M.Sc. (Statistics) (Semester II) (CBCS) Examination, 2017
THEORY OF TESTING OF HYPOTHESES
Day Date: Monday, 24-04-2017 Max. Marks: 70
Time: 10.30 AM to 01.00 PM
N.B. Attempt five questions
Q.No.(1) and Q.No.(2) are compulsory.
Attempt any three from Q.No.(3) to Q.No.(7).
Figures to the right indicate full mark
Q.1 Choose the correct alternatives: 05
In the most powerful test for testing simple full hypothesis
against simple alternative, the probability of
Both type I and type II error is minimum
Type I error is minimum
Type II error is minimum
Type I error is fixed at certain level and the probability of
type II error is minimum
In a MP level test the power of the test satisfies
Β
Uniform distribution
Has a MLR property
Belongs to one parameter exponential family
Both and
Neither nor
The probability density functions of X under H0 and H1 are
respectively given by 0 x 1 and 2x, 0 x
. Let X1, X2,….. Xn be iid observations from the population
of X. Then the MP level test for testing H0 against H1, 0

Is given by Σ
Is given by
Is give by
Does not exist
Let X following N Consider the testing problem H0
against H1 . Then
Both H0 and H1 are composite hypotheses
Both H0 and H1 are simple hypotheses
H0 is simple and H1 is composite
H0 is composite and H1 is simple
Q.1 Fill in the blanks 05
Maximum value of probability of type I error of a test is called
of test.
Neyman Pearson Lemma is used to construct tests.
Page 2 of 2
If the test function is
where
then the test is
The error of accepting the null hypothesis when the null
hypothesis is actually false is error.
UMA confidence intervals are obtained from tests.
Q.1. State whether following statements are true or false 04
NP lemma gives a method of testing simple hypothesis
against a composite hypothesis.
The exponential family of distribution is a proper subset of
family of distribution having the MLR property.
For unbiased tests, the probability of rejecting hypothesis H0
when it is false is bigger than that when it is true.
Test with Neyman-structure is similar test.
Q.2 Define the following
simple hypothesis
ii) Composite Hypothesis. Give illustrative examples
06
Write short notes on the following
Test of randomness
ii) Contingency tables
08
Q.3 State Neyman-Pearson lemma. Find the Neyman Pearson
size test of H0 against H1 based on a sample of
size n from f .
07
Let and be a MP size test for H0 against H1. Let
Show that is MP test for testing H1 against
H0 at level .
07
Q.4 Show that for a family having MLR property, there exists UMP
test for testing one sided hypotheses against one sided
alternative.
07
Let X1, X2,….. Xn be a random sample from obtain
UMP level test for testing H0
against H0
.
07
Q.5 Define shortest length confidence interval. Describe the pivotal
quantity method to obtain the shortest length confidence
interval.
07
Let X1, X2,….,Xn be a random sample from distribution.
Obtain the shortest length confidence interval for .
07
Q.6 Define likelihood ratio test (LRT). Show that LRT for testing
simple hypothesis against simple alternative is equivalent to
Neyman-Pearson test.
07
Let X1, X2,….,Xn be a random simple from distribution
where both and are unknown. Find the likelihood ratio test
of H0 against H1 .
07
Q.7 Define UMA confidence interval. Obtain one sided UMA
confidence interval for based on a sample of size n from
exponential distribution with mean .
07
State one sample and two sample U statistic theorems 07