05.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
FIFTH SEMESTER – NOVEMBER 2003
ST – 5501/STA506 – TESTING OF HYPOTHESIS
Max:100 marks
SECTION-A
Answer ALL questions.
(10x2=20 marks)
1. Define the Best critical region for testing a simple hypothesis against an alternative simple
hypothesis.
2. When do you say that a family of probability density functions has a monotone likelihood
ratio property?
3. Explain likelihood ratio criterion.
4. What is the test statistic used for testing the significance of correlation in a bivariate normal
population? What is the distribution of the statistic?
5. A coin is tossed 100 times. If, the number of heads obtained is 45, can you say the coin is
unbiased? justify.
6. What are non-parametric methods?
7. 400 oranges are taken from a large consignment and 40 are found to be bad. Find the limits
in which the percentage of bad oranges is likely to come at 95% confidence.
8. Define Type I error and Type II error.
9. Explain p-value.
10. Arrange the following sets of x and y values. How many runs you get?
x
40
50
48
60
y
45
51
52
55
SECTION-B
Answer any FIVE questions.
(5x8=40 marks)
11. Let X1, X2, .. Xn be a random sample from N(0,1). Verify whether uniformly most
powerful test exists for testing Ho :  = 1 vs H1 :   1 .
12. Let x have the probability mass function
M  N  M 
  

x   n  x 

P [X = x] =
x = 0, 1,2, ..M.,
N
 
n
It is decided to test Ho : M  M0 vs H1 : M > M0 based on a sample of size 1 from this
distribution. Derive the uniformly most powerful test.
13. In a locality 100 persons are randomly selected and asked about their educational
attainments. The results one as under
Education
Middle
High school College
Male
10
15
25
Female
25
10
15
Does education depend on sex?
1
14. It is decided to study the model
Xi =  +  (ci - c ) + ei
with E(xi) =  +  (ci - c) ,
V(Xi) = 2.

a random sample of size n = 10 yielded  = 67


 = 2.1  2 = 288 and  (ci  c) 2  241 .
Test H0:  = 0 vs H1 :   0 . Also find 95% confidence interval for .
15. Let X have the pdf f ( x ; ) = x (1-)1-x x = 0, 1 zero elsewhere. We test H0:  =
1
against
2
1
. Let X1, X2, X3, X4, X5 be a random simple from this distribution. Obtain the
2
2
.
uniformly most powerful test with  =
32
16. Explain Sign Test.
17. Explain how will you construct the confidence interval for ratio of variances, when there
are two random samples from 2 independent normal populations. suppose two independent
samples of sizes m = 15 and n = 10 yielded
H1:  <
x  3.6, S12  4.14,
m  15
y  13.6, S 22  7.26,
n  10
Find the 98% confidence interval for  22   12 .
18. It is known that the random variable X has a pdf
f (x ;  ) =
1
x
e
,
0 x  

0
, elsewhere.
It is decided to test H0:  = 2 Vs H1:  = 4.
If a random sample of size 2 is observed, find probability of Type I error and probability of
type II error.
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. a) State and prove Neyman – pearson theorem.
(12)
1
b) Let X have the pdf f( x, ) =
0<x<

Let Y1 < Y2 < Y3 < Y4 denote the order statistics of a random sample of size 4 from
this distribution.
1
We reject H0:  = 1 and accept H1:   1 if Y4 
or Y4  1. Find the Power function. (8)
2
20. a) Derive likelihood ration test for testify
H0: 1 =2 Vs
H1: 1  2
When the two random samples are drawn from two independent normal populations
with mean 1 and 2 and with common unknown variance.
(10)
2
b) Two independent samples of sizes 8 and 7 items respectively had the following values
Sample I
9
11
13
11
15
9
12
14
Sample II
10
12
10
14
9
8
10
Is the difference between means of samples significant. (Assume common variance)
(10)
21. a) Explain Run test for equality of distributions.
(12)
b) Apply the Mann-Whitney – Wilcoxon test for the following data to test
H0: Fx =Fy Vs H1: Fx = Fy.
X
4.3
5.9
4.9
3.1
5.3
6.4
6.2
3.8
7.5
5.8
Y
5.5
7.9
6.8
9
5.6
6.3
8.5
4.6
7.1
(8)
22. a) Explain the procedure of testing equality of two proportions. Also obtain the 95%
confidence interval for the difference in proportions.
(10)
b) Explain sequential probability ratio test
(10)
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3