University of Hyderabad (UoH) 2011 M.Sc Statistics Entrance for - Question Paper
V- oS'
Entrance Examination, 2011 M.Sc. (Statistics-OR)
Hall Ticket No.
Time : 2 hours Max. Marks. 75
Part A : 25 marks Part B : 50 marks
Instructions
1. Write your Booklet Code and Hall Ticket Number on the OMR Answer Sheet given to you. Also write the Hall Ticket Number in the space provided above.
2. There is negative marking.
3. Answers sure to be marked on the OMR answer sheet .
4. Please read the instructions carefully before marking your answers on the OMR answer sheet.
5. Hand over the question paper booklet and the OMR answer sheet at the end of the examination.
6. No additional sheets will l>e provided. Rough work can be done in the question paper itself/space provided at the end of the booklet.
7. Calculators are not allowed.
8. There are a total of 50 questions in Part A and Part B together.
9. The appropriate answer should be coloured in either a blue or black ball point or sketch pen. DO NOT USE A PENCIL.
Find the correct answer and mark it on the OMR sheet.
A right answer gets 1 mark and wrong answer gets 0.33 mark
1. The number of 4 digit numbers greater than 1000 in which all the numerals are distinct is
[A] Q2 x 8 x 7.
[B] 10 x 9 x 8 x 7.
[C] 102 x 9 x 8.
[D] 9 x 8 x 7 x 6.
2. Eight points are chosen on the circumference of a circle. How iriauy different chords can be drawn by joining any two points?
[A] 28.
[B] 8.
[C] 16.
[D] G4.
3. Let A and be two events with P{A U B) = 0.6 and P(A n B) = 0.3, then P(AC) + P(BC) is
[A] 0.8.
4. A and B\ arc independent events, so are A and 2, then
[A] A and B\ n B2 are always independent.
[B] if B\ and 2 are independent, then A and B\ fl2 are always independent.
[C] A and B\ U B2 are always independent.
[D] none of the above is true.
5. Three numbers are selected from the set {1,2,..., 10}, the probability that the selected numbers are such that the minimum is at most 5 lies in the interval
[A] (UJ.
[C] (,!].
6. If p = 0.1 and n = 5, then the corresponding binomial distribution is
[A] right skewed.
[B] left skewed.
[C] symmetric.
[D] bimodal.
7. A population is distributed according to the four standard blood types as follows A : 42%, 0:33%, B : 18%, AB : 7%
Assuming that the people chose their partners independently of the blood types, what is the probability that a randomly selected couple from this population will have same blood type?
[A] (0.42)2.
[B] (0.33)2.
[C] (0.18)2.
[D] (0.42)2 -I- (0.33)2 + (0.18)2 + (0.07)2.
8. FYom a box containing N\ white balls (all $like) and 7V2 blue balls (all alike), n balls are drawn randomly with replacement, the expected number of white balls in the sample is
[A]-
I*-'] Ni + Ni'
n[Ni+N-j)
9. The expected values of two random variables X\ and X2 are equal to //, whereas Var(Xi) > Var(X2), one can say that
[A] X\ and X2 are identically distributed.
[B] X\ is more likely to be near /z than X2.
[C] X2 is more likely to be near /z than Ai.
[D] P(X2 > Xx) = 1.
10. Let X denote the number of times a gambler has to throw two distinct fair dice till the sum of the numbers that show up is 8, then the random variable X is distributed as
[A] Binomial (100,5/36).
[B] Geometric (5/36).
[C] Binomial (100,5/36).
[D] Geometric (1/36).
11. 10% of shirts produced in a work shop are defective, the expccted number of defective shirts in a randomly selected sample of 100 shirts from this work shop is
[A] 1.
[B] 5.
[C] 10.
[D] 20.
12. If X has Poisson distribution with mean 10, then it is true that
[A] P(X = 7) > P{X = 8).
[B] P{X = 8) > P(X = 10).
[C] P{X = 10) > P(X = 6).
[D] P(X = 14) > P(X = 13).
13. Let X be a random variable having Poisson distribution with mean 2, also let Y = 2X 1, then the variance of Y is
[A] 3.
[B] 4.
[CJ 7.
14. A number is randomly chosen from the interval [0,1], the probability that its second decimal digit is 1 is
[A] 0.1.
[B] 0.2.
[C] 0.5.
[D] 0.9.
15. The height of men Ls normally distributed with mean /j, = 167 cm and standard deviation <7 = 3 cm. What is the percentage of the population of men that have height greater than 167 cm?
[A] 30%.
[B] 50%.
[C] 70%.
[D] 100%.
16. The probability density function of a random variable X is
el*l
fx(x) - , -OO < X < oo.
Then the moment generating functions Mx(t) and My(t) of X and Y = X respectively satisfy
[A] MY(t) = -Mxit) V* 6 H.
[B] My(t) = Mx(t) Vt ft.
[C] My{t) = Mx{t) only when > 0.
[D] My{t) = Mx{t) only when t < 0.
17. The median and standard deviation of 25 numbers are m and s respectively, 5 is added to the largest and 5 is subtracted from the smallest numbers. For thus new set of observations, the median mi and standard deviation si satisfy
[A] mi > m and Si i= s.
[B] mi = m and sx > s.
[C] mi = m and Sj < s.
[D] mi > m and Si > s.
18. The differences between every pair of real numbers Xi,X2, ...n. n > 2 are known, however X\yX2, ...,xn are not known, if m and s are the mean and standard deviation respectively of these numbers, it is true that
[A] both m and s can be determined.
[B] neither m nor s can be determined.
[C] s can be determined but not m.
[D] m can be determined but not s.
19 The correlation coefficient pxy between two random variables X and Y is so P-x,y is
[CIO.
20. The maximum likelihood estimator of any parameter 6
[A] is always unbiased for 9.
[B] is always a sufficient estimator for 9.
[C] has the smallest mean square error always.
[D] none of the above is correct.
21. Four random numbers where selected from the interval (0,0), they were 0.6, 0.49, 032, 0.83, the maximum likelihood estimate of 0 is
[A] 0.83.
[B] 0.56.
[C] 0.715.
[D] 0.575.
22. Let A = {1,2,3,4} and B = {6,7}. The number of different onto functions from A to B is
[C] 14.
23. The sequence {a,,} is defined as follows
Identify the correct answer
[A] The sequence {a} is not monolonic.
[B] lim On oc.
n*oo
[C] lim On = 1.
noo
[D] lim an = 0.
n- *oo
24. A and B are two n x n real matrices, if Rank (B) = n 3 and AB = 0, then for any n >7
[A] Rank (A)= Rank (B).
[B] Rank (A)<3.
[C] Rank (A)>n-3.
[D] Rank (A)> 3.
_ n
25. For any set of real numbers zi,X2, where n > 2, let E = i then
1=1
[A] <?>E.
[B] e* = E.
[C] e* = E.
[D] e* < E.
Find the correct answer and mark it on the OMR sheet.
A right answer gets 2 marks and wrong answer gets 0*66 >\a.irK
26. The probability that at least one of the events A, D occurs is 0.8 and the probability that at most one of them occurs is 0.5, the probability that exactly one of them occurs
[A] is 0.3.
[B] is 0.2.
[C] is 0.
[D] cannot be determined from the data given.
27. A fair coin is tossed three times, given that at least one of the first two tosses showed tails, the probability that exactly one head showed up
[A] is
[B] is J.
[C] is I
[D] cannot be determined.
28. The probability that among seven people no two were born in the same day of the week is
[B]
29. The coefficient of x in P(x) = 2x7bx+3 is selected from the set {7, -6,..., 0,1,..., 6,7} with probability each. The probability that P(x) has real roots is
[A] f5-
[B] I
[C] f.
[D] f-
P{X = 101 \X > 100) is [A}.
31. The random variable X takes values in the set {1,2,3,4,5} and it is given that
P(X > 2) + P(X > 3) -I- P(X > 4) = | and E(X) = so P(X = 5)
[A] cannot be determined from the data given.
[B] is J.
[C] is 1. ID] is j.
32. The amount of bread that a bakery sells in a day is a random variable with density function
0 < x < 3 3 < x < 6 otherwise
cx
/(x) = c(6 - x) 0
33. The cubic equation (x 3)(x 5)(x 8) + 1 has
what is c?
[A] some complex roots.
[B] all real roots and the largest is in the interval (5,6].
[C] all real roots and the smallest is in the interval (1,2]
[D] all real roots and the largest is in the interval (7,8].
34. Let X be a random variable having distribution function
0 if x < 0
x+g if 0 < x < 0.5
F{x) =
1 if x > 0.5
Then P(0 < X < 0.5)
[A] 5.
[B] |.
[C] }.
[D] I-
35. From a bag containing 5 white balls (all alike), take out at least one ball, that is you may take out 1,2,3,4 or 5 balls with equal probabilities, paint the selected balls blue and replace them, now take out 3 balls without replacement., the probability that one of them is blue Ls
[A] 0.9.
[B] 0.7.
[C] 0.6.
[D] 0.3.
36. The mean marks of a class of 60 students is m and variance is s2, the mean marks of another class of 40 students is m + 5 and variance is sa. Identify the correct statement regarding the mean marks M and variance S3 of all the 100 students
[A] M = m and S2 > s2.
[B] M = m and S2 = s2.
[C] M > m and S2 = s2.
[D] M > m and S2 > s2.
37. For the two sets of numbers given below
Ak ={7,7,9,12,14,16,18} and A2 ={8,8,8,8,8,8,19}
Let Ri, Mj and Si, i = 1,2, denote the range, median and standard deviation of the numbers of the sets Ai 1,2, respectively, it can be seen that
[A] R\ /?2 M\ M2 and S\ = S2-
[B] Hi = #2, M\ > M2 and S\ > S2.
[C] R\ R2i Mi > M2 and Si < S2-
[D] R\ = R-2, Mi > M2 and Si = S2.
|
X |
0 |
1 |
2 |
3 |
|
tJ * II |
Pi |
Pi |
P2 |
P2 |
where 0 < pi < 1/2. Prom the above distribution we have the following random sample of size 5 {0,2,2,3,1}. The maximum likelihood estimate of p\ is
[A] 0.1
[B] 0.2
[C] 0.3
[D] 0.4
39. Let |xj denote the largest integer < x, Vrr K, eg. [2.6GJ = 2, [3.5J = -4. The function / : 71 * H defined as
f{x) = z|xj
[A] is not continuous at any x Z \ {0}.
[B] is not continuous at any x Z.
[C] is continuous at every x 1Z but not differentiable at x Z.
[D] is continuous and differential >lc at every x 71.
40. Suppose a man has invested in the share market which earns 10% in the first year and 50% in the second year, what is his average rate of return?
[A] 40.
[B] 10.
[C] 28.45.
[D] 30.
41. X\ and X2 arc independent and idcntic-ully distributed random variable with finite mean /x and finite variance a2, defir. j U\ = and U2 XlX**, then
[A] E(Ui) = E{U2) and Var{Ux) = Var(U2).
[B] E{UX) = E{U2) and Var{Uy) > Var(U2).
[C] E{UX) > E(U2) and Var(/7X) = Var{U2).
[D] (i) = E{U2) and Var(Ui) < Var(U2).
42. Suppose Xi} X2, ...Xn is a random sample from the Exp( 1/A) population. Denote
n n
U = Xit and V = then an unbiased estimator for A2 is
km 1 fc=l
[A] IP 12.
[B] 2/i/2.
[D] 2/V.
43. Xi,X2, ...,Xn is a random sample from the N(fjL,fjL2) population, define U and V as in Question No: 42, the maximum likelihood estimator of /i is
[A )U.
[B] U\
[C] V.
[D] none of the above 3.
44. A die is rolled 60 times with resulting frequency distribution as given below
|
Faces |
1 |
2 |
3 |
4 |
5 |
G |
|
Frequency |
11 |
10 |
9 |
10 |
12 |
8 |
To test the hypothesis that the die is fair, the value of the x2 goodness-of-fit test statistic is
[B]l.
[C]2.
[D] 3.
45. Suppose X is the sample mean based on a random sample from a density
for 0 < x < 1, and zero otherwise, where 6 > 0, then the moment estimator of 9 is
|B] 1/X.
[C] (1 - xyx.
46. Let X be sample of size 1 from discrete uniform distribution over the set {1,2,6}, where 6 {3,4}. For testing the hypothesis H0 : 6 = 3 against Hi : 6 = 4, the size and the power of the test which rejects H0 if X 4, are respectively.
[A] 0 and
[B] 0 and
[C] \ and i.
[D] J and
47. Ci is the a, i = 1,2, level critical region to test the null hypothesis Hq against the alternative H\, which of the following is true
[A] if Qi < a2 then Ci C C2.
[B] if aj < a2 then C2 C C\.
[C] if c*i < a2 then Ci = C2.
[Dj if tti 7 a2 then Ci and C2 may be disjoint.
48. Let f{x) = x1/*. Then the maximum value of / is
[A] 1/e.
[D] 1.
49. Consider the linear programming problem
|
minimise |
i |
+pu2 |
|
subject to |
Ul |
> 0, u2 > 0 |
|
3ui |
+u2 > 3 | |
|
Ul |
+2w2 > 4 | |
|
Ui |
+6u2 > 6. |
For which value or values of p is there is no solution to this problem?
[A] only 2.
[B] only 0.
[C] any value less than 0.
[D] any value greater than 0.
3x + Ay < 12, x > 0, y > 1.
V'OSr
|
[A] |
3. |
|
[B] |
6. |
|
[C] |
8. |
|
[D] |
4. |
12
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Attachment: |
| Earning: Approval pending. |
