Exam Details
| Subject | statistics | |
| Paper | ||
| Exam / Course | m.sc | |
| Department | ||
| Organization | central university | |
| Position | ||
| Exam Date | 2017 | |
| City, State | telangana, hyderabad |
Question Paper
1. Bag1 contains 6 red, 5 blue and 4 green balls while Bag2 contains 6 green, 5 blue and 4 red balls, a ball is drawn from each of the bags, if every ball is equally likely to be drawn, the probability that a ball of the same colour will be drawn from both the bags is
a little more than 1/2.
equal to 1/4.
equal to 1/3.
a little less than 1/3.
2. If every arrangement of 10 balls numbered ... 10 in a row is equally likely, the probability that all the even numbers are before any odd number is
less than 0.004.
between 0.005 and 0.007.
between 0.008 and 0.01.
more than O.OU.
3. Arrange the numerals in row to get a 5 digit number, the number of such arrangements that are divisible by 3 is
5!.
4. From a set of 100 distinct objects what is the number of different non-empty subsets containing an even number of objects?
2^99
2^99
2^50
2^49
5. The probabilities of two events A and peA) and PCB) respectively are positive,further P(AIB) then
Nothing can be said definitely about
6. A and B are mutually exclusive events and peA) 0 PCB) so,
p(A n
7. A1, A2, and A3 are independent events each of which occur with the same probability then, the probability of at most one of A1, A2 and A3 occurring is
p^3
8. X is a non-negative discrete random variable for which P(X P(X j Vj ..., which of the following distributions fits this fact?
Geometric distribution.
Binomial distribution.
Poisson distribution.
Hypergeometric distribution.
9. The expected value of a random variable is-IO and its variance is 100, the value of the second moment is
10.
100.
200.
1000.
10. V the probability that the quadratic equation x^2 6x 0 will have complex roots is
1/2^10
1/2^10
1/2
1/10.
11. The probability distribution function of a Poisson random variable with a very large mean is to be approximated as a
Exponential random variable.
Negative Binomial random variable.
Normal random variable.
Hypergeometric random variable.
12. The 4th head did not occur till the 15th toss of a coin, so,
A Binomial random variable will be observed to be 15.
A Negative Binomial random variable will take a value that is at least 15.
A Binomial random variable will be observed to be more than 15.
A Negative Binomial random variable will take a value that is more than 15.
13. In garment workshop, shirts are stitched for export, about 10% of the shirts stitched here do not meet specifications and hence are called defective, from a lot of 1000 shirts made in a day, the number of defective shirts in a sample of 100 is a
Negative Binomial random variable.
Hypergeometric random variable.
Binomial random variable.
Poisson random variable.
14. The most appropriate diagram to represent data on grades achieved by students in a public exam is
Bar charts.
Histogram.
Stem and leaf plot.
Ogive.
15. 'Which of the following is not a measure of dispersion
Range.
Mean deviation about median.
Mode.
Mean deviation about mean.
16. The correlation coefficient PX,Y between the random variables X and Y is 0.8, then the correlation coefficient between X and U 20 .-3.2Y is
0.8.
-0.8.
1.
0
17. The correlational coefficient calculated based OIL n observations on the random variables X and Y was -0.3. which of the scatter plots given below reveals this correlation?
<img src='./qimages/16051-17.jpg'>
18. The average of n positive numbers is 100 and their product is 100000000, then,
n 2.
3.
n 4.
we can't say anything about n based on data given.
19. Consider the function
3/2
otherwise
It is a probability density function
It is not a probability density function because integral dx
It is not a probability density function because 0 for some values of x.
It is not a probability density function because f is not increasing in .T.
20. The heights of adult males in a certain population are normally distributed, the heights of half of them are more than 165cm., while the heights of of them are more than 183cm. the percentage of adult males whose heights are less than 147cm.
is less than 2%.
is 5%.
is more than 2%.
we CCll1't say based on data given.
