Exam Details
| Subject | statistics | |
| Paper | ||
| Exam / Course | m.sc | |
| Department | ||
| Organization | central university | |
| Position | ||
| Exam Date | 2013 | |
| City, State | telangana, hyderabad |
Question Paper
1. Probability that at most one of the events A and B occurs is 7/8 the probability of both A and B occurring
is 1/8. is 1/6.
is 1/4. can not be determined from the information given.
2. The mean and median of 10 distinct real numbers are IvI and m respectively,now remove the largest of these numbers let and denote the mean and the median respectively of these 9 numbers,then
JvJ' JvI, <m.
Ai, Ai, >m.
3. 5 balls out of 30 are red how many of the remaining 25 should be blue and how many green so that the number of different possible arrangements in a row is maximized?
9 blue 16 green and vice versa.
10 blue 15 green and vice versa.
11 blue 14 green and vice versa.
12 blue 13 green and vice versa.
4. The sum of 10 positive numbers is 10 then their product
is at least 10. is at most 1.
is equal to 2. is in the interval
5. From a bag containing 2 balls each of 5 different colours, 2 balls are to be drawn without replacement the probability that they will be of different colours is
7/8. 8/9. 9/10. 11/12.
6. The random variable X "" its median is
in the set 13, 14, 15}. in the set 17, 18}.
equal to 10. less than 10.
7. Xl, X 2,.· Xn is a random sample from the population. Let and denote the sample mean maximum and minimum respectively of the sample an unbiased estimator of +is
""
X.
None of the above.
8. What is the number of subsets of 3 numbers from the set ... 1O} that contain at least one multiple of 5
48. 56. 60. 64.
9. The random variable X is uniformly distributed over that is X "" then,
P(O
P(O 1).
P(l X P(O X
P(l X X
10. If most of the measurements in a large data set are of approximately the same magnitude except for a few measurements that are quite a bit larger, then
the mean is smaller than the median and the histogram is skewed with a long left tail.
the mean is larger than the median and the histogram is skewed with a long right tail.
the mean is larger than the median and the histogram is skewed with a long left tail.
the mean is equal to the median and the histogram is symmetric.
11. A sample of 99 distances has a mean of 24 feet and a median of 24.5 feet. Unfortunately, it has just been discovered that an observation which was erroneously recorded as 0U actually had a value of 35. If we make this correction to the data, then:
the mean remains the same, but the median is increased
the mean and median remain the same
the median remains the same, but the mean is increased
we do not know how the mean and median are affected without further calculations; but the variance is increased.
12. According to Chebychev's inequality what can one say about the variance of a random variable for which the probability that it takes values more than 10 away from the mean is 0.1
The variance is more than 10.
The variance is equal to 10.
The variance is less than 10.
Nothing definite can be said.
13. Many professional schools require applicants to take a standardized test. Suppose that 1000 students write the test, and you find that your mark of 63 (out of 100) was the 73rd percentile. This means:
At least 73% of the people got 63 or better.
At least 270 people got 73 or better.
At least 27% of the people got 73 or worse.
At least 270 people got 63 or better.
14. In hypothesis testing, is the probability of committing an error of Type II. The power of the test, 1 is then:
the probability of rejecting Ho when HI is true
(B)the probability of failing to reject Ho when Hl is true
the probability offailing to reject Ho when Ho is true
(D)the probability of failing to reject Ho.
15. The correlation coefficient provides a measure of the
extent to which changes in one variable cause changes in another variable.
strength of the linear association between two quantitative variables.
streneth of the linear association between two categorical variables.
I V
strengthof thelinearassociation.betweenaquantitativevariableandacategorical variable.
16. A fair coin is tossed three times. What is the probability that it lands on heads exactly once?
0.125 0.250 (C)0.333 0.375
17. Identify the correct statements from the following 3 claims. I. The standard error is computed solely from sample attributes. II. The standard deviation is computed solely from sample attributes. III. The standard error is a measure of central tendency.
I only
II only
and III only
None of the above
18. Suppose X and Yare independent random variables. The variance of X is equal to 16; and the variance of Y is equal to 9. Let Z X-Y. What is the standard deviation of
2.65. 5. 7. 25.
