Indian Economic Service - Indian Statistical Service Examination

STATISTICS II

 

1. Let x₁, x₂, …, xₙ be a random sample from a distribution for which mean μ and variance σ² exist. Then the consistent estimator of μ is

(a) x₍₁₎ = min (x₁, x₂, …, xₙ)

(b) x₍ₙ₎ = max (x₁, x₂, …, xₙ)

(c) Σᵢ₌₁ⁿ xᵢ

(d) (x₍ₙ₎ + x₍₁₎) / 2

2. Let x₁, x₂, …, xₙ be a random sample from the p.d.f. f(x) = θ / x²; 0 < θ ≤ x < ∞. Which one of the following is a sufficient statistic for θ?

(a) Mean of (x₁, x₂, …, xₙ)

(b) Median of (x₁, x₂, …, xₙ)

(c) Minimum of (x₁, x₂, …, xₙ)

(d) Minimum of (x₁², x₂², …, xₙ²)

3. Let x₁, x₂, …, xₙ be a random sample from the distribution having p.d.f. f(x; θ, β) = β e^(−β(x − θ)); x ≥ θ, β > 0. To test H₀ : β = 2, θ = 3 against H₁ : β = 4, θ = 2 the best critical region is given by

(a) x̄ ≥ 1/2 (8 + log 2 − (log k)/n)

(b) x̄ ≥ {1 + (log 2)/2 − (log k)/n}

(c) x̄ ≤ {1 + log 2 − (log k)/(2n)}

(d) x̄ ≤ {1 + (log 2)/2 − (log k)/(2n)}

where k > 0

4. Let x₁, x₂, …, xₙ be a random sample from N(μ, σ²) distribution, the parameter μ is known. Which of the following statements is/are correct?

I. The Cramer-Rao lower bound to the variance of an unbiased estimator of σ is 2θ²/n, where θ = σ².

II. An unbiased estimator of σ is Σᵢ₌₁ⁿ |xᵢ − μ|.

Select the answer using the code given below.

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

UPSC Previous year question paper Indian Economic Service - Indian Statistical Service Examination, 2026 exam

5. Let x₁, x₂, …, xₙ be a random sample from the distribution having p.d.f. f(x, θ) = e^(−(x − θ)), x > θ. Further, (x₍₁₎ − (log 40)/n, x₍₁₎) is 100β% confidence interval for θ, where

x₍₁₎ = min {x₁, x₂, …, xₙ}

What is the value of β?

(a) 0.90

(b) 0.92

(c) 0.95

(d) 0.975

6. Let x₁, x₂, …, xₙ be a random sample from N(θ, σ²) distribution, σ² is known, θ is unknown. Which of the following statements is/are correct?

I. The MLE of e^(2θ) is e^(2x̄) with bias e^(2σ²/n) − 1.

II. An unbiased estimator of e^(2θ) is e^(2x̄) − 2σ²/n.

Select the answer using the code given below.

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

7. Let x₁, x₂, …, xₙ be a random sample from the following probability distribution :

P(X = 1) = 2(1 − θ)/(2 − θ), P(X = 2) = θ/(2 − θ); θ ∈ (0, 1)

The moment estimate of θ is

(a) x̄

(b) 2(1 − x̄)

(c) 2(1 − 1/x̄)

(d) 2/(2 − x̄)

8. Let X ∼ f(x, θ). The most powerful test of size 0·10 for testing H₀ : f(x, θ) = 8x⁷, 0 < x < 1 against H₁ : f(x, θ) = 4x³, 0 < x < 1 has power

(a) √0.10

(b) ⁴√0.10

(c) ⁶√0.10

(d) ⁸√0.10

9. Let x₁, x₂, …, xₙ be a random sample from a distribution with p.d.f.

The sufficient estimator of θ is

(a) Σᵢ₌₁ⁿ xᵢ²

(b) Σᵢ₌₁ⁿ xᵢ + 1

(c) Σᵢ₌₁ⁿ xᵢ² + 1

(d) Σᵢ₌₁ⁿ xᵢ

10. A random sample x₁, x₂, …, xₙ is taken from a normal population with mean 0 and variance σ², that is, X ∼ N(0, σ²). The minimum variance bound (MVB) estimator for σ² is

