Page 583 - Probability and Statistical Inference

560 12. Large-Sample Inference

From (12.4.8), it is clear that we should look at the transformation

carefully and consider the asymptotic distribution of In

view of (12.4.1) with we can claim that
as → ∞.

That is, for large n, we consider the pivot

which is approximately N(0, 1). (12.2.9)
See Johnson and Kotz (1969, Chapter 4, Section 7) for a variety of other
related approximations.
For large n, one may use (12.4.9) to derive an approximate 100(1 − α)%
confidence interval for λ. Also, in order to test a null hypothesis H : λ = λ ,
0
0
for large n, one may use the test statistic

to come up with an approximate level α test against an appropriate alternative
hypothesis. The details are left out for brevity.
Example 12.4.5 A manufacturer of 3.5" diskettes measured the quality
of its product by counting the number of missing pulse (X) on each. We
are told that X follows a Poisson(λ) distribution where λ(> 0) is the un-
known average number of missing pulse per diskette and that in order to
stay competitive, the parameter λ should not exceed .009. A random sample
of 1000 diskettes were tested which gave rise to a sample mean . Is there
sufficient evidence to reject H : λ = .009 in favor of H : λ > .009 approxi-
1
0
mately at 5% level? In view of (12.4.10), we obtain

which exceeds z = 1.645. Hence, we reject H approximately at 5%
0
.05
level. Thus, at an approximate 5% level, there is sufficient evidence that
the defective rate of diskettes is not meeting the set standard.!

12.4.3 The Correlation Coefficient

Suppose that the pairs of random variables (X , X ) are iid bivariate nor-
1i
2i
mal, N (µ , µ , ρ), i = 1, ..., n(≥ 4). It will be clear later from
2 1 2