AUAT 2024 Question Paper M.Sc in Statistics

Entrance Exam
2024

QUESTION
PAPER

Question Booklet No. Question Booklet Series : A
AUAT — 2024
2-Year M. Sc. in Statistics (P14)
( TEST BASED ON MCQ )

Full Marks : 100 Duration : 2 Hours

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Date of Examination : _______________________________________

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AUAT-2024 /110- A

PART—I
( Core Subject )
1. Neyman Pearson Lemma provides 4. The characteristic equation of a
always matrix A is t2 – t – 1 = 0, then
[A] an unbiased test [A] A –1 exists but cannot be
determined from the data
[B] a minimax test
[C] a most powerful test [B] A–1 does not exist
[D] None of the above [C] A–1 = A – 1

2. It is observed that 50% of the emails [D] A–1 = A + 1
that we receive in Gmail are spam
emails. Someone has developed a 5. Which of the following statements is
software which can detect 99% of the correct for a positively skewed
spam emails correctly, however, the distribution?
probability that a non-spam Email is
detected as a spam email by the [A] Third Quartile (Q3) – Median <
software is 5%. Assume that an email Median – Third Quartile (Q1)
is detected as spam by the software,
then find the probability that it is not [B] Third Quartile (Q3) – Median =
a spam email. (Answer should be Median – Third Quartile (Q1)
correct up to three decimal places,
[C] Third Quartile (Q3) – Median >
error range : 0·005)
Median – Third Quartile (Q1)
[A] 0·178
[D] None of the above
[B] 0·782
[C] 0·048 6. Arithmetic Mean is _____ affected by
[D] 0·091 extreme values.

[A] not
3. Let X 1 , X 2 , X 3 and X 4 be
independently and identically [B] highly
2
distributed N (µ, σ ) random variables.
[C] less
X4 − X2
Define Y = and find the value [D] Nothing can be said
X 3 − X1

P (Y > 1) 7. Which of the following is not a basic
of . principle of experimental design?
P (Y < −1)
[A] Randomization
[A] 1
[B] 0·3 [B] Local control

[C] 2 [C] Replication
[D] 0·5
[D] Confounding

AUAT-2024 /110- A 2

8. In Mumbai, it is observed that the 12. Which one of the following can be
probability of selecting a smoker or a used both as a parametric test as well
male is 7/10. Also it is found that the as a non parametric test?
probability of finding a male smoker
[A] F-test
is 2/5 and a male, if a smoker is
already selected is 2/3. The [B] z-test
probability of selecting a non smoker, [C] t-test
if a male is first selected is
[D] Chi-square test
[A] 1/5
[B] 2/5 13. Suppose there are 11 boxes numbered
[C] 3/5 1, 2, ..., 11 and 10 identical balls
placed one each in the boxes 1 to 10.
[D] 4/5
In one operation a ball is chosen at
random and placed in the empty box.
9. Let I be the identity transformation of
Find the probability that the 5th box
the finite dimensional vector space V,
will remain empty after four such
then the nullity of I is
operations. (Answer should be correct
[A] dim(V) up to three decimal places, error
[B] 0 range : 0·005)
[C] 1 [A] 0·31
[D] dim(V) – 1 [B] 0·12
[C] 0·02
10. The 11th term of the sequence 1, 3,
9, 27, … is [D] 0·09
[A] 2 8
14. For which of the following
[B] 2 10 distributions mean and variance are
[C] 3 8 equal?
[D] 3 10 [A] Negative Binomial Distribution
[B] Binomial Distribution
11. In a large company manufacturing
computers, it is noted that 80% of [C] Normal Distribution
their computers are equipped with a [D] Poisson Distribution
Pen drive, 95% with a CD drive, and
75% with both. Determine the 15. Suppose X ~ Uniform (a,b). Then E(X)
probability that a randomly selected and V(X) are respectively
computer from the company is
equipped either with a Pen drive or a a (b )2
[A] and
CD drive but not both. (Answer should 2 12
be correct up to three decimal places, a +b (b − a )2
error range : 0·005) [B] and
2 12
[A] 0·25
a −b (b − a )4
[B] 0·75 [C] and
2 6
[C] 0·19
a +b (b − a )2
[D] 0·01 [D] and
2 6
AUAT-2024 /110- A 3 [ P.T.O.

