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1. Simple Linear Regression Model:
Y
i
=
β
0
+
β
1
X
i
+
±
i
,
i
= 1
, ..., n
where:
Y
i
is the value of the response variable in the
i
th trial
β
0
and
β
1
are parameters
X
i
is a known constant, namely, the value of the independent variable in the
i
th trial
±
i
is an error term with mean 0 and variance
σ
2
;
±
i
and
±
j
are uncorrelated for
i
6
=
j
.
(Usually, we also assume that
±
i
is normally distributed.)
The least squares estimates of
β
1
and
β
0
are respectively given by
ˆ
β
1
=
∑
n
i
=1
(
X
i

¯
X
)(
Y
i

¯
Y
)
∑
n
i
=1
(
X
i

¯
X
)
2
and
ˆ
β
0
=
¯
Y

ˆ
β
1
¯
X.
where
¯
X
=
∑
n
i
=1
X
i
/n
and
¯
Y
=
∑
n
i
=1
Y
i
/n
.
The ﬁtted (or estimated) regression line is given by
ˆ
Y
=
ˆ
β
0
+
ˆ
β
1
X
For the observations in the sample, we will call
ˆ
Y
i
:
ˆ
Y
i
=
ˆ
β
0
+
ˆ
β
1
X
i
the ﬁtted value for the
i
th observation (
i
= 1
, ..., n
).
A natural measure of the eﬀect of
X
in reducing the variation in
Y
, i.e., the uncertainty
in predicting
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This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.
 Spring '08
 ?
 Linear Regression

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