Geoscience Reference
In-Depth Information
The confidence limits produced by Equation (2.19) are based on the
assumption that the sample proportion is approximately normally distrib-
uted. A useful rule of thumb is that if
np
(1 −
p
) ≥ 5, then this assumption is
reasonable. If this is not the case, then alternative methods for calculating
confidence limits should be used, as discussed, for example, by Dixon and
Massey (1983).
EXAMPLE 2.4 Estimating the Proportion of Barren Sage Grouse Hens
A survey of sage grouse hens in an area in the state of Wyoming, USA,
found that of
n
= 120 randomly sampled hens,
r
= 39 were barren. This
gives an estimated proportion of =
ˆ
39/120 0.325
barren hens in the
population. Assuming that the population size
N
is large relative to the
sample size of 120, Equation (2.18) then gives the estimated standard
error of the proportion as
=
SE(
ˆ
) [0.325(1 0.325)/120 0.042
·
, and an
approximate 95% confidence interval for the population proportion of
barren hens is therefore 0.325 ± 1.96 × 0.0428 or 0.241 to 0.408.
In a random sample of 88 sage grouse hens taken in another area in
Wyoming, it was found that 15 were barren, so that =
p
=
−
=
ˆ
15/88 0.170
,
=
·
with
, again assuming that the pop-
ulation size is much larger than the sample size and therefore omitting
the finite population correction. A 95% confidence interval for the pro-
portion of barren hens in the second area is therefore 0.17 ± 1.96 × 0.0400
or 0.092 to 0.248.
A rough-and-ready procedure to compare two sample proportions
involves checking whether the confidence intervals overlap. Here, they
overlap slightly, so it appears that the proportion of barren hens might
not have differed for the two districts. However, a more accurate compar-
ison can be made by noting that the variance of the difference between
the proportions in two independent samples is the sum of the individual
variances, so that
SE( ˆ)
p
=
[0.170(1 0.170)/88
−
=
0.0400
Var(
ˆ
ˆ
)Var(
ˆ
)Var(
ˆ
)
1 2 1 2
. Therefore, the differ-
ence of 0.155 between the first proportion (0.325) and the second propor-
tion (0.170) is an estimate of the population difference with an estimated
standard error of
pp
−=
p
+
p
2
2
. On this basis, a 95% confi-
dence interval for the true population difference is 0.155 ± 1.96 × 0.0586 or
0.040 to 0.269. As this interval does not include zero, there is clear evidence
that the proportion of barren hens was different for the two districts.
{0.0428
+
0.0400 }
=
0.0586
2.8 Determining Sample Sizes for the
Estimation of Proportions
An approximate 100(1 − α)% confidence interval for a proportion
based on a sample size of
n
from a population of size
N
takes the form
