STATISTICS-PAGE 1
STATISTICS MCQ- PAGE 1
1. To test H0 : µ = 1 against H0 : µ ≠ 1 based on large sample, the test statistic Z has a value 2. Then p-value associated to the test is:
A:-P[|Z|<2]
B:-P[|Z|>2]
C:-P[Z<2]
D:-P[Z>2]
Ans: B
2. Let X and Y be random variables with Cov(X,Y)=-0.25, then which of the following is true:
A:-Var(X+Y)>Var(X-Y)
B:-Var(X+Y)<Var(X-Y)
C:-Var(X+Y)=Var(X-Y)
D:-None of these
Ans: B
3. The degrees of freedom associated to t-test for the difference of the means of two samples having sizes m, n based on large sample is:
A:-m+n-1
B:-m+n-mn
C:-m+n
D:-m+n-2
Ans: D
4. If F follows F(7,8), then 1/F follows:
A:-F(7,8)
B:-F(1,8)
C:-F(7,1)
D:-F(8,7)
Ans: D
5. The distribution function F(x) of a random variable X lies between:
A:-0 and 1
B:--1 and 1
C:-0 and ∞
D:-None of these
Ans: A
6. The probability mass function of a discrete random variable X is f(x)= x/10 for x=1,2,3,4 and 0 for other values of X. Let F(x) denote the distribution function of X. Then F(4)-F(3) is:
A:- 4/10
B:- 2/10
C:- 3/10
D:- 1/10
Ans: A
7. A continuous random variable X is symmetric about a real number a (a ∈ R) if the distribution function X-a is same as the distribution function of:
A:-a-X
B:-X+a
C:--X-a
D:--X+a
Ans: A
8. Let X be a random variable for which E(X) exists and A is any real number. Then E|X-A| is minimum if:
A:-A=E(X)
B:-A=Med(X)
C:-A=Mod(X)
D:-None of these
Ans: B
A:-P[|Z|<2]
B:-P[|Z|>2]
C:-P[Z<2]
D:-P[Z>2]
Ans: B
2. Let X and Y be random variables with Cov(X,Y)=-0.25, then which of the following is true:
A:-Var(X+Y)>Var(X-Y)
B:-Var(X+Y)<Var(X-Y)
C:-Var(X+Y)=Var(X-Y)
D:-None of these
Ans: B
3. The degrees of freedom associated to t-test for the difference of the means of two samples having sizes m, n based on large sample is:
A:-m+n-1
B:-m+n-mn
C:-m+n
D:-m+n-2
Ans: D
4. If F follows F(7,8), then 1/F follows:
A:-F(7,8)
B:-F(1,8)
C:-F(7,1)
D:-F(8,7)
Ans: D
5. The distribution function F(x) of a random variable X lies between:
A:-0 and 1
B:--1 and 1
C:-0 and ∞
D:-None of these
Ans: A
6. The probability mass function of a discrete random variable X is f(x)= x/10 for x=1,2,3,4 and 0 for other values of X. Let F(x) denote the distribution function of X. Then F(4)-F(3) is:
A:- 4/10
B:- 2/10
C:- 3/10
D:- 1/10
Ans: A
7. A continuous random variable X is symmetric about a real number a (a ∈ R) if the distribution function X-a is same as the distribution function of:
A:-a-X
B:-X+a
C:--X-a
D:--X+a
Ans: A
8. Let X be a random variable for which E(X) exists and A is any real number. Then E|X-A| is minimum if:
A:-A=E(X)
B:-A=Med(X)
C:-A=Mod(X)
D:-None of these
Ans: B
9. Who among the following is the real giant in the development of the theory of Statistics?
A:-I. Fisher
B:-Prof. R.A. Fisher
C:-P.C. Mahalanobis
D:-C.R. Rao
Ans: B
10. A suitable method of collecting data in cases where the informants are literate and spread over a vast area:
A:-Direct personal interview
B:-Mailed questionnaire method
C:-Sample method
D:-Primary method
Ans: B
11. The point of intersection of ogives correspond to:
A:-Mean
B:-Geometric mean
C:-Mode
D:-Median
Ans: D
12. In a ratio graph, the vertical scale starts with:
A:-0
B:--1
C:-1
D:-Any positive number
Ans: D
13. Out of 19 students appeared for a test only 10 students are qualified and their scores are respectively 36, 45, 58, 63, 39, 43, 47, 34, 41 and 50. The median mark of all students is :
A:-45
B:-39
C:-34
D:-41
Ans: C
14. The arithmetic mean and harmonic mean of certain data set are respectively 90 and 40. Then the geometric mean is :
A:-50
B:-60
C:-80
D:-Data is not sufficient
Ans: B
15. The arithmetic mean of two sample observations is greater than the smallest by their :
A:-Standard error
B:-Variance
C:-Range
D:-None of these
Ans: A
16. The harmonic mean of certain data set is 25 and if each observation is multiplied by 2. Then the harmonic mean of new data set is :
A:-25/2
B:-25
C:-100
D:-50
Ans: D
17. In Lorenz curve , the diagonal line y=x is known as:
A:-Coefficient of determination
B:-Line of unequal distribution
C:-Line of equal distribution
D:-Line of poverty
Ans: C
18. If 25% of the items in a distribution are less than 10 and 25% are more than 40, the quartile deviation is :
A:-25
B:-20
C:-15
D:-5
Ans: C
19. The standard deviation of the observations x and y is :
A:-Absolute value of (x-y)/2
B:-Absolute value of (x-y)
C:-(x-y)