(a) Σᵢ₌₁ⁿ xᵢ² / n

(b) Σᵢ₌₁ⁿ xᵢ / n

(c) Σᵢ₌₁ⁿ xᵢ / (n + 1)

(d) Σᵢ₌₁ⁿ xᵢ / (n − 1)

11. If 95% confidence interval for μ with Z_(α/2) = 1.96 is (400, 407.84) and the sample size is large, the value of the standard error of mean X̄ will be

(a) 2

(b) 4

(c) 10

(d) 20

12. Let x₁, x₂, …, xₙ be a random sample from a distribution with p.d.f. or p.m.f. f(x, θ), θ ∈ Θ. A statistic T = (x₁, x₂, …, xₙ) is said to be sufficient for θ, iff the conditional distribution of X, given T = t, is

(a) dependent on θ

(b) independent of θ

(c) dependent on θ²

(d) None of the above

13. Let x₁, x₂, …, xₙ be a random sample from a distribution F_θ for which mean and variance exist. Let μ be the mean and σ² be the variance of the distribution. Then the unbiased estimator of σ² is

(a) s² = (1/n) Σᵢ₌₁ⁿ (xᵢ − x̄)²

(b) s² = (1/(n − 2)) Σᵢ₌₁ⁿ (xᵢ − x̄)²

(c) s² = (1/(n − 1)) Σᵢ₌₁ⁿ (xᵢ − x̄)²

(d) s² = (2/(n(n − 1))) Σᵢ₌₁ⁿ (xᵢ − x̄)²

14. Neyman-Pearson fundamental lemma provides the most powerful (MP) test of its size for testing

(a) simple null hypothesis against composite alternative

(b) composite null hypothesis against simple alternative

(c) simple null hypothesis against simple alternative

(d) composite null hypothesis against composite alternative

15. Let x₁, x₂, …, xₙ be a random sample from a normal (μ, θ) and μ is known. Which of the following statements is/are correct?

I. T = Σᵢ₌₁ⁿ (xᵢ − μ)² is sufficient statistic for θ.

II. S₀² = (1/n) Σᵢ₌₁ⁿ (xᵢ − μ)² is sufficient statistic for θ.

Select the answer using the code given below.

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

16. Let T₀ be an MVU estimator of γ(θ) and T₁ an unbiased estimator with efficiency e_θ < 1. Which of the following statements is/are correct?

I. The unbiased linear combination of T₀ and T₁, i.e., U = a₀T₀ + a₁T₁; a₁ ≠ 0, a₀ + a₁ = 1 is MVU.

II. The unbiased linear combination of T₀ and T₁, i.e., V = a₀T₀ − a₁T₁; a₁ ≠ 0, a₀ − a₁ = 1 is MVU.

Select the answer using the code given below.

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

17. Let x₁, x₂, …, xₙ be a random sample from N(μ, σ²), the parameter μ being known. If the unknown (θ = σ²) parameter is σ² with the parametric space Θ = {θ | 0 < θ < ∞}, then E_θ (∂² log f_θ(x) / ∂θ²) is equal to

(a) −1/(20²)

(b) 20²

(c) −n/(20²)

(d) n/(20²)

18. Let x₁, x₂, …, xₙ be a random sample from N(μ, σ²), the parameter μ being known. The unknown (θ = σ²) parameter is σ² with parametric space Θ = {θ | 0 < θ < ∞} and S₀² = (1/n) Σᵢ₌₁ⁿ (xᵢ − μ)². Which of the following statements is/are correct?

I. Var(S₀²) is more than the CR lower bound.

II. Var(S₀²) coincides with the CR lower bound.

III. S₀² is an MVB estimator of θ.

Select the answer using the code given below.

