Agriculture Reference
In-Depth Information
considerations or because it will give an overall standard of precision higher than
originally considered. If the available funds are not sufficient to take the largest of
these calculated sample sizes, the median or mean of the calculated
n
s may be used
as a compromise. Otherwise, the desired standard of precision may be relaxed for
some items, to permit the use of a smaller value of
n
. In some cases, the
n
s required
for different items are so different that certain items must be excluded from the
design.
We must determine if the chosen value of
n
is consistent with the available
resources. To do this, we need to estimate the costs, labor, time, and materials
required to obtain the proposed sample size. It sometimes becomes evident that we
need to drastically reduce
n
. A difficult decision has to be made: whether to proceed
with a much smaller sample size, thus reducing the precision, or to abandon the
survey until more resources can be found.
It is important to control the relative precision (
c
) of the estimated population
total. In SRS with an estimator of the total
t
HT
, we require that
Pr
t
HT
t
j
j
t
HT
t
c
¼ Pr
ð
j
j ct
Þ ʱ;
ð
8
:
5
Þ
t
where
is a low value of probability, and
t
is the unknown population total (which
has similar considerations to those for the estimate of the unknown variance of the
population). We assume that
ʱ
t
HT
is normally distributed. Its standard error is
p
Var t
H
ðÞ
p
.
Hence
ct
¼
z
1
NS
y
,
U
n
¼
p
p
p
1
¼
z
NS
y
,
U
Var t
H
ðÞ
n
n
N
, where
z
is the value of the normal
p
deviate
corresponding
to
the
desired
confidence
probability,
and
S
y
,
U
¼
X
. Solving for
n
and using the estimator
S
y
¼
X
k2s
N
2
2
ð
y
k
y
Þ
ð
y
k
y
Þ
N
1
n
1
k
¼1
gives
z
2
N
2
S
y
c
2
t
2
n
¼
þ z
2
NS
y
:
ð
8
:
6
Þ
If we aim at just controlling the absolute precision, we obtain
z
2
N
2
S
y
n
¼
þ z
2
NS
y
:
ð
8
:
7
Þ
ʷ
2
It is common practice to control the coefficient of variation if we do not want to
make a distributional assumption such as the Gaussian law of the previous case. The
coefficient of variation is defined as
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