19. What is a statistical inference?
A decision,estimate,prediction,or generalization about the population based on information contained in a sample.
A statement made about a sample based on the measurements in that sample.
A decision, estimate, prediction or generalization about sample based on information contained in a population.
A set of data that characterizes some phenomenon.
6
20. Which of the following describes the Binomial random variable with parameters
The number of heads that show up when n coins are tossed one time each.
The number of heads that show up when a coin is tossed n times.
The number of heads that show up when a coin is tossed 2n times.
None of the above.
21. Tn is an unbiased estimator for a parameter Jl based on a random sample of size n
and is equal to an unbiased estimator for J.L2 based on the same sample is
Tn2 . Tn2 +Tn.
Tn 2 Tn 2
22. If a data set of numbers shows two modes,say ml and m2 which appear nl and n2 times respectively also suppose nl n2 it means that
Iml -m21 is less than the absolute difference between any other pair of numbers in the data.
nl and n2 are the highest frequencies.
some other number might appear more often than m2 but it is not close to
ml and m2 have the same largest frequencies.
23. X "V andY Cov.(X, is equal to
O. -1.
24. A is a n x n singular matrix and every row of the matrix B is a linear combination of rows of A the determinant of B
is less than 0 is more than 0
is equal to O. can not be determined from the given information.
25. The negation of the statement "Some books in the library are more than 100 years old" is
Some books in the library are not more than 100 years old.
No book in the library is more than 100 years old.
All the books in the library are more than 100 years old.
At least one book in the library is at least 100 years old.
Part-B
•
Questions may 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 may 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. The two events A and B have positive probabilities and P(AIB) then
P(BjA) P(BIA)
P(AIBC) P(ACIBC)
27. For n unknown positive numbers Xl, X2, ... X n the sums of every pair and the differences of every pair are given but it is not known as to of which pairs the sums and differences are,with this information we can determine
all the numbers. the mean of the n numbers.
the median of the n numbers. the variance of the n numbers.
28. The probability of heads showing up upon tossing a coin is twice the probability of tails showing up, if this coin is to be tossed 10 times, the probability of
all 10 tosses showing up heads is twice the probability of all 10 tosses showing up tails.
6 heads showing up is 16 times the probability of 6 tails showing up.
6 heads showing up is 4 times the probability of 6 tails showing up.
7 heads showing up is 16 times the probability of 7 tails showing up.
29. Regarding the number of different arrangements of 10 red (all alike) and 5 blue (all alike) in a row
more than (1/3yd of the arrangements have red balls at both the ends.
more than (1/3yd of the arrangements have blue balls at both the ends.
almost half the arrangements have balls of different colours at either end.
less than the arrangements have all the 5 blue balls in the first 10 places.
8
30. Given below are marks of 6 students in 2 subjects 51 and 52
I 51 42 596266 75 52
I52 466366 70 7956
The variances are the same for the marks in 51 and 52 and so are the mean deviations about the respective medians.
(B)The variances are the same but the mean deviation about the median in the marks in 51 is more than the marks in 82,
The correlation coefficient between the marks in the 2 subjects is +1.
(D)The correlation coefficient between the marks in the 2 subjects is not but is greater than O.
31. X is a discrete random variable for which P(X P(X 0.8, P(X 0.5, P(X P(X P(X ,so
P(X P(X 0.3
can not be determined. can be determined and is less than 3.
32. 10% of the students of a class weigh at least 65kg. 40% of them weigh at least 55kg. and less than 65kg. 35% of them weigh at least 45kg. and less than 55kg. 15% of them weigh at least 40kg. and less than 45kg.
The average weight of the students of this class is less than 45kg.
The average weight is more than 50kg.
The median weight is more than 55kg.
The median weight is about 45kg.
33. The probability of heads showing up upon tossing a certain coin is this coin is tossed 3 times, let Xi, be 1 or depending on the outcome of the ith toss being heads or tails respectively.
Xl X2+ X3 is a sufficient statistic for p.
X 12 xl+ Xj is a sufficient statistic for p.
X 1X 2X3 is a sufficient statistic for p.
X13 X 23 Xi is a sufficient statistic for p.
34. In general, which of the following statements are ttue?
The sample mean is more sensitive to extreme values than the median.