(a) I only

(b) III only

(c) II and III

(d) I and III

19. A batch of 19 students is examined in three subjects-Statistics (x₁), Mathematics (x₂) and Computer Programming (x₃). It has been given that r₁₃.₂ = 0.6. It is expected that x₂ will affect the correlation between x₁ and x₃ (i.e., the partial correlation between x₁ and x₃ after eliminating the effect of x₂ is zero). The value of test statistic for testing H₀ : ρ₁₃.₂ = 0 against H₁ : ρ₁₃.₂ ≠ 0 is

(a) 2.5

(b) 3

(c) 4

(d) 4.5

20. In order to test H₀ : θ = θ₀ against the alternative H₁ : θ ≠ θ₀, the likelihood ratio to be used as the test statistic is given by

(a) L(θ₀) / max (L(Θ))

(b) max (L(Θ)) / L(θ₀)

(c) min (L(θ₀) / L(Θ))

(d) min (L(Θ) / L(θ₀))

where Θ is the parametric space.

21. Let x₁, x₂, …, xₙ be a random sample from N(μ, σ²) distribution with μ known. Consider the following testing problems for N(μ, σ²) :

H₀ : σ² = σ₀² against H₁ : σ² < σ₀²

H₀ : σ² = σ₀² against H₁ : σ² > σ₀²

H₀ : σ² = σ₀² against H₁ : σ² ≠ σ₀²

The UMP level α test exists for

(a) I and II

(b) III

(c) I but not for II

(d) II but not for I

22. Let x₁, x₂, …, xₙ be a random sample from a Bernoulli population with parameter p, where 0 < p < 1. Which of the following is/are sufficient statistic(s) for p?

I. (1/n) Σᵢ₌₁ⁿ xᵢ

II. (1/n) Σᵢ₌₁ⁿ (xᵢ − x̄)²

III. Σᵢ₌₁ⁿ xᵢ

Select the correct answer using the code given below.

(a) I only

(b) III only

(c) I and II

(d) I and III

23. If T₁ is an MVUE of γ(θ), θ ∈ Θ and T₂ is any other unbiased estimator of γ(θ) with efficiency e < 1, then which of the following is/are correct?

I. T = (T₁ + T₂)/2 is MVUE of γ(θ)

II. Var(T) = Var(T₁/4)

III. T cannot be MVUE for γ(θ)

IV. T is unbiased for γ(θ)

Select the answer using the code given below.

(a) I

(b) IV only

(c) II and IV

(d) III and IV

24. If X ∼ Exp(λ), then based on a random sample of size n from this population, consider the following statements :

I. MLE and MME estimators of λ are equal to λ̂ = 1/x̄

II. MLE and MME estimators of λ are equal to λ̂ = x̄

III. MLE and MME of λ are unequal.

Which of the statements given above is correct?

(a) I

(b) II

(c) III

(d) Neither I nor II nor III

25. If T₁ and T₂ are two unbiased estimators of γ(θ) having the same variance σ² and ρ is the correlation between them, then which one of the following is correct?

(a) ρ = 2e − 1, where e is the efficiency of each estimator

(b) ρ ≥ 2e − 1, where e is the efficiency of each estimator

(c) ρ = 2e₁ − e₂, where e₁ is the efficiency of T₁ and e₂ is the efficiency of T₂

(d) ρ = e₁ + e₂ − 1, where e₁ is the efficiency of T₁ and e₂ is the efficiency of T₂

26. An estimator Tₙ is consistent for γ(θ) under which of the following?

I. Tₙ is unbiased for γ(θ)

II. V(Tₙ) ≤ 1

III. Tₙ →ᵖ γ(θ)

IV. Tₙ/n →ᵖ γ(θ)

Select the correct answer using the code given below.

(a) I and II

(b) II and III

(c) III only

(d) IV

27. Let X ∼ Bernoulli (P). Then the Fisher information based on a random sample of size one from this population is

(a) P(1 − P)

(b) 1/(P − P²)

(c) 1 − P

(d) 1/(1 − 2P + P²)

28. Let x₁, x₂, …, xₙ be a random sample from a distribution with p.d.f.

Then a sufficient statistic for θ is

(a) Σᵢ₌₁ⁿ xᵢ

(b) Σᵢ₌₁ⁿ xᵢ − n

(c) ∏ᵢ₌₁ⁿ xᵢ

(d) ∏ᵢ₌₁ⁿ (1/xᵢ) − n

29. The most powerful test in testing H₀ : θ = θ₀ against H₁ : θ = θ₁ maximizes

(a) variance

(b) power

(c) type I error

(d) type II error

30. Consider a random sample of size n from a Bernoulli (p) population. To test H₀ : p = 0.4 against H₁ : p = 0.6, the most powerful test depends on

