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2
2
¼ x 1 n 1 P
þ x 2 n 2 P
2
In particular when
k ¼
2
; χ
:
n 1 P
P
n 2 P
P
1
1
P ¼ x 1 þ x 2
Since
n 1 þ n 2 , on simplification
;
we get
2
ð
p 1 p 2
Þ
2
χ
¼
with 1 d
:
:
f
which is equivalent to
1
n 1 þ
1
n 2
P
P
1
p 1 p 2
P
τ ¼
s
for testing
H 0 : P 1 ¼ P 2 ¼ P ð
unknown
Þ
provided in test 6
:
1
n 1 þ
1
n 2
P
1
2
2
(b)
χ
Bartlett's Test
for Homogeneity of
essentially a
χ
test. We suppose that we
Variances
Proposed by Bartlett also known as
Bartlett's test for homogeneity of
variances. When experiments are repeated
in several seasons/places/years, pooling
the data becomes necessary; test for
homogeneity of variance is necessary
before pooling the data of such
experiments. When more than two sets
of experimental data are required to be
pooled, then homogeneity of variance
test through
have
k
samples (
k >
2) and that for the
i
th
sample of size
n i , the observed values are
x ij (
).
We also assume that all the samples are
independent and come from the normal
populations with means
j ¼
1, 2, 3, 4,
...
,
n i ;
i ¼
1, 2, 3,
...
,
k
μ 1 ; μ 1 ; μ 1 ; ...;
1 ; σ
2 ; σ
3 ; ...; σ
2
k
μ k
and variances
σ
.To
1
2
test the null hypothesis
H 0 : σ
¼ σ
¼
¼σ
2
k ¼ σ
ð
2
;
say
Þ
against the alterna-
tive hypothesis
H 1 is that these variances
are not all equal, the approximate test sta-
tistic is
statistic as mentioned ear-
lier does not serve the purpose. Bartlett's
test
F
for homogeneity of variances
is
P k
1 ðn i
e s
2
1
Þ
log
s i
2
2
2
8
<
9
=
χ
¼
where
k 1
4
X k
1
1
ðn i
1
1
þ
Þ
:
;
3
ðk
1
Þ
1
P k
1 ðn i
1
1
Þ
1 n i
2
1
n i
2
i ¼
s
1 x ij x i
¼
sample mean square for the
i
th sample
ði ¼
1
;
2
;
3
; ...kÞ
,
P k
1 ðn i
2
1
Þðs i Þ
2
and
s
¼
:
P k
1 ðn i
1
Þ
2
2
If cal
χ
> χ
α;k 1 ; H 0 is rejected.
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