aditya ari mustoha(k1310003) irlinda manggar a (k1310043) novita ening (k1310060) nur rafida...
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Aditya Ari Mustoha (K1310003)Irlinda Manggar A (K1310043)Novita Ening (K1310060)Nur Rafida Herawati (K1310061)Rini Kurniasih (K1310069)
Binomial, Poison, and Most Powerful Test
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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
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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
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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?
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Solution
Given :
a) Using Theorem 12. 4. 1
Reject to if
then
Thus, is Rejected
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b) Using Theorem 12.4.2
Reject to if
Then;
Since
Thus, is Rejected.
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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
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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
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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
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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.
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Thank You