Simple Random Sampling for Proportions
Often we are interested in estimating a population proportion pfor some
characteristic, such as the proportion of voters favoring some proposal, or the
proportion of an animal species having a particular genetic condition. To
estimate a proportion for a particular characteristic, we define the variable yi
for each sampled element to be equal to 1 if the element has the characteristic,
and 0 otherwise. Then our estimator bpfor the proportion pis just the sample
mean of y:
bp=y=
n
P
i=1
yi
n,
and since bp=y, the variance estimator can be obtained by using our
expression for b
V(y) and expressing it in terms of bpand bq= 1 −bp:
b
V(bp) = bpbq
n−1µN−n
N¶.
Sample Size selection for Proportions
Again, we can use the same approach that we used earlier to obtain
a sample size expression for estimating proportions, by using the expression
for the bound Band solving for the sample size n:
n=Npq
(N−1)(B2/4) + pq .
The question also arises as to what value to use for p, since we are trying to
estimate it. Here, however, it is easier, because if we do not have information
from previous studies, we can set p=.5 as a conservative value.
Examples