UPSC Statistics 2021 — Paper II

The directive 'solve' requires demonstrating complete analytical solutions for all five numerical problems. Structure your answer by addressing each sub-question sequentially: (1) Single Sampling Plan with OC curve illustration, (2) Assignment problem using Hungarian method, (3) Game theory problem via algebraic method for mixed strategies, (4) Markov chain periodicity and persistence proof, and (5) Reliability function derivation from piecewise hazard rate. Each solution must show method, calculation, and final interpretation.

• Single Sampling Plan: Define n (sample size) and c (acceptance number) with concrete example; sketch OC curve showing P(A) vs p with points at p=0, p=AQL, p=LTPD, p=1
• Assignment Problem: Apply Hungarian method—row reduction, column reduction, minimum lines to cover zeros, optimality check, and final assignment with minimum cost
• Game Theory: Verify no saddle point exists, formulate as LPP or use algebraic method for 2×2 subgames, find mixed strategy probabilities and game value
• Markov Chain: Compute P² to show period d=2, verify irreducibility, calculate stationary distribution to confirm persistent non-null states
• Reliability: Derive R(t) = exp(-∫h(u)du) giving R(t)=exp(-0.015t) for t≤200 and R(t)=exp(-3)exp(-0.025(t-200)) for t>200 with continuity at t=200

Solve this three-part numerical problem by first calculating the warranty failure probability using exponential distribution properties, then constructing and interpreting a c-chart for defect data with proper control limits, and finally computing producer's and consumer's risks using binomial distribution for the given sampling plan. Present each part sequentially with clear headings, showing all formulas, substitutions, and final interpretations.

• Part 1: Calculate total warranty period as 1.8 × 365 = 657 hours and use P(T ≤ 657) = 1 - e^(-657/2000) for exponential failure probability
• Part 2: Compute c-bar = 33/20 = 1.65, then UCL = 1.65 + 3√1.65 ≈ 5.50 and LCL = 0, identifying Item 6 (8 defects) as out of control
• Part 3: Define producer's risk α = P(reject | p=0.01) and consumer's risk β = P(accept | p=0.05) using binomial or Poisson approximation
• Correct application of Poisson approximation with λ₁ = 1 for AQL and λ₂ = 5 for LTPD to find α = 1 - P(X≤2; λ=1) and β = P(X≤2; λ=5)
• Numerical values: α ≈ 0.080 or 8% and β ≈ 0.125 or 12.5% (or precise binomial equivalents)

Solve this three-part numerical problem by allocating approximately 30% time to part (a) for deriving the Pollaczek-Khinchine formula, 30% to part (b) for the MODI method transportation problem, and 40% to part (c) for the two-phase simplex method. Begin each part with clear problem setup, show all computational steps systematically, and conclude with verified final answers. For (a), explain M|G|1 characteristics before derivation; for (b), ensure initial basic feasible solution before MODI optimization; for (c), complete Phase I before proceeding to Phase II.

• Part (a): Correct explanation of M|G|1 queuing system components (Poisson arrivals, General service time, single server) and derivation of Pollaczek-Khinchine formula for mean queue length Lq = λ²E(S²)/[2(1-ρ)] or equivalent forms
• Part (b): Correct initial basic feasible solution using VAM or NWCR method, followed by MODI (UV method) iterations with proper stepping stone paths until optimality is reached with minimum transportation cost
• Part (c): Proper conversion of inequalities to equations using slack, surplus and artificial variables; successful completion of Phase I to eliminate artificial variables; Phase II optimization yielding optimal solution
• Verification of supply-demand balance (total supply = total demand = 200) in part (b) before solving, and correct handling of ≥ constraint in part (c) with surplus and artificial variables
• Clear presentation of all simplex tableaus for part (c) showing entering and leaving variables, pivot operations, and final optimal value of objective function

Begin with the directive 'solve' for part (a), applying the simplex method or graphical method to find the optimal solution, then analyze sensitivity of Cⱼ coefficients. For part (b), apply the EOQ formula with given parameters (D=600, C₀=₹80, Cₕ=₹0.60) and calculate percentage cost variations. For part (c), construct X̄ and R control charts using given constants d₂=2.326 and d₃=0.864, computing center lines and control limits to assess process stability. Allocate approximately 30% time to (a), 25% to (b), and 45% to (c) given their mark distribution and computational complexity.

