Entrance Examination, 2010 M.Sc. (Statistics-OR)
IHall Ticket No.
Time 2 hours Part A 25 marks
Max. Marks. 75 Part B 50 marks
Instructions

1.
Write your Hall Ticket Number in 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 are 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 be 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.


PART A S-02:

•
Find the correct answer and mark it on the OMR answer sheet.

•
A right answer gets 1 mark and a wrong answer gets -0.33 mark.


1. 5 red balls( all alike) and 4 blue balls( all alike) are to be placed in two bags numbered 1 and 2. The number of ways in which this can be done is
2048,
512.
24.

30.
2. A company wants to reduce its sale13 force by dismissing 4 of its 10 sales people. In how many different ways can this be done?
40.
210,
5040.
151,200,
3. One page of a 500 page book is defective. It is decided to find the page, the
first 250 pages are non defective, the probability that the defective page is one of the pages numbered 251, 252, . 300
. 1
IS
1
IS
.I

IS
cannot be determined from the data given.
4. Two boxes contain 100 balls which are marked ... ,100. One ball is picked
up from each box, what is the probability that the two balls bear the same number?
1
100
2
100
1
1002
2

1002


5. 4girlsGI, G2 G3 G4 and 4 boys Bl ,B2,B3,B4 are randomly arranged in a row. The probabilities that all the girls are seated in such a way that there is no boy between any two girls is
[AJ


sr·

sr·
2.41.4!


6. Which measure of central tendency among the measures given below is influ­enced by all the values in the data set?
[Al Mean.
Median.

Mode.
Midrange.
7. If there is data on shoe sizes for adult males in India which are given as or which of the following is the most suitable measure of central tendency?
Arithmetic mean.
Geometric mean.
Median.
[OJ Mode.

8. The distribution of marks of 100 students in a class is as follows
Marks 0-30 30-45 45-55 55-80 80-95 95-100
Frequency 0 25 55 12 6 2

The distribution of marks is
uniform.
symmetric but not uniform. positively skewed.
negatively skewed.
r




9. Marks of 6 students in two courses G1 and G2 are given below.
Students
1 2 3 4 5 6
Course G1 40 41 42 43 44 45
Course G2 40 41.5 41.5 43.5 43.5 45

Let (Mi denote the mean and the standard deviation of the marks in Gi, 1,2 respectively. It can be seen that
M1 M2 and 81 82.
M1 M2 and 81 82.

10. The random variable X follows a Binomial distribution with parameters n 10 and p 0.5, i.e X rv 0.5). Define another random variable Y as Y then E(X is equal to
5.
10.
-5.
O.
11. A coin for which probability of heads showing up upon tossing is is tossed 30 times. Which of the following statements is correct?
5 heads are more likely to show up than 8 heads.
15 heads are more likely to show up than any other number of heads.
10 heads are more likely to show up than any other number of heads.
20 heads are more likely to show up than 15 heads.
12. X is a discrete random variable for which Then X is dis­tributed as
a Poisson random variable.
a Binomial random variable.
a Negative Binomial random variable.

None of the above.


S-OC6

13. The random variable X has exponential distribution with mean 5. P(X is
e-5.

eT.

e-1•
14. The correlation coefficient between two random variables X and Y is 1. Define two new random variables U and V as U 2X 7 and V 9Y 12. The correlation coefficient between U and V is
O.

1.
1.
15. Given below are measurements for six individuals on two random variables X and Y

We can say that the correlation coefficient between X and Y is
O.
almost-1.
almost +1.
between and
16. Non constant random variables X and Yare distributed in such a way that VeX vex then
X and Y are necessarily independently distributed.
cov(X, O.
VeX) must be equal to
X and Yare necessarily identically distributed.
17. In order to draw a representative sample from a population, it is divided into at least two subgroups and then a random sample is drawn from each of the subgroups. What is this type of sampling scheme called?
Aggregate sampling.
Cluster sampling.
Cross-sectional sampling.
Stratified sampling.


18. Based on a random sample of size n from the population to test H f.,l f.,lo against HI f.,l f.,lo, the most powerful level a test is Xn an where Xn is the sample mean. Which is the following statements is correct about the behavior of an as n changes for the same level of significance?
an remains the same no matter what n is.
an decreases as n increases.
an increases as n increases.
an increases up to some say no and then decreases.
19. Let X be a random variable taking values a with probabilities respectively. If is It, then a is
6,
7.
4.
5.
20. T is an unbiased estimator for a parameter and V An unbiased estimator for is

2
T2+T

T2

none of the above.
21. Xl ).X2 (naDd Xs be vectors in Let
alXI a2X2 a3X3 where ai E lR for 1,2,3. The choice of aI, a2 and a3 for which the set of vectors Xl, X2 and X4 do not form a basis is
rw_ 1 rw_ 1 rw_ 1

rw_ 1 rw_ 1 rw-O

11
al 2,02 2,a3
1-1 1
01 2,02 2,03

22. Consider the following linear programming problem
Maximize 5Xl 7X2
Subject to 2Xl 3X2 33 3Xl +X2 25 4Xl X2 32
Xl,X2
Which of the following is the optimal solution to the problem
Xl 6,X2 7.
Xl= 1O,x2 9.
Xl 7,X2 4.

Xl= 8,X2 O.
23. A is a 4 x 4 real non singular matrix, a matrix B is obtained in the following
way.
The first row of B is the sum of the first and second rows of
The second row of B is the sum of the second and third rows of
The third and fourth rows of B are the third and fourth rows of A respectively.
The rank of B is

l.
2.
3.
4.

00
1
24. The series
n.
n=l
diverges.
does not exist.
converges to a positive number.
converges to O.
25. The negation of the statement aj+l for all .. is
aj aj+l for all j .
aj aj+l for some ..
aj aj+l for at most one ....
aj aj+l for all ....



PartB
•
Find the correct answer and mark it on the OMR answer sheet.

•
A right answer gets 2 marks and a wrong answer gets -0.66 mark.


26. At least one of the events A or B will certainly occur, the probability of both occurring is The probability that at most one of them will occur
cannot be determined from the given information.
is O.
I
IS
.I

IS
27. A fair coin is thrown 5 times let Ei denote the event that the ith and the 2yh throw show heads for i respectively. Which of the following statements is correct?
Ell E3 are not independent.
Ell E2 are not independent.
E2, E3 are not independent.
EI, E2, E3 are jointly independent.
28. B is an event for which 0 1. Two events Al and A2 are equiprobable when B occurs as well as when B does not occur. Then
Al and A2 are necessarily independent.
Al and A2 are equiprobable.
Al and A2 are certainly mutually exclusive.
Al always occurs when A2 occurs.
29. The probability that the difference between two numbers drawn without re­placement from 2 ... 20 is at least 10 is in the interval



1).


30. 10 red balls all alike) and 10 blue balls( all alike) are randomly placed in a row. The probability that there are more red balls among the first 10 is
r
[AJ


10
(in
". .
10

[CJ



31. There are as many males as females in a population. fh of the females believe in a rumour while rh of the males believe it. A person is chosen randomly
from. this population and is found to believe the rumour. The probability that the person chosen was female is

[BJ :S.
153'
[DJ
32. The range of three positive integers is 3 and their variance is then
exactly two of them are equal.
the three integers are distinct. all are equal.
this can never happen.
33. The median marks of all the twenty girls of a class is 12 and that of all the eighty boys of the same class is 6. Which of the following statements on the median of all the 100 students of the class is correct?
It is certainly more than 6. It can be less than 6. It is certainly less than 12. It is certainly more than 12.
8


34. Anand tosses a fair coin(say 01) and stops when a head appears. Bharath does the same experiment with a fair coin (say C2). The probability that both of them tossed their respective coins the same number of times is


i.

35.
The expected value of a random variable whose probability density function is

36.
The expected value of a uniformly distributed random variable X is 5and p the variance of X


f cx 4 l U x 3 otherwise
is O.
does not exist.
is 1.
is 2.

is
• 25
IS
is 3.
cannot be determined from the given information.
37. Consider a distribution of data that is not necessarily bell-shaped. According to Chebyshev's theorem, the minimum percentage of data values lying within ±1.6 standard deviations from the mean would be
37.5%
39.0625%
60.9375%
62.5%
38. Xl'" and then it is true that
PI P2 P2 PI
P2 PI
PI P2
9



39. A fair die is rolled till 6 shows up. the first 16 throws were unsuccessful(i.e. 6 did not show up in any of the first 16 throws), the probability that the first 6 will show up in the 19th throw is




40. Ashok and Ravi playa game in which a toss a fair coin alternately and who ever gets the first heads is the winner and the game stops. If Ashok takes the first toss, the expected number of tosses that Ashok took till the game stopped
IS


1.
4.
41. The number of scratches in a compact disc is a Poisson random variable with mean 1. The probability that there are two scratches in two randomly selected compact discs is
2e-1
.
e-2. 2e-2
.
e-l.
42. Xn is a random sample from the 0 0 population. Let max ... ,Xn
is an unbiased as well as a sufficient estimator of O.
is neither an unbiased nor a sufficient estimator of O.
is an unbiased but not a sufficient estimator of
is not unbiased but is a sufficient estimator of
43.
The sum of 10 independent observations of a Bernoulli random variable X for which P(X P and P(X 0 P 1 was 7. The maximum likelihood estimate of P if p E is

44.
The value of is








O.
l.
2.
3.
e5x x2
45. Consider the function -00 x 00, which of the following statements is correct?
Its maximum value is 1 which is attained at x O.
Its maximum value is greater than 1 and is attained at x which is positive.
It has more than one maxima.
It is an increasing function.
46. Define the function f as follows
x f 0
x=o
then
f is continuous everywhere.
f is differentiable at x O.
f is continuous everywhere except at x O.
f attains a finite maximum.
47. T1 and T2 are two unbiased estimators based on the same sample for a param­eter but V(TI V(T2). If the observed values of TI and T2 are al and a2 respectively, we should
choose al to estimate 0 because it is very close to O.
choose a2 to estimate because it is closer to than al is.
choose al to estimate because it is more likely to be close to than a2.

Toss a fair coin, choose al if it shows heads otherwise choose a2 to estimate
48. Suppose the hypothesis Ho J.L J.Lo is accepted against HI J.L J.Lo at ex level of significance, (where J.L is the mean of a Normal random variable with variance 2). This means that
the data shows strong evidence in favour of Ho.
Ho can be accepted even if the data shows strong evidence in favour of
H2 J.L J.Lo·
the data shows strong evidence in favour of HI.

the sample mean is equal to J.Lo.
11



Figure Figure for Problems 49 and 50
The next two questions are based on the following linear programming problem and the figure given in Figure
Min Xl +X2
S.t. 4XI +X2
-Xl +X2 2 XI,X2 0
49. The feasible region of the linear programming problem is given by
[AJ the triangle ABF.
[BJ the triangle ADE.
[CJ the triangle FCE.

the quadilateral FBDE.
50. If constraint 4XI +X2 8 is dropped, which of the following is true?
[AJ The feasible region is unbounded and the optimal solution occurs at point D. [BJ The feasible region does not change. [CJ The feasible region is unbounded and the optimal value is -00. [DJ The feasible region does not change and the optimal solution is -2.