(a) Σᵢ₌₁ⁿ xᵢ

(b) x₍₁₎

(c) Σᵢ₌₁ⁿ (xᵢ − x̄)²

(d) Σᵢ₌₁ⁿ xᵢ²

31. If the SPRT of strength (α, β) and boundary points (A, B) terminates with probability 1, then which of the following is/are correct?

I. A ≤ (1 − β)/α

II. B ≥ β/(1 − α)

Select the answer using the code given below.

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

32. Under regularity conditions, the maximum likelihood estimators are asymptotically

I. unbiased

II. normal

III. efficient

Select the correct answer using the code given below.

(a) I and II only

(b) II and III only

(c) I and III only

(d) I, II and III

33. Let T = T(x₁, x₂, …, xₙ) be an estimator of γ(θ). Consider the following statements:

I. E_θ[T − γ(θ)]² = E_θ[T − E_θ(T)]² + [E_θ(T) − γ(θ)]²

II. The bias of T is E_θ(T) − γ(θ).

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

34. Let x₁, x₂, …, xₙ be a random sample from a population with probability density function f(x; θ) = θ x^(θ − 1); 0 < x < 1, θ > 0. Which of the following statements are correct?

I. ∏ᵢ₌₁ⁿ xᵢ is a sufficient estimator of θ.

II. Σᵢ₌₁ⁿ xᵢ is a sufficient estimator of θ.

III. n / Σᵢ₌₁ⁿ log xᵢ is a maximum likelihood estimator of θ.

Select the answer using the code given below.

(a) I and II only

(b) I and III only

(c) II and III only

(d) I, II and III

35. Let V = V(x₁, x₂, …, xₙ) be an unbiased estimator of γ(θ) and T be a sufficient statistic for γ(θ). Define Φ(t) = E[V|T = t]. Which of the following statements is/are correct?

I. Φ(t) is an unbiased estimator of γ(θ).

II. The variance of Φ(t) is less than or equal to variance of V.

III. Φ(t) is a function of sufficient statistic which is dependent on θ.

Select the answer using the code given below.

(a) I only

(b) II and III only

(c) I and II only

(d) I, II and III

36. Let x₁, x₂, …, xₙ be a random sample from N(μ, σ²); −∞ < μ < ∞, σ² > 0, where both μ and σ² are unknown. Which of the following statements are correct?

I. μ̂ = x̄ and σ̂² = (1/n) Σᵢ₌₁ⁿ (xᵢ − x̄)² maximize the likelihood function L(μ, σ²; x).

II. μ̂ = x̄ and S² = (1/(n − 1)) Σᵢ₌₁ⁿ (xᵢ − x̄)² maximize the likelihood function L(μ, σ²; x).

III. σ̂² = (1/n) Σᵢ₌₁ⁿ (xᵢ − x̄)² is an unbiased estimator of σ².

IV. S² = (1/(n − 1)) Σᵢ₌₁ⁿ (xᵢ − x̄)² is an unbiased estimator of σ².

Select the answer using the code given below.

(a) II and IV

(b) II and III

(c) I and III

(d) I and IV

37. Let T₁ and T₂ be two unbiased estimators for a parameter θ such that E_θ(Tᵢ²) < ∞; i = 1, 2. If the efficiency of T₁ relative to T₂ is e_θ(T₁/T₂) = Var_θ(T₂)/Var_θ(T₁), then which one of the following is correct?

(a) e_θ(T₁/T₂) > 1 implies T₁ is more efficient than T₂

(b) e_θ(T₁/T₂) < 1 implies T₁ is more efficient than T₂

(c) e_θ(T₂/T₁) > 1 implies T₁ is more efficient than T₂

(d) e_θ(T₂/T₁) < 1 implies T₂ is more efficient than T₁

38. Let x₁, x₂, …, xₙ be a random sample from

Then the MLEs of α and β are respectively given by

(a) α̂ = x̄ and β̂ = x̄ − 1

(b) α̂ = 1/x̄ and β̂ = 2/x̄

(c) α̂ = x₍₁₎ and β̂ = x₍ₙ₎

(d) α̂ = x₁ and β̂ = xₙ

39. Consider the following :

I. Bootstrap

II. Jackknife

III. Cross-validation

IV. Stratified sampling

How many of the above are resampling methods?

