BHU PET 2012 M.sc.Statistics

8/12/2019 BHU PET 2012 M.sc.Statistics

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12P/221/31 Question Booklet No .......l44 .To be filled up y the candidate y bluel black ball point pen

Roll No. LI_- ---- _...I.._- ---- _...I..I_- ---Roll No (Write the digits in words) , _ .....................................................................................Serial No of Answer Sheet ..............................................Day and Date ( Signature of Invigilator )

INSTRUCTIONS TO CANDIDATESse only blue/black ball-point pen in the space above and on both sides of the Answer Sheet)

1. Within 10 minutes of the issue of the Question Booklet, check the Question Booklet to ensure thatit contains all the pages in correct sequence and that no page/question is missing. In case of faultyQuestion Booklet bring it to the notice of the Superintendent/Invigilators immediately to obtain afresh Question Booklet.

2. Do not bring any loose paper, written or blank, inside the Examination Hall except the Admit Cardwithout its envelope.

3 A separate Answer Sheet is given. It should not be folded or mutilated. A second Answer Sheet shallnot be provided. Only the Answer Sheet will be evaluated.

4. Write your Roll Number and Serial Number of he Answer Sheet by pen in the space provided above.5. On the front page of the Answer Sheet, write by pen your Roll Number in the space providedat the top, and by darkening the circles at the bottom.. Also, wherever applicable, write the

Question Booklet Number and the Set Number in appropriate places.6. No overwriting is allowed in the entries of Roll No., Question Booklet No. and Set No. (if any) onOMR sheet and also Roll No. and OMR Sheet No. on the Question Booklet.7. Any change in the aforesaid entries is to be verified by the invigilator, otherwise it will be taken as

unfair means.8. Each question in this Booklet is followed by four alternative answers. For each question you are torecord the cOlTect option on the Answer Sheet by darkening the appropriatecircle in the corresponding

row of the Answer Sheet, by ball-point pen as mentioned in the guidelines given on the first pageof the Answer Sheet.

g For each question, darken only one circle on the Answer Sheet. If you darken more than one circleor darken a circle partially, the answer will be treated as incorrect.

10. :Vote that the answer once filled in ink cannot be changed. If you do not wish to attempt a question,leave all the circles in the corresponding row blank such question will be awarded zero mark).11. For rough work, ust : the inner back page of the title cover and the blank page at the end of thisBooklet.

12. Deposit only the OMR Answer Sheet at the end of the Test.13. You arc not permitted to leave the Examination Hall until the end of the Test.14. f a candidate attempts to use any form of unfair means, he/she shall be liable to such punishment

as the University may determine and impose on him/her.[No of Printed Pages: 54+2

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No. of Questions/"lIV'ii


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(I)3 ) _ ~

3 Variance of first n natural numbers is

I) (n;lf-1(3) 6

4) n n + 11I2n + I)6

4 f the geometric mean of positive numbers Xl X 2 ' , X is G the geometric mean of2Xl 22X2 2nXn is

1 ft\ """,3if X X , ,X n fiT 1.fl


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6 If the variance of XI_ X2. ', 10 is 10 and i = SXI + 4, i = 1. 2, ' , 10, then variance ofYl ,Y2, ' ,Y lO IS

x I , x2 , x lO CfiT '5fmUT 10 am Yi=5xi+4, i=1,2, , lO, i t Yl.Y2, ,YlO CfiTml (U] tiMr(I) 54 (2) 50 3) 250 (4) 254

7. Second and third central moments of a distribution are equal. What is the nature of thedistribution?(1) Symmetric (2) Asymmetric(3) Positively skewed (4) Negatively skewed

1) wrflrn 2 ) ~8. Four years ago, the average age of a family of four persons was 18 years. During this

period, a baby was born. Today, if the average age of family is still 18 years, what is theage of baby?1) 20 yearsIR ' 'f '['l ml\ ' fum it; 'lli\


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10. A man having to drive 90 km wishes to achieve an average speed ,of 30 lan/hr. For thefirst half he averages only 20 lan/hr. His average speed in the second half of thejourney in order to achieve the desired average should be1('" -..fu; fom 90 km t, 30 km/hr ffiI ir.fl I) 40 km/hr (2) 45 km/hr (3) 50 km/hr (4) 60 km/hr

11. The limiting form of a histogram when class intervals are made very small isI) frequency- polygon 2) frequency curve

(3) oglVe (4) pie diagram'If ; "" """"" m?: 'IR IT'i, til 3 Tlffi m "" l " oT


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' lH lln " fIii1< y iii ofR ""11,4" '- ~2) Y q-( x


