Introductory Biostatistics

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