16. The number of linearly independent 20. A committee of 6 persons is to be
eigenvectors of formed from a group of 7 men and 4
women. What is the probability that
1 1 0 0
the committee will have exactly two
2 2 0 0
0 0 3 0 women?
 
0 0 5 5
is [A] 0·158

[A] 4
[B] 0·889
[B] 3
[C] 1 [C] 0·455
[D] 2
[D] 0·965
17. The minimal polynomial of
2 1 0 0
0 21. Criteria to check a point estimator to
2 0 0
0 0 2 0 be good is/are
 
0 0 0 5
is [A] consistency
[A] (x – 2)
[B] unbiasedness
[B] (x – 2)(x – 5)
[C] (x – 2)3(x – 5) [C] efficiency
[D] (x – 2)2(x – 5)
[D] All of the above
18. Consider the C.D.F. F(x) of the random
variable X. Let M be the mode of the
distribution 22. Suppose a dice is tossed 120 times and
the following results are obtained :
F(x) = 1 – e–x – xe–x, if 0 < x < ∞ ,
= 0, otherwise
Number in the
top most face 1 2 3 4 5 6
Find the value of F(M).
Frequency 30 25 18 10 22 15
[A] 0·2642
[B] 0·1591
[C] 0·6710 Suppose it is given that the tabular
value of Chi-square is 11·7. Then the
[D] 0·1131
dice is
19. Suppose Xi ~ Binomial(l, p) independ-
[A] unbiased
ent for each i. The distribution of
 i =1 X i2 is
n

[B] biased
[A] Binomial(l, p)
[B] Binomial(n, p) [C] More information needed
[C] Poisson(n)
[D] Poisson(l ) [D] None of the above

AUAT-2024 /110- A 4

23. The exponential distribution is 26. If the regression coefficients bxy =0·25
characterized by and b yx = 0·64, then correlation
coefficient is
[A] loss of memory property
[A] 0·16
[B] new better than used property

[C] constant hazard property [B] 0·89

[D] non monotone hazard property [C] 0·40

24. Sample survey involves the following : [D] 0·30

(i) Sampling unit
27. Suppose a random number N
(ii) Size of samples
is chosen between 1, 2, ..., 20 with
(iii) Type of universe P(N = n) = C * 0·2 * (0·8)n for n = 1, 2,
..., 20. If it is given that the chosen
(iv) Source list number is divisible by 4, compute the
probability that the number is either
(v) Sampling procedure divisible by 3 or 5. (Answer should be
correct up to three decimal places,
Which one of the following sequences error range : 0·005)
is correct?

[A] (i), (ii), (iii), (iv), (v) [A] 0·9920

[B] (iii), (i), (iv), (ii), (v) [B] 0·3302

[C] (iii), (iv), (i), (ii), (v)
[C] 0·5137
[D] (iii), (v), (i), (iv), (ii)
[D] 0·1170
25. Which of the following is not true for
Horvitz Thomson estimator?
28. If A and B are invertible square
[A] It is used for unequal probability matrices of size n × n, then which of
sampling the following statement is not true?

[B] It is used strictly for sampling
with replacement [A] det(A + B) = det(A) + det(B)

[C] It is used both for sampling with [B] det(AB) = det(A) det(B)
and without replacement

[D] It is an unbiased estimator of the [C] det(kA) = kn det(A)
population parameter for the
population total [D] All of the above

AUAT-2024 /110- A 5 [ P.T.O.

29. If P(A) = 3/7, P(B) = 9/13 and 33. The radius of convergence of the
P ( A ∩ B ) = 4/13 . Find P ( A |B ) . power series  a n x n is R and k be a
positive integer. Then the radius of
[A] 4/7
convergence of the power series
[B] 4/9
 an x kn is
[C] 5/9
[A] R/k
[D] 2/9
[B] R

30. The distribution of test statistic used [C] Not depend on k
in sign test is 1
[D] R k
[A] Poisson
34. A is 5 × 5 matrix, all of whose entries
[B] Normal are 1, then
[C] Gamma [A] A is not diagonalizable
[D] Binomial [B] A is idempotent

