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

p
value
(i.e., 
probability value
), also known as
the 
observed 
or
exact level of significance
or the 
exact probability of committing a Type
I error.
More technically, the 
p
value is defined as 
the lowest significance level at which
a null hypothesis can be rejected.
To illustrate, let us return to our wages-education example. Given the null hypothesis
that the true coefficient of education is 0.5, we obtained a 
t
value of 3.2 in Eq. (5.7.4). What
is the 
p
value of obtaining a 
t
value of as much as or greater than 3.2? Looking up the 
t
table
given in 
Appendix D,
we observe that for 11 df the probability of obtaining such a 
t
value
must be smaller than 0.005 (one-tail) or 0.010 (two-tail).
If you use Stata or EViews statistical packages, you will find that the 
p
value of obtain-
ing a 
t
value of 3.2 or greater is about 0.00001, that is, extremely small. This is the 
p
value
of the observed 
t
statistic. This exact level of significance of the 
t
statistic is much smaller
than the conventionally, and arbitrarily, fixed level of significance, such as 1, 5, or 10 per-
cent. As a matter of fact, if we were to use the 
p
value just computed, and reject the null
hypothesis that the true coefficient of education is 0.5, the probability of our committing a
Type I error would be only about 1 in 100,000!
As we noted earlier, if the data do not support the null hypothesis, 
|
t
|
obtained under the
null hypothesis will be “large” and therefore the 
p
value of obtaining such a 
|
t
|
value will
be “small.” In other words, for a given sample size, as 
|
t
|
increases, the 
p
value decreases,
and one can therefore reject the null hypothesis with increasing confidence.
What is the relationship of the 
p
value to the level of significance 
α
? If we make the habit
of fixing 
α
equal to the 
p
value of a test statistic (e.g., the 
t
statistic), then there is no conflict
between the two values. To put it differently,