Page 443 - Probability and Statistical Inference

420 8. Tests of Hypotheses

Since the alternative hypothesis is µ > µ , here the phrase more extreme is
0
interpreted as “ > ” and hence the p-value is given by

which can be evaluated using the standard normal table.
If the alternative hypothesis is µ < µ , then the phrase more extreme will be
0
interpreted as “
> ” and hence the p-value is going to be

This probability can again be evaluated using the standard normal table.
A test with a “small” p-value indicates that the null hypothesis is less
plausible than the alternative hypothesis and in that case H is rejected.
0
A test would reject the null hypothesis H at the chosen level α
0
if and only if the associated p-value < α.
Example 8.4.4 (Example 8.4.1 Continued) Suppose that X , ..., X are iid
1
15
N(µ, σ ) with unknown µ ∈ ℜ but s = 9. We wish to test H : µ = 3.1 versus
2
2
0
H : µ > 3.1 at the 5% level. Now, suppose that the observed data gave =
1
5.3 so that Here, we have z = 1.645
α
and hence according to the rule given by (8.4.1), we will reject H at the 5%
0
level since z > z .
α
calc
On the other hand, the associated p-value is given by P{Z > 2.8402} ≈
2.2543 × 10 . If we were told that the p-value was 2.2543 × 10 in the first
–3
–3
place, then we would have immediately rejected H at any level α > 2.2543 ×
0
10 .
–3
8.4.2 Monotone Likelihood Ratio Property
We now define the monotone likelihood ratio (MLR) property for a family of
pmf or pdf denoted by f(x; θ), θ ∈ Θ ⊆ ℜ. In the next subsection, we exploit
this property to derive the UMP level α tests for one-sided null against one-
sided alternative hypotheses in some situations. Suppose that X , ..., X are iid
1 n
with a common distribution f(x; θ) and as before let us continue to write X =
(X , ..., X ), x = (x , ..., x ).
1 n 1 n
Definition 8.4.1 A family of distributions {f(x; θ): θ ∈ Θ} is said to have
the monotone likelihood ratio (MLR) property in a real valued statistic T =
T(X) provided that the following holds: for all {θ*, θ} ⊂ Θ and x ∈ χ, we
have