Agriculture Reference
In-Depth Information
1.
Test for Specified Values of Population Mean
with Known Population Variance from a
Normal Population
To test
known variance,
the appropriate test statistic
will be
τ ¼
x μ
0
H
0
:
μ ¼ μ
0
against
the alternative
σ
X
=
p :
hypotheses
H
1
:
ð
1
Þ μ 6¼ μ
0
; ð
2
Þ μ>μ
0
;
and
2
.
ð
3
Þ μ<μ
0
with known population variance
σ
The test
statistic under
the given null
For this problem,
hypothesis is
45
50
9
5
p
τ ¼
¼
949
¼
5
:
269
:
:
τ ¼
x μ
0
0
=
10
σ=
p ;
From the table of the standard normal variate,
we have
where
x
is the sample mean and this
τ
follows
.So
the test is significant at 1% level of significance.
Hence, we reject the null hypothesis, that is,
H
0
:
τ
0
:
01
ð¼
2
:
576
Þ jj
cal
ð¼
5
:
269
Þ
a standard normal distribution.
Conclusion:
(a) For
H
1
:
μ 6¼ μ
0
, reject
H
0
if the calculated
μ ¼ μ
0
. So the average student weight is
not 50 kg.
2.
value of
jj< τ
α=
2
, where
τ
α=
2
is the table
Test for Specified Value of Population Mean
with Unknown Population Variance from a
Normal Population
To test
2 level of significance
(e.g., 1.96 or 2.576 at 5 or 1% level of signif-
icance, respectively); otherwise accept it.
(b) For
value of
τ
at upper
α=
H
0
:
μ ¼ μ
0
against
the alternative
H
1
:
μ>μ
0
, reject
H
0
if the calculated
hypotheses
H
1
:
ð
1
Þ μ 6¼ μ
0
; ð
2
Þ μ>μ
0
;
and
value of
τ > τ
α
, where
τ
α
is the table value
ð
3
Þ μ<μ
0
with
unknown
population
level of significance (e.g.,
1.645 or 2.33 at 5 or 1% level of signifi-
cance, respectively); otherwise accept it.
(c) For
of
τ
at upper
α
2
.
The test statistic under the given null hypothesis
is
variance
σ
s=
n
p
H
1
:
μ<μ
0
, reject
H
0
if the calculated
t ¼ x μ
0
=
ð
Þ
,where(
n
1) is the
2
are the
sample mean and sample mean square,
value of
τ > τ
1
α
, where
is the table
τ
1
α
degrees of freedom, and
x
and
s
value of
τ
at lower
α
level of significance
n
1
n
i¼
(e.g.,
1.645 or
2.33 at 5 or 1% level
2
2
1
s
¼
1
xi x
ð
Þ
(an unbiased estimator
of
significance,
respectively); otherwise
accept it.
2
).
of population variance
σ
Conclusion
(a) For
Example 9.1.
A random sample drawn from
ten students of a college found that the average
weight of the students is 45 kg. Test whether
this average weight of the students be taken as
50 kg at 1% level of significance or not, given
that the weight of the students follows a nor-
mal distribution with variance 9 kg
2
.
H
1
:
μ 6¼ μ
0
, reject
H
0
if the calculated
value of
t > t
α=
2
;
n
1
or cal
t < t
ð
1
αÞ=
2
;n
1
¼
t
α=
2
;n
1
i
t
α=
2
;n
1
is the table value of
t
at upper
α=
2 level of
significance with (
:
:
jj
> t
2
;n
1
where
e
cal
n
1) d.f.; otherwise
accept it.
(b) For
H
1
:
μ>μ
0
, reject
H
0
if the calculated
value of
t > t
α;n
1
, where
τ
α;n
1
is the table
Solution.
Given that (1) the population is nor-
mal with variance 9, (2) sample mean (
value of
t
at upper
α
level of significance
)is45
and (3) population hypothetical mean is 50.
To test
x
with (
n
1)d.f.; otherwise accept it.
(c) For
H
1
:
μ<μ
0
, reject
H
0
if the calculated
H
0
: Population mean
μ ¼
50 against
value of
t < t
1
α;n
1
, where
t
1
α;n
1
is the
H
1
:
50.
This is a both-sided test. As the sample has
been drawn from a normal population with
μ 6¼
table value of
t
at lower
α
level of signifi-
n
cance with (
1)d.f.; otherwise accept it.
