81. No. e-DTN-L-TUA

" . STATISTICS
Pap'et-I

ITime Allowed Three Hours I Marks 300 I
INSTRUCTIONS
Each ques.tion is priryied both in Hindi and in English.
Answers, must-be written in ,the medium

specified in the Aq.mission Certificatei$sued to you; which must be stated clearly on the coyer .of theanswer:book In the space for the -purpose. No marks _will be given for the answers written in a 17l.edium other than that specified in Admfssion

Candidates: should,attempt Question Nos. 1 any-three of t1}.e. remaining questions ,selecting at least one question,from each -.Section-.
The number of car,ried by each question is indicated,cit-.the' of the question.
Assume, suitable" data. if-considered' nec..essary and indicate'the same .cleafly. Notq.tions and used ar.e. as
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Section-A
1. Of the under-5'children in a locality, 20% are malnourished but' not immunised, while 40% are immunised but not malnourished.-Jf 30% are neither immunised nor malnourished, what percentage of the under-5 children the locality are malnourished? 12
For a discrete random 'variable X ... Find:the probability mass,function ofX. 12
(c)Suppose every packet of, the biscuit NiCE contains a cOuPon bearing one of the letters I C E with equal probability. A customer who has all the 'four letters, wins a free packet. A housewife buys 8 such packets. What is the probability that she wins one free 'packet and two packets? 12
(d)Scores of 10 candidates of Civil Service Main Examination of a recent year Statistics-Papers J and II are as under
Serial" No. of Candidates 1 3 4 5
Score in Statistics I 98 172 58 106 185
Score in Statistics II 69 123 65 120 131

Serial No. of Candidates 7 8 9
Score in Statistics I 89 140 168 102
Score in Statistics II 92 96 120 75

Use sign test to.verify the hypothesis that candidates' in the above examination offering both the Papers ofStatistics ,scored higher on an averagein Paper J than in Paper II. 12

Obtain a sufficient statistic for 0 insampling. from <img src='./qimages/1052-1e.jpg'>
A random,yariable the.probability density function <img src='./qimages/1052-2a.jpg'> Derive the expression' for its characteristic function.
(B)theorem relating to the. characteristic function of a continuous random varible use this theorem to obtain the probability density function of a random variable having the characteristic function
(c)the random-variable X follows-uniform distributions, let be the order· stastics in a randomom sample of size.3 from the population. show that both and 1 unbaised for 0.20

3. a random sample(X1,X2, ..,·Xn is taken,the ;population with the
probability .density function.
0>0.
0,otherwise
where k is constant to be, suitably chosen. Derive the maXimum likelihood' estimator of 0 Also show that.
s2=1/3nsigman i=1xi2
is an unbiased estimator of the population variance. 20
It is desired to test if average percentage of family inC0me ,spent on food is the same ir k selected' blocks· of a district. 1. Assuming the underlying poPulations to be·normal and homoscedastic,indicate how you.would use likelihood ratio test for the purpose. Show that the resultant test; boils down to the usual F-fest. 20
The joint probability density function of Y is given by
<img src='./qimages/1052-3c.jpg'>
compute and
also derive the regression equation of y on x.
4. For-the :sequence of random variables P(Xi 2i P(Xi and P (Xi i ... Check if the sequence obeys the weak law of large numbers and the central limit theorem. 20
Define Wald's SPRT of strength f3) for a ·simple hypothesIs against a simple alternative. Derive the· SPRT procedure,of strength (alpha, beta) for h0:mew=mew0 against HI: mew mew1' .based on successive observations from.. n(mew,sigma2),sigma2 known.Indicate how the test. can be carried. out graphically. 20
The manufacturers of a certain brand of electrical switches claim that at least 80% of the bOxes, containing 12 such switches each, will no defective. Design a randomised test of exact size 0'05 based "on, the number of boxes, among a random sample of,8-'eXamined, containing at least one defective. 20

