Agriculture Reference
In-Depth Information
median
θ
, we are to test
H
0
:
θ ¼ θ
0
against
There are 10 (
)
minus (
) signs, and one observation is equal to
the median value and discarded. This
¼ r
) plus (+) signs and 9 (
¼ s
the
alternative
hypotheses
ð
1
Þ θ 6¼ θ
0
;
ð
2
Þ θ > θ
0
;
and
ð
3
Þθ < θ
0
.
r
is a bino-
If
θ
0
be the median, then there will be equal
number of observations below and above the
value
mial variate with parameter
(
m ¼ r
+
s ¼
20
1
¼
19) and
P
(
¼
1/2). Thus, testing of
θ
0
. We denote the observations greater
than the
H
0
: θ ¼
10 ft against
H
1
: θ 6¼
10 ft is equiva-
θ
0
with plus (+) signs and the
observations smaller than
lent to testing of
H
0
:
P ¼
½ against
H
1
:
P 6¼
1/2.
)
signs and ignore the sample values equal to
the median. Let item be
θ
0
with minus (
The critical region for
α ¼
0.05 (two-sided
r r
α=
2
and
r r
0
α=
2
where
r
test) is
is the num-
ber of plus signs and
r
α=
2
and
r
0
α=
2
are the smallest
and largest integer, respectively, such that
P
19
r
α=
2
plus (+) sign and
minus signs. Such that
r
+
s ¼ m n
. Distri-
bution of
r
is binomial with
probability ½. Thus, the above null hypothesis
becomes equivalent to testing
r
given
r
+
s ¼ m
!
1
2
!
1
2
19
19
19
x
α=
2 and
P
r
0
α=
2
0
19
x
H
0
:
P ¼
1/2,
where
x > θ
0
). One-tailed cumulative
binomial probabilities are given in Table
A.9
.
For H
1
:
P ¼ P
(
α=
2.
From the table we get
r
0
0
:
025
¼
r
0
:
025
¼
14 and
θ 6¼ θ
0
the critical
region for
α
4 for 19 distinct observations at
p ¼
1
=
2. For
0
α=
2
and
level of significance is given by
r r
this example we have 4/
14, which
lies between 4 and 14, so we cannot reject the
null hypothesis at 5% level of significance, that
is, we conclude that the median of the long jump
can be taken as 10 ft.
[If total number of signs, that is, plus (+) signs
plus the minus (
r ¼
10
<
r r
α=
2
where r
0
2
and
r
α=
2
are the largest
α=
that
P
r
0
α=
2
r¼
0
and
smallest
integer
such
!
m
α=
!
1
2
m
m
r
2 and
P
r¼α=
2
m
r
) signs (i.e.,
r
+
s ¼ m
), is
α=
2. For
H
1
:
θ > θ
0
, the critical region for
greater than 25, that is,
25, then
normal approximation to binomial may be used,
and accordingly the probability of
m ¼ r
+
s >
α
level of significance is given by
r r
α
;
r
α
where
is the smallest integer
such that
!
2
m
r
or fewer
P
r¼r
α
m
r
success will be
tested with the
statistic
α:
For
H
1
:
θ < θ
0
,
the
τ ¼
r
rþs
rs
2
¼
r s
2
rþs
4
q ¼
rþs
p
r þ s
.]
p
2
critical region for ^
α
level of significance is
0
α
0
α
given by
r r
where
r
is the larger integer
!
1
2
2.
Test of Randomness
(a)
α:
such that
P
r¼
0
m
r
One Sample Run Test
In socioeconomic or time series analysis,
the researchers often wants to know
whether the data point or observations
have changed following a definite pattern
or in a haphazard manner. One-sample run
test is used to test the hypothesis that a
sample is random. In other form, run test
is used to test the hypothesis that the given
sequence/arrangement is random.
A run is a sequence of letters (signs) of the
same kind bounded by letters (signs) of
Example 9.30.
Test whether the median long
jump (
) of a group of students is 10 ft or not at
5% level of significance from the following data
on long jumps.
Long jumps (feet): 11.2, 12.2, 10.0, 9, 13,
12.3, 9.3, 9.8, 10.8, 11, 11.6, 10.6, 8.5, 9.5, 9.8,
11.6, 9.7, 10.9, 9.2, 9.8
θ
Solution.
Let us first assign the signs to each of
the given observations as follows:
Long jump
10
9
11.2
12.2
10
9
13
12.3
10.8
11
9.3
9.8
Signs
+
+
+
+
+
+
Long jump
8.5
9.5
11.6
10.6
8.5
9.5
9.8
11.6
9.2
9.8
9.7
10.9