CUSAT CAT 2023 Question Paper Statistics

STATISTICS
(FINAL)

1. If A is a 3 × 3 idempotent matrix with determinant value 3 then the determinant value
2
of A is

(A) 3
(B) 9
(C) 27
(D) 81

2. Which of the following are linearly independent set of vectors?

(A) (1, 1, 0), (0, 1, 1), (1, 2, 1)
(B) (2, 0, 0), (0, 3, 0), (0, 0, 4)
(C) (1, 1, 1), (2, 2, 2), (3, 3, 3)
(D) (1, −1, 0), (−1, 1, 0), (0, 0, 0)

3. For a given orthogonal matrix A, we have the following statements

(i) A′ is also orthogonal
(ii) A−1 is also orthogonal
(iii) |A| is also orthogonal

(A) (i),(ii) and (iii) are true
(B) (i) and (ii) alone are true
(C) (ii) and (iii) alone are true
(D) (i) and (iii) alone are true

4. A box contains 3 blue and 2 red marbles while another box contains 2 blue and 5 red
marbles. A marble drawn at random from one of the boxes, if turns out to be blue,
what is the probability that it came from the first box?

2
(A)
5
3
(B)
5
4
(C)
7
3
(D)
12

5. The probability of getting identical faces while throwing three dice is

1
(A)
36
1
(B)
6
1
(C)
216
(D) 0

6. A five-figure number is formed by the digits 0, 1, 2, 3, 4 (without repetition). Find the
probability that the number formed is divisible by 4.

5
(A)
16
3
(B)
18
6
(C)
16
(D) None of the above

7. How many different words can be formed by permuting letters of the word
STATISTICS?

10!
(A)
2!3!3!
5!
(B)
2!3!3!
2!3!3!
(C)
10!
2!3!3!
(D)
5!

8. Given the two line of regression as, 3X – 4Y + 8 = 0 and 4X – 3Y = 1, the means of X
and Y are

(A) X = 4, Y = 5
(B) X = 4 3, Y = 5 4
(C) X = 3, Y = 4
(D) None of the above

9. Variance inflation factor is used to study

(A) normality
(B) linearity
(C) additivity
(D) multicollinearity

10. In a 5 × 5 Latin Square Design the degrees of freedom for error is

(A) 24
(B) 16
(C) 12
(D) 8

11. If the inner product of the columns of the design matrix X equal to 0 then such
experimental designs are said to be

(A) incomplete block designs
(B) non – orthogonal designs
(C) orthogonal designs
(D) linear designs

12. In a Randomized Block Design with 4 blocks and 5 treatments having one missing
observation the error degree of freedom will be

(A) 12
(B) 11
(C) 10
(D) 9

 1 1 1 1 
13. The value of lim n  2
+ 2
+ 2
+⋯ +  is
n→∞
 (n + 1) (n + 2) (n + 3) (2n) 2 

1
(A)
2
1
(B)
4
(C) 1
(D) ∞

14. Let f (x) = 0 for all x ≠ 0 and let f (0) = 2. Then, zero (0) is

(A) an essential discontinuity
(B) a removable discontinuity
(C) a jump discontinuity
(D) an end point discontinuity

15. Choose the CORRECT statement among the following

∞
xk
(A) ∑ converges uniformly on [−a, a], for every a ∈ℜ
k =0 k !

∞
xk
(B) ∑ converges uniformly on all of the real line
k =0 k !

∞
xk
(C) ∑ does not converge
k =0 k !

∞
xk
(D) ∑ converges, but not uniformly
k =0 k !

16. Let f : ℜ → ℜ be a continuous as well as periodic function. Then, it attains its

(A) supremum
(B) infimum
(C) supremum and infimum
(D) maximum and minimum

17. Which of the following statements is correct?

(A) If { f n } is a sequence of non-measurable functions and is fundamental in
measure, then some subsequence { f nk } is almost uniformly fundamental
(B) If { f n } is a sequence of non-measurable functions and is fundamental in
measure, then some subsequence { f nk } is fundamental in measure
(C) If { f n } is a sequence of measurable functions and is fundamental in measure,
then some subsequence { f nk } is almost uniformly fundamental
(D) If { f n } is a sequence of measurable functions and is fundamental in measure,
then some subsequence { f nk } is uniformly fundamental

18. If the rank of a square matrix A is less than its order, then

(A) | A | = | A |T
(B) A−1 exist
(C) | A | > 0
(D) A−1 does not exist

19. Which of the following is said to be orthogonal?

 0 2 3
 
(A)  − 2 0 5
 − 3 − 5 0
 
 cosθ − sin θ 
(B)  
 sin θ cosθ 

 1 2 3
 
(C)  4 5 6
7 8 9
 
1 2
(D)  
3 4

 1 2 3
 
20. Find the trace of the matrix  4 5 6 
7 8 9
 

(A) 45
(B) 15
(C) 11
(D) 19

21. If A is a real m × n matrix, then A′A is a

(A) m × n matrix
(B) n × n symmetric matrix
(C) m × n symmetric matrix
(D) n × n skew symmetric matrix

22. A square matrix B is said to be idempotent if

(A) B′B = I
(B) B 2 = B
(C) BB − 1 = I
(D) B = B′

23. The odds in favor of A winning a game of chess against B are 3 : 2. If 3 games are to
be played, then the odds against A losing the first 2 games to B is

(A) 44 : 81
(B) 81 : 44
(C) 4 : 21
(D) 21 : 4

24. A box contains two fair coins and a biased coin with probability for heads given as
0.2. A coin is chosen at random from the box and tossed three times. If two heads and
a tail are obtained, then the probability of the event that the chosen coin is fair is