• Part (a): Correct identification of feasible region vertices (0,0), (4,0), (4,3), (2,6), (0,6) and optimal solution at (2,6) with Z=36
• Part (a): Sensitivity analysis showing range for C₁ as [0, 7.5] and C₂ as [4, ∞) maintaining optimality of current basis
• Part (b): EOQ calculation as √(2×600×80/0.6) = 400 units; optimum period = 400/600 × 12 = 8 months
• Part (b): Total cost at EOQ = ₹240; cost at 480 units = ₹244 (1.67% increase); cost at 240 units = ₹264 (10% increase)
• Part (c): Grand mean X̄̄ = 44.2, average range R̄ = 5.8; X̄ chart limits: 44.2 ± 3×5.8/(2.326×√5) = 44.2 ± 3.34
• Part (c): R chart limits: UCL = 5.8×(1+3×0.864/2.326) = 12.27, LCL = 5.8×(1-3×0.864/2.326) = 0 (adjusted to 0)
• Part (c): Correct conclusion that Sample 3 (mean=37) and Sample 7 (mean=51) fall outside X̄ control limits, indicating process is not under statistical control

The directive 'explain' demands clear exposition with theoretical foundations and derivations. Allocate approximately 20% (10 marks) to each sub-part equally. For (a), present SURE model structure and FGLS estimation steps; for (b), enumerate NSSO functions with Indian statistical system context; for (c), derive GLS estimators with matrix algebra; for (d), clarify standardization rationale with formulae; for (e), complete life table calculations and interpret actuarial symbols. Structure: brief introduction, systematic part-wise treatment with equations, and concluding synthesis on statistical estimation methods.

• Part (a): SURE model specification with contemporaneous correlation across equations, disturbance covariance matrix structure, and FGLS two-step estimation (estimated Ω used for feasible estimator)
• Part (b): NSSO functions—conducting large-scale sample surveys (NSS, ASI), data dissemination, methodological research, coordination with state DES, and international reporting (SDGs)
• Part (c): GLS transformation with known heteroscedastic variances σᵢ², weighted least squares derivation, variance-covariance matrix of estimators, and efficiency comparison with OLS
• Part (d): Desirability of standard scores (comparability, norm-referenced interpretation), z-score formula, normalized scores (T-scores, stanines), and derivation steps
• Part (e): Life table completion—calculating nqx, npx, nLx, Tx, ex from given lx and ndx columns; explanation of q (mortality), p (survival), L (person-years), T (total), e (expectation)

The directive 'explain' demands clear exposition with logical flow and appropriate technical depth. Allocate approximately 30% time/words to part (a) on Box-Jenkins methodology, 30% to part (b) on index number calculations with explicit working, and 40% to part (c) covering price quotations and foreign trade publications. Structure: brief conceptual introduction for each part, detailed methodological explanation with formulae, worked calculations for (b), and specific Indian statistical system references for (c).

• Part (a): Identification of ARIMA model components (AR, I, MA), stationarity testing via ADF or KPSS, differencing procedures, ACF/PACF analysis for order identification, parameter estimation via maximum likelihood, diagnostic checking with Ljung-Box test, and forecasting with confidence intervals
• Part (b): Correct application of aggregative method (Laspeyres or Paasche formula with base year quantities as weights) and weighted relatives method (price relatives multiplied by base year expenditure weights), proper identification of missing data items from 'above data' reference, and meaningful economic interpretation of cost of living changes
• Part (c): Explanation of price quotation system (selection of representative items, specification of quality, choice of markets, timing of collection), role of NSSO and CSO in price data collection, and detailed coverage of DGCI&S publications (Monthly Statistics of Foreign Trade, Annual Report, commodity-wise/country-wise trade data, ITC-HS classification)
• Integration of Indian context: citing RBI's use of ARIMA for monetary policy forecasting, CPI-AL/RL/CPI-U/CPI-IW construction by Labour Bureau, and Ministry of Commerce's trade statistics dissemination
• Critical commentary in (b) on limitations of index numbers (substitution bias, quality changes, new goods) and in (c) on challenges in price quotation representativeness and trade data timeliness
• Correct mathematical notation: ∇^d for differencing, φ(B) and θ(B) operators, p/d/q orders, and index number formulae with proper subscripts