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lI1llT"l ~ i , . , if ( ~ ~ x _r2 "" 1P' II l R ~( I)


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26. The probability of two persons being born on the same day is

1( 1 ) - -49 2) 1I 365 (3) 17 (4) ~727. An unbiased coin with faces marked 1 and 2 is tossed two times. Let X be the number

obtained in the first toss and'Y be the maximum of the two numbers obtained. ThenP X o Y j is

' I f_ fu1 i1, '" 1 'IT 2 fu1m t,


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30. Let x be a continuous random variable with distribution function x (.) DefineZ = Fx{x . Then var {Z isl HT x ' WI< . , r ; ; , , h "i'f{ t f.mq;, oir- 'liml"T -. \ 11(I) (2) ~ (3) ~ (4) I2 12 13

31. Let P A ) ~ P I , P B ) ~ P 2 and P A B ) ~ P 3 . Then P B/A) isl HT ill; P A) PI, P B) ~ P2 am P AB) ~ P3 I f\ ' l" i j ' O l ~ r ; ; , , h it fu'I1 1 111 "lilI'h"1 ill; it wftq -. \ 11 " ill;' ftfu it Wfi(I) 2 (2) 3 (3) 4 (4) 5

33. et X be a random variable with.c.d.f. F(x)=l_e-J..x, O


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35. The cumulative distribution function of any random variable is1. always right continuousII. right discontinuous at countable number of pointsIII. monotone non-decreasingSelect the correct answer from the following :(l) None of the above three statements is always true(2) I and III are true, but II is false(3) U and III are true, but I is false(4) All the above three statements are true when the LV. is discreteM , ~ o ; ' 'fiT m '1iffi ;jRrr ~I. >$ ""'LII. qffi ~ 3 i l 'R *""\

f.i ' i fig rl >It\ OW[ I I m)m"$if q;,ffi ~ i\ >$ "'" 'ltl ~[21 I


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39. If X and Y are two random variables and a and b are constants; then coy (X +a, Y + b)will beII) ab coy IX, y) (2) COy IX, Y) +ab(3) COy X, Y) (4) None of the above

II) ab coy IX, Y) (2) COy X, Y) +ab(3) COy X, Y)

40. Starting from the origin, unit steps are taken to the right with probability p and to theleft with probability q ::: 1 - p). Assuming independent movements, the expectation ofthe distance moved from the origin after steps is'[


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42 If X - X. I and Y - X are independent r v. s, then the distribution of the variate (X Yis

3) X 2 with n, + n2) d.f. 4) All of the above

(2) (n2 ,n22(3) X 2, n, +n2) ' ' ' itfl (4) ;m)u; > >fi

43. Let6f(x)=,, x = l, 2 3, ...x x

= 0 elsewhere' 6exand M t) = L ~ Then which one of the following is correct?

x=llt

I) f ix is a p.rn.f. but M t) is not a rn.g.f.2) f ix is a p.rn.f. and M t) is a rn.g.f.3) f ix is not a p.rn.f. but M t) is a m.g.f.4) f ix is not a p.rn.f. and M t) is not a rn.g.f.

(347) 14

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44

6f ( x ) ~ - , - - - - , - x L 2, 3, ...n x= o.

6 xM t) L:, , tl mM if 1I -.iR-m "'" i?x ,}1t

I) f (x) "'" 1110",0"'0 t M t) ' ano ,o ,o i2) f ix ) "'" 1110 , ' t


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46. Let X be.a standard normal variate. Then Pr (X >196) isl Rl f.fi X 1 i'fi l fI ' I ' I; ' ~ I , ' R iI Pr IX > 196)

I) a 2) 0 025 (3) 005 (4) 0 9547 f random variable has exponential distribution wit parameter e then

PIX>a b/X>aj is .

(I) PIX>b j (2) PIX aj(3) PIX >a+bj (4) I -P IX >aj

48. The probability mass function or the negative binomial distribution wit parameters rand p is

4) All of the above

(4) aWm ll ft

(347) 16

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49 For an exponential distribution with probability density functiont m r ~ snfil fldl

f{xl",, e-:


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2Pf22 f3

52. If is a standard normal variate, then X 2 is a gamma variate with parameters2< Ii; X ' lTR'fi ' i l l ' ' ' t ill 2. X 2 ' IT ll. t ftm liT -.Frr2(1) I, 2.2 (2) 1 (3) 2. 2.2 2 (4) 1, 1

53. f X F m, n , the variable m X is distributed asn

< Ii; X - F (m, n), ill m X q;] -.Frrn(I) P m, n) 4) PI (m, n)

