PG-QP – 29 -4- *PGQP29*
PART – A
Questions 1 – 10 : Choose the most grammatically correct and meaningful option from those provided at the end of each sentence :
1. One who is incapable of committing a mistake is called A) sure B) certain C) definite D) infallible
2. The opposite of ‘curious’ is A) calm B) indifferent C) undemanding D) provoked
3. Their entire future ______on the result of this examination. A) depends B) depending C) resolved D) revolving
4. We haven’t had ______rest all of this week. A) many B) some C) any D) no
5. This seems to be the ______question to answer right now. A) hardly B) hardest C) very hard D) harder
6. If I ______the decision-maker, things would have been much easier. A) had been B) were C) was D) am
7. Everyone was in time for the show, ______? A) wasn’t he B) isn’t it C) weren’t they D) were they
8. He has ______all that he had in this gamble. A) lossed B) loosed C) losed D) lost
9. We should refrain ______overeating. A) from B) to C) in doingD) over
10. ______are we waiting for ? A) Why B) When C) What D) How *PGQP29* -5- PG-QP – 29
11. The highest rank in Indian Air Force is A) Air Chief Marshal B) Air Marshal C) Air Vice-Marshal D) Air Commander
12. Who was the first women to win the Jnanpith Award ? A) Mahadevi Varma B) Subhadra Kumar Chouhan C) Amrita Pritam D) Ashapurna Devi
13. Yuri Gagarin was a A) Nuclear Scientist B) Musician C) Playcont D) Cosmonaut
14. ‘People’s Daily’ is an important newspaper of A) China B) Russia C) Egypt D) Indonesia
15. Find the next term of the series POQ, SRT, VUW A) XYX B) XZY C) ZYX D) YXZ
16. Which of the following States has banned the slaughter of camels ? A) Gujarat B) Punjab C) Haryana D) Rajasthan
17. The first woman Judge of the Supreme Court of India was A) Leela Seth B) Cornelia Sorabjee C) Omna Kunjamma D) M. Fathima Beevi
18. Shelly travelled 9 km to the West and turned right and covered 7 km; then she turned left and covered 8 km; then she turned back and travelled 11 km, finally she turned right and travelled 7 km. How far is she from the starting point ? A) 2 km B) 6 km C) 10 km D) 15 km
19. The ______is an instrument used to measure distance travelled on foot. A) Phorometer B) Pedometer C) Micrometer D) Pachymeter PG-QP – 29 -6- *PGQP29*
20. In a row of 21 girls, when Monika was shifted by four places towards the right, she became 12th from the left end. What was her earlier position from the right end of the row ? A) 9th B) 10th C) 12th D) None of these
21. Micheal Corleone is a character from the book A) For Whom the Bells Toll B) Lord of the Rings C) Love Story D) Godfather
22. Who amongst the following are associated with communism ? (i) Karl Marx (ii) Ayn Sue-Khi (iii) Ayn Rend (iv) Mao Tse Tung The correct answer is A) only (i) B) (i) and (iii) C) (i) and (iv) D) only (iv)
