Loyola College B.Sc. Statistics Nov 2012 Statistical Methods Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2012

ST 1502/ST 1500 – STATISTICAL METHODS

 

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions:                                                                                       (10 x 2 = 20)

  1. Why is sampling necessary under certain conditions?
  2. A survey of 100 people is conducted and all are asked questions relating to the following characteristics:
  • marital status
  • salary
  • occupation
  • number of hours of television they watch per week

What type of data and measurement scales are applicable?

  1. List the requisites of a good measure of central tendency.
  2. What is meant by Kurtosis?
  3. State the principles of least squares.
  4. What is the general form of growth curves?
  5. Define rank correlation coefficient.
  6. Find the means of variables X and Y and the correlation coefficient given the following information:

Regression equation of Y on X:    3Y – X – 50 = 0

Regression equation of X on Y:    3Y– 2X –10 = 0

  1. Out of 900 persons, 300 were literates and 400 had travelled beyond the limits of their district.100 of the literates were among those who had not travelled. Is there any relation between literacy and travelling?
  2. What is meant by coefficient of colligation?

PART – B

Answer any FIVE questions:                                                                           (5 x 8 = 40 marks)

  1. The survey about colour preferences reported the age distribution of the people who responded.
Age group (years) 1-18 19-24 25-35 36-50 51-69 70-74
count 10 97 70 36 14 5

Draw ‘less than ogive’ curve and locate the median.

  1. Describe the various ways of classification of statistical data with suitable illustrations.
  2. The volumes of water (in litres) consumed by 12 elephants in one day are listed below:

66        90        68        94        86        96        70        138      90        120      92        102

Calculate the mean and variance and interpret the data.

  1. Describe the construction of Lorenz curve.
  2. What is skewness? Distinguish diagrammatically the different types of skewness.
  3. Calculate the sample coefficient of correlation between number of ovulated follicles

and number of eggs laid by pheasants. Data of 11 pheasants were collected:

Number of eggs 39 29 46 28 31 25 49 57 51 21 42
Number of follicles 37 34 52 26 32 25 55 65 40 25 45

 

  1. Fit a curve of the form y = abt for the following data observed on the growth of a fruitfly population
Time t (in days) 2 3 4 5 6 7 8 9
No.of flies y 110 116 122 128 134 141 148 155
  1. Describe the conditions for consistency of data when there are three attributes.

PART – C

Answer any TWO questions                                                                                 (2 x 20 = 40 marks)

  1. (a) What is meant by a questionnaire? Explain the precautions that must be taken

while drafting a questionnaire.                                                                             (12 marks)

(b) Distinguish between primary and secondary data.                                               (8 marks)

 

  1. (a) Establish the relationship between raw and central moments.                              (10 marks)

(b) The following frequency distribution is the weight in pounds of 57 children at a

day-care center:

Weight (in pounds) 10-19 20-29 30-39 40-49 50-59 60-69 70-79
No. of children 5 19 10 13 4 4 2

Calculate mean deviation about median.                                                               (10 marks)

 

  1. (a) What is meant by ‘curve fitting’? Give the normal equations to fit a second degree

parabola.                                                                                                                (10 marks)

(b) In a sample of 500 children, 200 came from higher income group and the rest

from lower income group. The numbers of delinquent c hildren in these groups

were 25 and 100 respectively. Calculate the coefficient of association between

delinquency and income group.                                                                            (10 marks)

 

  1. Potato chip lovers do not like soggy chips,so it is important to find characteristics of the production process that produce chips with an appealing texture. The following sample data on frying time(in seconds) and moisture content(%) were selected.
Frying time 65 50 35 30 20 15 10 5
Moisture content 1.4 1.9 3.0 3.4 4.2 8.1 9.7 16.3

Predict the moisture content of the chips if the frying time is 40 seconds.

 

 

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Loyola College B.Sc. Statistics Nov 2012 Statistical Mathematics – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – STATISTICSTHIRD SEMESTER – NOVEMBER 2012ST 3503/3501/3500 – STATISTICAL MATHEMATICS – II
Date : 02/11/2012 Dept. No.         Max. : 100 Marks                 Time : 9:00 – 12:00
PART-A  Answer ALL the questions: [10×2 =20]
1. Define upper sum and lower sums.2. Give an example of a function that is not Riemann integrable.    3. Define improper integrals.   4. State the comparison test for the improper integrals    5. Define variance-covariance matrix.     6. When do we say that a system of equations is homogenous?7. State the order and degree of the differential equation:  8. Obtain the Laplace transform of  t > 0. 9. Define characteristic equation and characteristic roots. 10. Write down the importance of Caley-Hamilton Theorem.

PART – BAnswer any FIVE questions:      [5×8 =40]11. Evaluate   from first principles.12. Show that every continuous function defined on a closed interval of the real line is       Riemann integrable.
13. Find the mean and variance of Beta distribution of I kind.
14. Discuss the convergence of gamma integral.
15. Find the covariance between X and Y whose joint p.d.f.  is  .
16. Evaluate  over the upper half of the circle with centre (0, 0) and radius 1.