54. For the geometric distribution [X = x 1 _1_. x = 1 2, ', the Chebyshev's inequality isx

I(I) P( IX-21>2] 2]< 2.2

1(2) PI IX-21>1]1]


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59. A sample of 16 items from an infinite population having SD 4, yielded total "scores as160. The standard error of sampling distribution of mean is' ""'" m. 1 F


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m-.i ilf; a om P ...... i t ' JIiIiR


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64 Let'(X\X 2 ) be a random sample from a gamma distribution 0 1,8). Then for testingH G : = 1 against HI : 8 = 2, a critical region'lH1 Ii ; WI m GILS) J.lo from N ~ a 2 ) when the population SD is known, theappropriate test is(I) (-test (2) Z -test (4) None of the aboveN (f, a 2) it H,: > o ' H 0 : fL 0 fLO ' WIW T ' fu1( WIW T


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_ fi ; wrll f i x , 8) i1 n ' ~ .,,R"" 'R 3l1 I1fur . R , , ~ , i ~ T t 3i'r< EIT) =01


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N (I'. ,,2) >I ,,2 >m t -.;0; .,I , ,-llT . I i i ~ . f , ~ ?(1) I 3it( II 12) I, II 3it( III13) III 3it( IV (4) I, H, III 3it( IV

70. sufficient condition for Tn to be consistent for f is1) E Tnj----)B s n t O J2) V T I l ) ~ 7 0 as n co

3) E Tn) - 0 or Til) t a as n - 0 00(4) l> Tn) --> 8 and V Tn) --> 0 as n --> 00

I) E(Tn )--> 8 ' n--> 00

(3) E(Tn) --> a V Tn) --> 0 ' n --> 00

(347) 24

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71 For a frequency distribution, a two-parameter normal distribution was fitted. Theobserved and expected frequencies in the various classes are given below :1('n 'I ''''' m ~ ~ 1('n -.r.l ~ ' ' ' ' ' m qi\ 'l'lT1 -.>if ifltf 3it< " " " ~ ' d ' d ' ~ -;fti\ fbserved frequency 6 14 20 19 16 6

f t f ~ ilH1=4lRdlExpected frequency 4 15 21 21 15 4

J l ~ ~ l a ilHJ:ilHfllThen to test the goodness of fit using the X '2 -statistic, what is/are the degree(s) offreedom of the X2 -statistic?X 2 " f d " r ~ m ' t l ~ -.i.i ~ X ' - " f d Q f ~


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73. What is the maximum likelihood estimator of p based on a single observation X fromBernoulli distribution with parameter E ~ J?1ITifeI P E [ ~ Iffi m i t , It..,ur X R :mmIt p W ITflrn, -w 7

2X +12) 7 3) X 17 (4) X7

74. Let X I X2 ' , xn be a random sample from f{x, eJ= e - X-B); x e 0< e< co and zerootherwise. Then a sufficient statistic for 9 is

75.

(347)

1) Max (xl,X2, ' ,XI' lJ3) LX (4) IT Xi

'tl RT fcfi (x ,8)=e-(x-e); x>9, 0


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76 If X is a sample mean from the binoIhial distribution b 1, p , then1) x is a sufficient statistics for p(2) x is an efficient statistics for p3) both I) and 2)4) None of 1) and 2)

3) 1) 31 R (2)


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79. Let X 1X 2 be a random sample from N{8,I . Then for testing Ho :8=9 againstHI: e >8 0 which of the following is UMP critical region?lIT' ill; NI8,1 " (X IX 2 ) ' ~ t HI :8>8 0 ~ il< H o : 8 = 8 o ~'ll\l\I"I ~ M -.;ifuq; i\ lIT UMP oTm?

80, Let'lFITilI;

X 1 =Jl+ElX = 2j.J. +E:2

Eland E2 are independent with same variance a 2 Then BLUE of l is

(2) X l +2X 25

81. f X is a binomial variate with parameters (5, 8), the UMVUE for '1'(8) =8 (1-8) is

(347)

.m; M ' X fuliI; " T"ffi (5,8) t i\ '1'(8) =8 (1-8) lIT W T'rn: l IWl 3FIh(UMVUE) oTm

(2) X2 -5X) /20(3) X (I-X)/20 (4) XIX -1 /20

29 P.T.D.)

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82 If Xl, X2. ', Xn are the values of random sample from a normal population N (Il, 0'2),then S2 =_1_2: x, _x 2 is a/an

n -I(1) unbiased est imator of t'f22) sufficient statistics for 0 2

(3) mean squared error consistent estimator of 0 24) All of the abovelfi; Xl, X2 ', Xn. SI< ;'II"'I 1 ' 1 Wlfu N ( ~ , a )


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,,;r.,--.1\ . I i i ~ . , 1hr.rr ~ .Iii,, ~ >rll t?I)


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88 In SRSWOR, probability of a particular set of n units selected as the sample, is (whereN is the population size)SRSWOR i j -3 Jl11'ft mr.


