Loyola College Question Paper 2006

Loyola College M.Sc. Visual Communication April 2006 Media Management Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – VISUAL COMMUNICATION

OZ 10

THIRD SEMESTER – APRIL 2006

                                                        VC 3800 – MEDIA MANAGEMENT

 

 

Date & Time : 28-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

 

Kindly note: This is an open book test. You are allowed to use your own notes and not allowed to borrow notes from your companions. Answers need to be your personal reflections and not reproduction of class notes or handouts. Reproduction will fetch you minimum marks/no marks.

PART A

Write short notes on ANY FIVE of the following not exceeding one page each.          (5×8=40)

  1. Attempt a profile of the work force in South Indian media industry – in terms of demographics and psychographics.
  2. Your comments on Gender and Leadership, especially in the media field.
  3. How will the new technology (digital) in filmmaking and distribution affect Recruitment and Development of work force in the film industry in India.
  4. Feedback matters and feedback is a ritual. How do you see the value of feedback in an organization?
  5. Two current challenges for any management – globalisation and work-force diversity.
  6. Briefly explain four parameters that will affect film scheduling in the Indian context.
  7. Arthashashtra is a great source of management thought. Defend or Criticize this statement.

PART B

Write essays on ANY TWO of the following not exceeding three pages each.   (2×15=30)

  1. Conflicts are one of the major sources of news making for the media. Conflicts do occur in the media industry (the episodes of Thangar Bachan and Kushboo). As a student of media management, what is your view on conflicts and conflict management skills?
  2. Nuances of `Theory i Management’ as proposed by Arindam Chaudhuri and your comments on his theory.
  3. `The business of business is business’. Express your views FOR and AGAINST this statement from the perspective of management and social responsibility.

PART C

The break-down of the film script and related paper works.                          (30 Marks)

 

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Loyola College M.Sc. Visual Communication April 2006 Information & Communication Technology Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – VISUAL COMMUNICATION

OZ 9

SECOND SEMESTER – APRIL 2006

                             VC 2954 – INFORMATION & COMMUNICATION TECHNOLOGY

 

 

Date & Time : 24-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

PART A

Answer the following in about 30 words                                                       10´3=30

 

  1. Explain the relevance of ICT.
  2. Define CDMA.
  3. Expand JPEG and MPEG.
  4. What is HD-DVD?
  5. What is Media Convergence?
  6. Define Information Flow.
  7. What is terrestrial transmission?
  8. Mention the usages of Flash Drive.
  9. What is GPRS?
  10. How important is Data Retrieval?

 

PART B

Answer any five questions in about 250 words                                                  5´8=40

 

  1. Explain digitalization and its impact.
  2.  Explain the various barriers in communication transfer.
  3. Comment on information transfer cycle.
  4. Explain online information retrieval with examples.
  5. Write about various information storage devices.
  6. What is DBMS? Explain its usages.
  7.  Illustrate the structure of coaxial cable.
  8. What is DTH? What are its advantages?

 

PART C

Answer all the questions in about 500 words                                             2´15=30

 

  1. What is e-marketing? Explain the scope of e-marketing and e-business in India               with examples.

Or

What is e- governance? Explain with examples.

 

  1. Explain the functions of ATMs with illustrations.

Or

Comment on the increasing relevance of the Information Superhighway in
Indian society.

 

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Loyola College M.Sc. Visual Communication Nov 2006 Quantitative Media Research Tools Question Paper PDF Download

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                      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – VISUAL COMMUNICATION

AI 10

THIRD SEMESTER – NOV 2006

VC 3875 – QUANTITATIVE MEDIA RESEARCH TOOLS

 

 

Date & Time : 06-11-2006/9.00-12.00      Dept. No.                                                      Max. : 100 Marks

 

 

 

  • Please read the questions carefully BEFORE you start answering them.
  • The questions are set to evaluate your comprehension of the different theories, methods and methodology and your creative ability to apply them in media research.
  • Reproduction of the class materials and pages from books will give you a low score. Instead, your sharp skill in critiquing the methods/methodology will be rated high.

 

PART A

Answer any FIVE questions in 100 words each:                                   (5 x 4 = 20)

 

  1. What is the difference between validity and reliability in media research?
  2. How do you differentiate research method from research methodology?
  3. Who is the diegetic narrator in narrative analysis?
  4. Write a note on the different types of variables in media research.
  5.  How will you explain hypothesis testing?
  6.  Briefly explain abductive strategy.
  7.  Give a description of Seven Day Diary in Audience Research?

 

PART B

Answer any FIVE questions in 350 words each:                                    (5 x 7 = 35)

 

  1. Explain the four traditions in the history of media research?
  2. Hermeneutics plays a vital part in media research. Discuss.
  3. How do you explain the postmodern turn to the tool of analysis in media research?
  4. What are the five main purposes of Content Analysis?
  5. Explain the three types of interviewing.
  6. Random sampling is a sample which is selected by choice. Explain.
  7. What is discourse analysis?

 

PART C

Answer any THREE questions in 800 words each:                               (3 x 15 = 45)

 

  1. Write an essay on the six levels of Empirical Research.
  2. What are the steps that a media research should follow in transferring qualitative data into quantitative data?
  3. Discuss with an example the descriptive and analytical survey methods.
  4. Critically explain the main quantitative methods that most audience research follows.
  5. Identify major elements of Survey Research in media and give your own analysis.

 

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Loyola College M.Sc. Visual Communication Nov 2006 Integrated Marketing Communication Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – VISUAL COMMUNICATION

AI 11

THIRD SEMESTER – NOV 2006

VC 3955 – INTEGRATED MARKETING COMMUNICATION

 

 

Date & Time : 27-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

 

 

  1. Write Short Notes on the following (100 words each): 10×3=30

 

  1. Crisis Management
  2. Audience research
  3. Media relations
  4. Contemporary advertising
  5. Budgeting
  6. Ad creativity
  7. USP
  8. Event management
  9. SWOT Analysis
  10. Marketing mix

 

 

  1. Answer any FIVE of the following questions (250 words each): 5×6=30

 

  1. What is TRP? Explain the importance of TRP in advertising.
  2. “Advertising as a marketing communication tool”-Explain.
  3. What is marketing research? Explain.
  1. Trace the development of advertising in India.
  2. Comment on digital advertising.
  3.  Explain the impact of advertising on children.
  4. Comment on New media advertising.

 

 

III. Answer any TWO of the following questions (1000 words each):       2×20=40

 

  1. Explain the role of a creative department in an ad agency.
  2.  What is campaign strategy? Plan a campaign for a new mobile product.
  3.  What is Integrated Marketing Communication? Explain.

 

 

 

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Loyola College M.Sc. Visual Communication Nov 2006 Contemporary Media Trends Question Paper PDF Download

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                     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – VISUAL COMMUNICATION

AI 12

THIRD SEMESTER – NOV 2006

VC 3956 – CONTEMPORARY MEDIA TRENDS

 

 

Date & Time : 27-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

 

 

  1. Answer any FOUR of the following in about 100 words each:

                                                                                                                 (4 ´ 5 = 20)

  1. Write about the advantages of ‘Digital film making’.
  2. What are the ‘Ethics’ that should be followed by print media?
  3. How effective are ‘Hoardings’ in the context of Advertisements?
  4. Discuss about TRP & “Tamil Television Serials”.
  5. Write about the merits in Media Convergence.
  6. Give some psychological reasons for ‘Cyber Crimes’.

 

  1. Answer any FOUR of the following in about 200 words each:                                                                         (4 ´ 10 = 40)
  2. What is ‘Demand and Supply’ trend? And what are the impacts it creates on       ‘quality’ media?
  3. Write about ‘Violence’ in Cine Media; and give some psychological reasons         for the success of action movies.
  4. Discuss the merits and demerits of ‘Virtual Reality’.
  5. Is ‘Media’ the ‘Stepping Stone’ to politics? – Discuss.
  6. Discuss some of the ‘Media Myths’ and their negative influence on youth.
  7. Media is not for the Marginalized people – Discuss.

 

III. Answer any TWO of the following in about 500 words each:                                                                                                                                      (2 ´ 20 = 40)

  1. Modern Trends in Cine, Tele, Print, Ad & Web Media – Discuss
  2. Write about the ‘Media Objectivity’ in the        present media.
  3. How powerful is Media? Discuss the Positive and Negative impact of

Media on Society.

