Research Methodology - Dr. Krishan K. Pandey

Sampling Fundamentals 167
σ
s
1⋅ 2
=
where X
2
2
2
2
1d
si 1 2d
si
2 1d 1 1⋅2i 2d 2 1⋅2i n σ + n σ + n X − X + n X − X
1⋅ 2=
d i d i
n X + n X
1 1 2 2
n + n
1 2
n + n
1 2
Note: (1) All these formulae apply in case of infinite population. But in case of finite population where sampling is done
without replacement and the sample is more than 5% of the population, we must as well use the finite
population multiplier in our standard error formulae. For instance, S.E. in case of finite population will be as
X
under:
SE
X
σpb g
= ⋅
n
N − n
N −1
b g
It may be remembered that in cases in which the population is very large in relation to the size of the sample,
the finite population multiplier is close to one and has little effect on the calculation of S.E. As such when
sampling fraction is less than 0.5, the finite population multiplier is generally not used.
(2) The use of all the above stated formulae has been explained and illustrated in context of testing of hypotheses
in chapters that follow.
ESTIMATION
σ
X
σ s
= =
n
Σd i 2
Xi− X
n −1
n
(ii) Standard error of difference between two sample means when σ p is unknown
σ X1 − X2
=
2
2
dX1i − X1i + dX2i − X2i
Σ Σ
1 1
⋅ +
n + n − 2
n n
1 2 1 2
In most statistical research studies, population parameters are usually unknown and have to be
estimated from a sample. As such the methods for estimating the population parameters assume an
important role in statistical anlysis.
The random variables (such as X and σ s
2 ) used to estimate population parameters, such as
μ and σ p 2 are conventionally called as ‘estimators’, while specific values of these (such as X = 105
2 or σ s = 2144 . ) are referred to as ‘estimates’ of the population parameters. The estimate of a
population parameter may be one single value or it could be a range of values. In the former case it
is referred as point estimate, whereas in the latter case it is termed as interval estimate. The