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Part 8
Linear Regression
1.
Simple Linear Regression Models
2.
Multiple Linear Regression Models
3.
F
Tests
4.
Prediction and Cautions
5.
Further Inference about Coefficients of
Individual Independent Variables
Section 8.1
Simple Linear Regression
Models
Example
(Reed Auto)
Reed Auto periodically has a special
weeklong sale. As part of the advertising
campaign Reed runs one or more television
commercials during the weekend preceding
the sale.
Data from a sample of 5 previous
sales are shown below.
Week
1 2 3 4 5
Number of TV Ads
1 3
2
1
3
Number of Cars Sold
14 24 18 17 27
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By looking at the
scatter diagram
, we can
observe that there exists a strong linear
relationship between the number of TV ads
(the
independent variable
,
explanatory
variable
or
predictor
) and the number of
cars sold (the
dependent variable
or
response variable
). We may draw more than
one straight line through the scatter diagram.
0
10
20
30
123
Number
of TV Ads
Number of Cars Sold
Number
of TV Ads
Number of Cars Sold
0
10
20
30
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Which straight line do you think is the best?
Simple linear regression model
Assumption
1.
For
i
=
1, 2, …,
n
,
Y
i
follows
N(
β
0
+
1
x
i
,
σ
2
),
where
0
and
1
are constants and
x
i
’s are
non
-random.
n
is called the number of
observations.
2.
Y
1
,
Y
2
, …,
Y
n
are independent.
Definition
For
n
pairs of data (
x
1
,
y
1
), (
x
2
,
Density
of
Y
1
Density
of
Y
2
Same shape
y
x
x
2
x
1
0
Regression line
y
=
0
+
1
x
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y
2
), …, (
x
n
,
y
n
), let
∑
=
=
n
i
i
x
n
x
1
1
,
∑
=
=
n
i
i
y
n
y
1
1
,
b
1
=
()
∑
∑
=
=
−
−
−
n
i
i
n
i
i
i
x
x
y
y
x
x
1
2
1
and
b
0
=
x
b
y
1
−
.
Then
y
=
b
0
+
b
1
x
is called the
estimated
regression equation
.
Let
i
y
ˆ
=
b
0
+
b
1
x
i
for
i
= 1, 2, …,
n
.
Theorem
1.
b
0
and
b
1
meet the
least squares criterion
.
x
0
x
i
y
(
x
i
,
y
i
)
i
i
y
y
ˆ
−
y
=
b
0
+
b
1
x
y
i
i
y
ˆ