 

 

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Loyola College M.Sc. Statistics April 2006 Testing Statistical Hypothesis Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 34

SECOND SEMESTER – APRIL 2006

                                         ST 2809 – TESTING STATISTICAL HYPOTHESIS

(Also equivalent to ST 2807/2802)

 

 

Date & Time : 21-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                            10 x 2 = 20

  1. Define test function and randomized test function.
  2. Let X be B(1, q), q = 0.2,0.4,0.5. For testing H: q = 0.2,0.4 Vs K: q = 0.5, a test is given by

f(x)   =  0.3,    x = 0

=  0.6,    x =  1.

Find the size of the test.

  1. Show that a UMP level a test is unbiased.
  2. Define MLR property and give an example.
  3. Show that a test with Neyman structure is similar.
  4. Describe Type I and Type II right censoring.
  5. Give two examples for multiparameter exponential family.
  6. Define location family and give an example.
  7. Describe likelihood ratio test.
  8. Explain UMA and UMAU confidence sets.

 

SECTION B                            Answer any five questions                           5 x 8 = 40

  1. Let X be DU{1,2,…, q }, q = 1,2. For testing H: q = 1 Vs K: q = 2,  find MP level a test using LP technique.
  2. Give an example of a testing problem for which UMP test does not exist.
  3. Given a random sample of size n from E(0, q ), q > 0, derive UMP level a test for testing H: q £ q 0 Vs K: q > q 0.Examine whether the test is consistent.
  4. If the power function of an unbiased test is continuous, show that the test is similar.

15.Given a random sample of size n from P( q ), q > 0, derive UMPU level a test for testing H: q = q 0 Vs K: q ¹ q 0.

16.Show that a statistic is invariant if and only if it is a function of a maximal invariant statistic.

17.Derive likelihood ratio test for testing H: q = q 0 Vs K: q > q 0 based on a random sample from E(0,q), q >0.

18.Explain shortest length confidence interval and illustrate with an example.

 

 

 

SECTION C                           Answer any two questions                         2 x 20 = 40

19 a).   State and establish the sufficient part of Neyman-Pearson lemma.

  1. b) Let X1,X2,…Xn denote a random sample of size n from E(q ,1), q e Examine if there exists UMP level a test for testing H: q = q 0 Vs K: q ¹ q 0.

20 a)  In the case of one-parameter  exponential family show that there exists UMP level a  test for testing one-sided hypothesis against one-sided alternative. State your assumptions.

  1. b) Derive UMPU level a test for testing H:  q1 £ q £ q2 Vs K: q < q1  or q > q                               based on a random sample from N(q , 1), q e R. Explain the determination of the constants.Is the test unique?

21 a)   Discuss the relation between similar tests and tests with Neyman structure.

  1. b) Let X1,X2,…Xn  be a random sample from P( ) and Y1,Y2,…Ym be a random sample from an independent  Poisson population P( ).Derive UMPU level a test for testing H:l £ m  Vs K:l > m. Determine the constants when  n = 2 and m = 1, X1 = 1, X2 = 2 and Y1 = 3.

22 a)   State and establish the asymptotic null distribution of the likelihood ratio statistic.

  1. b) For testing H:(X1 , X2 ) is BVN(q, q ,1,1, 0.5) Vs K: (X1 , X2 ) is BVN(q, q,1, 4, 0.5), derive UMPI level a test with respect to location transformations.

 

 

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Loyola College M.Sc. Statistics April 2006 Stochastic Processes Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2006

                                                   ST 3806 – STOCHASTIC PROCESSES

 

 

Date & Time : 17-04-2006/AFTERNOON   Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                           10 ´ 2 = 20

 

  1. Define stationary independent increment process.
  2. Show that the square of a stochastic matrix is stochastic.
  3. Suppose the one-step tpm is an identitiy matrix, show that the states are all recurrent.
  4. Find a stationary disribution of an MC with one-step tpm

 

P =       0.3    0.7

0.7    0.3

  1. For a Poisson process, find the covariance function.
  2. Describe Pure birth process.
  3. Define excess life and current life of a Renewal process.
  4. For a martingale { Xn , n = 0.1.2,…}, show that E(Xn) = E(Xn +1), n = 0,1,2,…
  5. Describe a branching process.
  6. Define a covariance stationary process and give an example.

 

SECTION  B                               Answer any five questions                       5 ´ 8 = 40

 

  1. For a stationary independent increment process, show that the variance of the marginal distribution is linear in the time parameter.
  2. Define periodicity and show that it is a class property.
  3. If the one step tpm of an irreducible finite state Markov Chain is symmetric, show that the stationary distribution is uniform.
  4. Describe a Poisson Process and derive its marginal distribution.
  5. For a linear growth process with immigration, find the average size of the population if the initial population is i units.
  6. Derive the generating function relations satisfied by a Branching process.
  7. If { X(t), t ³ 0} is a Brownian motion process, show that the distribution of (X(t1),X(t2)) is bivariate normal.
  8. If the interoccurrence distribution of a Renewal process is exponential, find the distributions of i ) current life and ii ) excess life.

 

 

 

 

 

 

SECTION  C                              Answer any two questions                      2 ´ 20 = 40

 

19 a). State and establish Chapman-Kolmogorov equations satisfied by a Markov Chain.

b).   Illustrate Basic limit theorem with an example.

20 a). Describe Birth-Death process. Derive Kolmogorov backward equations satisfied by the Birth-Death process.

  1. b) Describe telephone trunking model and find its stationary distribution.

 

21 .a) State and prove Elementary renewal theorem in Renewal theory.

  1. b) Find the renewal function associated with a renewal process having the interoccurrence distribution with pdf

 

f(x) = l2 x exp(-l x), x > 0, l > 0.

 

22 a) Let { Xn , n = 0.1.2,…}be a Branching process with X0 = 1. Find the mean and variance of  Xn in terms of those of the offspring distribution.

  1. b) Let { Xn , n = 0.1.2,…}be a covariance stationary process with zero mean and the covariance function Rx(v) . Find the best predictor of Xn+1 of the form aXn, where a is a real constant.

 

 

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Loyola College M.Sc. Statistics April 2006 Statistics For Competitive Examinations Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 49

FOURTH SEMESTER – APRIL 2006

                               ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

SECTION A

Answer ALL the Questions                                                            (40 ´1 = 40 Marks)

 

  1. If the difference between two numbers is 1.2, then the variance of them is

(A) 0.18          (B) 1.44        (C) 0.72       (D) 0.36

  1. To test the hypothesis that the variance of a normal distribution is 2, the test procedure used is

(A) Normal test        (B) Chi-square test    (C) F-test   (D) t-test

  1. To test which one of the following hypothesis, F-test is used?

(A) Goodness of fit (B) equality of means of two normal populations

(C)  Significance of correlation coefficient (D) Equality of Variances of two normal populations

  1. If X has Poisson distribution with 3P[X = 2] = 2P[X ≤ 1], then the expected value of X is

(A) 3               (B) -2/3           (C) 2           (D) 3/2

  1. X1, X2 and X3 are independent observations on a normal random variable with

mean μ and variance σ2.What is the efficiency of   (3X1+2X2+X3 ) / 6 as an estimator of  μ ?

(A) 6/7             (B) 1           (C) 1/3           (D) 1/12

  1. If E(Y/X) = α X + β and X has standard normal distribution, then E(Y) is

(A) 0               (B) 1                (C) β          (D) α

∞

  1. If P ( An) = 1, n=1, 2, 3… then the value of P (An) is

n=1

(A) 0               (B) 1                (C) 1/2       (D) ¼

  1. A random variable X has characteristic function

Φ (t) = (sin t)/t,    t ≠0

1          otherwise

Then, Var(X) is equal to

(A) 1        (B) 0       (C) 1/6        (D) 1/3

  1. If X1 and X2 are independent and identically distributed random variables with p(x) = qx.p, x = 0, 1, 2, 3… and (p+q) = 1, then the distribution of (X1+X2) is

(A) Geometric        (B) degenerate          (C) Negative Binomial    (D) Hyper- geometric

 

  1. In a random sample of size n from the distribution

dF (x) = e-x.dx,   0<x<∞,

the mean of the smallest sample value is

(A) 1/n      (B) 1/n2      (C) 0       (D) 1

  1. If the degrees of freedom for error in the analysis of Variance for a Latin square design is 30, the number of treatments is

(A) 5               (B) 6           (C) 7           (D) not possible to determine

  1. If a population consists of 10 units and the population Variance is 20, the Variance of the sample mean of a simple random sample pf size 4 without replacement is

(A) 5               (B) 2           (C) 20           (D) 3

  1. The number of simple random samples of size 4 that can be drawn without replacement from a population with 12 units is

(A) 124           (B) 495        (C) 11,880        (D) 48

  1. The standard deviation of a symmetric distribution is 4. The value of the fourth moment about the mean in order that the distribution be leptokurtic is

(A) greater than 768              (B) equal to 768           (C) equal to 256       (D) less than 48

  1. Given Maximize subject to

 

 

 

For what values of the above problem will have several optimum solutions?

