Page 572 - Probability and Statistical Inference

12. Large-Sample Inference 549

We use as an approximate pivot since its asymp-
totic distribution is N(0, 1) which is free from p.
Confidence Intervals for the Success Probability
Let us first pay attention to the one-sample problems. With preassigned α ∈
(0, 1), we claim that

which leads to the confidence interval

with approximate confidence coefficient 1 − α. Recall that z is the upper
a/2
100(α/2)% point of the standard normal distribution. See, for example, the
Figure 12.3.1.

A different confidence interval for p is given in Exercise 12.3.8.
Next, let us briefly discuss the two-sample problems. Suppose that the
random variables X , ..., X are iid from the i population having the
th
ini
i1
Bernoulli(p ) distribution where 0 < p < 1 is unknown, i = 1, 2. We suppose
i
i
that the X ’s are independent of the X ’s and denote the sample
2j
1j
mean obtained from the i population, i = 1, 2. In this case, one invokes the
th
following CLT: If n → ∞, n → ∞ such that n /n → δ for some 0 < δ < ∞,
1 2 1 2
then
For large sample sizes n and n , we should be able to use the random variable
1 2
as an
approximate pivot since its asymptotic distribution is N(0, 1) which is free
from p and p . With preassigned α ∈ (0, 1), for large n and n , we claim that
1 2 1 2

which leads to the confidence interval