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.

 

1. a) In a genetical experiment the frequencies observed in 4 different classes are 1997,
   906, 904,32. Theory predicts that these should be in the proportion
   and   respectively. Find the maximum likelihood estimate of the parameter q and
   also obtain the estimate of its variance.

 

1. b) The following table gives the number of flower heads each having exactly ‘x’ gall cells.

 

No.of gall cells in a flower head	1	2	3	4	5	6	7	8	9	10
No.of flower heads	287	272	196	79	29	20	2	0	1	0

Estimate the parameter ‘q’ 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)

 

2. 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.

• Obtain the MLE of

(ii) Obtain the MLE of the probability that a randomly selected family has atleast
3 boys.

 

1. b) The following data relates to the results of a genetical experiment:

Gene type	Relative frequencies	Observed frequencies
I		508
II		432
III		397
IV		518

 

Estimate the parameter q by the method of modified minimum Chi-Square.    (17+17)

 

3. 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 s _x² and s _y² respectively.  Obtain 98% Confidence interval for the ratio.

 

1. Obtain a 95% confidence interval for the parameter q 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

 

4. a) Let X₁, X₂, …,X₅ be a random sample of SAT mathematical scores assumed to be
   N (m₁,s²) and let Y₁, Y₂, …,Y₈ be an independent random sample of SAT verbal
   scores assumed to be N(m₂,s²). If the following data are observed find 90% confidence
   interval for m₁ – m₂ .

X₁ = 644;         X₂ = 493;         X₃ = 532;         X₄ = 462;         X₅ = 565.

Y₁ = 623;         Y₂ = 472;         Y₃ = 492;         Y₄  = 661;        Y₅ = 540;         Y₆ = 502;         Y₇ = 549;         Y₈ = 518.

 

81. b) Test for randomness of the following 14 observations at 5% level:
    4,     76.3,    85.6,    76.4,    88.4,    80.2,    85.6,    84.6,    78.3,    82.8,    88.1,    85.4,    87.7,    86.6.                                                                                        (20+14)

 

5. a) Let Y ~ B (200, p). To test H₀: p = 0.75 against H₁: p > 0.75. we observe Y and reject
   H₀ if Y150. 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.

 

64. b) Let X₁, X₂, …,X_n be a random sample from a normal distribution with mean ‘m’ and
    variance 64. Show that C = {(x_1, x₂, …, x_n):}  is a best critical region for testing
    H₀: m = 80 against H₁: m = 76. Find ‘n’ and ‘c’ so that  = 0.05 and b = 0.05
    (17+17)

 

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