(A) 1.0
(B) 0.11
(C) 0.78
(D) 0.89

25. A random variable is a continuous random variable, if

(A) its cumulative distribution function is a continuous function for all x ∈ℜ
(B) its density function is a continuous function for all x ∈ℜ
(C) its cumulative distribution function is a continuous function for some x ∈ℜ
(D) its density function is a continuous function for some x ∈ℜ

26. Choose the mode of the distribution whose density function is given by
12 x 2 (1 − x), 0 < x < 1
f ( x) = 
0, else where

1
(A)
2
1
(B)
4
2
(C)
3
3
(D)
4

27. A random variable X is said to be strictly larger than the random variable Y if
………………. for all real z

(A) P ( X > z ) ≥ P (Y > z )
(B) P ( X > z ) ≤ P (Y > z )
(C) P ( X > z ) ≥ P (Y < z )
(D) P ( X < z ) ≤ P (Y > z )

28. If X is a Uniform ( −π 2, π 2 ) random variable, the distribution of Y = tan X is

(A) Pareto
(B) Cauchy
(C) Laplace
(D) Weibull

29. Let X be a random variable such that its mean is 3 and variance is 4. The lower bound
for the probability P ( −2 < X < 8) is

(A) 0.84
(B) 0.16
(C) 0.04
(D) 0.96

30. Find the mean if the pairs of values and their frequencies x, f ( x ) , are (1, 1), (2, 2),
(3, 3), (4,4), (5,5), (6, 6), (7, 7), (8, 8), (9,9), and (10,10).

(A) 7
(B) 21
(C) 55
(D) 385

31. What is the value of skewness if the distribution is having mean, median and standard
deviation 100, 90, and 10 respectively?

(A) −3
(B) 0
(C) 1
(D) 3

32. A student travelled 100 km distance to college in his car at a speed of 20 km per hour
and return back to home at a speed of 30 km per hour. What is the average speed of
the entire journey?

(A) 25 km per hour
(B) 24 km per hour
(C) 24.49 km per hour
(D) 36 km per hour

33. What is the value of coefficient of quartile deviation when first, second and third
quartiles are 25, 50, and 75 respectively?

(A) 0.5
(B) 1
(C) 0.9
(D) 0.2

34. If X1, X 2 ,⋯ , X n are independent random variables, then the distribution function of
Y1 = min( X1, X 2 ,⋯ , X n ) is

n
(A) FY ( y ) = 1 − ∏ 1 − FX ( y ) 
1
i =1  i 
n
(B) FY ( y ) = ∏ 1 − FX ( y ) 
1
i =1  i 
n
(C) FY ( y ) = 1 − ∏ FX ( y )
1 i
i =1
n
(D) FY ( y ) = ∏ FX ( y )
1 i
i =1

35. Cauchy random variable X has the density function given by
a
f ( x) = 2 , −∞ < x < ∞, a > 0 . This density function is
a + x2

(A) symmetrical about x = 0 so that its median is 0
(B) symmetrical about x = a so that its median is a
(C) asymmetrical with median a
(D) positively skewed

36. Find the correlation coefficient if the two regression coefficients are 0.4 and 0.9
respectively.

(A) 0.36
(B) 0.55
(C) 0.60
(D) 0.65

37. A measure of statistical dispersion intended to represent the income inequality is

(A) Co-efficient of skewness
(B) Correlation coefficient
(C) Regression coefficient
(D) Gini’s coefficient

38. Failure to measure some of the units in the selected sample leads to

(A) Standard error
(B) Sampling error
(C) Non-sampling error
(D) Experimental error

39. What is the probability of a specified sample of 3 units selected from 10 units?

3
(A)
10
1
(B)
120
(C) 10C3
3
(D)
30

40. Which of the following statements is mostly considered in Neyman allocation of
sample size in different strata?

(A) Stratum size and stratum variation
(B) The cost in taking observations per sampling unit in the stratum
(C) Sampling fraction
(D) Total number of units in the population

41. The relative precision of systematic sampling with simple random sampling is given
by

( N − 1)
(A) [1 + ρ (n − 1)]
( N − n)
(1 − f )
(B) [1 + ρ (n − 1)]
( n)
( N − 1)
(C) [1 − ρ (n − 1)]
( N − n)
(1 − f )
(D) [1 − ρ (n − 1)]
( n)

42. A study provides the population size, variance and cost per unit of two strata are (400,
10, 4) and (600, 20, 9) respectively. Find the required sample size under optimum
allocation by considering the fixed variance which is equal to 1.

(A) 40, 30
(B) 100, 67
(C) 67,100
(D) 88,176

43. What will be the mean sum of squares due to error if the design having 4 treatments
each replicates 5 times, total sum of squares is 256, and treatment sum of square is
144?

(A) 7
(B) 48
(C) 112
(D) 55

44. What is the formula to calculate a missing value in a n × n latin square design if R, C,
th th th
and T be the total of known observations in the i row, j column, k treatment
respectively and S is the total of known observations?

n( R + C + T ) − 2S
(A)
(n − 1)(n − 2)
n( R + C + T )
(B)
(n − 1)(n − 2)
n( R + C + T ) − S
(C)
(n − 1)(n − 2)
n( R + C + T ) − 2 S
(D)
n2 − 1

3
45. What will be the error degrees of freedom in a 2 factorial experiment having b
randomized blocks?