 

17. a) Solve the differential equation   .    (b) Obtain the inverse Laplace transform of  .18. Find the characteristic roots and corresponding vectors of the matrix          .

PART – C Answer any TWO questions: [2×20 =40]
19. (a) State and Prove the first fundamental theorem on integral calculus.       (b) Derive the MGF of normal distribution. Hence find its mean and variance.  20. (a) Establish the relation between the Beta and Gamma integrals. Hence find the            value of β (3, 4).
(b) Solve the differential equation:   .
21. (a) Solve the following initial value problem using Laplace transform,            where y(0) = -2 and y'(0) = 5 :  .     (b) Let X and Y be two independent one parameter Gamma random variables with            parameters 1 and 2 respectively. Use ‘transformation of variables’ method to            obtain the distribution  of  .
22. (a) Show that if λ  is a characteristic root of A, then  λn is a characteristic root of An.
(b) Solve the system of equations using matrix inverse method.       .

 

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Loyola College B.Sc. Statistics Nov 2012 Probability And Random Variables Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034B.Sc. DEGREE EXAMINATION – STATISTICSFIRST SEMESTER – NOVEMBER 2012ST 1503/ST 1501 – PROBABILITY AND RANDOM VARIABLES
Date : 10/11/2012 Dept. No.         Max. : 100 Marks                 Time : 1:00 – 4:00                                               PART – AAnswer ALL the questions:                  (10 x 2 = 20 marks)1. Give the empirical definition of probability. Mention any one of its limitations.2. Define an event with reference to a random experiment. Give an example.3. Define conditional probability. When does this probability become Zero?   4. When do we say that an event A is statistically independent with respect to an event B? 5. Distinguish between pair wise independence and mutual independence.6. State multiplication law of probability for any two events. Hence, write down the law for      independent events.7. Find the E(X) of X whose pdf is f(x) =  ,  x >0.8. Write down the importance of Chebyshev’s inequality.9. Mention the advantages of generating functions.10. If X is random variable with mean 0 and variance 25, give an upper bound for the       probability P (│X – μ│ > 25 ).
PART – B
Answer any FIVE questions:          (5×8=40 marks)
11. Define Moment Generating Function (MGF). Show that MGF can be used to obtain the mean and        variance of a random variable.12. State and prove the addition theorem for two events. Hence, show that probability of a sure       event is One.   13. For any two events A and B, show that                                                                               P (AB C ) = P (A C) + P (B C) – P (AB C)14. If two dice are thrown, what is the probability that the ‘sum’ is (a) greater than 8 and (b) either 7 or 1115. If X is random variable with pdf f(x)=  , 2 ≤ x ≤4. Find the value of k.  Hence find Standard      Deviation of X.

 

16. Sixty per cent of the employees of the XYZ Corporation are college graduates. Of these, ten per cent       are in sales. Of the employees who did not graduate from the college, eighty per cent are in sales.     (i) What is the probability that an employee selected at random is in sales?     (ii) What is the probability that an employee selected at random is neither in sales nor a           college graduate?     17. Illustrate through an example that pair-wise independence does not imply mutual independence.18. The diameter of an electric cable is assumed to be a continuous random variable X with pdf f(x) = k        x(1-x) if  0  x  1 and 0, otherwise.  (i) Find the value of k. (ii) Obtain the distribution function of X.         (iii) Determine the value of constant c such that P[X< c] = P[X>c].
PART – C
Answer any TWO questions:          (2×20=40 marks)
19. (a) If two events A and B independent, show that their complements are independent of            each other.       (b) The probability of the wife who is 40 years old  living till 70 is   and the probability of              the husband who is now 50 living till 80 is  .  Find the probability that (a) only one will be alive for 30 years  (b) at least one will be alive for 30 years.
20. (a) State and prove Bayes theorem.       (b) Three identical urns contain the following proportion of balls. Urn1 : 2black, 1white.Urn2 : 1black, 2white.Urn3 : 2black, 2white.            An urn is selected at random and a ball is drawn. This ball turns out to be white. What is the            probability of drawing a white ball again if the first white ball drawn is not replaced.     21. (a) State and Prove Chebyshev’s Inequality.       (b) If X is a random variable with pdf f(x) =  ,  < x< , obtain an upper bound for the                probability P[│X – μ│ >  ]. Compare it with the actual probability.22. (a)  For any three events A. B and C, such B , and P(A) >0, show that  P (B A ) ≤ P (C A).(b) If A and B are two events such that P(A) =  and P(B) = , show that    (i) P (A )      and       (ii)     P (A )

 

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Loyola College B.Sc. Statistics Nov 2012 Mathematics For Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2012

MT 1101 – MATHEMATICS FOR STATISTICS

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

PART – A

         Answer ALL questions                                       (10 x 2 = 20)

