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The discussion above is an example of a hypothesis test for a population proportion. There are
three basic sets of hypotheses that can be tested. In all cases we make use of the central limit
theorem for proportions: if the sample size is large, then the sampling distribution of the sample
proportion is approximately normal with mean and s.d.
where is
the population proportion and n is the sample size.

The burden of proof is to show that , where is the reference value that is
initially assumed to be true. The test statistic for this test is
The pvalue is
, the area to the right of under the standard normal
density. Note that the s.d. used for the test statistic is not the same as what we used for
confidence intervals.

The burden of proof is to show that , where is the reference value that is
initially assumed to be true. The test statistic for this test is
The pvalue is
, the area to the left of under the standard normal
density. Note that evidence for the alternative hypothesis would be a sample proportion that is
less than in which case the test statistic would be negative.
 The previous two sets of hypotheses are examples of onesided tests  we only are interested
in detecting the possibility that the population proportion falls on one particular side of the
reference value. In the first case, we only are interested in showing that . The
possibility that might be less than instead of equal to is of no concern and
so these two are lumped together into the null hypothesis that . However, there are
situations in which three separate actions would be taken depending on whether ,
, or . In this case we must proceed in two steps. The first is to test
twosided hypotheses,
If the null hypothesis is not rejected, then we take the action associated with the hypothesis
. If the null hypothesis is rejected, then we take the action associated with the
hypothesis if
and we take the action associated with the hypothesis
if
. The test statistic is and the pvalue for this
twosided test includes both tail areas,
One important characteristic of a hypothesis test is its power function which represents the
probability that the test will reject the null hypothesis expressed as a function of the actual
population proportion. Suppose in the example above the level of significance was 5%. In this case
the null hypothesis, , would be rejected if
since
. This is equivalent to
If the actual population proportion is , then the central limit theorem tells us that
the samping distribution is approximately normal with mean and s.d.
.
Therefore
For example, if , then the probability this test will reject the
null hypothesis is
That is, if the population proportion is actually , then there is a 92.4% chance this test will end
up rejecting the null hypothesis. Note that the power function evaluated at the null hypothesis value (in this
example, 0.25) is equal to the level of significance of the test, . Power functions for the other
sets of hypotheses are obtained similarly.
Next: Hypothesis tests for a
Up: Statistical Decisions
Previous: Statistical Decisions
ammann
20171116