PUMDET 2023 Question Paper Statistics

Subject : STATISTICS

(Booklet Number)

Duration : 90 Minutes No. of Questions : 50 Full Marks : 100

INSTRUCTIONS
1. All questions are of objective type having four answer options for each. Only one option is
correct. Correct answer will carry full marks 2. In case of incorrect answer or any
combination of more than one answer, ½ mark will be deducted.
2. Questions must be answered on OMR sheet by darkening the appropriate bubble marked
A, B, C or D.
3. Use only Black/Blue ink ball point pen to mark the answer by complete filling up of the
respective bubbles.
4. Mark the answers only in the space provided. Do not make any stray mark on the OMR.
5. Write question booklet number and your roll number carefully in the specified locations of
the OMR Sheet. Also fill appropriate bubbles.
6. Write your name (in block letter), name of the examination centre and put your signature (as
is appeared in Admit Card) in appropriate boxes in the OMR Sheet.
7. The OMR Sheet is liable to become invalid if there is any mistake in filling the correct
bubbles for question booklet number/roll number or if there is any discrepancy in the
name/signature of the candidate, name of the examination centre. The OMR Sheet may also
become invalid due to folding or putting stray marks on it or any damage to it. The
consequence of such invalidation due to incorrect marking or careless handling by the
candidate will be sole responsibility of candidate.
8. Candidates are not allowed to carry any written or printed material, calculator, pen, docu-
pen, log table, wristwatch, any communication device like mobile phones, bluetooth devices
etc. inside the examination hall. Any candidate found with such prohibited items will be
reported against and his/her candidature will be summarily cancelled.
9. Rough work must be done on the question booklet itself. Additional blank pages are given in
the question booklet for rough work.
10. Hand over the OMR Sheet to the invigilator before leaving the Examination Hall.
11. Candidates are allowed to take the Question Booklet after examination is over.

Signature of the Candidate : ______________________________
(as in Admit Card)

Signature of the Invigilator : ______________________________

Statistics 

SPACE FOR ROUGH WORK

Statistics 2 

1. Consider the set A  [1,3)  [2, 4) and two real numbers a = 2, b = 1.
Which of the following statements is correct ?
(A) b is an interior point of A but not a
(B) a is an interior point of A but not b
(C) Both a and b are interior points of A
(D) None of these two points are interior points of A

2. The series

1
 n 2
n 1

(A) converges to 2 (B) converges to 1
(C) is divergent (D) converges to 2

3. The value of
2 2 2 2
xyzw
1 1 1 1 x  y  z  w dx dy dz dw
is
1
(A) 1 (B)
3
1 1
(C) (D)
4 2

4. For t > 0, the value of the integral


e
 tx 2
dx
0

is
2 
(A) (B)
t 2t
 1 
(C) (D)
t 2 t

5. For every positive integer n ≥ 1, define dn = number of divisors / factors of n, e.g., for
n = 8, there are 4 factors, namely, 1, 2, 4, 8 so d8 = 4. Then the radius of convergence of the


power series  d x is
n
n
n 0

(A) 0 (B) 1
(C) 2 (D) infinity

Statistics 3 

6. Let U1, U2, ………. be i. i. d. Uniform (0, 1) random variables.
 3 
Then lim P  U1  U 2  .......  U n  n 
n 
 4 
(A) does not exist (B) exists and equals 1
1
(C) exists and equals (D) exists and equals 0
2
7. Suppose X n : n  1 is a sequence of independent Uniform (0, 1) random variables. Let
Yn  min  X1 ,......, X n  . Then nYn converges in distribution to
(A) a N(0, 1) random variable
(B) a N(0, 2) random variable
(C) an exponential random variable with mean 1
(D) an exponential random variable with mean 2

8. Let X be a random variable which is symmetric about zero. Let F be the cumulative
distribution function of X. Which of the following statements is always true ?
(A) F(x) + F(–x) = 1 x 
(B) F(x) + F(–x) = 0 x 
(C) F(x) + F(–x) = 1 + P[X = x] x 
(D) F(x) + F(–x) = 1 – P[X = – x] x 

9. From 10 pairs of shoes in a cupboard, 4 shoes are chosen at random. The probability of
selecting at least one of the pairs is
97 95
(A) (B)
323 323
224 99
(C) (D)
323 323

10. For any two events A and B, define   P  A  Bc  ,   P  A  B  and   P  A c  . Then
which of the following relations always holds ?
1
(A) 2 + 2 + 2 = 1 (B) 2 + 2 + 2 <
3
1 1
(C) 2 + 2 + 2 = (D) 2 + 2 + 2 >
3 3

11. Suppose X has an exponential density with mean θ > 0. Define Y as follows :
Y = k if k ≤ X < k + 1, k = 0, 1, 2, …….. Then the distribution of Y is
(A) binomial (B) Poisson
(C) geometric (D) hypergeometric

Statistics 4 

12. Consider the quadratic equation x2 + 2Ux + V = 0 where U and V are chosen
independently and randomly from the set {1, 2, 3} with equal probabilities. Then the
probability that the equation has both roots real, equals
2 1
(A) (B)
3 2
7 2
(C) (D)
9 9

13. Suppose (X1, X2) follows a bivariate normal distribution with
E  X1   E  X 2   0, Var  X1   Var  X 2   2 and Cov (X1, X2) = −1. If
1 x  y 2/2
 x 
2 
ee dy,
dy, then P [X1 – X2 > 6] is equal to

(A) (–1) (B) (–3)
(C) ( 6 ) (D) (  6 )

14. Suppose X1, X2 and X3 are i. i. d. random variables each having a Bernoulli distribution
1  X1 0 
with probability . Consider the 2 × 2 matrix A    . Then P (A is invertible)
2  X2 X3 
equals
(A) 0.20 (B) 0.15
(C) 0.25 (D) 0.75

15. From the 6 letters A, B, C, D, E and F, 3 letters are chosen at random with replacement.
What is the probability that either the word BAD or the word CAD can be formed from the
chosen letters ?
1 1
(A) (B)
72 36
1 1
(C) (D)
18 126

16. Suppose X1, X2, …., Xn are i. i. d. Bernoulli(p) random variables, i.e. P(X1 = 1) = p and
n
P(X1 = 0) = 1 – p, 0 < p < 1. Let Y   X i and Z = max (X1, X2, …., Xn). Then
i 1

  and Z ~ Bernoulli 1 p 
(A) Y ~ Bernoulli p n n

(B) Y ~ Bernoulli 1  p   and Z ~ Bernoulli 1 p  
n n

(C) Y ~ Bernoulli 1 p   and Z ~ Bernoulli p 
n n

(D) Y ~ Bernoulli (p ) and Z ~ Bernoulli 1  1  p  
n n

Statistics 5 

17. Let the distribution function F of a random variable X be given by
0 if x  0
 x 1

Fx   if 0  x  1
 2
1 if 1  x

Then P  3  X   is
1
 2
1 3
(A) (B)
2 4
1
(C) 0 (D)
6

2 1 3 4 5
18. Let B   
0 1 1 2 3
Then rank  B B  is
T

(A) 2 (B) 5
(C) 1 (D) 3

19.  
Let A   a ij  be an n × n real matrix. Then which of the following is NOT true?
(A) If A is invertible, tr  A T A   0
(B) If tr  A T A   0 , then A is invertible
(C) If tr  A T A   n 2 , then a ij  1 for some i, j
(D) If tr  A T A   0 , then A is the zero matrix

20. The inverse of the matrix
 1 1 1 1 
 2 2 2 2 
 
 1 
1
0 0 
 2 2 
A 
 1 1

2
0 
 6 6 6 
 
 1 1 1 3 
 
 12 12 12 12 
(A) does not exist (B) is AT
(C) is (AT)2 (D) is A2

Statistics 6 

21. Let X1, X2 and X3 be independently and identically distributed Bernoulli(p) random
variables. Which of the following is NOT a sufficient statistic for p ?
(A) (X1, X1 + X2, X1 + X2 + 2X3) (B) (X1, X1 + X2, X3)
(C) (X1 + X3, X1 + X2 + 2X3) (D) X1 + 17X2 + X3

22. Let X1, X2, …., Xn be a random sample from Uniform (θ, θ + 1). Then which of the
following is NOT an MLE of θ ?
X 1  X  n   1 X 1  X n  12X 1  3X n  3 2X 1  3X  n   3
(A)A A B (B)B
2 2 5 5
X 1  X  n   1 X 1  X n  1 3X 1  2X n  2 3X 1  2X n   2
(C)C C D (D)D
2 2 5 5

23. Let Y1, Y2, Y3 and Y4 be uncorrelated random variables with common unknown variance
 2 and expectations given by
E  Y1   1  2  3  E  Y2  ,
E  Y3   1  2  E  Y4  ,
where 1, 2 and 3 are unknown parameters. Define
1 1
e1   Y1  Y2  and e2   Y3  Y4  . An unbiased estimator of 2 is
2 2
1 2 1 2
(A) (e1  e2 2 ) (B) (e1  e 22 )
2 2
1 2 1 2
(C) (e1  e2 2 ) (D) (e1  e 22 )
4 4

24. Let X and Y follow 2-distributions with the same degrees of freedom, independently.
 X 
Then E   is
XY
1 1
(A) (B)
4 2
3
(C) (D) 1
4

25. Let X1, X2, …., X10 be a random sample from Normal (0, 1). Let N be the number of X i’s
less than or equal to 0. Then Var(N) is
(A) 10 (B) 5
(C) 2.5 (D) 1.25

Statistics 7 

26. Let X be a random variable such that E  X   E  X 2   1 . Then E  X 3  is
(A) 0 (B) 1
(C) 2 (D) 3
n
27. ~ N(, 1) . Let X  1 / n   X i . Then Var  E  X1  X 2 | X   is
Let X1 , X 2 , ., X n iid
i 1

1 2
(A) (B)
n n
3 4
(C) (D)
n n
28. Let X1 , X 2 , ., X10 iid
~ Cauchy (0, 1). Let Ri be the rank of Xi for i = 1, 2, 3, …., 10.
Then
(A) E(R10 – R1) = 0 (B) E(R10 – R1) = 9
10
(C) E(R10 – R1) = (D) E(R10 – R1) does not exist

29. Consider the problem of testing
H0 : X ~ Poisson (2) vs. H1 : X ~ Exp (2).
We reject H0 against H1 based on a single observation x if and only if x is not a non-
negative integer. Then
(A) P(Type -I error) = 0.5 and P(Type -II error) = 0.5
(B) P(Type -I error) = 0 and P(Type -II error) = 0
(C) P(Type -I error) = 0 and P(Type -II error) = 0.5
(D) P(Type -I error) = 1 and P(Type -II error) = 0.5

30. Let x1  3, x 2  4, x 3  3.5, x 4  2.5 be the observed values of a random sample from the
probability density function
1 1  θ 
x
f x | θ    e  e  x , x  0, θ  0. Then the method of moment estimate of θ is
2 θ 
 
(A) 3.5 (B) 4
(C) 5.5 (D) 6.5

31. Suppose X1, X2, ., Xn are independent random variables and Xk ~ N(0, k2), k = 1, 2, …, n
where 2 is unknown. The maximum likelihood estimator for 2 is
1 n 1 n
(A) 
n k 1
Xk2 (B) 
n k 1

Xk  Xn
2

1 n Xk2 n n
(C) 
n k 1 k
(D)  Xk2 k
k 1 k 1

Statistics 8 

32. Suppose X = (X1, X2, X3)T has N3() distribution where  = (–3, 1, 4)T and
1 2 0
 
   2 5 0  . Then which of the following pairs of random variables are NOT
 0 0 2
 
independent ?
X1  X 2
(A) X2 and X3 (B) and X 3
2
5 X  X3
(C) X2 and X2 – X1 – X3 (D) X1 and 2
2 2

33. Let the random variable ‘X’ have binomial distribution with parameters 3 and θ. A test of
3 1
hypothesis H 0 :   against H1 :   rejects H0 iff X ≤ 1. Then the test has
4 4
5 27 5 18
(A) size = , power = (B) size = , power =
32 32 32 32
15 27 1 31
(C) size = , power = (D) size = , power =
32 32 32 32
34. The name of the header file to which puts () belongs is
(A) math.h (B) stdio.h
(C) iomanip.h (D) stdlib.h
35. Let X1 and X2 be independently and identically distributed as normal with mean θ and
variance unity. Then E  X | X1  1 will be
 1  1
(A) (B)
2 2
 
