Biostatistics

6.6 CONFIDENCE INTERVAL FOR THE DIFFERENCE BETWEEN TWO POPULATION PROPORTIONS 187
6.6 CONFIDENCE INTERVAL FOR
THE DIFFERENCE BETWEEN TWO
POPULATION PROPORTIONS
The magnitude of the difference between two population proportions is often of interest. We
may want to compare, for example, men and women, two age groups, two socioeconomic
groups, or two diagnostic groups with respect to the proportion possessing some characteristic
of interest. An unbiased point estimator of the difference between two population
proportions is provided by the difference between sample proportions, ^p 1 ^p 2 .Aswe
have seen, when n 1 and n 2 are large and the population proportions are not too close to 0 or 1,
the central limit theorem applies and normal distribution theory may be employed to obtain
confidence intervals. The standard error of the estimate usually must be estimated by
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^p
^s^p1 ^p 2
¼ 1 ð1 ^p 1 Þ
þ ^p 2ð1 ^p 2 Þ
n 1
n 2
because, as a rule, the population proportions are unknown. A 100ð1
confidence interval for p 1 p 2 is given by
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^p
ð^p 1 ^p 2 Þz 1 ð1 ^p 1 Þ
1 a=2 þ ^p 2ð1 ^p 2 Þ
n 1
n 2
aÞ percent
(6.6.1)
We may interpret this interval from both the probabilistic and practical points of view.
EXAMPLE 6.6.1
Connor et al. (A-17) investigated gender differences in proactive and reactive aggression in
a sample of 323 children and adolescents (68 females and 255 males). The subjects were
from unsolicited consecutive referrals to a residential treatment center and a pediatric
psychopharmacology clinic serving a tertiary hospital and medical school. In the sample,
31 of the females and 53 of the males reported sexual abuse. We wish to construct a 99
percent confidence interval for the difference between the proportions of sexual abuse in
the two sampled populations.
Solution:
The sample proportions for the females and males are, respectively, ^p F ¼
31=68 ¼ :4559 and ^p M ¼ 53=255 ¼ :2078. The difference between sample
proportions is ^p F ^p M ¼ :4559 :2078 ¼ :2481. The estimated standard
error of the difference between sample proportions is
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ð:4559Þ ð:5441Þ
^s^pF ^p M
¼
þ ð:2078Þ
ð:7922Þ
68
255
¼ :0655
The reliability factor from Appendix Table D is 2.58, so that our confidence
interval, by Expression 6.6.1, is
:2481 2:58 ð:0655Þ
:0791;:4171