(A) 2    (B) 3    (C) 6    (D) 1

 

  1. The objective function in the phase-I (when we use two phase simplex method) is formed by
  • summing all the variables
  • summing all the artificial variables

(C)   taking the product of artificial variables

  • subtracting the sum of artificial variables from the sum of other variables

 

  1. The following set of constraints require x artificial variables

 

 

where x is

(A) 0                (B) 1                (C) 2                (D) 3

 

  1. Given the following simplex table (associated with a maximization problem)

 

Basic   z          x1        x2        x3        x4        Solution

 

z          1          -4         -2         0          0          8

 

x3        0          4          3          1          0          1

 

x4        0          -1         1          0          1          2

 

The leaving and entering variables are

(A) x1, x3        (B) x1, x4        (C) x2,x3         (D) x2,x4

 

  1. An LPP has 4 variables and 2 constraints. How many sets of basic variables are possible?

(A) 10        (B) 6                (C) 3                (D) 20

  1. The power function associated with the UMPT for testing  against the      alternative in is always

(A) Strictly increasing in      (B) Strictly decreasing

(C) Periodic in                                 (D) can’t say

  1. Which of the following is the form of UMPT for testing  against the

alternative in

(A)               (B)

(C)                (D)

  1. Choose the correct statement

(A) Power functions of UMPTs are always monotone

(B) A UMPT is always UMPUT

(C) MPT’s are not unique

(É)All similar tests will have Neyman structure

  1. Choose the correct statement

(A) RR methods are not associated with sensitive attributes

(B)Yates Grundy estimator is non-negative under Midzuno scheme

(C) HTE can not be used under PPSWOR

(D)Balanced systematic sampling is not recommended for populations with linear

trend.

  1. Lahiri’s method

(A) is a PPS selection method involving a given number of attempts

(B) is a PPS selection method involving unknown number of attempts

(C) is an equal probability selection method involving a given number of attempts

(D) is an equal probability selection method involving unknown number of

attempts

 

  1. Ratio estimator is

(A) a particular case of regression estimator

(B) an unbiased estimator

(C)more suitable when y and x have high negative correlation

(D) more suitable when y and x have no correlation

 

  1. Random group method is due to

(A) Desraj       (B) Murthy      (C) Hartley-Ross         (D) Rao-Hartley-Cochran

 

  1. Randomised response methods are meant for

(A) homogeneous data

(B) heterogeneous data

(C) sensitive data

(D) stratified populations

 

  1. Which name is associated with shortest route problems

(A) Kuhn-Tucker

(B) Floyd

(C) Charnes

(D) Karmakar

 

  1. Which of the following functions is NOT continuous at 0?

(A) |x|              (B) ex               (C) x – [x]                   (D)Sin x

 

  1. A tosses a fair coin twice and B throws a fair die twice. Let

a = Probability of getting at least two heads

b = Probability that the sum of the numbers that show up is less than 6

Then

(A) a > b        (B) a < b         (C) a = b         (D) a + b > 1

 

31.The system of equations

2x + 4y – z = 3

x + 2y +2z = 2

x + (m2+1) y + 7z = 4m – 1

has infinitely many solutions if m equals

(A)0                 (B) – 1             (C) 1                (D)2

 

  1. The mean and variance of 8 items are 10 and 100 respectively. An observation 3

is deleted from the data. The variance of the remaining 7 observations is

(A)100             (B)106             (C)112             (D)120

 

  1. Let T: R3 ® R2 be defined as T(x, y, z) = (3x + y – z , x + 5z). The matrix

corresponding to this linear transformation is

(A)       (B)               (C)        (D)

 

  1. Which of the following is NOT true of a normal variable with mean 0?

(A) E(X2) = 1, E(X3) = 0                        (B) E(X2) = 1, E(X4) = 2

(C) E(X2) = 2, E(X4) = 12                      (D) E(X2) = 1/2 , E(X4) = 3/4

 

  1. If X and Y are uncorrelated random variables with equal means and variances,

then

(A) X + Y and X – Y are identically distributed

(B) X + Y and X – Y are independent

(C) X + Y and X – Y are negatively correlated

(D) X + Y and X – Y have equal variance

 

  1. In a bivariate dataset {(Xi, Yi), i =1, 2, …,n}, X assumed only two values namely 0 and – 1 and the correlation coefficient was found to be –0.3. Then , the correlation coefficient for the transformed data {(Ui, Vi), i =1, 2, …,n}, where Ui = 3 – 5 Xi2 and Vi = 2Yi –3  is

(A) –0.3           (B) 0.3             (C) 0                (D) cannot be determined

 

  1. If X1, X2,…, Xn is a random sample from U( 0, q), which of the following is a biased estimator of q?

(A) 2           (B) X(n)                        (C)X1 + Xn      (D) (n+1)X(n) / n

 

  1. The Cramer-Rao lower bound for estimating the parameter l of a Poisson distribution based on a random sample of size n is

(A) l               (B) nl              (C) l / n           (D) l1/2

 

  1. The lower control limit of a c-chart is 4. The upper control limit is

(A)16               (B)20               (C)24               (D) none of these

 

  1. In a 24 factorial experiment with 4 blocks the degree of freedom for Error Sum of Squares is

(A) 25              (B)35               (C)45               (D)55

SECTION  B

Answer any SIX questions                                                                    (6 X 10 = 60 Marks)

 

  1. Explain the procedure for solving a game theory problem graphically

 

  1. Show that family of Uniform densities binomial densities has

MLR in

  1. A sample has two strata with relative sizes and . He believes

that . For a given cost , show that (assuming stratum

sizes are large)

 

 

 

  1. The exponent of a bivariate normal density is given below:

– ⅔(x2+9y2-13x-3xy+60y+103)

Find μ1, μ2, σ1, σ2 and ρ.

 

  1. The number of accidents in a town follows a Poisson process with the mean of 2

accidents per day and the number of people involved in ith accident has the

distribution

P[X1=k] = 1/ 2k, k≥1.

Find the mean and variance of the number of people involved in accidents per

week.

 

  1. If Φ is a characteristic function, show that e λ (Φ -1) is a characteristic function for

all λ>0.

 

  1. (a)Let X­1, …,Xn, Xn+1 be a random sample from N(m, s2). Let M be the average of the first ‘n’ observations and S2 be the unbiased estimator of the population variance based on the first ‘n’ observations. Find the constant ‘k’ so that the statistic k( M – Xn+1) /S follows a t- distribution.

 

(b) Let X have a Poisson distribution with parameter q. Assume that the

unknown q is a value of a random variable which follows Gamma distribution

with parameters a = a / ( 1- a) and  p = r, where ‘r’ is a positive integer. Show

that the marginal distribution of X is Negative Binomial.                          (5 + 5)

 

  1. (a) Let X1,…,Xn be a random sample from Poisson distribution with parameter l. Starting with the initial estimator X12 – X1­ for l2, use Rao-Blackwellization to get an improved estimator by conditioning on the sufficient statistic S Xi. State whether the resulting estimator is UMVUE and justify.

(b) Let X1, …, Xn be a random sample from N(m, s2). Obtain an unbiased and

consistent estimator of s4.                                                                           (6 + 4)

 

  1. (a) Derive an expression for E(Mean Treatment Sum of Squares) in LSD.

(b) Consider four quantities T1, …,T4 and let T1 – 2T2 + T3 be a contrast. Find

two other contrasts so that all the three are mutually orthogonal.              (7 + 3)

 

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Loyola College M.Sc. Statistics April 2006 Statistical Process Control Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 48

FOURTH SEMESTER – APRIL 2006

                                           ST 4803 – STATISTICAL PROCESS CONTROL

 

 

Date & Time : 22-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION A  

Answer all the questions                                                                             10 x 2 = 20

  1. Discuss the statistical basis underlying the general use of 3 – sigma limits on control charts.
  2. Define rational subgroup concept.
  3. How is lack of control of a process determined by using control chart techniques?
  4. What is process capability ratio (PCR)?
  5. Why is the np chart not appropriate with variable sample size?
  6. Explain an attribute single sampling plan.
  7. What purpose does an OC curve serve?
  8. Define AOQ.
  9. Define a). Specification limit. b). Natural tolerance limit.
  10. Explain the concept of TQM.