[C] A is nilpotent
31. Consider the following statements :
S1: Sum of the two singular n × n [D] the minimal polynomial and the
matrices may be non-singular. characteristics polynomial of A
are not equal
S2: Sum of the two non-singular
n × n matrices may be singular.
35. A set of linear equations is represented
Which of the following statements is by the matrix equation Ax = b. The
correct? necessary condition for the existence
of a solution for this system is
[A] S1 and S2 are both true
[A] A must be invertible
[B] S1 is true, S2 is false

[C] S2 is true, S1 is false [B] b belongs to the column space of A

[D] S1 and S2 both are false [C] b does not belong to the column
space of A

32. Sum of the eigenvalues of the matrix [D] None of the above
 2 0 0 −1
 0 1 0 0 36. The most appropriate average to be
 0 0 3 0  is
  used to compute the average rate of
 −1 0 0 4 growth in population is
[A] –10 [A] Arithmetic Mean
[B] 24 [B] Harmonic Mean
[C] 22 [C] Geometric Mean

[D] 10 [D] Median

AUAT-2024 /110- A 6

41. If a hypothesis is rejected at 0·6 level
 nπ

37. The sequence sin 2  n = 1, ∞ of significance, then
 n
[A] it will be rejected at any level of
[A] is divergent significance
[B] converges to 1 [B] it must be rejected at 0·5 level of
[C] converges to 0 significance

[D] None of the above [C] it may be rejected at 0·5 level of
significance

38. M : R 4 → R 3 , where [D] it cannot be rejected at 0·5 level
of significance
1 2 3 1
M =  1 3 5 −2 
 3 8 13 −3  42. Let A and B be square matrices such
 
that AB = I, then zero is an eigenvalue
Then the dimension of kernel of M is of
[A] 2 [A] A but not B
[B] 1 [B] B but not A
[C] 3 [C] Both A and B
[D] 4 [D] Neither A nor B

39. Under frequentist set up, a parameter is
43. If A and B are non-zero square
[A] a random quantity matrices, then AB = 0 implies

[B] always 0 [A] A and B are diagonal

[C] always 1 [B] A and B are singular

[D] a non-random quantity [C] B is singular

[D] A is singular
2
40. Suppose X ∼ N (µ, σ ). Then which of
the following is not true? 44. Suppose for a statistical test α and β

[A] The distribution of X is symmetric are the type I and type II errors
respectively. Then the power of the test
and E ( X ) = µ
is
[B] The distribution of X has median [A] α
at x = µ
[B] 1 – α
[C] The odd order moments are of odd
values [C] 1 – β

[D] For the distribution of X, the [D] β
coefficient of skewness is zero

AUAT-2024 /110- A 7 [ P.T.O.

45. Which one of the following statistics 48. Let X be a random variable with M.G.F.
is unaffected by outliers? 2
given by M (t ) = e 4t +8t . Calculate
E(X2).
[A] Mean
[A] 16
[B] Interquartile range
[B] 12

[C] Standard deviation
[C] 32

[D] Range [D] 8

49. Let the lifetime of a electric bulb is
46. Goodman Kruskal’s Gamma measure
denoted by a random variable X with
is used to find the association
p.d.f. f(x). If the mean of the lifetime is
between
M1 and the median of the lifetime is
M2
[A] two nominal attributes M2, find , where
M1

[B] one ordinal and one nominal x
attributes 1 −
f (x ) = e α , if 0 < x < ∞, α > 0
α
= 0, otherwise
[C] two ordinal attributes
[A] 0·7980
[D] None of the above
[B] 0·3329

47. Condition for applying the Central [C] 0·6931
Limit Theorem (CLT) which
approximates the sampling [D] 0·3129
distribution (with sample size n) of the
mean with a normal distribution is
50. Suppose X ~ Binomial (2m, 1/2), then
which of the following is not true?
[A] n < 30
[A] Mean = m
[B] n > 30
[B] Median = m

[C] n = 15 [C] Mode = 2m

[D] Mode = m
[D] n > 60

AUAT-2024 /110- A 8

51. A hypothesis test is being performed for 54. The correlation coefficient is
a process in which a type I error will be dependent on the change of
costly, but type II error will be relatively
inexpensive and unimportant. Which
[A] scale value only
of the following would be the best
choice for α in this test?
[B] origin value only

[A] 0·10
[C] both origin and scale values

[B] 0·05
[D] neither the origin value nor the
scale value
[C] 0·01

[D] 0·50 55. Pearson’s correlation coefficient
measures

52. In general line, diagram is used to [A] cubic relationship between two
represent variables

[A] spatial series data [B] linear relationship between two
variables

[B] time series data
[C] curvilinear relationship between
two variables
[C] Both [A] and [B]
[D] None of the above
[D] Neither [A] nor [B]

56. If there is linear trend present in the
population, then which of the
53. Eigenvalues of a real symmetric
following methods is the most efficient
matrix are always
sampling technique?