(A) 7(b – 1)
(B) 7b – 1
(C) 8b – 1
(D) b–1

46. Find the missing value in a randomized block design if 40 and 90 are the treatment
and block total of known observations in the missing row and column respectively,
200 is the sum of known observations in the design having 5 treatments and 4 blocks.

(A) 38
(B) 18
(C) 18.9
(D) 30

3
47. The contrast for estimating the ABC effect in 2 factorial experiments is

(A) (abc) − (bc) + (ac) − (c) + (ab) + (b) − (a) − (1)
(B) (abc) + (bc) + (ac) − (c) − (ab) − (b) − (a) + (1)
(C) (abc) + (bc) − (ac) + (c) − (ab) − (b) + (a) − (1)
(D) (abc) − (bc) − (ac) + (c) − (ab) + (b) + (a) − (1)

48. With the following diagram, find the optimum coordinates of the given linear
programming problem, Min Z = 200x1 + 400x2,

(A) (50,100)
(B) (0, 0)
(C) (100,100)
(D) (250, 50)

49. Find the optimum assignment from the given optimum table in an assignment
problem.

T1 T2 T3 T4 T5
D1 15 0 20 15 0
D2 15 15 0 10 0
D3 15 0 20 15 5
D4 0 15 20 0 5
D5 5 0 10 0 0

(A) D1 → T5 ; D2 → T3 ; D3 → T2 ; D4 → T4 ; D5 → T1
(B) D1 → T1 ; D2 → T2 ; D3 → T3 ; D4 → T4 ; D5 → T5
(C) D1 → T2 ; D2 → T5 ; D3 → T3 ; D4 → T1 ; D5 → T4
(D) D1 → T5 ; D2 → T3 ; D3 → T2 ; D4 → T1 ; D5 → T4

50. How many ways of the tour plan if a salesman has to visit 6 cities?

(A) 720
(B) 120
(C) 5040
(D) 6

51. What will be the value of the game with the following payoff matrix?

Player B
 −1 2 −2 
Player A  
 6 4 −6 

(A) −2
(B) −1
(C) −6
(D) 6

52. Find the critical path of the following network

(A) 1→2→3→4→5
(B) 1→2→4→5
(C) 1→3→4→5
(D) 1→4→5

53. Basic feasible solution of a LPP is defined as

(A) a solution which satisfies the non-negative
non restrictions
(B) a basic solution which satisfies the non-negative
non restrictions
(C) a basic solution which optimize the objective function
(D) a solution which satisfies the constraints and non-negative
non restrictions

54. The process capability index C pk is defined by

USL − µˆ µˆ − LSL 
(A) C pk = min  , 
 3σˆ 3σˆ 

USL − µˆ LSL − µˆ 
(B) C pk = min  , 
 3σˆ 3σˆ 
USL − µˆ LSL − µˆ 
(C) C pk = min  , 
 6σˆ 6σˆ 
USL − µˆ µˆ − LSL 
(D) C pk = min  , 
 6σˆ 6σˆ 

55. Which of the following are the performance measures of sampling plans?

(A) Operating characteristic function and average outgoing quality
(B) Acceptable quality level and limiting quality level
(C) Producer’s quality level and consumer’s quality level
(D) Point of control, acceptable quality level and limiting quality level

56. Under the assumption of normal distribution, the usual three-sigma limits imply that
the type I error probability is

(A) α = 0.001
(B) α = 0.0027
(C) α = 0.0456
(D) α = 0.0

2
57. The minimum variance bound for unbiased estimator of variance (σ ) of a random
sample of n independent normal variate is

σ2
(A)
n
n
(B)
σ2
2σ 4
(C)
n
n
(D)
( 2σ 4 )

58. The Fisher information measure of n independent random observations from the
1
exponential distribution with mean , is
θ

−n
(A)
θ
n
(B)
θ2
1
(C)
θ2
−1
(D)
θ2

2
59. If Xi follows bin(n, θ) and t = Σxi, (i = 1, 2…, k), then the UMVUE of θ is

t 2 −1
(A)
nk − 1
t2 − t
(B)
nk − 1
t 2 −1
(C)
nk ( nk − 1)
t (t − 1)
(D)
nk (nk − 1)

60. Which of the following is NOT true?

(A) A complete sufficient statistic is minimal sufficient statistic
(B) A statistic is said to be ancillary if its distribution depends on the θ
(C) If S is complete sufficient statistic for θ, then any ancillary statistic A is
independent of θ
(D) A minimal sufficient statistic may not be complete sufficient statistic

61. Let X be the mean of a random sample of size n from normal distribution
with mean µ and variance 25. What is the approximate value of n such that
P( X − 1 < µ < X + 1) = 0.95?

(A) 10
(B) 49
(C) 2401
(D) 96

62. Which of the following is NOT true?

(A) Consistent estimator is unique
(B) An unbiased estimator need not to be consistent
(C) Unbiased estimator may not be unique
(D) T 1 and T 2 are functionally related if T 1 and T 2 are sufficient statistics

2
63. Let Xi (i = 1, 2, …, k) be iid variates from N(µ,σ ) and µ is known. What distribution
2
of pivot for constructing the confidence interval for the parameter σ is used?

