## Loyola College B.Sc. Statistics April 2003 Statistical Methemattics – I Question Paper PDF Download

LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

B.Sc. DEGREE EXAMINATION – STATISTICS

second SEMESTER – APRIL 2003

ST   2500/  STA  501 statistical methemattics I

23.04.2003

9.00 – 12.00                                                                                          Max: 100 Marks

SECTION A                      (10 ´ 2 = 20 Marks)

Answer ALL questions.  Each carries TWO marks.

1. What is ‘permutation of indistinguishable objects’? State its value in factorial notation.
2. If A, B,C are events, give the set theoretic notation for the following
• Exactly one of the three events occurs
• None of the three events occur
1. If P(AÈB) =7/8, P(AÇB) = 1/6, P(Ac) =3/8, find P(B).
2. Define a bounded function and give an example.
3. Define a monotonic sequence and give an example.
4. “The following is not a cumulative distribution function (c.d.f)” -Justify this statement:

1. “The series diverges” – Justify
2. “The function     is not a probability generating function (p.g.f)” –
3. State the Leibnitz test for alternating series.
4. Find (0) if f(x) = x |x|, xÎÂ.

Section B                       (5 ´ 8 = 40 Marks)

# Answer any FIVE.  Each carries EIGHT marks

1. (a) Consider the construction of four-letter words from the word “CHEMISTRY”,

How many of these words begin and end with consonants? How many begin

with T  end with  a vowel?

• If there are 4 persons from Tamilnadu, 2 from Karnataka, 5 from Orissa and

3 from Kerala, find the number of ways they can be seated in a row such that

persons from the state sit together. Find the number of arrangements if they

have to be seated around a circular table.                                         (4+4)

1. State the relevant theorem and solve the  following problem  using it:

A box contains 10 black and 6 white marbles.  Three balls are drawn at random

one by one without replacement.  Find the probability that

• (i) First two draws give black and third draw gives            white

(ii)  First and third drawn give balls of the same colour while the second

draw gives the other colour.

1. Consider the experiment of tossing a fair coin indefinitely until a Head

occurs.  Write down the sample space of the experiment.  If X is the number

of  tosses to get the first Head, find the probability mass function (p.m.f.)

and c.d.f. of X.

1. Show that the function and log x are continuous functions on (0, ¥).
2. (a)  Show  by using first  principles that

(b)    Show that                                                          (4 + 4)

1. Test the convergence of the following series and state the test which you use in each:

(a)           (b)                                         (4+ 4)

1. Find the values of ‘s’ for which f(s) =   is a p.g.f.   Find the probability

distribution  for which it is the   p.g.f.  Hence or otherwise find the mean of the distribution.

1. Obtain the expansion of the exponential function and define the Poisson distribution.

SECTION C                      (2 ´ 20 = 20 Marks)

Answer any TWO.  Each carries twenty marks.

1. (a) State the  Binomial theorem for positive integer index.  Find term which

contains y10 in the expansion of (3x2y-xy2/2)8

(b)   State and prove Baye’s theorem.

(c)   Three machines produce respectively 40%, 40% and 20% of the total

production in a factory.  Of their output 3%, 5% and 4% respectively are

defective  items. If an item is selected at random from the entire lot, what is

the probability that it is a defective?  Given that a selected item is defective,

what is the probability that it was produced by the second machine.  (7+7+6)

1. (a) Show that the product of two continuous functions is continuous.

(b) Consider the following c.d.f. of a r.v.X

Identify the type of the distribution.  Also, find P(X =3), P(2<X £ 4),

P(0 £ X <3/2) and P(X ³ 5/2)                                                            (8+12)

1. (a) Discuss the convergence of the Geometric series for all possible

variations in x.  Find the value of ‘a’ for which the sequence  defines

a probability distribution on the set of positive integers.  Find the p.g.f and

hence the variance of the distribution.

(b)    Examine the applicability and validity of Rolle’s   theorem for the function

(13+7)

1. (a) Investigate the extreme values of  f(x) = 2x5 – 104 +10x3 + 8
• Define Binomial distribution. Find the moment generating function (m.g.f.)

and hence its mean and variance.                                                        (10 +10)

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## Loyola College B.Sc. Statistics April 2003 Basic Sampling Theory Question Paper PDF Download

LOYOLA  COLLEGE (AUTONOMOUS), CHENNAI-600 034.

B.Sc. DEGREE  EXAMINATION  – STATISTICS

FOURTH SEMESTER  – APRIL 2003

ST   4500/ sta  504   BASIC SAMPLING   THEORY

11.04.2003

9.00 – 12.00                                                                                           Max: 100 Marks

SECTION A                                  (10 ´ 2 = 20 Marks)

Answer ALL the questions.  Each carries TWO marks.

1. Define a population and mention the assumptions made regarding population size in sampling Theory.
2. Explain ‘Sampling frame’ of a population.
3. What are the constraints for carrying out a census?
4. Distinguish between a statistic and a parameter.
5. Derive the expression for the mean square error of an estimator T in estimating q in terms of the variance and bias of the estimator T.
6. In SRSWOR, find the probability of selecting rth population unit in ith
7. Explain the cumulative total method of selecting a PPS sample.
8. Write all possible linear systematic samples of size n = 6 given N = 24.
9. Explain stratified random sampling.
10. Name any three allocation schemes used in stratified sampling.

SECTION B                                                (5 ´ 8 = 40 Marks)

Answer any FIVE questions.  Each carries EIGHT marks.

1. Consider a population containing 4 units with values 7,4,10,5. Two units are drawn one by one without replacement with equal probabilities.  Verify whether or not the statistic T =(3y1+ 4y2) /5 is unbiased for the population mean.
2. Show that, under usual notations, in SRSWOR

1. Show that Lahiri’s method of selection is a PPS selection.
2. Examine whether the estimator is unbiased for the population total under PPSWR, where yi is the value of ith drawn unit (i = 1,2,….,n).
3. Deduce is SRSWR using expressions for available in PPSWR.
4. Letdenote the sample mean of only distinct units under SRSWR.

Find

1. Show that in Balanced systematic sampling, the sample mean coincides with

population mean in the presence of linear trend.

1. In stratified sampling, under proportional allocation, derive Vand find

an unbiased estimator of V.

SECTION C                                (2 ´ 20 = 40 Marks)

Answer any TWO questions.  Each carries TWENTY marks.

1. (a) Illustrate that an estimator T can become biased under a sampling scheme

even though T is unbiased under another sampling scheme..                         (8)

(b) A population contains 5 units and it is known that   (12)

Under the usual notations, (i) compare  with

(ii) Find the values of a for which  is less efficient than.(10)

1. (a) consider the following data:

 Plot number No. of plants No. of roses 1 20 56 2 32 65 3 14 30 4 19 47 5 22 37 6 7 28

Assume that a PPSWR sample of size 2 is drawn.  Compute Vfor the

data given.  Compare it with V.                                                       (10)

• Derive an unbiased estimator of Vin PPSWR scheme.                  (10)

1. (a) Describe centered systematic sampling.                                                 (5)
• Explain in detail the principal steps involved in the planning and

execution of  a sample survey.                                                               (15)

• (a) If  Co is the overhead cost and  Ch is the cost of collecting  information per

unit in stratum h, then the total  cost of the survey is given by Co+

In optimum allocation, determine the values of n1, n2,……,nL  for which

V  is minimum subject to a fixed total cost.                                          (12)

• Show that if Yi = a + bi, i =1,2,..,N, then ,

where a stratified sample of size n is drawn as follows:

Divide the population into n groups of k units each.  And draw 1 unit from each group where the first stratum contains the first k units, the second

stratum contains the next k units and so on.                                             (8)

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## Loyola College B.Sc. Statistics Nov 2003 Testing Of Hypothesis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST – 5501/STA506 – TESTING OF HYPOTHESIS

05.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

1. Define the Best critical region for testing a simple hypothesis against an alternative simple hypothesis.
2. When do you say that a family of probability density functions has a monotone likelihood ratio property?
3. Explain likelihood ratio criterion.
4. What is the test statistic used for testing the significance of correlation in a bivariate normal population? What is the distribution of the statistic?
5. A coin is tossed 100 times. If, the number of heads obtained is 45, can you say the coin is unbiased? justify.
6. What are non-parametric methods?
7. 400 oranges are taken from a large consignment and 40 are found to be bad. Find the limits in which the percentage of bad oranges is likely to come at 95% confidence.
8. Define Type I error and Type II error.
9. Explain p-value.
10. Arrange the following sets of x and y values. How many runs you get?

x          40        50        48        60

y          45        51        52        55

SECTION-B

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

1. Let X1, X2, .. Xn be a random sample from N(0,1). Verify whether uniformly most powerful test exists for testing Ho :  q = q1 vs H1 : q ¹ q1 .
2. Let x have the probability mass function

P [X = x]  =  x = 0, 1,2, ..M.,

It is decided to test Ho : M £ M0 vs H1 : M > M0  based on a sample of size 1 from this distribution.  Derive the uniformly most powerful test.

1. In a locality 100 persons are randomly selected and asked about their educational attainments. The results one as under

Education

Middle            High school     College

Male                   10                       15                   25

Female                25                        10                  15

Does education depend on sex?

1. It is decided to study the model

Xi = a + b (ci  – ) + ei

with E(xi) = a + b (ci – ,    V(Xi) = s2.

a random sample of size n = 10 yielded  = 67

= 2.1    = 288 and .

Test H0: b = 0 vs H1 : b ¹ 0 .  Also find 95% confidence interval for b.

1. Let X have the pdf f ( x ; q) = qx (1-q)1-x x = 0, 1 zero elsewhere. We test H0: q = against H1: q < .   Let X1, X2, X3, X4, X5 be a random simple from this distribution. Obtain the uniformly most powerful test with a =
2. Explain Sign Test.
3. Explain how will you construct the confidence interval for ratio of variances, when there are two random samples from 2 independent normal populations. suppose two independent samples of sizes m = 15 and n = 10 yielded

Find the 98% confidence interval for s |  s.

1. It is known that the random variable X has a pdf

f (x ; q ) =

0        ,    elsewhere.

It is decided to test H0: q =    Vs H1: q = 4.

If a random sample of size 2 is observed, find probability of Type I error and probability of

type II error.

SECTION-C

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

1. a) State and prove Neyman – pearson theorem. (12)
2. b) Let X have the pdf f( x, q) = 0 < x < q

Let Y1 < Y2 < Y3  < Y4  denote the order statistics of a random sample of size 4 from

this distribution.

We reject H0: q = 1 and accept H1: q ¹ 1  if Y4 £  or Y4 ³ 1.  Find the Power function. (8)

1. a) Derive likelihood ration test for testify

H0: m1 =m2  Vs       H1: m1 ¹ m2

When the two random samples are drawn from two independent normal populations with mean m1 and m2 and with common unknown variance.                                                 (10)

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

Sample I          9          11        13        11        15        9          12        14

Sample II        10        12        10        14        9          8          10

Is the difference between means of samples significant. (Assume common variance)                                                                                                                                             (10)

1. a) Explain Run test for equality of distributions.                   (12)
2. b) Apply the Mann-Whitney – Wilcoxon test for the following data to test

H0: Fx =Fy  Vs  H1: Fx = Fy.

X         4.3       5.9       4.9       3.1       5.3       6.4       6.2       3.8       7.5       5.8

Y         5.5       7.9       6.8         9        5.6       6.3       8.5       4.6       7.1                   (8)

1. a) Explain the procedure of testing equality of two proportions. Also obtain the 95%

confidence interval for the difference in proportions.                                               (10)

1. b) Explain sequential probability ratio test           (10)

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## Loyola College B.Sc. Statistics Nov 2003 Statistical Mathematics – II Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST-3500/STA502 – STATISTICAL MATHEMATICS – II

04.11.2003                                                                                                           Max:100 marks

9.00 – 12.00

SECTION-A

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

1. If P* is a partition of [a , b] finer than the partition P, state the inequality governing the upper sums lower sums of a function f corresponding to P and P*.
2. Find .
3. State the first Fundamental Theorem of Integral calculus.
4. Solve: .
5. “The function f(x,y) =   xy/(x2+y2) ,    (x,y)  ¹(0,0)

0                ,   (x, y) = (0, 0)

does not have double limit as (x, y)   – verify.

1. State the rule for the partial derivative of a composite function of two variables.
2. Define Gamma distribution.
3. Write down the Beta integral with integrand involving Sine and Cosine functions.
4. Define a symmetric matrix.
5. Find the rank of the matrix .

SECTION-B

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

1. Evaluate (a) . (4+4)

(b)

1. If f(x) = kx2 , 0 < x¸< 2 , is the probability density function (p.d.f) of X, find (i) k

(ii) P[X<1/4],  (iii) P,  (iv) P[X >1].

1. Solve the non-homogeneous differential equation:

(y – x – 3) dy = (2x + y +6) dx

1. For the function          xy(x2 – y2) / (x2 + y2)  ,    (x,y)  ¹(0,0)

f(x,y) =

0                                ,      (x, y) = (0, 0)

Show that fx (x, 0) = fy (0,y) = 0 , fx (0, y) = -y , fy (x, 0) = x.

1. Find the mean and variance of Beta distribution of II kind stating the conditions for their existence.
2. If f(x,y) = e-x-y , x > 0, y > 0, is the joint p.d.f of (x, y),  find the joint c.d.f. of (x, y).  Verify that the second order mixed derivative of the joint c.d.f.is indeed the joint p.d.f.
3. Establish the reversal law for Transpose of product of matrices. Show that the operations of Inversion and Transpositions are commutative.
4. Find the inverse of using Cayley – Hamilton Theorem.

SECTION-C

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

1. a) Show that, if fÎ R [a, b] then f2 Î R [ a, b].
2. b) If f(x) = c.e-x, x > 0, is the p.d.f. of X, find (i) c (ii) E(X), (iii) Var (X).
3. c) Discuss the convergence of (8+6+6)
4. a) Investigate for extreme values of the function

f (x, y) = x3 + y3 – 12x – 3y + 5, x, y Î R.

1. b) Define joint distribution function for bivariate case and state its properties. Establish

the property which gives the probability P[x1 < X £ x2,  y1 < Y £ y2] in terms of the

joint distribution function of (X, Y).                                                               (10+10)

1. If x + y ,   0 < x, y < 1

f (x, y) =

0        ,   otherwise

is the joint p.d.f of (x, y),  find the means and variances  of X and Y and covariance

between X and Y.  Also find  P [ Y < X] and the marginal p.d.f’s of X and Y.

1. a) By partitioning into 2 x 2 submatrices find the inverse of
2. b) Find the characteristic roots and any characteristic vectors for the matrix

(10 + 10)

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# B.Sc. DEGREE EXAMINATION  –  STATISTICS

First SEMESTER  – NOVEMBER 2003

# ST 1500/ STA  500  STATISTICAL  METHODS

07.11.2003                                                                                        Max: 100 Marks

9.00 – 12.00

section A

Answer ALL questions                                                            (10 ´ 2 = 20 Marks)

1. Give the definition of statistics according to Croxton and Cowden.
2. Comment on the following: “ Sample surreys are more advantageous than census”.
3. Give an example for

(i) Quantitative continuous data     (ii)  Discrete time series data

1. Prove that for any two real numbers ‘a’ &’b’ , A.M £M.
2. Mention any two limitations of geometric mean.
3. From the following results obtained from a group of observations, find the standard deviation. S(X-5) = 8 ;  S(X-5)2 = 40;  N = 20.

1. For a moderately skewed unimodal distribution, the A.M. is 200, the C.V.

is 8 and the  Karl Pearson’s coefficient of skewness is 0.3.  Find the mode

of the distribution.

1. Given below are the lines of regression of two series X an Y.

5X-6Y + 90 = 0

#### 15X -8Y-130 = 0

Find the values of .

1. Write the normal equations for fitting a second degree parabola.
2. Find the remaining class frequencies, given (AB) = 400;

(A) = 800; N=2500; (B) = 1600.

### SECTION – B

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

1. Explain any four methods of collecting primary data.
2. Draw a histogram and frequency polygon for the following data.
 Variable Frequency Variable Frequency 100-110 11 140-150 33 110-120 28 150-160 20 120-130 36 160-170 8 130-140 49

Also determine the value of mode from the histogram.

1. Calculate arithmetic mean, median and mode from the following

frequency distribution.

 Variable Frequency variable Frequency 10-13 8 25-28 54 13-16 15 28-31 36 16-19 27 31-34 18 19-22 51 34-37 9 22-25 75 37-40 7

1. The number of workers employed, the mean wages (in Rs.) per month and standard deviation (in Rs.) in each section of a factory are given below. Calculate the mean wages and standard deviation of all the workers taken together.

 Section No. of workers employed Mean Wages (in Rs.) Standard  deviation (in Rs.) A 50 1113 60 B 60 1120 70 C 90 1115 80

1. Calculate Bowley’s coefficient of skewness from the following data.

 Variable frequency 0-10 12 10-20 16 20 -30 26 30- 40 38 40 -50 22 50-60 15 60- 70 7 70 -80 4

1. Calculate Karl Person’s coefficient of correlation from the following data.
 X 44 46 46 48 52 54 54 56 60 60 Y 36 40 42 40 42 44 46 48 50 52

1. Explain the concept of regression with an example.
2. The sales of a company for the years 1990 to 1996 are given below:

 Year 1990 1991 1992 1993 1994 1995 1996 Sales (in lakhs of  rupees) 32 47 65 88 132 190 275

Fit an equation of the from Y = abfor the above data and estimate the

sales for the year 1997.

SECTION C

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

1. a) Explain (i) Judgement sampling (ii) Quota sampling and

(iii) Systematic sampling methods with examples.

1. (i) Draw a blank table to show the distribution of personnel  in a

manufacturing concern according to :

• Sex: Males and Females.
• Salary grade: Below Rs.5,000; Rs.5,000 -10,000;

Rs.10,000 and above.

• Years: 1999 and 2000
• Age groups: Below 25, 25 and under 40, 40 and above

(ii) Draw a multiple bar diagram for the following data:

 Year Sales (in’000Rs.) Gross Profit Net profit 1992 120 40 20 1993 135 45 30 1994 140 55 35 1995 150 60 40

(10+5+5)

1. a) (i)  An incomplete distribution is given below

 Variable 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Frequency 10 20 f1 40 f2 25 15

##### Given the median value is 35 and the total frequency is 170, find

the missing frequencies f1 and f2.

• Calculate the value of mode for the following data:
 Marks 10 15 20 25 30 35 40 Frequency 8 12 36 35 28 18 9

1. b) Explain any two measures of dispersion.                                       (7+7+6)
2. a) The scores of two batsman A and B is 10 innings during a certain season are:

 A 32 28 47 63 71 39 10 60 96 14 B 19 31 48 53 67 90 10 62 40 80

Find which of the two batsmen is consistent in scoring.

1. Calculate the first four central moments and coefficient of skewness from the

following distribution.

 Variable frequency Variable Frequency 25-30 2 45-50 25 30-35 8 50-55 16 35-40 18 55-60 7 40-45 27 60-65 2

(10+10)

1. a) From the following data obtain the two regression equations and calculate

the correlation coefficient.

 X 60 62 65 70 72 48 53 73 65 82 Y 68 60 62 80 85 40 52 62 60 81

1. b) (i)   Explain the concept of Kurtosis.

(ii)   In a co-educational institution, out of 200 students 150 were boys.

They took an examination and it was found that 120 passed, 10 girls

had failed. Is there any association between gender and success in the

examination?                                                                 (10+5+5)

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## Loyola College B.Sc. Statistics Nov 2003 Estimation Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST-5500/STA 505/S 515 – ESTIMATION THEORY

03.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

1. State the problem of point estimation.
2. Define ‘bias’ of an estimator in estimating a parametric function.
3. Define a ‘Uniformly Minimum Variance Unbiased Estimator’ (UMVUE).
4. Explain Cramer-Rao lower bound.
5. Define completeness and bounded completeness.
6. Examine if is complete.
7. Let X1, X2 denote a random sample of size 2 from B(1, q), 0<q<1. Show that X1+3X2 is sufficient of q.
8. Give an example where MLE is not unique.
9. Define BLUE
10. State Gauss – Markoff theorem.

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 .
2. If Tn is asymptotically unbiased with variance approaching zero as n , show that Tn is consistent.
3. Show that UMVUE is essentially unique.
4. Show that the family of Binomial distributions is complete.
5. State and establish Lehmann – Scheffe theorem.
6. State and prove Chapman – Robbin’s inequality.
7. Give an example where MLE is not consistent.
8. Describe the linear model in the Gauss – Marboff set-up.

SECTION-C

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

1. a) Let X1, X2,….., Xn (n > 1) be a random sample of size n from P (q), q > 0. Show that the class of unbiased estimator of q is uncountable.
2. b) Let X1, X2,….., Xn denote a random sample of size n from a distribution with pdf

f(x) ;q) =

0              ,   other wise.

Show that X(1) is a consistent estimator of q.                                                          (10+10)

1. a) Obtain CRLB for estimating q, in the case of

f  based on random sample of size n.

1. b) State and establish factorization theorem in the discrete case. (8+12)
2. a) Explain the method of maximum likelihood.
3. b) Let X1, X2, …., Xn denote a random sample of size n from N (. Obtain MLE of q = (.                                                                                           (5+15)
4. a) Let Y = Ab + e be the linear model where E (e) = 0. Show that a necessary and sufficient condition for the linear function  of the parameters to be linearly estimable is that rank (A) = rank .
5. b) Explain Bayesian estimation procedure with an example. (10+10)

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## Loyola College B.Sc. Statistics Nov 2003 Distribution Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST-4501/STA503 – DISTRIBUTION THEORY

31.10.03                                                                                                          Max:100 marks

9.00-12.00

SECTION-A

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

1. Let f(x,y) = e

0          else where.

Find the marginal p.d.f of X.

1. Let the joint p.d.f of X1 and X2 be f(x1,y1) = and x2 = 1, 2.

Find P(X2 = 2).

1. If X ~ B (n, p), show that E
2. If X1 andX2 are stochastically independent, show that M (t1, t2) = M (t1, 0) M (0, t2), ” t1, t2.
3. Find the mode of the distribution if X ~ B .
4. If the random variable X has a Poisson distribution such that P (X = 1) = P (X = 2),

Find p (X = 4).

1. Let X ~ N (1, 4) and Y ~ N (2, 3). If X and Y are independent, find the distribution of

Z = X -2Y.

1. Find the mean of the distribution, if X is uniformly distributed over (-a, a).
2. Find the d.f of exponential distribution.
3. Define order statistics based on a random sample.

SECTION-B

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

1. Let f(x­1, x2) = 12

0        ;   elsewhere

Find P .

1. The m.g.f of a random variable X is

Show that P (= .

1. Find the mean and variance of Negative – Binomial distribution.
2. Show that the conditional mean of Y given X is E (Y÷X=x)for trinomial
3. Find the m.g.f of Normal distribution.

1. If X has a standard Cauchy distribution, find the distribution of X2. Also identify its

distribution.

1. Let (X, Y) have a bivariate normal distribution. Show that each of the marginal

distributions is normal.

1. Let Y1, Y2 , Y3 andY4 denote the order statistics of a random sample of size 4 from a

distribution having a p.d.f.

f(x) =    2x  ;  0 < x < 1

0   ;    elsewhere   .   Find p .

SECTION-C

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

1. Let x (X1, X2) be a random vector having the joint p.d.f.

f (x1, x2) =         2  ;   0 < x1 < x2 <1

0  ;   elsewhere

(i) Find the correlation between x1 and x2                                                                        (10)

(ii) Find the conditional variance of x1 / x2                                                                      (10)

1. a) Find the mean and variance of hyper – geometric distribution. (10)
2. b) Let X and Y have a bivarite normal distribution with

Determine the following probabilities

1. i) P (3 < Y <8) ii) P (3 < Y< 8 ½X =7)                                                       (10)
2. i) Derive the p.d.f of ‘t’ – distribution with ‘n’ d.f (10)
3. ii) If X1 and X2 are two independent chi-square variate with n1 and n2f. respectively,

show that                                                                          (10)

1. i) Let Y1, Y2 and Y3 be the order statistics of a random sample of size 3 from a

distribution having p.d.f.

1      ;      0 < x < 1

f (x) =

0     ;       elsewhere.

Find the distribution of sample range.                                                             (10)

ii)Derive the p.d.f of  F variate with (n1, n2) d.f.                                                   (10)

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## Loyola College B.Sc. Statistics Nov 2003 Computational Statistics – I Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST-5503/STA508 – COMPUTATIONAL STATISTICS – I

10.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

1. a) In a survey conducted to estimate the cattle population in a district containing 120

villages, a simple random sampling of 20 villages was chosen without replacement.

The cattle population in the sampled villages is given as follows: 150, 96, 87,101, 56,

29, 120, 135, 141, 140,  125, 131, 49, 59, 105, 121, 85, 79, 141, 151.  Obtain an

unbiased estimator of the total cattle population in the district and also estimate its

standard error.                                                                                                                (14)

1. b) The data given in the table represents the summary of farm wheat census of all the

2010 farms in a region.  The farms were stratified according to farm size in acres into

seven strata.  (i) Calculate the sampling variance of the estimated area under wheat for

the region from a sample of 150 farms case (a) If the farms are selected by the method

of SRS without stratification.  Case (b): The farms are selected by the method of SRS

within each stratum and allocated in proportion to 1) number of farms in each stratum

(Ni).  And 2) product of Ni Si . Also calculate gain in efficiency resulting from case (b)

1 and 2 procedures as compared with unstratified SRS.

 Stratum number Farm Size (in acres No.of farms (Ni) Average Area under wheat Standard deviations (sI) 1 2 3 4 5 6 7 0-40 41-80 81-120 121-160 161-200 201-240 More than 240 394 461 391 334 169 113 148 5.4 16.3 24.3 34.5 42.1 50.10 63.8 8.3 13.3 15.1 19.8 24.5 26.0 35.2

(20)

1. c) Consider a population of 6-units with values 1,5,8,12,15 and 19. Writ down all possible

samples of size 3 without replacements from the population and verify that the sample

mean is an unbiased estimator of the population mean.  Also i) calculate the sampling

variance and verify that it agrees with the formula of variance of the sample mean.  (ii)

Verify that the sampling variance is less than the variance of the sample mean from

SRSWOR.                                                                                                                                    (14)

1. d) Five samples were collected using systematic sampling from 4-different pools located in a

region to study the mosquito population, where the mosquito population exhibits a

fairly steady raising trend.  i] Find the average mosquito population in all 4-poolss

ii] Find sample means iii] Compare the precision of systematic sampling, SRSWOR and

stratified sampling.

 Pool no Systematic Sample Number 1        2       3        4       5 I II III IV 2        5       6        8        10 4        8      10      11       13 8       10     11      13       14 16      18     19     20       22

(20)

1. a) The following is a sequence of independent observations on the random variable X with the

density function

f(x ; q1, q2)  = .

The observations are 1.57  0.37  0.62  1.04   0.21  1.8   1.03    0.49   0.81  0.56.  Obtain the maximum likelihood estimates of q1 and q2 .                                                                   (15)

1. b) Obtain a 95% confidence interval for the parameter l of the Poisson distribution based

on the following data:

No. of blood corpuscles :                     0         1         2         3         4          5

No. of cells                  :          142       156      96        27        5          1                     (12)

1. c) Find a 99% confidence interval for m if the absolute values of the random sample of 8

SAT scores (scholastic Aptitude Test) in mathematics assumed to be N(m, s2) are 624,

532,565,492, 407,  591, 611 and 558.                                                                           (7)

(OR)

1. d) The following data gives the frequency of accidents in Chennai City during 100 weeks.

No of accidents:          0          1          2          3          4          5

No. of weeks:              25        45        19        5          4          2

If P(X = x) =

x = 0 ,1, 2,….

estimate the parameters by the method of moments.                                                 (12)

1. e) The following is a sample from a geometric distribution with the parameter p. Derive a

95% confidence interval for p.

x:         0          1          2          3          4          5

f:          143      103      90        42        8          14                                            (5)

1. f) An absolute sample of 11 mathematical scores are assumed to be N (m, s2). The

observations are 26, 31, 27,28, 29, 28, 20, 29, 24, 31, 23.

Find a 99% confidence interval fo s.                                                                          (7)

1. 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.  Ten sets of paired

test results are

 Lot number Company Test Results (X) Vendor Test Results (Y) 1 2 3 4 5 6 7 8 9 10 0.50 0.58 0.90 1.17 1.14 1.25 0.75 1.22 0.74 0.80 0.79 0.71 0.82 0.82 0.73 0.77 0.72 0.79 0.72 0.91

Let D = X – Y and let mD denote the median of the differences.

Test  H0 : mD = 0  against  H1 : mD > 0  using the sign test.    Let a = 0.05 approximately.                                                                                                                              (14)

1. b) Freshmen in a health dynamics course have their percentage of body fat measured at the

beginning (x) and at the end (y) of the semester.  These measurement are given for 26

students in Table below.  Let m equal the median of the differences, x – y.  Use the

Wilcoxon statistic to test the null hypothesis H0 : m = 0 against the alternative

hypothesis H1 : m > 0 at an approximate a = 0.05 significance level.

 X Y 35.4 28.8 10.6 16.7 14.6 8.8 17.9 17.8 9.3 23.6 15.6 24.3 23.8 22.4 23.5 24.1 22.5 17.5 16.9 11.7 8.3 7.9 20.7 26.8 20.6 25.1 33.6 31.9 10.5 15.6 14.0 13.9 8.7 17.6 8.9 23.6 13.7 24.7 25.3 21.0 24.5 21.9 21.7 17.9 14.9 17.5 11.7 10.2 17.7 24.1 20.4 21.9

(20)

(OR)

1. A certain size bag is designed to hold 25 pounds of potatoes. A former fills such bags in the field.  Assume that the weight X of potatoes in a bag is N (m,9).  We shall test the null hypothesis Ho : m = 25 against the alternative hypothesis H1 : m < 25.  Let X1,X2 , X3, X4 be a random sample of size 4 from this distribution, and let the critical region for this test be defined by , where  is the observed value of .

(a) What is the power function of this test?.  In particular, what is the significance

level of this test?  (b) If the random sample of four bags of potatoes yielded the values

= 21.24,  = 24.81 , = 23.62, = 26.82,would you accept or reject Ho using this test?  (c) What is the p-value associated with the  in part (b) ?                                             (20)

1. (d) Let X equal the yield of alfalfa in tons per acre per year. Assume that X is N (1.5, 0.09).

It is hoped that new fertilizer will increase the average yield.  We shall test the null

hypothesis Ho: m = 1.5 against the alternative hypothesis H1: m > 1.5.  Assume that the

variance continues to equal s2 = 0.09 with the new fertilizer.  Using , the mean of a

random sample of size n, as the test statistic, reject Ho if  ≥ c.  Find n and c so that

the power function bf(m)  =  P ( ≥ c) is such that

a  =  bf (1.5)  =  0.05  and  bf (1.7)  =  0.95.                                                                (14)

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## Loyola College B.Sc. Statistics Nov 2003 Basic Sampling Theory Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST  4500 / STA 504 – BASIC SAMPLING THEORY

01.11.2003                                                                                                           Max:100 marks

9.00 – 12.00

SECTION-A

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

1. Explain sampling frame and give two examples.
2. If there are two unbiased estimators for a parameter then show that one can construct, uncountable number of unbiased estimators.
3. If T is an estimator for , then show that MSE (T)  =  V(T)  +  [B(T)]2 .
4. Explain Lottery method for drawing random numbers.
5. Show that probability of including the ith population unit (i =  1, 2, …, N) when a SRSWOR of size n is drawn from a population containing N units is .
6. Find the probability of selecting ith population unit in cumulative total method.
7. Examine whether the estimator is unbiased for the population total under PPSWR.
8. Show that the sample mean under SRSWOR is more efficient than under SRSWR.
9. Explain Linear Systematic Sampling Scheme.
10. When do we use Neyman allocation?

SECTION-B

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

1. Examine the validity of the following statement using a proper illustration :

“property of unbiasedness depends on the sampling scheme under use”.

1. Prove that, under usual notations, in SRSWOR, P[yi =
2. What is PPS sampling? Describe cumulative total method?
3. Deduce expressions for , V() and v () under SRSWR using the expressions for , V() and v () available under PPSWR.
4. Prove that is an unbiased estimator for population mean under stratified random sampling.  Derive .
5. Derive the formula for Neyman allocation.
6. Prove that the sample mean coincide with the population mean in Centered Systematic Sampling, when there is linear trend in the population.
7. a) List all possible Balanced Systematic Samples if N = 40,  n= 8.
8. b) List all possible Circular Systematic Samples if N = 7, n = 3.

SECTION-C

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

1. a) Describe the principal steps involved in the planning and execution of a survey. (14)
2. b) Let denote the sample mean of only distinct units under SRSWR. Find E

and V .                                                                                                     (6)

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

with .  Find the values of a for which

is less efficient than .                                                 (12)

1. b) Show that Lahiri’ s method of selection is a PPS selection. (8)
2. a) Show that an unbiased estimator of V() is

(10)

1. b) Derive values of nh such that Co + is minimum for a given value of

.                                                                                                                                  (10)

1. a) Compare and assuming Nh is large for all h = 1,2, …., L.

(12)

1. b) A sampler has 2 strata. He believes that S1 and S2 can be taken as equal. For a given

cost c = c1 n1 + c2 n2,   show that   =   .                      (8)

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## Loyola College B.Sc. Statistics Nov 2003 Applied Stochastic Process Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

# ST-5400/STA400 – APPLIED STOCHASTIC PROCESS

12.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

1. Define a stochastic process with an example.
2. Define Bernoulli process with an example.
3. Give an example of a continuous time, discrete space stochastic process.
4. When do you say a process has stationary independent increments?
5. Define a Markov Chain.
6. Explain a doubly stochastic matrix.
7. Define the periodicity of a state i of a Markov Chain.
8. When the state i is said to be recurrent?
9. Explain the one dimensional random walk with an example.
10. Define a Martingale.

SECTION-B

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

1. Let {Xn, n=0, 1, 2, …} be a sequence of iid r.v’s with common distribution P(Xo = i) = pI, iÎS, pI>0, . Prove that {Xn, n = 0. 1,2, ….} is a Markov chain.
2. Prove that the state i of a Markov chain is recurrent iff .
3. Show that the state ‘0’ of a one-dimensional symmetric random walk is recurrent.
4. Explain a counting process with an example.
5. Explain a Poisson process with an example.
6. If {X1 (t), t Î (o, ¥,} and {X2 (t), t Î (o, ¥,} one two independent Poisson processes with parameters l1 and l2, Show that the distribution of X1 (t) / (X1 (t) + X2(t) = n follows B

.

1. Explain in detail the generalization of a Poisson process.
2. Let {Xt, t Î T} be a process with stationary independent increments when T – 0,1,2….. show that the process is a Markov process.

SECTION-C

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

1. a) Show that a communication is an equivalence relation.
2. b) Let{Xn, n ≥ 0} be a Markov Chain with state space S = {0,1,2,3,4} and transition

probability matrix      p=.     Find the equivalence classes.

1. Show that for a Poisson process with distribution of x(t) is given by P{x(t) = m} =

1. Explain the two dimensional symmetric random walk. Also prove that the state ‘0’ is recurrent.
2. a) Let Y = 0 andY1, Y2 .. be iid r.r’s with E(Yk) = 0 and E(Yk2) = s2 ” k = 1,2, ….

Let Xn = .  Show that {Xn, n≥ 0 } is a martingale w.r.t {Yn, n ≥ 0}.

1. b) Write short notes on Renewal process.

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# B.Sc. DEGREE EXAMINATION  –  STATISTICS

Fifth  SEMESTER  – NOVEMBER 2003

### ST 5502/STA 507 APPLIED STATISTICS

07.11.2003                                                                                        Max: 100 Marks

1.00 – 4.00

SECTION A

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

1. Distinguish between weighted and unweighted Index numbers.
2. What do you mean by splicing of Index numbers?
3. How do you eliminate the effect of trend from time series and measure seasonal variations?
4. Distinguish between seasonal variations and cyclical fluctuations.
5. Given the data: rxy =0.6 rxz = 0.4, find the value of ryz so that Ryz , the coefficient of multiple correlation of x on y and z, is unity.

1. Explain briefly the significance of the study of multiple correlation in statistical analysis.

1. Define Vital statistics. What is the importance of these statistics?
2. What are crude and standardised death rates? Why is comparison on the basis of standardised death rates more reliable?

1. Write a short rote on De-Facto and De-Jure enumeration.
2. Give that the complete expectation of life at ages 35 and 36 for a particular group are respectively 21.39 and 20.91 years and that the number living at age 35 is 41,176, find the number that attains the age 36.

SECTION B

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

1. An enquiry into the budget of the middle class families in a certain city in

India gave the following information.

 Expenses on Food Fuel Clothing Rent Misc. 40% 10% 18% 20% 12% Prices (2001) (in Rs.) 2250 600 1000 1500 700 Price (2003) 2500 900 1100 1600 800

What changes in cost of living figures of 2003 as compared with that

of 2001 are seen?

1. Obtain the trend of bank clearance by the method of moving averages by

assuming a 5 -yearly cycle:

 Year 1991 92 93 94 95 96 Bank clearance (in crores) 53 79 76 66 69 94 Year 1997 98 99 2000 01 02 Bank clearance (in crores 105 87 79 104 97 92

Also, draw original and trend lines on the graph and compare them.

1. Production of a certain commodity is given below:

 Year 1999 2000 2001 2002 2003 Production (in lakh tons) 7 9 10 7 5

Fit a parabolic curve of second degree to the production.

Estimate the production for 2004.

1. The following means, standard deviations and correlations are found for

X1= seed hay crop in kgs. per acre, X2 = spring rainfall in inches,

X3 = Accumulated temperature above 42°F.

r12  = 0.8

r13  = – 0.4

r23   = – 0.56

Number of years of data = 25

Find the regression equation for hay crop on spring rainfall

and accumulated temperature.

1. a) It is possible to get: r12 = 0.06, r23 = 0.8 and r13 =  -5 from a set of

experimental data?                                                                                     (3)

1. If all the correlation coefficients of zero order on a set of p variates are

equal to  then show that every partial correlation coefficient of the sth

order is                                                                                             (5)

1. a) Given the age returns for the two ages x = 9 years and x +1 = 10 years with

a few life-table values as l9 = 75,824, l10 = 75,362, d10 = 418 and

T10 = 49,53,195. Give the complete life-table for the ages of persons.       (5)

1. b) In what way, does the construction   of an abridged life-table differ

from a complete life-table?                                                                          (3)

1. What are the current research developments and landmarks in

agricultural statistics?

1. Explain in detail the different methods of measuring National Income.

SECTION C

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

1. a) Using the following data, construct Fisher’s Ideal Index number

and show how it satisfies  Time Reversal and Factor Reversal tests:

## Commodity

Base year Current year
Price Quantity Price Quantity
A 6 50 10 56
B 2 100 2 120
C 4 60 6 60
D 10 30 12 24
E 8 40 12 36

(12)

1. What are Index numbers? How are they constructed? Discuss the

applications of Index numbers.                                                                 (8)

1. Calculate the seasonal variation indices by the method of link relatives for

the following figures.

 Year Quarterly cement  production in 1000 tons Q1 Q2 Q3 Q4 1998 45 54 72 60 1999 48 56 63 56 2000 49 63 70 65 2001 52 65 75 73.5 2002 63 70 84 66
1. For the following set of data:
2. Calculate the multiple correlation coefficientand the partial correlation coefficient .
3. Test the significance of both population multiple correlation coefficient and partial population correlation coefficient at 5% level of significance.

 Y 10 17 18 26 35 8 X1 8 21 14 17 36 9 X2 4 9 11 20 13 28

(10+10)

1. The population and its distribution by sex and number of births in a

town in 2001 and survival rates are given in the table below.

 Age group Males Females Male births Females births Survival rate 15  -19 6145 5687 65 60 0.91 20 – 24 5214 5324 144 132 0.90 25 – 29 4655 4720 135 127 0.84 30 – 34 3910 3933 82 81 0.87 35 – 39 3600 2670 62 56 0.85 40 – 44 3290 3015 12 15 0.83 45 – 49 2793 2601 3 3 0.82

From the above data, calculate

1. i) Crude Birth Rate
2. ii) General fertility  rate

iii)   Age specific fertility  rate

1. iv) Total fertility rate
2. v) Gross reproduction rate and
3. vi) Net reproduction rate; assuming no mortality.           (2 +2 + 4 + 2 + 5 +5)

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