(a) One

(b) Two

(c) Three

(d) All the four

40. The critical region W and consequently a test for testing H₀ : θ = θ₀ against H₁ : θ = θ₁ is unbiased if

(a) P(x ∈ W|H₁) ≥ P(x ∈ W|H₀)

(b) P(x ∈ W|H₀) ≥ P(x ∈ W|H₁)

(c) P(x ∈ W|H₀) = P(x ∈ W|H₁)

(d) P(x ∈ W|H₁) > P(x ∈ W|H₀)

41. Let x₁, x₂, …, xₙ be independent random variables each having the distribution P(xᵢ = k) = 1/N; k = 1, 2, …, N; N ∈ Z, the set of positive integers. If x₍ₙ₎ = max (xᵢ) and x₍₁₎ = min (xᵢ) then the sufficient statistic for the parameter N is

(a) x₍₁₎

(b) x₍ₙ₎

(c) x₍₁₎/2

(d) None of the above

42. Let x₁, x₂, …, xₙ constitute a random sample of size n from the probability density function f(x; θ) = θ e^(−θx), 0 < x < ∞, θ > 0. Consider the following statements :

I. The estimator of θ with method of moments is x̄

II. Both maximum likelihood estimator and moment estimator of θ are same.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

43. Let x₁, x₂, …, xₙ be independent and identically distributed b(1, p) i.e., Bernoulli random variables with parameter p, 0 < p < 1. Define T = Σᵢ₌₁ⁿ xᵢ. The UMVUE for Ψ(p) = p(1 − p) is given by

(a) T(1 − T)/(n(n − 1))

(b) T(T − 1)/(n(n − 1))

(c) T/(n(n − 1))

(d) T(n − 1 − T)

44. Consider the following statements :

I. If a sufficient statistic t for θ exists, then the maximum likelihood estimator will be a function of the sufficient statistic t.

II. If θ̂ is the maximum likelihood estimator of θ, then θ̂/(1 + θ̂) is the maximum likelihood estimator of θ/(1 − θ).

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

45. Let x₁, x₂, …, xₙ be a random sample from a distribution with probability density function f(x; θ) = (1/π) · 1/(1 + (x − θ)²), −∞ < x < ∞, θ is real. Then the minimum variance bound estimator of θ

(a) sin (1/n) Σᵢ₌₁ⁿ xᵢ

(b) sin Σᵢ₌₁ⁿ xᵢ

(c) sin (1/n) Σᵢ₌₁ⁿ xᵢ²

(d) does not exist

46. Let x₁, x₂, …, xₙ be a random sample from a distribution with probability density function

f(x; α) = (1/Γ(α)) x^(α − 1) e^(−x); 0 ≤ x < ∞, α > 0

Then the estimator of α obtained by method of moments is

(a) x̄/2

(b) x̄

(c) 1/x̄

(d) 2x̄

47. In an SPRT with n, α and β, which one of the following is correct?

(a) n → fixed constant, α → fixed constant, β → fixed constant

(b) n → random variable, α → fixed constant, β → fixed constant

(c) n → fixed constant, α → random variable, β → random variable

(d) n → random variable, α → random variable, β → fixed constant

where the symbols have their usual meanings.

