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Unformatted text preview: M4056 Hypothesis Testing III. November 1, 2010 A. Review A hypothesis H is an assertion about a (population) parameter. If the parameter is , then H is of the form: H , where is a subset of the set of all values that the parameter may have. A test of H is a statement T about a sample vector X of the form: T vector X A. Here, A is a subset of sample space; it is called the acceptance region. When T is true, we say that the test result is accept H . The complement of A is called the rejection region. When vector X lies in this region, we say the test result is reject H . Comment. The test is dependent on the sample, vector X . Of course, the test only yields a decision when some specific data vectorx are inserted. But we are often interested in how the test behaves under repeated applications. For example, the power function of T is ( ) := P ( vector X negationslash A ) , the probability of rejecting H expressed as a function of . (We will study below.) Note that in the definition of , the test itself is treated as a -indexed family of discrete random variables. If is fixed, then each of the two possible values of T (accept or reject) has a probability, and these two numbers sum to 1. Error types: Type I: reject a true H . ( H is true, T is false.) Type II: accept a false H . ( H is false, T is true.) B. A Classical Example Sir R. A. Fishers Lady Tasting Tea is a famous didactic example. A lady at a party asserts that she is capable of distinguishing the order in which milk and tea are poured into a cup by taste alone. The statisticians at the party devise an experiment. The lady is given 8 cups, four of each kind. The cups are presented in random order, and the lady is asked to taste them and divide them into two groups of four according to kind. (Note that theretaste them and divide them into two groups of four according to kind....
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