B.Sc Corporate Statistics Question Paper 2004

Loyola College B.Sc. Statistics April 2004 Testing Of Hypothesis Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5501/STA 506 – TESTING OF HYPOTHESIS

06.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. Define a simple hypothesis and a composite hypothesis.
  2. Let X1, X2,…, Xn be a random sample from N (m, s2).

Write the distributions of   i)            ii) .

  1. Define uniformly most powerful critical region.
  2. Explain Type – I error and Type – II error.
  3. Explain the likelihood ratio principle.
  4. What do you mean by non- parametric methods?
  5. When do we need the randomised test?
  6. Find the number of runs in the sequence.

x yyy   xxx y  x y xxx  yy  xxxxx

  1. Explain the term confidence interval.
  2. What is a p – value?

 

SECTION – B

 

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

 

  1. Let X have pdf of the form f (x, q) = q xq-1, 0 < x < 1, zero elsewhere, where q Î {q ½q = 1,2}.  To test Ho: q = 1 vs H1 : q = 2, a random sample of size 2 is chosen.  The critical region is C = { (x1, x2) ½  < x1 x2}.  Find Type I error and Type II error.
  2. Verify whether UMPT exists for testing

Ho: q = q‘ Vs H1: q ¹ q‘

when the random sample X1, X2, …, Xn is from N (q , 1).

  1. Explain Wilcoxon’s Test.
  2. The theory predicts the proportion of beans in the 4 groups A, B, C and D should be

9 : 3 : 3 : 1.  In an experiment among 1600 beans, the number in the 4 groups were 882,

313, 287, 118.  Does the experimental results support the theory?

  1. Explain how will you test for regression coefficients b and a in

yi = a + b (ci – ),    i = 1, 2, … n

  1. Explain the t-test for equality of means of two independent Normal populations.
  2. In a random sample of 500 men from a particular district of Tamil Nadu, 300 are found to be smokers. In one of 1000 men from another district, 550 are smokers.  Do the data indicate that the two districts are significantly different with respect to the prevalence of smoking among men?
  3. Derive the distribution of number of runs.

SECTION – C

 

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

 

  1. a) State and prove Neyman – Pearson lemma.          (10)
  2. b) Explain monotone likelihood ratio property (MLR) and its use in testing the

hypothesis.                                                                                                                  (10)

 

  1. a) Derive the likelihood ratio test for testing the equality of two variances of two normal

populations N (q1, q3­) and N (q2, q4­), q1, q2 unspecified.                                             (12)

  1. b) Two independent samples of 8 and 7 respectively had the following values of the

variables.

Sample I          9          11        13        11        15        9          12        14

Sample II        10        12        10        14        9          8          10

Do the population variances differ significantly?                                                       (8)

 

  1. a) Explain Man-Whitney – Wilcoxon Test.           (10)
  2. b) Explain Sign-Test (10)

 

  1. a) Test the hypothesis that there is no difference in the quality of the 4 kinds of tyres A,

B, C and D based on the data given below:

                                                                                     Tyre Brand
A B C D
Failed below 40,000 kms 26 23 15 32
Lasted from 40,000 to 60,000 kms 118 93 116 121
Lasted more than 60,000 kms 56 84 69 47
  1. b) Let X1, X2, …, Xn be a random sample from N (q, 100). Find n and c if

Ho­: q = 75 vs H1 :  q = 78.  Given P [ X Î C ½Ho] = .05   and P [X Î C ½H1] = .90.

C = { (x1, x2, …,xn) ½  ≥ c}  is the Best critical region.

 

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Loyola College B.Sc. Statistics April 2004 Statistical Process Control Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

SIXTH SEMESTER – APRIL 2004

ST 6602/STA 602 – STATISTICAL PROCESS CONTROL

07.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

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

 

  1. Explain Statistical Process Control.
  2. What are control limits?
  3. Describe Total Quality Management (TQM).
  4. Define ‘Chance’ and ‘Assignable’ causes of variation.
  5. Discuss rational subgroup concept.
  6. What information is provided by the operating characteristic curve of a control chart?
  7. Define Process Capability Ratio (PCR).
  8. Define Average Run Length Concept.
  9. Why is np-chart not appropriate with variable sample size?
  10. Explain Demerit Scheme.

 

SECTION – B

 

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

 

  1. What is quality? What are different dimensions of quality?

 

  1. 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 state is 1 -b.

Find the following:

  1. The expected number of subgroups analyzed before the shift is detected.
  2. The probability that the shift is not detected on m -th subsequent sample.

 

  1. Samples of n = 8 items each are taken from a manufacturing process at regular intervals. A quality characteristic is measured and  and R values are calculated for each sample.  After 50 samples, we have

Assume that the quality characteristic is normally distributed.  Compute control limits for

the and R control charts.

 

  1. Statistical monitoring of a quality characteristic uses both an and S charts.  The charts are to be based on the standard values m = 200 and s = 10, with n = 4.  Find 3 – sigma control limits for the S-chart and – chart.

 

  1. Distinguish between c and u charts.

 

 

  1. In designing a fraction non-conforming chart with center line at p = 0.20 and 3 – sigma control limits, what is the sample size required to yield a positive LCL?

 

  1. Define the terms i) Specification Limits and ii) Natural Tolerances with an illustration.

 

  1. How is lack of control of a process determined using control chart technique?

 

 

SECTION – C

 

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

 

  1. a) Explain ‘Stem – amd – Leaf plot’ with an illustration.

 

  1. b) A normally distributed quality characteristic is monitored by a control chart with L –

sigma control limits.  Develop a general expression for the probability that a point will

plot outside the control limits when the process is really in control.                    (10+10)

 

  1. Suppose each automobile produced on an assembly between 9 a.m. and 10 a.m. is examined for paint blemishes on the left front door, with the results given in the following Table. These blemishes may not be easily seen with the naked eye, but a trained inspector with a special light source and magnifying glasses can spot them.

 

Automobile No. 1            2         3          4         5         6          7         8        9       10

 

0.84    0.62    0.84     1.08    0.62     0.84     1.08    0.84   1.08    0.62

 

 

3         2         4          5         4          2          12       6        7          4
Surface Area in Sq.Mt.
No. of Blemishes

 

For the above case, state:

  1. Which control chart (s) would be appropriate to use for ongoing SPC?
  2. Why do you suggest the chart (s) in a) ?
  3. What assumptions are you making in suggesting the chart (s) in a) ?      (8+6+6)

 

  1. a) Mention the theoretical base of p-chart and set up its control limits:

 

  1. Explain the procedure of obtaining the OC curve for a p-chart with an illustration.

(8+12)

  1. Write short notes on:-
  2. Six – Sigma quality

 

  1. Single Sampling Plan.      (10+10)

 

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Loyola College B.Sc. Statistics April 2004 Statistical Methods Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2004

ST 1500/STA 500 – STATISTICAL METHODS

17.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION -A

 

Answer ALL questions.                                                                               (10 ´ 2 = 20 marks)

 

  1. Give an example for primary and secondary data.
  2. What is meant by judgement sampling?
  3. Mention the difference between histogram and bar diagram?
  4. What are the characteristics of a good measure of central tendency?
  5. For a frequency distribution, the mean and mode were found to be 15 and 24 respectively. Find the median of the distribution.
  6. Comment on the following: “The mean deviation of a frequency distribution about an origin is minimum, when the origin is the mean”.
  7. Define: Skewness
  8. Give an example for positive correlation.
  9. Find the regression equation of y on x given the following information:
  10. Check the consistency of the following data:
  • = 400;  (AB) = 250;     (B) = 550;       N = 1,200.

 

SECTION – B

 

Answer any FIVE questions.                                                                                     (5 ´ 8 = 40 marks)

 

  1. Explain the different types of classification with examples.

 

  1. The following data relate to the monthly expenditure of two families A and B:
Items of expenditure Expenditure (in Rs.)
Family A Family B
Food 1600 1200
Clothing   800   600
Rent   600   500
Fuel   200   100
Miscellaneous   800   600

Represent the above data by a percentage bar diagram.

 

  1. Calculate Q1, Q2, P3 and P20 from the following data:

Class Interval: 0-5      5-10     10-15   15-20   20-25

 

Frequency     :   7        18         25        30         20

 

 

 

  1. Calculate mean deviation about median and its coefficient from the following data:

 

Class Frequency Class Frequency
0-10   5 40-50 20
10-20   8 50-60 14
20-30 12 60-70 12
30-40 15 70-80  6

 

  1. Explain the concepts of correlation and regression through an example.

 

  1. Ten competitors in a beauty contest are ranked by 3 judges in the following order:

Judge 1:          1          6          5          10        3            2        4            9        7          8

Judge 2:          3          5          8            4        7          10        2            1        6          9

Judge 3:          6          4          9            8        1            2        3          10        5          7

Use the rank correlation coefficient to determine which pair of judges has the nearest approach to common tastes in beauty.

 

  1. Find Yule’s coefficient of association between literacy and unemployment from the following data:

Total Adults:               10,000

Literate:                         1,290

Unemployed:                1,390

Literate unemployed:       820

Comments on the results.

 

  1. Fit a straight line trend for the following time series.

Year           :    1990    1991    1992    1993    1994    1995    1996

Production

of steel

(in tonnes) :       60      72         75        65        80        85        95

Estimate the production for the year 1997.

 

SECTION – C

 

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

 

  1. i) Explain the various method of collecting primary data.
  2. ii) Draw ‘less than’ and ‘more than’ Ogive curves for the following data:
Profit

(in lakh)

 Number of Companies Profit

(in lakh)

 Number of companies
10-20   6 60-70 16
20-30   8 70-80  8
30-40 12 80-90  5
40-50 18 90-100  2
50-60 25

 

Also find the value of the median.                                                                            (10+10)

 

 

 

 

 

  1. i) Calculate mode ( by grouping method) from the following data:

Class interva l: 10-20  20-30   30-40   40-50   50-60   60-70   70-80   80-90

 

Frequency     :      5         9         13        21        20        15         8         3

 

  1. ii) From the prices of shares of 2 firms X and Y given below, find out which is more stable

in value.

Firm X:            35        54        52       53       56       58        52        50        51        49

 

Firm Y:            108      107      105      105      106      107      104      103      104      101

(10+10)

  1. i) Explain the concept of kurtosis.

 

  1. ii) Calculate the first four central moments from the following frequency distribution.

x:         2          3          4          5          6

f:          1          3          7          3          1

 

iii) Calculate Karl pearson’s coefficient of correlation, for the following data and interpret

its value.

X:        48        38        17        23        44

Y:        45        20        40        25        45                                                             (5+7+8)

 

  1. i) From the following data, obtain the regression equations of Y on X and X on Y.

 

Aptitude scores(X):  60    62       65        70        72        48        53        73        65        82

Productivity

Index(Y)               :   68    60       62        80        85        40        52        62        60        81

Also estimate the productivity index, when test score is 92 and the test score when productivity index is 75.

 

 

  1. ii) Explain any two methods of studying the association between attributes. (15+5)

 

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Loyola College B.Sc. Statistics April 2004 Statistical Mathematics – II Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2004

ST 3500/STA 502 – STATISTICAL MATHEMATICS – II

21.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION -A

 

Answer ALL questions.                                                                              (10 ´ 2 = 20 marks)

 

  1. Define a Skew-Symmetric matrix and give an example.
  2. Define an Orthogonal matrix. What can you say about its determinant?
  3. Find the rank of .
  4. State a necessary and sufficient condition for R-integrability of a function.
  5. Is convergent?
  6. If f(x) = C x2, 0 < x < 1, is a probability density function (p.d.f), find ‘C’.
  7. Give an example of a homogeneous differential equation of first order.
  8. Distinguish between ‘double’ and ‘repeated’ limits.
  9. State any two properties of a Bivariate distribution function.
  10. State the rule of differentiation of a composite function of two variables.

 

SECTION -B

 

Answer any FIVE questions.                                                                                  (5 ´ 8 = 40 marks)

 

  1. Define ‘upper triangular matrix’. Show that the product of two upper triangular matrices is an upper triangular matrix.
  2. Find the inverse of A = using Cayley- Hamilton theorem.
  3. Find a)     b)
  4. State and prove first Fundamental Theorem of Integral Calculus.
  5. If X has p.d.f f(x) = x2/18, -3 £ x £ 3,  find the c.d.f of X.  Also, find P(< 1),

P (X < -2)

  1. Solve: .

 

 

 

  1. Show that the mixed derivative of the following function at the origin are different:

 

 

f (x, y) =

 

 

  1. Define Gamma integral and Gamma distribution.

find the mean and variance of the distribution.

 

SECTION – C

 

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

 

  1. a) Find the inverse of using sweep-out process or partitioning

method.

  1. b) Find the characteristic roots and any characteristic vector associated with them for the

matrix.

(10+10)

  1. a) Test the convergence of: (i) (ii)   (iii) .
  2. b) Define Lower and Upper sum in the context of Riemann integration. Show that lower

sums increase as partitions become finer.                                                           (12+8)

  1. a) Investigate the maximum and minimum of

f(x,y) = 21x – 12x2 – 2y2 + x3 + xy2

  1. b) If f(x,y) = e-x-y, x,y > 0, is the p.d.f of (X, Y),  find the distribution function.     (12+8)
  2. a) Change the order of integration and evaluate: .
  3. b) Define Beta distributions of I and II kinds.

Find the mean and variance of Beta distribution of I kind                                  (10+10)

 

 

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Loyola College B.Sc. Statistics April 2004 Statistical Mathematics – I Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – NOVEMBER 2003

ST-2500/STA501 – STATISTICAL MATHEMATICS – I

01.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

  1. Define ‘permutation with indistinguishable objects’.  State its value in factorial notation.
  2. If A, B, C are events, write the set notation for the following: (i) A or B but not C occur  (ii) None of the three events occur.
  3. If A and B are independent events, show that A and BC are independent.
  4. What are the supremum and infinum of the function f(x) = x – [x], x ÎR
  5. Define probability mass function (p.m.f) of a discrete random variable and state its properties.
  6. “The series is divergent”  – justify.
  7. ” The function f (s) = is not a probability generating function” (p.g.f)” – justify.
  8. For what value of ‘a’ does the sequence {4an}define a probability distribution on the set of positive integers?
  9. Find f ‘(3) for the function f(x) = , xÎ
  10. Define radius of convergence of a power series.

 

SECTION-B

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

 

  1. In how many ways 3 Americans, 4 French, 4 Germans and 2 Indians be seated in a row so that those of the same nationality are seated together? Find the number  of ways, if they are seated around a circular table.
  2. State and prove the Addition Theorem of Probability for two events. Extend it for three events.
  3. a) Show that limit of a convergent sequence is unique.
  4. b) Define monotonic sequence with an example.
  5. Consider the experiment of tossing a fair coin indefinitely until a head appears. Let X = Number of tosses until first Head.  Write down the p.m.f. and c.d.f of X.
  6. Discuss the convergence of the Geometric series for variations in ‘a’.
  7. Investigate the extreme values of the function f(x) = 2x3-3x2-36x+10, xÎ
  8. Show that the series is divergent.
  9. Define Binomial distribution and obtain its moment Generating function (m.g.f). Hence find its mean and variance.

 

 

 

SECTION-C

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

 

  1. a) Establish the theorem on Total probability.
  2. b) Establish Baye’s theorem
  3. c) Three machines produce 50%, 30% and 20% of the total products of a factory. The percentage of defectives manufactured by these machines are 3%, 4% and 5% of their total output. If an item is selected at random from the items produced in the factory, find the probability that the item is defective.  Also given that a selected item is defective. What is the probability that it was produced by the third machine?                       (6+6+8)
  4. a) Show by using first principle that the function f(x) = x2 is continuous at all points of R.
  5. b) Identify the type of the r.v. whose c.d.f. is

0,        x < 0

f(x) =      x/3,    0 £ x < 1

2/3,    1 £ x < 2

1,       2 £  x

 

Also find P (X = 1.5), P(x < 1),  P (1£ x £ 2), P (x ³ 2) P (1 £ x < 3).                     (8+12)

  1. a) Test the convergence of (i) (ii)

For each case, state the ‘test’ which you use.

  1. b) Identify the probability distribution for which f (s) =is the p.g.f.  Find

the Mean and Variance of the distribution.                                                           (10+10)

  1. a) Verify the applicability and validity of Mean Value theorem for the function

f(x) = x (x-1) (x-2),  xÎ[0,1/2].

  1. b) Obtain the expansion of the Exponential function and hence define Poisson

distribution.                                                                                                            (10+10)

 

 

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Loyola College B.Sc. Statistics April 2004 Numerical Methods Using ‘C’ Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2004

ST-4202/STA202 – NUMERICAL METHODS USING ‘C’

14.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION -A

 

Answer ALL questions                                                                               (10 ´ 2 = 20 marks)

 

  1. What is an assignment statement? What is the relationship between an assignment statement and an expression statement?
  2. Mention the difference between pre-increment and post-increment operator.
  3. Give the syntax of ternary operator.
  4. What is the output of the following program segment?

for (i = 0, j = 0; i < = 5; i ++, j ++)

print f (” %d\ n”, i+j);

  1. Mention the difference between break and continue statement.
  2. What is an array?
  3. Comment on the following statement:

” scanf () and get char() functions are equivalent”.

  1. Find the value of f (10) using Lagrange’s linear interpolation for the following function:

x:   7    19

f:  15    35.

 

  1. Evaluate using Trapezoidal rule where f(x) is defined as

 

x: 0.8 0.9 1.0 1.1 1.2
f: 0.71736 0.78333 0.84147 0.92314 0.96356

 

  1. Give the steps to find the root of an equation by Newton-Raphson method.

 

SECTION – B

 

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

 

  1. a) Explain the data types available in C.
  2. b) What are type qualifiers? (6+2)

 

  1. Explain the concept of branching and looping with examples.

 

  1. Explain the syntax of the following:
  2. a) printf() b) toupper() c) pow()           d) floor()

 

 

 

 

 

  1. Give the output of the following program segments:
  2. a) i = x = 0; b) x = 0;

do                                                   for (i = 0; i<5; i++)

{                                                                 {

if (i% 5 = =0)                                  for (j = 0; j<i;j++)

{                                                                 {

x++;                                                switch (i+j-1)

printf(“%d \n”, x);                         {

}                                                     case -1:

i++;                                                case 0: x = x+1;

}while (i<20);                                             break;

printf (“%d’1′ x);                             case 1:

case 2:

case 3:  x = x+2;

break;

default:  x = x+3;

break;

}

printf (“%d\n”, x);

}

}                                                                         (3+5)

 

  1. a) Write a program to accept an integer and display it in reverse order.

 

  1. b) Write a program using function to find the factorial of a given positive integer. (4+4)

 

  1. Find the determinant of the following matrix by pivotal condensation method.

 

 

 

  1. Write a program to solve the equation x2+3x-5=0 using bisection method.

 

  1. Write a program to find the largest eigen vector by the method of iteration.

 

SECTION – C

 

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

 

  1. a) Explain the structure of a C program.
  2. b) Write a program to find the maximum among ‘n’ given integers.
  3. c) Explain the concept of arrays with an example. (6+6+8)

 

  1. a) Write a program to find the roots of a quadratic equation of the form ax2+bx+c=0.
  2. b) Write a program to find the sum of 2 square matrices of order mxm.
  3. c) What are user defined functions? Explain the method of declining and defining a user

defined function.                                                                                                  (6+6+8)

 

 

 

  1. a) Explain the Gauss-elimination method to solve a given set of simultaneous equation

and write a program for the same.

 

  1. b) Evaluate using simpson’s 1/3 rule by taking h as 0.1. (12+8)

 

  1. a) Write a program using Lagrange’s interpolation method to evaluate f(y) where ‘f’ is a function defined as

x    :  x1    x2   ……..xn

f(x):  f1  f2  ……..fn

Assume that x1, x2…..xn,   f1,  f2,……fn are known values.

 

  1. b) Find the inverse of the following matrix by Gauss-Jordon elimination method.

 

A =                                                                                         (12+8)

 

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Loyola College B.Sc. Statistics April 2004 Mathematical Statistics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – MATHEMATICS

FOURTH SEMESTER – APRIL 2004

ST-4201/STA 201- MATHEMATICAL STATISTICS

07.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

SECTION -A

 

Answer ALL questions                                                                               (10 ´ 2 = 20 marks)

  1. If the MGF of a random variable X is , write the mean and variance of X.
  2. If the random variable X has a Poisson distribution such that Pr [X = 1] = Pr [X = 2], find Pr [X = 0].
  3. Define the mode of a distribution.
  4. Express the central moment in terms of the raw moments.
  5. The MGF of a chi-square distribution with n degrees of freedom is ___________ and its variance is ____________.
  6. Write any two properties of a distribution function.
  7. There are 2 persons in a room. What is the probability that they have different birth days assuming 365 days in the year?
  8. Define an unbiased estimator.
  9. Explain Type I error.
  10. If the MGF of a random variable X is M (t), express the MGF of Y = aX + b in terms of M(t).

 

SECTION – B

 

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

 

  1. State and prove Baye’s theorem.
  2. State and prove Chebyshev’s inequality.
  3. Obtain the mode of Poisson distribution.
  4. Derive the pdf of t – distribution.
  5. If the random variable X is N , obtain the MGF of X. Derive the mean and variance.
  6. Let X and Y have the joint pdf

(X, Y)             :      (0, 0)    (0, 1)  (1, 0)    (1, 1)    (2, 0)    (2, 1)

 

P [X=x, Y=y]       :

Find i) the marginal density functions and ii) E [X ½ Y = 0], E[Y ½ X = 1]

 

  1. Let the random variables X and Y have the joint pdf

x+y    0<x<1,  0<y<1

f (x, y) =

  • else where,

Find the correlation coefficient between X and Y.

  1. Let X1,  X2 be a random sample from N (0, 1).  Obtain the pdf of  .

SECTION – C

Answer TWO questions                                                                               (2 ´ 20 = 40 marks)

 

  1. a) Show that Binomial distribution tends to Poisson distribution under certain conditions (to be stated).                                                                                                       (8)
  2. b) Show that, for a Binomial distribution

.

Hence obtain .                                                                                          (10+2)

 

  1. a) Discuss any five properties of Normal distribution.            (10)
  2. b) Of a large group of men , 5% are under 60 inches in height and 40% are between 60 and 65 inches. Assuming Normal distribution find the mean and variance. (10)

 

  1. a) Obtain the MLE of and  in N (,) based on a random sample of size n.   (10)
  2. b) State and prove Neyman- Pearson theorem. (10)

 

  1. a) Four distinct integers are chosen at random and without replacement from the first 10

positive integers.  Let the random variable X be the next to the smallest of these 4

numbers.  Find the pdf of X.                                                                                        (8)

  1. b) Obtain the MGF of (X, Y) if the pdf is

f(x,y) =  p, 0

 

Hence obtain E (X), Var(X) and Cov (X,Y).                                                    (5+2+2+3)

 

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Loyola College B.Sc. Statistics April 2004 Estimation Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2004

ST 5500/STA 505/S 515 – ESTIMATION THEORY

03.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

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

 

  1. Define ‘bias’ of an estimator in estimating a parametric function.
  2. Explain ‘consistent estimator’
  3. Describe ‘efficiency’ of an estimator.
  4. Define ‘Uniformly Minimum Variance Unbiased Estimator’.
  5. Define Cramer – Rao lower bound (CRLB).
  6. Explain bounded completeness.
  7. Define complete sufficient statistic.
  8. Let X1, X2 denote a random sample of size 2 from B(1, q), 0< q <1. Show that X1 + 3X2 is sufficient for q.
  9. State Chapman – Robbins inequality.
  10. Give an example where MLE is not unique.

 

SECTION – B

Answer any FIVE questions.                                                                                   (5 ´ 8 = 40 marks)

 

  1. Show that the sample variance is a biased estimator of the population Variance. Suggest an UBE of s2.
  2. State and derive Cramer – Rao inequality.
  3. Let T1 and T2 be two unbiased estimators of a parametric function with finite variances. Obtain the best unbiased linear combination of T1 and T2.
  4. State and establish Rao – Blackwell theorem.
  5. Give an example of an UMVUE which does not take values in the range of the parametric function.
  6. State and prove Bhattacharya inequality.
  7. State and prove invariance property of MLE.
  8. Describe the method of moments and illustrate with an example.

 

SECTION – C

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

 

  1. a) If Tn is consistent for y (q) and g is continuous, show that g (Tn) is consistent for

g (y (q)).

  1. b) Show that UMVUE is essentially unique. (10+10)

 

  1. a) Give an example to show that bounded completeness does not imply completeness.
  2. State and establish factorization theorem in the discrete case.      (10+10)

 

  1. a) Explain the method of maximum likelihood.
  2. b) Let X1, X2, …, Xn denote a random sample of size n from N (m, s2). Obtain MLE of

q = (m, s2).                                                                                                              (5+15)

 

  1. a) Describe the method of minimum chi-square and method of modified minimum chi-

square.

  1. b) Obtain the estimate of p based on a random sample of size n from B (1, p),0 < p <1 by the method of i) Minimum chi-square and ii) Modified Minimum Chi-Square. (10+10)

 

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Loyola College B.Sc. Statistics April 2004 Design And Analysis Of Experiments Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

SIXTH SEMESTER – APRIL 2004

ST 6600/STA 600 – DESIGN AND ANALYSIS OF EXPERIMENTS

02.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

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

 

  1. Define orthogonal contrasts with an example.
  2. State Cochran’s theorem
  3. Briefly explain the term ‘Local control’.
  4. Give the missing value formula for the RBD with one missing observation.
  5. Explain orthogonal Latin Square Design.
  6. Give any two differences between RBD and LSD.
  7. State any two advantages of factorial design.
  8. Define a symmetric BIBD.
  9. Explain Experimental unit and Treatments.
  10. Define Affine Resolvable Design.

 

SECTION – B

 

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

 

  1. Explain the difference between ‘Randomization’ and ‘Replication’ with a suitable example.
  2. In CRD, show that , with usual notations.
  3. Develop in detail the analysis of variance of Randomised block design.
  4. Explain the concept of ANOVA. When do you perform critical difference?
  5. Complete the following table
Source of Variance d.f. Sum of Squares Mean Sum of Squares F-ratio
Columns 5 – – –
Rows – 4.20 – –
Treatments – – 2.43 –
Error – – 0.65
Total – 39.65 –

The columns as representing schools, the row as classes, the treatments as methods of

teaching Spelling and the observations as grades based on 100 points.  Test the hypothesis

that the treatment effects are equal to zero, showing all steps in the general procedure.

  1. Distinguish between ‘complete confounding’ and ‘partial confounding’
  2. Explain YATE’S method of computing the sum of squares due to main effects and interaction effects in the case of 22 factorial design.
  3. State and prove the parametric conditions of a BIBD.

 

SECTION – C

 

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

 

  1. a) Estimate a single missing observation in LSD.
  2. b) Develop the analysis of variance in the case of LSD. when one observation is missing

in the design.                                                                                                           (8+12)

 

  1. a) Derive the analysis of two-way classification with m-observations per cell by stating

all the effects, ANOVA  and conclusion.                                                                   (12)

  1. b) Consider the results given in the following table for an experiment involving six

treatments in four randomised blocks. The treatments are indicated by numbers within

parentheses.

 

Yield for a randomised block experiment treatment and yield.

Blocks
1 (1)

24.7

 (3)

27.7

 (2)

20.6

 (4)

16.2

 (5)

16.2

 (6)

24.9
2 (3)

22.7

 (2)

28.8

 (1)

27.3

 (4)

15.0

 (6)

22.5

 (5)

17.0
3 (6)

26.3

 (4)

19.6

 (1)

38.5

 (3)

36.8

 (2)

39.5

 (5)

15.4
4 (5)

17.7

 (2)

31.0

 (1)

28.5

 (4)

14.1

 (3)

34.9

 (6)

22.6

Test whether the treatments differ significantly at 5% level of significance.             (8)

 

  1. Explain in detail the analysis of 32 factorial design by stating all the effects, ANOVA and Inference.                                                                                                                  (20)

 

  1. a) Construct a BIBD with the following parameters

v = 5, b= 20, r = 8, k = 2, l = 2                                                                                 (10)

  1. b) For a BIBD, show that i) bk = rv  and       ii) l (v – 1) = r (k – 1)                            (10)

 

 

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Loyola College B.Sc. Statistics April 2004 Basic Sampling Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2004

ST-4500- BASIC SAMPLING THEORY

02.04.2004                                                                                                           Max:100 marks

9.00 – 12.00

 

SECTION -A

 

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

 

  1. Define Sampling frame. Give two examples.
  2. If there are two unbiased estimators for a parameter, then show that one can construct uncountable number of unbiased estimators for that parameter.
  3. If T is an estimator for , then show that MSE (T) = V (T) +
  4. Explain Lottery method of drawing simple random sample of size n.
  5. Show that the probability of selecting a given subset consisting of ‘n’ units of the population of N units is .
  6. Find the probability of selecting ith population unit in a given draw in PPS sampling.
  7. Examine whether the estimator is unbiased for the population total under PPSWR.
  8. Prove that the sample mean is a more efficient estimator of population mean under SRSWOR than under SRSWR.
  9. Explain circular systematic Sampling Scheme.
  10. Compute the number of units to be sampled for each stratum under proportional allocation scheme, when the total sample size is 40 and there are 4 strata of sizes 40, 30, 60, and 70.

 

SECTION – B

 

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

 

  1. Examine the validity of the following statement using a proper illustration: ‘An unbiased estimator under one method of sampling can become a biased estimator under another method of sampling’.

 

  1. Show that, in SRSWOR,

cov (yi , yj)  = –

 

  1. Prove that sample mean is unbiased for population mean in SRSWOR by using the probability of selecting a subset of the population as a sample.

 

  1. What is PPS sampling? Describe cumulative total method.

 

  1. Derive the variance of Hansen Hurwitz estimator for population total.

 

  1. Derive E (and Var , where denotes the sample mean based on only distinct units under SRSWR.

 

  1. Write a descriptive note on centered Systematic Sampling.

 

  1. Derive the formula for under Neyman allocation.

 

SECTION – C

 

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

 

  1. a) Derive the variance of sample mean, V(in SRSWOR by using probabilities of inclusion.                                                                                                                             (10)
  2. b) Describe Lahiri’s method. Show that Lahiri’s method of selection is a PPS selection.

(10)

  1. (a) A population contains 5 units and it is known that

Compare   Find the values of for which

is less efficient than .                                                           (10)

  1. b) Derive the expressions for in SRSWR using the expressions for

available in PPSWR.                                                                                  (10)

 

  1. a) Show that when irrespective of the random start ‘r’ .                                                                                                      (10)
  2. b) In Linear systematic Sampling, when N is not a multiple of n, explain the undesirable situations encountered using suitable illustrations. (10)

 

  1. a) Compare Vand assuming is large for all h = 1, 2, … , L.                                                                                                                            (10)
  2. b) A sampler has two strata. He believes that stratum standard deviations are the same. For a give cost C = C1n1+C2n2, show that

.                                                            (10)

 

 

 

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Loyola College B.Sc. Statistics April 2004 Applied Stochastic Processes Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5400/STA 400 – APPLIED STOCHASTIC PROCESSES

17.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION -A

 

Answer ALL questions.                                                                              (10 ´ 2 = 20 marks)

 

  1. Define a Stochastic Process.
  2. What is ‘State Space’ of a Stochastic Process?
  3. Define ‘Counting Process’.
  4. Explain ‘Independent Increments’.
  5. Define ‘Markor Process’.
  6. Define ‘Transition Probability Matrix’.
  7. Define ‘accessibility’ of a state from another.
  8. If  is a stochastic matrix,

fill up the missing entries in the matrix.

  1. Define ‘Aperiodic’ Markov chain.
  2. Write down the postulates of ‘Pure Birth Process’.

 

SECTION – B

 

Answer any FIVE questions.                                                                                     (5 ´ 8 = 40 marks)

 

  1. State the classifications of Stochastic Processes based on time and state space. Give an example for each type.
  2. Show that a sequence of independent random variables is a Markov Chain (M.C).
  3. If and the TPM is

,  find P (X2 = 2).

 

  1. Show that ‘Communication’ is an equivalence relation.

 

 

 

 

 

 

 

  1. Classify the states of a M.C. whose TPM is

 

0        1        2     3      4

 

  1. Describe a one-dimensional Random walk and write down its TPM.

 

  1. State and prove any one property of a Poisson Process.

 

  1. Write brief notes on: (a) Stochastic and Doubly Stochastic Matrices; (b) Extensions of Poisson Process.

 

SECTION – C

 

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

 

  1. a) Let { X(t) : t Î T} be a process with stationary independent increments where

T =  {0,1,2, ….}.   Show that the process is a Markov Process.

  1. b) If {Xn : n = 1,2,3, …} is a sequence of i.i.d, r.v.s and Sn = n = 1,2,…., show that

{Sn} is an M.C.                                                                                                  (10+10)

 

  1. a) Define ‘recurrent’ and ‘transisiant’ states. State (without proof) a necessary and

sufficient condition for a state to be recurrent.

  1. b) Describe the two-dimensional random walk. Discuss the recurrence of the states.

(6+14)

  1. State the posulates of a Poisson Process and derive the distribution of X(t).

 

  1. a) Define a ‘Martingale’.

If Yo = 0, Y1, Y2, …., are i.i.d with E (Yn) = 0, V (Yn) = s2, show that:

  1. b) Xn = is a Martingale with respect to {Yn}
  2. c) Xn = – n s2 is a Martingale with respect to {Yn}.                               (3+7+10)

 

 

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Loyola College B.Sc. Statistics April 2004 Applied Statistics Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5502/STA 507 – APPLIED STATISTICS

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

  1. What is the purpose of constructing index numbers?
  2. How do you select base period while constructing index numbers?
  3. Distinguish between seasonal variations and cyclical fluctuations.
  4. What do you understand by the term moving average? How is it used in measuring trend?
  5. Given the following values:

r23 = 0.4,  r13 = 0.61,   r12  = 0.7

Find the partial correlation coefficient r12.3.

  1. Define multiple correlation and give an example.
  2. Distinguish between crude and specific death rates.
  3. Describe the significance and importance of a life table.
  4. What are De-Jure and De-Facto enumeration in population census?
  5. Write a brief note on National Institute of Agricultural Marketing.

 

SECTION – B

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

 

  1. Calculate price index using Fisher’s ideal formula from the following data:
2002 2003
Commodity Price Quantity Price Quantity
A 10 50 12 60
B 8 30 9 32
C 5 35 7 40
  1. A textile worker in Chennai earns Rs.3500 per month. The cost of living index for a particular month is given as 136.  Using the following data, find out the amounts he spent on house rent and clothing:
Group: Food Clothing House rent Fuel and lighting Misc.
Expenditure: 1400 – – 560 630
Group index: 180 150 100 110 80
  1. Fit a curve of the type Y = abX to the following data and estimate for 2004.

Year:                     1999         2000           2001        2002            2003

Population:            132            142            157          170              191

(in 1000 tons)

  1. Describe one method each of i) eliminating the effect of trend from a time series and ii) measuring the seasonal variations.
  2. In a trivariate distribution, it was found:

r12 = 0.7           s1 = 3

r23 = 0.4           s2 = 4

r31 = 0.61         s3 = 5

Find the regression equation of X1 on X­2 and X3, when the variables are measured from their means.

 

 

  1. Compute gross reproduction rate and net reproduction rate from the data given below:
Age-group Female Population Female births Survival rate
15-19 13,000 300   0.9
20-24   9,000 630 0.89
25-29   8,000 480 0.88
30-34   7,000 280 0.87
35-39   6,000 150 0.85
40-44   5,000   35 0.83
  1. Write an elaborate note on population census.
  2. Explain in detail the developments in Fisheries and point out the welfare programmes available for Traditional Fishermen.

 

SECTION – C

 

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

 

  1. a) By giving suitable examples, explain
  2. Splicing of index numbers
  3. Deflating of prices and income          (4+4)
  4. b) Show that Fisher’s formula satisfies both time reversal and factor reversal tests using

the following data:

              Base year        Current year
Commodity Price Quantity Price Quantity
A 4 3 6 2
B 5 4 6 4
C 7 2 6 2
D 2 3 1 5

(6+6)

  1. Compute seasonal indices by the ratio to moving average method from the following data:
                                                Year
Current production in 1000 tons Quarter 2000 2001 2002 2003
I 75 86 90 100
II 60 65 72 78
III 54 63 66 72
IV 59 80 85 93

 

  1. a) Calculate the multiple correlation coefficient of X1 on X2 and X3 from the following

data:

X­1: 5 3 2 4 3 1 8
X2: 2 4 2 2 3 2 4
X3: 21 21 15 17 20 13 22

(12)

  1. b) For the problem in (a), test the significance of the population multiple correlation at 5%

level of significance.                                                                                                      (8)

 

  1. a) Define vital statistics. What is the importance of these statistics?                              (5)
  2. b) Distinguish between Age specific fertility rate and General fertility rate. (5)
  3. c) Given the age returns for the two ages x = 9 years and x+1 = 10 years with a few life – tale values as = 75,824, = 75,362, d10 = 418 and T10 = 49,53,195. Give the complete life-table for two ages of persons.                                                      (10)

 

 

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Loyola College B.Sc. Statistics Nov 2004 Estimation Theory Question Paper PDF Download

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2004

ST 5500/STA 505/S 515 – ESTIMATION THEORY

25.10.2004                                                                                                           Max:100 marks

9.00 – 12.00 Noon

 

SECTION – A

 

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

 

  1. State the problem of point estimation.
  2. Define ‘bias’ of an estimator in estimating a parametric function.
  3. Define a ‘Consistent estimator’.
  4. Define ‘efficiency of an estimator.
  5. Explain ‘Uniformly Minimum Variance Unbiased Estimator’.
  6. What is Cramer – Rao lower bound?
  7. Define ‘bounded completeness’.
  8. Examine if {N (0, s2), s2 > 0} is complete.
  9. Let X1, X2 denote a random sample of size 2 from B (1, q), 0 < q < 1. Show that X1 + 3X2 is sufficient for q.
  10. Explain BLUE.

SECTION – B

 

Answer any FIVE questions.                                                                          (5 ´ 8 = 40 marks)

 

  1. Show that the sample variance is a biased estimator of the population variance s2. Suggest an UBE of s2.
  2. If Tn is a consistent estimator of j (q), show that there exists infinitely many consistent estimators of j (q).
  3. State and derive Cramer – Rao inequality.
  4. Show that UMVUE is essentially unique.
  5. Give an example to show that bounded completeness does not imply completeness.
  6. Show that the sample mean is a complete sufficient statistic in the case of P (q), q > 0.
  7. State and establish Lehmann – Scheffe theorem.
  8. State and prove ‘Invariance property’ of MLE.

 

SECTION – C

 

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

 

 

  1. a) Let f (x;q) =

 

  • ,  otherwise

 

Based on a random sample of size n, suggest an UBE of

  1. q when s is known and
  2. s when q is known          (5+5)

 

  1. b) Obtain CRLB for estimating q, in the case of f(x; q) = x Î R, q Î R,

based on a random sample of size n.                                                                          (10)

  1. a) State and establish factorization theorem in the discrete case.

 

  1. b) Obtain a sufficient statistic for q = (m, s2) based on a random sample of size n from

N (m, s2), m Î R, s2 > 0.                                                                                     (12 + 8)

 

  1. a) Explain the method of maximum likelihood.

 

  1. b) Let X1, X2, …, Xn denote a random sample of size n from N (m, s2). Obtain MLE of

q  = (m, s2).                                                                                                           (5 + 15)

 

  1. a) Describe the method of minimum chi-square and the method of modified minimum

chi-square.

 

  1. b) Describe the linear model in the Gauss – Markov set – up.

 

  1. c) Let y = Ab + e be the linear model where E (e) = 0. Show that a necessary and

sufficient condition for the linear function  b of the parameters to be linearly

estimable is that rank (A) = rank .                                                           (10 + 6 + 4)

 

 

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