Section-b
5. The p levels of a.factor A· are the only
levels the experimenter is interested in, while the q lvel s of the factor B chosen from a random sample. from the totality of levels.sigma critical regions. one.observation per cell Ai x B j is taken on the response variable. Write down the appropriate linear model, the hypotheses ·to:be. tested, the,test statistics and the level alpha critical regions, 12
Assuming the model
xi= m1+.eli
Yi m2 +e2i
zi= mi+m2+e3i,
where eli ,e2i and e3i are the errors, obtain least square estimates "of m1 and 12
days are.to .be .selected, from the 366 calendar days of :2012 for recording the-maximum daily ·temperature in a
metropolitan citY. (Using,the-following " set of random number's; select" these days, 'month, -and the for each, giving the procedure
in detail 12
6503 0085 3822 2193 5392
4635 0495 3296 1348
Suggest the cofounding scheme of a 2 4 -design to be carried out m 4 "replicates each comprising 'incomplete blocks of size such that full information IS :'retained' on the main effects, each of 'the first-order and second-order interaction is is confounded only In one replicate, while the third­order; interaction is confounded' in two replicates. Give the complete layout of one such replicate indicating factorial effects confounded in it. 12
Suppose, with uSual notation
r12 r13 0 .29
r14 -0·62 r23 =0·41
r24 -0·24 r34 =-0·52
For, predicting the variable x4 has already been included it t the' regression equation. Which" one of x2 .and x3 is
'worth 'including m the regression equation in to· x4i 12 ....
6. Give 'the outline or-the procedure, using ANOVA: technique, for testing the response variable y ·is dependent on a fIXed set of variables ... xk, based on an observed set of data on these variables. 20
(b)Determine the: sample size such that .estimated. value; based on this sample, the 802 level in air in 1006. petrol pumps in a large town, differs ,,from the" true value at most -SJ,Lg fu 3 With probability 0·95, assuming,the' standard deviation to be 35 mewg and 5%points of a standard normal! variate" are" 1·960 and 1-645 respectively.] .
(c)Introduce the concepts of confounding, total confounDing,. partial confounding and' balanced, confounding In the text of; a factorial design. A 3 2 -design-has conducted using 4 replicates each,comprising 3 incomplete blocks size 3 each_ In two of the replicates, AB has' beeI1 'confounded, :while' in 'remaining two,AB2 has been, confounding. Write down the ,ANOVA table:."indicate 'how the sums of squares due to different .factorial effects' will be calculated.
7.(a)Show that a necessary· and .sufficient '-condition for xp. to distributed as, that verY linear function Y is distributed . uni variate normal 1,segma1). 20
(b)the-total population. size: a district in ,2001., census ,is available. The .population sizes of randoJ;Il, sample of villages from' the district indiVidually in :both ,2001 and 2011 censuses are also available. Suggest a suitable estimator of the total population of the district-;in 11; based these,figures the. ratio of estimation. Derive an expression fOT the bias; of the estimator and deduce the condition under which the bias is negligible. 20
Construct a pair of orthogonal. Latin squares ·with symbols .C and y respectively..Super image one onto the other. 'Numbering. the small squares from 1 to_ 9i use-this arrangement construct, a. baced
.incomplete block design with parameters v b .r k and A=1. 20
8. ,Does the observed correlation between two variables and X2 always imply causal relationship between them? Give an example. Derme partial correlation coefficient r12.3 l,between ¥l arid X2 eliminating' from, each .the effect of X3 . Why is r12.3 mqre ,suitable.for t1?-rowing light on the net correlation between Xl f and X2? Based on the observed data set xij and j derive the expression for r12.3. 20
(b)derive the regression equation of on Xk, when Xl and jointly' 'follow trinominal distribution' 'with parameters p1 and p2 . tube lights. of a .certain brand are graded as 111,11, If or I according as they fail during the first 6000 hours and the next hours 'survive' the fJIst 12000 hours respectively. 12 such new tube lights are. put life-testing simultaneously. 50f them are found be grade III. many of "the remaining are expected be grade 20
(C).in a planned toWn, "there are N blocks each containing',M residential simple random, sample of n ;'blocks is drawn, and the numbers' of ,working women in all the residential buildings to these selected blocks are t counted. Based; on these figures,.... suggest an unbiased (to .shown) estimatoI:of the total.number of working women·in the town, and derive the standard error of the' estimator." 20