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Unformatted text preview: ∙ I used the invnormal function in Stata to compute − 1 .80 . For the entire expression, we can use di : . di (4*invnormal(.80) 6.6)^2 99.330822 38 EXAMPLE : Suppose we are sampling from a count distribution – and we do not specify a particular class of distributions. Let 0 be the mean and 2 0 the variance. Consider the null and alternative: H : 2 H 1 : 2 ≠ (where the null holds for the Poisson distribution but others, too). The null is composite because it can hold for any positive value of . Of course the alternative is composite, too. 39 ∙ A test statistic can be based on the ratio S 2 / X ̄ or maybe the difference S 2 − X ̄ , where S 2 is the sample variance. In either case, a rejection rule of the form T c L or T c U is called for. Or, we might use a symmetric rejection rule,  T  c . In any case, we reject H if S 2 is sufficiently less than or sufficiently greater than X ̄ . 40 ∙ Without specifying the population distribution, we can only use asymptotic analysis to derive a general test statistic in this example – so we hold off for now. ∙ Another way to generate a composite null (and alternative) is H : 2 ≤ H 1 : 2 so the null is that either the Poisson variance holds or the variance is underdispersed (relative to the Poisson distribution). The alternative is overdispersion. 41 Summary of Classical Hypothesis Testing 1. Choose the null and alternative. 2. Choose the size of the test (maximum probability of Type I error) you are willing to tolerate. 3. Choose a test statistic (often based on whether its distribution can be calculated, at least under the null hypothesis). 4. Choose a rejection rule based on the null and size of the test. 42 6 . Finite Sample Properties of Tests ∙ In the previous section we covered the example of testing H : ≤ against H 1 : 0 in a Normal ,1 population. The test statistic T n X ̄ and rejection rule T c for a critical value c make intuitive sense. But how do we formalize the notion that we have the “best” test? 43 Uniformly Most Powerful Test ∙ As mentioned earlier, once we have chosen a size , we would like our test to provide the highest power against all alternatives. DEFINITION : Consider a class of size tests with generic power function , ∈ Θ . Suppose a test has power function ∗ such that ∗ ≥ , all ∈ Θ 1 Then the test is said to be uniformly most powerful ( UMP ). 44 ∙ UMP tests often exist for onesided alternatives. They can often be derived by applying the NeymanPearson Lemma (see Casella and Berger)....
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 Fall '12
 Jeff
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, alternative hypotheses

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