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Chapter 12: Hypothesis Tests – Large Samples
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difference is due to the fact that the corresponding population proportions
are not the same or that the proportions are from different populations. In
this section, we will discuss problems in which we compare two population
proportions.
Figure 12- 18
shows the experimental display of the sampling populations
with respective proportions
and
. Independent samples of size
and
are selected from the respective populations and the respective number of
successes observed are
and
.
Figure 12- 18:
Experimental Display for Sampling from
the two populations with respective proportions
p
1
and p
2
We will assume large samples when performing the hypothesis tests for
comparing the two population proportions. Here, large sample is assumed
when
> 5,
> 5,
> 5 and
> 5.
Since we will be testing for the difference between two population
proportions, we first will establish properties of the sampling distribution for
the difference between the two proportions. The point estimate for the
difference between the two population proportions is the difference between
the two sample proportions. So if
and
are the respective population
proportions and we let
̂
and
̂
to be the corresponding sample
proportions, then the point estimate for
-
will be
̂
-
̂
where
̂
and
̂
. Observe from
Figure 12-18
that
and
are the number of
successes fromthe respective independent samples of size
and
.
Now as the sample sizes
and
increase, the shape of the distribution of
the differences of the sample proportions obtained from any population will