D:-None of these
Ans: A
20. The distribution of mortality rates with respect to the age after ignoring the accidental deaths will give:
A:-Positively skewed distribution
B:-Negatively skewed distribution
C:-Symmetric distribution
D:-None of these
Ans: A
21. Which one of the following is true for a discrete distribution?
A:-β2 > 1
B:-β2 > 3
C:-β2 < 3
D:-β2 > 2
Ans: A
22. The sum of squares of deviations is least when measured from :
A:-Median
B:-Mean
C:-Mode
D:-None of these
Ans: B
23. The axiomatic approach to probability was proposed by:
A:-Karl Pearson
B:-Laplace
C:-A. Kolmogorov
D:-A.N. Kolmogorov
Ans: D
24. 10 persons are seated on 10 chairs at a round table. The probability that two specified persons are sitting next to each other is:
A:- 2/10
B:- 1/10
C:-2/9
D:-1/9
Ans: C
25. Which of the following statement is most correct:
A:-P(AB) ≤ P(A)
B:-P(AB) ≤ P(B)
C:-P(AB) ≤ min(P(A),P(B))
D:-P(AB) ≤ max(P(A),P(B))
Ans: C
26. A random sample of 10 different observations is given. How many samples of {(x, y): x < y} can be formed is:
A:-45
B:-90
C:-60
D:-30
Ans: A
27. If P(A)=P(B)=P(C)=0.5, P(AB)=P(AC)=P(BC)=0.2 and P(ABC)=0.1, then P(A-B-C) is :
A:-0.15
B:-0.20
C:-0.10
D:-0
Ans: B
28. The probability of choosing a square of dimension 2 from a chess board of dimension 8 is:
A:- 1/64
B:- 2/64
C:- 4/64
D:-None of these
Ans: D
29. If A and B are exhaustive and equally likely events with P(AB)=0.2, then P(A) is:
A:-0.6
B:-0.4
C:-0.8
D:-None of these
Ans: B
30. A problem in statistics is given to 3 students A, B and C whose chances of solving it are 1/2, 3/4, and 1/4 respectively. The probability that exactly one solves the problem is:
A:-19/32
B:-29/32
C:- 3/32
D:-13/32
Ans: D
31. Which of the following statement is true ?
A:-Disjoint events are independent
B:-Independent events may be disjoint
C:-Both options 1 and 2
D:-None of these
Ans: B
32. Five events are said to be mutually independent if they have to satisfy ........... conditions:
A:-26
B:-30
C:-28
D:-32
Ans: A
33. Two friends decided to meet between 2pm and 3pm with the proviso that one waits the other for at most 20 minutes. The chance of their meeting is:
A:-1/9
B:-2/9
C:-4/9
D:-5/9
Ans: D
34. Bayes' formula is used to obtain the probabilities of:
A:-Posterior events
B:-Likelihood events
C:-Prior events
D:-None of these
Ans: A
35. The distribution which holds the property non correlation of random variables implies independence is:
A:-Bivariate normal
B:-Bivariate exponential
C:-Bivariate Cauchy
D:-None of these
Ans: A
36. The mean sum of squares is obtained by dividing the sum of squares by:
A:-Size of the sample
B:-Degrees of freedom
C:-Squared degrees of freedom
D:-Squared sample size
Ans: B
37. The method of moment estimator for Θ in a uniform distribution over [-Θ, Θ ] with sample mean 10 and sample variance 4 is:
A:-2√3
B:-24
C:-10
D:-0
Ans: A
38. The degrees of freedom associated to error sum of squares in one-way ANOVA having n observations and k treatments is:
A:-n-1
B:-k-1
C:-n-k
D:-k+1
Ans: C
39. The sum of all two digit numbers formed using the digits 1, 2, 3 and 4 if each digit is used exactly once is:
A:-110
B:-284
C:-330
D:-None of these
Ans: C
40. The moment generating function M(t) of a random variable X exists at:
A:-Any real value of t
B:-t=0
C:-Neighborhood of zero
D:-Deleted neighborhood of zero
Ans: C
41.Francis Galton is pioneered in the study of:
A:-Biometry
B:-Genetics
C:-Regression
D:-Correlation
Ans: C
42. The correlation coefficient of the bi variate data: (1,10), (2,9), (3,8) and (4,7) is
A:-1
B:--1
C:-0.6
D:-None of these
Ans: B
43. Let r(x, y)=0.8. Then the explained variation in y due to x is:
A:-80%
B:-64%
C:-81%
D:-70%
Ans: B
44. If both regression coefficients are positive, then their sum is always:
A:- ≥ 1
B:-Lies between 1 and 2
C:- ≥ 2
D:-None of these
Ans: D
45. The line of best fit can be obtained by the principle of:
A:-Least squares
B:-Moments
C:-Mixed moments
D:-Minimum chi-sqare
Ans: A
46. Probable error is used to test:
A:-Observed correlation coefficient
B:-Regression coefficients
C:-Rank correlation
D:-Consistency
Ans: A
47. Let X be the number of successes follow B(n,p), then the distribution of failures follow:
A:-B(n,p)
B:-B(n, 1-p)
C:-B(2n, 1-p)