23. In covering a distance of 30 km, Abhay takes 2 hours more than Sameer. If Abhay doubles his speed, then he would take 1 hour less than Sameer. Abhay’s speed is A) 5 kmph B) 6 kmph C) 6.25 kmph D) 7.5 kmph
24. The dog breed “Pug” is associated with the advertisement of A) Airtel B) Vodafone C) Aircel D) BSNL
25. “Anokhi” is a brand associated with A) Hand Block Prints B) Pottery C) Antique Furniture D) Organic Food *PGQP29* -7- PG-QP – 29
PART – B
26. Let a be a non-zero real number. Consider the sequence {xn) where a2 xl = 2a and xn = xl – x for n≥ 2. Then xn for n ≥ 2, can be expressed in terms n -1 of a as
n n +1 A) x = a B) x = a n n +1 n n
n n −1 C) x = a D) x = a n n −1 n n
m n 27. Consider the set X = { + : m, n are natural numbers}. Then n m
A) X is neither bounded below nor bounded above
B) X is bounded above and sup X = 4
C) X is bounded below and inf X = 2
D) X is bounded below and inf X = 1
⎧ n ⎫ 28. Consider the sequence ⎨ ⎬ . This sequence ⎩2n ⎭
A) is bounded but not convergent
B) is not bounded
C) is convergent and converges to 1
D) is convergent and converges to 0. PG-QP – 29 -8- *PGQP29*
∞ 1 3 29. The series ∑ =+Sin Cos + n11 2n 2 n 1 1 A) converges to Sin 1 B) converges to Sin 1 2 1 C) converges to Cos 1 D) converges to Cos 1 2 ⎛ π ⎞ Cos⎜ Cosx ⎟ ⎝ 2 ⎠ 30. The lim exists and its value is x→0 Sin (Sin x) A) –1 B) 0 C) 1 D) 2 1 31. Consider the function f(x) = for real numbers x≠ 0. Denote by e1/ x +1 lim lim − f(x) the left hand limit and + f(x) the right hand limit x→0 x→0 of f(x) at x = 0. Then
lim lim A) − f(x) exists but not + f(x) x→0 x→0
lim lim B) + f(x) exists but not − f(x) x→0 x→0
lim lim C) both − f(x), + f(x) exist and their values are not equal x→0 x→0
lim lim D) both − f(x), + f(x) exist and their values are equal x→0 x→0
32. Let f : 3 → 3 be given by
⎪⎧x2 −1, if x isirrational f(x) = ⎨ ⎩⎪ 0, if x isrational Then f is continuous A) only at x = – 1 and x = 1 B) only at x = – 1 C) only at x = 1 D) at no point of 3 *PGQP29* -9- PG-QP – 29
33. Let f :3 → 3 be a function given by
⎧4x, if x ≤ 0 ⎪ f(x) = ⎨ax 2 + bx + c, if 0 < x <1 ⎪ ⎩⎪3 − 2x, if x ≥1
Then f is differentiable on 3 if A) a = b = 3 and c = l B) a = – 3, b = 4 and c = 0 C) a = – 3, b = – 4 and c= 0 D) a = 3, b= – 4 and c = 0
34. The polynomial p(x) = x13 + 7x3 – 5 has A) no real root B) two real roots C) only one real root D) more than two real roots
π x Sin x 35. Value of the integral ∫ dx is + 2 0 1 Cos x
π2 π2 π2 A) π 2 B) C) D) 2 3 4
36. Let f be a continuous function on [0, 1] such that f (x) > 0 for all x ∈[0, 1]. Then the 1 f(x) value of ∫ + − dx is 0 f(x) f(1 x)
1 1 1 A) 1 B) C) D) 2 3 4 PG-QP – 29 -10- *PGQP29*
37. Let f : 32→ 3 be a function of two real variables given by
⎧x ⎪ , y ≠0 f(x, y) = ⎨y ⎩⎪ 0, y =0
Determine which one of the following facts about f is not correct ?
