where the coe‰cient is 1.96 ifn 1 þn 2 is large; otherwise, atcoe‰cient is
used with approximately
df¼n 1 þn 2 2
4.3 ESTIMATION OF PROPORTIONS
The sample proportion is defined as in Chapter 1:
p¼
x
n
wherexis the number of positive outcomes andnis the sample size. However,
the proportionpcan also be viewed as a sample meanx, wherexiis 1 if theith
outcome is positive and 0 otherwise:
p¼
P
xi
n
Its standard error is still derived using the same process:
SEðpÞ¼
s
ffiffiffi
n
p
with the standard deviationsgiven as in Section 2.3:
s¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pð 1 pÞ
p
In other words, the standard error of the sample proportion is calculated from
SEðpÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pð 1 pÞ
n
r
To state it more formally, the central limit theorem implies that the sampling
distribution ofpwill be approximately normal when the sample sizenis large;
the mean and variance of this sampling distribution are
mp¼p
and
sp^2 ¼
pð 1 pÞ
n
respectively, wherepis the population proportion.
160 ESTIMATION OF PARAMETERS