Standard Sampling Methods and Analyses - Introduction to Ecological Sampling - page 24

One possible procedure in this case is to use poststratification, whereby
a simple random sample of n is taken from the entire population, and the
sampled units are classified into K strata. Then, the stratified sampling esti-
mator y s = Σ N i y i / N is used to estimate the population mean.
Poststratification is almost as precise as stratified sampling with propor-
tional allocation providing that the sample size is larger than about 20 in
each of the strata. Furthermore, the stratified sampling equation for the vari-
ance of y s is still approximately correct.

2.9.2 Stratified Sampling for Proportions

Stratified sampling can also be used with the estimation of proportions. If ˆ i
is the sample proportion in stratum i , then this is an unbiased estimator of
the stratum proportion p i with estimated variance

·

{ }{ }

ˆ (1 ˆ )/ 1/

Var( p ) = p pn nN − − . (2.29)

i i i i i i

An unbiased estimator of the overall population proportion p is then

K

∑

p NpN ˆ = ˆ /

ii

, (2.30)

i = 1

with estimated variance

K

∑

·· ,

Var( ˆ) p NpN

2

Var( ˆ )/

2

= (2.31)

i i

i = 1

· · . An approximate 100(1 − α)%
confidence interval for the true population proportion is given by

SE( ˆ ) ar( ˆ )

and estimated standard error p = p

·

ˆ

{ E( ˆ )}
/2

pz p ± , (2.32)

α

where z α/2 is the value that is exceeded with probability α/2 with the stan-
dard normal distribution.
In practice, if the cost to sample a unit is the same for all strata, then the
gain from stratified random sampling over simple random sampling is small
unless the proportions vary greatly with the strata. However, these equa-
tions might still be useful for surveys that include the estimations of means
and proportions at the same time. Then, the value of stratification might be
most obvious with the estimation of the mean values, but still the sample
must be treated as stratified for estimating the required proportions.