48. Let f(x; θ) = (1/√(2π)) e^(−((x − θ)²)/2); −∞ < x < ∞, −∞ < θ < ∞. The sequential probability ratio test is used for testing H₀ : θ = θ₀ against H₁ : θ = θ₁ (θ > θ₀). Then we reject H₀ if

(a) Σᵢ₌₁ᵐ xᵢ ≤ (1/(θ₁ − θ₀)) log(β/(1 − α)) + m(θ₀ + θ₁)/2

(b) Σᵢ₌₁ᵐ xᵢ ≥ (1/(θ₁ − θ₀)) log((1 − β)/α) + m(θ₀ + θ₁)/2

(c) Σᵢ₌₁ᵐ xᵢ = (1/(θ₁ − θ₀)) log(β/(1 − α)) − m(θ₀ + θ₁)/2

(d) Σᵢ₌₁ᵐ xᵢ ≥ (1/(θ₁ − θ₀)) log((1 − β)/α) − m(θ₀ + θ₁)/2

49. Consider the following statements :

I. Neyman-Pearson lemma gives a general method for testing simple null hypothesis against composite alternative hypothesis.

II. Likelihood ratio test gives a general method for testing simple or composite null hypothesis against simple or composite alternative hypothesis.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

50. For testing H₀ : μ = μ₀ against H₁ : μ ≠ μ₀, the 90% confidence interval for μ of normal distribution, when σ² is unknown, is (360, 400). Then the value of sample mean is given by

(a) 380

(b) 390

(c) 400

(d) 420

51. Let A be the matrix of 6 columns with rank 3 and y be a non-null vector. The number of linearly independent solutions of the consistent equation Ax = y is

(a) 3

(b) 4

(c) 5

(d) 6

52. Consider the following statements under the linear model Y = Xβ + ε and G be a generalized inverse of X′X :

I. X G X′ is invariant to G

II. X G X′ is symmetric only when G is symmetric.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

53. Consider the following statements :

I. The rank of a generalized inverse of a matrix A does necessarily have the same rank as A.

II. The rank of a generalized inverse of a matrix A is the same as the rank of A, if and only if it is a reflexive inverse.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

54. Let Y₁, Y₂, Y₃ and Y₄ be uncorrelated observations with common unknown variance σ². Let

E(Y₁) = β₁ + β₂ + β₃ = E(Y₂)

E(Y₃) = β₁ − β₂ = E(Y₄)

where β₁, β₂ and β₃ are unknown parameters. Define e₁ = (1/√2)(Y₁ − Y₂) and e₂ = (1/√2)(Y₃ − Y₄). An unbiased estimate of σ² is

(a) (1/2)(e₁² − e₂²)

(b) (1/2)(e₁² + e₂²)

(c) (1/4)(e₁² + e₂²)

(d) e₁² + e₂²

55. Consider Gauss-Markov linear model as E(Y₁) = β₁ + β₂, E(Y₂) = β₁ − αβ₂, E(Y₃) = 2β₁ − β₂. For what value of α, BLUEs of β₁ and β₂ (i.e., β̂₁ and β̂₂) are uncorrelated?

(a) −2

(b) −1

(c) 1

(d) 2

56. Let Y₁, Y₂ and Y₃ be uncorrelated observations with common variance and expectations given by E(Y₁) = β₁, E(Y₂) = β₂ and E(Y₃) = β₁ + β₂, where β₁ and β₂ are unknown parameters. The BLUE of (β₁ + β₂) is

(a) Y₃

(b) Y₁ + Y₂

(c) (Y₁ + Y₂ + 2Y₃)/3

(d) (Y₁ + Y₂ + Y₃)/3

57. Model-I: Y = β₀ + β₁X and Model-II: Y = β₀ + β₁X + β₂X² are fitted for the data set (Xᵢ, Yᵢ); i = 1, 2, …, n.

Denote β̂₀ and β̂₁ as least square estimates of β₀, β₁ from Model-I and β₀*, β₁*, β₂* be the least square estimates from Model-II.

Define P = Σᵢ₌₁ⁿ (Yᵢ − β̂₀ − β̂₁Xᵢ)² and Q = Σᵢ₌₁ⁿ (Yᵢ − β₀* − β₁*Xᵢ − β₂*Xᵢ²)².

Consider the following statements:

I. P ≥ Q

II. It can happen that Q = 0 but P > 0.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

58. Using the ordinary least squares method, Student-I fits the model Ŷᵢ = α̂₀ + α̂₁Xᵢ (that is Y on X) and Student-II fits the model X̂ᵢ = β̂₀ + β̂₁Yᵢ (that is X on Y) for the given data set (Yᵢ, Xᵢ); i = 1, 2, …, n. Which one of the following pairs is a possible value of (α̂₁, β̂₁)?