The directive 'derive' demands rigorous mathematical proofs and derivations. Allocate approximately 30% time to part (a) on female birthrate estimation using correlation structure, 30% to part (b) covering IQ definition, measurement procedures, aptitude-intelligence distinction, and reliability calculations using Spearman-Brown prophecy, and 40% to part (c) on force of mortality derivations and life table relationships. Structure with clear section headings, state assumptions explicitly, show step-by-step derivations, and conclude with precise final expressions.

• Part (a): Derivation of female birthrate formula using uniform distribution assumption, correlation structure between c(x,t) and f(x,t), and proper substitution of T_f(t) and G_f(t) with algebraic manipulation of (β-α) terms
• Part (b): Precise definition of IQ (Mental Age/Chronological Age × 100 or deviation IQ), Stanford-Binet and Wechsler procedures, distinction between aptitude (specific potential) and intelligence (general ability) tests
• Part (b): Application of Spearman-Brown prophecy formula n = r₂(1-r₁)/r₁(1-r₂) to find required test length for reliability 0.95, and calculation of new reliabilities for doubled/tripled lengths and 135 items
• Part (c): Definition of μₓ as instantaneous death rate and derivation of qₓ = (1/lₓ)∫₀¹ μₓ₊ₜ lₓ₊ₜ dt using relationship between force of mortality and survival function
• Part (c): Proof of μₓ = (1/eₓ⁰)[1 + (deₓ⁰/dx)] using complete expectation of life definition eₓ⁰ = Tₓ/lₓ and differentiation with respect to age
• Correct handling of Indian demographic context: mention of SRS (Sample Registration System) data for fertility estimation and applicability to Indian population studies

The directive 'explain' demands clear exposition with causal reasoning and illustrative examples. Allocate approximately 40% of time/words to part (a) given its 20 marks weight, covering autocorrelation definition, consequences, and both heteroscedasticity tests with step-wise procedures. Devote roughly 30% each to parts (b) and (c): for (b), construct the identification analysis using order and rank conditions for both restriction sets; for (c), explain Leslie matrix structure, fertility/survival parameters, and iterative projection mechanics with Indian demographic application. Structure as: definitional clarity → methodological exposition → worked application → concluding synthesis.

• Part (a): Precise definition of autocorrelation (correlation between error terms across observations) and distinction from heteroscedasticity; enumeration of consequences including inflated t-statistics, inefficient OLS estimates, and misleading R²
• Part (a): Goldfeld-Quandt test: correct procedure of ordering observations, splitting samples, computing F-ratio of residual variances, and interpretation against critical values
• Part (a): Glesjer test: auxiliary regression of absolute/squared residuals on explanatory variables, test statistic derivation, and comparison with Goldfeld-Quandt in terms of power and applicability
• Part (b): Application of order condition (K-k ≥ m-1) and rank condition for identification under restriction set (i) γ₁₂=0, γ₂₁=0, showing both equations are identified
• Part (b): Analysis under restriction set (ii) γ₁₁=0, γ₁₂=0, demonstrating identification failure for equation 1 due to rank deficiency
• Part (c): Leslie matrix structure: age-specific fertility rates (Fᵢ) in first row, survival probabilities (Pᵢ) on sub-diagonal, zeros elsewhere; matrix dimensions matching age classes
• Part (c): Population projection technique: iterative multiplication n(t+1) = L × n(t), stable population properties, intrinsic growth rate extraction, and application to Indian census projections