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'i'fi-I lI'IT 'i'fi-II t l


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91 If population size N := 100, sample size n = 12, then the ratio of variances of samplemean in SRSWR and SRSWOR is'If ; m N ~ 1 0 0 n ~ 1 2 m,


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95. or a n -factorial experiment in r replicates the. sum of square for the effect A in theANOVA table is

[A[(I) 8r[A](2) 2r

[A](3) 4r[A](4) 6r

96. In a randomised block design with 5 blocks 6 plots each if MSB 10 MST 15 TotalSS 140, ~ h n MS is"""


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99. In a 2 3-factorial experiment, th treatment effect

100.

11.' 2 3 ;;0 ,,41 3lfmeiM i\ 0'RlR 'is due to

1) AB

. :. [ abc) + ab) + c) + (1) be) b) ac) a) J

(2) AC 3) BC (4) A

Let A B C D be four treatments then which one of the following can be considered aslayout of LSD?_ flI; A. B. C. D ;m q;wo t it f. R if it ..:t. 11.' q>f ,,31m _ "l1' ' 7

'A B C D A B C DIc D A B B A C D( I) (2)jE C D A C A B DIiDc A B C D B A CrA C B D D B C AD A C B B C A D(3) D B A C (4) C D B ClA D B A A B C D

101. Ifthe degree of freedqrn for error SS in a LSD is 30, then the order ofthe design is

(347)

mit q>f i\ ': q>f m t """'" W 30 t, it ' 3lT< iR(I) 4x 4 (2) 5x 5 (3) 6 x 6 4) 7x7

37 P.T.O.)

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102. Match List I with List II and select the correct answer using the code given below thelists :

347)

List IA Replication is usedB Randomisation is usedC, Randomisation andreplication are usedD. Replication and local

control are usedCode

BC DI) (d) (e) (b) (a)2) (e) d) (a) b)

(3) (d) (e) (a) b)(4) (e) (d) (b) (a)

List I I{al For validity of estimate of errorb) For diminution of errorel To achieve the independence of errord) To estimate the experimental error

f ll-I q;) f ll-u I ,,;n f ll 'iii WI 'l1l. F 'fiT m -.;< 00< :~ I ~ I I

A. 'r m if 31"Tffi a) ~ 'iii


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103. For flxed effect model

Y i j = J . 1 + t i + ~ j + e y , i=l 2 .u; j=l,2, .bwhat is the linear unbiased estimate of t 1 - t 4 ?

(I) - -Y Y.n 2) iiI -ii.. 3) Y (4) - -Yl . Y104. The probability of rejecting a lot having as the process average defectives is known as

(1) consumer s risk (2) type II error3) producer s risk (4) All of the above

"-" BTl f.mIlIil l 11 Wl P -.i\ 31\41 .. -.;f.\ fil ,ill..d' ;;n;f\ omit t(I) oq>iImT (2) ~ I l I ; R fil Wl

(4) ;;Wu; Pol regarding bip.omial proportion,consider the following values of the OC-function L {pI :

(347)

i\; 3 ~ " " ' ' ' ,ill..d, 3f:l' 1d ' I i l ~ SPRT i\; it ~ a " , ~ q ' d i\; it Ho: P ~ PoHI: P ~ P I > Po iI; ' I i l ~ -.;f.\ iI; (OC) ' L p) iI;

R",Rlf@d l f.i\ ' It flr


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106 The graph of the proportion of defectives in the lot against average sample number is1) ~ C c u r v e (2) ASN curve (3) Power curve 4) All of the above

2) ASN 'Iji; 3) Iji; (4) iTffiT am 3 , ' G ' l , , ~ f 4)


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p i\; ~ " ' : l q ' d i\; fero. 3-funn ~(1) U e L ~ p , + ) 3 ~ q ' , e L ~ p '


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115 In case of multiplicative model the sum of seasonal- indices is(1) 100 times the number of seasons(2) zero(3) 100(4) 4002 * 1lflw- < i\ G>T1 ii .;16 1. liT .iPT

(2) PI(3) 100 (4) 400

116 f the index number of 1990 to the base 1980 is 250 the index number for 1980 to thebase 1990 is

1980 311"" '1


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123 If P j and P2 are the population at two census conducted at an interval of five years,then formula for the growth rate of population isl1l; lfu ' if; 'R ~ " " H , a i \ il J lll ~ , < i , 1 P, 1('i P, m.