(a) (−0.3, 0.6)

(b) (0.7, 3.5)

(c) (0.5, −0.5)

(d) (−1.5, −0.3)

59. Let λ′β be an estimable function of β in the model Y = Xβ + ε, where E(Y) = Xβ and X is an n × p matrix of rank k < p ≤ n. Let β̂ be any solution to the normal equations X′Xβ̂ = X′Y and let r be any solution to X′Xr = λ. Consider the following statements for the two estimators λ′β̂ and r′X′Y :

I. λ′β̂ is equal to r′X′Y for any β̂ or r.

II. λ′β̂ and r′X′Y are invariant to the choice of any β̂ or r.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

60. In the Gauss-Markov linear model Y_(n×1) = X_(n×k)β_(k×1) + ε_(n×1), E(ε) = 0 with rank r(X) = r < k, the set

{l ∈ ℝⁿ : E(l′Y) = 0}

is

(a) not a subspace

(b) a subspace of ℝⁿ with dimension k − r

(c) a subspace of ℝⁿ with dimension n − k

(d) a subspace of ℝⁿ with dimension n − r

61. Which of the following divisions are parts of the National Statistical Office of the Ministry of Statistics and Programme Implementation?

I. National Accounts Division (NAD)

II. Field Operations Division (FOD)

III. Infrastructure and Project Monitoring Division (IPMD)

IV. Data Informatics and Innovation Division (DIIID)

Select the correct answer using the code given below.

(a) I, II and III

(b) I, II and IV

(c) I, III and IV

(d) II, III and IV

62. Which one of the following is not an objective of the International Comparison Programme (ICP)?

(a) To produce purchasing power parities (PPPs)

(b) To produce comparable price level indexes (PLIs) of economies

(c) To monitor exchange rate of different currencies

(d) To convert measures of Gross Domestic Product (GDP) of different economies into a common currency

63. Consider the following statements :

I. The Ministry of Statistics and Programme Implementation is releasing the Consumer Price Index (CPI) with revised Base 2024 = 100.

II. The Ministry of Statistics and Programme Implementation is releasing the National Accounts Estimates with revised Base period 2023-2024.

Which of the statements given above is/are correct?

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

64. Which of the following are the new initiatives taken for the Census 2027?

I. Data will be collected using mobile applications.

II. Option to self-enumerate will be provided to the public.

III. Caste enumeration will be included.

Select the correct answer using the code given below.

(a) I and II only

(b) II and III only

(c) I and III only

(d) I, II and III

65. Which of the following are correct regarding 'Swachh Survekshan'?

I. It is the world's largest urban cleanliness survey.

II. It is conducted every five years.

III. It is conducted by the Ministry of Statistics and Programme Implementation.

IV. It is conducted by the Ministry of Housing and Urban Affairs.

Select the answer using the code given below.

(a) I and IV only

(b) II and III only

(c) I, II and III

(d) I, II and IV

66. Which of the following statements are correct regarding 'Agriculture Census'?

I. The latest Agriculture Census has 2021-2022 as reference period.

II. It is conducted every five years.

III. It is conducted by the Ministry of Statistics and Programme Implementation.

IV. It is conducted by the Ministry of Agriculture and Farmers' Welfare.

Select the answer using the code given below.

(a) I and IV only

(b) II and III only

(c) I, II and III

(d) I, II and IV

67. Which of the following components has the highest share in the expenditure component of Gross Domestic Product (revised base)?

(a) Gross Fixed Capital Formation (GFCF)

(b) Government Final Consumption Expenditure (GFCE)

(c) Private Final Consumption Expenditure (PFCE)

(d) Exports (X)

68. Why are official statistics considered 'public good'?

I. They are financed from general tax revenue.

II. Their use by one person does not affect the use by others.

III. They are used by public officials but private persons are denied access to them.

Select the correct answer using the code given below.

