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  
