C. S. (MAlN) EXAM 2009

STATISTICS

Paper II

I Time Allowed Three Hours j I Maximum Marks 300 I
INSTRUCTIONS

Each question is printed both in Hindi and
in English.

Answers ,nust be written in the medium
specified in the Admission Certificate issued
to you, which must be stated clearly on the
cover of the answer-booh in the space provided
for the purpose. No marhs will be given for
the answers written in a medium other than
that specified in the Admission Certificate.

Candidates should attempt Questions no. i
and 5 which are compulsory, and any three
of the remaining questions selecting at least
one question from each Section.

Assume suitable data if considered necessary
and indicate the same clearly.

Symbols notations used carry · usual
meaning, unless otherwise indicated.
Charts I figures, wherever required, are to be
drawn in the answer booh itself and not on
separate graph sheet.

The number of marks carried by each
question is indicated at the end of the
question.

SECTION A

1. Attempt any five sub-parts of the following ·

Describe classification of states in a Markov chain. Define n-step transition probability. Prove the following Chapman Kolmogorov equation for the transition probabilities<br><br> <img src='./qimages/57-1a.jpg'>
What are the goals achieved through sensitivity analysis If in the following linear programming problem (L.P.P.) Maximize z 3x 1 2x2 5x3 subject to 3x 1 2x2 x3<=430 3x 1 2x3 s 460 x1 4x2 420 x1, Xz, X3 the profit coefficient of x1 is reduced to 2 from determine whether the original solution remains 12 optimal. 12
In a life test with replacement, 35 heaters were put into continuous operation and the first five failures occurred after 250, 380, 610, 980 and 1250 hours. Assuming the exponential model, construct a 95 percent confidence interval for the mean life of this kind of heater. Test the manufacturer's claim that the rriean life of these heaters is at least 5000 hours at 5o/o level of significance. Given 2 2 x 0 .025 20·48, x0 .975 3·25, X 3 ·9 5 18·31
Describe a CUSUM control chart. Compare this chart with a Shewhart chart with respect to performance. How is a V-mask useful in CUSUM12 charts? 12
What is the effect of lengthening of a test on its reliability and validity A given test has a reliability coefficient of 0·8 and st andard deviation 20. What is the estimated reliability coefficient of this test in a group in which standard deviation is 15
Explain feasible solution and basic solution in the context of linear programming problem (L.P.P.). Prove that the objective function of a L.P.P. assumes its minimum at an extreme point of the convex set generated by the set of all feasible solutions to the problem, and if it assumes its minimum at more than one extreme point then it takes on the same value for every convex combination of those particular points. 12

2. Show that an assignment problem (A.P.) .lS the special case of a transportation problem. Describing the steps of the method you use to solve an A.P., find the solution of the following A.P. <img src='./qimages/57-2a.jpg'> 20
Define transient and persistent states in a Markov chain. <img src='./qimages/57-2b.jpg'>

What do you mean by acceptance sampling plans Between a single sampling plan with n 20, c 2 and a double sampling plan with n1 10, c1 n2 10, c2 can it be said that the second inspection scheme is more economical than the first? Give reasons justifying your answer. 20

3. What is replacement problem How would you determine the replacement policy of items whose maintenance cost increases with time and money carries a rate of interest r per year An auto owner finds from his past record that the cost per year of an auto whose purchase cost is Rs. 60,000 is as given below <img src='./qimages/57-3a.jpg'> Determine at what time is its replacement due. 15

Let A a ta 1 e }.t be the Weibull failure density function. Given the censored sample iJ ... corresponding to a total sample of size n from Weibull distribution, show that the estimating equations for A and a are respectively <img src='./qimages/57-3b.jpg'> 25
Derive an expression for finding optimum inventory level in a single-period probabilistic model without set-up cost with continuous demand and discrete replenishment unit. A baking firm makes a profit of Rs. 5·00 per kg of cakes sold on the same day it baked and disposes all unsold cakes at a loss of Rs. 1·20 per kg. The demand follows rectangular distribution between 2000 and 3000. Find optimum quantity to be baked daily. 20

4. Describe simulation and Monte Carlo simulatio n. What IS the role of random numbers In simulation Describe how · would you select a random sample from a population having exponential distribution . 15
Explain the princip les which are used to find the solution of a rectangular game which does not possess a saddle point. Find the value of the game and mixed str ategies of the players for th e following game <img src='./qimages/57-4b.jpg'> 25
Explain assignable and chance causes of variation. Describe the principles on which a control chart is based . Describe ex, cr) chart and mention its25 advantages and disadvantages over ex, chart . 20

SECTION B

5. Attempt any five sub-parts of the following
What do you mean by 'seasonal index' m the analysis of a time series Discuss the differ ent steps involved in the computation of the seasonal indices using link relative method.
Discuss the problem of autocorrelation and its consequences when one applies least square 12 method for estimating the parameters. 12
Differ entiate between total fertility rate gross reproduction rate and net reproduction rate (NRR). Does TFR strictly conform to our idea of a measure for reproduction How does NRR indicate the growth of population 12
Wr ite a note on the collection, compil ation and publication of demographic data by National Sample Survey Organization. 12
(e)What do you mean by a life table The values of lx 1n a life table are given as follows
age 102 103 104 105 106 107 108
lx 97 59 32 15 6 2 0
Calculate remaining entries of the life table for x 102 and find the probability that a person of exact age 102 years will die between ages 103 and 107. 12

Explain clearly the concepts of rehability and validity of scores in psychological and educational experiments. Explain the relation of validity coefficient to error of measurement. 12

6. What do you understand by agricultural statistics, trade and price statistics What are the present Indian official statistical systems for collecting these data 20
What do you understand by cost of living index number Explain how would you construct it. The following table gives the group index numbers and the weights of different heads of expenditure in the calculation of cost of living index number except the index for the group 'fuel and lighting' <img src='./qimages/57-6b.jpg'> If the overall cost of living index is 193, find th e index number of fuel and lighting group. 20

Explain the terms percentile score, summationscore as a function of ability, Z-score and T-score. Describe how T-scores are to be found when we are given test scores in the form of a frequency table. 20

7. Discuss the differences between direct and indirect methods of standardization of death rates. Calculate the crude and standardized death rate of the year 1990 by direct and indirect methods of standardization by taking 1980 population as standard <img src='./qimages/57-7a.jpg'> 25

What are the tests to be satisfied by a good index number Examine ho,v far they are met using Paasche's index number and Fisher's ideal index number. Usin g Laspeyres' and Paasche's index numbers, the price index numbers for the year 1970 with 1966 as base year are 121 and 144 respectiv ely. What would . be the value of Fisher's index number 20
What is a time series What do you mean by the analysis of a time series Describe briefly different components of a time ser ies. Which com ponent of the time series is mainly applicable in the following cases Fall of death rate due to advances in science Fire in a fa ctory Sales of new year greeting cards 15

8. Prove the following px=ex/1+ex+1 mx=ux+1/x 20
Define generalized least squares estimator. Obtain mean and sampling variance for the estimator . Show that a generalized least square estimator is equivalent to a tw o-step procedure- estim ator. 30
Describe the test-retest method for estimating the reli ability of a test and discuss its merits and demerits. 10

Note English version of the Instructions is printed on
the front cover of this question paper.