SECTION B  

Answer any five questions                                                              5 x 8= 40

  1. What are the dimensions of quality? Explain.
  2. A quality characteristic is monitored by a control chart designed so that the probability that a certain out of control condition will be detected on the first sample following the shift to that is 1 – b. Find the following:

a). The probability that the out of control condition will be detected on the second sample following the shift.

b). The expected number of subgroups analyzed before the shift is detected.

  1. A control chart for the fraction non-conforming is to be established using a CL of p = 0.10. What sample size is required if we wish to detect a shift in the process fraction non-conforming to 0.20 with probability 0.50?
  2. Explain the method of constructing control limits for X – bar and R charts when the sample sizes are different for various subgroups.
  3. In designing a fraction non-conforming chart with CL at p =0.20 and 3-sigma control limits, what is the simple size required to yield a positive LCL? What is the value of n necessary to give a probability of .50 of detecting a shift in the process to 0.26?
  4. Estimate process capability using X – bar and R charts for the power supply voltage data . If specifications are at 350 + 5 V, calculate PCR, PCRk and PCRkm. Interpret these capability ratios.
Sample # 1 2 3 4 5 6 7 8 9 10
X1 6 10 7 8 9 12 16 7 9 15
X2 9 4 8 9 10 11 10 5 7 16
X3 10 6 10 6 7 10 8 10 8 10
X4 15 11 5 13 13 10 9 4 12 13
  1. Find a single sampling plan for which p1 = 0.05, a = 0.05 p2 = 0.15 and b = 0.10.
  2. What are chain sampling and skip-lot sampling plans?

 

SECTION C

Answer any two questions                                                                2 X 20 = 40

 

  1. a) Distinguish between c and u charts. Explain the situations where c and u charts are applicable and are the limits obtained for these charts.
  2. b) Find 0.900 and 0.100 probability limits for a c-chart when the process average is equal to 16 non- conformities.                                                                                              (14+6)
  3. a) Write a detailed note on the moving average control chart.
  4. b) What are modified control charts?. Explain the method of obtaining control limits for modified control charts.                                                                                                         (8+12)
  5. a) Outline the procedure of constructing V-mask.
  6. b) What is Exponentially Weighted Moving Average control chart?.           (15+5)
  7. a) Write a detailed note on six-sigma quality.
  8. b) Explain with an illustration the method of obtaining the probability of acceptance for a triple sampling plan.                                                                                                               (10 + 10)

 

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Loyola College M.Sc. Statistics April 2006 Statistical Computing – I Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 29

FIRST SEMESTER – APRIL 2006

                                                 ST 1812 – STATISTICAL COMPUTING – I

 

 

Date & Time : 22-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

 

Answer any THREE questions

 

  1. a) Find a G- inverse of the matrix

A =

 

  1. b) Check whether the following vectors are linearly dependent:
  2. i) X¢  = (1, -1, 2),       Y¢ = (2, 0, -1),             Z¢ = (0, -2, -5).
  3. ii) X¢ = (1, 0,0), Y¢ = (0, 1, 0), Z¢ = (0, 0, 1).                            (20+14)

 

  1. a) The following data relates to the results of an experiment the relative frequencies for 4 different types of genes are expected to be and
    where 0 < < 1.

The frequencies observed were 508, 432, 397 and 518 respectively.  Estimate

the parameter q by the method of maximum likelihood and find the estimate of

the  standard error of the estimator.

 

  1. b) The scores of 17 students are given by the following table. Assuming that this

is a sample from normal population whose variance is s2, obtain

  1. a 95% confidence interval for s
  2. a 99% confidence interval for s

 

Scores:

(Out of 100)         45     65     68     77     95     69     56         72        75

38     68     72     65     42     66     55         62

(14+20)

 

  1. a) Below are given two random samples drawn from different normal populations:

Sample 1:    10        6          16        17     13     12     8       14     15        9

Sample 2:    7          13        22        15     12     14     18     8       21        23 10

 

Obtain a 99% confidences limits for the difference of means of the 2 populations.

 

  1. b) Fit a normal distribution to the following data

 

C.I: 60-65 65-70 70-75 75-80 80-85 85-90 90-95 95-100
Frequency 3 21 150 335 325 135 26 4

(20+14)

 

  1. a) Fit a multiple regression model of Y on X1 and X2 for the following data. Estimate Y when X1 = 1350 sq.ft and X2 = 2 years                                      (20)

 

  1. b) Also, test the significance of the population multiple correlation coefficient at

5% level of significance.                                                                                (14)

 

FLAT PRIZE IN LAKHS

(Y)

 FLAT SIZE IN SQ.FT

(X1)

 AGE OF THE FLAT IN YEARS        (X2)
12.3 1050 1
15 1200 1
14.8 1180 3
11 950 2
10.3 900 3
16.9 1300 3
18 1400 3
6 450 4
5.2 480 5
4.6 420 4
18 1450 6
9.3 850 3
12.2 1020 7

 

  1. a) Use Step-wise regression analysis to identify the most significant independent

variable(s) and comment on your finding regarding the significance of

population regression coefficients for the data given in question-4                (20)

  1. b) Compute the condition index for the data in question-4 and examine whether

the multi-co linearity problem is present in the data or not.                          (14)

 

 

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Loyola College M.Sc. Statistics April 2006 Sampling Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2006

                                                          ST 2810 – SAMPLING THEORY

 

 

Date & Time : 24-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section A  (10 x 2 =20)

 

Answer ALL the questions. Each carries TWO marks.

  1. Define : Midzuno sampling design
  2. State the identity which relates sample size of a sampling design with its first order inclusion probabilities
  3. Give the formula for unbiased estimator of under Warner’s RR
  4. Define balanced systematic sampling
  5. Mention the situations in which product and ratio estimators can be used instead of .
  6. When do you recommend “Two phase sampling”?
  7. Name an estimator which uses selection probabilities.
  8. Give any two limitations of “Linear Systematic Sampling “
  9. Name any one sampling-estimating strategy in which no unbiased estimator for variance of estimator can be found.
  10. Write the variance of Yates corrected estimator under LSS when there is a linear trend in the population

Section B (5 x 8 = 40)

 

Answer any Five. Each carries Eight  marks.

  1. Prove the following identities : and verify the same in the case of following sampling design

 

 

  1. From a population containing N units a sample of n units is drawn using SRS and from the drawn sample a subsample of n’ units. Suggest an unbiased estimator for the population total based on the subsample and obtain its variance
  2. Describe modified systematic sampling and show that under the model
  3. Describe Desraj ordered estimator and obtain an unbiased estimator of

 

  1. Explain proportional allocations (1) for a given cost (2) for a given sample size. Derive the variance of under the above cases assuming simple random sampling is used in all strata.

 

  1. Explain Warner’s randomized response model in detail.

 

  1. Define product estimator . Obtain an expression for its bias under simple random sampling and hence develop an unbiased estimator for the population total.
  2. Derive the approximate mean square error of estimators in the class also obtain the minimum mean square error in the class.

 

Section C  (2 x 20 =40)

 

Answer any TWO. Each carries TWENTY marks

 

  1. Define : Horvitz-Thompson estimator. Show that it is unbiased for the population total and derive its variance in Yates-Grundy form

 

  1. Derive the first and second order inclusion probabilities under Midzuno sampling scheme and show that under this design the Yates-Grundy estimator is non-negative

 

  1. Develop Yates corrected estimator under Linear Systematic Samping

 

  1. Develop Hartley-Ross ratio type unbiased estimator under simple random sampling.

 

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Loyola College M.Sc. Statistics April 2006 Reliability Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 53

FOURTH SEMESTER – APRIL 2006

                                                        ST 4955 – RELIABILITY THEORY

 

 

Date & Time : 29-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section-A (10×2=20 marks)

Answer ALL the questions. Each question carries TWO marks.

  1. Define the terms: (a) Reliability function R(t)

(b) Hazard function r(t)

  1. In the usual notation, show that MTBF = R*(0)
  2. If n components functioning independently and having equal reliabilities are operating in parallel, find the reliability of the entire system.
  3. Comment on the following statement: Series and parallel systems are particular cases of an (m, n) system.
  4. What are (a) parallel-series and (b) series-parallel systems?
  5. Define a coherent structure and give two examples.
  6. Define (a) Minimal path vector (b) Minimal cut vector.
  7. What do you mean by (i) the number of critical path vectors of component i and (ii) relative importance of component i?
  8. Give an example of a set of random variables that are not associated.
  9. What is a cumulative damage shock model?

