Agriculture Reference
In-Depth Information
Table 6.3
Selection of random numbers for SRSWR and
SRSWOR scheme using direct method
Table 6.4
Selection of random numbers for SRSWR and
SRSWOR scheme using remainder approach
Selected random
numbers
SRSWR SRSWOR
12 12 12
4 04 4
36 36 36
80 - -
36 36 -
32 32 32
95 — —
63 - -
78 — —
18 18 18
94 — —
11 11 11
87 — —
45 45 45
15 15 15
32 32 -
71 — —
77 - -
55 - -
95 - —
27 - 27
33 - 33
— means random number drawn from random number
table is more than 48.
Selected random
numbers
SRSWR SRSWOR
Random numbers taken
from the table
6.7
Random
numbers
Remainder when
divided by 48
12
12
12
12
4
04
04
4
36
36
36
36
80
32
32
32
36
36
36
-
32
32
32
-
95
47
47
47
63
15
15
15
78
30
30
30
18
18
18
18
94
46
-
46
11
11
-
11
If a sample of
n
units is drawn from a population
N
n
ordered
possible samples and the probability of getting any
sample is
of
N
units with SRSWR, then there are
N
n
. If a sample is drawn with
SRSWOR, then there are
N
C
n
P
(
s
)
¼
1/
ð
Þ
unordered possible
=
N
C
n
samples and
. It is further to be
noted that the probability of getting
i
PðsÞ¼
1
ð
Þ
th
unit at
r
th
draw is
P [
i
ðÞ¼
1
=N;
i ¼
1, 2, 3,
...
,
N
and
r ¼
for both SRSWR and SRSWOR.
In case a sample of
1, 2, 3,
...
,
n
n
units is drawn from a
population of
N
units with SRSWR and the sam-
ple values are (
y
1
,
y
2
,
y
3
,
...
,
y
n
), then the sample
Method 2 (Remainder Approach)
To reduce time and labor, the commonly used
“remainder approach” is employed to avoid the
rejection of random numbers. The greatest two-
digit number, which is a multiple of 48, is 96, and
we consider two-digit numbers from 01 to 96,
rejecting the numbers greater than 96 and 00.
By using two-digit random numbers as above, we
prepare Table
6.4
.
The random samples of size 10 with replacement
and without replacement consist of the variety num-
bers 12, 4, 36, 32, 36, 32, 47, 15, 30, and 18 and 12,
4, 36, 32, 47, 15, 30, 18, 46, and 11, respectively.
In the direct approach, as many as 22 random
numbers were selected to obtain a random sam-
ple of ten only. On the contrary, only 12 random
numbers were sufficient in the second “remain-
der approach” to draw the sample of 10. Thus,
with the help of the remainder approach, one can
save time as well as labor in drawing a definite
simple random sample.
P
n
i¼
1
y
i
1
n
mean
y ¼
is an unbiased estimator of the
Y;
; EðyÞ¼Y
population mean
, and a
sampling variance of the sample mean, that is,
VðyÞ
that is
2
is the population
variance. It is also observed that the sample mean
square (
2
is equal to
σ
σ
where
n
2
) is an unbiased estimator of the popu-
lation variance,
s
2
2
that
is
Eðs
Þ¼σ
where
1
n
2
. Thus, the standard error
2
1
n
s
¼
i¼
1
ðy
i
yÞ
ðyÞ¼
σ
of the sample mean is given by SE
and
p
ðyÞ¼
s
the estimated SE
p
:
2
n
EðyÞ¼Y; VðyÞ¼
N n
N
1
σ
For SRSWOR
2
n
¼
N n
N
S
2
¼ S
2
n
;Es
f Þ
S
2
¼ð
1
