Page 551 - Probability and Statistical Inference

528 11. Likehood Ratio and Other Tests

With α = .10 and 6 degrees of freedom, one has t 6,.05 = 1.9432. Since |t |
calc
exceeds t 6,.05 , we reject the null hypothesis at 10% level and conclude that the
job performance scores before and after training appear to be significantly
correlated. !

11.4.3 Test for the Variances
With fixed α ∈ (0, 1), we wish to construct a level α test for the null hypoth-
esis H : σ = σ against the upper-, lower-, or two-sided alternative hypoth-
1
0
2
esis H . The methodology from Section 11.3.2 does not apply. Let us denote
1

Observe that (Y , Y ) are iid bivariate normal, N (ν , ν , ρ*), i = 1, ...,
1i 2i 2 1 2
n(≥ 3) where ν = µ + µ , ν = µ – µ ,
1 1 2 2 1 2
and Cov(Y , Y ) = so that ρ* = ( )/(τ τ ). Of
1i 2i 1 2
course all the parameters ν , ν , ρ* are unknown, (ν , τ ) ∈ ℜ × ℜ , l =
+
1
2
l
l
1, 2 and –1 < ρ* < 1.
Now, it is clear that testing the original null hypothesis H : σ = σ is
0
2
1
equivalent to testing a null hypothesis H : ρ* = 0 whereas the upper-, lower-
0
, or two-sided alternative hypothesis regarding σ , σ will translate into an
2
1
upper-, lower-, or two-sided alternative hypothesis regarding ρ*. So, a level
α test procedure can be derived by mimicking the proposed methodologies
from (11.4.11)-(11.4.13) once r is replaced by the new sample correlation
coefficient τ* obtained from the transformed data (Y , Y ), i = 1, ..., n(≥ 3).
1i 2i
Upper-Sided Alternative Hypothesis
We test H : σ = σ versus H : σ > σ . See the Figure11.4.5. Along the
2
1
0
2
1
1
lines of (11.4.12), we can propose the following upper-sided level α test:
Lower-Sided Alternative Hypothesis
We test H : σ = σ versus H : σ < σ . See the Figure 11.4.6. Along
0 1 2 1 1 2