Section-B (5×8=40 marks)

Answer any FIVE questions

  1. Obtain the reliability function, hazard rate and the system MTBF for Weibull   distribution with the parameters λ and α.
  2. Suppose that gi(t) is the density function for Ti, the time to failure of ith component in a standby system with three independent and identical components and is given by gi(t) = λ e-λt, i = 1, 2, 3; t>0. Obtain the system failure time density function and hence find its expected value.
  3. What is a series system? Obtain the system failure time density function for a series system with n independent components. Suppose each of the n independent components has an exponential failure time distribution with constant failure rate λi, i = 1, 2, 3, …, n. Find the system reliability.
  4. Let Φ be a coherent structure. Show that

Φ(x Ц y) ≥ Φ(x) Ц Φ(y)

Further, show that the equality holds for all x and y if and only if the structure is parallel.

  1. Given the structure Φ, define the dual of the structure Φ. Also, show that the minimal path sets for Φ are the minimal cut sets for ΦD.
  2. Consider a coherent system with three components having the structure function Φ(x1, x2, x3) = x1. (x2 Ц x3)

Determine the number of critical path vectors of each component. Also determine       the relative importance of each component. Are components 2 and 3 equally important?

  1. When do you say that a set of random variables T1, T2,… , Tn are associated? Show that a set consisting of a single random variable is associated.
  2. Let the density of exist. Show that F is DFR if and only if r(t) is decreasing       in t.

Section-C (2×20 = 40 marks)

Answer any TWO questions. Each carries TWENTY marks

19.a. What is a series- parallel system of order (m, n)? Write down the system reliability and system failure rate of the same.                          (10 marks)

  1. Assuming that the components have identical constant failure rate λ, obtain MTBF of the series- parallel system. (10 marks)

20.a. Define the terms. (i) System availability.

(ii) Steady state availability.                                (4 marks)

  1. A system consists of a single unit, whose lifetime X and repair time Y are independent random variables with probability density functions f (.) and g (.) respectively. Assume that initially at time zero, the unit just begins to operate. Determine the reliability, availability and steady state availability of the system.                                                                                                  (4+6+6 marks)

21.a. Let Φ be a coherent structure. Show that

Φ(x .y) ≤ Φ(x) .Φ(y)

Also, show that the equality holds for all x and  y if and only if the structure is series.                                                                                      (10 marks)

  1. Let h be the reliability function of a coherent system. Show that h (p Ц p’) ≥ h (p) Ц h (p’) for all 0 ≤ p, p’ ≤ 1                                     (10 marks)

22.a.  Show that the order statistics Y1:n, Y2::n,…,Yn:n corresponding to n independent random variables are associated.                                                    (10 marks)

  1. Examine whether Gamma distribution G (λ, α) is IFR or DFR. (10 marks)

 

 

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Loyola College M.Sc. Statistics April 2006 Probability Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 31

SECOND SEMESTER – APRIL 2006

                                                       ST 2805 – PROBABILITY THEORY

(Also equivalent to ST 2800)

 

 

Date & Time : 24-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

Section-A

Answer ALL questions                                                                   (10 ´ 2 = 20 marks)

  1. With reference to tossing a regular die once and noting the outcome, identify completely all the elements of the probability space (Ω, A, P).
  2. Show that the limit of any convergent sequences of events is an event.
  3. Let F(x) = P [X < x], x Є R. Prove that F (.) is continuous to the left.
  4. Write down any two properties of the distribution function of a random vector (X, Y).
  5. If X is a random variable with P[X = (-1) k2k] = 1/2k, k = 1, 2, 3…, examine whether E[X] exists.
  6. If X2 and Y2 are independent, are X and Y independent?
  7. Define almost sure convergence and convergence in probability for a sequence of random variables.
  8. If Φ is the characteristic function (CF) of a random variable X, find the CF of (2X+3).
  9. Let {Xn, n = 1, 2, …} be a sequence of independent and identically distributed (iid) n

N (μ, σ2) random variables. Define Yn = 1/n   Σ X2 k, n = 1, 2, 3,… Examine                                                                                  K=1

whether Kolmogorov strong law of large numbers (SLLN) holds for
{Yn, n = 1, 2, 3…}

  1. State Lindeberg – Feller Central limit theorem.

 

Section – B

Answer any FIVE questions                                                       (5 × 8 = 40 marks)

  1. Define the distribution function F(x) of a random variable X. State and establish its defining properties.
  2. State and prove Minkowski’s inequality
  3. State and prove Borel Zero- one law.
  4. Find var(Y), if the conditional characteristic function of Y given X=x is

[1+ (t2 /x)]-1 and X has frequency function f(x) = 1/x2   for x ≥ 1

=   0      otherwise

    1. Show that convergence in probability implies convergence in distribution.

 

  1. Define convergence in quadratic mean for a sequence of random variables.

X is a random variable, which takes on positive integer values.

Define Xn =   (n+1) if X=n

=    n     if X = (n+1)

=    X    otherwise

Show that Xn converges to X in quadratic mean.

  1. Show that Xn → X in probability if and only if every subsequence of {Xn} contains a further subsequence, which converges almost surely.
  2. Let {Xn, n ≥ 1} be a sequence of independent random variables such that Xn has uniform distribution on (-n, n). Examine whether the central limit theorem holds for the sequence {Xn, n ≥ 1}.

Section-C

Answer any TWO questions.                                                          (2 × 20 = 40 marks)

  1. a. Define the probability distribution of a random variable. Show that the probability distribution of a random variable is determined by its distribution function.(8 marks)
  2. Show that the vector X = (X1, X2, …, Xp) is a random vector if and only if Xj,

j = 1, 2,… , p is a real random variable.                                                (8 marks)

  1. If X is a random variable with continuous distribution function F, obtain the probability distribution of F(X).                                        (4 marks)

20.a. Show that convergence in quadratic mean implies convergence in probability. Illustrate by an example that the converse is not true.                               (8 marks)

  1. State and prove Kolmogorov zero-one law.                                         (12 marks)

21.a.  State and prove Kolmogorov three series criterian for almost sure convergence of the series    ∞

Σ Xn of independent random variables.         (12 marks)

n=1

  1. Let {Xn} be a sequence of normal variables with E (Xn) = 2 + (1/n) and

var (Xn) = 2 + 1/n2, n= 1, 2, 3 … Examine whether the sequence converges in distribution.                                                                              (8 marks)

22.a. State and prove the continuity theorem for a sequence of characteristic functions.

(12 marks)

  1. Let {Xk} be a sequence of independent random variables with

P [Xk = kλ]  = P [Xk = -kλ] = 1/2, k = 1, 2, 3… Show that central limit theorem holds for

λ ≥ -1/2.                                                                                                  (8 marks)

 

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Loyola College M.Sc. Statistics April 2006 Measure And Probability Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 26

FIRST SEMESTER – APRIL 2006

                                       ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

Date & Time : 25-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Part A

Answer all the questions.                                                                            10 ´ 2 = 20

 

  1. Define set of all real numbers as follows. Let An = ( -1/n, 1] if n is odd and

An = ( -1, 1/n] if n is even. Find lim sup An and lim inf An.

  1. Explain Lebesgue-Stieltjes measure with an example.
  2. Define counting measure with an example.
  3. State Borel- Cantelli Lemma.
  4. If h is B– measurable function, show that | h | is also B-measurable

function.

  1. What is induced probability space?
  2. If random variable X takes only positive integral values, show that
    E(X) = P[ X ³ n].
  3. Define convergence in r-th mean.
  4. If Xn  X and g is continuous, show that g(Xn)  g (X).
  5. State Levy’s theorem.

Part B

Answer any five questions.                                                                     5 ´ 8 = 40

 

  1. If { Ai , i ³ 1) is a sequence of subsets of a set W, show that

Ai = (A i  – A i – 1).

  1. Show that countable additivity of a set function with m(f) = 0 implies finite additivity of a set function.
  2. Prove that every finite measure is a s – finite measure. Is the converse true? Justify.
  3. Let f be B-measurable and if f = 0 a.e. [m],  show that f dm = 0.
  4. State and establish  additivity theorem of integral.

 

 

 

  1. State and establish Minkowski’s inequality.
  2. Show that Xn  X implies Xn  X. Is the converse true? Justify.
  3. If XnX, show that (Xn2 + Xn) (X2 + X).