(A) t - distribution
(B) F - distribution
(C) normal distribution
(D) chi-square distribution

64. A test φ is unbiased for testing H 0 : θ ∈ Θ0 vs H1 : θ ∈ Θ1 if and only if its power
function βφ(θ) satisfies

(A) βφ(θ) ≥ α, for θ ∈ Θ0
(B) βφ(θ) ≥ α, for θ ∈ Θ1
(C) βφ(θ) ≤ α, for θ ∈ Θ1
(D) βφ(θ) = α, for θ ∈ Θ0

65. If the powers of the most powerful tests at level α and α′ be β and β′, for testing a
simple hypothesis H0 against a simple alternative H1, then always

(A) β > β′
(B) β < β′
(C) β ≠ β′
(D) β = β′

66. The observed values are x1 = −1 and x2 = 3, in a test function

1, if x1 + x2 > 1

ϕ ( x) = 1 4, if x1 + x2 = 1
0, if x + x < 1
 1 2

(A) reject H0
(B) accept H0
(C) neither reject nor accept H0
(D) not sufficient information given

67. Find the size of the test for testing H0 : θ = ¼ against H1 : θ > ¼ by taking random
sample of size 10 and rejecting H0 iff Σ xi > 8 (for all i = 1, 2…,10), when X is a
Bernoulli random variable with parameter θ

31
(A)
410
410
(B)
31
10
1
(C)  
4
1
(D)
410

68. Let the random sample of sizes 20 and 20 drawn from the two independent
populations respectively, then the mean of Wald-Wolfowitz run test statistic (under
normal approximation) is

(A) 11
(B) 20
(C) 19
(D) 21

69. The likelihood ratio test statistic for testing H0 : θ = θ0 against H1 : θ ≠ θ0 based on a
2(θ − x)
sample of size 1 from the density f (x, θ) = 2
, 0 < x < θ is
θ

2 x (θ0 − x )
(A)
θ02
2 (θ0 − x )
(B)
θ02
4 x (θ0 − x )
(C)
θ02
4 x (θ − x )
(D)
θ2

70. If X and Y are the random variables having the joint density function given by
f ( x, y ) = x + y , 0 < x < 1, 0 < y < 1, then the covariance of X and Y is equal to

1
(A)
11
1
(B)
12
1
(C)
144
1
(D)
72

71. The random variables X and Y whose joint probability density function given by
f ( x, y ) = e− y , 0 < x < y < ∞ are

(A) independent
(B) pairwise independent
(C) dependent
(D) mutually independent

72. If {An } is a non-decreasing sequence of events, then

∞ 
(A) (
n→∞ )
P lim An = P  ∪ Ai 
 i =1 
∞ 
(B) (
n→∞ )
P lim An ≤ P  ∪ Ai 
 i =1 
∞ 
(C) (
n→∞ )
P lim An = P  ∩ Ai 
 n=1 
∞ 
(D) (
n→∞ )
P lim An ≥ P  ∩ Ai 
 n=1 

73. If X 1, X 2 ,… , X n are mutually independent random variables, each having finite mean
1 n
µ and standard deviation σ, and if Sn = ∑ X i , then the law of large numbers states
n i =1
that

S 
(A) lim P  n − µ < ε  = 0
n→∞
 n 
S 
(B) lim P  n − µ ≥ ε  = 0
n→∞
 n 
S 
(C) lim P  n − µ ≥ ε  = 1
n→∞
 n 
S 
(D) lim P  n − µ < ε  > 1
n→∞
 n 

74. Test of homogeneity of variances of several normal populations is done by

(A) Sequential Probability Ratio Test
(B) Bartlett test
(C) Sign test
(D) Kruskal-Wallis test

75. If the Laspeyre’s and Paasche’s price index are 144 and 121 respectively, then the
Fisher’s price index is

(A) 132.5
(B) 132
(C) 265
(D) 204.5

76. Which organization provides data for national income?

(A) Central Statistical Organization
(B) Indian Census Organization
(C) National Sample Survey Organization
(D) National Population Commission

77. A census method in which persons are counted wherever they are at the date of census
is known as

(A) De facto method
(B) De jure method
(C) Slip system
(D) Pure census

π
2
∫ sin x cos x dx =
5
78.
0

1
(A)
3
1
(B)
6
2
(C)
3
3
(D)
2

π
2
79. ∫ sin x + 1 cos x dx =
0

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

1
80. The value of ∫ x(1 − x) 4 dx is
0

1
(A)
12
1
(B)
30
1
(C)
24
1
(D)
20

2 dw
81. If w = x + 2y + z and x = cos t, y = sin t , z = t, then is
dt

(A) sin t + cos t + 2t
(B) –sin t – cos t +2t
(C) –sin t + 2cos t + 2t
(D) sin t + 2 cos t + 2t

dy
82. Let y = u , u = v3 + 1, v = sin x, then =
dx

3
(A) sin x cos x
2
3 sin 2 x cos x
(B)
2 sin 3 x + 1
3 sin x cos x
(C)
2 sin 4 x + 1
3 cos x
(D)
2 sin 3 x + 1

∞ ∞
n!
83. The series of positive terms ∑ an = ∑ n is
n =0 n=0 n

(A) Convergent
(B) Divergent
(C) Equal to 1
(D) Equal to 0

84. If a function f ( x ) is twice differentiable and has a minimum value, then

(A) f ''( x) < 0
(B) f ''( x) > 0
(C) f ''( x) = 0
(D) f ''( x) is a constant

 cos θ sin θ 
85. The matrix A =  is
 − sin θ cos θ 

(A) Singular
(B) Orthogonal
(C) Skew Symmetric
(D) Negative semi definite

1 −2   −3 3 
86. If A − 2 B =   and 2 A − 3B =   then B is equal to
3 0   1 −1

 −5 7 
(A)  −5 1 
 
 −5 7 
(B)  0 1
 
 −5 7 
(C)  −5 −1
 
 5 −7 
(D)  −5 1 
 

87. If A and B are two possible outcomes of an experiment and if
P ( A) = 0.4, P ( AC ∩ B C ) = 0.3 then for what value of P ( B ), A and B are mutually
exclusive?