  1. If y=log(2x+3) then find
  2. If with respect to ,
  3. Define implicit function.
  4. Find the value of.
  5. Define Homogeneous function.
  6. If then find .
  7. Evaluate:
  8. Solve
  9. Write the two properties of definite integrals.
  10. Evaluate:

 

PART – B

    Answer any FIVE questions                                (5 x 8 = 40)

  1. Differentiate: .
  2. If with respect to, find
  3. Examine whether the following functions are odd or even functions.
  1. Show that , when x is positive.
  2. State and prove Euler’s theorem.
  3. Evaluate:
  4. Evaluate the rational algebraic function .
  5. Evaluate:

 

PART – C

    Answer any TWO questions                                  (2 x 20 = 40)

  1. a). If then find

b). Differentiate.                                                                                (10+10)

  1. a). Find the limit of the function

b). Determine the maxima and minima of the function

(10+10)

  1. a). Evaluate:

b). Using by parts method find out the value of                                  (10+10)

  1. a).Evaluate:

b). Evaluate  using reduction formula.                                                      (10+10)

 

 

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Loyola College B.Sc. Statistics Nov 2012 Estimation Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5504 – ESTIMATION THEORY

 

 

 

Date : 01/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL the questions:                                                                                                      (10 x 2 = 20)

  1. Define Unbiasedness.
  2. If T is an unbiased estimator of θ, show that T2 is a biased estimator for θ2.
  3. Define Efficiency.
  4. Let X1, X2, …, Xn be a random sample from a population with pdf

f( x, θ) = θ x θ – 1, 0 < x < 1, θ > 0.

Show that  is sufficient for θ.

  1. Define BLUE.
  2. What is meant by prior and posterior distribution?
  3. Define sufficiency.
  4. Write down the normal equation associated with a simple regression model.
  5. Define Completeness.
  6. Define MVB.

PART- B

Answer any FIVE questions:                                                                                                    (5 x 8 = 40)

  1. State and prove the sufficient condition for an estimator to be consistent.
  2. Let X1, X2, …, Xn be a random sample from a Bernoulli distribution:

 

Show that  is a complete sufficient statistics for θ.

  1. Mention the properties of MLE.
  2. A random sample X1, X2, X3, X4, X5 of size 5 is drawn from a normal population with unknown mean µ. Consider the following estimators to estimate µ:

(i)    (ii)   (iii)  where λ is

such that t3 is an unbiased estimator of m.

  1. Find λ.
  2. Are t1 and t2 unbiased?
  3. Which of the three is the best estimator?

 

  1. State and prove Cramer – Rao Inequality.
  2. Samples of sizes n1 and n2 are drawn from two populations with mean T1 and T2 and common variance σ2. Find the BLUE of l1T1 + l2T2.
  3. Prove that if T1 and T2 are UMVUE of g(θ) then T1 = T2 almost surely.
  4. Obtain the UMVUE of the parameter l for the poisson distribution based on a random sample of size n.

PART – C

Answer any TWO questions:                                                                                                   (2 x 20 = 40)

  1. a) State and prove Rao – Blackwell theorem.
  2. b) Show that if the MLE exists uniquely then it is a function of the sufficient statistic.
  3. a) State and prove the necessary and sufficient for a parametric function to be linearly

estimable.

  1. b) Prove that the MLE of α of a population having density function: 0 < x < α

for a sample of size one  is 2x, x being the sample value. Show also that the estimate is

biased.

  1. a) State and prove factorization theorem.
  2. b) Let (X1, X2, …, Xn) be a random sample from N(µ, σ2) . Obtain the Cramer – Rao

lower bound for the unbiased estimator of m.

  1. a) Explain the method of moments.
  2. b) Let X1, X2, …, Xn be a random sample from Bernoulli distribution b(1, θ). Obtain the

Bayes estimator for θ by taking a suitable prior.

 

 

 

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Loyola College B.Sc. Statistics Nov 2012 Econometrics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2012

ST 4207/4204 – ECONOMETRICS

 

 

Date : 07/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

Section –A

Answer all the questions:                                                                                                     (10 x 2 = 20)

 

  1. Mention any two property of variance.
  2. Write a note on interval estimation.
  3. Define BLUE

 

  1. Obtain ESS from the following data given that RSS = 133.

 

Y

 

10 14 17 20 25 30 19 27
  1. Define hypothesis
  2. What is Multiple Regression? Give an Example.
  3. Give the formula for Durbin Watson d – statistic.
  4. What do you mean by bench mark category?
  5. State the reasons under which Multicollinearity
  6. Define lagged variables.

 

                          Section –B                                             

Answer any five questions:                                                                                                      ( 5 x 8 = 40)

 

  1. A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart or a red card.
  2. The diameter of an electric cable, say X, is assumed to be a continuous random variable with p.d.f:
  • Check that is p.d.f.
  • Determine a number b such that P ( X < b ) = P ( X > b ).
  1. Explain in detail the Goals of Econometrics.
  2. Derive least square estimators for simple linear regression model.
  3. Explain in detail Variance Inflation Factor.
  4. From the following data estimate d-statistic and test for autocorrelation.

et : 0.6, 1.9, -1.7, -2.2, 1.3,3.2, 0.2,0.8, 2.1, -1.5, -1.1

(Given dL = 1.45 and du = 1.65)

  1. What are dummy variables? Explain its usefulness in regression analysis with

example.

 

 

 

  1. Find the value of R2 for the following data:
Y 12 8 9 6 8
X1 8 10 4 3 6
X2 10 12 6 5 7

 

 

                       

                

Section – C

 Answer any two questions:                                                                                                    ( 2 x 20= 40)

 

  1. Two random variable X and Y have the following joint probability density function:

 

Find (i) Marginal probability density functions of  X and Y

  • Conditional density functions
  • Var ( X) and Var ( Y)
  • Covariance between X and Y.

 

  1. Consider the following data on  X and Y

 

X 50 42 71 35 61 45 53 45 38 41 63 34 41
Y 145 123 155 120 150 130 155 120 135 160 165 115 120

 

  1. Estimate the equations of Y on X
  2. Test the significance of the parameters at 5% level of significance.
  3. Given the following data the estimated model is . Test the problem of heteroscedasticity with the help of park test.

 

X 1 2 3 4 5 6
Y 2 2 2 1 3 5

 

  1. Fit a linear regression model for the given data by the use of dummy variables
Aptitude score 4 9 7 3 5 8 9 5 6 8
Education qualification UG PG UG HSC PG UG PG HSC UG PG

 

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Loyola College B.Sc. Statistics Nov 2012 Applied Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5506/ST 5502 – APPLIED STATISTICS

 

 

 

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

Time : 9:00 – 12:00

 

PART – A

Answer ALL the questions:                                                                                 (10X2=20 Marks)

 

  1. Mention any two uses of index numbers.
  2. Define Laspeyre’s and Paasche’s index numbers.
  3. Explain the concept of business cycle.
  4. Mention the different types of time series models for the component combinations.
  5. State any two uses of vital statistics.
  6. Define total fertility rate.
  7. Define partial correlation coefficient.
  8. Given r12 = 0.77, r13= 0.72, r23= 0.52, find R23
  9. What is forest statistics?
  10. Define national income.

 

PART – B

 

Answer any FIVE questions                                                                                (5X8=40 Marks)

  1. Explain the tests to be satisfied by a good index number. Show that Fisher’s Index number is an Ideal Index number.
  2. What is meant by (a) splicing (b) deflating and (c) base shifting of index numbers?
  3. Explain link relative method to measure seasonal fluctuations.
  4. Explain fitting of a second degree parabola by the method of least squares.
  5. What is life table? Briefly outline the uses of life table.
  6. Define gross and net reproduction rates. Discuss the steps for estimating the net reproduction rate.
  7. Discuss the methods of national income estimation.
  8. Write short notes on (i) De Facto method (ii) De Jure method.

P.T.O]

PART – C

Answer any TWO questions:                                                                              (2X20=40 Marks)

  1. (a) Discuss the problems and precautions in the construction of an index number.

(b) What are the uses of consumer price index number? Calculate the CPI using the following data:

Items Index Number Weight
Food 352 48
Fuel 220 10
Clothing 230 8
Rent 160 12
Miscellaneous 190 15

 

  1. (a) Briefly explain the components of time series.

(b) Explain Ratio to Moving Average method for determining seasonal index.

 

  1. (a) Explain the various mortality rates used in vital statistics and discuss their relative merits.

(b) Estimate the standardized death rates for the two countries from the data given below:

Age group (in years) Death Rate per 1000 Standardised

Population (in lakhs)

Country A Country B
0 – 4 20.00 5.00 100
5 – 14 1.00 0.50 200
15 – 24 1.40 1.00 190
25 – 34 2.00 1.00 180
35 – 44 3.30 2.00 120
45 – 54 7.00 5.00 100
55 – 64 15.00 12.00 70
65 – 74 40.00 35.00 30
75 and above 120.00 110.00 10

 

  1. (a) Write short notes on (i) Central Statistical Organisation

(ii) National Sample Survey Organisation

(b) Define (i) Partial Regression (ii) Multiple Correlation (iii) Multiple Regression

 

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Loyola College B.Sc. Statistics Nov 2012 Applied Stochastic Processes Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Section-A

Answer all the  questions:                                                                                                  (10×2=20 marks)

 

  • Give an example for one and two dimensional Stochastic Processes.
  • Define Time space with an example.
  • Define Null recurrence.
  • What is meant by Periodicity?
  • Briefly explain the term random walk.
  • Define the term communication of the states.
  • What is meant by absorbing state?
  • What is meant by TPM?.
  • Define Markov Chain.
  • What is meant by Birth process

 

Section-B

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

 

11)Discuss in detail the classifications of the Stochastic Processes.

12) Distinguish between Symmetry and Transitivity of communication with an example.

13) Discuss in detail  any two applications of Stochastic modeling. .

14) Explain the Gambler’s Ruin problem with an example.

15)Discuss the applications of stationary distribution with suitable illustration.

16) Discuss in detail the higher order transition probabilities with suitable illustration.

17) A white rat is put into the maze consisting of 9 compartments. The rat moves through the

compartment at random. That is there are k ways to leave a compartment. The rat chooses each of the

move with probability1/k.

  1. a) Construct the Maze

b)The Transition probability matrix

18) Discuss the Social Mobility problem.

 

Section-C

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

 

19a) Show that a Markov Chain is fully determined, when its initial distribution and one step transition

probabilities of the Markov chain are known.

 

19b) State and prove Chapman-Kolmogrov equation.

 

 

 

 

20) Sociologist often assumes that the social classes of a successive generation in a family can be regarded as a Markov chain. The TPM of such model is as follows.

Son’s Class
Lower Middle Upper
Lower 0.4 0.5 0.1
Father’s Class Middle 0.05 0.7 0.25
Upper 0.05 0.5 0.45

Find

  1. What proportion of people are lower class in the long run?
  2. What proportion of people are middle class in the long run?
  • What proportion of people are upper class in the long run?

21a) Explain the one dimensional random walk problem with the TPM .

 

21b) If the  probability of a dry day (state-0) following a rainy day (state-1)is 1/3, and that of a rainy day following a dry day is  ½.   Find i) Probability that May 3 is a dry day given that May first is a dry day. ii) Probability that May 5 is a rainy day given that May first is a dry day..

 

22) Write short notes on the following

 

  1. a) Poisson Process
  2. b) Irreducible Markov Chain

 

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Loyola College B.Sc. Statistics Nov 2012 Distribution Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2012

ST 4502/4501 – DISTRIBUTION THEORY

 

 

 

Date : 7/11/2012               Dept. No.                                        Max. : 100 Marks

Time : 1.00 – 4.00

 

PART – A

 

Answer ALL questions:                                                                                            (10 x 2 = 20 marks)

 

  1. Suppose that two dimensional continuous random variable (X, Y) has joint p.d.f. given by

Find E (xy).

  1. Prove that sum of squares of deviations is minimum when the deviations taken from mean.
  2. If X1 and X2 are independent Poisson variates with parameters l1 and l2 find the distribution

of X1 + X2.

  1. Under what conditions Binomial tends to poisson distribution?
  2. Define MGF of a random variable.
  3. State the properties of normal distribution.
  4. Identify the distribution of sum of n independent exponential variates.
  5. Write the pdf of the Laplace distribution.
  6. Obtain the distribution of when X has F(n1, n2).
  7. Define Stochastic convergence.

PART – B

 

Answer any FIVE questions:                                                                                         (5 x 8 = 40 marks)

 

  1. The two dimensional random variable (x,y) has the joint density function,

Find marginal density function of x, y and mean of x, y.

 

  1. Find the recurrence relation for the moments of binomial distribution with parameters n and p.
  2. Explain memory less property. Prove that Geometric distribution has this property.
  3. Derive the distribution of k th order statistic.
  4. Find the moment generating function of Gammma distribution. Hence find the mean and variance.
  5. Derive the mean and variance of Beta distribution.
  6. State and prove central limit theorem for for iid random variables.
  7. Define chi-square variate. Find its probability density function using moment generating function.

PART – C

Answer any TWO questions:                                                                                   (2 x 20 = 40 marks)

 

  1. a) Find the marginal distribution of X and conditional distribution X given Y=y in a bivariate

normal distribution.

  1. b) State and prove the additive property of poisson distribution.
  2. a) Prove that for a Normal distribution all odd order central moments vanish and find the

expression for even order moments.

 

  1. b) Derive the pdf of t-distribution.

 

  1. a) Define the Hyper – geometric distribution. Find its mean and variance.

 

  1. b) Show that t-distribution tends to standard Normal distribution as

 

  1. Identify the distribution of sample mean and sample variance. Also prove that they are

independently distributed.  Assume the parent population is Normal.

 

 

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Loyola College B.Sc. Statistics Nov 2012 Computational Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5507/5503 – COMPUTATIONAL STATISTICS

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer any THREE of the following questions:                                             

 

  • (a) A study of randomly selected motor-cycle accidents and drivers who use cellular phones provided the following data. Based on the following data, does it appear that use of cellular phones affects driving safety? (15)

 

Had Accidents               Had no Accidents

 

Cell phones used                                                    23                                   282

 

Cell phones not used                                              46                                   407

 

(b)  Find an α level Likelihood Ratio Test of     against based on a sample of size 10 from, where both µ and σ2  are unknown. If the observed value of sample mean and variance are 0.6 and 0.36 respectively, should the hypothesis H0  be accepted or rejected?       (18)

 

2) a) From the following informations, Compare the precision of Systematic Sample, Simple Random Sampling and Stratified sampling.

