Statistics 2022 Paper I 50 marks Compulsory Solve

Q5

(a) Define general linear model with usual assumptions. If y₁ = β₁ + u₁, y₂ = –β₁ + β₂ + u₂, y₃ = –β₂ + u₃, where u₁, u₂, u₃ are mutually independent random variables with mean zero and variance σ², then find the least square estimators of β₁ and β₂. (10 marks) (b) Given X ~ N₃(μ, Σ), where μ = (2, 4, 3)' and Σ = ⎛8 2 3⎞ ⎜2 4 1⎟ ⎝3 1 3⎠ (i) find the regression function of X₁ on X₂ and X₃, and (ii) compute the conditional variance of X₁ given X₂ and X₃. (10 marks) (c) What is a uniformity trial ? Explain how it can be used to determine optimum shape and size. (10 marks) (d) In a 2⁶ – factorial experiment, the key block is given as : (1), ab, cd, ef, ace, abef, abcd, bce, cdef, acf, ade, abcdef, bde, bcf, adf, bdf. Identify the confounded effects. (10 marks) (e) If the coefficients of variation of x and y are equal and the correlation coefficient between x and y is ρ = 2/3, compute the efficiency of ratio estimator relative to the mean of a simple random sample. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) सामान्य रैखिक निदर्श को प्रचलित कल्पनाओं सहित परिभाषित कीजिए । यदि y₁ = β₁ + u₁, y₂ = –β₁ + β₂ + u₂, y₃ = –β₂ + u₃, जहाँ u₁, u₂, u₃ परस्पर स्वतंत्र यादृच्छिक चर हैं जिनका माध्य शून्य तथा प्रसरण σ² है, तो β₁ और β₂ के न्यूनतम वर्ग आकलकों को ज्ञात कीजिए । (10 अंक) (b) दिया गया है कि X ~ N₃(μ, Σ), जहाँ μ = (2, 4, 3)' और Σ = ⎛8 2 3⎞ ⎜2 4 1⎟ ⎝3 1 3⎠ (i) X₁ का X₂ और X₃ पर समाश्रयण फलन ज्ञात कीजिए, और (ii) X₂ और X₃ के दिए होने पर X₁ के सप्रतिबंध प्रसरण की गणना कीजिए । (10 अंक) (c) एकसमानता परीक्षण क्या है ? व्याख्या कीजिए कि इसका उपयोग इष्टतम आकृति और आकार ज्ञात करने के लिए कैसे किया जा सकता है । (10 अंक) (d) एक 2⁶ – बहु-उपादानी प्रयोग में, मुख्य खंडक इस प्रकार दिया गया है : (1), ab, cd, ef, ace, abef, abcd, bce, cdef, acf, ade, abcdef, bde, bcf, adf, bdf. संकीर्ण प्रभावों की पहचान कीजिए । (10 अंक) (e) यदि x और y के विचरण गुणांक समान हैं और x और y के बीच सहसंबंध गुणांक ρ = 2/3 है, तो अनुपात आकलक की दक्षता सरल यादृच्छिक प्रतिदर्श के माध्य के सापेक्ष परिकलित कीजिए । (10 अंक)

Evaluate your answer to this question → See all 2022 Statistics questions

Directive word: Solve

This question asks you to solve. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

This is a computational-cum-descriptive question requiring precise derivations and calculations across five sub-parts. Allocate approximately 20% time to part (a) for matrix formulation of GLM and LSE derivation, 20% to part (b) for multivariate normal conditional distributions, 15% to part (c) for explaining uniformity trials with agricultural field trial context, 25% to part (d) for systematic identification of confounded effects in 2⁶ factorial, and 20% to part (e) for ratio estimator efficiency computation. Begin each part with clear statement of method, show all computational steps, and conclude with boxed final answers.