∂f ∂f A) Partial derivatives and at (0, 0) exist ∂x ∂y B) Directional derivative of f at (0, 0) in the direction of the vector (a, b), where ab≠ 0, does not exist C) f is bounded on some neighbourhood of (0, 0) D) f is not continuous at (0, 0)
2 2 38. The value of the double integral ∫∫(x + y ) dxdy, where S is the square S
with vertices (0, 0), (1, 1), (2, 0), (1, – 1) is
1 2 4 8 A) B) C) D) 3 3 3 3
⎛ ⎞ ⎜ 4 2 1 3⎟ ⎜ ⎟ ⎜ 6 3 4 7⎟ ⎜ ⎟ 39. The rank of the matrix ⎜ 2 1 0 1⎟ is ⎜ ⎟ ⎝ 2 1 1 2⎠
A)4 B)3 C)2 D)1 *PGQP29* -11- PG-QP – 29
40. The general solution of y′′− 2y′ + y = x + ex is 1 A) y = c x + c ex + x + x2ex + 2, c , c are arbitrary constants 1 2 2 l 2 1 B) y = (c x + c )ex + x + x2ex + 2, c , c are arbitrary constants 1 2 2 l 2 x C) y = (c1x + c2)e + x + 2, cl, c2 are arbitrary constants 1 D) y = (c + c x)ex + x + x2ex, c , c are arbitrary constants 1 2 2 l 2
41. Eight people P1, P2, ... , P8, are arranged randomly in a line, the probability
that Pl and P2 are not next to each other is 3 1 1 3 A) B) C) D) 4 4 8 8
42. Of all row arrangements of 6 boys and girls, the number of arrangements with at least two boys is A) 47 B) 37 C) 57 D) 27
43. A class contains 10 boys and 20 girls of which half the girls have brown eyes, the probability that a person chosen at random is a boy or has brown eyes is
3 7 5 4 A) B) C) D) 9 6 6 6
44. The mean of 16 numbers is 27 and their median is 21. Suppose the largest number is increased by 6 and the smallest number is reduced by 9, the mean and median of the modified numbers are, respectively A) 26.81, 21 B) 27.00, 19.75 C) 26.81, 19.75 D) 27.00, 21
45. A coin is weighted so that heads is twice as likely to appear as tails. The probability of head and tail are, respectively,
2 1 1 2 1 3 3 1 A) , B) , C) , D) , 3 3 3 3 4 4 4 4 PG-QP – 29 -12- *PGQP29*
1 46. The probability that a men will live 10 more years is and the probability that his 4 1 wife will live 10 more years is . The probability that neither will be alive in 10 3 years is 1 1 1 3 A) B) C) D) 4 2 12 4
47. In a certain college, 4% of the men and 1% of the women are taller than 1.8 m. Further, 60% of the students are women. Now if a student is selected at random and is taller than 1.8 m., then the probability that the student is a women is A) 0.36 B) 0.29 C) 0.27 D) 0.34
3 5 3 48. Let A and B be events with P(A) = , P(B) = and P(A ∪ B) = . The conditional 8 8 4 probability of A given B, P(A | B) is 2 3 3 4 A) B) C) D) 5 5 4 5
49. Let A and B be events with P(A) = 0.7, P(B) = 0.4 and P(Ac ∪ Bc) = 0.75. The probability P(A ∪ B) is A) 0.75 B) 0.35 C) 0.85 D) 0.95
2 1 50. Let A and B be events with P(Ac) = and P(A ∩ B) = . The probability P(A∩ Bc) is 3 4 2 3 1 7 A) B) C) D) 9 4 12 12
51. Let A and B be events with P(A | B) = 0.75, P(A | Bc) = 0.6 and P(B) = 0.25. The probability P(Bc|Ac) is 1 A) at most 4 1 1 B) more than but strictly less than 4 2 1 C) more than 2 D) cannot be determined from the given information *PGQP29* -13- PG-QP – 29
52. For a random variable X, the value of b that minimizes E(|X – b|) is A) The mean of the random variable X. B) The first absolute moment (E(|X|))of the random variable X. C) The median of the random variable X. D) The modal value of the random variable X.
