critical
critical
t
for one-sided test, and t
for two-sided test.
critical
critical
n K ,
n K ,
/ 2
t
t
K is the number
of the parameters in the model.
Then compare the test statistic with the test critical values.
At last, reject H
0
if the absolute value of t-statistic is larger than the critical value, i.e., t-statistic>
c
ritical
.
Example: A consultant believes that on other similar neighbourhood households will spend an
additional $7.50 on food per $100 additional income.
1. Set up the hypothesis
0
2
1
2
H :
7.5
against
H :
7.5
(2.28)
2. Find the test statistic
2
2
2
2
7.5
10.21
7.5
1.29
(
)
2.09
2.09
b
b
t
se b
(2.29)
3 Find the critical value at
5
and
( 0.025,40
2 )
( 0.975,40
2 )
.
0.0 .
t
2.024
t
2.024

27
4.
Since -2.204 <1.29 < 2.204 we do not reject the null hypothesis that =7.5. The sample data are
consistent with the conjecture households will spend an additional $7.50 on food per additional
$100 income increase.
The following summarizes the rejection regions for one sided test.
a) When testing the null hypothesis
H
0
:
β
k
=
c
against the alternative hypothesis
H
1
:
β
k
<
c
, reject
the null hypothesis and accept the alternative hypothesis if
t
≤
t
(1-
α
;
N
-2).
Figure 8 Rejection region for a one-tail test of
H
0
:
β
k
=
c
against
H
1
:
β
k
<
c
b) When testing the null hypothesis
H
0
:
β
k
=
c
against the alternative hypothesis
H
1
:
β
k
>
c
, reject
the null hypothesis and accept the alternative hypothesis if
t
≥
t
(1-
α
;
N
-2) .
Figure 9 Rejection region for a one-tail test of
H
0
:
β
k
=
c
against
H
1
:
β
k
>
c
c) When testing the null hypothesis
H
0
:
β
k
=
c
against the alternative hypothesis
H
1
:
β
k
≠
c
, reject
the null hypothesis and accept the alternative hypothesis if
t
≤
t
(1-
α
;
N
-2)
or
t
≥
t
(1-
α
;
N
-2) .

28
Figure 10 Rejection region for a test of
H
0
:
β
k
=
c
against
H
1
:
β
k
≠
c
12.2 The p-value
It has become standard practice to report the probability value of the test (p-value for an
abbreviation) when reporting the outcome of statistical hypothesis tests. If we have the p-
value of a test,
p
, we can determine the outcome of the test by comparing the p-value to the
chosen level of significance, α, without looking up or calculating the critical values.
p-Value rule states that Reject the null hypothesis when the
p-
value is less than, or equal to,
the level of significance α. That is, if
p
≤ α then reject
H
0
. If
p
> α then do not reject
H
0
.
If t is the calculated value of the t-statistic, then:
if H1: β
K
> c
p = probability to the right of t
if H1: β
K
< c
p = probability to the left of t
if H1: β
K
≠ c
p = sum of probabilities to
the right of |t| and to the left of
–
|t|
Figure 11 The p-value for a two-tail test of significance
2
2
7.5
10.21
7.5
1.29
se
2.09
b
t
b
20
.
0
29
.
1
29
.
1
38
38
t
P
t
P
p

29
If we choose α =0.05, α =0.10 or even α =0.20, we will fail to reject the null hypothesis,
because p> α.

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