21. X1 and X2 are independent standard normal random variables x1^2 x2^2 follows:
Chi square distribution with 1 degree of freedom.
Exponential distribution with mean 1
Exponential distribution with mean 2
None of the above.
22. A sample survey is to be done to estimate the average milk consumption per family in a city, to draw the most representative sample it was decided to stratify the families and then select samples from each stratum, on the basis of which of the following criteria should the families be stratified.
The last digit of the cell phone number of the head of the family.
The height of the head of the family.
The month of birth of the head of the family.
the total income of the family.
23. In a hypothesis testing problem of Ho against If} based on a sample, type error occurs when
the sample is such that it falls in the complement of the critical region when H1 is true.
the sample is such that it falls in the critical region when the Ho is true.
the sample is such that it falls in the critical region when Ho is false.
the sample is such that it falls in the critical region when neither Ho nor H1 is true.
24. The function
x<0
1
continuous at all xER
continuous everywhere but not differentiable at some points.
decreasing in x.
not continuous at one point.
25. Which of the following is equivalent to the statement 'Ashok did not solve all the problems'
Ashok did not solve any problem.
Ashok did not solve at least one problem.
Ashok solved at least one problem.
Ashok solved at most one problem.
Questions have more than one correct option. • For the answer to be right all the correct options have to be marked on the OMR sheet. • No credit will be given for partially correct answers. • Questions have only one correct option. • Find the correct answers and mark them on the OMR sheet. Correct answers (marked in OMR sheet) to a question get 3 marks and zero otherwise.
26. Which of the following arc random experiments?
Put paper in fire and see what happens to it.
Ask a child to place 10 balls, 1 of which is red and the rest blue in a line and observe the position of the red ball.
Place 10 distinct objects in three distinct boxes and observe which object is in which box.
From a large basket of mangoes take out 5 and report their total weight.
27. X1 and X2
P(Xl P(X2 0).
P(-1.5 Xl P(-3.5 X2
P(X1 P(X2 1).
E(X2).
28. For two random variables X and V(X
is never less than either or
is never less than either or if X and Y are uncorrelated or positively correlated.
is always less than one of and
can be less than both and only if X and Y are negatively correlated.
29. In a certain country, 70% of the households have incomes less than the average income, those households among the highest 10% earners are considered upper class
the median income is less than the average income.
the median income is more than the average income.
About a third of the households with more than average incomes are in the upper class.
Less than a quarter of the households with more than the median income are in the upper class.
30. The probability distribution of a random variable X is P(X ...,
c is equal to 2/3.
c is equal to 4/3.
The expected value of X does not exist.
The variance of X does not exist.
31. Identify the common properties of the random variables with the following pdfs
x 1
o e.w
1/sqrt(2x) -00 x 00
1/2 -00 x 00
All of them have the same mean.
The third moment of all of them is 0
X and of all of them are identically distributed.
All of the have the same variance.
32. Regarding a simple random sample of size n without replacement from a population of size identify the correct statements
Sample raw moments are unbiased estimators of the corresponding raw moments of the population.
Every collection of n population units is equally likely to be the selected sample.
The second central moment of the sample is not an unbiased estimator of the second central moment of the population.
Every unit of the population is equally likely to be in the selected sample.
33. The probability with which a coin shows heads upon tossing is the random variable X1 takes the values 1 and 0 if the outcome of the first toss is heads or tails respectively; another random variable X2 is defined in the same way based on the second toss.
X1+X2/2 is an unbiased estimator of p.
2X1 -X2 is also an unbiased estimator of but not the most efficient.
X1 -X2 is a sufficient statistic of p.
X1 +X2 is a sufficient estimator, but X1-X2 is not a sufficient statistic for p.
34. Band C are three events and if p(AnB) p(AnC) p(BnC) then p(A n B n
has to be zero.
can be 2/5.
can not be 1/6.
can be 1/4.
35. If X and Y are independent real valued random variables, then
V(X
for every y E R
2X and -3Y are also independent random variables.