53. A bag contains 100 slips numbered 1,2,...,100. Two slips are drawn without replacement. Let E be the event that the sum of the two numbers on the drawn slips is and let F denote the event that the sum is an odd number. Then A) Ec ⊂ F B) P (E) < P(F) C) P(E)=P(F) D) P(E) > P(F)
54. In a locality, there are 100 families with two children each. The expected number of families among them where both the children are girls is A)0 B)25 C)45 D)85
55. For a random variable X, which one of the following is correct ? A) | E (X) | ≥ E | X | B) | E (X) | ≤ E | X | C) | E (X) | = ( E | X | )2 D) | E (X) | = 2 E | X |
56. Let a random variable X has Poisson distribution with parameter 2, then E (X(X - 1)(X – 2)) is A) 27 B) 8 C) 1 D) 3
57. Let a random variable X follows standard normal distribution, then P(X2> 0) is
1 1 A)= 1 B)= 0 C)= D) < 2 2 58. Let a random variable X has exponential distribution with parameter θ > 0, then for any s, t > 0, P(X > s + t | X > s) is A) = P(X < t) B) = P(X > s) C) = P(X > t) D) = P(X < s)
59. Let a random variable X has normal distribution with mean μ and variance σ 2 and let Φ (.) denote the distribution function of the standardized normal distribution, then the probability P(μ – kσ ≤ X ≤ μ + kσ), k > 0 is A) 1 – 2Φ (k) B) Φ (k) – 1 C) 1 – Φ (k) D) 2Φ (k) – 1 PG-QP – 29 -14- *PGQP29*
60. Let X1, X2, X3 be independent and identically distributed random variables with σ2 variance and let Yl = Xl + X2 , Y2 = X2 + X3. Then the correlation coefficient
between Yl and Y2 is 1 1 A) 0 B) C) D) 1 4 2
61. Let fX(x) be the probability density function of a continuous random variable X. Then which of the following statements is wrong ?
A) fX(x) cannot be negative
B) fX(x) may be larger than 1
C) P(X = x) = fX(x) D) Exactly one of the statements A), B) and C) is wrong
62. Let a random variable X has Poisson(λ ) distribution and P(X = 0) = P(X = 1), then A) E(X) = 1 and Var(X) = 2 B) E(X) = 2 and Var(X) = 1 C) E(X) = 1 and Var(X) = 1 D) E(X) = 2 and Var (X) = 2
63. The probability density function of a random variable X, is given by
⎧ 1 1 − x ⎪ e θ , x >0, θ >0 f (x)=⎨θ X ⎪ ⎩ 0, otherwise
Then the value of f (E(X)) is θ e 1 A) θ B) C) D) e θ θe
64. Let X be a random variable with probability mass function P(X = x) = (1– p)x–1p, 0 < p < 1, x = 1, 2, . . . . then the distribution function FX(x) is A) l – (l – p) x B) l – (l– p)x – l C) (1– p)x D) (l – p)x – l *PGQP29* -15- PG-QP – 29
θ 65. T1 and T2 are independent unbiased estimators for and Var(T2 ) = Var(Tl ). Consider T + T 2T + T two more estimators forθ, T = 1 2 , T = 1 2 . Then 3 2 4 3 θ A) Neither T3 nor T4 is unbiased for θ B) Both T3 and T4 are unbiased estimators for and T3 is more efficient than T4 C) Among T1, T2, T3 and T4; T1 is the most efficient D) Among T1, T2, T3 and T4; T4 is the most efficient 66. Let X be a random variable with mean 1 and standard deviation 3. Then the probability P( – 4 ≤ X ≤ 6) is
16 16 9 9 A) B) at least C) D) atmost 25 25 25 25
67. Let X1, X2 be independent and identically distributed Poisson(λ ) random variables, + ⎛⎛ ⎞X1 X2 ⎞ ⎜⎜ 1 ⎟ ⎟ then E⎜ ⎟ is ⎝⎝ 2 ⎠ ⎠
