Agriculture Reference
In-Depth Information
2
2
2
and the conclusion is drawn accordingly. But
if the
S ¼ R
1
R
j
þ R
2
R
j
þþ R
8
R
j
7, then a table has been prepared
for different critical values of
N
2
2
2
¼
24
24
ð
Þ
þ
22
27
ð
Þ
þþ
19
27
ð
Þ
S
associated
¼
664
:
with
W
. If the observed
S
the value shown
in table, then
sets of ranking are inde-
pendent may be rejected at the particular level
of significance.
H
0
:
K
S
664
W ¼
¼
¼
0
:
439
:
1
12
K
1
12
2
2
6
2
88
2
NN
1
1
Example 9.38.
In a heptathlon event, eight
athletes were judged by six judges, and the fol-
lowing table shows the rank provided by each
judge to all the athletes. Test whether the judges
independently worked or not.
2
As
N >
7, we can use the approximate
χ
¼
KðN
1
ÞW
with
ðN
1
Þ
d
:
f
:
Here
N ¼
8,
2
K ¼
6, and
W ¼
0.439,
χ
¼ KðN
1
ÞW ¼
6
ð
8
1
Þ
0
:
439
¼
18
:
438;
table value of
cal
. Thus, the null hypothesis
of independence of judgment is rejected, and we
conclude that
0
:
05
;
9
¼
16
:
919
< χ
χ
Athletes
12345678
Ranks
Judge 1 24357816
Judge 2 32456718
Judge 3 43567821
Judge 4 54376821
Judge 5 45687321
Judge 6 64875132
W
is significant at 5% level of
significance.
Judges
9.5.2 Two-Sample Test
1.
Paired-Sample Sign Test
The sign test for one sample mentioned above
can be easily modified to apply to sampling
from a bivariate population. Let a random
sample of
n
pairs (
x
1
,
y
1
), (
x
2
,
y
2
), (
x
3
,
y
3
),
...
,
(
Solution.
There are six sets of ranking for eight
objects. Kendall's coefficient of concordance can
very well be worked out followed by the test of
H
0
:
x
n
,
y
n
) be drawn from a bivariate population.
Let
). It is
assumed that the distribution function of dif-
ference,
d
i
¼ x
i
y
i
(
i ¼
1, 2, 3,
...
,
n
W ¼
0. For this purpose we make the follow-
ing table:
d
i
, is also continuous. We want to test
H
0
: Med(
D
)
¼
0,
that
is,
P
(
D >
0)
¼ P
(
D
<
0)
¼
12
=
. It is to be noted that Med(
D
)is
Athletes
12 3 4 5 6 7 8
X
Y
not necessarily equal to Med (
) - Med(
), so
Judges
that
H
0
is not that Med(
X
)
¼
Med(
Y
), but the
Judge 1
2 4
3
5
7
8
1
6
Med(
0. Like the one-sample sign test,
we assign plus (+) and minus (
D
)
¼
Judge 2
3 2
4
5
6
7
1
8
) signs to the
difference values which are greater and lesser
than zero, respectively. We perform the one-
sample sign test as given in the previous sec-
tion and conclude accordingly.
Judge 3
4 3
5
6
7
8
2
1
Judge 4
5 4
3
7
6
8
2
1
Judge 5
4 5
6
8
7
3
2
1
Judge 6
6 4
8
7
5
1
3
2
Total rank (
R
j
) 4 2 9 8 8 5 1 9
2
9 25
4
121
121
64
256
64
R
j
R
j
Example 9.39.
Ten students were subjected to
physical training. Their body weights before and
after the training were recorded as given below.
Test whether there is any significant effect of
training on body weight or not.
N
j¼
1
R
j
¼
R
j
¼
R
1
þ R
2
þþR
8
8
1
N
¼
27
:
