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Entrance Examination: M.Sc. Statistics - OR, 2017

Hall Ticket Number IIIIIIIII
Time : 2 hours Part A : 25 marks
Max. Marks. 100 Part B : 75 marks

Instructions

1. Write your Hall Ticket Number on the OMR Answer Sheet given to you. Also write the Hall Ticket Number
in the space provided above.

2. Answers are to be marked on the OMR answer sheet.

3. Please read the instructions carefully before marking your answers on the OMR answer sheet.

4. Hand over the OMR answer sheet after the examination.

5. There are plain sheets in the booklet for rough work, no additional sheets will be provided.

6. Calculators are not allowed.

7. There are a total of 50 questions in Part A and Part B together.

8. Each question in Part - A has only o~e correct option and there is negative marking of 0.33.

9. There is no negative marking in Part - B. Some questions have more than one correct option.
All the correct options have to be marked in the OMR answer sheet, otherwise zero marks will
be credited.

10. The appropriate answer(s) should be coloured with either a blue or a black ball point or a sketch pen. DO
NOT USE A PENCIL.

11. THF; MAXIMUM MARKS FOR THIS EXAMINATION IS 100

12. Given below are the meanings of some symbols that may have appeared in the question paper:
JR- The set of all real numbers,E(X)-Expected value of the random variable X,
V(X)-Variance of the random variable X, Cov.(X, Y)-Covariance of the random variables X and Y, PX,Y de-
notes the correlation coefficient between X and Y, iid-independent and identically distributed, pdf-probability
density function, B(n,p) and N(ll,cr 2 ) denote respectively,the Binomial and the Normal distributions with
the said parameters.Rank(A) and det(B) mean rank and determinant of the matrices A and B respectively.

13. This book contains 10 pages including this page and excluding pages for the rough work. Please check that
your paper has all the pages.
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Part - A

• Find the correct answer and mark it on the OMR sheet. Each correct answer gets
1 (one) mark and wrong answer gets -0.33 marks.

1. Bag1 contains 6 red, 5 blue and 4 green balls while Bag2 contains 6 green, 5 blue and 4 red balls, a
ball is drawn from each of the bags,' if every ball is equally likely to be drawn, the probability that
a ball of the same colour will be drawn from both the bags is

(A) a little more than 1/2.

(B) equal to 1/4.
(C) equal to 1/3.

(D) a little less than 1/3.

2. If every arrangement of 10 balls numbered 1,2, ... , 10 in a row is equally likely, the probability that
all the even numbers are before any odd number is
(A) less than 0.004. (B) between 0.005 and 0.007.
(C) between 0.008 and 0.01. (D) more than 0.011.

3. Arrange the numerals 1,2,3,4,5 in 8, row to get a 5 digit number, the number of such arrangement:)
that are divisible by 3 is
(A) ¥. (B) ~. (C) ¥. (D) 5!.

4. From a set of 100 distinct objects what is the number of different non-empty subsets containing an
even number of objects?
(C) 250 -- 1.

5. The probabilities of two events A and B, peA) and PCB) respectively are positive,further P(AI B) >
peA), then
(A) P(BIA) > PCB). (B) P(BIA) < PCB).
(C) P(BIA) = PCB). (D) Nothing can be said definitely about P(BIA).

6. A and B are mutually exclusive events and 0 < peA) < 1, 0 < PCB) < 1, so,
(A) peA) ~ P(BC). (B) peA n B) = P(A)P(B).
(C) P(AC) < PCB). (D) P(BIA) = PCB).

7. AI, A2, and A3 are independent events each of which occur with the same probability p, then,
the probability of at most one of AI, A2 and A3 occurring is
(A) 3p(1--p)2.

N-+
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8. X is a non-negative discrete random variable for which P(X = j) > P(X = j + 1), Vj = 0,1,2, ... ,
which of the following distributions fits this fact?

(A) Geometric distribution.

(B) Binomial distribution.
(C) Poisson distribution.
(D) Hypergeometric distribution.

9. The expected value of a random variable is -10 and its variance is 100, the value of the second
moment is
(A) 10. (B) 100. (C) 200. (D) 1000.

10. V '" B(10, 1/2), the probability that the quadratic equation x 2 + 6x + V = 0 will have complex
roots is
(A) zk. (B) 1- zk. (C) ~. (D) 1/10.

11. The probability distribution function of a Poisson random variable with a very large mean is to be
approximated as a

(A) Exponential random variable.