(A) 0.5
(B) 0.4
(C) 0.3
(D) 0.2

88. Let P be the probability function that assigns the same weight to each of the
points of the sample space, Ω = {1, 2, 3, 4}. Consider the events E = {1, 2}, F = {1, 3}
and G = {3, 4}. Then which of the following statement(s) is/are true?

(i) E and F are independent
(ii) E and G are independent
(iii) F and G are independent
(iv) E, F and G are independent

(A) (i) and (ii) only
(B) (i) and (iii) only
(C) (ii) and (iii) only
(D) All (i), (ii) and (iii)

89. The probabilities that an item has defects I, II and III are respectively 0.3, 0.2 and 0.1.
The probability that an item would not contain any one of these defects, is not less
than

(A) 0.121
(B) 0.901
(C) 0.400
(D) 0.504

90. From integers 1 to 40, both inclusive, one integer is chosen at random. What is the
probability that it is divisible by 4 or 5?

2
(A)
5
1
(B)
20
9
(C)
20
5
(D)
108

91. An urn contains ' a ' white and ' b ' black balls. All the balls are drawn from it in
succession. What is the probability that the second drawn ball is white?

a
(A)
(a + b − 1)
(a − 1)
(B)
(a + b − 1)
a
(C)
( a + b)
(a + b − 1)
(D)
( a + b)

2
92. Let A and B be two independent events with P ( A) + P ( B ) = 1, P ( A ∩ B ) = and
9
P ( B ) < P ( A) . Then P ( A) equals

1
(A)
3
1
(B)
2
2
(C)
3
3
(D)
4

93. Let X1 and X 2 be independent standard normal variates. Let U1 and U 2 be
independent and identically distributed U (0,1) variates, independent of X1 and X 2 .
X1U1 + X 2U 2
Define Z = . Then which of the following is/are CORRECT?
U12 + U 22

(i) E (Z ) = 0
(ii) V ( Z ) = 1
(iii) Z is Cauchy
(iv) Z ~ N (0,1)

(A) (i) and (ii) only
(B) (iii) only
(C) (i), (ii) and (iv) only
(D) (i), (ii) and (iii) only

x
94. The probability density function (pdf) of X is f ( x) = + k ; 0 < x < 3. Then, the value
6
of k is

1
(A)
3
1
(B)
6
1
(C)
12
1
(D)
2

1
95. The pdf of X is f ( x) = e−|x| , −∞ < x < ∞. The value of P ( X > −2) is
2

1
(A)
(
2 1 − e −2 )
(B) e−2
1
(C) 1 − e−2
2
e −2
(D)
2

96. If the pdf of a random variable X is f ( x) = θ e−θ x , x > 0, then the pdf of Y = e−θ X is

(A) f ( y) = θ y, θ < y < 1

(B) f ( y ) = θ yθ −1, θ < y < 1
(C) f ( y ) = 1, 0 < y < 1
(D) f ( y ) = θ log y , 0 < y < 1

97. For a hypergeometric distribution having its probability mass function
 M  N − M 
 x  n − x 
Px =    , the actual lower and upper limits of x are respectively
N
 
n

(A) 0 and ∞
(B) 0 and M
(C) n and M
(D) 0 and min(M, n)

98. A Poisson distribution with mean λ > 0 is

(A) positively skewed and platykurtic
(B) positively skewed and leptokurtic
(C) negatively skewed and platykurtic
(D) negatively skewed and leptokurtic

4
99. If X is uniformly distributed with mean 1 and variance , then the range of X is given
3
by

(A) (0, 2)
(B) (−1, 3)
(C) (−3, 5)
(D) (−2, 1)

100. Let X be a discrete random variable with moment generating function,
2 3
1 3  3 1 
M x (t ) =  + et   + et  . Then, which one of the following is CORRECT?
4 4  4 4 

9
(A) E( X ) =
4
15
(B) V (X ) =
32
27
(C) P ( X ≥ 1) =
1024
3
(D) P ( X = 5) =
1024

2t (1+t )
101. Let the random variable X have moment generating function, M x (t ) = e . Then
P ( X ≤ 2) is

1
(A)
4
1
(B)
2
1
(C)
3
2
(D)
3

102. The second moment about origin of beta distribution of the first kind having
parameters ‘a’ and ‘b’ is

a( a − 1)
(A)
(a + b)(a + b − 1)
a( a + 1)
(B)
(a + b)(a + b + 1)
a(a + 1)
(C)
b(b + 1)
2
 a +1 
(D)  
 a+b

2
 
− x +1 
1  3 
103. The pdf of a normal distribution is given by f ( x) = e ; −∞ < x < ∞ . Find
3 π
the values of its characteristics given in Group A by matching them with those given
in Group B

Group A Group B
243
(a) Mode (i)
4
(b) Variance (ii) −1
9
(c) Fourth central moment (iii)
2

(A) (a) – (i), (b) – (ii), (c) – (iii)
(B) (a) – (i), (b) – (iii), (c) – (ii)
(C) (a) – (iii), (b) – (i), (c) – (ii)
(D) (a) – (ii), (b) – (iii), (c) – (i)

104. If X ~ N (0,1) , the values of E (sin X ) and E (cos X ) are respectively

(A) 0 and 1
(B) 1 and 0
1
(C) 0 and
e
1
(D) and 1
e

105. Let X i' s(i = 1, 2,3) be independent N (0,1) variates. If ( X1 + kX 3 , X 2 + kX 3 ) follows
bivariate normal distribution with correlation coefficient 0.25, then the absolute value
of will be

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

106. If the random variables X, Y and Z have the variance, σ x2 = 10 , σ y2 = 14 and σ z2 = 20 ;
cov( X , Y ) = 1, cov( X , Z ) = −3 and cov(Y , Z ) = 2, then what is the covariance
between U = X + 4Y + 2Z and V = 3 X − Y − Z .