Strata 1 2 3 4 5 6
I 28 32 33 35 37 39
II 15 16 17 21 22 25
III 2 3 4 7 9 9
IV 5 7 9 12 14 15
V 25 22 21 17 17 23

 

b).        A sample of 40 students is to be 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+ 17)

  • (a) Compute index number for the given data using the following methods (i)

Laspeyre’s  method,  (ii) Passche’s method and (iii) fisher’s ideal formula                                   (8)

 

 

Item (Rs.) Base year Current year
Price (in Rs) Quantity Price (in Rs) Quantity
Food 12 20 20 22
Rent 40 10 42 12
Clothing 8 50 12 50
Fuel 20 20 24 22
Others 16 20 25 20

 

(b) Change the base year 2000 to 2003 and rewrite the series of index numbers in the

following data:

 

Year 2000 2001 2002 2003 2004 2005 2006 2007 2008
Index 100 115 120 122 125 128 130 135 140

(5)

 

(c) Calculate the seasonal indices by the method of least squares from the following data:

(Multiplicative model)                                                                                          (20)

Exports of cotton textiles (million Rs.)
Year I II III IV
2001 71 68 79 71
2002 76 69 82 74
2003 74 66 84 80
2004 76 73 84 78
2005 78 74 86 82

 

 

 

 

 

 

 

 

 

  • (a) A Vendor of milk products produces and sells low fat dry milk to a company that uses it to produce baby formula. In order to determine the fat content of the milk, both the company and the vendor take a sample from each lot and test it for fat content in percent. 10 sets of paired results are :

 

 

Lot no. 1 2 3 4 5 6 7 8 9 10
Company test results(X) 0.5 0.58 0.9 1.17 1.14 1.25 0.75 1.22 0.74 0.80
Vendor test result (Y) 0.79 0.71 0.82 0.82 0.73 0.77 0.72 0.79 0.72 0.91

 

Test  against, using a paired t test with the differences. Let.  (D=X-Y)   (20)

 

(b)  Let  be a random sample from. Test  against.   Find the Uniformly Most Powerful Test.                      (13)

 

 

 

 

 

 

 

 

 

  • (a)  The following  are the weight gains (in pounds) of two random samples of young Indians fed on two different diets but otherwise kept under identical conditions:

 

Diet I 16.3 10.1 10.7 13.5 14.9 11.8 14.3 10.2
Diet II 21.3 23.8 15.4 19.6 12 13.9 18.8 19.2
Diet I 12 14.7 23.6 15.1 14.5 18.4 13.2 14
Diet II 15.3 20.1 14.8 18.9 20.7 21.1 15.8 16.2

 

Use U test at 0.01 level of significance to test the null hypothesis that the two population samples are identical against the alternative hypothesis that on the average the second diet produces a greater gain in weight.                                                         (16)                   

 

(b) The following are the speeds at which every fifth passenger car was timed at a certain

checkpoint: 46, 58, 60, 56, 70, 66, 48, 54, 62, 41, 39, 52, 45, 62, 53, 69, 65, 67, 76,

52, 52, 59, 59, 67, 51, 46, 61, 40, 43, 42, 77, 67, 63, 59, 63, 63, 72, 57, 59, 42, 56, 47,

62, 67, 70, 63, 66, 69 and 73. Test the null hypothesis of randomness at the 0.05 level

of significance.                                                                                                      (17)

 

 

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Loyola College B.Sc. Statistics Nov 2012 Basic Sampling Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2012

ST 3504/3502/4500 – BASIC SAMPLING THEORY

 

 

 

Date : 05/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL the questions                                                                                        (10×2=20 Marks)

 

  1. What is meant by sampling frame?
  2. What is pilot survey?
  3. Define simple random sampling with replacement.
  4. Define unbiased estimator of a parameter.
  5. Distinguish between SRSWR and SRSWOR.
  6. Explain stratified random sampling.
  7. Write any two advantages of stratified sampling.
  8. Define Lahiri’s method.
  9. Define linear systematic sampling.
  10. Write down the merits of systematic sampling.

 

PART – B

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

 

  1. What are the advantages of sampling over census method?
  2. List out the dangers in using statistical packages.
  3. Derive any two properties of sample mean in SRSWR.
  4. Prove that in stratified sampling, sample mean is an unbiased estimator of population mean.

Also find its variance.

  1. Write a descriptive note on cluster sampling.
  2. Explain ‘Lottery Method’ of selecting a simple random sample.
  3. Explain the advantages and disadvantages of systematic sampling.
  4. Explain cumulative total method of PPS selection.

 

PART – C

Answer any TWO questions                                                                                      (2×20=40 Marks)

 

  1. (a) What are non-sampling errors? Explain its sources.

(b) Write a note on simple random sampling of attributes.

