Page 573 - Probability and Statistical Inference

550 12. Large-Sample Inference

for (p - p ) with approximate confidence coefficient 1 − α.
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Tests of Hypotheses for the Success Probability
Let us first pay attention to the one-sample problems. With preassigned α
∈ (0, 1), we test a null hypothesis H : p = p with an approximate level α. The
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0
specific alternative hypotheses will be given shortly. Here, 0 < p < 1 is a
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fixed. We consider the test statistic

for large n when p = p .
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Upper-Sided Alternative Hypothesis
Here we test H : p = p versus H : p > p . See, for example, the Figure
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12.3.2. We can propose the following upper-sided approximate level a test:
Lower-Sided Alternative Hypothesis
Here we to test H : p = p versus H : p < p . See, for example, the Figure
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12.3.3. We can propose the following lower sided approximate level α test:
Two-Sided Alternative Hypothesis
Here we test H : p = p versus H : p < p . See, for example, the Figure
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1
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12.3.4. We can propose the following two sided approximate level a test:

Remark 12.3.2 In the case of the upper- or lower-sided alternative
hypothesis, one should in fact prefer to use the corresponding randomized
UMP tests discussed in Chapter 8 instead of the proposed approximate
tests given by (12.3.23) and (12.3.24). On the other hand, when n is
large, finding the cut-off points for implementing the UMP test may be-
come too cumbersome. Hence, the approximate tests in proposed (12.3.23)
and (12.3.24) are also used in practice whenever n is large.
Next, let us briefly address the two-sample problems. With preassigned
α ∈ (0, 1), we test a null hypothesis H : p = p with the approximate level
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0
1
α. The specific alternative hypotheses will be given shortly. Under the null
hypothesis H we have p = p = p where 0 < p < 1 is unknown and
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