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126. The death rate of babies under one month is - mown as(1) neonatal mortality rate(3) maternal mortality rate

'(2) infant mortality rate(4) foetal death rate

.."" fur( 'Rl f


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130 The passenger and the trrun in queueing system are(1) customer and server(3) both server'IT'fi 3itt f;


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133 GivenMin -3xl 2 X 2subject to

XI-X2?>:1 Yl-3x\ 2x2 2 : -2, Y2Xl 2 : 0, X unrestricted in sign

The optimal solution isI) XI 010, x 0-11 2) XI 01, x202

4) X l o x 20 - 1

Xl - x2 '2': 1, Yl-3x l 2x2 ; ; -2, Y2XI 0 , x2 Fro it '

134 The component useful for long-term forecasting is(I) trend (2) seasonal 3) cyclical

(I) (2) q iolft. 3) ..tllR.fiI,

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135. The component useful for short- term forecasting is(I) cyclical (2) trend

I) (2) JI< ftr136. The missing value for the following data

xy

52

105

IS?

208

by me binomial expansion method is' JmRUl AA1:IU. WI 1 H -.T>n.(I) 7 (2) -7

3) seasonal 4) irregular

(4) ",f.I"liId

(3) 3 (4) 25/3137. f Y ~ f x ) and the values of f ix for given x are f I ) ~ 1 4 . J 1 2 ) ~ I 2 , f 5 ) ~ 6 and

n 8 1 ~ 2 1 , then f 7) isY ~ f x ) am x i\; fu1( f ix i\; ir ~ f l l ) ~ 1 4 . J 2 ) ~ 1 2 . J 1 5 ) ~ 6 am f 8 ) ~ 2 1 , it

flT -.T>n(I) 2 2) 12 3) 8 4) 10

138. f z is constant then x maY:be

347)

(I) constant3) Both I) and 2)

~ y z 3 ffi to it x -.T>n1) _31 11 am (21 *il

2) at equal intervals4) None of the above

(2) WIR ''''''


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139. The relationship between u of Stirling s formula and v in Bessel s formula forinterpolation is

(I) u=v+1 2) u = v - l I(3) u = v - -2I4) u = v -2

I.q-O. If the temperature of three dates of June, 1994 were as follows~ ~ 1994


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143 Given Yo, Yl. Y2 Y3 corresponding to values xo, XI' x 2 X3 for functionY - = [ xl. a :;: x ;; b. Let f ix) is a polynomial of degree 3. Then by Simpson s three-eightrule J J f i x dx i

31) J ~ h I Y o +3y, +3Y2 +Y3]8

13) J=-h I2yo +4Yl +2Y213

144. The value of bon (ax n + bx n:- 1 + c) is

1) n 2) an

(4) None of the above

12) J=-h IYo +4Yl +Y2]3

3) (n-1) 4) b(n - l )

145 Ii the observed values of x and function U x arex 2

1986

1508

102The interpolating function U)( IS

993

1) x 3 -4 x 2 +80x+ l02 2) x 3 18x 2 +80x+294{3J x 3 -18x 2 +80x+102 {4} None of the above

347) 51 P. T. D.)

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x 2198

6150

1) x 3 _4x 80x l023) x -18x 80x 102

8102

993

2) x 3 -18x 80x 294

146. In Picard s method given initial value problem

347)

nth approximation

y dy f ix, y) with ylxo ~ Y odx

1) Yn = y o + I x f {x ,xn dxx 2) Yn Yo + IX f ix, y) dxx(3) Yn Yo + IX f ix, Yn Ji dxx 4) None of the above

y dy f ix, y) with ylxo Yodx

i2I Yn Yo + J f ix, y)dx(3) Yn ~ Y o I x flx,Yn_Jidxx

52

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147 Parabolic method Of estimation is good for(1) interpolation(2) extrapolation(3) interpolation as well as extrapolation(4) None of the above


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1 ..if


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1. *' flI&.\ iii 10 fIR?: iii "'R" ~ ~ 1 l I # f< ; if I1 l1; ~ 3i'n: q ; ] ~ *'W1 ;@ ~ I ~ -.rf.\ 'R ' ' 'li1a-f.rttPJ' > q;] ~ '1'r>

'lR1 ' '8. --: fum" """"..., "0; .. .. ' ~ '1', JrFf


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