  1. (a) If the population consists of linear trend, then prove that

.

(b) Compare

  1. (a) Derive the variance of unbiased estimator for mean per element under cluster sampling in

terms of the intra cluster correlation.

(b) Prove that is unbiased for in SRSWOR.

  1. Define systematic sampling. Obtain the sampling variance of the mean and

compare with that of SRSWOR and stratified sampling.

 

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Loyola College B.Sc. Statistics Nov 2012 Advanced Statistical Methods Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.COM. DEGREE EXAMINATION – COMMERCE

THIRD SEMESTER – November 2012

ST 3202- ADVANCED STATISTICAL METHODS

 

 

 

Date :9/11/2012                   Dept. No.                                        Max. : 100 Marks

Time : 9.00 – 12.00

PART A                                           (10 X 2 = 20 marks)

     Answer ALL questions:         

                                  

  1. Define independence of attributes.
  2. What are the types of non- probability sampling?
  3. State the Axioms of probability
  4. State addition theorem on probability.
  5. State any four properties of Poisson distribution.
  6. What is meant by probable error? Mention its uses.
  7. Differentiate between Small Samples and Large Samples.
  8. What is meant by analysis of variance?
  9. Explain the various types of control charts.
  10. Distinguish between the control limits and tolerance limits.

 

PART  B                                              (5 X 8 = 40 Marks)

     Answer any FIVE questions:

 

  1. State and prove Baye’s theorem.

 

  1. The result of a certain survey shows that out of 50 ordinary shop of small size 35 are managed by men

of which 17 are in cities, 12 shops in villages are run by women. Can it be inferred that shops run by

women relatively more in villages than in cities ?

 

  1. Five men in a company of 20 are graduates, if 3 men are picked out from this 20 at random, what is the

probability that (i) all are graduates (ii) at least one is a graduate.

 

  1. An Automatic Machine fills in tea in sealed tins with Mean Weight of tea 1 kg. and S.D. 1gm . A

random sample of 50 tins was examined and it was found that their Mean Weight was 999.50 gms. Is

the machine working properly .

  1. The following data is collected on two characteristics:
Smokers Non-Smokers
Literate 83 57
Illiterate 45 68

Based on this test whether there is relation between the habit of smoking and literacy.

 

 

 

16 . An IQ test was administered to 5 persons before and after they were trained. The results are given

below:

Candidates

 

I II III IV V
IQ before training 110 120 123 132 125
IQ after training 120 118 125 136 121

 

 

 

 

 

Test whether there is any change in IQ after the training programme. Use 5% level of  significance.

  1. State the advantages and disadvantages of statistical quality control.

 

 

  1. The following table gives the number of defective items found in 20 successive samples of 100 items

each

2    6   2   4   4   15   0   4   10   18   2   4   6   4   8   0   2   2   4   0

 

Comment whether the process is under control. Suggest suitable control limits for the future.

 

PART   C                                   (2 X 20  =  40 Marks)

Answer any TWO questions

 

19.(a) A number of school-children were examined for the presence or absence of certain

defects of which three chief descriptions were noted; A-development defects;

B-nerve signs; C low nutrition. Given the following ultimate frequencies, find the

frequencies of the classes defined by the presence of the defects.

(ABC) = 57; (aBC) = 78

(ABg) = 281; (aBg) = 670

(AbC) = 86; (abC) = 65

(Abg) = 453; (abg) = 8310                                                                                                     (10)

19 . (b)  Two boxes contain 12 white and 18 black and 15 white and 25 black balls respectively.  One  box

was taken at random and a ball was taken from the same.  It is a black ball. What is the probability

that it is from the (i) first box (ii) second box.                                                                                (10)

  1. (a) If 3% of the electric bulbs manufactured by a company are defective, find the probability that in a

sample  of 100 bulbs exactly five bulbs are defective (e-3 = 0.0498).                                             (10)

 

  1. (b) The average daily sales of 500 branch offices was Rs.150,000 and the standard deviation

Rs.15,000. Assuming the distribution to be normal, find how many branches have sales between

  • 1,20,000 and Rs.1,45,000
  • 1,40,000 and Rs.1,60,000                                                                                   (10)

.

21.(a) The sales manager of a large company conducted a sample survey in states A and B taking 400

Samples in each case. The results were as follow

State A               State B

Average sales             Rs.2500               Rs.2200

Standard Deviation      Rs.400                Rs.550

Test whether the average sales is the same in the two states. Test at 1% level.                                  (10)

 

  1. (b) Value of a Variety in two samples are given below:
Sample I 5 6 8 1 12 4 3 9 6 10
Sample II 2 3 6 8 1 10 2 8 * *

 

 

 

Test the significance of the difference between the two sample means.                                     (10)

 

  1. The following table gives the fields of 15 samples of plot under three varieties of seed.
A B C
20 18 25
21 20 28
23 17 22
16 15 28
20 25 32

 

 

 

 

 

