Research Methodology - Dr. Krishan K. Pandey

64 Research Methodology
n / N σ = n / N σ = ......... = n / N σ
1 1 1 2 2 2
k k k
where σ1, σ2,
... and σ k denote the standard deviations of the k strata, N , N ,…, N denote the
1 2 k
sizes of the k strata and n , n ,…, n denote the sample sizes of k strata. This is called ‘optimum
1 2 k
allocation’ in the context of disproportionate sampling. The allocation in such a situation results in
the following formula for determining the sample sizes different strata:
n
i
n ⋅ Ni
σi
=
N σ + N σ + + N σ
1 1 2 2 ...
We may illustrate the use of this by an example.
k k
for i = 1, 2, … and k.
Illustration 1
A population is divided into three strata so that N 1 = 5000, N 2 = 2000 and N 3 = 3000. Respective
standard deviations are:
σ = 15, σ = 18 and σ = 5.
1 2 3
How should a sample of size n = 84 be allocated to the three strata, if we want optimum allocation
using disproportionate sampling design?
Solution: Using the disproportionate sampling design for optimum allocation, the sample sizes for
different strata will be determined as under:
Sample size for strata with N 1 = 5000
n1 =
84b5000gb15g b5000gb15g + b2000gb18g + b3000gbg 5
= 6300000/126000 = 50
Sample size for strata with N 2 = 2000
n 2
Sample size for strata with N 3 = 3000
n 3
=
=
84b2000gb18g b5000gb15g + b2000gb18g + b3000gbg 5
= 3024000/126000 = 24
84b3000gbg 5
b5000gb15g + b2000gb18g + b3000gbg 5
= 1260000/126000 = 10
In addition to differences in stratum size and differences in stratum variability, we may have
differences in stratum sampling cost, then we can have cost optimal disproportionate sampling design
by requiring
n1
n2
nk
= = ...
=
N σ C N σ C N σ C
1 1 1
2 2 2
k k k