(A) −76
(B) 82
(C) −82
(D) 76

107. Let X1 , X 2 , X 3 and X 4 be four uncorrelated random variables each with variance
σ 2 then the correlation coefficient between U = X1 + X 2 + X 3 and V = X1 + X 2 + X 4

(A) 1
1
(B)
3
2
(C)
3
(D) 0

108. If only two observations on ( X , Y ), ( x1 , y1 ) and ( x2 , y2 ) are available, then which of
the following is an appropriate estimator of the cov( X , Y ) ?

(A) (1 2 ) ( x1 − x2 )( y1 − y2 )
(B) (1 4 ) ( x1 − x2 )( y1 − y2 )
(C) ( x1 − x2 )( y1 − y2 )
(D) (1 2 ) ( x1 − y1 )( x2 − y2 )

109. If σ X2 , σ Y2 and σ X2 −Y are the variance of X, Y and X – Y respectively, then the
correlation coefficient between X and Y is

σ X2 + σ Y2 − 2σ X2 −Y
(A)
2σ X σ Y

σ X2 + σ Y2 − σ X2 −Y
(B)
2σ X σ Y

σ X2 + σ Y2
(C)
2σ X σ Y

σ X2 − σ Y2
(D)
2σ X σ Y

110. Let X1, X 2 ,..., X n be identically independently distributed random variables each
having a uniform distribution on (0, 1). Consider the histogram of these values with k
equally spaced intervals given by {( ai , bi ] , i = 1, 2,..., k } where ai = (i − 1) k and
bi = ai + (1 k ) . Let Ni be the number of values in the interval ( ai , bi ] . Then the
covariance N1 and N k is

(A) 0
−n
(B)
k2
n
(C)
k2
1
(D)
2

111. The minimum value of V (Y − aX ) for all the value of ' a ' is given by

V (Y )
(A) ρ2
V (X )
V (X )
(B) ρ2
V (Y )
(C) ρ 2V (Y )
(D) (1 − ρ 2 )V (Y )
n
112. For n pairs of observations, the maximum possible value of ∑ ∑ di2 where di is
i =1
th
the difference between the two ranks of i object, is

(A) ( )
n n2 − 1

n ( n 2 − 1)
(B)
2

(C)
(
n n2 − 1 )
6

(D) (
n n2 − 1 )
3

113. If x = 4 y + 5 and y = kx + 4 are the lines of regression of x on y and that of y on x
respectively, then which one of the following is true?

(A) k ≥1
(B) k ≤ −1
(C) −1 ≤ k ≤ 0
(D) 0 ≤ k ≤ (1 4 )

114. The two regression lines associated with ( X , Y ) are 8 x − 10 y + 66 = 0 and
40 x − 18 y = 214. The correlation coefficient between X and Y is

(A) −0.6
(B) +0.6
(C) −0.36
(D) +0.36

1
115. If the joint pdf of (X, Y) is f ( x, y ) = x 2e − y (1+ x ) , x > 0, y > 0, then the regression
3
equation of Y on X is

1
(A) y=
(1 + x)
x2
(B) y=
(1 + x)
(C) y = 1+ x
x2
(D) y=
2(1 + x)2

116. The standard error of the sample correlation coefficient rxy is equal to

1− r2
(A)
n−2

(1 − r 2 )
(B)
( n − 2)

(C) 1− r 2
n−2
1− r2
(D)
n−2

117. A simple random sample of n units is drawn from population containing N units with
replacement. Let T1 be the mean of all the units in the sample and T2 the mean of
distinct units in the sample. Which one of the following assertions about T1 and T2 is
correct?

(A) T2 is a biased estimator of the population mean
(B) Both T1 and T2 are unbiased estimators of the populations mean and
V (T1 ) ≥ V (T2 )
(C) Both T1 and T2 are unbiased estimators of the populations mean and
V (T1 ) = V (T2 )
(D) Both T1 and T2 are unbiased estimators of the populations mean and
V (T1 ) < V (T2 )

118. In simple random sampling without replacement, the probability that two particular
units are selected in the sample is

n(n − 1)
(A)
N ( N − 1)

n2
(B)
N2
1
(C)
N ( N − 1)
N ( N − 1)
(D)
n +1

119. From a population of 200 units, a simple random sample without replacement has
been obtained as (4, 3, 5, 8, 9, 7). An unbiased estimate of the population total is

(A) 3600
(B) 6
(C) 1200
(D) 1800

120. In a simple random sample of 4 from 16 districts in a state has standard deviation of 45.
Then, the standard error of mean is

(A) 21.5
(B) 22.5
(C) 19.5
(D) 11.25

121. Suppose there are k strata of N = kM units each with size M. Draw a sample of size
ni denote by yi , the sample mean of the study variable selected in the i th stratum,
1 k 1 n ny
i = 1, 2,...k . Define ys = ∑ yi and yw = ∑ i i . Which of the following is
k i =1 n i =1 n
necessarily true?

