CS MAINS STATISTICS PAPER II
SECTION
1.(a)Describe a CUSUM control chart..Compare ,this chart' with Shewhart chart with respect to performance. ,How is a V-mask useful in CUSUM charts?.
1.(b)Let ...} be a Markov chain on the state with' initial probability values qc and one step transition probability. matrix given by.. <img src='./qimages/299-1-b-1.jpg'> Compute <img src='./qimages/299-1-b-2.jpg'>
Discuss the estimation of parameter of exponential distribution in case of time censoring. 10
To a single serVer queuing system, the arrivals of customers are according to a Poisson process at the rate of 2 customers per hour. The service time has exponential distribution with mean service time of 12 minutes per customer. The steady state probability distribution (pJ. for the number of
customers is given below:
n 0 1 2 3 5 6 and above
Pn 0·0& 0·1 OA 0·2 0·2 0·02 0
Compute
the expected number of customers in the system and in the queue.
the mean waiting time in the, system and in the queue. 10
Describe a two-person zero-sum game and explain mini-max, max-mini and net values of a game.10
Explain the concept of sequential sampling plan for attributes and obtain the' average sampling number function of a sequential sampling plan for testing the hypothesis
Ho: beta =beta0 against the alternative hypothesis.
HI beta beta1 in sampling from a population with density function beta). 15
State and prove the necessary and sufficient condition for the existence of a saddle point in a twoperson zero-sum game.. Find the value of the game <img src='./qimages/299-2-b.jpg'>
Define' and explain the following terms: Reliability function Reliability of a series system 'Reliability of a parallel system The mean life of a component is 100 hours. If you want to build a parallel system having a mean life of 200 hours, how many components would be required considering a constant hazard model? 15
For an queuing system, derive the set of steady-state difference equations and hence find the solution of the system. In a supermarket, the average. arrival rate of customers is 5 every 30 minutes. The average time it takes to list and calculate the customers purchases at the cash desk is 4·5 minutes and this time is exponentially distributed.
How long will the customers expect to wait for service at the cash desk
What is the probability that the cashier is working?
Describe the principles on which a control chart is based. What is the concept behind the u,se
3-a limits in statistical quality control?
Control on measurements of pitch diameter of threads in aircraft, fittings is checked with
5 successive items measured at reglJlar intervals. 8 such samples are given below:
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(Values are expressed in units of 0·0001 inch)
Sample Measurement on each item of5 items per hour
I 46 45 44 43 42
2 41 41 45 42 41
3 40 40 43 42 . 40
4 42 43 43 42 .45
5 42 44 47 47 45
6 39 46 45 42 41
46 44 41 40 45.
8 40 45 43 40 39
Construct the X R chart and 'comment on. the process control.. 20
3.(c)Discuss the inventory model with instantaneous stochastic demand with continuous replenishment unit and obtain the condition for finding the optimum inventory level., 15
4.(a)What do you understand by the term "Sensitivity Analysis" Discuss the effect of variation in the resource vector b in the linear programming problem with set of constraints AX b. . In the problem
Maximize .Z -XI 2x2-x3)
subject to 3xI +x2
-xI 4x2 +x3
x2 +x3 4
0
discuss the effect of discrete change in the component bz so as to maintain the ,optimality of the current' optimum solution. 20
4.(b)Explain !he terms.
Acceptance quality level
'LTPD
Consumers' risk
Producers' ·risk
OC curve 10
4.(c)Suppose that the purchaser and the ender agree. that a defective rate of 10% in a 19t is satisfactory. For the sampling plan n c determine the probability that a lot with this per cent (i.e. defective, will be accepted. Also determine the said probability if the sampling plan happens to be N 100, n C 1. Which of these two,sampling pains seems better and why? 20

SECTION
5.(a)Let sigma1, sigma1+1' ... and sigma be independent variables with zero mean .and unit variance. Considering u1 a sigma sigma1,-infinity, show that the process is stationary with correlogram <img src='./qimages/299-5-a.jpg'>
5.(b)what is validity of Test Scores? J:low you will calculate validity? Also make comparison with the concept of reliability. 10
5.(c)Define index· numbers. Discuss their uses and limitations. 10
5.(d)Describe a method of fitting trend by logistic curve.
Why the gross reproduction rate and net reproduction rate are considered as refined measures of fertility? .
Assuming that the ratio of female babies to. total birth is 48·8 per cent, compute the gross reproduction rate for the following data: 10
Age group 16-20 21-25 26-30 31-35 36-40 41-45 46-50
Fertility rate per thousand women 19 173 253 201 157 67 9
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Fill in the blanks in· a portion of the life table given below: 20
Age in years lx dx qx Px Lx Tx E2x;
4 95000 500 4850300
5 400
6.(b)Following are two sets of indices one with 1960 as base and the other as 1970 as base:
Year 1960 1961 1966. 1968 1970 1971 1972 1973
Index 100 110 115 125 150 ... ... ...
Index ... ... ... ... 100 105 120 130
Splice the new series ·to old series, so as to. have a continuous series from 1960. Also splice the old series to new series so as to have a continuous series from 1960. 15
Given below is the data regarding deaths in two districts. On the basis of the given data, calculate the standardised death rates. Give your comments. . 15
Age Range Population Number of deaths Age Distribution of DistrictA DistrictB District A District B Standard 1000
0-10 2000 1000 50 20 206
10-55 7000 3000 75 30 583
55 and above 1000 2000 . 25 40 211
For estimating simultaneous linear system of equatiors, how would you use the following methods of estimation and why?
Indirect least squares method:
Two-stages least squares .(2 SLS) method. . 20
What is meant by intelligence quotient (I.Q.) Describe type procedure and tests for measuring ·it. 15
Elucidate ARlMA representation of.a time series. For an autoregressive scheme x2+1 aXt b sigmat+1 show that the correlogram is given by 7k ak, where k is the order of the serial correlation. Also, show that the correlogram isdeqeasing exponentially for 0 a 1. 15
Explain the time reversal and factor reversal tests of index numbers. Examine whether Laspeyre's and Paasche's index numbers satisfy these tests. 15
·8.(b) What do you mean by stable and quasi-stable populations? Stating the ·basic assumptions of stable population theory, derive three basic equations which provide information about the intrinsic growth .. rate, birth rate and age distribution of the population. 20
Explain the terms
summation score as a function of ability
percentile scores
Z-score.
Describe how T-scores are to be found we are'given test scores fn the form of a frequency table. 15
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