| Post Name | Statistician |
|---|---|
| Department | KIRTADS Category No : 255/2022 |
| Question Paper Code | 021/2023/OL |
| Category Code | 255/2022 |
| Date of Test | 12 April 2023 |
| Exam Type | Online Examination |
| Total Questions | 100 |
1
If f is convex and EX is finite, then f(EX) ≤ Ef(X) is known as
Answer
C. Jensen's inequality
2
If X and Y are uniformly distributed on (0, 1), then E(|X − Y|) is
Answer
Question Cancelled
3
Let X be a random variable with characteristic function given by Φ(u), then the value of Var(Sin x) + Var(Cos x) =
Answer
C. 1 − |Φ(1)|²
4
If probability of hitting a target is 0.5, what is the probability of hitting the target on 6th attempt
Answer
B. (0.5)⁶
5
Buses travel on every half hours in day times in a remote area. What is the probability that a man reaching bus stop in day time will have to wait for 20 minutes ?
Answer
D. 0.33
6
What is the probability of success in an event with both mean and variance are found 4, assuming binomial distribution ?
Answer
Question Cancelled
7
The points of inflexion of a normal curve are
Answer
A. μ±σ
8
The characteristic function for Cauchy distribution is
Answer
B. e^(−|t|)
9
Xᵢ = 1 if the iᵗʰ outcome is a success with p(success) = p; Xᵢ = 0 if the iᵗʰ outcome is a failure with p(failure) = q. Then the distribution of the random variable Sₙ = ΣXᵢ, where Xᵢ are independent, is ___________ as n tends to infinity.
Answer
C. Asymptotically normal
10
A fair die is thrown 600 times find the lower bound for probability of getting the number of sixes in between 80 and 120 assuming binomial distribution.
Answer
D. 19/24
11
If an unbiased estimator and a sufficient statistic exist for T, then the minimum variance estimator for T is always a function of
Answer
B. Sufficient Statistic
12
If x₁, x₂, ..., xₙ are independent observations from a normal population such that E(xᵢ)=μ and V(xᵢ)=σ², for i=1,2,...,n. Then _____ is a least square estimate of σ².
Answer
D. [1/(n−1)]∑ᵢ₌₁ⁿ(xᵢ−x̄)²
13
If Tₙ is a sequence of estimates such that E(Tₙ) → θ and Var(Tₙ) → 0 as n → ∞, then Tₙ is:
Answer
Question Cancelled
14
An estimator is said to be sufficient statistic for a parametric function T(θ) if it contain all the information that is in the
Answer
B. Sample
15
________ do not always exist.
Answer
A. Unbiased estimator
16
The graphical shape of t-distribution is based on
Answer
B. degrees of freedom
17
The MLE of a parameter α of a population having density α − x), 0 < x < α is
Answer
Question Cancelled
18
The method of moments were first invented by
Answer
B. Karl Pearson
19
Difference between expected value of an estimator and the corresponding parameter is
Answer
D. bias
20
The consistency of estimators is identified by comparing
Answer
Question Cancelled
21
To test the hypothesis involving proportions, both np and nq should be
Answer
A. greater than 5
22
Power of the test is the probability of
Answer
B. correct decision
23
A critical region which is most powerful when a hypothesis is tested against a series of alternatives is called
Answer
C. UMPCR
24
Neyman-Pearson Lemma is used to find ______ for testing simple against a simple .
Answer
B. Best Critical region
25
Degrees of freedom for a chi square test of independence with contingency table of order m n is
Answer
D. (m − 1)(n − 1)
26
Kruskal Wallis test is a nonparametric analogue of
Answer
C. One-way ANOVA
27
As the value of Chi-square near to zero in testing goodness of fit, we say that
Answer
C. good fit
28
Statistical hypothesis under test are called
Answer
D. Null hypothesis
29
Which of the following is not a non parametric test ?
Answer
A. F
30
A statistical test to determine whether there is non random association two categorical variables is
Answer
B. Fisher's exact test
31
In an RCBD with 5 treatments each replicated in 4 blocks, the d.f. for error is
Answer
C. 12
32
Analysing a CRD, if the F value in ANOVA Table is less than one, what is shows
Answer
A. Design is wrong
33
If we want to identify the best nutritious food for weigh gain in chicken after trying three such foods and measured weekly for one year, which analysis you prefer ?