(a) I and II only

(b) II and III only

(c) I and III only

(d) I, II and III

69. Consider the following pairs :

Index — Released by

I. CPI-IW — Labour Bureau, Ministry of Labour and Employment

II. CPI-AL/RL — National Sample Survey Office, Ministry of Statistics and Programme Implementation

III. CPI-U, R, Combined — National Statistical Office, Ministry of Statistics and Programme Implementation

IV. WPI — Office of the Economic Advisor, Ministry of Commerce and Industry

How many of the pairs given above are correctly matched?

(a) One

(b) Two

(c) Three

(d) All the four

70. Consider the following statements regarding the National Statistical Commission (NSC):

I. NSC is statutory body created through an Act passed by the Parliament in 2003.

II. NSC has a part-time Chairperson, part-time Members and an ex officio Member.

III. The Chief Statistician of India (CSI) and Secretary to the Government of India in the Ministry of Statistics and Programme Implementation is the Secretary to NSC.

Which of the statements given above is/are correct?

(a) I and II

(b) II and III

(c) III only

(d) II only

71. As per the constitutional provisions in India, the subject matter of collection of statistics comes under the

I. Union List

II. State List

III. Concurrent List

Which of the above is/are correct?

(a) I only

(b) II only

(c) I and III only

(d) I, II and III

72. The Sustainable Development Goals (SDGs) framework has been developed by which one of the following agencies?

(a) The United Nations to track various socio-economic indicators for gender awareness

(b) The United Nations to measure progress across nations in respect of designated goals, targets and indicators by 2030

(c) The World Health Organization to carry out surveys and census for improving quality of life

(d) The World Health Organization to measure progress across nations in respect of designated goals, targets and indicators by 2030

73. Which of the following are included in the secondary sector in the estimation of GDP?

I. Manufacturing

II. Construction

III. Forestry and Fishing

IV. Mining and Quarrying

Select the correct answer using the code given below.

(a) I and III

(b) I and II

(c) I and IV

(d) II and III

74. Which of the following is/are correctly defined?

I. Neonatal Mortality Rate (NMR) is defined as

NMR = (Number of infant deaths of less than 29 days during the year) / (Number of live births during the year) × 1000

II. Post-Neonatal Mortality Rate (PNMR) is defined as

PNMR = (Number of infant deaths of less than one year during the year) / (Number of live births during the year) × 1000

Select the answer using the code given below.

(a) I only

(b) II only

(c) Both I and II

(d) Neither I nor II

75. Which of the following are correct regarding the Population Census 2027?

I. It will be the 15th Census in the country.

II. It will be the 8th after the Independence.

III. The reference date will be 1st March, 2027.

IV. The reference date will be 1st February, 2027.

Select the answer using the code given below.

(a) I and II

(b) I and III

(c) II and III

(d) II and IV

76. In general, inflation is calculated by which one of the following?

(a) Producer Price Index

(b) Consumer Price Index

(c) Wholesale Price Index

(d) A Weighted Aggregative Price Index

77. Which one of the following divisions under the Ministry of Statistics and Programme Implementation deals with eSAKSHI portal?

(a) Administrative Statistics and Policy Division (ASPD)

(b) Household Survey Division (HSD)

(c) Data Informatics and Innovation Division (DIIID)

(d) Members of Parliament Local Area Development Scheme (MPLADS) Division

78. Consider the following data :

Commodity — Base period price — Current period price — Percentage change

P — 20 — 40 — 100

Q — 50 — 60 — 20

R — 40 — 50 — 25

S — 20 — 30 — 50

What will be the simple aggregative price index for the above data?

(a) 138·50

(b) 140·00

(c) 141·25

(d) 145·75

79. It is given that Σp₁q₀ = 2070, Σp₀q₀ = 1660, Σp₀q₁ = 1470 and Σp₁q₁ = 1790. The Fisher's Price Index is close to

(a) 115

(b) 119

(c) 123

(d) 127

80. In which one of the following price index numbers, weights are the arithmetic mean of the base year quantities and current year quantities?

(a) Walsch Price Index

(b) Dorbish-Bowley Price Index

(c) Fisher's Ideal Index

(d) Marshall-Edgeworth Price Index

Indian Economic Service - Indian Statistical Service Examination STATISTICS II

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