[A] positive
[A] Cluster sampling

[B] real and imaginary [B] Stratified sampling

[C] negative [C] Systematic sampling

[D] real [D] Simple random sampling

AUAT-2024 /110- A 9 [ P.T.O.

57. Suppose X ~ Exp(1), then Y = e–x will 61. Consider the following data :
follow :
X ‾3 ‾2 ‾1 0 1 2 3
[A] Bin(2, 0·5) Y 9 4 1 0 1 4 9

[B] Poisson(0·2)
[A] The Pearson’s correlation
coefficient between X and Y is 0
[C] Rectangular(0, 1)
[B] The Pearson’s correlation
[D] Bin(0, 1) coefficient between X and Y is 1

[C] The Pearson’s correlation
58. A coin is tossed five times in
coefficient between X and Y is 0·5
succession. What is the probability of
getting at least four heads?
[D] The Pearson’s correlation
coefficient between X and Y is
[A] 1/2
–0·5

[B] 1/16
62. I arrive home from a feast and attempt
[C] 3/4 to open my front door with one of the
three keys in my pocket. (You may
[D] 3/16 assume that exactly one key will open
the door and that if I use it I will be
successful.) Find the expected
59. Which of the following is not a number of tries that I will need if I
measure of dispersion? take the keys at random from my
pocket but drop any key that fails onto
[A] Range
ground. (Answer should be correct up
to three decimal places, error range :
[B] Mean Deviation
0·005)

[C] Quartile [A] 4

[D] Standard Deviation [B] 3

[C] 1
60. Suppose the probability of injury on
each individual parachute jump is [D] 2
0·05. Then the greatest lower bound
for probability of landing safely on both
of two jumps is 63. Which one of the following plots helps
to detect outliers?
[A] 0·09
[A] Histogram
[B] 0·1 [B] Line Diagram

[C] 0·9 [C] Boxplot

[D] 0·01 [D] Bar Diagram
AUAT-2024 /110- A 10

64. For a regression model with 68. Suppose that P (A ) = 0.25, P(B − A) = 0.3
continuous response, the error should
follow the _____ distribution. a n d P (C − (A ∪ B )) = 0.1. D e t e r m i n e
P ( A ∪ B ∪ C ). (Answer should be
[A] Cauchy
correct up to three decimal places,
[B] Double Exponential error range : 0·005)

[C] Normal [A] 0·85

[D] Binomial
[B] 0·65

65. The odds are 2 to 1, that when A and
[C] 0·15
B play tennis, A wins. Suppose A and
B play 2 matches. What is the
probability that A wins at least 1 [D] 0·35
match?

[A] 1/9 69. For perfect correlation, both the
regression lines
[B] 3/8
[A] will be perpendicular to each
[C] 1
other
[D] 8/9
[B] will coincide
66. For a normally (standard) distributed
random variable, the probability that [C] makes an angle of π / 6
it will lie between –3 to 3 is

[A] 0·95 [D] makes an angle of π / 4

[B] 0·9973
70. Suppose X 1, X 2 , X 3, X 4 , X 5 be 5
[C] 0·65 independent random variables with
f(x) = 2x, 0 <x < 1. Let X(1), X(2), X(3),
[D] 0·51
X(4) and X(5) be the corresponding
order statistics. Then P(X(4) < 1) is
67. For a multivariate normal distribution,
the marginals follow [A] 0·0245

[A] univariate normal distribution
[B] 0·0156
[B] any symmetric distribution
[C] 0·0378
[C] bivariate normal distribution

[D] None of the above [D] 0·0498

AUAT-2024 /110- A 11 [ P.T.O.

