1
/
1
2
2
k
n
R
k
R
F
Rejection Region:
F > F(k, n-(k+1)
Using the p-value approach:
Reject H
0
if p
value <
α
where the p value = P(F(k, n
–
(k+1)) > F)
Conditions:
1)
The error component
ε
is normally
distributed.
2)
The mean of
ε
is zero.
3)
The variance of
ε
is
σ
2
, a constant
for all values of x.
4)
The errors associated with different
observations are independent.
Multiple Regression Analysis
Inference in Multiple Regression
Testing the Utility of a Multiple Regression Model: The Global F
–
Test
Ho:
β
1
=
β
2
= … = β
k
= 0
Ha:
At least one of the
β
parameter is nonzero.
1
/
1
2
2
k
n
R
k
R
F
Rejection Region:
F > F(k, n-(k+1)
Using the p-value approach:
Reject H
0
if p
value <
α
where the p value = P(F(k, n
–
(k+1)) > F)
Conditions:
1)
The error component
ε
is normally
distributed.
2)
The mean of
ε
is zero.
3)
The variance of
ε
is
σ
2
, a constant
for all values of x.
4)
The errors associated with different
observations are independent.
Multiple Regression Analysis
Testing the Utility of a Multiple Regression Model: The Global F
–
Test
Example
:
For the electrical usage example,
Ho:
β
1
=
β
2
= … = β
k
= 0
Ha:
At least one of the
β
parameter is nonzero.
n = 10,
k = 2 gives n
–
(k+1) = 7 degrees of freedom.
At
α
= 0.05, we
reject
Ho:
β
1
=
β
2
= 0 if F
0.05
(2,7) = 4.74
From the computer printout, computed F = 53.88.
Since computed F >
4.74, we conclude that at least one of the model coefficient
β
1
and
β
2
is
nonzero.
1
/
1
2
2
k
n
R
k
R
F
Multiple Regression Analysis
Estimating and Testing Hypotheses about Individual Parameters
The hypothesis for making inferences about particular
β
parameters is as
follows:
Ho:
β
j
=
0 (j = 1, 2, … k)
H
1
:
β
j
>
0
Test Statistics:
Degrees of freedom:
(n
–
[k + 1])
Rejection Region:
t > t
α
(n
–
[k+1])
^
^
j
s
t
Multiple Regression Analysis
Test of an Individual Coefficient in the Multiple Regression Model
The hypothesis for making inferences about particular
β
parameters is as
follows:
a) Right-tailed test
Ho:
β
j
=
0 (j = 1, 2, … k)
H
1
:
β
j
>
0
Test Statistics:
Degrees of freedom:
(n
–
[k + 1])
Rejection Region:
t > t
α
(n
–
[k+1])
^
^
j
s
t