− λ λ −λ λ A) e 2 B) e C) e D) e2
68. Let X denote the number of successes in 10 independent Bernoulli trials whose 1 probability of success is , then E(X (10 – X)) is 4
135 125 150 185 A) B) C) D) 8 6 9 7 69. A discrete random variable X takes values – 1, 2, 3 with probabilities
P(X = –1) = 0.2, P(X = 2) = p2, P(X = 3) = p3 and E(X) = 2.2. Then p2 and p3 are, respectively A) 0.2 and 0.6 B) 0.4 and 0.4 C) 0.8 and 0 D) 0 and 0.8
70. If X is uniformly distributed over the interval ( –2, 2), then the probability P(|X – 0.5| > 1.5) is
1 1 1 2 A) B) C) D) 4 3 2 3 PG-QP – 29 -16- *PGQP29*
71. Suppose that a random variable X has probability density function
⎪⎧2x, 0 < x <1, f (x) = ⎨ X ⎩⎪ 0, otherwise
The probability density function of Y = e–x is given by − 2 log y 1 A) g (y) = , < y < 1 Y y e 2 log y 1 B) g (y) = , < y < 1 Y y e log y 1 C) g (y) = , < y < 1 Y 2y e 1 D) g (y) = – 2log y, < y < 1 Y e
72. Let a random variable Z has standard normal distribution and P (– 1.5 < Z < 1.5) = 0.8664, P(Z < 0.29) = 0.6141. Based on the given information the probability P(– 1.5 < Z < 0.29) A) cannot be found B) is equal to 0.5473 C) is at least 0.6141 D) is at least 0.8664 μ σ2 73. Suppose X1 and X2 are each independently distributed with distribution N( , ), 2 ⎛ X − X ⎞ ⎜ 1 2 ⎟ then ⎜ ⎟ has ⎝ 2σ ⎠
A) N(0, 2σ 2) distribution B) chi-square distribution with one degree of freedom C) t-distribution with one degree of freedom D) chi-square distribution with two degrees of freedom *PGQP29* -17- PG-QP – 29
74. Let X be a random variable whose probability mass function is x 1 ⎛ 2 ⎞ P(X = x) = ⎜ ⎟ , x = 0, 1, 2, .... 3 ⎝ 3 ⎠
Then P(X = l1|X > 10) is
10 10 1 2 ⎛ 1 ⎞ ⎛ 2 ⎞ A) B) C) ⎜ ⎟ D) ⎜ ⎟ 3 3 ⎝ 3 ⎠ ⎝ 3 ⎠
75. A candidate can appear in certain competitive examination for a maximum of four attempts. The probability of success for a particular candidate in an attempt is 0.3. Assuming that the outcome of an attempt is independent of the outcome in other attempts. The expected number of attempts for this candidate is A) 2.533 B) 2.431 C) 2.413 D) 2.353
76. Suppose that a random variable X has normal distribution with meanμ = 3 and variance σ2 = 4. If P(X > c) = 2P(X ≤ c) and from the standard normal tables P(Z ≤ – 0.43) = 0.333, then c A) = 3.15 B) = 2.19 C) = 2.14 D) = 3.23
77. Let X be a continuous random variable with probability density function f (x) and
distribution function F(x). Let Xl, X2, . . . , Xn be a random sample of X and let X(l) ≤ X(2) ≤ ... ≤ X(n) be the corresponding order statistics. The probability density functions of X(1) and X(n) are, respectively
A) (n – 1)[1– F (x)]n –1 f(x) , (n – l)[F(x)]n–1 f(x)
B) (n – 1)[1– F (x)]n –1 f(x) , n[F(x)]n–1 f(x)
C) n[1– F (x)]n –1 f(x) , (n – l) [F(x)]n–1 f(x)
D) n[1– F (x)]n –1 f(x) , n[F(x)]n–1 f(x) PG-QP – 29 -18- *PGQP29*
78. Let X1, X2, X3 be a random sample from a distribution with probability density function
⎪⎧2x, 0 < x <1 f (x) = ⎨ X ⎩⎪ 0, otherwise
Let X(1) = min{X1, X2, X3} and M denote the median of the distribution,