D:-None of these
Ans: B
48. Let X follows B(n,p) is positively skewed if :
A:-p<1/2
B:-p>1/2
C:-p=1/2
D:-o<p<1
Ans: A
49. Correlation coefficient between the number of successes and failures in B(n,p) is:
A:-1
B:--1
C:-0
D:-None of these
Ans: B
50. Referring to Question 50, E(X/U=3) is:
A:-1
B:-2/3
C:-5/3
D:-6/5
Ans: D
51. Which of the following statement about B(n,p) is always true?
A:-It is under dispersed
B:-It is over dispersed
C:-Neither option1 nor option 2
D:-Both options 1 and 2 depend on values of p
Ans: A
52. If X follows U(0,1), then Var(1-X) is:
A:- 1/12
B:-1/6
C:-1/2
D:-1/4
Ans: A
53. The maximum height of N(0,1) curve is :
A:-e
B:-√e
C:- 1/√π
D:- 1/√2Π
Ans: D
54. As the scale parameter of normal curve increases, the distribution retains symmetry and becomes:
A:-Flatter
B:-Peaked
C:-Neither 1 nor 2
D:-None of these
Ans: A
55. If X and Y are independent N(0,1) random variates, then P(X<Y) is :
A:-1/2
B:-0
C:-1.96
D:-1.65
Ans: A
56. The Normal curve has an area about .......within one unit of SD from mean:
A:-65%
B:-68%
C:-33%
D:-67%
Ans: B
57. The square of t distribution is an F distribution for:
A:-2 df
B:-1 df
C:-n df
D:-None of these
Ans: B
58. The ratio of two independent N(0,1) variates is a:
A:-t1
B:-t2
C:-tn
D:-χ 2
Ans: A
59. The random variable X has mean 5 and variance 9. Then P[|X-5|>4] is:
A:-> 9/16
B:->4/9
C:-< 9/16
D:-<4/9
Ans: C
60. The statistical error associated to the statement "An innocent person is proved as guilty" is :
A:-Type 1 error
B:-Type 2 error
C:-Power
D:-Critical region
Ans: A
A:-I. Fisher
B:-Prof. R.A. Fisher
C:-P.C. Mahalanobis
D:-C.R. Rao
Ans: B
10. A suitable method of collecting data in cases where the informants are literate and spread over a vast area:
A:-Direct personal interview
B:-Mailed questionnaire method
C:-Sample method
D:-Primary method
Ans: B
11. The point of intersection of ogives correspond to:
A:-Mean
B:-Geometric mean
C:-Mode
D:-Median
Ans: D
12. In a ratio graph, the vertical scale starts with:
A:-0
B:--1
C:-1
D:-Any positive number
Ans: D
13. Out of 19 students appeared for a test only 10 students are qualified and their scores are respectively 36, 45, 58, 63, 39, 43, 47, 34, 41 and 50. The median mark of all students is :
A:-45
B:-39
C:-34
D:-41
Ans: C
14. The arithmetic mean and harmonic mean of certain data set are respectively 90 and 40. Then the geometric mean is :
A:-50
B:-60
C:-80
D:-Data is not sufficient
Ans: B
15. The arithmetic mean of two sample observations is greater than the smallest by their :
A:-Standard error
B:-Variance
C:-Range
D:-None of these
Ans: A
16. The harmonic mean of certain data set is 25 and if each observation is multiplied by 2. Then the harmonic mean of new data set is :
A:-25/2
B:-25
C:-100
D:-50
Ans: D
17. In Lorenz curve , the diagonal line y=x is known as:
A:-Coefficient of determination
B:-Line of unequal distribution
C:-Line of equal distribution
D:-Line of poverty
Ans: C
18. If 25% of the items in a distribution are less than 10 and 25% are more than 40, the quartile deviation is :
A:-25
B:-20
C:-15
D:-5
Ans: C
19. The standard deviation of the observations x and y is :
A:-Absolute value of (x-y)/2
B:-Absolute value of (x-y)
C:-(x-y)
D:-None of these
Ans: A
20. The distribution of mortality rates with respect to the age after ignoring the accidental deaths will give:
A:-Positively skewed distribution
B:-Negatively skewed distribution
C:-Symmetric distribution
D:-None of these
Ans: A
21. Which one of the following is true for a discrete distribution?
A:-β2 > 1
B:-β2 > 3
C:-β2 < 3
D:-β2 > 2
Ans: A
22. The sum of squares of deviations is least when measured from :
A:-Median
B:-Mean
C:-Mode
D:-None of these
Ans: B
23. The axiomatic approach to probability was proposed by:
A:-Karl Pearson
B:-Laplace
C:-A. Kolmogorov
D:-A.N. Kolmogorov
Ans: D
24. 10 persons are seated on 10 chairs at a round table. The probability that two specified persons are sitting next to each other is:
A:- 2/10
B:- 1/10
C:-2/9
D:-1/9
Ans: C
25. Which of the following statement is most correct:
A:-P(AB) ≤ P(A)
B:-P(AB) ≤ P(B)
C:-P(AB) ≤ min(P(A),P(B))
D:-P(AB) ≤ max(P(A),P(B))
Ans: C
26. A random sample of 10 different observations is given. How many samples of {(x, y): x < y} can be formed is:
A:-45
B:-90
C:-60
D:-30
Ans: A
27. If P(A)=P(B)=P(C)=0.5, P(AB)=P(AC)=P(BC)=0.2 and P(ABC)=0.1, then P(A-B-C) is :
A:-0.15
B:-0.20
C:-0.10
D:-0
Ans: B
28. The probability of choosing a square of dimension 2 from a chess board of dimension 8 is:
A:- 1/64
B:- 2/64
C:- 4/64
D:-None of these
Ans: D
29. If A and B are exhaustive and equally likely events with P(AB)=0.2, then P(A) is:
A:-0.6
B:-0.4
C:-0.8
D:-None of these
Ans: B
30. A problem in statistics is given to 3 students A, B and C whose chances of solving it are 1/2, 3/4, and 1/4 respectively. The probability that exactly one solves the problem is:
A:-19/32
B:-29/32
C:- 3/32
D:-13/32
Ans: D
31. Which of the following statement is true ?
A:-Disjoint events are independent
B:-Independent events may be disjoint
C:-Both options 1 and 2
D:-None of these
Ans: B
32. Five events are said to be mutually independent if they have to satisfy ........... conditions:
A:-26
B:-30
C:-28
D:-32
Ans: A
33. Two friends decided to meet between 2pm and 3pm with the proviso that one waits the other for at most 20 minutes. The chance of their meeting is:
A:-1/9
B:-2/9
C:-4/9
D:-5/9
Ans: D
34. Bayes' formula is used to obtain the probabilities of:
A:-Posterior events
B:-Likelihood events
C:-Prior events
D:-None of these
Ans: A
35. The distribution which holds the property non correlation of random variables implies independence is:
A:-Bivariate normal
B:-Bivariate exponential
C:-Bivariate Cauchy
D:-None of these
Ans: A
36. The mean sum of squares is obtained by dividing the sum of squares by:
A:-Size of the sample
B:-Degrees of freedom
C:-Squared degrees of freedom
D:-Squared sample size
Ans: B
37. The method of moment estimator for Θ in a uniform distribution over [-Θ, Θ ] with sample mean 10 and sample variance 4 is:
A:-2√3
B:-24
C:-10
D:-0
Ans: A
38. The degrees of freedom associated to error sum of squares in one-way ANOVA having n observations and k treatments is:
A:-n-1
B:-k-1
C:-n-k
D:-k+1
Ans: C
39. The sum of all two digit numbers formed using the digits 1, 2, 3 and 4 if each digit is used exactly once is:
A:-110
B:-284
C:-330
D:-None of these
Ans: C
40. The moment generating function M(t) of a random variable X exists at:
A:-Any real value of t
B:-t=0
C:-Neighborhood of zero
D:-Deleted neighborhood of zero
Ans: C
41.Francis Galton is pioneered in the study of:
A:-Biometry
B:-Genetics
C:-Regression
D:-Correlation
Ans: C
42. The correlation coefficient of the bi variate data: (1,10), (2,9), (3,8) and (4,7) is
A:-1
B:--1
C:-0.6
D:-None of these
Ans: B
43. Let r(x, y)=0.8. Then the explained variation in y due to x is:
A:-80%
B:-64%
C:-81%
D:-70%
Ans: B
44. If both regression coefficients are positive, then their sum is always:
A:- ≥ 1
B:-Lies between 1 and 2
C:- ≥ 2
D:-None of these
Ans: D
45. The line of best fit can be obtained by the principle of:
A:-Least squares
B:-Moments
C:-Mixed moments
D:-Minimum chi-sqare
Ans: A
46. Probable error is used to test:
A:-Observed correlation coefficient
B:-Regression coefficients
C:-Rank correlation
D:-Consistency
Ans: A
47. Let X be the number of successes follow B(n,p), then the distribution of failures follow:
A:-B(n,p)
B:-B(n, 1-p)
C:-B(2n, 1-p)
D:-None of these
Ans: B
48. Let X follows B(n,p) is positively skewed if :
A:-p<1/2
B:-p>1/2
C:-p=1/2
D:-o<p<1
Ans: A
49. Correlation coefficient between the number of successes and failures in B(n,p) is:
A:-1
B:--1
C:-0
D:-None of these
Ans: B
50. Referring to Question 50, E(X/U=3) is:
A:-1
B:-2/3
C:-5/3
D:-6/5
Ans: D
51. Which of the following statement about B(n,p) is always true?
A:-It is under dispersed
B:-It is over dispersed
C:-Neither option1 nor option 2
D:-Both options 1 and 2 depend on values of p
Ans: A
52. If X follows U(0,1), then Var(1-X) is:
A:- 1/12
B:-1/6
C:-1/2
D:-1/4
Ans: A
53. The maximum height of N(0,1) curve is :
A:-e
B:-√e
C:- 1/√π
D:- 1/√2Π
Ans: D
54. As the scale parameter of normal curve increases, the distribution retains symmetry and becomes:
A:-Flatter
B:-Peaked
C:-Neither 1 nor 2
D:-None of these
Ans: A
55. If X and Y are independent N(0,1) random variates, then P(X<Y) is :
A:-1/2
B:-0
C:-1.96
D:-1.65
Ans: A
56. The Normal curve has an area about .......within one unit of SD from mean:
A:-65%
B:-68%
C:-33%
D:-67%
Ans: B
57. The square of t distribution is an F distribution for:
A:-2 df
B:-1 df
C:-n df
D:-None of these
Ans: B
58. The ratio of two independent N(0,1) variates is a:
A:-t1
B:-t2
C:-tn
D:-χ 2
Ans: A
59. The random variable X has mean 5 and variance 9. Then P[|X-5|>4] is:
A:-> 9/16
B:->4/9
C:-< 9/16
D:-<4/9
Ans: C
60. The statistical error associated to the statement "An innocent person is proved as guilty" is :
A:-Type 1 error
B:-Type 2 error
C:-Power
D:-Critical region
Ans: A
61. In which experimental design, there are three experimental errors ?
62. In a latin square design, if there are six treatments then what will be the number of replications in the experiment ?
63. The ratio of standard deviation to mean expressed in percentage is termed as :-
64:-Which of the following is the simplest measure of dispersion?
A:-Mean
B:-Median
C:-Range
D:-Mode
Ans: C
65:-Which of the following is a non parametric test?
A:-ANCOVA
B:-Correlation
C:-Chi-square
D:-MANCOVA
Ans: C
- Randomized Block Design
- Split Plot Design
- Strip Plot Design
- Latin Square Design
62. In a latin square design, if there are six treatments then what will be the number of replications in the experiment ?
- 5
- 6
- 7
- 8
63. The ratio of standard deviation to mean expressed in percentage is termed as :-
- Standard error
- Critical difference
- Coefficient of variation
- Mean deviation
64:-Which of the following is the simplest measure of dispersion?
A:-Mean
B:-Median
C:-Range
D:-Mode
Ans: C
65:-Which of the following is a non parametric test?