Answer
C. Repeated Measures ANOVA
34
Replication is used in Design of Experiments to
Answer
D. To estimate the error
35
In a uniformity trial to estimate the fertility gradient of an experimental material which method is suitable ?
Answer
C. Fairfield Smith's Variance law
36
When standard deviation is proportional to mean, the transformation used before doing ANOVA is
Answer
C. Logarithmic
37
The method used to split the factorial experiment in to two in the case of large number of treatments is
Answer
A. Defining contrast
38
A symmetric BIBD exists for even value of v, where v is the number of treatments, r the number of replications and is the no. of pairs, only if r- is a
Answer
B. a perfect square
39
In a resolvable BIBD with number of treatments v, number of replications r, the number of blocks b should be
Answer
C. ≥ v + r − 1
40
In a breeding trial with large number of lines, where RBD or BIBD can't be considered
Answer
A. Lattice Design
41
In the case of study of estimation of milk production from cows in Kerala, which type of sampling design is better with limited resources and time ?
Answer
Question Cancelled
42
Unit of sample selected is known as
Answer
B. Sample unit
43
Proximity to the value obtained by the repeated application of the sampling procedure is
Answer
D. precision
44
Ratio estimator is more efficient when
Answer
A. ρ >
45
If there is perfect correlation with y and x so that y = a + bx, then in SRS the estimator Nȳ will be superior to ratio estimator X R̂ if
Answer
D. [x̄² v(1/x̄³)]/sₓ² > (b²/a²)(1−f)/n
46
Intra cluster correlation coefficient ρ is calculated using the formula
Answer
A. [1/(M−1)](σ_b²−σ_w²)/σ²
47
If c′ is the cost of observing x and c is the cost of observing y, then double sampling for the ratio method is superior to single-phase sampling if
Answer
B. ρ² > 4cc′/(c+c′)²
48
In SRSWOR sample mean square is an unbiased estimate of population
Answer
B. Mean square
49
Sampling Variance __________ due to post stratification.
Answer
A. increases
50
In medical practices, there are some diseases which are yet to be under research, person having such diseases can be identified be some references. Such type of sampling is known as
Answer
D. Snowball Sampling
51
Let X be a random variable with pdf f(x)=kx³e⁻²ˣ, x>0. The value of K is
Answer
B. 8/3
52
Let X and Y be two independent geometric random variables. The conditional distribution of X|X + Y is
Answer
A. Uniform
53
Consider a random experiment with two possible outcomes success and failure. Experiment is repeating n times independently. Let X denote the no. of success with constant probability p. Distribution of the no. of failure is
Answer
D. Binomial
54
Which of the following statements is/are correct about Cauchy distribution ? i. For a Cauchy random variable, central moments exist ii. For a Cauchy random variable, mean does not exist iii. For a Cauchy random variable, characteristic function exists
Answer
C. Only (ii and iii)
55
Let X and Y are two independent exponential random variables with parameters 2 and 3. Which of the following is correct ?
