Page 581 - Probability and Statistical Inference

558 12. Large-Sample Inference

.91661, and hence p lies between .5362 and .6297. We report that between
53.62% and 62.97% of the voters in the town favor the proposed computer-
ization. !

In Examples 12.4.1-12.4.3, the sample size n was so large that
the results obtained after variance stabilization looked nearly
identical to what was found in Section 12.3.1.

Example 12.4.4 (Examples 12.3.5 and 12.4.1 Continued) Suppose that in
a random sample of 30 voters, 18 indicated that they favored the proposed
computerization. Let p denote the proportion of voters in the town who fa-
vored the proposed computerization. Then, Since n = 30 is
large, in view of (12.4.4), an approximate 90% confidence interval for

will be which reduces to .88608 ±
.15017. In other words, we conclude that lies between .73591
and 1.03625, and hence p lies between .45059 and .74046. We report that
among the voters, between 45.06% and 74.05% favor the proposed comput-
erization.
On the other hand, in view of (12.3.18), an approximate 90% confidence
interval for p will be which simplifies to .6±.14713.
We conclude that among the voters, between 45.29% and 74.71% in the
town favor the proposed computerization. Now, one can feel some difference
in the two approximations. !
For moderate values of n, an approximate 100(1 − α)% confidence inter-
val for p based on (12.4.4) is expected to fare a little better than that based on
(12.3.18). An approximate 100(1 − α)% confidence interval for
p would be (b , b ) where
L U
which are based on (12.4.4). Another
approximate 100(1 − α)% confidence interval for p would be (a , a ) where
L U
which are based on
(12.3.18). Since a , b may fall below zero or a , b may exceed unity, instead
U
L
U
L
of the intervals (a , a ) and (b , b ) we may respectively consider the confi-
L
U
U
L
dence interval estimators
In Table 12.4.1, we exhibit these two respective intervals based on
simulated data from the Bernoulli(p) population with 10 replications when