The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

(5.7.3)
11
as shown diagrammatically in Figure 5.3. 
In practice, there is no need to estimate Eq. (5.7.2) explicitly. One can compute the
t
value in the middle of the double inequality given by Eq. (5.7.1) and see whether it lies
between the critical 
t
values or outside them. For our example,
t
=
0
.
7240
−
0
.
5
0
.
0700
=
3
.
2
(5.7.4)
which clearly lies in the critical region of Figure 5.4. The conclusion remains the same;
namely, we reject
H
0
.
Pr [
β
∗
2
−
t
α/
2
se (
ˆ
β
2
)
≤ ˆ
β
2
≤
β
∗
2
+
t
α/
2
se (
ˆ
β
2
)]
=
1
−
α
Density
f
(
2
)
β
Critical 
region
2.5%
b2
= 0.7240
lies in this 
critical region
2.5%
b
2
β
0.3460
0.5
0.6540
ˆ
2
β
FIGURE 5.3
The 95% confidence
interval for 
ˆ
β
2
under
the hypothesis that 
β
2
=
0.5.
11
In Sec. 5.2, point 4, it was stated that we 
cannot
say that the probability is 95 percent that the fixed
interval (0.5700, 0.8780) includes the true 
β
2
. But we can make the probabilistic statement given in
Eq. (5.7.3) because 
ˆ
β
2
, being an estimator, is a random variable.
guj75772_ch05.qxd 07/08/2008 12:46 PM Page 116

---

Chapter 5
Two-Variable Regression: Interval Estimation and Hypothesis Testing
117
Notice that if the estimated 
β
2
(
= ˆ
β
2
) is equal to the hypothesized 
β
2
, the 
t
value in Equa-
tion 5.7.4 will be zero. However, as the estimated 
β
2
value departs from the hypothesized 
β
2
value, 
|
t
|
(that is, the absolute 
t
value; 
note: t
can be positive as well as negative) will be in-
creasingly large. 
Therefore, a “large” 
|
t
|
value will be evidence against the null hypothesis.
Of
course, we can always use the 
t
table to determine whether a particular 
t
value is large or small;
the answer, as we know, depends on the degrees of freedom as well as on the probability of
Type I error that we are willing to accept. If you take a look at the 
t
table given in 
Appendix D
(Table D.2), you will observe that for any given value of df the probability of obtaining an
increasingly large 
|
t
|
value becomes progressively smaller. Thus, for 20 df the probability of
obtaining a
|
t
|
value of 1.725 or greater is 0.10 or 10 percent, but for the same df the probabil-
ity of obtaining a 
|
t
|
value of 3.552 or greater is only 0.002 or 0.2 percent.
Since we use the 
t
distribution, the preceding testing procedure is called appropriately
the