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

(5.4.4)
The interpretation of this interval is:
If we establish 95 percent confidence limits on 
σ
2
and if we maintain a priori that these limits will include the true 
σ
2
, we will be right in the
long run 95 percent of the time.
Pr
(
n
−
2)
ˆ
σ
2
χ
2
α/
2
≤
σ
2
≤
(
n
−
2)
ˆ
σ
2
χ
2
1
−
α/
2
=
1
−
α
f
(
χ
2
)
χ
2
Density
95%
2.5%
2.5%
21.9200
3.8157
χ
2
0.025
χ
2
0.975
FIGURE 5.1
The 95% confidence
interval for 
χ
2
(11 df ).
6
For proof, see Robert V. Hogg and Allen T. Craig, 
Introduction to Mathematical Statistics,
2d ed.,
Macmillan, New York, 1965, p. 144.
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Chapter 5
Two-Variable Regression: Interval Estimation and Hypothesis Testing
113
5.5
Hypothesis Testing: General Comments
Having discussed the problem of point and interval estimation, we shall now consider the
topic of hypothesis testing. In this section we discuss briefly some general aspects of this
topic;
Appendix A
gives some additional details.
The problem of statistical hypothesis testing may be stated simply as follows: 
Is a given
observation or finding compatible with some stated hypothesis or not?
The word “compati-
ble,” as used here, means “sufficiently” close to the hypothesized value so that we do not re-
ject the stated hypothesis. Thus, if some theory or prior experience leads us to believe that
the true slope coefficient 
β
2
of the wages-education example is unity, is the observed
ˆ
β
2
0.724 obtained from the sample of Table 3.2 consistent with the stated hypothesis? If it is, we
do not reject the hypothesis; otherwise, we may reject it.
In the language of statistics, the stated hypothesis is known as the