LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
SUPPLEMENTARY EXAMINATION – JUNE 2008
ST 5501 - TESTING OF HYPOTHESIS
Date : 27-06-08
Time : 9.00 – 12.00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL the questions
(10 x 2 = 20 marks)
1. Define level of significance and power of a test.
2. What do you understand by a randomized test? Give an example.
3. Define one – parameter exponential family. Give an example.
4. What is MLR property?
5. Define SPRT procedure.
6. Show that the likelihood ratio test is a function of every sufficient statistic for .
7. Suppose X N (0, 1). For testing Ho:  = o against H1:  = 1 based on a random sample of size
15, which test would you use?
8. Write the statistic for testing Ho: 2 =  o2 against H1: 2   02 based on a random sample of size n
from Normal N (, 2),  unknown.
9. What is a distribution free test?
10. Write down the Kolmogorov – Smirnov two sided statistic for one sample problem.
PART – B
Answer any FIVE questions
(5 x 8 = 40 marks)
11. Let X P( ). To test the hypothesis Ho:  = 1 against H1:  = 2 consider the test  (x) = 1 if x >
3 and = 0 if x  3 based on a random sample of size 1. Find the size and power of .
12. State and prove Neyman – Pearson fundamental lemma.
13. Verify whether  N (4, 2 ),   R   is a one – parameter exponential family.
14. Let X B (1, p ). Develop SPRT for testing Ho: p = 1/4 against H1: p = ½ based on a random
sample of size n.
15. Let X B(n, p ). Obtain the likelihood ratio test for testing Ho: p  po against
H1: p  po.
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16. Develop the test procedure for testing Ho: 1 = 2 against H1: 1  2 based on random samples of
size n and m from independent N (1, 2) and N (2, 2) 2 unknown, populations.
17. Describe Sign test for location.
18. Describe Mann-Whitney Wilcoxon test.
PART – C
Answer any TWO questions
(2 x 2 = 40 marks)
19. a) Let X B (2, p ) . Let Ho: p = 1/2 H1 : p = 3/4 . Based on a random sample of
size 3, find the Best critical region of level  = 0.05.
b) Show that U (0, ),  0 family of distributions possesses MLR property.
20. Let X N (  ,  2 ). For testing Ho:  = o against H1:   o, obtain the likelihood
ratio statistic based on a random sample of size n, when 2 is unknown.
21. a) A die is rolled 120 times with the following results.
Result:
1
2
3
4
5
Frequency:
20
30
20
25
15
6
10
Test the hypothesis that the die is fair at 5% level.
b) Develop the procedure for testing the equality of two variances from two
independent normal populations when the respective means are known.
22. In an automotive safety test, the average tyre pressure in a random sample of 81 tyres
was found to be 26 pounds per square inch, and the standard deviation was 1.8
pounds per square inch. Find the 90% confidence interval for the population mean.
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