Answer
C. Min(X, Y) ~ exponential (5)
56
Let X₁, X₂, ..., Xₙ be a set of n independent observations taken from U(0,2). Let X₍₁₎, X₍₂₎, ..., X₍ₙ₎ be the ordered sample observations. Then the joint distribution of X₍₁₎, X₍₂₎, ..., X₍ₙ₎ is
Answer
D. n!/2ⁿ
57
Ratio of squares of two independent standard normal variate is
Answer
B. F distribution with (1, 1) degrees of freedom
58
Suppose , and are three independent standard normal variates. Then the distribution of is
Answer
D. t distribution with 3 degrees of freedom
59
The mark of students of a class is normally distributed with mean 35 and standard deviation 5. The probability of a student mark between 25 and 45 is
Answer
C. 0.9544
60
The coefficient of variation of Poisson distribution with mean 16 is
Answer
A. 25%
61
Let X be (μ, Σ) with Σ = Which of the following statements is/are true ? (i) and are independent (ii) and are not independent (iii) ( , ) and are independent (iv) ( , ) and are not independent
Answer
C. Only (ii and iii)
62
Sum of independent products of multivariate normal random vectors is
Answer
A. Wishart Distribution
63
Let X be (μ, Σ) with μ'= [1 -2 1] and Σ = . Find the distribution of − −
Answer
C. N(4, 14)
64
Let X₁, X₂, ..., X₁₀ be a random sample of size 10 from N₅(μ,Σ). Then the distribution of 10(X̄−μ)′Σ⁻¹(X̄−μ) is
Answer
A. χ²₅
65
Let and are independently distributed with W(Σ, 2) and W(Σ, 5) respectively. Then the distribution of + is
Answer
D. W(Σ, 7)
66
Multivariate Behrin Fisher's problem is
Answer
C. Testing the equality of means of two multivariate populations when the covariance matrices are not equal
67
Consider the testing of equality of means of two multivariate normal populations N(μ⁽¹⁾, Σ) and N(μ⁽²⁾, Σ). Then the relationship between Hotelling T² and Mahalanobis D² is
Answer
B. T² = [N₁N₂/(N₁ + N₂)]D²
68
Suppose X ~ (μ, Σ) and S is the sample covariance matrix of a sample of size N. then the Statistic N( − μ)′ ( −μ) is
Answer
C. Hotelling
69
Suppose random variables X₁ and X₂ have covariance matrix [[5, 2], [2, 2]]. What proportion of total population variance is explained by the first principal component?
Answer
B. 6/7
70
Consider the orthogonal factor model X = M + LF + ε. Which statements are true? (i) The covariance matrix of specific factors is diagonal. (ii) Common and specific factors are uncorrelated. (iii) The variance of the i-th variable is its communality.
Answer
A. Only (i and ii)
71
A Markov chain {Xₙ, n ≥ 0} has states 0, 1, 2, transition matrix P = [[1/3, 2/3, 0], [1/4, 1/2, 1/4], [0, 1/3, 2/3]], and P(X₀ = i) = 1/3 for i = 0, 1, 2. Find P[X₃ = 1, X₂ = 2, X₁ = 1, X₀ = 0].
Answer
B. 1/54
72
Suppose a Markov chain { } have a state space {1, 2, 3, 4} and transition probability matrix is given by . Which of the following statement is/are true ? (i) Markov chain { } is irreducible (ii) Markov chain { } is aperiodic (iii) Markov chain { } does not have a stationary distribution
Answer
D. Only (i and ii)
73
A Markov chain has state space {1, 2, 3, 4} and transition matrix P = [[1/2, 1/2, 0, 0], [1/2, 0, 1/2, 0], [1/2, 0, 0, 1/2], [0, 0, 0, 1]]. Find lim(n→∞) (Pⁿ)₁₁.
Answer
C. 0
74
Let be a Markov chain having a state space {0, 1, 2} and transition probability matrix is given by P = . Which of the following states are ergodic ?
Answer
D. None of the states
75
Let be a Markov chain having a state space {1, 2, 3, 4, 5} and transition probability matrix is given by Which of the following statements is/are correct ? i. 1 and 4 are in the same communicating class. ii. 2 and 3 are in the same communicating class. iii. 2 and 5 are in the same communicating class. iv. 3 and 4 are in the same communicating class.
Answer
A. Only (i and iii)
76
The interval between two successive occurrence of a Poisson process N(t), t ≥ 0 having parameter λ follows
Answer
B. Exponential with mean 1/λ
77
Suppose { } is a random walk on real line. A unit is moving from state i to i + 1 with probability ½ and also from state i to i-1 with probability ½. Then random walk is (i) Recurrent (ii) Null recurrent (iii) Aperiodic (iv) Ergodic
Answer
B. Only (i and ii)
78
In a birth and death process, if the birth rate is λ and the death rate is μ, and death rate is greater than birth rate, then the probability of ultimate extinction is
Answer
B. 1
79
Consider a queuing model M/M/1 with interarrival time λ and service time μ. Which of the following statements is/are correct ? (i) Steady state distribution of Queue length exist if λ < μ (ii) Steady state distribution of Queue length exist if λ > μ (iii) Steady state distribution is given by Geometric distribution
Answer
A. Only (i and iii)
80
Suppose N(t) is a renewal process generated by exponential random variables , , .. . , with parameter λ and H(t) is a renewal function. Which of the following statements is/are true ? (i) N(t) is a Poisson process (ii) Renewal function H(t) is (iii)
Answer
D. All of the above (i, ii and iii)
81
Consider the simple linear regression model y = + + , Which of the following is a wrong statement ?
