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MTH/STA 562
COMMON LARGESAMPLE STATISTICAL TESTS
In the preceding section, we have laid out a ground work for hypothesis testing which
statistic, and rejection region. In practice, the null hypothesis is usually stated so as to
specify an exact value of the population parameter
, namely
H
0
:
=
&
0
, where
&
0
is a
. However, there are in general three di/erent kinds of alternative hy
potheses associated with this null hypothesis; they are uppertail alternative
H
a
:
& > &
0
,
lowertail alternative
H
a
:
& < &
0
, and twotailed alternative
H
a
:
6
=
&
0
. In the present
section, we shall formally develop the testing procedure with respect to each of the above
three alternatives for hypothesis testing on the basis of large samples.
For any single population, the sample mean
Y
and sample proportion
b
p
are known to be
respective unbiased estimators of the population mean
±
and population proportion
p
, with
standard errors given as
²
Y
=
²
p
n
and
²
b
p
=
r
p
(1
p
)
n
;
respectively, where
²
is the population standard deviation and
n
is the sample size. Similarly,
for the sake of comparing two populations, the di/erence between two sample means,
Y
1
Y
2
,
is an unbiased estimator for the di/erence between two population means,
±
1
±
2
, with
standard error
²
Y
1
Y
2
=
s
²
2
1
n
1
+
²
2
2
n
2
;
whereas the di/erence between two sample proportions,
b
p
1
b
p
2
, is an unbiased estimator for
the di/erence between two population proportions,
p
1
p
2
, with standard error
²
b
p
1
b
p
2
=
s
p
1
(1
p
1
)
n
1
+
p
2
(1
p
2
)
n
2
;
where
²
2
1
and
²
2
2
are respective population variances and
n
1
and
n
2
are respective sample
sizes. In view of the
Central Limit Theorem
, it follows that each of the above four unbiased
estimators has an approximately normal distribution when samples are large. If the unknown
population parameter
is set to be one of the four population parameters,
±
,
p
,
±
1
±
2
, or
p
1
p
2
, then for large samples, the random quantity
Z
=
b
²
b
(10.3.1)
has approximately a standard normal distribution.
1
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View Full Document Consider the problem of testing the hypothesis that the unknown population parameter
&
0
on the basis of a large random sample
Y
1
,
Y
2
,
,
Y
n
.
Throughout this section, it is assumed that the unbiased estimator
b
has an approximately
normal distribution with mean
and standard error
±
b
=
r
V ar
b
±
. This is a reasonable
(10
:
3
:
1)
will
serve as (at least approximately) a test statistic in the following procedures for hypothesis
testing on the basis of large samples.
I. UpperTail Test
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This note was uploaded on 04/20/2008 for the course MTH 562 taught by Professor Cheng during the Spring '08 term at Creighton.
 Spring '08
 Cheng

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