Agriculture Reference
In-Depth Information
0
1
Var
q
Var
θ θ
z
θ
@
A
¼ Pr
θ θ
Pr
q
z
¼
ʱ
ð
8
:
4
Þ
θ
The variance of the estimator
θ
decreases as
n
increases, so the above inequality
q
Var
θ
will be satisfied if it is possible to choose
n
large enough so that
z
c
.
Although statistical theory may sometimes help, it is usually common practice to
substitute this value into the appropriate variance (or coefficient of variation)
formulas after having decided on the standard error (or coefficient of variation of
a sample estimator), together with an estimation of the population variance
(or coefficient of variation). Then, we solve the resultant equation for
n
.
Equation (
8.4
) contains some parameters that are unknown properties of the
population, which must be estimated to provide specific results. It is worth noting
that the coefficient of variation can be used to estimate the variability, as it tends to
be more stable over time and location than the variance.
We can use different methods to estimate
Var θ
using the estimator
V θ
.
Firstly, it is possible to draw a preliminary SRS of size
n
1
n
, and then calculate
the estimate
v θ
of
Var θ
. This solution is not commonly used because it slows
the survey process. Secondly, a researcher can use sample quantities obtained
through a pilot survey. This is most likely the best method if the pilot survey is
itself a SRS. Unfortunately, in practice, the pilot survey is restricted to a part of the
population that is convenient to handle or that will reveal the magnitude of certain
problems. Therefore, we should exercise caution when using this method to esti-
mate
Var θ
, because of the selective nature of the pilot. Finally, we can use a
previous sample of the same or a similar population, or other available data.
However, practitioners are faced with other problems when determining an
adequate sample size. For example, data are often needed for certain major sub-
divisions of the population, and the desired error limits are set for each subdivision.
In this case, we calculate
n
separately for each subdivision, adding the values
together to find the total
n
. Furthermore, an investigator often measures several
variables at once and there may be a number of goals for the survey. This can lead to
a large number of items. If there is a desired degree of precision for each item, the
computations can lead to a series of conflicting values of
n
, one for each item.
Methods for reconciling these values must be developed. One method is to specify
margins of error for the variables that are considered of primary interest in the
survey, separately estimate the sample size needed for each of these important
items, and use these for estimating the final sample size. When the single item
estimations of
n
have been completed, we consider the situation as a whole. If the
different required values of
n
are all reasonably close, and the largest falls within
the limits of the budget, this
n
is selected. More commonly, there are significant
variations amongst the
n
s. In this case, the final sample size could also be the largest
calculated sample size. However, this may not be appropriate because of budgetary
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