PART—II
( Islamic History and Culture, General English & General Knowledge )
71. Revelation to Prophet Muhammad 76. Prophet Muhammad (PBUH) was born
(PBUH) started with the word ‘Iqra’ in
which means
>¤ã ³åÒà´¶ƒ (Îà.)-&¹ \@µÑ‚à> Òº
>¤ã ³åÒà´¶ƒ (Îà.)-&¹ A¡àìá "¤t¡ão¢ šøt¡¸àìƒÅ Ç¡¹ç¡
[A] Jordan
ÒìÚìá ‘ÒüA¡¹à’ Ŧ [ƒìÚ, ™à¹ "=¢ Òº
[B] Jeddah
[A] pray
[B] prostrate [C] Mecca
[C] read [D] Medina
[D] write
77. In Islam, legal matters are part of
72. Which of the following cities is not a Òüκàì³ "àÒü> Î}yû¡à”z [¤ÈÚà[ƒ A¡ãìι "”z®¢å¡v¡û¡?
sacred city for Muslims?
[A] Fikr
[>ì´•¹ ëA¡à>ô ÅÒ¹[i¡ ³åÎ[º³ìƒ¹ A¡àìá šåo¸Ñ‚à> >Ú?
[B] Zikr
[A] Jerusalem
[B] Mecca [C] Shariah
[C] Cairo [D] Taharat
[D] Medina
78. Hajj is performed in the month of
73. What does Islam say about acquiring
ë™ ³àìÎ Ò\ šà[ºt¡ ÒÚ, t¡à Òº
knowledge?
[A] Dhul Qadah
`¡à> "\¢ì>¹ ¤¸àšàì¹ ÒüκೠA¡ã ¤ìº?
[A] Obligatory for men only [B] Dhul Hijjah
[B] Obligatory for women only [C] Shaban
[C] Obligatory for rich people only [D] Shawwal
[D] Obligatory for all
79. Rashidun Caliphate lasted for _____
74. Which one is not part of the five pillars years.
of Islam?
¹à[Ńå> [Jºàó¡ìt¡¹ γÚA¡àº _____ ¤á¹¡ú
[>ìW¡¹ ëA¡à>[i¡ Òüκàì³¹ še¡Ñzì´±¹ ³ì‹¸ "”z®¢å¡v¡û¡
[A] 20
>Ú?
[B] 30
[A] Profess Allah as the only God
[B] Pray five times daily [C] 40
[C] Make a pilgrimage to Mecca [D] 50
[D] Read the Qur’an everyday
80. Who accompanied Prophet
75. The divine book revealed to Prophet Muhammad (PBUH) during migration?
Muhammad (PBUH) is ëA¡à>ô ¤¸[v¡û¡ [Ò\¹ìt¡¹ Î³Ú >¤ã ³åÒà´¶ƒ (Îà.) ëA¡ ÎU
>¤ã ³åÒà´¶ƒ (Îà.)-&¹ [>A¡i¡ "¤t¡ão¢ 'Åã Kø씂¹ >à³ [ƒìÚ[áìº>?
Òº [A] Abu Hurayra
[A] Torah [B] Hazrat Ali
[B] Qur’an
[C] Khadija
[C] Injeel
[D] Abu Bakr
[D] Zabur

AUAT-2024 /110- A 12

81. What are the companions of the 86. Which gates are closed during the
Prophet called? month of Ramadan?
>¤ã¹ ÎàÒà¤ã샹 A¡ã ¤ºà ÒÚ? ¹³\à> ³àìÎ ëA¡à>ô ƒ¹\àP¡ìºà ¤Þê¡ =àìA¡?
[A] Sahabah [A] Gates of Hell
[B] Friends [B] Gates of Heaven

[C] Ansar [C] Gates of Light
[D] Gates of Mosque
[D] None of the above
87. What is the Shari’ah?
82. What to say when anyone sneeze?
ŹãÚt¡ A¡ã?
ëA¡l¡ü Òòà[W¡ [ƒìº A¡ã ¤ºìt¡ ÒÚ? [A] A religious school
[A] Yarhamukallah [B] A native dance
[B] Ya Allah [C] A style of calligraphy
[C] Alhamdulillah [D] A form of Muslim law
[D] La Ilaha Illallah
88. The Kabah is located in
83. What does Salat mean? A¡à¤àQ¹ _____ ët¡ "¤[Ñ‚t¡¡ú
Îàºàt¡ ³àì> A¡ã? [A] Mecca
[B] Medina
[A] Fasting
[C] Istanbul
[B] Giving to the poor
[D] Iraq
[C] Praying
89. Are women allowed to go to mosques
[D] Pilgrimage to offer prayers?

