Agriculture Reference
In-Depth Information
Year
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
Production (m.t)
13.12
13.00
13.30
14.36
10.01
12.22
13.99
8.98
13.99
11.86
Signs
Year
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
Production (m.t)
13.55
13.28
15.80
12.69
14.45
14.18
13.03
13.81
13.48
13.10
Signs
+
Year
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Production (m.t)
15.91
15.52
15.44
14.60
18.20
17.22
15.45
19.64
21.12
19.78
Signs
+
+
+
+
+
+
+
+
+
Year
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
Production (m.t)
20.00
19.51
21.10
21.03
19.79
21.79
21.18
21.18
22.27
21.81
Signs
+
+
+
+
+
+
+
+
+
+
As the sample size is large, one should apply
randomness test through normality test of the
number of runs,
r
. Here
r ¼
6, so the number of
runs is found to be 6.
namely, in six ways. But in only four of these
ways would there be a turning point. Hence,
the probability of turning points in a set of
three values is 4/6
¼
2/3.
Let
U
n
be a set of
observations and let us define a marker vari-
able
U
1
,
U
2
,
U
3
,
...
,
EðrÞ¼
2
þ
1
¼
40
=
2
þ
1
¼
21 and Var
ðrÞ
¼
nðn
X
i
by
2
Þ
40
ð
40
2
Þ
380
39
¼
Þ
¼
Þ
¼
:
9
744
4
ðn
1
4
ð
40
1
X
i
¼
1 when
U
i
<U
iþ
1
>U
iþ
2
and
U
i
>U
iþ
1
<U
iþ
2
6
EðrÞ
Var
6
21
9
¼
15
p
p
τ ¼
¼
122
¼
4
:
806
:
3
:
:
ðrÞ
744
¼
0 otherwise
8; I ¼
1
;
2
;
3
; ...; n
ð
2
Þ:
Let
the level of significance be
α ¼
0
:
05.
Hence, the number of turning points
p
is then
Thus,
jj¼
4
:
806
>
1
:
96
ð
the critical value at
p ¼
P
n
2
α ¼
; hence, the null hypothesis of
randomness is rejected. We conclude that the
production of the state has not changed in ran-
dom manner.
(b)
0
:
05
Þ
i¼
1
x
i
,
then we have
EðpÞ¼
P
n
2
2
i¼
1
Eðx
i
Þ¼
3
ðn
2
Þ
2
ðÞ¼E
P
n
2
2
40
n
2
144
nþ
131
90
and
Ep
i¼
1
ðx
i
Þ
¼
Test of Turning Points
In the test of turning points to test the
randomness of a set of observations, count
peaks and troughs in the series. A “peak” is a
value greater than the two neighboring values,
and a “trough” is a value which is lower than
of its two neighbors. Both the peaks and
troughs are treated as turning points of the
series. At least three consecutive observations
are required to find a turning point, let
2
¼
ð
16
n
29
Þ
90
2
Var
,
the number of observations, increases the
distribution of
ðpÞ¼Ep
ðÞEðpÞ
ð
Þ
.As
n
tends to normality. Thus,
for testing the null hypothesis
p
H
0
: Series is
τ ¼
pEðpÞ
p
Var
random the test statistic,
~
N
ðpÞ
(0,1) and we can conclude accordingly.
Example 9.32b.
The area ('000ha) under cotton
crop in a particular state of India since 1971 is
given below. Test whether the area under jute has
changed randomly or not.
U
1
,
U
2
,
U
3
. If the series is random then these three
values could have occurred in any order,
and