 

Part C

Answer any two questions.                                                                   2 ´ 20 = 40

 

  1. a). State and establish extended monotone convergence theorem.

b). State and establish basic integration theorem.                                         ( 12 + 8)

  1. a). Let l (A) = dm;  A in the s – field Á, where fdm exists; thus l is a signed measure on Á. Show that l+(A) = f +dm, l– (A) = f – dm and |l|(A) = |f|dm.

b). State and establish Jordan – Hohn decomposition theorem.                    (8 + 12)

  1. a). If hdm exists and C є R, show that Chdm = Chdm.

b). Let X be a random variable defined on the space (W, A, p) and E |X|k < µ, k>0, Show that nk P[|X|>n] ® 0 as n ® µ.                                                         (10 + 10)

  1. a). Show that Xn  X implies Xn   X. Is the converse true? Justify.
    b). State and establish Lindberg Central limit theorem.                              (10 + 10)

 

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Loyola College M.Sc. Statistics April 2006 Industrial Statistics Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 46

FOURTH SEMESTER – APRIL 2006

                                                     ST 4801 – INDUSTRIAL STATISTICS

 

 

Date & Time : 25-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer all the questions.                                                                                  10 ´ 2 = 20

  1. Discuss the logic and statistical basis underling the general use of 3 – sigma limits on control charts.
  2. Mention the theoretical basis of p – chart and set up its control limits.
  3. How is lack of control of a process determined using control chart techniques?
  4. What is process capability ratio (PCR)?
  5. Write down the control limits of a coefficient of variation chart.
  6. What is an average run length (ARL)?
  7. Explain an attribute single sampling plan.
  8. Define AOQ for a single sampling plan.
  9. Discuss the concepts of chance and assignable causes of variability.
  10. Write a short note on multivariate control chart.

 

PART – B
Answer any five questions.                                                                     5 ´ 8 = 40

 

  1. Explain the method of constructing control limits for X-bar and R charts when the sample sizes are different for various subgroups.
  2. A control chart indicates that the current process fraction non-conforming is 0.02. If 50 samples are inspected each day, what is the probability of detecting a shift in the fraction non-conforming to 0.04 on the first day after shift? By the end of the third day following the shift?
  3. Write a detail note on the moving average control chart.
  4. Consider a modified control chart with CL at m = 0 and s =1. If n = 5, the tolerable fraction non-conforming d = 0.00135 and the control limits are at 3, sketch the OC curve of the chart.
  5. Design a cumulative sum control chart to detect a shift of D = 0.75 that has L(0) = 400. Is it possible to find a cumulative sum control chart for detecting this shift that has L(0.75) £ 12? What is L(0) for this chart?
  6. For the double sampling plan N = 120, n1 = n2 = 13, c1 =0 and c2 = 1, obtain Pa, ASN, AOQ and ATI when the submitted lot has the fraction non – conforming at p = 0.18.
  7. What are acceptances and rejection lines of a sequential sampling plan for attributes? How are the OC and ASN values obtained for this plan?
  8. What are modified control charts? Explain the method of obtaining control limits.

 

PART – C
Answer any two questions.                                                                   2 ´ 20 = 40

 

  1. a). Distinguish between c and u charts. Explain the situations where c and u charts are

applicable and how are the limits obtained for these charts.

b). Explain the procedure of obtaining the OC curve for a p – chart with an

illustration.

(10 + 10)

  1. a). Define the terms i). Rational subgroups ii). Specification limits

iii). Natural tolerance limits iv). Probability limits.

b). Explain the relevance of non-parametric method in quality control procedures with

an illustration.                                                                                              (10 + 10)

  1. a). What purpose does a cumulative sum chart serve?

b). Outline the procedure of constructing a V-mask.                                         (5 + 15)

  1. a). What are continuous sampling plans and mention a few situation where these

plans are applied

b). Explain with an illustration the method of obtaining the probability of acceptance

for a triple sampling plan.                                                                            (10 + 10)

 

 

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Loyola College M.Sc. Statistics April 2006 Estimation Theory Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2006

                                                        ST 2808 – ESTIMATION THEORY

 

 

Date & Time : 19-04-2006/FORENOON     Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

 

Answer  ALL  questions.  Each  carries TWO  marks.     (10 x 2 =  20 marks)

 

  1. If the class of unbiased estimators of  a parametric function is neither empty nor singleton, then show that the class is uncountable.
  2. Prove or disprove the uniqueness of UMVUE.
  3. If  δ  is a UMVUE and bounded, then  show that any polynomial in δ  is also a UMVUE.
  4. State Chapman – Robbin’s  inequality.
  1. Suppose δ  is sufficient for Р  and  Р0  С  Р,  then show that δ  is sufficient for Р0.
  1. Let S be a sufficient statistic. If  likelihood equivalence of x and y in the support A of the random variable X implies S(x) = S(y)   x , y  Є  A , then show that S is minimal sufficient.
  2. Let X 1  , X2   be a random sample from N ( θ ,1),   θ Є R . Verify whether or not   (X1  , X 2 ) is complete.
  3. Give two examples of a  Location-Scale family of distributions .
  4. Define Ancillary Statistic and give an example.
  5. Give an example of  M- estimator Tn  of  θ  which can be thought of as a weighted

average of the sample values with the weights depending on the data.

 

PART – B

 

Answer  any FIVE  questions.  Each  carries EIGHT marks.     (5 x 8 =  40 marks)

 

  1. Let X be a discrete random variable with  pdf  pθ(x)  =  θ  if  x  =  -1 and

pθ(x)  =  (1 – θ )2θx   if  x  =  0,1,2,…, where 0 < θ <  1. Find the class U0 of

unbiased estimators of ‘0’ and hence find the class Ug of unbiased estimators of

g (θ)  =  θ , 0< θ < 1.

  1. Give an example where only constant estimable parametric functions have

UMVUE.

  1. Give an example of a UMVUE whose variance is greater than

Chapman-Robbins’s  Lower Bound.

    1. Let X1 ,…,Xn   be a random sample of size n from N(θ , 1) ,  θ Є R. Using Fisher information, show that   α ixi  is sufficient iff  αi  are equal for all i .
    2. Let X1 ,… ,Xn   be a random sample of size n from U (0,θ), θ  > 0. Then show that S() = X(n)  is minimal sufficient.
    3. Show that a complete sufficient statistic is minimal sufficient if it exists.

 

  1.  Let X1 ,… ,Xn   be a random sample of size n from B (m,θ),  m known and

θ unknown.   Show that the joint distribution of  (X1 ,…,Xn  ) belongs to an

exponential family. Hence find the mgf of   Xi.

  1. Let X ~ N (θ , 1 ), θ Є R , and let the prior distribution of θ be N ( 0 , 1 ).

Find the Bayes estimator of θ when the loss function is

  • Squared error
  • Absolute error.

 

 

PART – C

 

Answer  any TWO  questions.  Each  carries TWENTY marks.     (2 x 20 =  40 marks)

 

19(a). State and prove a necessary and sufficient condition for an estimator in the class ug

to  be a  UMVUE.  (10)

19(b). Derive Chapman – Robbin’s inequality, using covariance inequality. (10)

 

20(a). Give an example of a family which is not boundedly complete.(10)

20(b). Let X1  ,….., Xn   be a random sample from N(μ ,σ2 ), μ Є R, σ  > 0. Show that the distribution of ( X1 ,…..,Xn  ) belongs to two-parameter exponential family. Hence by using Basu’s theorem, establish the independence of   and s2. (10)

21(a). Prove that  δ*  is  D-Optimal estimator of g(θ)  iff each component of δ*  is a            UMVUE.  (14)

21(b). Let X1  ,… , Xn  be a random sample from N(μ ,σ2 ), μ Є R, σ  > 0. Obtain Jackknife estimator of variance   σ2  .    (6)

 

22(a). State and prove Lehmann – Scheffe Theorem for convex loss function. (8)

22(b). Let  have the p.d.f    f ( – ξ ).  If   δ  is a location equivariant estimator ,  then

show that the bias ,  risk and variance of  δ  do not depend of  ξ  .  (6)

22( c )  Let X1  ,… ,Xn  be a random sample from N(ξ  ,1 ) ,   ξ Є R .  Find the MRE

estimator of  ξ   when the loss is squared error.  (6).

 

 

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Loyola College M.Sc. Statistics April 2006 Computational Statistics – III Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 41

THIRD SEMESTER – APRIL 2006

                                            ST 3803 – COMPUTATIONAL STATISTICS – III

 

 

Date & Time : 02-05-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Answer THREE questions choosing one from each section.

