Transcript Document

LECTURE 8
BUS 173
SIMPLE LINEAR REGRESSION



Simple Linear Regression Model
Least Squares Method
Coefficient of Determination

Model Assumptions
Testing for Significance
Using the Estimated Regression Equation
for Estimation and Prediction
Computer Solution

Residual Analysis: Validating Model Assumptions



SIMPLE LINEAR REGRESSION MODEL
 The equation that describes how y is related to x and
an error term is called the regression model.
 The simple linear regression model is:
Y = b0 + b1x +e
where:
b0 and b1 are called parameters of the model,
e is a random variable called the error term.
SIMPLE LINEAR
REGRESSION EQUATION

Positive Linear Relationship
E(Y)
Regression line
Intercept
b0
Slope b1
is positive
x
Simple Linear Regression Equation

Negative Linear Relationship
E(Y)
Intercept
b0
Regression line
Slope b1
is negative
x
Simple Linear Regression Equation

No Relationship
E(Y)
Regression line
Intercept
b0
Slope b1
is 0
x
Estimated Simple Linear Regression Equation

The estimated simple linear regression equation
yˆ  b0  b1 x
• The graph is called the estimated regression line.
• b0 is the y intercept of the line.
• b1 is the slope of the line.
• yˆ is the estimated value of Y for a given x value.
LEAST SQUARES METHOD
Slope for the Estimated Regression Equation
b1
( x  x )( y  y )


 (x  x )
i
i
2
i
Least Squares Method

y-Intercept for the Estimated Regression Equation
b0  y  b1 x
where:
xi = value of independent variable for ith
observation
yi = value of dependent variable for ith
_ observation
x = mean value for independent variable
_
y = mean value for dependent variable
n = total number of observations
Simple Linear Regression

Example: Kako Auto Sales
Kako Auto periodically has
a special week-long sale.
As part of the advertising
campaign Kako runs one or
more television commercials
during the weekend preceding the sale. Data from a
sample of 5 previous sales are shown on the next slide.
Simple Linear Regression

Example: Kako Auto Sales
Number of
TV Ads
1
3
2
1
3
Number of
Cars Sold
14
24
18
17
27
ESTIMATED REGRESSION EQUATION

Slope for the Estimated Regression Equation
b1
( x  x )( y  y ) 20



5
4
(x  x )
i
i
2
i

y-Intercept for the Estimated Regression Equation
b0  y  b1 x  20  5(2)  10
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Estimated Regression Equation
yˆ  10  5x
SCATTER DIAGRAM AND TREND LINE
30
Cars Sold
25
20
y = 5x + 10
15
10
5
0
0
1
2
TV Ads
3
4
COEFFICIENT OF DETERMINATION
Relationship Among SST, SSR, SSE
SST
=
SSR
+
SSE
2
2
2
ˆ
ˆ
(
y

y
)

(
y

y
)

(
y

y
)
 i
 i
 i i
where:
SST = total sum of squares
SSR = sum of squares due to regression
SSE = sum of squares due to error
Coefficient of Determination

The coefficient of determination is:
r2 = SSR/SST
where:
SSR = sum of squares due to regression
SST = total sum of squares
THANK YOU
End of Presentation