LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034                            LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION – STATISTICSTHIRD SEMESTER – NOV 2006ST 3810 – STATISTICAL COMPUTING – II

Date & Time : 30-10-2006/9.00-12.00   Dept. No.                                                    Max. : 100 Marks
(i) Choose either 1 or 2(ii) 3 is compulsory(iii) Choose either 4 or 5(iv) 1. Compare the performances of SRS-HTE strategy and SRS-HARTELY ROSS UNBIASED RATIO TYPE ESIMATOR strategy in estimating  total population during the year 2006  using the  following population data assuming the sample size is 2 (Treat 2004 data as auxiliary information)
Area       : 1 2 3 4
Population in 2004    : 37 36 48 51 (in ‘000)
Population in 2006    : 41 49 51 57 (in ‘000)
2. Certain characteristics associated with a few recent US presidents are listed below:
President Birth region Elected first time Party Prior congressional experience Served as vice presidentReagen Midwest Yes Republican No NoCarter South Yes Democrat No NoFord Midwest No Republican Yes YesNixon West Yes Republican Yes YesJohnson South No Democrat yes Yes Define suitable binary variables to convert the above data into categorical data. Form clusters using   single and complete linkage methods with suitable similarity measure. Draw dendograms and compare your results.
3. (a) It is decided to estimate the proportion of students in a college having the habit of indulging in malpractice during examinations. Two random experiments were deviced. Device 1 when conducted will result in either the question “Do you indulge in copying during examinations ? “  or “Do you know the first prime minister of India ?” with probabilities 0.4 and 0.6 respectively. Device 2 also results in one of those two questions with probabilities 0.45 and 0.55. The following is the data collected from 2 independent SRSWRs of sizes 10 and 15. Responses from the first and second samples which used device 1and device 2 are
yes,no,yes,yes,no,no,yes,yes,no,no
and no, no, yes,yes,no,yes,yes,no,yes,no,no,no,yes,yes,no
Estimate the proportion of students in the college who got the habit of using unfair means during exams and also estimate the variance of your estimate.
(b)  Given the normal distribution Np , where
=        and      =
(i) Find the distribution of CX, where C = (1, -1 , 1 )(ii) Find the conditional distribution of       [X1, X2] X3 = 190;  [ X1,  X3] X2 = 160 ;   X1 [ X2 = 150 , X3 = 180]
4.    (a)  A certain genetic model suggests that the probabilities of a particular  trinomial               distribution are respectively P1 = p 2,  P2 =  2p(1-p) and P 3 = (1-p2) , 0 < p < 1.                If x1, x2  and x3 represent the respective frequencies in  n independent trials, how                 we would check on the adequacy of the genetic model given x1 = 25 ,  x2 = 35              and x3 = 40.       (b) The following table gives the probabilities and the observed frequencies in 4              phenotypic classes AB, Ab, aB, ab in a genetical experiment.  Estimate the               parameter  by the method of maximum likelihood and find the standard error.
Class        :    AB   Ab   aB   ab Prabability  :         Frequency  :   102   25 28   5      (16+17)
5. (a)  A markov chain with state space   has tpm given by           Find      (i)  equivalence classes. (ii)  recurrent and transient states (iii) mean recurrence time for recurrent states (iv) periodicity of the states.
(b) A Markov chain with state space  Obtain the steady state distribution of the Markov chain.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 26

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3810 – STATISTICAL COMPUTING – II

 

 

 

Date : 07-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer ALL the questions:                                                                         ( 5 x 20 = 100 )

 

1).(a) Let X ~ N₄    , compute

b). Two independent samples observation are drawn from a bivariate normal distribution with common population variance matrix. Test whether the two groups have the same population mean vector.

 

Group A

Age	55	58	59	60	62	65	68
Bp	120	125	130	100	105	120	116
Glucose	140	145	155	158	162	170	180

 

Group B

Age	59	62	58	57	56	69	65	62
Bp	100	126	95	100	105	110	115	120
Glucose	145	155	148	142	143	160	159	156

 

2). (a) Let X be normally distributed according as  N₃ ( ,

with .

Find conditional distribution (X₁ | X₂   = 8, X₃ = 5).

b). Find the maximum likelihood estimator of the 2 x  1   mean vector   and  2 x 2 covariance matrix  based on the random sample  from the bivariate normal population.

c). Income in excess of Rs. 2000 of people in a city is distributed as exponential  20 people were selected and their incomes are shown below

2200	3250	8000	8500	9500
2500	4500	6200	6000	8100
3000	7500	2100	7200	3700
2750	10000	9000	8600	97500

 

Obtain the point estimate of the expected income of a person in this city by maximum likelihood method. Obtain the estimate of its variance.

