Page 589 - Probability and Statistical Inference

566 12. Large-Sample Inference

benefits. In a random sample of 2000 working women in the city, we ob-
served 20% with employer-provided health benefits. At an approximate 5%
level, test the validity of the critic’s claim.
12.3.10 A die has been rolled 100 times independently and the face 6 came
up 25 times. Suppose that p stands for the probability of a six in a single trial.
At an approximate 5% level, test a null hypothesis H : p = 1/6 versus an
0
alternative hypothesis H : p ≠ 1/6.
1
12.3.11 (Exercise 12.3.10 Continued) A die has been rolled 1000 times
independently and the face 6 came up 250 times. Suppose that p stands for
the probability of a six in a single trial. At an approximate 5% level, test a null
hypothesis H : p = 1/6 versus an alternative hypothesis H : p ≠ 1/6. Also,
0
1
obtain an approximate 99% confidence interval for p.
12.3.12 We looked at two specific brands (A and B) of refrigerators in the
market and we were interested to compare the percentages (p and p ) requir-
A
B
ing service calls during warranty. We found that out of 200 brand A refrigera-
tors, 15 required servicing, whereas out of 100 brand B refrigerators, 6 re-
quired servicing during the warranty. At an approximate 5% level, test a null
hypothesis H : p = p versus an alternative hypothesis H : p ≠ p .
0 A B 1 A B
12.3.13 (Exercise 12.3.12 Continued) Obtain an approximate 90% confi-
dence interval for p − p .
A B
12.3.14 Suppose that X , ..., X are iid Poisson(λ) random variables where
n
1
0 < λ < ∞ is the unknown parameter. The minimal sufficient estimator of λ is
the sample mean which is denoted by One observes immediately that
is also asymptotically standard normal. Hence, for large n,
we have ≈ 1 − α so that

≈ 1 − α. Now, solve the quadratic equation in
λ to derive an alternative (to that given in (12.3.33)) approximate 100(1 −
α)% confidence interval for λ.
12.3.15 Derive an approximate 100(1−α)% confidence interval for λ −λ
1 2
in the two-sample situation for independent Poisson distributions.
12.3.16 Derive tests for the null hypothesis H : λ = λ in the two-sample
0
1
2
situation for independent Poisson distributions with approximate level α, when
the alternative hypothesis is either upper-, lower-, or two-sided respectively.
12.3.17 (Sign Test) Suppose that X , ..., X are iid continuous random
1
n
variables with a common pdf f(x; θ) which is symmetric around x = θ
where x ∈ ℜ, θ ∈ ℜ. That is, the parameter θ is the population median
assumed unknown, and f is assumed unknown too. The problem is to test