(A) ys is unbiased but yw is not unbiased for the population mean
(B) ys is not unbiased but yw is unbiased for the population mean
(C) Both ys and yw are unbiased for the population mean
(D) Neither ys nor yw is unbiased for the population mean

122. In the following ANOVA table for Completely Randomized Design (CRD) find the
Error Sum of Squares where b is some positive value.

Source Df SS MSS
Treatment 4 -- 3b
Error -- -- b
Total 9 425

(A) 125
(B) 25
(C) 106.25
(D) 47.22

123. Control chart for the number of defectives is

(A) c-chart
(B) p- chart
(C) np- chart
(D) R-chart

124. Which of the following is NOT true for R-chart?

(A) R and standard deviation fluctuate together in case of small samples
(B) R is easily calculable
(C) R- charts are economical
(D) R- chart can identify small shifts in the data

125. When there is no defective in the lot, the OC function for p = 0 is

(A) L (0) = 0
(B) L (0) = 1
(C) L (0) = ∞
(D) L (0) = 0.5

126. The producer’s risk is probability of

(A) rejecting a good lot
(B) accepting a good lot
(C) rejecting a bad lot
(D) accepting a bad lot

127. Let X1, X 2 ,..., X n be random samples from Uniform (θ1,θ2 ) and let
X (1) , X (2) ,..., X ( n ) denote their ordered values. Which of the following statement is
correct?

(A) X (1) and X ( n ) are sufficient for θ 2 and θ1 respectively

(B) ( X (1) , X (n) ) is jointly sufficient for (θ1,θ2 )
(C) X (1) is sufficient for both θ1 and θ 2

(D) X ( n ) is sufficient for both θ1 and θ 2

128. Let X1, X 2 ,..., X n be a random sample from Poisson distribution with mean µ. Then
the MLE of µ is

(A) ∑ Xi n
∑( Xi − µ )
2
(B)

(C) ∑ X i2
(D) Median of X1 ,..., X n

129. If T1 and T2 be two unbiased estimators of a parameter θ, then the efficiency of T1
with respect to T2 is

(A) V (T1 ) + V (T2 )
V (T2 )
(B)
V (T1 )
(C) V (T1 ) − V (T2 )
V (T1 )
(D)
V (T2 )

130. Let X1 and X 2 be sample means based on independent random samples drawn from
2
normal distribution with means µ1 and µ2 respectively and common variance σ . If
2
Sp denote the pooled sample variance, then 100 (1 − α)% for confidence for (µ1 − µ2)
is

1 1 
(A) ( X 1 − X 2 ) ± Zα 2 S P2  + 
 n1 n2 

1 1 
(B) ( X1 − X 2 ) ± tα 2 S P2  + 
 n1 n2 

1 1 
(C) ( X1 − X 2 ) ± Fα 2 S P2  + 
 n1 n2 

(D) ( X 1 − X 2 ) ± Zα 2 S P2 ( n1 + n2 )

131. ( )
Let X1, X 2 ,..., X n be a random sample from N µ , σ 2 , σ 2 is known. To test
H0 : µ = µ 0 against H1 : µ = µ 1 , we use

(A) Z – test
(B) t – test
(C) F – test
(D) Chi square test

132. In the usual notation, the confidence interval for the population variance σ 2 is

( n − 1) s 2 ( n − 1) s 2
(A) ≤σ2 ≤
χα2 χ12−α
2 2

ns 2 ns 2
(B) ≤σ2 ≤
χα2 χ12−α
2 2

(n − 1) s 2 (n − 1) s 2
(C) ≤σ2 ≤
χα2 χ12−α

( n − 1) s 2 ( n − 1) s 2
(D) ≥σ2 ≥
χα2 χ12−α
2 2

133. If [62, 75] is the 95% confidence interval for the mean of a population based on 10
observations then to test H : Mean = m against K : Mean < m, the correct test
procedure is

(A) Reject H if m ∈ [62,75]
(B) Reject H if m lies outside the interval [62,75]
(C) Reject H if m < 62
(D) Reject H if m > 75

134. Student t-test is used for testing the H0 : ρ = 0 against H1 : ρ > 0, ρ being the
r
population correlation coefficient for which the test statistic t = n − 2 and r
1− r2
being the sample correlation coefficient. The hypothesis is rejected if

(A) r is positive and t ≥ tα ,n−2
(B) r is negative and t ≥ tα ,n−2
(C) r is positive and t = tα ,n−1
(D) r is negative and t = tα ,n

2 2
135. Let X1, X 2 ,..., X n be a random sample from N(µ , σ ), σ is unknown. To test
H0 : µ = µ 0 Vs. µ = µ 1 , we use

(A) Z – test
(B) t – test
(C) F – test
(D) χ 2 − test

136. Every UMP critical region is necessarily

(A) biased
(B) a null set
(C) an infinite set
(D) unbiased

137. Suppose ( X1, X 2 ) ∼Bivariate Normal distribution with E ( X1 ) = E ( X 2 ) = 0;
x −y2
1
V ( X1 ) = V ( X 2 ) = 2 and cov ( X1, X 2 ) = −1. If φ ( x) = ∫ e 2 dy , then
2π −∞
P  X1 − X 2 > 6 is equal to

(A) φ ( −1)
(B) φ ( −3)
(C) φ ( 6)
(D) φ (− 6)

138. Let (X, Y ) have bivariate normal distribution. Which of the statements are correct?

(i) X and Y are independent if and only if the correlation coefficient between them
is zero
(ii) Every linear combination of X and Y is a normal variate
(iii) The regression equation of Y on X and regression equation X on Y are linear
and homoscedastic

(A) (i) and (iii) only
(B) (ii) and (iii) only
(C) (i) and (ii) only
(D) All (i), (ii) and (iii) are correct

