LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – STATISTICSTHIRD SEMESTER – APRIL 2012ST 3814/3810 – STATISTICAL COMPUTING – II
Date : 26-04-2012 Dept. No.   Max. : 100 Marks    Time : 1:00 – 4:00

Answer any THREE questions:  a) For three state Markov chain with states {0,1,2} and transition probability matrix                 P =    [■(1/3&0&2/3@0&1/2&1/2@1/2&1/4&1/4)]  ,                            Find the mean recurrence transients of states 0, 1, 2.                                                (22)                b). Let {Xn, n=0,1,2,…} be a Markov chain with state space {0,1,2} and one step transition          probabilities            (12)   P =    [■(0.6&0.1&[email protected]&0.5&[email protected]&0.2&0.6)]                         Find (i) P2  (ii)   (iii) P[X2 = 0] given X0 takes the values 0, 1, 2 with probabilities 0.3,                                   0.4, 0.3 respectively         2.  Let X ~ B ( 1, θ ); θ = 0.1, 0.2, 0.3. Examine if UMP level 0.05 test exists for H : θ = 0.2  Vs                      K : θ = 0.1, 0.3. Otherwise find UMPU 0.05 test. (34)
In a population with N = 5, the Yi values are 9,10,1 1,12,13. Enlist all possible samples of size n = 2, with SRSWOR and verify that E (s2) = S2.Also Calculate the standard error of the sample mean.  (34)            4. a) Given the normal distribution Np (μ, ∑)                   µ =    ,   ∑ =                     Find the conditional distribution of X1 and X2 given X3 = 205 (16)

 

 

b) The distances between pairs of five objects are given below:                               1     2      3     4      5                       ■(1@2@3@4@5)  (■(0&&&&@9&0&&&@3&10&0&&@11&6&8&0&@12&7&4&10&0))                      Apply the Single Linkage Algorithm to carry out clustering of the five objects.           (18)
(a) Consider a population of 5 units with values 1,2,3,4,5. Write down all possible samples of (without replacement) from this population and verify that sample mean is an unbiased estimate of the population mean. Also calculate its sampling variance and verify that it agree with the formula for the variance of the sample mean, and this variance is less than the variance obtained from sampling with replacement.     (13)       (b) A sample of 30 students is to be drawn from a population consisting of 410 students belonging to            two colleges A and B. The means and standard deviations of their marks are given below:  Total no. of students(Ni) Mean Standard deviation(σi)College X 230 40 14College Y 180 25 9

How would you draw the sample using proportional allocation technique? Hence obtain the variance of estimate of the population mean and compare its efficiency with simple random sampling without replacement.   (21)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2012

ST 3814 – STATISTICAL COMPUTING – II

 

 

Date : 06/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer any THREE questions:

All carry equal marks          

                                                           

1. From the following transition probability matrix,

0        1      2        3        4     5

 

1. State the state space
2. Find the equivalence class

• Find the states which are recurrent or transient

1. Determine the periodicity of the states
2. Find the stationary distribution

 

2. Suresh has scored 97% in an entrance exam. It is decided to estimate the number of candidates who have scored more than Mr.Suresh. The marks scored by the candidates are displayed in 5 boards. The following is the relevant data,

 

Board No	No.Of Candidates
1	30
2	15
3	20
4	25
5	10

Guided by the contents of the boards it is decided to use the sampling design,

 

 

 

 

 

Estimate the number of candidates who have scored more than Mr. Suresh and also compute the estimated variance of the estimate assuming the set {1,3,5} is the sampled set. Find the true variance of the estimator.

 

 

3. a). The data  below are obtained from a small artificial population which exhibits a fairly study raising trend. Each column represents a Systematic sample and the rows are the strata. Compare the precision of Systematic sampling, Simple random sampling and Stratified sampling.

	Systematic Sampling Number
Strata	1	2	3	4	5	6	7	8	9	10
I	28	32	33	33	35	34	37	39	40	40
II	15	16	17	17	21	20	22	25	26	24
III	2	3	3	4	7	6	9	9	10	8
IV	5	7	8	9	12	11	14	15	15	16

(17 M)

 

b).        A sample of 40 students is to drawn from a population of two hundred students belonging

to A&B localities. The mean & standard deviation and their heights are given below

 

 

Locality	Total No.Of People	Mean (Inches)	S.D(Inches)
A	150	53.5	5.4
B	50	62.5	6.2

1. Draw a sample for each locality using proportional allocation
2. Obtain the variance of the estimate of the population mean under proportional allocation.

(16 M)

 

 

 

4. a) If X₁ and X₂ be 2 observations from f ( x, θ )= θ X^a-1 ,0 < X < 1. To test  H₀ : θ = 1 Vs H₁ : θ = 2, the critical region in C = {(X₁, X₂ )|3/4x₁ < x₂ } . Find the significance level and power of the test. Draw the power curve.                                                                                                (18 M)

 

1. b) Let X ~ B ( 1, θ ); θ = 0.1, 0.2, 0.3. Examine if UMP level 0.05 test exist for H : θ = 0.1 Vs K : θ = 0.2, 0.3. (15 M)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5. Perspiration from 20 healthy females were analyzed. Three components X₁ = Sweat rate , X₂ = Sodium content  and X₃ = Potassium  content  were measured and the results are given below:

 

Individual           X₁                 X₂                     X₃

1                   3.8                48.6                  9.4

2                5.8                   65.2                 8.1

3                  3.9                       47.3                 11.0

4                  3.3                       53.3                 12.1

5                 3.2                       55.6                  9.8

6                 4.7                       37.1                  8.0

7                 2.5                       24.9                 14.1

8                            7.3                       33.2                  7.7

9                 6.8                       47.5                  8.6

10                 5.5                      54.2                11.4

11               4.0                   37.0              12.8

12               4.6                   58.9              12.4

13               3.6                   27.9                9.9

14               4.6                   40.3                8.5

15               1.6                   13.6              10.2

16               8.6                   56.5                7.2

17               4.6                   71.7                8.3

18                           6.6                   52.9             11.0

19               4.2                   44.2             11.3

20               5.6                   41.0               9.5

 

Test  the hypothesis  H₀ : µ^´  = [ 6  ,  52  , 12 ]  against H₁ : µ^´  ≠ [ 6  ,  52  , 12 ]  at 1% level of

significance.

 

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