Key points expected

  • Part (a): Correct matrix formulation of GLM y = Xβ + u with assumptions E(u)=0, Var(u)=σ²I; proper construction of design matrix X and derivation of LSE β̂ = (X'X)⁻¹X'y yielding β̂₁ = (y₁ - y₂)/2 and β̂₂ = (y₁ + y₂ + 2y₃)/2
  • Part (b)(i): Correct partitioning of Σ into Σ₁₁, Σ₁₂, Σ₂₁, Σ₂₂ and computation of regression coefficients β = Σ₁₂Σ₂₂⁻¹ for E(X₁|X₂,X₃) = μ₁ + Σ₁₂Σ₂₂⁻¹(x₂-μ₂, x₃-μ₃)'
  • Part (b)(ii): Computation of conditional variance Var(X₁|X₂,X₃) = Σ₁₁ - Σ₁₂Σ₂₂⁻¹Σ₂₁ using Schur complement
  • Part (c): Definition of uniformity trial as trial with uniform treatment to assess field variability; explanation of how coefficient of variation and soil heterogeneity index guide selection of plot shape (long narrow for fertility gradient) and size (balancing variance reduction vs cost)
  • Part (d): Systematic identification of confounded effects by finding generalized interaction of defining contrasts; recognition that key block corresponds to I = ABCDEF or equivalent 6-factor interaction confounding
  • Part (e): Application of ratio estimator efficiency formula RE = (1-ρ²)/(Cₓ²/Cᵧ² + 1 - 2ρCₓ/Cᵧ) with Cₓ = Cᵧ yielding simplified computation; final numerical answer for efficiency

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly writes matrix form y = Xβ + u for (a) with explicit X matrix; properly partitions Σ for (b); defines uniformity trial with agricultural context for (c); identifies 2⁶ factorial structure for (d); states ratio estimator assumptions for (e)Partially correct setup with minor matrix dimension errors or missing some assumptions; incomplete Σ partitioning; vague uniformity trial definition; recognizes factorial but misses confounding structure; states efficiency formula with errorsIncorrect model specification, wrong matrix dimensions, or missing crucial assumptions; fails to partition multivariate normal; no agricultural context; misunderstands factorial design; wrong efficiency formula
Method choice20%10Uses normal equations/LSE principle for (a); applies conditional distribution theory for multivariate normal in (b); employs CV and heterogeneity analysis for (c); uses generalized interaction method for (d); applies correct relative efficiency derivation for (e)Correct broad method but inefficient approach or missing optimization steps; some confusion between regression and conditional variance methods; basic uniformity explanation without optimization linkage; trial-and-error for confounding; correct formula but messy derivationWrong method entirely (e.g., MLE instead of LSE, simple linear regression instead of conditional MVN); no method for uniformity optimization; random guessing for confounded effects; incorrect efficiency comparison base
Computation accuracy20%10Accurate matrix inversion (X'X)⁻¹ for (a); correct Σ₂₂⁻¹ and Schur complement for (b); correct numerical values for regression coefficients and conditional variance; accurate identification of all confounded effects in (d); precise efficiency value for (e)Minor arithmetic errors in matrix operations; correct method but wrong final numbers in one part; partial identification of confounded effects; correct formula substitution but calculation error in final efficiencyMajor computational errors in matrix algebra; wrong inverse or determinant; completely wrong conditional variance; fails to identify confounding structure; nonsensical efficiency value
Interpretation20%10Interprets LSE as BLUE under Gauss-Markov for (a); explains regression surface geometry for (b); links uniformity trials to Indian agricultural experiments (e.g., IARI field trials) for (c); explains practical consequences of confounding for (d); interprets efficiency value for survey design decisions in (e)Basic interpretation without theoretical depth; mentions BLUE but doesn't explain; superficial uniformity application; notes confounding without explaining aliasing consequences; states efficiency >1 or <1 without practical meaningNo interpretation of results; fails to connect to statistical theory or practical applications; meaningless or wrong interpretation of efficiency; ignores consequences of confounding
Final answer & units20%10Clear boxed final answers: explicit β̂₁, β̂₂ formulas for (a); numerical regression function and conditional variance for (b); specific shape/size recommendations for (c); complete list of confounded effects for (d); exact efficiency ratio for (e) with statement of superiority/inferiorityFinal answers present but poorly formatted; one missing answer; incomplete list of confounded effects; efficiency without comparison statement; missing units where relevantMissing or wrong final answers; incomplete solutions; no conclusion on which estimator is better in (e); fails to identify any confounded effects

Practice this exact question

Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.

Evaluate my answer →

More from Statistics 2022 Paper I