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Less reliable variance estimates could be doubled (as a safety factor) before applying these equa-
tions. In many cases, the variance estimate may be so poor as to make the sample size computation
just so much statistical window dressing.
When sampling is without replacement (as it is in most forest sampling situations) the sample size
estimates given above apply to populations with an extremely large number (
N
) of units so that the
sampling fraction (
n/N
) is very small. If the sampling fraction is not small (say
n/N
= 0.05), then the
sample size estimates should be adjusted. This adjusted value of
n
is
n
n
a
=
n
N
1
+
Suppose that we plan to use quarter-acre plots in a survey and estimate the variance among plot
volumes to be
s
2
= 160,000. If the error limit is
E
= 5000 feet per acre, we must convert the variance
to an acre basis or the error to a quarter-acre basis. To convert a quarter-acre volume to an acre basis
we multiply by 4, and to convert a quarter-acre variance to an acre variance we multiply by 16. Thus,
the variance would be 2,560,000 and the sample-size formula would be
2
t
(,
2 560 000
500
,
)
2
n
=
=
t
(.)
10 24
2
(
)
Alternatively, we can leave the variance alone and convert the error statement from an acre to a
quarter-acre basis (
E
= 125). Then, the sample-size formula is
2
t
(
160 000
125
,
)
2
n
=
=
t
(.), as before
10 24
2
(
)
7.11.2 s
tratiFied
r
andom
s
ampling
In
stratified random sampling
, a population is divided into subpopulations (strata) of known size,
and a simple random sample of at least two units is selected in each subpopulation. This approach
has several advantages. For one thing, if there is more variation between subpopulations than within
them, the estimate of the population mean will be more precise than that given by a simple random
sample of the same size. Also, it may be desirable to have separate estimates for each subpopula-
tion (e.g., in timber types or administrative subunits). In addition, it may be administratively more
efficient to sample by subpopulations.
■
EX AMPLE 7.7
Problem:
A 500-acre forested area was divided into three strata on the basis of timber type. A
simple random sample of 0.2-acre plots was taken in each stratum, and the means, variances, and
standard errors were computed by the formulas for a simple random sample. These results, along
with the size (
N
h
) of each stratum (expressed in number of 0.1-acre plots), are shown in Table 7.5.
Solution:
The squared standard error of the mean for stratum
h
is computed by the formula given
for the simple random sample
2
s
n
n
N
2
h
h
h
h
s
=
1
−
x
h
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