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

October 30, 2017 by 01 prakash

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.

What is ‘permutation of indistinguishable objects’? State its value in factorial notation.
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

If P(AÈB) =7/8, P(AÇB) = 1/6, P(A^c) =3/8, find P(B).
Define a bounded function and give an example.
Define a monotonic sequence and give an example.
“The following is not a cumulative distribution function (c.d.f)” -Justify this statement:

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

Section – B (5 ´ 8 = 40 Marks)

Answer any FIVE. Each carries EIGHT marks

(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)

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.

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.

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

(b) Show that (4 + 4)

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

(a) (b) (4+ 4)

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.

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.

(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.

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)

(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)

(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)

(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

October 30, 2017 by 01 prakash

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.

Define a population and mention the assumptions made regarding population size in sampling Theory.
Explain ‘Sampling frame’ of a population.
What are the constraints for carrying out a census?
Distinguish between a statistic and a parameter.
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.
In SRSWOR, find the probability of selecting r^th population unit in i^th
Explain the cumulative total method of selecting a PPS sample.
Write all possible linear systematic samples of size n = 6 given N = 24.
Explain stratified random sampling.
Name any three allocation schemes used in stratified sampling.

SECTION –B (5 ´ 8 = 40 Marks)

Answer any FIVE questions. Each carries EIGHT marks.

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.
Show that, under usual notations, in SRSWOR

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

Find

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

population mean in the presence of linear trend.

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.

(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)

(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)

(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 C_ois the overhead cost andC_h is the cost of collecting information per

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

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

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

Show that if Y_i= a + b_i, 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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST – 5501/STA506 – TESTING OF HYPOTHESIS

05.11.2003 Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions. (10×2=20 marks)

Define the Best critical region for testing a simple hypothesis against an alternative simple hypothesis.
When do you say that a family of probability density functions has a monotone likelihood ratio property?
Explain likelihood ratio criterion.
What is the test statistic used for testing the significance of correlation in a bivariate normal population? What is the distribution of the statistic?
A coin is tossed 100 times. If, the number of heads obtained is 45, can you say the coin is unbiased? justify.
What are non-parametric methods?
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.
Define Type I error and Type II error.
Explain p-value.
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)

Let X_1,X₂, .. X_n be a random sample from N(0,1). Verify whether uniformly most powerful test exists for testing H_o : q = q₁ vs H₁ : q ¹ q₁ .
Let x have the probability mass function

P [X = x] = x = 0, 1,2, ..M.,

It is decided to test H_o : M £ M₀vs H₁ : M > M₀ based on a sample of size 1 from this distribution. Derive the uniformly most powerful test.

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?

It is decided to study the model

X_i = a + b (c_{i –}) + e_i

with E(x_i) = a + b (c_i – , V(X_i) = 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.

Let X have the pdf f ( x ; q) = q^x (1-q)^1-x x = 0, 1 zero elsewhere. We test H₀: q = against H₁: q < . Let X_1, X₂, X₃, X₄, X₅ be a random simple from this distribution. Obtain the uniformly most powerful test with a =
Explain Sign Test.
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.

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)

a) State and prove Neyman – pearson theorem. (12)
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)

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)

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)

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

H₀: F_x =F_y Vs H₁: F_x = F_y.

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)

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

confidence interval for the difference in proportions. (10)

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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

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)

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^*.
Find .
State the first Fundamental Theorem of Integral calculus.
Solve: .
“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.

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

SECTION-B

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

Evaluate (a) . (4+4)

(b)

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].

Solve the non-homogeneous differential equation:

(y – x – 3) dy = (2x + y +6) dx

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

f(x,y) =

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

Show that f_x (x, 0) = f_y (0,y) = 0 , f_x (0, y) = -y , f_y (x, 0) = x.

Find the mean and variance of Beta distribution of II kind stating the conditions for their existence.
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.
Establish the reversal law for Transpose of product of matrices. Show that the operations of Inversion and Transpositions are commutative.
Find the inverse of using Cayley – Hamilton Theorem.

SECTION-C

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

a) Show that, if fÎ R [a, b] then f² Î R [ a, b].
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).
c) Discuss the convergence of (8+6+6)
a) Investigate for extreme values of the function

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

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

the property which gives the probability P[x₁ < X £ x_2, y₁ < Y £ y₂] in terms of the

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

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.

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

(10 + 10)

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Loyola College B.Sc. Statistics Nov 2003 Statistical Methods Question Paper PDF Download

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034

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)

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

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

Prove that for any two real numbers ‘a’ &’b’ , A.M £M.
Mention any two limitations of geometric mean.
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.

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.

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 .

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

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

SECTION – B

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

Explain any four methods of collecting primary data.
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.

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

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

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

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

Explain the concept of regression with an example.
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 = ab^Xfor the above data and estimate the

sales for the year 1997.

SECTION – C

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

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

(iii) Systematic sampling methods with examples.

(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)

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 f_2.

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

b) Explain any two measures of dispersion. (7+7+6)
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.

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)

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

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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

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

03.11.2003 Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions. (10×2=20 marks)

State the problem of point estimation.
Define ‘bias’ of an estimator in estimating a parametric function.
Define a ‘Uniformly Minimum Variance Unbiased Estimator’ (UMVUE).
Explain Cramer-Rao lower bound.
Define completeness and bounded completeness.
Examine if is complete.
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.
Give an example where MLE is not unique.
Define BLUE
State Gauss – Markoff theorem.

SECTION-B

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

Show that the sample variance is a biased estimator of the population variance. Suggest an UBE of .
If T_n is asymptotically unbiased with variance approaching zero as n , show that T_n is consistent.
Show that UMVUE is essentially unique.
Show that the family of Binomial distributions is complete.
State and establish Lehmann – Scheffe theorem.
State and prove Chapman – Robbin’s inequality.
Give an example where MLE is not consistent.
Describe the linear model in the Gauss – Marboff set-up.

SECTION-C

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

a) Let X₁, X₂,….., X_n (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.
b) Let X₁, X₂,….., X_n 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)

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

f based on random sample of size n.

b) State and establish factorization theorem in the discrete case. (8+12)
a) Explain the method of maximum likelihood.
b) Let X₁, X₂, …., X_n denote a random sample of size n from N (. Obtain MLE of q = (. (5+15)
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 .
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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2003

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)

Let f(x,y) = e

0 else where.

Find the marginal p.d.f of X.

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

Find P(X₂ = 2).

If X ~ B (n, p), show that E
If X₁ andX₂ are stochastically independent, show that M (t₁, t₂) = M (t₁, 0) M (0, t₂), ” t_1,t₂.
Find the mode of the distribution if X ~ B .
If the random variable X has a Poisson distribution such that P (X = 1) = P (X = 2),

Find p (X = 4).

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

Z = X -2Y.

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

SECTION-B

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

Let f(x₁, x₂) = 12

0 ; elsewhere

Find P .

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

Show that P (= .

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

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

distribution.

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

distributions is normal.

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)

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)

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

Determine the following probabilities

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

show that (10)

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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST-5503/STA508 – COMPUTATIONAL STATISTICS – I

10.11.2003 Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.

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)

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

(N_i). And 2) product of N_i S_i. 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 (N_i)

Average Area under wheat

Standard deviations (s_I)

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)

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)

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

III

2 5 6 8 10

4 8 10 11 13

8 10 11 13 14

16 18 19 20 22

(20)

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)

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)

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)

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)

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)

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)

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)

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

0.73

0.77

0.72

0.79

0.72

0.91

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

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

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.

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)

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 H_o : m = 25 against the alternative hypothesis H₁ : m < 25. Let X₁,X₂ , X_3, 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 H_ousing this test? (c) What is the p-value associated with the in part (b) ? (20)

(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 H_o: 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 H_o if ≥ c. Find n and c so that

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

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

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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – NOVEMBER 2003

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)

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

SECTION-B

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

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

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

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

SECTION-C

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

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

and V . (6)

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)

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

(10)

b) Derive values of n_hsuch that C_o + is minimum for a given value of

. (10)

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

(12)

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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST-5400/STA400 – APPLIED STOCHASTIC PROCESS

12.11.2003 Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions. (10×2=20 marks)

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

SECTION-B

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

Let {X_n, n=0, 1, 2, …} be a sequence of iid r.v’s with common distribution P(X_o = i) = p_I, iÎS, p_I>0, . Prove that {X_n, n = 0. 1,2, ….} is a Markov chain.
Prove that the state i of a Markov chain is recurrent iff .
Show that the state ‘0’ of a one-dimensional symmetric random walk is recurrent.
Explain a counting process with an example.
Explain a Poisson process with an example.
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

Explain in detail the generalization of a Poisson process.
Let {X_t, 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)

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

probability matrix p=. Find the equivalence classes.

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

Explain the two dimensional symmetric random walk. Also prove that the state ‘0’ is recurrent.
a) Let Y_o = 0 andY₁, Y₂ .. be iid r.r’s with E(Y_k) = 0 and E(Y_k²) = s² ” k = 1,2, ….

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

b) Write short notes on Renewal process.

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

October 30, 2017 by 01 prakash

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034

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)

Distinguish between weighted and unweighted Index numbers.
What do you mean by splicing of Index numbers?
How do you eliminate the effect of trend from time series and measure seasonal variations?
Distinguish between seasonal variations and cyclical fluctuations.
Given the data: r_xy =0.6 r_xz= 0.4, find the value of r_yzso that R_yz, the coefficient of multiple correlation of x on y and z, is unity.

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

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

Write a short rote on De-Facto and De-Jure enumeration.
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)

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?

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.

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.

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.

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

experimental data? (3)

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 s^th

order is (5)

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)

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

from a complete life-table? (3)

What are the current research developments and landmarks in

agricultural statistics?

Explain in detail the different methods of measuring National Income.

SECTION – C

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

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
Commodity	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)

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

applications of Index numbers. (8)

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

For the following set of data:
Calculate the multiple correlation coefficientand the partial correlation coefficient .
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)

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

i) Crude Birth Rate
ii) General fertility rate

iii) Age specific fertility rate

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

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Answer any FIVE. Each carries EIGHT marks

FIFTH SEMESTER – NOVEMBER 2003

ST – 5501/STA506 – TESTING OF HYPOTHESIS

THIRD SEMESTER – NOVEMBER 2003

ST-3500/STA502 – STATISTICAL MATHEMATICS – II

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

ST 1500/ STA 500 STATISTICAL METHODS

15X -8Y-130 = 0

SECTION – B

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

FIFTH SEMESTER – NOVEMBER 2003

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

FOURTH SEMESTER – NOVEMBER 2003

ST-4501/STA503 – DISTRIBUTION THEORY

FIFTH SEMESTER – NOVEMBER 2003

ST-5503/STA508 – COMPUTATIONAL STATISTICS – I

FOURTH SEMESTER – NOVEMBER 2003

ST 4500 / STA 504 – BASIC SAMPLING THEORY

FIFTH SEMESTER – NOVEMBER 2003

ST-5400/STA400 – APPLIED STOCHASTIC PROCESS

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI– 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

ST 5502/STA 507 APPLIED STATISTICS

Commodity