84. What is the meaning of ³[Òºà샹 [A¡ ³Î[\ìƒ >à³à\ šØl¡à¹ ">å³[t¡ ëƒ*Úà
‘Astaghfirullah’? ÒÚ?
[A] No, women are not allowed in the
‘"àÑzàK[ó¡¹ç¡ÀàÒ’-&¹ "=¢ A¡ã?
mosques
[A] I ask Allah for Forgiveness [B] Women can only listen to prayers
[B] We are for Allah [C] Yes, women can offer prayers in
mosques, provided there are
[C] Accept our prayer
separate facilities and provision
[D] None of the above [D] Women can pray in the mosques,
standing right beside the men
85. What is the first duty to become a
Muslim? 90. In which language, was the Holy
Qur’an revealed?
³åκ³à> Ò*Ú๠šø=³ A¡t¢¡¤¸ A¡ã?
‘š[¤y A塹"à>’ ëA¡à>ô ®¡àÈàÚ "¤t¡ão¢ ÒìÚ[áº?
[A] Salat
[A] Arabic
[B] Sawm
[B] Greek
[C] Hajj
[C] Latin
[D] Shahadah
[D] Hebrew
AUAT-2024 /110- A 13 [ P.T.O.

91. Fill in the blanks with the appropriate 96. In which of the following States,
words. Rawatbhata Nuclear Plant is
The _____ workers will come forward situated?
and protest against the _____ of the ¹à*Úàt¡®¡ài¡à šà¹³ào[¤A¡ ëA¡@ƒø [>´•[º[Jt¡ ëA¡à>ô ¹àì\¸
organization.
"¤[Ñ‚t¡?
[A] deprived, issues
[A] Rajasthan
[B] depraved, policies
[B] Gujarat
[C] derived, interests
[C] Madhya Pradesh
[D] depreciated, management
[D] Tamil Nadu

92. Find the word opposite in meaning to
the underlined word. 97. Who invented the railway engine?

Rahim’s foolishness will drive me to 빺*ìÚ Òü[g> ëA¡ "à[¤ÍHþ๠A¡ì¹>?
despair. [A] Charles Babbage
[A] Hopefulness [B] Isaac Newton
[B] Desperation [C] James Watt
[C] Destruction [D] George Stephenson
[D] Repulsed
98. Who is said to be the father of the
93. Fill in the blanks with the correct Indian Space Programme?
alternatives.
®¡à¹t¡ãÚ ³ÒàA¡àÅ A¡³¢Îè[W¡¹ \>A¡ A¡àìA¡ ¤ºà ÒÚ?
The farmer placed his shovel _____ the
[A] Abdul Kalam
wall and sat _____ to rest.
[B] Rakesh Sharma
[A] beside, over
[B] against, down [C] Vikram Sarabhai

[C] by, along [D] Homi Bhabha

[D] on, in
99. Which of the following was Indian’s
first mapping satellite?
94. Identify the synonym of the word
‘ABJECT’. [>ìW¡¹ ëA¡à>[i¡ ®¡à¹ìt¡¹ šø=³ ³¸à[š} θàìi¡ºàÒüi¡ [áº?
[A] Outrageous [A] CARTOSAT-1
[B] Fantastic [B] Aryabhata
[C] Devastated [C] Bhaskara-II
[D] Deplorable [D] INSAT-1A

95. Fill in the blank with the appropriate 100. The super computer ‘PARAM’ was
collective noun. developed by
A _____ of cards. Îåšà¹ A¡[´šl¡üi¡à¹ ‘PARAM’ ít¡[¹ A¡ì¹>
[A] pile [A] TATA
[B] bundle [B] IIT-Kharagpur
[C] deck [C] IIT-Kanpur
[D] bunch
[D] C-DAC
AUAT-2024 /110- A 14

SPACE FOR ROUGH WORK

AUAT-2024 /110- A 15 [ P.T.O.

SPACE FOR ROUGH WORK

AUAT-2024 /110- A 16 NNN24(141)—12×4

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