Matakuliah : I0174 – Analisis Regresi
Tahun
: Ganjil 2007/2008
Regresi Linear Sederhana
Pertemuan 01
Regresi Linier Sederhana
Simple Linear Regression
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Materi
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Types of Regression Models
Determining the Simple Linear Regression Equation
Measures of Variation
Assumptions of Regression and Correlation
Residual Analysis
Measuring Autocorrelation
Inferences about the Slope
Chapter Topics
(continued)
• Correlation - Measuring the Strength of the Association
• Estimation of Mean Values and Prediction of Individual Values
• Pitfalls in Regression and Ethical Issues
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Purpose of Regression Analysis
• Regression Analysis is Used Primarily to Model Causality and
Provide Prediction
– Predict the values of a dependent (response) variable based on values of at
least one independent (explanatory) variable
– Explain the effect of the independent variables on the dependent variable
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Types of Regression Models
Positive Linear Relationship
Negative Linear Relationship
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Relationship NOT Linear
No Relationship
Simple Linear Regression Model
• Relationship between Variables is Described by a Linear Function
• The Change of One Variable Causes the Other Variable to Change
• A Dependency of One Variable on the Other
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Simple Linear Regression Model
(continued)
Population regression line is a straight line that
describes the dependence of the average value
(conditional mean) of one variable on the other
Population
Slope
Coefficient
Population
Y Intercept
Dependent
(Response)
Variable
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Random
Error
Yi      X i   i
Population
Regression
Y |X
Line
(Conditional Mean)

Independent
(Explanatory)
Variable
Simple Linear Regression Model
Y
(Observed Value of Y) = Yi
(continued)
     X i   i
 i = Random Error

Y | X      X i

(Conditional Mean)
X
Observed Value of Y
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Linear Regression Equation
Sample regression line provides an estimate of
the population regression line as well as a
predicted value of Y
Sample
Y Intercept
Yi  b0  b1 X i  ei
Ŷ  b0  b1 X
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Sample
Slope
Coefficient
Residual
Simple Regression Equation
(Fitted Regression Line, Predicted Value)
Linear Regression Equation
• b0 and b1 are obtained by finding the values of
minimize the sum of the squared residuals

n
i 1
Yi  Yˆi
  e
2
n
i 1
• b0 provides an estimate of  
• b1 provides an estimate of 
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2
i
(continued)
b0 and b1that
Linear Regression Equation
Yi  b0  b1 X i  ei
Y
ei
(continued)
Yi      X i   i
b1
i

Y | X      X i

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b0
Observed Value
Yˆi  b0  b1 X i
X
Interpretation of the Slope
and Intercept
  E Y | X  0 is the average value of Y when the value of X is
•
zero
change in E Y | X 
• 1 
measures the change in the average value
change in X
of Y as a result of a one-unit change in X
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Interpretation of the Slope
and Intercept
•
(continued)
b  Eˆ Y | X  0  is the estimated average value of Y when the
value of X is zero
change in Eˆ Y | X 
• b1 
is the estimated change in the average
change in X
value of Y as a result of a one-unit change in X
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Simple Linear Regression: Example
You wish to examine
the linear dependency
of the annual sales of
produce stores on their
sizes in square footage.
Sample data for 7
stores were obtained.
Find the equation of
the straight line that
fits the data best.
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Store
Square
Feet
Annual
Sales
($1000)
1
2
3
4
5
6
7
1,726
1,542
2,816
5,555
1,292
2,208
1,313
3,681
3,395
6,653
9,543
3,318
5,563
3,760
Scatter Diagram: Example
Annua l Sa le s ($000)
12000
10000
8000
6000
4000
2000
0
0
1000
2000
3000
4000
S q u a re F e e t
Excel Output
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5000
6000
Simple Linear Regression Equation: Example
Yˆi  b0  b1 X i
 1636.415  1.487 X i
From Excel Printout:
C o e ffi c i e n ts
I n te r c e p t
1 6 3 6 .4 1 4 7 2 6
X V a ria b le 1 1 .4 8 6 6 3 3 6 5 7
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Annua l Sa le s ($000)
Graph of the Simple Linear Regression Equation:
Example
12000
10000
8000
6000
4000
2000
0
0
1000
2000
3000
4000
S q u a re F e e t
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5000
6000
Interpretation of Results: Example
Yˆi  1636.415  1.487 X i
The slope of 1.487 means that for each increase of
one unit in X, we predict the average of Y to
increase by an estimated 1.487 units.
The equation estimates that for each increase of 1
square foot in the size of the store, the expected
annual sales are predicted to increase by $1487.
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Chapter Summary
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•
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•
•
•
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Introduced Types of Regression Models
Discussed Determining the Simple Linear Regression Equation
Described Measures of Variation
Addressed Assumptions of Regression and Correlation
Discussed Residual Analysis
Addressed Measuring Autocorrelation
Chapter Summary
(continued)
• Described Inference about the Slope
• Discussed Correlation - Measuring the Strength of the Association
• Addressed Estimation of Mean Values and Prediction of Individual
Values
• Discussed Pitfalls in Regression and Ethical Issues
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