A:-ANCOVA
B:-Correlation
C:-Chi-square
D:-MANCOVA
Ans: C
66:-Local control is a device to maintain
A:-homogeneity within blocks
B:-homogeneity among blocks
C:-both (1) and (2)
D:-neither (1) nor (2)
Ans: A
67:-In a linear model `quadY_{ij}=alpha_i+e_{ij},` `quadj=1,2,...,n_i;` `quadi=1,2,...,k,` consider
(i) `quadalpha_1-3alpha_2+alpha_3+alpha_4`
(ii) `quadalpha_1+3alpha_2-alpha_3-alpha_4`
(iii) `quadalpha_1+3alpha_2-2alpha_3-2alpha_4`
Then which of the following is correct?
A:-(i) and (ii) are linear contrasts
B:-(i) and (iii) are linear contrasts
C:-(ii) and (iii) are linear contrasts
D:-(i), (ii) and (iii) are linear contrasts
Ans: B
68:-While analyzing the data of a `quadkxxk` Latin Square Design, the degrees of freedom in the ANOVA is
A:-`quadk^2-1`
B:-`quadk-1`
C:-`quadk^2-2k+1`
D:-`quad(k-1)(k-2)`
Ans: D
69:-In a split plot design with factor `quadA` at 3 levels in main plots, factor `quadB` at 3 levels in sub-plots and 3
replications, the degrees of freedom for sub-plot error is
A:-27
B:-12
C:-8
D:-4
Ans: B
70:-If the interactions `quadAB` and `quadBC` are confounded with incomplete blocks in a `quad2^n` factorial
experiment, then automatically confounded effect is
A:-`quadA`
B:-`quadC`
C:-`quadAC`
D:-`quadABC`
Ans: C
71:-Which among the following is a consistent estimator of the population mean when samples are from the Cauchy population?
A:-Sample mean
B:-Sample median
C:-Sample variance
D:-None of these
Ans: B
72:-If the regularity conditions of the CR inequality are violated then the least attainable variance will be
A:-equal to the CR bound
B:-greater than the CR bound
C:-less than the CR bound
D:-zero
Ans: C
73:-A method to obtain the UMVUE is by using
A:-Rao-Blackwell Theorem
B:-Baye's Theorem
C:-Neymann-Pearson Theorem
D:-Lehmann-Scheffe Theorem
Ans: D
74:-A complete-sufficient statistic for `p` in the Bernoulli distribution
`(x, p) = p^x (1-p)^x; x=0, 1.
= 0 `"otherwise"` is
A:-The first order statistic `X_{(1)}`
B:-The `n` `"^{th}"` order statistic `X_{(n)}`
C:-`sum_(i=1)^n X_i`
D:-`X_{(n)}-X_{(1)}`
Ans: C
75:-The least square estimators are
A:-Unbiased
B:-BLUE
C:-UMVUE
D:-All these
Ans: D
76:-A 95% confidence interval for λ, when a large sample is taken from a Poisson population with parameter λ is
A:-`bar x ``+- 1.65``sqrt(( bar x)/(n)) `
B:-`lambda+- 1.65 sqrt((lambda)/(n)) `
C:-`bar x+- 1.96sqrt((bar x)/(n)) `
D:-`lambda +- 1.96sqrt((lambda)/(n)) `
Ans: C
77:-The minimum Chi-squared estimators are not necessarily
A:-Unbiased
B:-Consistent
C:-Efficient
D:-Asymptotically normal
Ans: A
78:-Which one of the following statements is true?
A:-Even if the UMP test does not exist, a UMPU test may exist
B:-Even if the UMPU test does not exist, a UMP test may exist
C:-A UMP test exists only if a UMPU test exists
D:-A UMPU test exists only if a UMP test exists
Ans: A
79:-In paired `t` test the two random variables should be
A:-Paired and uncorrelated
B:-Unpaired and correlated
C:-Both paired and correlated
D:-Neither paired nor correlated
Ans: C
80:-With usual notations, the criterion for acceptance in SPRT is
A:-`lambda_m <= ((1-beta))/(alpha) `
B:-`lambda_m >= ((1-beta))/(alpha) `
C:-`lambda_m <=(beta)/((1-alpha)) `
D:-` lambda_m >= (beta)/((1-alpha)`
Ans: C
81:-The Poisson process with parameter λ is a renewal counting process for which the unit lifetimes have ________distribution with common parameter λ.
A:-Poisson
B:-Exponential
C:-Uniform
D:-Geometric
Ans: B
82:-Let `{X_n, n = 0, 1, 2...}` be a Branching process and the corresponding offspring distribution has a pgf `P (s)
=(2)/(3) +(s+s^2)/(6) ` . Find the probability of extinction of the process
A:-0
B:-0.25
C:-0.66
D:-1
Ans: D
83:-Let `{X_n}` be a renewal process with `mu = E (X_1) <oo ` and if `M (t)` is the renewal function, then
`lim_(t->oo) (M (t))/(t) = .....`
A:-`(1)/(mu) `
B:-`mu `
C:-`(t)/(mu) `
D:-`(mu)/(t) `
Ans: A
84:-If `X_i`'s are independent Poisson variates with respective parameters `lambda_i`, for `i = 1, 2, ...k`, then the
conditional distribution of `X_1, X_2,... X_k` given their sum `sum_(i=1)^k X_i = n` is a ___________ distribution with
parameters ___________ and _____________.
A:-Binomial with parameters `n` and ` (1)/(k)`
B:-Binomial with parameters `k` and `(1)/(n)`
C:-Multinomial with parameters `n` and `(1)/(k)`
D:-Multinomial with parameters `k` and `(1)/(n)`
Ans: C
85:-If ` (X_1, X_2)` is a Bivariate normal random vector with parameters `(mu_{X1}, mu_{X2}, sigma ^2_X_1,sigma^2_X_2, rho`), when `sigma^2_X_1 = sigma^2_X_2 ` and `rho = 0` , the density function is called
A:-Elliptical Normal
B:-Circular Normal
C:-Symmetrical Normal
D:-Uniform Normal
Ans: B
86:-If the random vector `X` follows Multivariate Normal distribution with mean vector 0 and dispersion matrix `I`
and `Q_i = X^' A_i X` are quadratic forms of rank `r_i` such that `sum_(i=1)^k A_i = I_p` , then a necessary and sufficient
condition for `Q_i`'s to be distributed as independent chi-square random variables with `r_i` d.f is that
A:-` sum_(i=1)^k r_i=k`
B:-`sum_(i=1)^k r_i = p`
C:-`sum_(i=1)^k r_i=0`
D:-`sum_(i=1)^k r_i = kp`
Ans: B
87:-The relationship between partial correlation coefficients `r_{ij.k},` multiple correlation
coefficients `R_{i.jk}`and simple correlation coefficients `r_{ij}` is
A:-`R^2_1.23 = 1+ (1-r^2_12) (1 - r^2_13.2)`
B:-`R^2_1.23 = 1 - (1- r^2_12) (1- r^2_13.2)`
C:-`R^2_1.23 = 1 + (1-r^2_12)// (1-r^2_13.2)`
D:-`R^2_1.23 = 1- (1-r^2_12) // (1-r^2_13.2)`
Ans: B
88:-Hotelling's `T^2` statistic and Mahalnobis `D^2` statistic are connected by the relationship
A:-`D^2 = ((N_1 N_2))/((N_1+N_2)) T^2`
B:-`D^2 =((N_1 N_2))/((N_1-N_2)) T^2`
C:-`D^2 = ((N_1-N_2))/((N_1 N_2)) T^2`
D:-`D^2 = ((N_1 + N_2))/((N_1 N_2)) T^2`
Ans: D
89:-In principal component analysis the variances of the Principal Components are the __________ of the covariance matrix.
A:-diagonal elements
B:-eigen values
C:-normalized elements
D:-non-zero elements
Ans: B
90:-For discriminating between two populations R.A. Fisher suggested the linear discriminant function `X'l` for which
A:-` ("(mean difference)"^2)/("variance")`
B:-`("mean difference")^2/("A.M.")`
C:-`("mean difference")/("median")`
D:-`("variance")/("mean difference")`
Ans: A
91:-Assume that the time to failure `(T)` for a certain bulb has an exponential distribution `f ((t)/(lambda))` with parameter `lambda >0` with the prior pdf `g (lambda)` of `lambda` is an exponential distribution with parameter 2. Then the posterior pdf of `lambda` given `T = t` is
A:-`(2)/(t+2)`
B:-`(lambda)/(e^lambda (t+2))`
C:-`(lambda e^(lambda (t+2)))/((t+2)^2)`
D:-`(lambda (t+2)^2)/(e^(lambda (t+2)))`
Ans: D
92:-The basic elements of statistical decision theory is
A:-a space Ω `= {ul theta}` of all possible states of nature
B:-an action space `A = {a}` of all possible courses of action
C:-a loss function `L (ul theta, a)` giving the incurred loss when action `a` is taken and the state is ` ul theta`
D:-all these
Ans: D
93:-When there is no censoring for the life length `T`, the general formula of a survival function is
A:-`hat {S (t)} = (" # of individuals with" T >= t)/("total sample size")`
B:-`hat {S (t)} = (" # of individuals with" T <= t)/("total sample size")`
C:-`hat {S (t)} = (" # of individuals with" T = t)/("total sample size")`
D:-`hat {S (t)} = (" # of individuals with" T = 0)/("total sample size")`
Ans: A
94:-The Cox's Proportional Hazard Model (Cox's PH Model) with explanatory variables ` ul X = (X_1, X_2, ... X_p), beta_i` their regression coefficients and `h_0 (t)` a base line hazard, is `h (t, ul X) =`
A:-`e^{h_0 (t) sum_(i=1)^p beta_i X_i}`
B:-`log h_0 (t) + sum_(i=1)^p beta_i X_i`
C:-`h_0 (t) e sum_(i=1)^p beta_i X_i`
D:-`e^(h_0 (t)) sum_(i=1)^p beta_i X_i`
Ans: C
95:-When an inspection lot contains no defectives the OC function `L (p)` is
A:-`L (p) = 1`
B:-`L (p) = oo`
C:-` L (p) = 0`
D:-None of these
Ans: A
96:-"Simple random sampling" is the technique of drawing a sample in such a way that each unit of the population has
A:-distinct and dependent chance of being included in the sample
B:-distinct but independent chance of being included in the sample
C:-an equal but dependent chance of being included in the sample
D:-an equal and independent chance of being included in the sample
Ans: D
97:-In SRSWR with usual notations, the standard error of the sample mean `quadbary` is
A:-`quadS({N-n}/{Nn})^{1/2}`
B:-`quadS({N-1}/{Nn})^{1/2}`
C:-`quadS(1-{n}/{N})^{1/2}`
D:-`quadS/n(1-{1}/{N})^{1/2}`
Ans: B
98:-The formulae for optimum allocation in various strata in stratified sampling were first derived by
A:-Tschuprov
B:-Cochran
C:-Lahiri
D:-Neymann
Ans: A
99:-The ratio estimator of population mean is unbiased if sampling is done according to
A:-PPSWR
B:-PPSWOR
C:-SRSWR
D:-SRSWOR
Ans: A
100:-The cluster sampling is more efficient when
A:-the variation within clusters in more
B:-the variation between clusters is less
C:-both (1) and (2)
D:-neither (1) nor (2)
Ans: C
A:-homogeneity within blocks
B:-homogeneity among blocks
C:-both (1) and (2)
D:-neither (1) nor (2)
Ans: A
67:-In a linear model `quadY_{ij}=alpha_i+e_{ij},` `quadj=1,2,...,n_i;` `quadi=1,2,...,k,` consider
(i) `quadalpha_1-3alpha_2+alpha_3+alpha_4`
(ii) `quadalpha_1+3alpha_2-alpha_3-alpha_4`
(iii) `quadalpha_1+3alpha_2-2alpha_3-2alpha_4`
Then which of the following is correct?
A:-(i) and (ii) are linear contrasts
B:-(i) and (iii) are linear contrasts
C:-(ii) and (iii) are linear contrasts
D:-(i), (ii) and (iii) are linear contrasts
Ans: B
68:-While analyzing the data of a `quadkxxk` Latin Square Design, the degrees of freedom in the ANOVA is
A:-`quadk^2-1`
B:-`quadk-1`
C:-`quadk^2-2k+1`
D:-`quad(k-1)(k-2)`
Ans: D
69:-In a split plot design with factor `quadA` at 3 levels in main plots, factor `quadB` at 3 levels in sub-plots and 3
replications, the degrees of freedom for sub-plot error is
A:-27
B:-12
C:-8
D:-4
Ans: B
70:-If the interactions `quadAB` and `quadBC` are confounded with incomplete blocks in a `quad2^n` factorial
experiment, then automatically confounded effect is
A:-`quadA`
B:-`quadC`
C:-`quadAC`
D:-`quadABC`
Ans: C
71:-Which among the following is a consistent estimator of the population mean when samples are from the Cauchy population?
A:-Sample mean
B:-Sample median
C:-Sample variance
D:-None of these
Ans: B
72:-If the regularity conditions of the CR inequality are violated then the least attainable variance will be
A:-equal to the CR bound
B:-greater than the CR bound
C:-less than the CR bound
D:-zero
Ans: C
73:-A method to obtain the UMVUE is by using
A:-Rao-Blackwell Theorem
B:-Baye's Theorem
C:-Neymann-Pearson Theorem
D:-Lehmann-Scheffe Theorem
Ans: D
74:-A complete-sufficient statistic for `p` in the Bernoulli distribution
`(x, p) = p^x (1-p)^x; x=0, 1.
= 0 `"otherwise"` is
A:-The first order statistic `X_{(1)}`
B:-The `n` `"^{th}"` order statistic `X_{(n)}`
C:-`sum_(i=1)^n X_i`
D:-`X_{(n)}-X_{(1)}`
Ans: C
75:-The least square estimators are
A:-Unbiased
B:-BLUE
C:-UMVUE
D:-All these
Ans: D
76:-A 95% confidence interval for λ, when a large sample is taken from a Poisson population with parameter λ is
A:-`bar x ``+- 1.65``sqrt(( bar x)/(n)) `
B:-`lambda+- 1.65 sqrt((lambda)/(n)) `
C:-`bar x+- 1.96sqrt((bar x)/(n)) `
D:-`lambda +- 1.96sqrt((lambda)/(n)) `
Ans: C
77:-The minimum Chi-squared estimators are not necessarily
A:-Unbiased
B:-Consistent
C:-Efficient
D:-Asymptotically normal
Ans: A
78:-Which one of the following statements is true?
A:-Even if the UMP test does not exist, a UMPU test may exist
B:-Even if the UMPU test does not exist, a UMP test may exist
C:-A UMP test exists only if a UMPU test exists
D:-A UMPU test exists only if a UMP test exists
Ans: A
79:-In paired `t` test the two random variables should be
A:-Paired and uncorrelated
B:-Unpaired and correlated
C:-Both paired and correlated
D:-Neither paired nor correlated
Ans: C
80:-With usual notations, the criterion for acceptance in SPRT is
A:-`lambda_m <= ((1-beta))/(alpha) `
B:-`lambda_m >= ((1-beta))/(alpha) `
C:-`lambda_m <=(beta)/((1-alpha)) `
D:-` lambda_m >= (beta)/((1-alpha)`
Ans: C
81:-The Poisson process with parameter λ is a renewal counting process for which the unit lifetimes have ________distribution with common parameter λ.
A:-Poisson
B:-Exponential
C:-Uniform
D:-Geometric
Ans: B
82:-Let `{X_n, n = 0, 1, 2...}` be a Branching process and the corresponding offspring distribution has a pgf `P (s)
=(2)/(3) +(s+s^2)/(6) ` . Find the probability of extinction of the process
A:-0
B:-0.25
C:-0.66
D:-1
Ans: D
83:-Let `{X_n}` be a renewal process with `mu = E (X_1) <oo ` and if `M (t)` is the renewal function, then
`lim_(t->oo) (M (t))/(t) = .....`
A:-`(1)/(mu) `
B:-`mu `
C:-`(t)/(mu) `
D:-`(mu)/(t) `
Ans: A
84:-If `X_i`'s are independent Poisson variates with respective parameters `lambda_i`, for `i = 1, 2, ...k`, then the
conditional distribution of `X_1, X_2,... X_k` given their sum `sum_(i=1)^k X_i = n` is a ___________ distribution with
parameters ___________ and _____________.