SECTION – A

  1. A Scientist studied the relationship of size and shape for painted turtles. The following table contains their measurements on 10 females and 10 male turtles.  Test for equality of the two population mean vectors.

 

FEMALE

MALE
Length (x1) Width (x2) Height (x3) Length (x1) Width (x2) Height (x3)
98 81 38 93 74 37
103 84 38 94 78 35
103 86 42 96 80 35
105 86 42 101 84 39
109 88 44 102 85 38
123 92 50 103 81 37
123 95 46 104 83 39
133 99 51 106 83 39
133 102 51 107 82 38
133 102 51 112 89 40
  1. Given the following trivariate Normal distribution with mean vector and variance covariance matrix.

 

  1. Obtain the conditional distribution of X1 and X2 given X3 = 10

(15 marks)

  1. Obtain the distribution of CX where

(6 marks)

  • Find the correlation matrix for the data of Female turtles given in question No.1. Find whether the correlations are significant.                    (13 marks)

 

SECTION – B

Answer any ONE question

  1. a) Use two-phase method to solve following linear programming problem:

Max Z =

Sub. To

 

(18.5 marks)

 

  1. b) Solve the following transportation problem:

 

9   10  11                                                                              (15 marks)

  1. a) Solve the following game graphically:

B

(18.5 marks)

  1. b) Patients arrive at a clinic according to a Poisson distribution at a rate of 30

patients per hour.  The waiting room does not accommodate more than 14

patients.  Examination time per patient is exponential with mean rate 20 per hour.

  1. Find the effective arrival rate at the clinic.
  2. What is the probability that an arriving patient will not wait? Will find a vacant seat in the room?
  • What is the expected waiting time until a patient is discharged from the clinic?                                                              (15 marks)

SECTION – C

 

Answer any ONE question

  1. a) Suppose the one step transition probability matrix (tpm) is given as below: Find Poo(2), foo(n), f13(u) and f33(u).

 

 

(17 marks)

  1. b) For a three state Markov Chain with states {0,1,2} and tpm.

P = ,   Find m0, m1 and m2.

(17 marks)

  1. a) An infinite Markov Chain on the set of non-negative integers has the transition

function as follows:

Pko =    and Pk1 k+1 =

 

  • Find whether the chain is positive recurrent, null recurrent or transient.
  • Find stationary distribution if it exists.
  1. b) For a Branching process with off-spring distribution given by p(0) = p (3) =

Find the probability of extinction, when   (i) X0 = 1    and    (ii)  X0 > 1.

(17+17 marks)

 

 

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Loyola College M.Sc. Statistics April 2006 Computational Statistics – II Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 44

THIRD SEMESTER – APRIL 2006

                                             ST 3807 – COMPUTATIONAL STATISTICS – II

 

 

Date & Time : 02-05-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Answer any three questions and each question carries 33.5 marks.

  1. The following data relates to three body measurements taken on boys and girls of the same community. Test whether boys and girls differ from each other on the basis of their measurements assuming the variance – covariance matrices of these two data sets are equal.

 

Boys (centimeter)
Serial Number Height Chest Mid Upper Arm
1

2

3

4

5

6

 78

76

92

81

81

84

 60.6

58.1

63.2

59.0

60.8

59.5

 16.5

12.5

14.5

14.0

15.5

14.0

 

 

Girls
Serial Number Height Chest Mid Upper Arm
1

2

3

4

5

6

7

8

9

 

 80

75

78

75

79

78

75

64

80

 58.4

59.2

60.3

57.4

59.5

58.1

58.0

55.5

59.2

 

 14.0

15.0

15.0

13.0

14.0

14.5

12.5

11.0

12.5

 

  1. a) For the girls data given in question number 1 obtain

i). sample correlation matrix and test for the significance of the correlations.

ii). Partial correlation r12..3

iii). Multiple correlation R1.23 and test its significance.

  1. b)   Draw a Q-Q plot for the girls data on chest measurement in question number 1 to find whether the data is from a normal distribution.

 

  1. a). The following frequencies with the corresponding probabilities observed in a genetical experiment are given below :

 

Cell Number 1 2 3 4
Probabilities (1+q) / 4 (1 – q) / 4 (1 – q) / 4 (1 + q) / 4
Frequencies 1997 906 904 32

Obtain the Maximum Likelihood Estimator of q and the variance of this estimator.

 

b). Obtain the estimator by the method of modified minimum chi-square for the data in question 3. a).

c). Estimate the parameter q assuming a truncated Poisson distribution truncated at 0 for the given data

X 1 2 3 4 5 6 7 8
Frequency 60 50 40 25 10 8 4 3

 

  1. a) Classify the states of the Markov chain having the following transition probability    matrix (tpm)

 

Find Pn and   lim  Pn

n→∞

  1. Consider the Markov chain with state space {0, 1, 2, 3} and one step tpm

 

where q = (1-p), 0<p<1. Comment on the nature of the states.

  1. a) Consider a Markov Chain having the state space {0, 1, 2} and transition matrix

 

 

  • Show that the Markov chain is irreducible.
  • Obtain the period for this Markov chain
  • Obtain lim Pdn.

n→∞

 

  1. b)   An infinite Markov chain on the set of non-negative integers has the transition   matrix as follows:

pk0 = (k+1) / (k+2)       and    Pk, k+1 = 1 / (k+2)

 

  • Find whether the chain is positive recurrent, null recurrent or transient.
  • Find the stationary distribution, in case it exists.

 

 

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Loyola College M.Sc. Statistics April 2006 Applied Regression Analysis Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 28

FIRST SEMESTER – APRIL 2006

                                            ST 1811 – APPLIED REGRESSION ANALYSIS

 

 

Date & Time : 27-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

SECTION – A

Answer ALL the Questions                                                                     (2´10 = 20 marks)

  1. Define ‘Residuals’ of a linear model.
  2. What is Partial F- test.
  3. What are the two scaling techniques for computing standardized regression coefficients.
  4. Define ‘Externally Studentized Residuals’.
  5. Stae the variance stabilizing tramsformation if V(Y) is proportional to [E(Y)]3.
  6. What is FOUT in Backward selection process.
  7. How is the multicollinearity trap avoided in regression models with dummy variables.
  8. State any one method of detecting multicollinearit.
  9. Give an example of a polynomial regression model.
  10. Give the motivation for Generalized Linear Models.

SECTION – B

Answer any FIVE Questions                                                                   (5´8 = 40 marks)

  1. Fill up the missing entries in the following ANOVA for a regression model with 5 regressors and an intercept:
Source d.f S.S Mean S.S. F ratio
Regression

Residual

 ?

14

 ?

?

 40.5

?

 13.5

——-
Residual ? ? ——– ——-

Also, test for the overall fit of the model.

 

  1. The following table gives the data matrix corresponding to a model
    Y = b0+b1X1+b2X2+b3X3. Suppose we wish to test H0: b2 = b3. Write down the restrcited model under H0 and the reduced data matrix that is used to build the restricted model.

1    2   -3    4

1   -1    2    5

1    3    4    -3

1   -2   1     2

X =     1    4    5   -2

1   -3    4    3

1    2    3     1

1    1    2     5

1    4   -2    2

1   -3    4    2

  1. Explain how residual plot are used to check the assumption of normality of the errors in a linear model.

 

  1. Discuss ‘Generalized Least Squares’ and obtain the form of the GLS estimate.

 

  1. Explain the variance decomposition method of detecting multicollinearity and derive the expression for ‘Variance Inflation Factor’.
  2. Discuss ‘Ridge Regression’ and obtain the expression for the redge estimate.

 

  1. Suggest some strategies to decide on the degree of a polynomial regression model.

 

  1. Describe Cubic-Spline fitting.

SECTION – C

Answer any TWO Questions                                                                 (2 ´ 20 = 40 marks)

  1. Build a linear regression model with the following data and test for overall fit . Also, test for the individual significance of X1 and of X2.