then P(X(l) > M) 3 5 1 2 A) = B) = C) = D) = 8 9 8 9
79. Suppose that a bivariate random vector (X, Y) has the joint probability mass function as follows : 2 P(X = 0, Y = 10) = P(X = 0, Y = 20) = , 18 3 P(X = 1, Y = 10) = P(X = 1, Y = 30) = , 18 4 4 P(X = 1, Y = 20) = , and P(X = 2, Y = 30) = . 18 18
Then P(Y > l0 | X = 1) is
3 7 4 7 A) B) C) D) 10 10 13 13
80. Suppose that a bivariate random vector (X, Y) has the joint probability density function
− − ⎪⎧2e xe 2y , 0 Then P(X < Y) is 2 1 1 3 A) B) C) D) 3 3 4 4 *PGQP29* -19- PG-QP – 29 81. Suppose the moment generating function of a continuous random variable X is given λ by for t < λ , then Var(X) is λ − t 2 2 1 2 A) λ2 B) λ C) λ D) λ2 2 82. Let Xi be a random variable distributed N(i, i ), i = 1, 2, 3. Assume that the random variables X1, X2 and X3 are independent. Then (X −1)2 2 1 ⎡(X − i)2 ⎤ 3 ⎢ i ⎥ ∑i2=2 ⎣⎢ i ⎦⎥ A) has F-distribution with 2 and 1 degrees of freedom B) has F-distribution with 1 and 2 degrees of freedom C) has t-distribution with 3 degrees of freedom D) has chi-square distribution with 2 degrees of freedom 83. Let X1, X2 be a random sample of size 2 from a distribution with probability density function 1 ⎧ − x ⎪1 2 = e , x > 0, f (x) ⎨2 X ⎪ ⎩ 0, otherwise X 1 Then the distribution of X is 2 A) chi-square distribution with 2 degrees of freedom B) t-distribution with 2 degrees of freedom C) F-distribution with 2 and 2 degrees of freedom D) F-distribution with 1 and 1 degrees of freedom PG-QP – 29 -20- *PGQP29* 84. Suppose that the two dimensional continuous random variable (X, Y) has joint probability density function ⎧ xy ⎪x2 + , 0 ≤ x ≤1, 0 ≤ y ≤ 2 f (x, y) = ⎨ X,Y 3 ⎩⎪ 0, elsewhere Let B = {X + Y ≥ 1}, then P(B) = 35 7 65 43 A) B) C) D) 62 72 72 65 85. Suppose that the two-dimensional random variable (X, Y) has the joint probability density function ⎪⎧2, 0 < x < y <1 f (x, y) = ⎨ X,Y ⎩⎪0, elsewhere The correlation coefficient is A) 0.3 B) 0.7 C) 0.4 D) 0.5 86. Let X1, X2, . . . , Xn be independent and identically distributed random variables σ 2 having variance , then Cov(Xi – X , X ), i = 1, 2, . . . , n A)= 1 B)= 0 C)= – 1 D)= 0.5 87. Let X and Y be two random variables having finite second moments, then A) [E(XY)]2 = E[X2]E[Y2] B) [E(XY)]2 ≥ E[X2]E[Y2] C) [E(XY)2] = E[X2Y2] D) |[E(XY)]|2 ≤ E[X2]E[Y2] 88. Let X1, X2, X3 be independent and identically distributed N( θ, l). Then E( X |X1) is 1 2 2 1 A) X + θ B) X + θ 3 1 3 3 2 3 1 2 2 1 C) X + θ D) X + θ 3 3 3 3 1 3 *PGQP29* -21- PG-QP – 29 89. Consider two random variables X and Y having a joint distribution function FX,Y(x, y) and marginal distribution function of X and Y are respectively FX (x), F(y). Then, for all x, y, ≤ ≤ F (x) F (y) A) FX (x) + FY(y) –1 FX,Y(x, y) X Y F (x) F (y) ≤ ≤ B) X Y FX,Y(x, y) FX (x) + FY(y) –1 F (x) F (y) C) FX (x) + FY(y) –1= FX,Y(x, y)= X Y 1 D) F (x) + F (y) –1≤ F (x, y)≤ F (x) F (y) X Y X,Y 2 X Y λ 90. Let X1, X2, . . . , Xn be a random sample from a Poisson population with parameter . Which one of the following statistics is an unbiased estimator for λ 2? 