Agriculture Reference
In-Depth Information
9
Formulation and Testing of Hypothesis
The main objective of the researchers is to study
the population behavior to draw the inferences
about the population, and in doing so, in most
of the cases the researcher uses sample
observations. As samples are part of the popula-
tion, there are possibilities of difference in sample
behavior from that of population behavior. Thus,
the process/technique is of knowing accurately
and efficiently the unknown population behavior
from the statistical analysis of the sample behav-
ior—known as
of estimation, namely,
point estimation
and
interval estimation
.
9.1.1 Point Estimation
Suppose
x
1
,
x
2
,
x
3
,
...
,
x
n
is a random sample
from a density
f
(
x
/
θ
), where
θ
is an unknown
parametric. Let
t
be a function of
x
1
,
x
2
,
x
3
,
...
,
x
n
and is used to estimate
θ
t
, then
is called a
.
Problems in statistical inference:
statistical inference
point estimator of
. Among the many estimators
based on sample observations, a good estimator
is one which is (a)
θ
(a)
Estimation
: Scientific assessment of the pop-
ulation characters from sample observations.
unbiased
, (b)
consistent
, and
(c)
efficient and sufficient
.
The details of these properties and their esti-
mation procedures are beyond the scope of this
book.
Testing of hypothesis
(b)
: Some information or
some hypothetical values about the popula-
tion parameter may be known or available
but it is required to be tested how far these
information or hypothetical values are
acceptable or not acceptable in the light of
the information obtained from the sample
supposed to have been drawn from the
same population.
9.1.2 Interval Estimation
In contrast to the point estimation method, in
interval estimation, we are always in search of a
probability statement about the unknown para-
meter
θ
of the population from which the sample
has been drawn. If
9.1
Estimation
x
n
be a random
sample drawn from a population, we shall be
always in search of two functions
x
1
,
x
2
,
x
3
,
...
,
Let (
x
n
) be a random sample drawn
from a population having density
f
(
x
/
θ
) where
θ
is an unknown parameter. Estimation is the prob-
lem of estimating the value
x
1
,
x
2
,
x
3
,
...
,
u
1
and
u
2
such that the probability of
θ
remains in the
interval (
u
1
,
u
2
) is given by a value say 1
α
.
with the help of the
sample observations. There are mainly two types
θ
Thus,
. This type of
interval, if exists, is called confidence interval
Pu
1
θ u
2
ð
Þ ¼
1
α