 

3) (a)  The biologist who studies the spiders was interested in comparing the lengths of female and male green, lynx spiders. Assume that the length X of the female spider is approximately distributed as       and the length Y of the male spider is approximately distributed as . Find an approximately 95 % confidence interval for () using 30 observation of X.

 

 

5.2	4.7	5.75	7.5	6.45	6.55	4.7	4.8	5.95	5.2
6.35	6.95	5.7	6.2	5.4	6.2	5.85	6.8	5.65	5.5
5.65	5.85	5.75	6.35	5.75	5.95	5.9	7	6.1	5.8

 

and the 30 observation of Y ,

 

8.25	9.95	5.9	6.55	8.45	7.55	9.8	10.9	6.6	7.55
8.1	9.1	6.1	9.3	8.75	7	7.8	8	9	6.3
8.35	8.7	8	7.5	9.5	8.3	7.05	8.3	7.95	9.6

Where measures are  in millimeters.

 

(b) .Given below are the qualities of 10 items ( in proper units) produced by two processors A and B.Test whether the variability of the quantity may be taken to be the same for the two processors

Processor A	33	37	35	36	35	34	34	35	33	33
Processor B	38	35	37	38	33	32	37	36	35	37

 

4). (a). For a 3 state Markov Chain with state {0, 1, 2,} and TPM

find the mean recurrence times.

 

• From the following population of 10 clusters compare the following sampling designs for the estimation of the population total

(i)  Select 5 clusters by SRSWTR method

(ii) Draw an SRSWTR of 8 clusters and select a SRSWTR of size 2 from each

cluster and comment upon your results

 

C luster No.		Values    of the	variates
1	345	123	345	456
2	256	345	367	345
3	321	145	456	256
4	267	235	387	478
5	378	378	367	245
6	409	254	390	346
7	236	378	342	234
8	265	456	234	290
9	234	321	345	456
10	267	149	456	345

 

 

5) A sample survey was conducted with the aim of estimating the total yield of paddy. The area is divided into three strata and from each stratum, 4 plots are selected using SRSWTR. From the data given below,  calculate an estimate of the total yield along with an estimate of its variance.

 

Stratum No.	Total No. of Plots	Yield of Paddy for 4 Plots in the sample ( Kgs )
I	200	120	140	160	50
II	105	140	80	200	140
III	88	110	300	80	130

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2010

    ST 3814  – STATISTICAL COMPUTING – II

 

 

 

Date : 03-11-10                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Note :    SCIENTIFIC CALCULATOR IS ALLOWED FOR THIS PAPER

 

Answer any THREE questions                                                                               

1 a). Let {X_n, n=0,1,2,…} be a Markov chain with state space {0,1,2} and one step transition                 probabilities                                                                                                                                 (12)

P =

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

b). Let {X_n, n=0,1,2,…} be a Markov chain with state space {0,1,2, 3, . . .} and transient function p_xy ,          where p₀₁= 1 and for x = 1, 2, 3,. . .                                                                                              (22)

• Find f₀₀⁽ⁿ⁾ , n = 1, 2, 3 ,. . .
• Find mean recurrence time of state 0.
• Show that the chain is irreducible. Is it Ergodic?
• Find for x = 0 ,1, 2 . . . whenever it exists
• Find the stationary distribution, if it exists.

2  a) Consider two independent samples of sizes n₁= 10 , n₂ = 12 from two tri-variate normal

populations with equal variance-covariance matrices. The sample  mean vectors and the pooled

variance- covariance matrix are

,     and

Test whether the mean vectors of the two populations are equal                                                       (16)

 

 

1. b) The distances between pairs of five objects are given below:

1     2      3     4      5

 

Apply the Single Linkage Algorithm to carry out clustering of the five objects.                                (18)

 

3. 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)

 

 

4. In a population with N = 4, the Y_i values are 11,12,1 3,1 4,15. 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)

 

 

 

5 (a) Marks secured by over one lakh students in a competitive examination were displayed in 39           display boards. In each board marks of approximately 3000 students were given. Kiran, a student         who scored 94.86 marks wanted to know how many candidates have scored more than him. In         order to estimate the number of student who have scored more than him, he took a SRS of 10         boards and counted the number of students in each board who have scored more than him. The        following is the data collected.

13, 28, 5, 12, 0, 34, 14, 41, 25 and 6.

Estimate the number of student who would have scored more than Kiran and also estimate the        variance of its estimate.                                                                                                                          (13)

 

(b) A sample of 30 students is to be drawn from a population consisting of 230 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	150	25	7
College Y	80	50	32

 

 

 

 

 

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      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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