(B) Negative Binomial random variable.
(C) Normal random variable.
(D) Hypergeometric random variable.

12. The 4th head did not occur till the 15th toss of a coin, so,

(A) A Binomial random variable will be observed to be 15.

(B) A Negative Binomial random variable will take a value that is at least 15.
(C) A Binomial random variable will be observed to be more than 15.
(D) A Negative Binomial random variable will take a value that is more than 15.

13. In garment workshop, shirts are stitched for export, about 10% of the shirts stitched here do not
meet specifications and hence are ~alled defective, from a lot of 1000 shirts made in a day, the
number of defective shirts in a sample of 100 is a

(A) Negative Binomial random variable.

(B) Hypergeometric random variable.
(C) Binomial random variable.
(D) Poisson random variable.

N--=7
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14. The most appropriate diagram to represent data on grades achieved by students in a public exam is
(A) Bar charts. (B) Histogram.
(C) Stem and leaf plot. (D) Ogive.

15. \Vhich of the following is not a measure of dispersion

(A) Range. (B) I-.lean deviation about median.
(C) Mode. (D) Mean deviation about mean.

1 G. Tlw correlation coefficient PX,Y between the random variables X and Y is 0.8, then the correlation
coeffi(,ient between X and U = 20 .- 3.2Y is
(A) O.S. (B) -0.8. (C) 1. (D) O.

17 TIl(' corrPlat ion ('odlicicnt calcnlated based OIl 71 observations on the random variables X and Y
was· 0.:). which of the scatter plots given below reveals this correlation?

l l

18. The average of n positive numbers is 100 and their product is 100000000. thCIl,
(A) n = 2. (B)n=:3.
(C) II > 4. (D) w(~ can't say anything about n based Ull data given.

;!'r(r· 1) 0< .1' ::: 2

19. Consider the functioll f(.I:) -= (2)'"
{ dSt'whCTf

(A) It is a probability density function

(B) It is not a probability dcnsity function because j~: f(:r)d:r # I.

(C) It is not a probability density function because f(x) < 0 for some values of x.

(D) It is not a probability density function because f is not increasing in .1;.

20. The heights of adult males in a certain population are normally distributed, the heights of half of
them are more than 165cm., while the heights of 5% of them are more than 183cm., the percentctge
of adult males whose heights are less than 147cm.
(A) is less than 2%. (B) is 5%.
(C) is more than 2%. (D) we can't say based on data given.
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2l. Xl and X 2 arc independent standard normal random variables N(O, 1), xf + xi follows:

(A) Chi square distribution with 1 degree of freedom.

(B) Exponential distribution with mean 1

(C) Expollent.ial distribution with mean 2

(D) None of the above.

22. A salllple survey is to be done to estimate the average milk consumptioll per family in a city, to
draw the most representative sample it was decided to stratify the families and then select samples
from each stratulIl, on the b,u;is of which of the following criteria should the families be stratified.

(A) The last digit of the cdl phone uumber of the head of the family.

(B) ThE' height of the head of the family.

(C) TIll' mouth of birth of the head of the family.

(D) t he total illcome of the falllily.

2:). III a hypothesis testing problem of Ho against Ih bFl,sed on a sample. type - 2 error oectln:; Wl)(,ll

(A) the sample is such that it falls ill the complement of the critical region wlwn ill is true.

(B) fhp sample is such that it falls in the critical region when HIe Ho is true.

(C) tbc sample is such that it falls in the critical regioIl when fIn is falsc.
(D) the sample is sHch that it falls in the critical region when neither Jl o ))or Ih is true.

0 1:<0
21. TIl(' functioll F(:r) = lS
{ 1 .T > 0

(A) contiullolls it t all 1> E' JR..

(B) continuolls (-'wry where but not ditferentiablp at some points.

(C) decrea,sing in ;1:.

(D) not continuous at olle point.

')~
.... d. Which of the following is equivalent to the statement 'Ashok did not solve all the problems'

(A) Ashok did not solve any problem.

(B) Ashok did not solve at least one problem.

(C) Ashok solved at least one problem.

(D) Ashok solved at most one problem.
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Part - B

• Questions (26)-(37) have more than one correct option.

• For the answer to be right all the correct options have to be marked on the OMR
sheet.
• No credit will be given for partially correct answers.

• Questions (38)-(50) have only one correct option.

• Find the correct answers and mark them on the OMR sheet. Correct answers
(marked in OMR sheet) to a question get 3 marks and zero otherwise.

