HYPOTHESIS TESTING
We have already described how the sample information is used to estimate the
parameters of the underlying population distributions. Now, we intend to use our
sample information to test existing presumptions regarding the values of parame
ters. More generally, we are concerned with testing hypotheses regarding the state
of nature.
The initial presumptions, which are formed without the benefit of the statistical
evidence, constitute the null or prior hypothesis, denoted by
H
0
. The alternative
hypothesis, denoted by
H
1
, is what we shall assert if we reject
H
0
in the light of
the statistical evidence.
Together,
H
0
and
H
1
ought to comprises all possibilities. Thus, the set Ω of
all states of nature is partitioned as Ω =
{
Ω
0
∪
Ω
1
;Ω
0
∩
Ω
1
=
∅}
in a manner
corresponding the hypotheses. The decision to maintain the null hypotheses after
the evidence has been reviewed will be denoted by
d
0
; and the decision to reject it
in favour of the alternative hypothesis will be denoted by
d
1
.
The procedure for testing an hypothesis will depend upon our forming a test
statistic
x
from a random sample. The decision will depend upon the value of the
statistic. Let
S
=
{
C
∪
C
C
;
C
∩
C
C
=
∅}
be the sample space comprising all possible
values of
x
. This is partitioned into the critical region
C
and its complement, which
is the noncritical region
C
C
. Then, the decision rule is as follows:
x
∈
C
C
=
⇒
d
0
,
x
∈
C
=
⇒
d
1
.
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 Spring '12
 D.S.G.Pollock
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, Type I and type II errors

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