
373.84
R 
 0.85
440.00
2
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 18
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 19
Tests of Significance
Coefficient of Correlation
r  R 2 withthe sign of b€
1  r  1
r  0.85  0.92
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 20
Chapter 4 Appendix
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Slide 21
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 22
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 23
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 24
Getting Started
• Install the Analysis ToolPak add-in from
the Excel installation media if it has not
already been installed
• Attach the Analysis ToolPak add-in
– From the menu, select Tools and then
Add-Ins...
– When the Add-Ins dialog appears, select
Analysis ToolPak and then click OK.
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
Copyright  2007 by Oxford University Press, Inc.
Slide 25
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 26
Entering Data
• Data on each variable must be entered
in a separate column
• Label the top of each column with a
symbol or brief description to identify
the variable
• Multiple regression analysis requires
that all data on independent variables
be in adjacent columns
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Copyright  2007 by Oxford University Press, Inc.
Slide 27
Example Data
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Slide 28
Running the Regression
• Select the Regression tool from the
Analysis ToolPak dialog
– From the menu, select Tools and then
Data Analysis...
– On the Data Analysis dialog, scroll down
the list of Analysis Tools, select
Regression, and then click OK
– The Regression tool dialog will then be
displayed
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Slide 29
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 30
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 31
Select the Data Ranges
• Type in the data range for the Y variable
or select the range on the worksheet
• Type in the data range for the X
variable(s) or select the range on the
worksheet
• If your ranges include the data labels
(recommended) then check the labels
option
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 32
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Slide 33
Select an Output Option
• Output to a selected range
– Selection is the upper left corner of the
output range
• Output to a new worksheet
– Optionally enter a name for the worksheet
• Output to a new workbook
• And then click OK
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Slide 34
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 35
Regression Output
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Slide 36
Multiple Regression Data
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Slide 37
PowerPoint Slides Prepared by Robert F. Brooker, Ph.D.
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Slide 38
Regression Output
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Slide 39