Binomial, Poison, and Most Powerful Test

1. Binomial, Poison, and Most Powerful Test Aditya Ari Mustoha (K1310003) IrlindaManggar A (K1310043) NovitaEning(K1310060) NurRafidaHerawati(K1310061) RiniKurniasih(K1310069)

2. Binomial Test Theorem 12. 4. 1 Let be an observed random sample from , and let , then a) Reject if to b) Reject if to c) Reject if to

3. 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 c) Reject if to

4. 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?

5. Solution Given : Using Theorem 12. 4. 1 Reject to if then Thus, is Rejected

6. b) Using Theorem 12.4.2 Reject to if Then; Since Thus, is Rejected.

9. Poisson Test Theorem 12.5.1 Let be an observed random sample from , and let , then a) Reject if to b) Reject if to c) Reject if to

10. Example : Suppose that the number of defects in a piece of wire of lengthtyards 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 100-yard pieces of wire. Test versus at significance level α = 0.0103

13. Most Powerful Test Definition 12.6.1 A test ofversusbased on a critical region C, is said to be a most powerful testofsize if 1) and, 2) for any other critical ragion C of size [ that is ] Theorem 12.6.1Neymanpearson Lemma Suppose that have joint pdf . Let And let be the set Where is a constant such that Then is a most powerful region of size for testing versus

14. Example 3: Condider a distribution with pdfif 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.

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