Download LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034. M.Sc. DEGREE EXAMINATION

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI  600 034.
M.Sc. DEGREE EXAMINATION  STATISTICS
SECOND SEMESTER  APRIL 2003
ST 2804 / S 819  COMPUTATIONAL STATISTICS  II
30.04.2003
1.00  4.00
Max : 100 Marks
Answer any three questions.
01. a) In a genetical experiment the frequencies observed in 4 different classes are 1997,
1 1 1
,
,
906, 904,32. Theory predicts that these should be in the proportion
4
4
4
1
and
respectively. Find the maximum likelihood estimate of the parameter  and
4
also obtain the estimate of its variance.
b) The following table gives the number of flower heads each having exactly ‘x’ gall cells.
No.of gall cells in
1
2
3
4
5
6
7
8
9
10
a flower head
No.of flower
287
272
196
79
29
20
2
0
1
0
heads
Estimate the parameter ‘’ by the method of maximum likelihood and also obtain its
asymptotic variance by assuming the frequency distribution of gall cells to be of truncated
Poisson type.
(14+20)
02. a) A survey of 200 families with four children each revealed the following data :
No.of boys
0
1
2
3
4
No.of families
8
48
76
54
14
Assume that X ~ B (4, p) where ‘p’ is the probability of a child being a boy.
(i) Obtain the MLE of p.
(ii) Obtain the MLE of the probability that a randomly selected family has atleast
3 boys.
b) The following data relates to the results of a genetical experiment:
Gene type
Relative frequencies
Observed frequencies
I
508

2
II
432
(1   )
2
III
397
(1   )
2
IV
518

2
Estimate the parameter  by the method of modified minimum ChiSquare.
(17+17)
03. a) Let X be the concentration in parts per billion of chromium in the blood of healthy
person and Y be the same measurement done on a person with some disease. 8
healthy persons and 10 persons with disease were taken up for studying and the
following observations were obtained.
X
15
23
12
18
9
28
11
10
Y
25
20
35
15
40
16
10
22
18
33
It is believed that X and Y have normal distributions with variance  x2 and  y2
respectively. Obtain 98% Confidence interval for the ratio
 x2
.
 y2
b) Obtain a 95% confidence interval for the parameter  of Poisson distribution based on
the following data:
(17+17)
No.of blood corpuscles
0
1
2
3
4
5
No.of Cells
142
156
96
27
5
1
04. a) Let X1, X2, …,X5 be a random sample of SAT mathematical scores assumed to be
N (1,2) and let Y1, Y2, …,Y8 be an independent random sample of SAT verbal
scores assumed to be N(2,2). If the following data are observed find 90% confidence
interval for 1  2 .
X1 = 644;
X2 = 493;
X3 = 532;
X4 = 462;
X5 = 565.
Y1 = 623;
Y2 = 472;
Y7 = 549;
Y8 = 518.
Y3 = 492;
Y4 = 661;
Y5 = 540;
Y6 = 502;
b) Test for randomness of the following 14 observations at 5% level:
81.4,
76.3,
85.6,
85.4,
87.7,
86.6.
76.4,
88.4,
80.2,
85.6,
84.6,
78.3,
82.8,
88.1,
(20+14)
05. a) Let Y ~ B (200, p). To test H0: p = 0.75 against H1: p > 0.75. we observe Y and reject
H0 if Y  150. Use the normal approximation to compute the level and power function
of the test for values of p starting from 0.75 at intervals of 0.02 up to 0.85.
b) Let X1, X2, …,Xn be a random sample from a normal distribution with mean ‘’ and
variance 64. Show that C = {(x1, x2, …, xn): x  c } is a best critical region for testing
H0:  = 80 against H1:  = 76. Find ‘n’ and ‘c’ so that  = 0.05 and  = 0.05
approximately.
(17+17)
+ + + + +