2 1 n 2 1 n − A) ( X ) B) X ( X – 1) C) ∑ = X D) ∑ = X (X 1) n i 1 i n i 1 i i 91. Let a random variable T has binomial distribution with parameters n and p, then []()T ()n−T E r n−r ( n ) r − n−r = A) = r p (1 p) , r 0, 1,...,n − B) = pr (1− p)n r , r =0,1,...,n − C) = pn r (1− p)r , r =0, 1,...,n ( n ) n− r − r = D) = r p (1 p) ,r 0,1,...,n 92. Let X1, X2,. . . , Xn be a random sample from a uniform distribution over ( θ, θ + 1). Which of the following is a consistent estimator of θ? 1 1 1 1 A) X + B) X − C) X + D) X − 2 2 4 4 PG-QP – 29 -22- *PGQP29* θ 93. Let X1, X2,. . . , Xn denote a random sample from a uniform distribution over (– , 0), θ > 0. Let X(l) = min{X1, X2,. . . , Xn} and X(n) = max{X1, X2,. . . , Xn}. The maximum likelihood estimator of θ is A) X(n) B) – X(n) C) X(1) D) – X(1) 94. Let X1, X2, . . . , Xn be a random sample from a normal distribution with meanθ and variance unity. Then which one of the following is a sufficient statistic ? n ∑ X 2 n 2 i =1 i A) ∑ = X B) i 1 i n −1 n ∑ = X 2 C) i 1 i D) ()∑n X n i =1 i 95. The probability density function of a random variable X, is given by −θ ⎪⎧θe x, x > 0,θ> 0 f (x) = ⎨ X ⎩⎪ 0, otherwise Then which one of the following is the UMVU estimator of θ, n 2 n n ∑ = X n ∑ = X A) i 1 i , n >1 B) i 1 i , n >1 n −1 n −1 n 2 − ∑ X (n 1) > i =1 i > , n 1 C) , n 1 D) ∑ n X n −1 i =1 i 96. Let X1, X2, . . . , Xn be a random sample from N(θ, l). The Cramer-Rao lower bound for the variance of an unbiased estimator of θ2 is θ2 4 θ2 θ2 θ2 A) B) C) D) n n 4n 2n *PGQP29* -23- PG-QP – 29 μ 97. Let X1, X2,. . . , X10 be a random sample from a normal distribution with mean and 1 10 variance σ 2. Suppose that = 10.48 and S = ∑ (X − X)2 = 1.36. From the X 9 i=1 i μ tables of t-distribution, t9, 0.95 = 1.83. Then a confidence interval for with the confidence coefficient 1 – α = 0.90 is A) (9.65,11.41) B) (10.35,11.31) C) (10.35,11.41) D) (9.65,11.31). 98. Let X be a single observation from a probability density function −θ ⎪⎧θe x, x > 0, θ> 0 f (x) = ⎨ X ⎩⎪0, otherwise 1 α Let (X, 2X) be a confidence interval for the mean θ . Then the confidence coefficient 1– 1 1 1 1 − − − − − 2 − 1 A) = e–1– e 2 B) = 2e 2 C) = e 2 – e–1 D) = e e 2 99. In hypothesis testing, β is the probability of committing an error of Type II. The power of the test, 1 –β is A) the probability of rejecting H0 when H1is true B) the probability of failing to reject H0 when H1 is true C) the probability of failing to reject H0 when H0 is true D) the probability of failing to reject H0 μ σ 2 μ σ 2 100. Let X1, X2, . . . , Xn be a random sample from N ( , ) where both and are 2 μ μ σ 2 σ unknown. We wish to test the null hypothesis H0 : = 0, = 0 against the 2 μ μ μ σ 2 σ . alternative H1 : = 1 > 0, = 0 The Neyman-Pearson fundamental lemma leads to the critical region, for some constant k, of the MP size-α test ∑n X < k ∑n X 2 > k A) i =1 i B) i=1 i ∑ n X > k ∑ n X 2 < k C) i=1 i D) i=1 i –––––––––––––– PG-QP – 29 -24- *PGQP29* SPACE FOR ROUGH WORK