26. Which of the following are random experiments?

(A) Put paper in fire and see what happens to it.

(B) Ask a child to place 10 balls, 1 of which is red and the rest blue in a line and observe the
position of the red ball.
(C) Place 10 distinct objects in three distinct boxes and observe which object is in which box.

(D) From a large basket of mangoes take out 5 and report their total weight.

27. Xl rv U« -)3,)3]) and X 2 rv N(O, 1),so,

(A) P(XI > 0) = P(X2 ~ 0). (B) P( -1.5 < Xl ~ --1) = P(--3.5 < X 2 ::; --3).
(C) P(XI > 1) > P(X2 > 1). (D) E(Xl) = E(X2).

28. For two random variables X and Y, VeX + Y)

(A) is never less than either VeX) or V(Y).
(B) is never less than either VeX) or V(Y) if X and Yare uncorrelated or positively correlated.

(C) is always less than one of VeX) and V(Y).

(D) can be less than both VeX) and V(Y) only if X and Yare negatively correlated.

29. In a certain country, 70% of the households have incomes less than the average income, those
households among the highest 10% earners are considered upper class

(A) the median income is less than the average income.

(B) the median income is more than the average income.
(C) About a third of the households with more than average incomes are in the upper class.

(D) Less than a quarter of the households with more than the median income are in the upper
class.
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30. The probability distribution of a random variable X is P(X = j) = j(/+2) , j = 1,2, ... ,

(A) c is equal to 2/3.

(B) c is equal to 4/3.
(C) The expected value of X does not exist.
(D) The variance of X does not exist.

31. Identify the common properties of the random variables with the following pdfs

h(x) = {!o -1
e.w
<x< 1 fa(x) = ~e-Ixl, -00 < x < 00

(A) All of them have the same mean.

(B) The third moment of all of them is 0
(C) X and -X of all of them are identically distributed.
(D) All of the have the same variance.

32. regarding a simple random sample of size n without replacement from a population of size N, identify
the correct statements

(A) Sample raw moments are unbiased estimators of the corresponding raw moments of the popu-
lation.
(B) Every collection of n population units is equally likely to be the selected sample.
(C) The second central moment of the sample is not an unbiased estimator of the second central
moment of the population.
(D) Every unit of the population is equally likely to be in the selected sample.

33. The probability with which a coin shows heads upon tossing is p, the random variable Xl takes the
values 1 and 0 if the outcome of the "first toss is heads or tails respectively; another random variable
X2 is defined in the same way based on the second toss.

(A) Xl!X2 is an unbiased estimator of p.

(B) 2XI - X 2 is also an unbiased estimator of p, but not the most efficient.
(C) Xl - X2 is a sufficient statistic of p.
(D) Xl + X2 is a sufficient estimator, but X I - X 2 is not a sufficient statistic for p.

34. A, Band C are three events and if P(A) = P(B) = P( C) = 2/3, p(AnB) = p(AnC) = p(BnC) =
1/2, then peA n B n C)
(A) has to be zero. (B) can be 2/5.
(C) can not be 1/6. (D) can be 1/4.

N-~
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:{5. If X and Yare independtmt real valued random variables, then

(A) E(XY) = E(.X,")E(Y).

(B) VeX + Y) = VeX) + V(Y).

(C) V(XIY = y) = E(X) for every y E R
(D) 2X and-3Y are also independent random variables.

36. Consider the data given below on the students of a university

NlL1nbe7' of f email' students: 2()()()

Number of mall' st1Ld('nts: 4()()()

Number of /I'male stu.dents residing In hostels: 1600

Number of male students residing in hostels: :3000

from this da.ta one can see tha.t

(A) only Oll(, third of the students are female.

(B) only one third of the hostel residcuts are female.

(C) a larger proportiull of female students stay in hostels than malp students.

(D) lllore than half of the students are hostel residents.

:n. A is a G x 5 real ll1airix whose 5th ro\\' is the sum of the first and sccond rows, let AT d(mot(~ its
transpose, then certainly

(A) the rank of A is cqual to .1.

(B) the nUlk uf iff is at most 4.

(C) the determinant of AT A i" pqnal to O.

(D) the rank of-A is 1es" than 5.

38. X is a Poisson random variablp with mean A, if E((2X1 l)(X - 1)) =:c 0, then
(A) VeX) = 1/2. (B) A = l.
(C) A = 2. (D) A can not be uniquely determined.