Answer
Question Cancelled
82
Consider the simple linear regression model y = + X+ , Which of the following estimates are not unbiased ?
Answer
D. Maximum likelihood estimates of
83
Consider the simple linear regression model y=β₀+β₁X+ε where ε's are uncorrelated with mean zero and variance σ². What is the value of Cov(β̂₀,β̂₁)?
Answer
A. −X̄σ²/S_XX where S_XX=Σ(Xᵢ−X̄)²
84
Consider a multiple linear regression model Y=Xβ+ε with hat matrix H=(hᵢⱼ). The variance of ith residual is
Answer
C. σ²(1−hᵢᵢ)
85
Consider a multiple linear regression model Y=Xβ+ε, ε's are uncorrelated with mean zero and variance σ² where the regression matrix X is [[1,0],[2,−1],[1,2]] and β′=(β₀,β₁). Find variance of β̂₁
Answer
D. σ²/5
86
Which of the following is/are polynomial regression models ? (i) y = + X + (ii) (iii)
Answer
B. Only (i and ii)
87
Suppose collinearity is present in a regression model then which of the statements is/are correct ? (i) Least Square estimate of Regression coefficients have large variances (ii) Eigenvalues of matrix X′ X is large (iii) Estimated coefficients have large variance inflation factor
Answer
A. Only (i and iii)
88
In the log-linear regression model ln y=β₀+β₁x+ε, the multiplicative change in y associated with a unit increase in x is estimated by
Answer
B. e^(β̂₁)
89
Which of the following link function is suitable for Poisson Regression model ?
Answer
B. Log link function
90
Consider the multiple linear regression model Y = Xβ + ε, where V(ε) = σ²V. Which statement is correct?
Answer
D. Generalised least square estimate is minimum variance unbiased
91
Let f be a Reimann integrable function on a closed interval [a, b]. Then which of the following statements is true ?
Answer
C. is Reimann integrable over [a, b]
92
Suppose f is defined on [0,1] by f(t) = 1 for 0 ≤ t ≤ 1/3; f(t) = 2 for 1/3 < t < 2/3; and f(t) = 3 for 2/3 ≤ t ≤ 1. Let α = f and β(t) = t². Then which of the following is not true?
Answer
A. f is integrable with respect to α
93
Evaluate the Integral ∫₀¹ x²/√(1−x³) dx
Answer
Question Cancelled
94
The series 1 + 1/2ᵖ + 1/3ᵖ + ..... is divergent when
Answer
C. 0 < P ≤ 1
95
Consider f(x,y) = xy/(x² + y²) if (x,y) ≠ (0,0), and f(x,y) = 0 if (x,y) = (0,0). Which of the following statements is true?
Answer
B. Partial derivatives ∂f/∂x and ∂f/∂y exist at (0,0)
96
Find the determinant of A = [[1, 6, 1, 0, 0, 0], [5, 6, 4, 0, 0, 0], [2, 3, 4, 0, 0, 0], [3, 4, 3, 5, 0, 0], [2, 3, 2, 2, 1, 3], [1, 5, 1, 5, 4, 6]].
Answer
C. 1710
97
Suppose A is an idempotent matrix of order n. Which statements are true? (i) Aʳ = A for every positive integer r. (ii) A − I is idempotent. (iii) I − A is idempotent. (iv) Rank(A) = trace(A).
Answer
D. Only (i, iii and iv)
98
Suppose A = find the eigenvalues of
Answer
B. 1, 81, 256
99
Consider the matrix A = . What is the Geometric multiplicity and algebraic multiplicity of eigenvalue one ?
Answer
A. 1 and 2 respectively
100
Suppose A = , which of the following statements is/are true ? (i) A is positive definite matrix (ii) A is positive semi definite matrix (iii) A is non negative definite matrix (iv) Quadratic form corresponding to matrix A is
Answer
Question Cancelled
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