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526 11. Likehood Ratio and Other Tests

Now, one rejects H if and only if Λ is small. Thus, we decide as follows:
0

or equivalently

where k(> 0) is a generic constant. Note that r /(1 - r ) is a one-to-one func-
2
2
2
tion of r . It is easy to see that this test rejects H when r is sizably different
0
from zero.

Figure 11.4.4. Two-Sided Student’s t Rejection Region
n-2
Now, recall from (4.6.10) that has the Student’s t dis-
tribution with n – 2 degrees of freedom when ρ = 0. This is why we as-
sumed that n was at least three. The derivation of this sampling distribution
was one of the earliest fundamental contributions of Fisher (1915). See the
Figure 11.4.4. Now, from (11.4.10), a level α LR test can be expressed as
follows:

Upper-Sided Alternative Hypothesis
We test H : ρ = 0 versus H : ρ > 0. See the Figure 11.4.5. Along the lines
0
1
of (11.4.11), we can propose the following upper-sided level α test: