Page 478 - Probability and Statistical Inference

9. Confidence Interval Estimation 455

9.2.1 in the symmetric case. But, any “optimality” property in the skewed
unimodal case does not easily translate into something nice and simple under
the scale parameter scenario. For skewed pivotal distributions, in general, we
can not claim useful and attractive optimality properties when a (1 – α) con-
fidence interval is constructed with the tail area probability ½α on both sides.

Convention: For standard pivotal distributions such as
Normal, Student’s t, Chi–square and F, we customarily
assign the tail area probability ½α on both sides in order
to construct a 100(1 – α)% confidence interval.

9.2.5 Using Confidence Intervals in the Tests of Hypothesis

Let us think of testing a null hypothesis H : θ = θ against an alternative
0
0
hypothesis H : θ > θ (or H : θ < θ or H : θ ≠ θ ). Depending on the nature
1
0
0
1
0
1
of the alternative hypothesis, the rejection region R respectively becomes up-
per or lower or two-sided. The reader has observed that a confidence interval
for θ can also be upper or lower or two-sided.
Suppose that one has constructed a (1 – α) lower confidence interval
estimator J = (T (X), ∞) for θ. Then any null hypothesis H : θ = θ will be
1
0
0
L
rejected at level α in favor of the alternative hypothesis H : θ > θ if and only
1
0
if θ falls outside the confidence interval J .
0 1
Suppose that one has constructed a (1 – α) upper confidence interval
estimator J = (–∞, T (X)) for θ. Then any null hypothesis H : θ = θ will be
U
0
0
2
rejected at level α in favor of the alternative hypothesis H : θ < θ if and only
1
0
if θ falls outside the confidence interval J .
2
0
Suppose that one has constructed a (1 – α) two–sided confidence interval
estimator J = (T (X), T (X)) for θ. Then any null hypothesis H : θ = θ will
0
0
U
L
3
be rejected at level a in favor of the alternative hypothesis H : θ ≠ θ if and
0
1
only if θ falls outside the confidence interval J .
0 3
Once we have a (1 – α) confidence interval J, it is clear that any
null hypothesis H : θ = θ will be rejected at level α as long as
0
0
θ ∉ J. But, rejection of H leads to acceptance of H whose
0
0
1
nature depends upon whether J is upper-, lower- or two-sided.
Sample size determination is crucial in practice if we wish to have
a preassigned “size” of a confidence region. See Chapter 13.