Agriculture Reference
In-Depth Information
An observed frequency distribution in a
sample may often be supposed to arise
from a true binomial, Poisson, normal, or
some other known type of distribution in
the population. In most cases in practice,
the hypothesis that we want to test is
Solution.
H
0
: The data fit well with the theoretical expec-
tation against
H
1
: The data do not fit well with the theoretical
expectation.
H
0
:
The sample has been drawn from a parent
population of certain type.
This hypothesis may be tested by compar-
ing the observed frequencies in various
classes with those which would be given
by the assumed theoretical distribution.
Usually the parameters of this distribution
may not be known but will have to be
estimated from the sample. The test statis-
tic under
Let the level of significance be
α ¼
0.05. The
test statistic is given by
X
n
2
ð
O
i
E
i
Þ
2
χ
¼
at
ðk
2
1
Þ
d
:
f
:
E
i
i¼
1
¼
14
:
896 at 5 d
:
f
:
H
0
is
(Obs.
exp.)
2
/exp.
Class
Obs.
Exp.
1
2
1
1
2
X
k
X
k
2
ðO
i
n P
i
Þ
ðO
i
e
i
Þ
2
4
7
1.285714
2
χ
¼
¼
n P
0
e
i
3
26
28
0.142857
i¼
1
i¼
1
4
60
56
0.285714
5
62
58
0.275862
2
distribution with (
which is
χ
k s
1)
6
29
31
0.129032
d.f. where
1) is the number of
parameters of this distribution to be
estimated from the sample and
P
i
is the
estimated probability that a single item in
the sample falls in the
s
(
<k
7
4
9
2.777778
8
4
1
9
0
From the table, we have
χ
¼
11
:
07, that
:
05
;
5
2
th class which is a
function of the estimated parameters. We
reject
i
is, the calculated value of
χ
is greater than the
2
. So the test is significant and the
data do not fit well with the expected distribution.
(e)
table value of
χ
2
2
H
0
if cal
χ
> χ
α;ks
1
, and otherwise
χ
Test for Independence of Attributes
A large number of statistical theories are
available which deal with quantitative
characters.
2
we accept it.
Example 9.26.
While analyzing a set of data
following frequency (observed and expected) are
obtained by a group of students. Test whether the
data fitted well with the expected distribution
or not.
2
test is one of the few avail-
able statistical tools dealing with qualita-
tive characters (attributes).
χ
2
test for
independence of attributes work out the
association or independence between the
qualitative characters like color, shape,
and texture which cannot be measured
but grouped.
χ
Class
Observed frequency
Expected frequency
2-4
2
1
4-6
4
7
2
test for independence of
attributes helps in finding the dependence
or independence between such qualitative
characters
χ
6-8
26
28
8-10
60
56
10-12
62
58
from their
joint
frequency
12-14
29
31
distribution.
Let us suppose two attributes
14-16
4
9
A
and
B
16-18
4
1
grouped into
r
(
A
1
,
A
2
,
...
,
A
r
) and