(20)

 

Test using analysis of variance whether there is a significant difference in the average yield of seeds

 

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Loyola College B.Sc. Statistics Nov 2012 Actuarial Statistics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2012

ST 5404 – ACTUARIAL STATISTICS

 

 

Date : 10/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Section – A

Answer all the questions:                                                                                                     ( 10 x 2 = 20)

 

  1. The amount with compound interest of a certain principal at 5% p.a. is Rs. 3969. Find the principal when period is 2 years.
  2. What is meant by discount?
  3. What is the effective rate p.a. corresponding to a nominal rate of 8 % p.a. convertible monthly?
  4. Evaluate v9 s13 @ 9 %
  1. Define an annuity.
  1. Show that
  2. What is perpetuity due?
  3. Define q
  4. Give the expression for e
  5. Write a short note on term assurance.

Section – B

Answer any five questions:                                                                                                 ( 5 x 8 =40)

 

  1. The amounts for a certain sum with compound interest at a certain rate in two years and in three years are Rs. 8820 and Rs. 9261 respectively. Find the rate and sum.
  1. A has taken a loan of Rs. 2000 at a rate of interest 4% p.a. payable half-yearly. He repaid Rs. 400 after 2 years, Rs. 600 after a further 2 years and cleared all outstanding dues at the end of 7 years from the commencement of the transaction. What is the final payment made by him?
  2. The cash purchase price of a bike is Rs. 10,000. A company however offers instalment plan  under an immediate payment of Rs. 2000 is to be made and a series of 5 equal half-yearly payments made thereafter, the first installment being payable at the end of 6 months. If the company wishes to realize a rate of interest of 12 % convertible half-yearly in the transaction, calculate the half-yearly instalment.
  1. Calculate the present value of a deferred annuity payable for 10 years certain, the first payment falling due at the end of 6 years from the present time. The annuity is payable at the rate of Rs. 100 p.a. for the first 5 years and Rs.200 p.a. thereafter.

Given (a5 = 4.3295,  a10 = 7.7217,  a15 = 10.3797)

  1. Derive the formula for accumulated value and present value of annuity certain due.
  2. Using the LIC ultimate table find the following probabilities:
    1.  a life aged 35 dies within 12 years.
    2.  a life aged 40 dies not earlier than 12 years and not later than 15 years.
    3.  a life aged 2 survives 12 years
  1. a life aged 52 will not die between ages 65 and 70

 

  1. What are the points to be borne in mind in deciding
  • Period of investigation?
  • Period of selection?
  • Method to be used for investigation?
  1. Derive an expression for A x:n.

Section – C

Answer any two questions:                                                                                                 ( 2 x 20 =40)

 

  1. a) A has right to receive an amount of Rs.1000 at the end of 12 years from now. This right has been sold to B for a present value calculated at the rate of 8% p.a. The money thus received was invested by A in deposit account at 9% p.a. payable half yearly. After 8 years the account had to be closed and A then invested the amount available at 6% p.a. in another bank. How has A gained or lost in this transaction, as at the end of 12 years?

 

  1. b) Derive an expression to find the present value for the following variable annuities:
  2. Increasing annuity
  3. Immediate Increasing Perpetuity
  • Increasing annuity due
  1. Increasing Perpetuity due
  2. a) A loan of Rs. 3000 is to be repaid with interest at 6% p.a. by means of an immediate annuity for 10 years. Find the level payment. What will be the interest and principal contained in the 5th instalment? What will be the principle outstanding immediately after the 8th payment is made?

( 10 + 10)

  1. b) In lieu of a single payment of Rs. 1000, at the present moment a person agrees to receive 3 equal payments at the end of 3 years, 6 years and 10 years respectively. Assuming a rate of interest of 6% p.a. what should be the value of each of the 3 payments? ( 10 + 10)

 

 

  1. a) Write down expression for probability  in the under mentioned cases:

(i)  Life aged 25 dies between ages 60 and 65

(ii)  Of the two life aged 25 and 30, at least one life dies before attaining age 70

(iii)  Of three lives aged 40, 40 and 45, exactly two lives survive 10 years

(iv) Life aged 28 survives 12 years and dies in the 13th, or 14th year.

 

  1. b) Fill up the blanks in the following portion of a life table:
       Age  X lx dx qx px
10 1000000 0.00409
11 0.00370
12 0.99653
13 0.99658
14 0.00342

 

( 10 + 10)

  1. a) A person aged 30 years has approached a life office for special type of policy providing for the following benefits:
  • 1000 on death during the first 5 years
  • 2000 on death during the next 5 years
  • Survival benefit of Rs. 500 at the end of the 5th year
  • Further payment of Rs. 2000 on survivance to 20 years.
  • An annuity of Rs. 200 per annum payable in his life time, the first such payment falling due along with the survival benefits of Rs. 2000.

 

  1. b) Derive the expression for Ax and (IA) x : n. ( 10 + 10)

 

 

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