Geoscience Reference
In-Depth Information
taken with the study design, with well-documented and controlled sampling
protocols. This is of particular importance when the sampling for a study is
carried out by a number of different people. They need careful training in all
aspects of the data collection, with an emphasis on ensuring that everyone
uses consistent sampling and measurement methods.
2.7 Estimation of Population Proportions
In a common situation, it is necessary to estimate the proportion,
p
say, of
units in a population that have a particular characteristic. For example, if
the units are trees, then there might be interest in the proportion of trees in
a particular size class. In this situation, the population proportion can be
estimated by the proportion observed in a simple random sample.
Let
r
denote the number of units with the characteristic of interest in a
random sample of
n
. Then, the sample proportion is
prn
ˆ
/
, and it can be
shown that this has a sampling variance of
{
}{
}
Var(
ˆ
)
pppn nN
=−
(1)/1/
−
,
(2.16)
and therefore a standard error (standard deviation) of
{
}{
}
SE(
ˆ
)
p
=
p pn nN
(1)/1/
−
−
.
(2.17)
Equations (2.16) and (2.17) include the finite population correction factor
(1 −
n
/
N
), where
N
is the size of the population being sampled. If
N
is large
relative to
n
or if
N
is unknown, then this factor is usually set equal to 1.
Estimated values for the variance and standard error can be obtained by
replacing the population proportion in Equations (2.16) and (2.17) with the
sample proportion
ˆ
. Thus,
·
{
}{
}
SE(
ˆ
)
ˆ
(1
ˆ
)/ 1/
p
=
p pn nN
−
−
.
(2.18)
This creates little error unless the sample size is quite small (say, less than
20). Using this estimate, an approximate 100(1 − α)% confidence interval for
the true proportion is
/2
·
ˆ
SE(
ˆ
)
pz p
±
,
(2.19)
α
where, as before,
z
α/2
is the value from the standard normal distribution that
is exceeded with probability α/2.
