06.04.2004
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
FIFTH SEMESTER – APRIL 2004
ST 5501/STA 506 - TESTING OF HYPOTHESIS
Max:100 marks
SECTION - A
(10  2 = 20 marks)
Answer ALL questions
1. Define a simple hypothesis and a composite hypothesis.
2. Let X1, X2,..., Xn be a random sample from N (, 2).
n (X  )
nS 2
Write the distributions of i) 2
ii)
.


3. Define uniformly most powerful critical region.
4. Explain Type - I error and Type - II error.
5. Explain the likelihood ratio principle.
6. What do you mean by non- parametric methods?
7. When do we need the randomised test?
8. Find the number of runs in the sequence.
x yyy xxx y x y xxx yy xxxxx
9. Explain the term confidence interval.
10. What is a p - value?
SECTION - B
Answer any FIVE questions
(5  8 = 40 marks)
11. Let X have pdf of the form f (x, ) =  x-1, 0 < x < 1, zero elsewhere, where   { 
= 1,2}. To test Ho:  = 1 vs H1 :  = 2, a random sample of size 2 is chosen. The critical
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region is C = { (x1, x2)  < x1 x2}. Find Type I error and Type II error.
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12. Verify whether UMPT exists for testing
Ho:  = ' Vs H1:   '
when the random sample X1, X2, ..., Xn is from N ( , 1).
13. Explain Wilcoxon's Test.
14. The theory predicts the proportion of beans in the 4 groups A, B, C and D should be
9 : 3 : 3 : 1. In an experiment among 1600 beans, the number in the 4 groups were 882,
313, 287, 118. Does the experimental results support the theory?
15. Explain how will you test for regression coefficients  and  in
yi =  +  (ci - c ), i = 1, 2, ... n
16. Explain the t-test for equality of means of two independent Normal populations.
17. In a random sample of 500 men from a particular district of Tamil Nadu, 300 are found to
be smokers. In one of 1000 men from another district, 550 are smokers. Do the data
indicate that the two districts are significantly different with respect to the prevalence of
smoking among men?
18. Derive the distribution of number of runs.
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SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
19. a) State and prove Neyman - Pearson lemma.
b) Explain monotone likelihood ratio property (MLR) and its use in testing the
hypothesis.
(10)
(10)
20. a) Derive the likelihood ratio test for testing the equality of two variances of two normal
populations N (1, 3) and N (2, 4), 1, 2 unspecified.
(12)
b) Two independent samples of 8 and 7 respectively had the following values of the
variables.
Sample I
9
11
13
11
15
9
12
14
Sample II
10
12
10
14
9
8
10
Do the population variances differ significantly?
(8)
21. a) Explain Man-Whitney - Wilcoxon Test.
b) Explain Sign-Test
(10)
(10)
22. a) Test the hypothesis that there is no difference in the quality of the 4 kinds of tyres A,
B, C and D based on the data given below:
Tyre Brand
A
B
C
D
Failed below 40,000 kms
26
23 15
32
Lasted from 40,000 to 60,000 kms
118 93 116 121
Lasted more than 60,000 kms
56
84 69
47
b) Let X1, X2, ..., Xn be a random sample from N (, 100). Find n and c if
Ho:  = 75 vs H1 :  = 78. Given P [ X  C Ho] = .05 and P [X  C H1] = .90.
C = { (x1, x2, ...,xn)  x ≥ c} is the Best critical region.
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