∙
An unbiased estimator of the sampling variance of
X
̄
,
Var
X
̄
2
/
n
,
is
S
2
/
n
.
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∙
A consistent estimator of
is
S
, the sample standard deviation. When
we replace
with
S
in
sd
X
̄
/
n
we get the
standard error
of
X
̄
:
se
X
̄
S
n
∙
Notice that what we call the sample standard deviation,
S
, is an
estimator of the population standard deviation,
. The standard error of
X
̄
is an estimator of
sd
X
̄
/
n
.
74

∙
Now again consider testing
H
0
:
0
against any of the three alternatives. (Remember, if
H
1
:
0 then the
null is effectively
H
0
:
≤
0; if
H
1
:
0 the null is effectively
H
0
:
≥
0.)
∙
When we replace
with
S
in the test statistic we get
T
X
̄
se
X
̄
n
X
̄
S
,
which is easily computed given a random sample
X
i
:
i
1,...,
n
.
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