Statistics Review - Mathematics and Statistics for the Environment - page 128

7.12.2 s ample s ize
An appropriate table can be used to estimate the number of units that would have to be observed in
a simple random sample in order to estimate a population proportion with some specified precision.
Suppose, for example, that we wanted to estimate the germination percent for a population to within
±10% (or 0.10) at the 95% confidence level. The first step is to guess about what the prop or tion of
seed germination will be. If a good guess is not possible, then the safest course is to guess p = 0.59,
as this will give the maximum sample size.
Next, pick any of the sample sizes given in the appropriate table (e.g., 1 0, 15, 20, 30, 50, 100, 250,
and 1000) and look at the confidence interval for the specified value of p . Inspection of these limits
will tell whether or not the precision will be met with a sample of this size or if a larger or smaller
sample would be mor e appropriate.
Thus, if we guess p = 0.2, then in a sample of n = 50 we would expect to observe (0.2)(50) = 10,
and the table says tha t t he 95% confidence limits on p would be 0.10 and 0.34. Since the upper limit
is not within 0.10 of p , a larger sample would be needed. Fo r a sample of n = 100 the limits are
0.13 to 0.29. Because both of these values are within 0.10 of p , a sample of 100 would be adequate.
If the table indicates the need for a sample of over 250, the size can be approximated by

41
2

( ( p − p ) ,for 95% confidence

n ≈

E

or,

20 ( ( p 1 − p ) ,for 99% confidence

n ≈

2

3 E

where E is the precision with which p is to be estimated (expressed in same for as p , either percent
or decimal).

7.13 CLUSTER SAMPLING FOR ATTRIBUTES
Simple random sampling of discrete variables is often difficult or impractical. In estimating tree
plantation survival, for example, we could select individual trees at random and examine them,
but it wouldn't make much sense to walk down a row of planted trees in order to observe a single
member of that row. It would usually be more reasonable to select rows at random and observe all
of the trees in the selected row.
Seed viability is often estimated by randomly selecting several lots of 100 or 200 seeds each and
recording for each lot the percentage of the seeds that germinate. These are examples of cluster
sampling ; the unit of observation is the cluster rather than the individual tree or single seed. The
value attached to the unit is the proportion having a certain characteristic rather than the simple fact
of having or not having that characteristic. If the clusters are large enough (say, over 100 individuals
per cluster) and nearly equal in size, the statistical methods that have been described for measure-
ment variables can often be applied. Thus, suppose that the germination percent of a seedlot is esti-
mated by selecting n = 10 sets of 200 seed each and observing the germination percent for each set:

Set 1 2 3 4 5 6 7 8 9 10 Sum

Germination percent ( p ) 78.5 82.0 86.0 80.5 74.5 78.0 79.0 81.0 80.5 83.5 803.5

then the mean germination percent is estimated by

∑ 803 5
10

p

.

p = = = 80 35 .%

n