Hypothesis Test of a Proportion, Given a Small Sample
In the
previous lesson, we showed how
to conduct a hypothesis test for a proportion when the sample
included at least 10 successes and 10 failures. This requirement
serves two purposes:
 It guarantees that the sample size will be at least 20 when
the proportion is 0.5.
 It ensures that the minimum acceptable sample size
increases as the proportion becomes more extreme.
When the sample does not include at least 10 successes and 10
failures, the sample size will be too small to justify the
hypothesis testing approach presented in the previous lesson.
This lesson describes how to test a hypothesis about
a proportion when the sample size is small, as long as the sample includes
at least one success and one failure. The key steps are:

Formulate the hypotheses to be tested. This means stating the
null hypothesis and the
alternative hypothesis.

Determine the sampling
distribution of the proportion. If the sample proportion is the outcome
of a binomial
experiment, the sampling distribution will be binomial. If it is the
outcome of a hypergeometric
experiment, the sampling distribution will be hypergeometric.

Specify the significance
level. (Researchers often set the significance level equal to 0.05 or
0.01, although other values may be used.)

Based on the hypotheses, the sampling distribution, and the significance level,
define the region of
acceptance.

Test the null hypothesis. If the sample proportion falls within the region
of acceptance, accept the null hypothesis; otherwise, reject the null
hypothesis.
The following hypothesis testing examples illustrate how this works. The first example involves a
binomial experiment; and the second example, a hypergeometric experiment.
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Example 1: Sampling With Replacement
Suppose an urn contains 30 marbles. Some marbles are red, and the rest are
green. A researcher hypothesizes that the urn contains 15 or more red marbles.
The researcher randomly samples five marbles,
with replacement, from the urn. Two of the selected marbles are red,
and three are green. Based on the sample results, should the researcher accept
or reject the hypothesis. Use a significance level of 0.20.
Solution: There are five steps in conducting a hypothesis test, as
described in the previous section. We work through each of the five steps below:
Example 2: Sampling Without Replacement
The Acme Advertising company has 25 clients. Account executives at Acme claim
that 80 percent of these clients are very satisfied with the service they
receive. To test that claim, Acme's CEO commissions a survey of 10 clients.
Survey participants are randomly sampled,
without replacement, from the client population. Six of the ten sampled
customers (i.e., 60 percent) say that they are very satisfied. Based on the
sample results, should the CEO accept or reject the hypothesis that 80 percent
of Acme's clients are very satisfied. Use a significance level of 0.10.
Solution: There are five steps in conducting a hypothesis test, as
described in the previous section. We work through each of the five steps below: