Agriculture Reference
In-Depth Information
n
h
j¼
1
y
hj
and
1
n
h
selection or repeat the process. So it is better to
generate more random numbers than what is
actually required and transform these into values
using Copy, Paste Special, Values command
immediately.
2
1
n
h
1
n
h
2
y
h
¼
s
h
¼
j¼
1
y
hj
y
h
:
Unbiased estimator for the population mean
Y
and the population total
Y
are given by
X
L
h¼
1
W
h
y
h
^
Y ¼ y
st
¼
6.2.1.3 Stratified Sampling
For a heterogeneous population, stratified
random sampling is found to be better compared
to SRS and simple random sampling with vary-
ing probability.
In stratified random sampling method, a pop-
ulation of size “
and
Y ¼ Ny
st
;
and their estimated variances are given by
X
L
h¼
1
W
h
2
n
n
h
Þ
s
V y
s
ðÞ¼
VNy
st
V y
s
ð ;
2
1
2
ð
f
h
ð
Þ¼N
” is divided into subpopu-
lations (called strata), which are homogeneous
within and heterogeneous among themselves.
Random samples are drawn from each stratum
separately. Age, gender, educational or income
status, geographical location, soil fertility pat-
tern, stress level, species of fish, etc., are gener-
ally used as stratifying factors. The efficiency of
the stratified random sampling design, relative
to the simple random sampling design, will be
high only if an appropriate stratification tech-
nique is used.
Number of strata
N
f
h
¼
n
h
where
N
h
:
Allocation of Sample Size to Different Strata
Three methods of allocation of sample size to
various strata in the population are given below:
1.
Equal allocation
: Total sample size is divided
equally among the strata, that is, sample
n
h
to be
selected from
h
th stratum such that
n
h
¼ n
/
L
.
2.
Proportional allocation
: In proportional allo-
cation,
n
h
1
(proportional to )
N
h
,
that is,
: Stratification can be done to
the extent such that the strata are not too many in
number because too many strata do not necessar-
ily bring down the variance proportionately.
Let there be
N
h
¼ nW
h
;
h ¼
1, 2, 3,
...
,
P
.
3.
Optimum allocation
: This is based on minimi-
Vðy
st
Þ
C
0
. The
zation of
under a given cost
simplest
cost
function is of
the
form
P
C ¼ C
0
þ
P
n¼
1
C
h
n
h
, where
strata in a population with mean
C
0
is the over-
Y
2
, and let
and variance
σ
N
h
be the sizes of
h
th
head cost,
C
h
is the cost of sampling a unit
stratum with mean
Y
h
and variance
2
h
σ
(
h ¼
1, 2,
from the
is the total cost.
To solve this problem, we have
h
th stratum, and
C
3,
...
,
P
) so that
N ¼ N
1
,+
N
2
+
......
.+
N
L
. One
can write
p
C
h
Þ
W
h
S
h
=
n
h
¼ C
0
C
0
ð
;
X
P
L
h¼
1
W
h
S
h
p
C
h
1
W
h
Y
h
;
Y ¼
h¼
¼
P
h¼
h
þ
P
h¼
1
W
h
Y
h
Y
2
2
2
σ
1
W
h
σ
ð
Þ
;
h ¼
...
L
S
2
h
1, 2, 3,
,
, where
is the population
mean square for the
th stratum.
A particular case arises when
h
C
h
¼ C
00
, that
is, if the cost per unit is the same in all the strata.
In this case,
where W
h
¼
N
N
:
Let us take a random sample of size
n
by
selecting
n
h
individuals from
h
th stratum such
that
P
h¼
1
n
h
¼ n
2
h
be the sample
mean and sample mean square for the nth stratum
where
. Let
y
h
and
s
W
h
S
h
L
h¼
1
W
h
S
h
N
h
S
h
L
h¼
1
N
h
S
h
n
h
¼ n
¼ n
:
