C. S. (MAIN) EXAM 2009

STATISTICS

Paper-I

Time Allowed: Three Hours J Maximum Marks 300]

INSTRUCTIONS

Each question is printed both in Hindi and
in English.

Answers must he written in the medium
specified in the Admission Certificate issued
to you, which must be stated clearly on the
cover of the an swer-book in the space
provided for the purpose. No marks will be
given for the answers written in a medium other
than that specified in the Admission Certificate.

Candidates should attempt Questions 1 and 5
which are compulsory, and any three of the
remaining questions selecting at least one
question from each Section.
Assume suita ble data if.considered necessary
and indicate the same clearly.

The number of marks carried by each question
is indicated at the end of the question.
Notations and symbols used are as usual.

SECTION-A

l. Answer any FIVE parts of the following 12 x5=60

In a two-tier competitive examination, only candidates with a minimum score of x0 in the preliminary examination are allowed to sit for the main examination. The score distrib ution in the preliminary examination can be ass umed to be normal with meanµ and s.d. cr. Derive the expressions for the mean and the variance of scores in the preliminary examination of the candidates sitting for the main exatnihation.

2 sets of n cards, both numbered from I to are randomly matched. Find and where X is the number of matches realised.
State (without proof) Tchebychev's inequality. How many times a fair coin must be tossed in order that the relative propor tion of heads lie between 0·4 and 0·6 with probability at least 0·9 (Upper and I points of standard normal variate are l ·645 and l ·282 respectively.)

State the invariance pr.operty of maximum likelihood estimator (m.l.e.). Use this to obtain the m.l.e. of 1/8 in sampling from <img src='./qimages/56-1d.jpg'>


0What is a size-a randomised test For testing the null hypothesis that at most 50% of the 500 ml. packs of hair oil of a certain brand are underweight, against the alternative that the percentage is more, it is desired to examine 8 such packs chosen at random. Use the following information to design a randomised test of size 0·05 <img src='./qimages/56-1e.jpg'> where is the probability mass function of a binomial variate with parameters m p 0·5.
A group of20 short distance runners were subjected to a month-long training. Discuss how you would examine if the training was at all effective based on their timings to clear 100 meters before and after the training, clearly mentioning the underlying assumptions, if any, the null and the alternative hypotheses, as also the size-a critical region.

2. X 1 and X2 are i.i.d. random variables with common probability mass function qxp, x ..... . 0 p q I p 0 otherwise. Obtain the prob ability distribut ion of Y max(XI' X2). Also derive 20

In sampling from cr2 µ unknown, derive 100(1 shortest confidence interval for cr2 • 2 0

Discuss how you would graphically perform the sequential probability ratio test for a simple hypothesis regarding the mean o f a normal population with known standard deviation against a simple alternative. 2 0

Define probability generating function of a discrete random variable. Let X be a random variable with p.m.f. <img src='./qimages/56-3a.jpg'> 20

For the p.d.f. <img src='./qimages/56-3b.jpg'> derive Rao-Cramer lower bound of variance ofan unbiased esti1J1ator of 8. 20

Show that the likelihood ratio test, for the hypothesis that the means of K independent normal populations with common variance are identical, boils down to F-test. 2 0

4. The daily demand (in kg) offish ofa certain variety upto 12 noon with a retailer follows exponential law with mean 2 5. He makes a profit of Rs. 2 0 on an average for each kg he is able to sell by 12 noon, and incurs a loss o f Rs. 1 0 on an average for each kg that remains unsold after 12 noon. What is the optimum daily stock he should procure? 2 0

For the Pareto distribution with distribution function <img src='./qimages/56-4b.jpg'> derive the p.d.f. Also obtain its harmonic mean and vanance. 2 0

Define most powerful test, and unbiased test o f a simple hypothesis against a simple alternative. Show that a most powerful test is necessarily unbiased. 2 0

SECTION-B

5. Answer any FIVE of the following parts
12x5=60

A committee of 5 is to be formed from amongst 86 officers, numbered serially from l to 86. Draw a circular systematic sample for this purpose, giving the procedure you h ave followed in detail. You may use the following set of random nwnbers: 1349 0417 9311 9787 1284 0769 8422 1077.

Examine, after deducing necessary result, if the following set of correlation coefficients are internally consistent r 12 0 ·62, r 13 0 ·55 and r 23 -0·42 .

Locate 5 points at random on a rectangular paddy field measuring 30 m x 20 m for conducting a crop-cutting experiment. The points are to be located using rectangular co-ordinates to nearest decimeters with one corner of the plot as origin. The following set of random numbers may be used:
2292 2933 6125 2464 1038 3163 3569 7155
2029 2538 7080 3027 6215 3125 5856 9543

For pps sampling without replac ement, give the Horvitz-Thompson estimator of the population total. Show that it is unbiased. Also obtain its vanance.

Suggest a balanced confounded design of a 2 4 experiment 1n 3 replicates, each replicate cont aining 4 incomplete blocks of size 4 each, retaining fu ll information on the main effects and the 2nd order interactions. Give the complete lay-out of one such replicate indicating the fa ct orial effects confounded in it.

Define a BIBD. Show that for a BIBD with parameters k and lambda <img src='./qimages/56-5f.jpg'>

6. The vector variable X fo llows p-variate normal distribution with mean vector µ and dispersion <img src='./qimages/56-6a.jpg'> 20

Based on a set of data y .. j n. I I i discuss how you would use the technique of Analysis of Var iance for testing the hypothe sis that the regression equation of y on x is linear. 20

Use the fo llowing set of random numbers to obtain the lay-out of a randomised block design with 5 treatments and 4 blocks, giving the outline of the procedure you have fo llowed 571 1 7343 7539 3684 9397 5335 403 1 1486 2588 3301 0553 2427 3598 2580 70 17 9176 20

7. The joint p.d.f. of X and Y is given by <img src='./qimages/56-7a.jpg'> Find the corre lation coefficient between X and as also the regression equation of Y on X. 20

A rural b lock i n d i strict was divided into 3 strata. The following table gives some relevant information <img src='./qimages/56-7b.jpg'> A sample of 1 0 villages is to be drawn. How many villages should be selected from different strata using proportional allocation, optimum allocation, and allocation proportional to area under ctop 20

3 factors, each at 2 levels, are to be tested in a single experiment using-r randomised blocks. Using standard notation, give the expressions for the sums o f squares due to different factorial effects. Also give the ANOVA table. 20

8. You are to estimate the proportion of school-going students (Classes V to XII) using mobile phones in a metropolitan c i ty. Sugges t a two- stage sampling scheme. Using a suitable notation, give the estimator to be used, and check if the esti mator i s unbiased. 20

Write down the random effects model for a set of two-way classified data with one observation per' cell, stating the underlying assumptions completely. What are the hypotheses th_at can b e tested using the ANOVA technique Give the ANOVA t a b le with explic it expressions f or the different sums of squares, and the test statistics for the above hypotheses. 20

Compare c om p le t e ly Ran domised . Design, Randomised Block Design and Latin Square Design in terms of f lexib i l ity f or number of treatments and number of replicates per treatment on the one hand, and error-control on the other. 2 0


Note English version of the In structions is printed on the
front cover of this question paper.
20