36. Consider the data given below on the students of a university
Number of female students: 2000 Number of male students: 4000 Number of female students residing In hostels: 1600
Number of male students residing in hostels 3000
from this data one can see that
only one third of the students are female.
only one third of the hostel residents are female.
a larger proportion of female students stay in hostels than male students.
more than half of the students are hostel residents.
37. A is a 5 x 5 real matrix whose 5th row is the sum of the first and second rows. let A^T denote its transpose, then certainly
the rank of A is equal to 4.
the rank of A^T is at most 4.
the determinant of A^T A is equal to 0.
the rank of is less than 5.
38. X is a Poisson random variable with mean if then
1/2.
A 1.
A 2.
A can not be uniquely determined.
39. The value of the integral x^4 dx (limits 0 to 00)is equal to
15/4.
5/2.
3/4.
1/2.
40. Let p be the probability that a coin will show heads upon tossing, further let X denote the number of heads in n tosses of this coin, an unbiased estimator for p^2
is X^2 .
does not exist.
is X^2.
is X(X
41. Let mo, Mo and So denote the mean, median and standard deviation respectively of 15 distinct numbers, further suppose the difference between the median and the largest number smaller than it is 5 and the difference between the median and the smallest number greater than it is 4. Now add 2 to each of the 5 smallest numbers in this list and subtract 2 from each of the larger 5 numbers, denote by TIll, .M1 and 51 the mean, median and the standard deviation respectively of this new set of numbers,
m1 mo; M1 Mo; S1 So
m1 mo; M1 Mo; S1 So
m1 mo; M1 Mo; S1 So
m1 mo; M1 Mo; S1
42. A statistic Tn to estimate a parameter e based on a random sample of size n of a certain random variable is such that for 90% of the samples of size n of that random variable the value of Tn is more than e+10 and for the remaining 10% of the samples of size n the value of Tn is equal to e. This statistic Tn is
a good estimator for e as it has low variance.
a good estimator for e as it seems to be unbiased
is not a good estimator for e as it is more likely to overestimate e.
is not a good estimator for e as it is more likely to underestimate e.
43. X1, ... Xn is a random sample from an unbiased estimator for is
<img src='./qimages/16051-43.jpg'>
44. Let p be the probability of a coin showing up heads when tossed, the hypothesis Ho p Po is to be rejected in favour of H1 P Po if the number of heads X that show up in 10 tosses of this coin is at least 8.
the power of this test is equal to the size of this test for any P1 Po·
for some values of P1 Po, the power of this test is more than its size.
the power of this test is more than its size for every P1 Po·
the power of this test is less than its size for every P1 Po·
45. The sum of squares and the products of every pair of n non-negative real numbers X1,... Xn are known, however X1, ... ,Xn are not known, based on this information
both the mean and the standard deviation of these numbers can be determined.
neither the mean nor the standard deviation of these numbers can be determined.
the mean can not be determined but the standard deviation of these numbers can be determined.
the mean can be determined but the standard deviation of these numbers can not be determined.
46. There are 3 True or False questions in an exam, if a candidate knows the answer she/he answers it correctly, otherwise a guess is made and the probability of getting it right is an examiner assumes that every candidate knows no answer, 1 answer, 2 answers, 3 answers with equal probabilities, a candidate answered two of the three questions correctly, what is the probability that this candidate knew the answer to only one of them?
1/11.
2/11.
3/11.
4/11.
47. There are 2 red and 2 blue balls in a bag, balls are to be removed one by one, the probability that the second ball to be drawn will be red is
2/3.
1/2.
1/3.
1/4.
48. X is a non-constant real valued random variable whose expected value is less than and PX,X^2 0
X and are independent.
O.
0.
49. The time to complete a one year project by an organization is a random variable with probability cx2 0 X 1 density function the probability that a project will get completed o e.w within 9 months(3/4 of a year) is
less than 1/3 .
very close to 1/2.
almost 3/4.
almost 1.
50. are 5 independent observations of a random variable X whose probability density function is
A^2 xe^-Ax
0 o.w
A>0 the maximum likelihood estimate of A
based on the given sample
2
2.5
3.75
5
a little more than 1/2.
equal to 1/4.
equal to 1/3.
a little less than 1/3.
2. If every arrangement of 10 balls numbered ... 10 in a row is equally likely, the probability that all the even numbers are before any odd number is
less than 0.004.
between 0.005 and 0.007.
between 0.008 and 0.01.
more than O.OU.
3. Arrange the numerals in row to get a 5 digit number, the number of such arrangements that are divisible by 3 is
5!.
4. From a set of 100 distinct objects what is the number of different non-empty subsets containing an even number of objects?
2^99
2^99
2^50
2^49
5. The probabilities of two events A and peA) and PCB) respectively are positive,further P(AIB) then
Nothing can be said definitely about
6. A and B are mutually exclusive events and peA) 0 PCB) so,
p(A n
7. A1, A2, and A3 are independent events each of which occur with the same probability then, the probability of at most one of A1, A2 and A3 occurring is
p^3
8. X is a non-negative discrete random variable for which P(X P(X j Vj ..., which of the following distributions fits this fact?
Geometric distribution.
Binomial distribution.
Poisson distribution.
Hypergeometric distribution.
9. The expected value of a random variable is-IO and its variance is 100, the value of the second moment is
10.
100.
200.
1000.
10. V the probability that the quadratic equation x^2 6x 0 will have complex roots is
1/2^10
1/2^10
1/2
1/10.
11. The probability distribution function of a Poisson random variable with a very large mean is to be approximated as a
Exponential random variable.
Negative Binomial random variable.
Normal random variable.
Hypergeometric random variable.
12. The 4th head did not occur till the 15th toss of a coin, so,
A Binomial random variable will be observed to be 15.
A Negative Binomial random variable will take a value that is at least 15.
A Binomial random variable will be observed to be more than 15.
A Negative Binomial random variable will take a value that is more than 15.
13. In garment workshop, shirts are stitched for export, about 10% of the shirts stitched here do not meet specifications and hence are called defective, from a lot of 1000 shirts made in a day, the number of defective shirts in a sample of 100 is a
Negative Binomial random variable.
Hypergeometric random variable.
Binomial random variable.
Poisson random variable.
14. The most appropriate diagram to represent data on grades achieved by students in a public exam is
Bar charts.
Histogram.
Stem and leaf plot.
Ogive.
15. 'Which of the following is not a measure of dispersion
Range.
Mean deviation about median.
Mode.
Mean deviation about mean.
16. The correlation coefficient PX,Y between the random variables X and Y is 0.8, then the correlation coefficient between X and U 20 .-3.2Y is
0.8.
-0.8.
1.
0
17. The correlational coefficient calculated based OIL n observations on the random variables X and Y was -0.3. which of the scatter plots given below reveals this correlation?
<img src='./qimages/16051-17.jpg'>
18. The average of n positive numbers is 100 and their product is 100000000, then,
n 2.
3.
n 4.
we can't say anything about n based on data given.
19. Consider the function
3/2
otherwise
It is a probability density function
It is not a probability density function because integral dx
It is not a probability density function because 0 for some values of x.
It is not a probability density function because f is not increasing in .T.
20. The heights of adult males in a certain population are normally distributed, the heights of half of them are more than 165cm., while the heights of of them are more than 183cm. the percentage of adult males whose heights are less than 147cm.
is less than 2%.
is 5%.
is more than 2%.
we CCll1't say based on data given.
21. X1 and X2 are independent standard normal random variables x1^2 x2^2 follows:
Chi square distribution with 1 degree of freedom.
Exponential distribution with mean 1
Exponential distribution with mean 2
None of the above.
22. A sample survey is to be done to estimate the average milk consumption per family in a city, to draw the most representative sample it was decided to stratify the families and then select samples from each stratum, on the basis of which of the following criteria should the families be stratified.
The last digit of the cell phone number of the head of the family.
The height of the head of the family.
The month of birth of the head of the family.
the total income of the family.
23. In a hypothesis testing problem of Ho against If} based on a sample, type error occurs when
the sample is such that it falls in the complement of the critical region when H1 is true.
the sample is such that it falls in the critical region when the Ho is true.
the sample is such that it falls in the critical region when Ho is false.
the sample is such that it falls in the critical region when neither Ho nor H1 is true.
24. The function
x<0
1
continuous at all xER
continuous everywhere but not differentiable at some points.
decreasing in x.
not continuous at one point.
25. Which of the following is equivalent to the statement 'Ashok did not solve all the problems'
Ashok did not solve any problem.
Ashok did not solve at least one problem.
Ashok solved at least one problem.
Ashok solved at most one problem.
Questions have more than one correct option. • For the answer to be right all the correct options have to be marked on the OMR sheet. • No credit will be given for partially correct answers. • Questions have only one correct option. • Find the correct answers and mark them on the OMR sheet. Correct answers (marked in OMR sheet) to a question get 3 marks and zero otherwise.
26. Which of the following arc random experiments?
Put paper in fire and see what happens to it.
Ask a child to place 10 balls, 1 of which is red and the rest blue in a line and observe the position of the red ball.
Place 10 distinct objects in three distinct boxes and observe which object is in which box.
From a large basket of mangoes take out 5 and report their total weight.
27. X1 and X2
P(Xl P(X2 0).
P(-1.5 Xl P(-3.5 X2
P(X1 P(X2 1).
E(X2).
28. For two random variables X and V(X
is never less than either or
is never less than either or if X and Y are uncorrelated or positively correlated.
is always less than one of and
can be less than both and only if X and Y are negatively correlated.
29. In a certain country, 70% of the households have incomes less than the average income, those households among the highest 10% earners are considered upper class
the median income is less than the average income.
the median income is more than the average income.
About a third of the households with more than average incomes are in the upper class.
Less than a quarter of the households with more than the median income are in the upper class.
30. The probability distribution of a random variable X is P(X ...,
c is equal to 2/3.
c is equal to 4/3.
The expected value of X does not exist.
The variance of X does not exist.
31. Identify the common properties of the random variables with the following pdfs
x 1
o e.w
1/sqrt(2x) -00 x 00
1/2 -00 x 00
All of them have the same mean.
The third moment of all of them is 0
X and of all of them are identically distributed.
All of the have the same variance.
32. Regarding a simple random sample of size n without replacement from a population of size identify the correct statements
Sample raw moments are unbiased estimators of the corresponding raw moments of the population.
Every collection of n population units is equally likely to be the selected sample.
The second central moment of the sample is not an unbiased estimator of the second central moment of the population.
Every unit of the population is equally likely to be in the selected sample.
33. The probability with which a coin shows heads upon tossing is the random variable X1 takes the values 1 and 0 if the outcome of the first toss is heads or tails respectively; another random variable X2 is defined in the same way based on the second toss.
X1+X2/2 is an unbiased estimator of p.
2X1 -X2 is also an unbiased estimator of but not the most efficient.
X1 -X2 is a sufficient statistic of p.
X1 +X2 is a sufficient estimator, but X1-X2 is not a sufficient statistic for p.
34. Band C are three events and if p(AnB) p(AnC) p(BnC) then p(A n B n
has to be zero.
can be 2/5.
can not be 1/6.
can be 1/4.
35. If X and Y are independent real valued random variables, then
V(X
for every y E R
2X and -3Y are also independent random variables.
36. Consider the data given below on the students of a university
Number of female students: 2000 Number of male students: 4000 Number of female students residing In hostels: 1600
Number of male students residing in hostels 3000
from this data one can see that
only one third of the students are female.
only one third of the hostel residents are female.
a larger proportion of female students stay in hostels than male students.
more than half of the students are hostel residents.
37. A is a 5 x 5 real matrix whose 5th row is the sum of the first and second rows. let A^T denote its transpose, then certainly
the rank of A is equal to 4.
the rank of A^T is at most 4.
the determinant of A^T A is equal to 0.
the rank of is less than 5.
38. X is a Poisson random variable with mean if then
1/2.
A 1.
A 2.
A can not be uniquely determined.
39. The value of the integral x^4 dx (limits 0 to 00)is equal to
15/4.
5/2.
3/4.
1/2.
40. Let p be the probability that a coin will show heads upon tossing, further let X denote the number of heads in n tosses of this coin, an unbiased estimator for p^2
is X^2 .
does not exist.
is X^2.
is X(X
41. Let mo, Mo and So denote the mean, median and standard deviation respectively of 15 distinct numbers, further suppose the difference between the median and the largest number smaller than it is 5 and the difference between the median and the smallest number greater than it is 4. Now add 2 to each of the 5 smallest numbers in this list and subtract 2 from each of the larger 5 numbers, denote by TIll, .M1 and 51 the mean, median and the standard deviation respectively of this new set of numbers,
m1 mo; M1 Mo; S1 So
m1 mo; M1 Mo; S1 So
m1 mo; M1 Mo; S1 So
m1 mo; M1 Mo; S1
42. A statistic Tn to estimate a parameter e based on a random sample of size n of a certain random variable is such that for 90% of the samples of size n of that random variable the value of Tn is more than e+10 and for the remaining 10% of the samples of size n the value of Tn is equal to e. This statistic Tn is
a good estimator for e as it has low variance.
a good estimator for e as it seems to be unbiased
is not a good estimator for e as it is more likely to overestimate e.
is not a good estimator for e as it is more likely to underestimate e.
43. X1, ... Xn is a random sample from an unbiased estimator for is
<img src='./qimages/16051-43.jpg'>
44. Let p be the probability of a coin showing up heads when tossed, the hypothesis Ho p Po is to be rejected in favour of H1 P Po if the number of heads X that show up in 10 tosses of this coin is at least 8.
the power of this test is equal to the size of this test for any P1 Po·
for some values of P1 Po, the power of this test is more than its size.
the power of this test is more than its size for every P1 Po·
the power of this test is less than its size for every P1 Po·
45. The sum of squares and the products of every pair of n non-negative real numbers X1,... Xn are known, however X1, ... ,Xn are not known, based on this information
both the mean and the standard deviation of these numbers can be determined.
neither the mean nor the standard deviation of these numbers can be determined.
the mean can not be determined but the standard deviation of these numbers can be determined.
the mean can be determined but the standard deviation of these numbers can not be determined.
46. There are 3 True or False questions in an exam, if a candidate knows the answer she/he answers it correctly, otherwise a guess is made and the probability of getting it right is an examiner assumes that every candidate knows no answer, 1 answer, 2 answers, 3 answers with equal probabilities, a candidate answered two of the three questions correctly, what is the probability that this candidate knew the answer to only one of them?
1/11.
2/11.
3/11.
4/11.
47. There are 2 red and 2 blue balls in a bag, balls are to be removed one by one, the probability that the second ball to be drawn will be red is
2/3.
1/2.
1/3.
1/4.
48. X is a non-constant real valued random variable whose expected value is less than and PX,X^2 0
X and are independent.
O.
0.
49. The time to complete a one year project by an organization is a random variable with probability cx2 0 X 1 density function the probability that a project will get completed o e.w within 9 months(3/4 of a year) is
less than 1/3 .
very close to 1/2.
almost 3/4.
almost 1.
50. are 5 independent observations of a random variable X whose probability density function is
A^2 xe^-Ax
0 o.w
A>0 the maximum likelihood estimate of A
based on the given sample
2
2.5
3.75
5