139. X and Y are two independent random variables with some of the joint probabilities
given as

X
1 2 3 Total
0 .08 -- -- --
Y
1 -- -- -- 0.2
Total -- 0.2 --

Then, P ( X = 3, Y = 1) is

(A) 0.10
(B) 0.12
(C) 0.14
(D) 0.16

xy 3
140. The joint distribution function of X and Y is F ( x, y ) = , 0 < x < 2; 0 < y < 1. The
2
marginal pdf of Y is

(A) fY ( y ) = 4 y 3 , 0 < y < 1
(B) fY ( y ) = 3 y 2 , 0 < y < 1
(C) fY ( y ) = 2 y,0 < y < 1
(D) fY ( y ) = y 3 2, 0 < y < 1

141. If X is a random variable with µ and variance σ 2 and k is any positive number, then
which one of the following is not the Chebyshev’s inequality?

(A) P { X − µ ≥ kσ } ≤ 1 k 2

(B) P { X − µ < kσ } ≤ 1 − 1 k 2 ( )
(C) P { X − µ > k} ≤ σ 2 k 2

(D) P { X − µ < k} ≥ σ 2 k 2

142. Let X1, X 2 ,... X n be identically independently distributed Exp(1) variate and
n
Sn = ∑ X i . Using the central limit theorem, the value of lim P ( Sn > n ) is
n→∞
i =1

(A) 0
1
(B)
3
1
(C)
2
(D) 1

143. Let X1 , X 2 ,... X n be a random sample from the uniform distribution on (0, 2) and
M n = max ( X1 , X 2 ,... X n ) . Then which one of the following statement is not true?

(A) M n → 2 almost surely
(B) M n → 2 in probability
(C) M n → 2 in distribution
(D) ( M n − 2) n converges in distribution to a normal random variable

144. Let X1 , X 2 ,... X n be iid variates with mean µ and variance σ 2 and as n → ∞

( X12 + X 22 + ... + X n2 ) n 
P
→ C for 0 ≤ C < ∞. The value of C is

(A) σ 2 + µ 2
(B) σ 2
(C) µ 2
(D) 1

145. Let { X n } , n ≥ 1 be a sequence of identically independently distributed random
variables having mean = n and variance = 2n. Let Z be a standard normal variate.
  3  
n→∞   4 
(
Then using the central limit theorem, lim  P  X n > n  + P X n > n + 2 2n 

)
equals

(A) 1 + P [ Z ≤ 2]
(B) 1 − P [ Z ≤ 2]
(C) P [ Z ≤ 2]
(D) 2 − P [ Z ≤ 2]

146. A appeared in three tests of the value 20, 50 and 30 marks respectively. He obtained
75% and 60% marks in the first two test respectively. If the aggregate marks is 60%,
his percentage of marks in the third test is

(A) 55%
(B) 50%
(C) 65%
(D) 45%

147. The average of n numbers is z. if the number x is replaced by the number x ' the
average becomes z ' . The relation among n, z , z ', x , x ' is

z '− z 1
(A) =
x '− x n
z − z' 1
(B) =
x '− x n
x '− x 1
(C) =
z '− z n
x − x' 1
(D) =
z' n

n 1
 
148. If ∑  log xi − log 2  = 0 , then the geometric mean of n observations,
i =1  2 
x1, x2 ,..., xn is

(A) 2
(B) log 2
(C) 2
(D) 4

149. The sum and the sum of squares of 20 observations were found as 790 and 36500
respectively. Later on, it was detected that one observation had wrongly been taken as
35 instead of 45. The correct value of the variance is

(A) 220
(B) 245
(C) 248
(D) 265

n 4 n 2
150. Let ∑ ( X i − X ) = 100 and ∑ ( X i − X ) = 30 . Which of the following may be a
i =1 i =1
possible value of n?

(A) 10
(B) 9
(C) 8
(D) 7

FINAL ANSWER KEY
Subject Name: STATISTICS
SI No. Key SI No. Key SI No. Key SI No. Key SI No. Key
1 A 31 D 61 D 91 C 121 A
2 B 32 B 62 A 92 C 122 A
3 B 33 A 63 D 93 C 123 C
4 B 34 A 64 B 94 C 124 D
5 A 35 A 65 D 95 C 125 B
6 A 36 C 66 A 96 C 126 A
7 A 37 D 67 A 97 D 127 B
8 A 38 C 68 D 98 B 128 A
9 D 39 B 69 C 99 B 129 B
10 C 40 A 70 C 100 A 130 B
11 C 41 A 71 C 101 B 131 A
12 B 42 D 72 A 102 B 132 A
13 A 43 A 73 B 103 D 133 C
14 B 44 A 74 B 104 C 134 A
15 A 45 A 75 B 105 D 135 B
16 C 46 D 76 A 106 C 136 D
17 C 47 D 77 B 107 C 137 D
18 D 48 C 78 B 108 B 138 D
19 B 49 D 79 C 109 B 139 C
20 B 50 B 80 B 110 B 140 B
21 B 51 A 81 C 111 D 141 D
22 B 52 A 82 B 112 D 142 C
23 D 53 B 83 A 113 D 143 D
24 D 54 A 84 B 114 B 144 A
25 A 55 A 85 B 115 A 145 D
26 C 56 B 86 C 116 B 146 B
27 A 57 C 87 C 117 D 147 A
28 B 58 B 88 B 118 A 148 D
29 A 59 D 89 C 119 C 149 D
30 A 60 B 90 A 120 B 150 A