The sample range is more sensitive to extreme values than the standard deviation.
The sample standard deviation is a measure of spread around the sample mean,
(D)The sample standard deviation is a measure of central tendency around the median.
35. There are 10 girls and 30 boys in M.Sc.(Mathematics) while in M.Sc.(Statistics) there are 4 girls and 16 boys, 2 students are randomly selected from both the classes,then
the probability of selecting both girls is more from M.Sc.(Statistics).
the probability of selecting one girl and one boy is more from M.Sc.(Mathematics).
the probability of selecting both boys is more from M.Sc.(Statistics).
the probability of girls outnumbering boys among the the 4 selected is more than 1/10.
36. Consider the following real valued function defined on R
x<O
x>l
The function f is
non-decreasing. not continuous at x O.
continuous and differentiable everywhere. not continuous at x 1.
37. Xl, X2, X3 are independent and identically distributed random variables with mean e and finite variance the two statistics Xl 3 andy.:= Xl+2X2-X 3 are both
" I3 22
unbiased estimators for eand also
2 V(Y2). V(Y2).
Y2 is a better estimator for ethan YI because has a smaller denominator.
YI isa better estimator for than Y2 because YI is less likely to take values far from than Y2•
The next 13 questions have only one correct option.
38. The number of accidents per day on the Hyderabad -Benguluru highway is a Poisson random variable with parameter .. on 10 randomly chosen days the number of accidents were observed as an unbiased estimate of eA is
eO.8 0.8 2
39. Of 5 girls and 35 boys of a class 5 were absent on a particular day and a class representative was randomly selected from among those present.
The probability that a girl was selected as class representative is 1/8.
The probability that a girl was selected as class representative is 1/16.
The expected number of girls present is 2.
The expected number of absentee boys is less than 4.
40. lim
n
does not exist.
41. is a random sample from X that has the following probability distribution
P(X P(X P(X P(X p 1/2 the Maximum Likelihood Estimate of p based on the given sample is
1/2. 1/3. 1/4. 1/6.
42. The random variable X has Poisson distribution and its 2nd raw moment is equal to 3 times the 1st raw moment so
P(X P(X 0). P(X 2P(X 0).
P(X 0). The statements are wrong.
43. A fair coin is tossed till the first heads shoes up suppose all first 10 tosses showed tails,then the probability that heads will show up on or before the 20th toss is
Si2 1;24
11
44. X is a normally distributed random variable with mean 22 and variance 16 that is X 16) and suppose Y then
I"V I"V
P(X 30) P(Y 30).
P(X 34) P(Y 34).
P(X 30) P(Y 30).
P(X 28) P(Y 29).
45. At least one of the 3 events C will certainly occur, the probability that exactly one of the 3 will happen is 0.3 and the probability that exactly 2 of the 3 events will happen is 0.6 ,from this information
the probability that all the 3 events will happen can be determined and is equal to 0.1.
the probability that all the 3 events will happen can not be determined.
the probability that' none of the 3 events will happen can be determined and is equal to 0.1.
the probability that at least one of the 3 events does not occur can not be determined.
46. The probability distribution of a random variable X is P(X Pk, k ... and -kb if Po 1/9,Pl P2 4/27 what can you say about the sequence of
Pk-l
probabilities ..
They are in Arithmetic Progression They are in Geometric Progression.
They are terms in a certain binomial expansion.
Neither of and is correct.
B-8
47. The a level critical region for the hypothesis testing problem Ho vs. HI: is C where is the mean of a normal random variable with variance 2 if the problem is changed to Ho vs. HI where ,what can we say about the probability of Type error with the same critical region
It is equal to a. It is less than a.
It is more than Q. Nothing definite can be said.
48. Xl rv exp(3) and X2 rv exp(6) let XOl and X02 denote the medians of Xl and X2 respectively then
XOI X02· XOI
XOl 2X02. XOl X02 +3.
49. The non -constant random variables X and Y have the following property: 2V(X it follows that
X and Yare independently distributed.
X and Yare uncorrelated but not independently distributed.
X and Yare negatively correlated.
X and Yare positively correlated.
50. The proportion of work that can be completed in a project in the first year is a random variable with density function
cx2(1 0 x 1
f
o o.W
The probability that more than of the work will be completed in the first year is
more than 3/4.
more than 1/2 but less than 2/3.
more than 1/3 but less than 1/2.
more than 1/4 but less than 1/3.
is 1/8. is 1/6.
is 1/4. can not be determined from the information given.
2. The mean and median of 10 distinct real numbers are IvI and m respectively,now remove the largest of these numbers let and denote the mean and the median respectively of these 9 numbers,then
JvJ' JvI, <m.
Ai, Ai, >m.
3. 5 balls out of 30 are red how many of the remaining 25 should be blue and how many green so that the number of different possible arrangements in a row is maximized?
9 blue 16 green and vice versa.
10 blue 15 green and vice versa.
11 blue 14 green and vice versa.
12 blue 13 green and vice versa.
4. The sum of 10 positive numbers is 10 then their product
is at least 10. is at most 1.
is equal to 2. is in the interval
5. From a bag containing 2 balls each of 5 different colours, 2 balls are to be drawn without replacement the probability that they will be of different colours is
7/8. 8/9. 9/10. 11/12.
6. The random variable X "" its median is
in the set 13, 14, 15}. in the set 17, 18}.
equal to 10. less than 10.
7. Xl, X 2,.· Xn is a random sample from the population. Let and denote the sample mean maximum and minimum respectively of the sample an unbiased estimator of +is
""
X.
None of the above.
8. What is the number of subsets of 3 numbers from the set ... 1O} that contain at least one multiple of 5
48. 56. 60. 64.
9. The random variable X is uniformly distributed over that is X "" then,
P(O
P(O 1).
P(l X P(O X
P(l X X
10. If most of the measurements in a large data set are of approximately the same magnitude except for a few measurements that are quite a bit larger, then
the mean is smaller than the median and the histogram is skewed with a long left tail.
the mean is larger than the median and the histogram is skewed with a long right tail.
the mean is larger than the median and the histogram is skewed with a long left tail.
the mean is equal to the median and the histogram is symmetric.
11. A sample of 99 distances has a mean of 24 feet and a median of 24.5 feet. Unfortunately, it has just been discovered that an observation which was erroneously recorded as 0U actually had a value of 35. If we make this correction to the data, then:
the mean remains the same, but the median is increased
the mean and median remain the same
the median remains the same, but the mean is increased
we do not know how the mean and median are affected without further calculations; but the variance is increased.
12. According to Chebychev's inequality what can one say about the variance of a random variable for which the probability that it takes values more than 10 away from the mean is 0.1
The variance is more than 10.
The variance is equal to 10.
The variance is less than 10.
Nothing definite can be said.
13. Many professional schools require applicants to take a standardized test. Suppose that 1000 students write the test, and you find that your mark of 63 (out of 100) was the 73rd percentile. This means:
At least 73% of the people got 63 or better.
At least 270 people got 73 or better.
At least 27% of the people got 73 or worse.
At least 270 people got 63 or better.
14. In hypothesis testing, is the probability of committing an error of Type II. The power of the test, 1 is then:
the probability of rejecting Ho when HI is true
(B)the probability of failing to reject Ho when Hl is true
the probability offailing to reject Ho when Ho is true
(D)the probability of failing to reject Ho.
15. The correlation coefficient provides a measure of the
extent to which changes in one variable cause changes in another variable.
strength of the linear association between two quantitative variables.
streneth of the linear association between two categorical variables.
I V
strengthof thelinearassociation.betweenaquantitativevariableandacategorical variable.
16. A fair coin is tossed three times. What is the probability that it lands on heads exactly once?
0.125 0.250 (C)0.333 0.375
17. Identify the correct statements from the following 3 claims. I. The standard error is computed solely from sample attributes. II. The standard deviation is computed solely from sample attributes. III. The standard error is a measure of central tendency.
I only
II only
and III only
None of the above
18. Suppose X and Yare independent random variables. The variance of X is equal to 16; and the variance of Y is equal to 9. Let Z X-Y. What is the standard deviation of
2.65. 5. 7. 25.
19. What is a statistical inference?
A decision,estimate,prediction,or generalization about the population based on information contained in a sample.
A statement made about a sample based on the measurements in that sample.
A decision, estimate, prediction or generalization about sample based on information contained in a population.
A set of data that characterizes some phenomenon.
6
20. Which of the following describes the Binomial random variable with parameters
The number of heads that show up when n coins are tossed one time each.
The number of heads that show up when a coin is tossed n times.
The number of heads that show up when a coin is tossed 2n times.
None of the above.
21. Tn is an unbiased estimator for a parameter Jl based on a random sample of size n
and is equal to an unbiased estimator for J.L2 based on the same sample is
Tn2 . Tn2 +Tn.
Tn 2 Tn 2
22. If a data set of numbers shows two modes,say ml and m2 which appear nl and n2 times respectively also suppose nl n2 it means that
Iml -m21 is less than the absolute difference between any other pair of numbers in the data.
nl and n2 are the highest frequencies.
some other number might appear more often than m2 but it is not close to
ml and m2 have the same largest frequencies.
23. X "V andY Cov.(X, is equal to
O. -1.
24. A is a n x n singular matrix and every row of the matrix B is a linear combination of rows of A the determinant of B
is less than 0 is more than 0
is equal to O. can not be determined from the given information.
25. The negation of the statement "Some books in the library are more than 100 years old" is
Some books in the library are not more than 100 years old.
No book in the library is more than 100 years old.
All the books in the library are more than 100 years old.
At least one book in the library is at least 100 years old.
Part-B
•
Questions may 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 may 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. The two events A and B have positive probabilities and P(AIB) then
P(BjA) P(BIA)
P(AIBC) P(ACIBC)
27. For n unknown positive numbers Xl, X2, ... X n the sums of every pair and the differences of every pair are given but it is not known as to of which pairs the sums and differences are,with this information we can determine
all the numbers. the mean of the n numbers.
the median of the n numbers. the variance of the n numbers.
28. The probability of heads showing up upon tossing a coin is twice the probability of tails showing up, if this coin is to be tossed 10 times, the probability of
all 10 tosses showing up heads is twice the probability of all 10 tosses showing up tails.
6 heads showing up is 16 times the probability of 6 tails showing up.
6 heads showing up is 4 times the probability of 6 tails showing up.
7 heads showing up is 16 times the probability of 7 tails showing up.
29. Regarding the number of different arrangements of 10 red (all alike) and 5 blue (all alike) in a row
more than (1/3yd of the arrangements have red balls at both the ends.
more than (1/3yd of the arrangements have blue balls at both the ends.
almost half the arrangements have balls of different colours at either end.
less than the arrangements have all the 5 blue balls in the first 10 places.
8
30. Given below are marks of 6 students in 2 subjects 51 and 52
I 51 42 596266 75 52
I52 466366 70 7956
The variances are the same for the marks in 51 and 52 and so are the mean deviations about the respective medians.
(B)The variances are the same but the mean deviation about the median in the marks in 51 is more than the marks in 82,
The correlation coefficient between the marks in the 2 subjects is +1.
(D)The correlation coefficient between the marks in the 2 subjects is not but is greater than O.
31. X is a discrete random variable for which P(X P(X 0.8, P(X 0.5, P(X P(X P(X ,so
P(X P(X 0.3
can not be determined. can be determined and is less than 3.
32. 10% of the students of a class weigh at least 65kg. 40% of them weigh at least 55kg. and less than 65kg. 35% of them weigh at least 45kg. and less than 55kg. 15% of them weigh at least 40kg. and less than 45kg.
The average weight of the students of this class is less than 45kg.
The average weight is more than 50kg.
The median weight is more than 55kg.
The median weight is about 45kg.
33. The probability of heads showing up upon tossing a certain coin is this coin is tossed 3 times, let Xi, be 1 or depending on the outcome of the ith toss being heads or tails respectively.
Xl X2+ X3 is a sufficient statistic for p.
X 12 xl+ Xj is a sufficient statistic for p.
X 1X 2X3 is a sufficient statistic for p.
X13 X 23 Xi is a sufficient statistic for p.
34. In general, which of the following statements are ttue?
The sample mean is more sensitive to extreme values than the median.
The sample range is more sensitive to extreme values than the standard deviation.
The sample standard deviation is a measure of spread around the sample mean,
(D)The sample standard deviation is a measure of central tendency around the median.
35. There are 10 girls and 30 boys in M.Sc.(Mathematics) while in M.Sc.(Statistics) there are 4 girls and 16 boys, 2 students are randomly selected from both the classes,then
the probability of selecting both girls is more from M.Sc.(Statistics).
the probability of selecting one girl and one boy is more from M.Sc.(Mathematics).
the probability of selecting both boys is more from M.Sc.(Statistics).
the probability of girls outnumbering boys among the the 4 selected is more than 1/10.
36. Consider the following real valued function defined on R
x<O
x>l
The function f is
non-decreasing. not continuous at x O.
continuous and differentiable everywhere. not continuous at x 1.
37. Xl, X2, X3 are independent and identically distributed random variables with mean e and finite variance the two statistics Xl 3 andy.:= Xl+2X2-X 3 are both
" I3 22
unbiased estimators for eand also
2 V(Y2). V(Y2).
Y2 is a better estimator for ethan YI because has a smaller denominator.
YI isa better estimator for than Y2 because YI is less likely to take values far from than Y2•
The next 13 questions have only one correct option.
38. The number of accidents per day on the Hyderabad -Benguluru highway is a Poisson random variable with parameter .. on 10 randomly chosen days the number of accidents were observed as an unbiased estimate of eA is
eO.8 0.8 2
39. Of 5 girls and 35 boys of a class 5 were absent on a particular day and a class representative was randomly selected from among those present.
The probability that a girl was selected as class representative is 1/8.
The probability that a girl was selected as class representative is 1/16.
The expected number of girls present is 2.
The expected number of absentee boys is less than 4.
40. lim
n
does not exist.
41. is a random sample from X that has the following probability distribution
P(X P(X P(X P(X p 1/2 the Maximum Likelihood Estimate of p based on the given sample is
1/2. 1/3. 1/4. 1/6.
42. The random variable X has Poisson distribution and its 2nd raw moment is equal to 3 times the 1st raw moment so
P(X P(X 0). P(X 2P(X 0).
P(X 0). The statements are wrong.
43. A fair coin is tossed till the first heads shoes up suppose all first 10 tosses showed tails,then the probability that heads will show up on or before the 20th toss is
Si2 1;24
11
44. X is a normally distributed random variable with mean 22 and variance 16 that is X 16) and suppose Y then
I"V I"V
P(X 30) P(Y 30).
P(X 34) P(Y 34).
P(X 30) P(Y 30).
P(X 28) P(Y 29).
45. At least one of the 3 events C will certainly occur, the probability that exactly one of the 3 will happen is 0.3 and the probability that exactly 2 of the 3 events will happen is 0.6 ,from this information
the probability that all the 3 events will happen can be determined and is equal to 0.1.
the probability that all the 3 events will happen can not be determined.
the probability that' none of the 3 events will happen can be determined and is equal to 0.1.
the probability that at least one of the 3 events does not occur can not be determined.
46. The probability distribution of a random variable X is P(X Pk, k ... and -kb if Po 1/9,Pl P2 4/27 what can you say about the sequence of
Pk-l
probabilities ..
They are in Arithmetic Progression They are in Geometric Progression.
They are terms in a certain binomial expansion.
Neither of and is correct.
B-8
47. The a level critical region for the hypothesis testing problem Ho vs. HI: is C where is the mean of a normal random variable with variance 2 if the problem is changed to Ho vs. HI where ,what can we say about the probability of Type error with the same critical region
It is equal to a. It is less than a.
It is more than Q. Nothing definite can be said.
48. Xl rv exp(3) and X2 rv exp(6) let XOl and X02 denote the medians of Xl and X2 respectively then
XOI X02· XOI
XOl 2X02. XOl X02 +3.
49. The non -constant random variables X and Y have the following property: 2V(X it follows that
X and Yare independently distributed.
X and Yare uncorrelated but not independently distributed.
X and Yare negatively correlated.
X and Yare positively correlated.
50. The proportion of work that can be completed in a project in the first year is a random variable with density function
cx2(1 0 x 1
f
o o.W
The probability that more than of the work will be completed in the first year is
more than 3/4.
more than 1/2 but less than 2/3.
more than 1/3 but less than 1/2.
more than 1/4 but less than 1/3.