Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

244 Some Elementary Statistical Inferences

n=n 1 +n 2 be the total sample size. Our estimator ofp 1 −p 2 is the difference in
sample proportions, which, of course, is given byX−Y. We use the traditional
notation and write ˆp 1 and ˆp 2 instead ofXandY, respectively. Hence, from the
above discussion, an interval such as (4.2.9) serves as an approximate confidence
interval forp 1 −p 2. Here,σ 12 =p 1 (1−p 1 )andσ^22 =p 2 (1−p 2 ). In the interval,
we estimate these by ˆp 1 (1−pˆ 1 )andˆp 2 (1−ˆp 2 ), respectively. Thus our approximate
(1−α)100% confidence interval forp 1 −p 2 is

pˆ 1 −ˆp 2 ±zα/ 2

√

pˆ 1 (1−pˆ 1 )

n 1

+

pˆ 2 (1−pˆ 2 )

n 2

. (4.2.14)

Example 4.2.5.Kloke and McKean (2014), page 33, discuss a data set from the
original clinical study of the Salk polio vaccine in 1954. At random, one group of
children (Treated) received the vaccine while the other group (Control) received a
placebo. LetpcandpTdenote the true proportions of polio cases for control and
treated populations, respectively. The tabled results are:

Group No. Children No. Polio Cases Sample Proportion

Treated 200,745 57 0.000284

Control 201,229 199 0.000706

Note that ˆpC>pˆT. The following R segment computes the 95% confidence interval
forpc−pT:
prop.test(c(199,57),c(201229,200745))
The confidence interval is (0. 00054 , 0 .00087). All values in this interval are positive,
indicating that the vaccine is effective in reducing the incidence of polio.

EXERCISES

4.2.1.Let the observed value of the meanXand of the sample variance of a random

sample of size 20 from a distribution that isN(μ, σ^2 ) be 81.2 and 26.5, respectively.

Find respectively 90%, 95% and 99% confidence intervals forμ.Notehowthe

lengths of the confidence intervals increase as the confidence increases.

4.2.2.Consider the data on the lifetimes of motors given in Exercise 4.1.1. Obtain

a large sample 95% confidence interval for the mean lifetime of a motor.

4.2.3.Suppose we assume thatX 1 ,X 2 ,...,Xnis a random sample from a Γ(1,θ)
distribution.

(a)Show that the random variable (2/θ)

∑n

i=1Xihas aχ

(^2) -distribution with 2n
degrees of freedom.
(b)Using the random variable in part (a) as a pivot random variable, find a
(1−α)100% confidence interval forθ.
(c)Obtain the confidence interval in part (b) for the data of Exercise 4.1.1 and
compare it with the interval you obtained in Exercise 4.2.2.