parameters
0
β
,
1
,
2
,….
.,
p
, and σ.
The sample has
n
observations.
Perform the multiple regression procedure on the data from the
n
observations.
0
1
2
,
,
,......
,
p
b
b b
b
denote the estimators of the population parameters
0
,
1
,
2
,….
.,
p
Another notation is
j
b
, the
j
th estimator of
j
, the
j
th population parameter,
where
j =
0, 1, 2, ….,
p,
and
p
is the number of explanatory variables in the
model.
For the ith observation, the predicted response is:
$
0
1 1
2
2
....
i
i
i
p ip
y
b
b x
b x
b x
=
+
+
+
+
The ith residual, the difference between the observed and predicted response
is:
i
e
= observed response – predicted response =
$
i
i
y
y

The method of least squares minimizes:
2
1
( )
n
i
i
e
=
∑
, or
2
(
)
i
i
y
y

∑
$
The parameter
2
σ
measures the variability of the response about the
regression equation.
It is estimated by:
2
2
1
i
s
e
n p
=
∑
 
The quantity
1 is the degree of freedom associated with
2
s
.
To determine/confirm which explanatory variables have strong relationships,
look at the slope tests and the ANOVA.
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