Agriculture Reference
In-Depth Information
As the samples are large, so
ð
s
n
1
s
n
2
Þ N
So the test is nonsignificant and the null
hypothesis cannot be rejected. We conclude that
the variabilities (measured in terms of s.d.) in
two populations are the same.
5.
.
To test the equality of two standard deviations
σ
1
and
2
1
2
2
2
n
2
2
n
1
þ
σ
σ
σ
1
σ
2
;
σ
2
,
that
is,
for null hypothesis
Test for Significance of Specified Population
Proportion
To test the hypothesis
H
0
: σ
1
¼ σ
2
.
The test statistic is
P
0
is the specified population proportion value
with respect to a particular character), the
test statistics would be
H
0
: P ¼ P
0
(where
ð
s
n
1
s
n
2
Þ
σ
1
2
2
s
τ ¼
:
n
1
þ
σ
2
2
n
2
2
P P
0
P
0
ð
r
τ ¼
Nð
0
;
1
Þ:
σ
1
2
and
σ
2
2
are unknown, and for
In practice,
1
P
0
Þ
n
large samples,
σ
2
2
are replaced by the
corresponding sample variances. Hence, the
test statistic becomes
σ
1
2
and
Example 9.21.
Assumed that 10% of the
students fail in mathematics. To test the assump-
tion, randomly selected sample of 50 students
were examined and found that 44 of them are
good. Test whether the assumption is justified or
not at 5% level of significance.
s
n
1
s
n
ð Þ
s
τ ¼
s
;
2
n
1
2
n
2
n
1
þ
s
2
2
n
2
which is a standard normal variate.
Solution.
The null hypothesis will be that the
proportion of passed students is 0.9,
that
is,
Example 9.20.
The following table gives the
features of two independent random samples
drawn from two populations. Test whether the
variability of the two are the same or not.
H
0
: P ¼
0
:
9, against the alternative hypothesis
H
1
: P 6¼
0
:
9. The test statistic is
P P
0
P
0
ð
1
P
0
Þ
n
0
:
90
0
88
0
:
r
r
τ ¼
¼
¼
0
:
41716
:
:
90
0
:
10
Sample
Size
Mean yield (q\ha)
S.D.
50
1
56
25.6
5.3
2
45
20.2
4.6
The calculated value of
jj
is less than the
tabulated value of
at upper 2.5% level of sig-
nificance. So the test is nonsignificant and we
cannot reject the null hypothesis. That means
we conclude that the 10% of the students failed
in mathematics.
6.
τ
Solution.
Let the variability be measured in
terms of standard deviation. So under the given
condition we are to test
H
0
:
The standard deviations are equal against
Test
for Equality
of Two Population
H
1
:
The standard deviations are not equal
;
that is,
Proportions
To test the equality of two population propor-
tions, that is,
H
0
: σ
1
¼ σ
2
against
H
1
: σ
1
6¼ σ
2
:
Let the level of significance be
H
0
: P
1
¼ P
2
¼ P
(known), the
α ¼
0
:
05
:
appropriate test statistic is
Under the above null hypothesis, the test sta-
tistic is
p
1
pð Þ
Pð
r
τ ¼
:
1
PÞ
n
1
þ
Pð
1
PÞ
n
2
s
n
1
s
n
2
s
n
1
2
2
5
6
5
:
3
4
:
τ ¼
s
Nð
0
;
1
Þ; ¼
r
3
2
112
þ
6
2
90
:
4
:
n
1
þ
s
n
2
2
If the value of
P
is unknown, we replace
P
2
n
2
¼
n
1
p
1
þ n
2
p
2
n
1
þ n
2
^
by its unbiased estimator
P
0
:
7
¼
6971
¼
1
:
004
:
0
:
based on both the samples. Thus,