Y:  12.8    13.9    15.2     18.3     14.5     12.4

X1:    2          3        5          5          4          1

X2:       4          2        5          1          2          3

 

  1. (a)Decide whether “Y =b0 + b1X” or “Y2 = b0 + b1X” is the more appropriate model for the following data:

X:    1      2       3      4

Y:  1.2   1.8    2.3   2.5

 

(b)The starting salary of PG students selected in campus interviews are given below

along with the percentage of marks they scored in their PG and their academic

stream:

Salary  (in ‘000 Rs) Stream Gender % in PG
12

8

15

12.5

7.5

6

10

18

14

 Arts

Science

Commerce

Science

Arts

Commerce

Science

Science

Commerce

 Male

Male

Female

Male

Female

Female

Male

Male

Female

 75

70

85

80

75

60

70

87

82

It is believed that there could be a possible interaction between Stream and % in

PG and between Gender and % in PG. Incorporate this view and create the data

matrix. (You need not build the model).                                                      (10+10)

  1. Based on a sample of size 16, a model is to be built for a response variable with four regressors X1, …,X4. Carry out the Forward selection process to decide on the significant regressors, given the following information:

SST = 1810.509, SSRes(X1) = 843.88, SSRes(X2) = 604.224, SSRes(X3) = 1292.923, SSRes(X4) = 589.24, SSRes(X1, X2) = 38.603, SSRes(X1,X3) = 818.048,           SSRes(X1,X4) = 49.84, SSRes(X2,X3) = 276.96, SSRes(X2,X4) = 579.23,          SSRes(X3,X4) = 117.14, SSRes(X1,X2,X3) = 32.074, SSRes(X1,X2, X4) = 31.98, SSRes(X1,X3,X4) = 33.89, SSRes(X2,X3,X4) = 49.22, SSRes(X1,X2,X3,X4) = 31.91.

 

  1. (a) Obtain the likelihood equation for estimating the parameters of a logistic regression model.

(b) If the logit score (linear predictor) is given by –2.4 + 1.5 X1 + 2 X2, find the estimated P(Y = 1) for each of the following combination of the IDVs:

X1:  0       1.5        2       3       -2      -2.5

X2:  1         0       1.5     -1        2       2.5                                    (12+8)

 

 

 

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Loyola College M.Sc. Statistics April 2006 Applied Regression Analysis Question Paper PDF Download

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 52

FOURTH SEMESTER – APRIL 2006

                                            ST 4954 – APPLIED REGRESSION ANALYSIS

 

 

Date & Time : 27-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION –A

Answer ALL the Questions                                                                   (10 X 2 = 20 marks)

 

  1. State the statistic for testing the overall fit of a linear model with ‘k’ regressors.
  2. Define ‘Extra Sum of Squares’.
  3. Define ‘Studentized’ Residuals.
  4. What is a ‘Variance Stabilizing Transformation’?
  5. State the consequence of using OLS in a situation when GLS is required.
  6. Define “Variance Inflation Factor’.
  7. Give the form of the Ridge Estimate when a constant ‘l’ is added to the diagonal elements of X’X.
  8. What is a hierarchical polynomial regression model?
  9. Mention the components of a ‘Generalized Regression Model’ (GLM).
  10. Define ‘Sensitivity’ of a Binary Logit Model.

 

SECTION – B

Answer any FIVE Questions                                                                   (5 X 8 = 40 marks)

 

  1. The following table gives the data on four independent variables used to build a linear model with an intercept for a dependent variable.
X1 X2 X3­ X4
2

1

5

4

-2

3

2

-3

2

1

 -1

4

2

3

-2

3

2

3

2

1

 3

2

-3

1

4

-1

2

5

-2

-3

 5

3

4

-1

2

1

4

1

3

-2

If one wishes to test the hypothesis H0: b1 = b3, b2 = 2b4, write down the reduced

data matrix and the restricted model under H0. Briefly indicate the test procedure.

 

  1. Depict the different possibilities that occur when the residuals are plotted against the fitted values. How are they interpreted?

 

  1. Define ‘Standardized Regression Coefficient’ and discuss any one method of scaling the variables.

 

  1. Decide whether “Y= b0 + b1X” or “Y1/2 = b0 + b1X” is the more appropriate model for the following data:
X 1 2 3 4
Y 3.5 4.7 6.5 9.2

 

  1. Discuss the issue of ‘multicollinearity’ and its ill-effects.

 

Eigen Values

of X’X

 Singular

Values of X

 Condition

Indices

 Variance decomposition Proportions

X1        X2          X3         X4        X5        X6
3.4784

2.1832

1.4548

0.9404

0.2204

0.0725

 ?

?

?

?

?

?

 ?

?

?

?

?

?

 0.0003  0.0005  0.0004  0.0004       ?      0.0350

?     0.0031  0.0001  0.3001  0.0006  0.0018

0.0004       ?      0.0005  0.0012  0.0032  0.2559

0.0011  0.6937  0.5010  0.0002  0.7175       ?

0.0100  0.0000       ?      0.0003  0.0083  0.2845

0.8853  0.3024  0.4964       ?      0.2172  0.0029
  1. Fill up the missing entries in the following table and investigate the presence of collinearity in the data, indicating which variables are involved in collinear relationships, if any.

 

  1. Explain ‘Cubic Spline’ fitting.

 

  1. Describe the components of a GLM. Show how the log link arises naturally in modeling a Poisson (Count) response variable.

 

SECTION – C

 

Answer any TWO Questions                                                                 (2 X 20 = 40 marks)

 

  1. The observed and predicted values of a response variable (based on a model using 25 data points) and the diagonal elements of the ‘Hat’ matrix are given below:
Yi 16.68   11.50   12.03   14.88  13.75   18.11   8.00   17.83   79.24   21.50   40.33   21.0   13.5
Yi^ 21.71   10.35   12.08   9.96    14.19   18.40   7.16   16.67   71.82   19.12   38.09  21.59 12.47
hii 0.102   0.071   0.089   0.058  0.075   0.043   0.082  0.064  0.498   0.196   0.086  0.114 0.061

 

Yi 19.75   24.00   29.00   15.35   19.00   9.50    35.10   17.90   52.32   18.75   19.83   10.75
Yi^ 18.68   23.33   29.66   14.91   15.55   7.71    40.89   20.51   56.01   23.36   24.40   10.96
hii 0.078   0.041   0.166   0.059   0.096   0.096   0.102   0.165   0.392   0.041   0.121   0.067

 

Compute PRESS statistic and R2prediciton. Comment on the predictive power of the

underlying model.

 

  1. (a) In a study on the mileage performance of cars, three brands of cars (A, B and C) and two types of fuel (OR and HG) were used. The speed of driving was also observed and the data are reported below:

 

 

 

Mileage(Y) 14.5  12.6  13.7  15.8  16.4  13.9  14.6  16.7  11.8  15.3  16.8  17.0 15.0  16.5
Speed

Car

Fuel

   45     60     50      60    55     52     59    50      40     53     62     56    62    55

A       B      C       B     A      A       C     A       B      B      C      C      A     B

OR    HG    OR   HG   HG   OR    HG  OR    OR    HG   HG    OR  HG   OR

 

Create the data matrix so as to build a model with an intercept term and interaction terms between Fuel and Driving Speed and also between Car-type and Driving Speed.

(You need not build any model).

 

(b) Discuss GLS and obtain an expression for the GLS estimate.                (14 + 6)

 

  1. Based on a sample of size 15, a linear model is to be built for a response variable Y with four regressors X1,…,X4. Carry out the Forward Selection Process to decide which of the regressors would finally be significant for Y, given the following information:

SST = 543.15,  SSRes(X1) = 253.14,  SSRes(X2) = 181.26,  SSRes(X3) = 387.88, SSRes(X4) = 176.77,         SSRes(X1,X2) = 11.58,         SSRes(X1,X3) = 245.41,         SSRes(X1,X4) = 14.95,      SSRes(X2,X3) = 83.09,         SSRes(X2,X4) = 173.77,      SSRes(X3,X4) = 35.15,      SRes(X1,X2,X3) = 9.62,        SSRes(X1,X2,X4) = 9.59, SSRes(X1,X3,X4) = 10.17, SSRes(X2,X3,X4) = 14.76,    SSRes(X1,X2,X3,X4) = 9.57

 

  1. The laborers in a coal-mine were screened for symptoms of pneumoconiosis to study the effect of “number of years of work” (X) on the laborers’ health. The response variable ‘Y’ defined as ‘1’ if symptoms were found and ‘0’ if not. The data on 20 employees are given below:
Y    0    1    1    0    1    1    0    0    0    1    1     1    0    0    1    0    0     1    1    1
X   10  30  28  14  25  35  15  12  20  24  33   27  13  12  18  17  11   28  32  30

 

The logit model built for the purpose had the linear predictor (logit score) function as – 4.8 + 0.1 X. Construct the Gains Table and compute the KS statistic. Comment on the discriminatory power of the model.

 

 

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