Binomial, Poison, and Most Powerful Test Aditya Ari Mustoha Irlinda Manggar A Novita Ening Nur Rafida Herawati Rini Kurniasih (K1310003) (K1310043) (K1310060) (K1310061) (K1310069) Binomial Test Theorem 12. 4. 1 Let from , and let a) Reject if to b) Reject if to Reject if to c) be an observed random sample , then Theorem 12. 4. 2 Suppose that and and denotes a binomial CDF. Denotes by s an observe value of S. a) Reject if to b) Reject if to Reject if to c) Example A coin is tossed 20 times and x = 6 heads are observed. Let p = P(head). A test of versus of size at most 0.01 is desired. a) Perform a test using Theorem 12.4.1 b) Perform a test using Theorem 12.4.2 c) What is the power of a size test of for the alternative ? d) What is the -value for the test in (b)? That is, what is the observed size? Solution Given : a) Using Theorem 12. 4. 1 Reject to if then Thus, is Rejected b) Using Theorem 12.4.2 Reject to if Then; Since Thus, is Rejected. Poisson Test Theorem 12.5.1 Let , and let be an observed random sample from , then a) Reject if to Reject if to Reject if to b) c) Example : Suppose that the number of defects in a piece of wire of length t yards is Poison distributed , and one defect is found in a 100-yard piece of wire. a) Test against with significance level at most 0.01, by means of theorem 12.5.1 b) What is the p-value for such a test? c) Suppose a total of two defects are found in two 100yard pieces of wire. Test at significance level α = 0.0103 versus Most Powerful Test Definition 12.6.1 A test of versus based on a critical region C, is said to be a most powerful test of size 1) if and, 2) is for any other critical ragion C of size ] Theorem 12.6.1 Neyman pearson Lemma Suppose that have And let Where joint pdf . Let be the set is a constant such that Then is a most powerful region of size versus for testing [ that Example 3: Condider a distribution with pdf if and zero otherwise. a) Based on a random sample of size n = 1, find the most powerful test of against with . b) Compute the power of the test in a) for the alternative c) Derive the most powerful test for the hypothesis of a) based on a random sample of size n. Thank You