A:-Binomial with parameters `n` and ` (1)/(k)`
B:-Binomial with parameters `k` and `(1)/(n)`
C:-Multinomial with parameters `n` and `(1)/(k)`
D:-Multinomial with parameters `k` and `(1)/(n)`
Ans: C
85:-If ` (X_1, X_2)` is a Bivariate normal random vector with parameters `(mu_{X1}, mu_{X2}, sigma ^2_X_1,sigma^2_X_2, rho`), when `sigma^2_X_1 = sigma^2_X_2 ` and `rho = 0` , the density function is called
A:-Elliptical Normal
B:-Circular Normal
C:-Symmetrical Normal
D:-Uniform Normal
Ans: B
86:-If the random vector `X` follows Multivariate Normal distribution with mean vector 0 and dispersion matrix `I`
and `Q_i = X^' A_i X` are quadratic forms of rank `r_i` such that `sum_(i=1)^k A_i = I_p` , then a necessary and sufficient
condition for `Q_i`'s to be distributed as independent chi-square random variables with `r_i` d.f is that
A:-` sum_(i=1)^k r_i=k`
B:-`sum_(i=1)^k r_i = p`
C:-`sum_(i=1)^k r_i=0`
D:-`sum_(i=1)^k r_i = kp`
Ans: B
87:-The relationship between partial correlation coefficients `r_{ij.k},` multiple correlation
coefficients `R_{i.jk}`and simple correlation coefficients `r_{ij}` is
A:-`R^2_1.23 = 1+ (1-r^2_12) (1 - r^2_13.2)`
B:-`R^2_1.23 = 1 - (1- r^2_12) (1- r^2_13.2)`
C:-`R^2_1.23 = 1 + (1-r^2_12)// (1-r^2_13.2)`
D:-`R^2_1.23 = 1- (1-r^2_12) // (1-r^2_13.2)`
Ans: B
88:-Hotelling's `T^2` statistic and Mahalnobis `D^2` statistic are connected by the relationship
A:-`D^2 = ((N_1 N_2))/((N_1+N_2)) T^2`
B:-`D^2 =((N_1 N_2))/((N_1-N_2)) T^2`
C:-`D^2 = ((N_1-N_2))/((N_1 N_2)) T^2`
D:-`D^2 = ((N_1 + N_2))/((N_1 N_2)) T^2`
Ans: D
89:-In principal component analysis the variances of the Principal Components are the __________ of the covariance matrix.
A:-diagonal elements
B:-eigen values
C:-normalized elements
D:-non-zero elements
Ans: B
90:-For discriminating between two populations R.A. Fisher suggested the linear discriminant function `X'l` for which
A:-` ("(mean difference)"^2)/("variance")`
B:-`("mean difference")^2/("A.M.")`
C:-`("mean difference")/("median")`
D:-`("variance")/("mean difference")`
Ans: A
91:-Assume that the time to failure `(T)` for a certain bulb has an exponential distribution `f ((t)/(lambda))` with parameter `lambda >0` with the prior pdf `g (lambda)` of `lambda` is an exponential distribution with parameter 2. Then the posterior pdf of `lambda` given `T = t` is
A:-`(2)/(t+2)`
B:-`(lambda)/(e^lambda (t+2))`
C:-`(lambda e^(lambda (t+2)))/((t+2)^2)`
D:-`(lambda (t+2)^2)/(e^(lambda (t+2)))`
Ans: D
92:-The basic elements of statistical decision theory is
A:-a space Ω `= {ul theta}` of all possible states of nature
B:-an action space `A = {a}` of all possible courses of action
C:-a loss function `L (ul theta, a)` giving the incurred loss when action `a` is taken and the state is ` ul theta`
D:-all these
Ans: D
93:-When there is no censoring for the life length `T`, the general formula of a survival function is
A:-`hat {S (t)} = (" # of individuals with" T >= t)/("total sample size")`
B:-`hat {S (t)} = (" # of individuals with" T <= t)/("total sample size")`
C:-`hat {S (t)} = (" # of individuals with" T = t)/("total sample size")`
D:-`hat {S (t)} = (" # of individuals with" T = 0)/("total sample size")`
Ans: A
94:-The Cox's Proportional Hazard Model (Cox's PH Model) with explanatory variables ` ul X = (X_1, X_2, ... X_p), beta_i` their regression coefficients and `h_0 (t)` a base line hazard, is `h (t, ul X) =`
A:-`e^{h_0 (t) sum_(i=1)^p beta_i X_i}`
B:-`log h_0 (t) + sum_(i=1)^p beta_i X_i`
C:-`h_0 (t) e sum_(i=1)^p beta_i X_i`
D:-`e^(h_0 (t)) sum_(i=1)^p beta_i X_i`
Ans: C
95:-When an inspection lot contains no defectives the OC function `L (p)` is
A:-`L (p) = 1`
B:-`L (p) = oo`
C:-` L (p) = 0`
D:-None of these
Ans: A
96:-"Simple random sampling" is the technique of drawing a sample in such a way that each unit of the population has
A:-distinct and dependent chance of being included in the sample
B:-distinct but independent chance of being included in the sample
C:-an equal but dependent chance of being included in the sample
D:-an equal and independent chance of being included in the sample
Ans: D
97:-In SRSWR with usual notations, the standard error of the sample mean `quadbary` is
A:-`quadS({N-n}/{Nn})^{1/2}`
B:-`quadS({N-1}/{Nn})^{1/2}`
C:-`quadS(1-{n}/{N})^{1/2}`
D:-`quadS/n(1-{1}/{N})^{1/2}`
Ans: B
98:-The formulae for optimum allocation in various strata in stratified sampling were first derived by
A:-Tschuprov
B:-Cochran
C:-Lahiri
D:-Neymann
Ans: A
99:-The ratio estimator of population mean is unbiased if sampling is done according to
A:-PPSWR
B:-PPSWOR
C:-SRSWR
D:-SRSWOR
Ans: A
100:-The cluster sampling is more efficient when
A:-the variation within clusters in more
B:-the variation between clusters is less
C:-both (1) and (2)
D:-neither (1) nor (2)
Ans: C