binomial, poison, and most powerful test

Aditya Ari Mustoha (K1310003)Irlinda Manggar A (K1310043)Novita Ening (K1310060)Nur Rafida Herawati (K1310061)Rini Kurniasih (K1310069)

Binomial, Poison, and Most Powerful Test

Theorem 12. 4. 1 Let be an observed random sample from , and let

, then

a) Reject if tob)

Reject if toc)

Reject if to

Binomial Test

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

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 12.4.1b) Perform a test using Theorem 12.4.2c) 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?

SolutionGiven :

a) Using Theorem 12. 4. 1Reject to ifthen

Thus, is Rejected

b) Using Theorem 12.4.2 Reject to if Then; Since

Thus, is Rejected.

Theorem 12.5.1 Let be an observed random sample from , and let , then

a)

Reject if tob)

Reject if toc)

Reject if to

Poisson Test

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 100-yard pieces of wire. Test versus at significance level α = 0.0103

Definition 12.6.1 A test of versus based on a critical region C, is said to be a most powerful test of size if

1) and,

2) for any other critical ragion C of size [ that is ]

Theorem 12.6.1 Neyman pearson 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

Most Powerful Test

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