39. The value of the integral 1°C :t

A e- 2x dx is equal to

(A) 15/4. (B) 5/2. (C) 3/4. (D) ]/2.

40. Let p be the probability that a coin will show heads upon tossing, further let X denote the number
of heads in n tosses of this coin, an unbiased estimator for p2
(A) is X2 . (B) does not exi"t.
(C) is :;&X2. (D) is n(L1)X(X - 1).
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41. Let rna, Ala and So denote the mean, median and standard deviation respectively of 15 distinct
numbers, fmther suppose the difference between the mpdian and the largest number smaller than it
is 5 and the difference between the median and the smallest number greater than it is 4. Now add
2 to each of the 5 smallest numbers in this list and subtract 2 from each of the larger 5 numbers.
denote by rn1, .Ml and 5, the mean, median and the standard deviation respectively of this new set
of numbers,

(A) nl, > rna; Ail = Aio; 8, > So

(B) Tnl = rna; M, -Aio; 8 1 < So

(C) rn1 < (110; lvI, < J'vJo: 8 1 < So

(D) rnl - rna:. Ah = Aio; 8 1 =--= So

42. A statistic 1~) to estimate a parameter e based on a random sample of size n of a certain random
variable is such thaL for 90% of the samples of size n of that randolll variable the value of Tn is mon~
thau e +- 10 a11(l for the remaiuillg 10% of the samples of size n , the vaille of Tn is equal to e. This
statistic ]~1 is

(A) a good estimator for e as it has low variallce.

(B) a good estimator for e as it seems to be unbiased

(C) is uot a good estimator for e m; it is lIlore likely to overestimate O.

(D) is llot a good estimator for e as it is more likely to underestimate e.

·13. Xl, .... Xn is a random sample from N(Ji. 0'2), an unbiased estimator for 0''2 + /1'2 is
(A) ~ 2::;'-1 (Xi - S)'2. (B) IIS'2 t 2::;'_1 xI (C) lLR2. (D) ~ 2:::'~1 X;'

-1-1. Let P be the probability of a coin showing up heads when tossed, the hypot.hesis flo : P = Po is t.o
l)c rcjected in favour of III : P > Po if the \lumber of heads-X that show up in 10 tosses of this coin
is aL least 8.

(A) the power of this test is equal to the size of this test for any Pl > Po·

(B) for some values of p, > Po, the power of this test is more than its size.

(C) the power of this test is more than its size for every PI > PO·

(D) the power of this test is less than its size for every PI > PO·
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45. The sum of squares and the products of every pair of n non-negative real numbers Xl, . .. , Xn are
known, however Xl, ... , Xn are not known, based on this information

(A) both the mean and the standard deviation of these numbers can be determined.
(B) neither the mean nor the standard deviation of these numbers can be determined.
(C) the mean can not be determined but the standard deviation of these numbers can be deter-
mined.
(D) the mean can be determined but the standard deviation of these numbers can not be deter-
mined.

46. There are 3 True or False questions in an exam, if a candidate knows the answer she/he answers it
correctly, otherwise a guess is made and the probability of getting it right is 1/2, an examiner assumes
that every candidate knows no answer, 1 answer, 2 answers, 3 answers with equal probabilities, a
candidate answered two of the three questions correctly, what is the probability that this candidate
knew the answer to only one of them?
(A) 1/11. (B) 2/11. (C) 3/11. (D) 4/11.

47. There are 2 red and 2 blue balls in a bag, balls are to be removed one by one, the probability that
the second ball to be drawn will be red is
(A) 2/3. (B) 1/2. (C) 1/3. (D) 1/4.

48. X is a non-constant real valued random variable whose expected value is less than 0, and PX,X2 = 0

(A) X and - X are independent.

(B) E(X3) > O.
(C) E(X3) > E(X)3.
(D) E(X3) < O.

49. The time to complete a one year project by an organization is a random variable with probability
cx2 (1 - x) 0 < X < 1
density function f(x) = , the probability that a project will get completed
{ o e.w
within 9 months(3/4 of a year) is
(A) less than 1/3 . (B) very close to 1/2. (C) almost 3/4. (D) almost 1.

50. 5/3,5/6,5/12,5/6,15/12 are 5 independent observations of a random variable X whose probability

A2xe-AX X ~ 0
density function is fx(x) = , A > 0, the maximum likelihood estimate of A
{ O O.w
based on the given sample
(A) 2. (B) 2.5. (C) 3.75. (D) 5.