Research Methodology - Dr. Krishan K. Pandey

176 Research Methodology
σ p = standard deviation of the popultion (to be estimated from past experience or on the basis of
a trial sample). Suppose, we have σ p = 48
. for our purpose.
If the difference between μ and X or the acceptable error is to be kept with in ±3 of the sample
mean with 95% confidence, then we can express the acceptable error, ‘e’ as equal to
b g b g
2 2
p
e = z⋅
n
σ or 3 196 48 .
= .
n
196 . 48 .
Hence, n = = 9834 . ≅ 10.
2 bg 3
In a general way, if we want to estimate μ in a population with standard deviation σ p with an
error no greater than ‘e’ by calculating a confidence interval with confidence corresponding to z, the
necessary sample size, n, equals as under:
z σ
n =
2
e
All this is applicable whe the population happens to be infinite. Bu in case of finite population, the
above stated formula for determining sample size will become
n =
2 2
2 2*
p
2 2 2
σ p
z ⋅ N ⋅σ
b g
N− 1 e + z
* In case of finite population the confidence interval for μ is given by
σ b g
X ± z
p
×
n
N −n
bN − 1g
where bN −ng bN − 1 g is the finite population multiplier and all other terms mean the same thing as stated above.
If the precision is taken as equal to ‘e’ then we have
σ p bN −ng
e = z ×
n N − 1
b g
σ
2 2 p N −n
or e = z ×
n N − 1
b g
or
2
e N − 1
z
=
σ p N z
−
n
σ p n
n
or
2 2 2
e bN − 1g
+ z σ p
2 2
z σ p N
=
n
2
2 2 2 2
2 2
z ⋅ σ p ⋅ N
or n =
e N − 1 + z σ
or
n =
b g
2 2 2
p
2
z ⋅
2
N ⋅σ
p
2 2 2
N − 1 e + z σ p
b g
This is how we obtain the above stated formula for determining ‘n’ in the case of infinite population given the precision
and confidence level.