Geoscience Reference
In-Depth Information
ˆ
{
ˆ
(1
ˆ
)/ }{1/}
ˆ
pz ppnnN
±
−
−
. Hence, a confidence interval of
pd
requires
α
/2
[{
ˆ
(1
ˆ
)/ }{1/}
that
dz ppnnN
=
−
−
. Solving for
n
then gives
/2
α
nzppdzppN
2
ˆ(1
ˆ)/{
2
2
ˆ(1
ˆ)/ }
=
−
+
−
.
(2.20)
α
/2
α
/2
To use this equation, a likely value for
ˆ
is guessed or a value from a previ-
ous sample is used. If
N
is very large or is unknown, then the second term in
the denominator can be ignored and the equation simplifies to
2
ˆ
(1
ˆ
)/
2
nzppd
=
−
.
(2.21)
α
/2
If there are no prior data and it is not possible to guess what
ˆ
might be, then
the worst possible case can be assumed, which is that
ˆ = 0.5. Substituting
this value into Equation (2.20) gives a sample size that will be larger than
necessary unless
ˆ
does happen to equal 0.5.
EXAMPLE 2.5 Determining the Sample Size for Sampling Sage
Grouse Hens
To illustrate the use of Equation (2.20), consider again the survey of
female sage grouse hens in an area in Wyoming discussed previously.
Suppose that it is desirable to estimate the proportion of barren hens
such that a 90% confidence interval will consist of the sample propor-
tion ± 0.05. The estimate from the first of two surveys was =
ˆ
0.325
, and
it will be assumed that there are about 1000 hens in the area that was
sampled. Substitution into Equation (2.20) gives
n
= 1.64
2
× 0.325(1 − 0.325)/{0.05
2
+ 1.64
2
× 0.325(1 − 0.325)/1000} = 190.9,
say, 191. This is the appropriate sample size. In a practical application,
this would probably be rounded to
n
= 200 to give an estimate that is
likely to be slightly better than what is required.
2.9 Stratified Random Sampling
It can be argued that simple random sampling leaves too much to chance,
particularly when the sample size is small. It might, for example, be clear that
the number of sampled units in different geographical areas does not match
the population sizes in those areas, with parts of the population under-
sampled and other parts oversampled. One way to overcome this potential
problem, while keeping the advantages of random sampling, is to use strati-
fied random sampling. To this end, the units in the population are divided
