Inferential Statistics 379
that the test statistic might become even more extreme than t(x). This prob-
ability is equal to the p-value.
If t(x) is a value pretty close to the median of the distribution of t(X),
then the chance of obtaining a more extreme value, which refutes the null
hypothesis more strongly, might be fairly feasible. Then, the p-value will
be large. However, if, instead, the value t(x) is so extreme that the chances
will be minimal under the null hypothesis that, in some other test run we
obtain a value t(X) even more in favor of the alternative hypothesis, this
will correspond to a very low p-value. If p is less than some given signifi-
cance level α, we reject the null hypothesis and we say that the test result
is significant.
We demonstrate the meaning of the p-value in Figure C.3. The hori-
zontal axis provides the state space of possible values for the statistic t(X).
The figure displays the probability, given that the null hypothesis holds,
of this t(X) assuming a value greater than c, for each c of the state space,
and in particular also at t(x) (i.e., the statistic evaluated at the observation
x). We can see that, by definition, the value t(x) is the boundary between
the accept ance region and the critical region, with t(x) itself belonging to the
acceptance region. In that particular instance, we happened to choose a test
with ΔA = (–∞, t(x)] and ΔC = (t(x), ∞).
figURe C.3 Illustration of the p-Value for Some Test δ with Acceptance Region ΔA =
(–∞, t(x)] and Critical Region ΔC = (t(x), ∞)
ΔCpPθ
0
(t(X) ≤ c)Pθ
0
(t(X) < t(x))
t(x) ΔA c