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 to

b)

Reject if to

c)

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 12.4.1

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.

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

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