(C) (D)
2 2
36. The diagnostic rating scale based on a mammogram to detect breast cancer as definitely
normal, probably normal equivocal, probably abnormal, definitely abnormal uses
(A) Ratio scale (B) Interval scale
(C) Nominal scale (D) Ordinal scale

37. For a given bivariate data set (xi, yi), i = 1, 2, …, n; the squared correlation coefficient (r2)
between x2 and y is found to be 1. Which of the following statement is most appropriate ?
(A) In the (x, y) scatter diagram, all points lie on a straight line.
(B) In the (x, y) scatter diagram, all points lie on a curve y = x 2.
(C) In the (x, y) scatter diagram, all points lie on a curve y = a + bx 2, a > 0, b > 0.
(D) In the (x, y) scatter diagram, all points lie on a curve y = a + bx 2, a and b are any real
numbers.

Statistics 9 

38. Suppose your data produces the regression of y on x as y = 10 + 3x. Scale y by multiplying
by 0.9 and do not scale x. The new intercept and slope estimates will be respectively
(A) 10, 3 (B) 9, 3
(C) 9, 2.7 (D) 11.11, 3.33

39. The data on annual income is summarised in the following frequency distribution table :
Annual Income (in ’000 `) 80 - 100 100-120 120-140 140-160 160-180
No. of families 5 10 15 10 5
The value of m3 (third order central moment) comes out as
(A) 2.25 (B) 0
(C) – 2.25 (D) 1

40. For a set of 8 observations if 3 and 23 are the minimum and maximum values respectively,
then the minimum and maximum possible values of the variance are respectively
(A) 25 and 100 (B) 50 and 150
(C) 20 and 80 (D) 50 and 80

41. Suppose a finite population consists of N units numbered 1, 2, ….., N. Two units are drawn
using SRSWOR and let (X, Y) be the numbers of the sampled units. If Z = min (X, Y),
then E(Z) is
N 1 N
(A) (B)
6 6
N ( N 1) N 1
(C) (D)
6 6

42. Consider the following model
(y1, y2, y3, y4)T = (1, 1, 1, 0)T 1 + (1, 1, 0, 1)T 2 + (e1, e2, e3, e4)T,
(e1, e2, e3, e4)T ~ N4(0, 2I4)
Then among the following, which is most preferable for deriving a confidence interval for
1  2 ?
(A) y3 – y4 (B) y3 – y4 + y1 – y2
(C) y1 – 2y2 + y3 (D) y1 – y2

43. In a 22 factorial experiment the main effect A is given by
1 1
(A) {(ab)  (a )  (b)  (1)} (B) {(ab)  (a )  (b)  (1) }
2 2
1 1
(C) {(ab)  (a )  (b)  (1)} (D) {(ab)  (a)  (b)  (1)}
2 2

Statistics 10 

44. If the degrees of freedom for the error sum of squares in a Latin square design is 6, then the
order of the design is
(A) 3 × 3 (B) 4 × 4
(C) 5 × 5 (D) 6 × 6

45. In a randomized block design with 4 blocks and 5 treatments having one missing value, the
error degrees of freedom will be
(A) 12 (B) 11
(C) 10 (D) 9

46. A simple random sample of size 3 is drawn with replacement from a population of size
N (> 7). The probability that the sample contains exactly 2 distinct units is
(A) 1/N2 (B) (N – 1) (N – 2)/N2
(C) 3(N – 1)/N2 (D) (N – 1)/N

47. Which of the following is NOT a division of the Central Statistics Office (CSO) ?
(A) National Accounts Division (B) Demography Division
(C) Social Statistics Division (D) Economic Statistics Division

48. Given that the Total Fertility Rate (TFR) is 2620 per thousand and sex-ratio at birth is 1
male to 1 female, then the Gross Reproduction Rate (GRR) will be
(A) 1.000 (B) 1.310
(C) 2.620 (D) 5.240

49. How many moving average values are feasible when 3-year moving averages are calculated
from the values for the years 2010 to 2022 ?
(A) 10 (B) 11
(C) 12 (D) 13

50. If μ and σ are the process mean and standard deviation respectively, then the control limits
+3 are known as
(A) modified control limits (B) natural control limits
(C) specified control limits (D) artificial control limits
_______________

Statistics 11 

SPACE FOR ROUGH WORK

Statistics 12 