Agriculture Reference
In-Depth Information
Year
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
Area
326
435
446
457
404
423
496
269
437
407
Year
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
Area
461
367
419
370
335
441
479
538
504
610
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Area
506
439
464
534
731
518
424
415
427
500
Year
2001
2002
2003
2004
2005
2006
2007
2008
200
2010
Area
573
493
475
508
516
620
642
642
614
613
From the given information, one has (1) num-
ber of observations
¼
40 and (2) number of
turning points
distribution. Let
x
n
be a random
sample from a population of distribution func-
tion
x
1
,
x
2
,
x
3
,
...
,
p ¼
20. The null hypothesis is
), and the sample cumulative distribu-
tion function is given as
F
(
x
given as
H
0
: The series is random.
We have the expectation of turning point (
F
n
(
x
)where
F
n
(
x
)is
p
),
defined as
is the number
of observations equal to or less than
F
n
ðxÞ¼k=
where
k
2
2
76
x
.Now
EðpÞ¼
3
ðn
2
Þ¼
3
ð
40
2
Þ¼
3
¼
25
:
33 and
for fixed value of
) is a statistic since
it depends on the sample, and it follows a bino-
mial distribution with parameter (
x
,
F
n
(
x
the variance
n
F
x
,
(
)). To
16
n
29
90
16
40
29
90
611
90
¼
var
ðpÞ¼
¼
¼
6
:
789
:
test both-sided goodness of fit
for
H
0
:
F
x
¼ F
0
(
x
x
(
)
) for all
against the alternative
Thus, the test statistic
hypothesis
, the test statistic
is
D
n
¼
Sup
x
F
n
ðxÞF
0
ðxÞ
H
1
:
FðxÞ 6¼ F
0
ðxÞ
½
j
j
. The distribu-
τ ¼
p EðpÞ
:
¼
:
20
33
6
25
5
33
tion of
D
n
does not depend on
F
0
so long
F
0
is
p
Var
¼
p
606
¼
2
:
045
:
2
:
continuous. Now if
F
0
represents the actual
distribution function of
ðpÞ
:
789
x
,
then one would
expect very small value of
D
n
; on the other
is standard normal variate,
and the value of standard normal variate at
P ¼
We know that the
τ
hand, a large value of
D
n
is an indication of the
deviation of distribution function from
F
0
.The
0.05 is 1.96. As the calculated value of
Table
A.10
decision is taken with the help of
.
jj>
96, so the test is significant, we reject the
null hypothesis. We conclude that at 5% level
of significance, there is no reason to take that the
area under jute in West Bengal has changed
randomly since 1961.
Note
1
:
Example 9.33.
The following data presents a
random sample of the proportion of insects killed
by an insecticide in ten different jars. Assuming
the proportion of insect killing varying between
(0,1), test whether the proportion of insect killed
follow rectangular distribution or not.
Proportion of insect killed: 0.404, 0.524,
0.217, 0.942, 0.089, 0.486, 0.394, 0.358, 0.278,
0.572
: Data for the above two examples have been
taken from the various issues of the economic
review published by the Government of West
Bengal.
3.
Kolmogorov-Smirnov One Sample Test
χ
2
test for goodness of fit is valid under certain
assumptions like large sample size. The paral-
lel test to the
Solution.
If
F
0
(
x
) be the distribution function of
a rectangular distribution over the range [0,1],
then
2
tests which can also be used
under small sample conditions is Kolmogorov-
Smirnov one-sample test. We test the null
hypothesis that the sample of observations
χ
H
0
:
F
(
x
)
¼ F
0
(
x
). We know that
x
1
,
F
0
ðxÞ¼
0if
x <
0
x
2
,
...x
n
has come from a specified popula-
tion distribution against the alternative hypo-
thesis that the sample has come from other
x
3
,
¼ x
if 0
x
1
¼
1if
x >
1
