Agriculture Reference
In-Depth Information
Solution.
Under the given condition, the null
hypothesis is
H
0
: P
1
¼ P
2
against the alternative
hypothesis
p
1
pð Þ
τ ¼
s
:
1
n
1
þ
1
n
2
^
^
P
1
P
H
1
: P
1
6¼ P
2
.
That means there exists no significant differ-
ence between the two proportions against the
existence of significant difference.
Let
Example 9.22.
From two large samples of 500
and 600 of electric bulbs, 30 and 25%, respec-
tively, are found to be defective. Can we con-
clude that the proportions of defective bulbs are
equal in both the lots?
the level of significance be
α ¼
0.05,
and the test
statistic for
the above null
hypothesis is
p
1
pð Þ
Pð
P ¼
n
1
p
1
þ n
2
p
2
500
0
:
3
þ
600
0
:
25
3
11
τ ¼
s
;
where
n
1
þ n
2
¼
¼
500
þ
600
1
n
1
þ
1
n
2
PÞ
1
ð Þ
3
11
0
:
3
0
:
25
05
0
:
02697
¼
0
:
nτ ¼
s
¼
:
:
1
8539
8
11
1
500
þ
1
600
Since the calculated value of
jj
is less than
is distributed as
χ
2
with
k
d.f. as
n
1
,
n
2
,
n
3
,
the tabulated value of
(1.96),wehaveto
accept the null hypothesis and conclude that
proportions of damaged bulbs are equal in both
the lots.
7.
τ
...
,
n
k
!1
.
To test
H
0
: P
1
¼ P
2
¼ P
3
¼¼P
k
¼
P ð
known
Þ
against all alternatives,
the
test statistic is
2
χ
Test
The
(
)
2
2
test is one of the most important
and widely used tests in testing of hypo-
thesis.Itisusedbothinparametricandin
nonparametric tests. Some of the impor-
tant uses of
χ
X
k
x
i
n
i
P
n
i
Pð
2
χ
¼
p
with
k
d
:
f
:;
1
PÞ
i¼
1
2
2
and if cal
χ
> χ
α;k
;
we reject
H
o
:
2
test are testing equality of
proportions, testing homogeneity or signifi-
cance of population variance, test for good-
ness of fit, tests for association between
attributes, etc.
(a)
χ
In practice
will be unknown. The unbi-
ased estimate of
P
P
is
P ¼
x
1
þ x
2
þþx
k
n
1
þ n
2
þþnk
:
Then the statistic
2
χ
Test for Equality of k
(
2)
Population
0
@
1
A
2
Proportions
Let
X
k
x
i
n
i
P
n
i
Pð
X
k
be independent ran-
dom variables with
X
1
,
X
2
,
X
3
,
...
,
2
χ
¼
q
X
i
~
B
(
n
i
,
P
i
),
i ¼
1, 2,
PÞ
1
i¼
1
3,
...
,
k
;
k
2. The random variable
(
)
2
2
with (
is asymptotically
χ
k
1) d.f.
X
k
X
i
n
i
P
i
n
i
P
i
ð
1
P
i
Þ
2
2
If cal
χ
>χ
α;
k
1
, we reject
H
0
: P
1
¼
p
P
2
¼ P
3
¼¼P
k
¼ P ð
unknown
Þ
.
i¼
1
