UPSC Statistics 2023 — Paper II

This multi-part question requires solving five distinct problems: (a) discuss SQC concepts with industrial applications, (b) derive reliability functions from given failure rate, (c) solve LP graphically, (d) construct and analyze game matrix, and (e) compute Markov chain probabilities. Allocate approximately 15-18 minutes per part, with extra attention to (c) and (e) where computational errors are common. Begin each part clearly labeled, show all derivation steps, and conclude with boxed final answers.

• Part (a): Define SQC as statistical methods for maintaining quality standards; explain need (mass production complexity, cost reduction, customer satisfaction) and utility (process control, acceptance sampling); classify variations into chance causes (random, inherent) and assignable causes (identifiable, correctable)
• Part (b): Derive R(t) = exp(-∫Z(t)dt) = (t+1)e^(-t); compute MTTF = ∫R(t)dt = 2; obtain conditional survival R(x|t) = R(t+x)/R(t) and MRL = ∫R(x|t)dx
• Part (c): Use constraint x₃ = x₁ + x₂ to reduce to 2-variable problem; minimize 10x₁ + 11x₂; identify feasible region vertices from intersection of x₁+x₂≥11, x₁-x₂≤5, 7x₁+12x₂≥35; optimal solution at (3,8) with value 118
• Part (d): General payoff matrix [aᵢⱼ] where i=1,...,m strategies for A, j=1,...,n for B; specific matrix with entries +7 for (H,H) and (T,T), -3 otherwise; apply Max Min: maximize minimum row payoff; saddle point exists if Max Min = Min Max
• Part (e): Construct transition matrix P from flea movement probabilities (typically given in diagram); compute P² and extract P[X₂=3|X₀=1] = (P²)₁₃ using Chapman-Kolmogorov equations

Explain the theoretical foundations of p-charts in part (a), then solve the numerical problems in (b) and (c) with systematic working. Allocate approximately 25-30% time to (a) as it requires conceptual elaboration, 30-35% to (b) for control chart construction and interpretation, and 35-40% to (c) as it carries the highest marks and involves multi-step replacement policy optimization. Present calculations in tabular format where possible and conclude with clear managerial recommendations.

• Part (a): Definition of control chart for fraction defective (p-chart), construction steps using sample proportion p̂ = d/n, and identification of Binomial distribution as the theoretical basis with Normal approximation for large samples
• Part (b): Calculation of center line (CL = p̄), control limits UCL/LCL = p̄ ± 3√[p̄(1-p̄)/n], plotting of 24 sample points, identification of Day 11 as out-of-control, and revised limits after removing assignable cause
• Part (c): Computation of expected failures Np, N·q·p₂, etc., individual replacement cost, group replacement cost for each policy period, and determination of optimal replacement interval at minimum average cost per week
• Correct handling of variable control limits when sample sizes differ (though here n=50 constant), and recognition that p-chart is appropriate for attribute data with varying sample sizes
• Economic interpretation: trade-off between individual replacement flexibility and group replacement economies of scale, with explicit cost comparison across weeks 1-5

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) Two-Phase method as it requires extensive tableau iterations, 35% to part (c) EOQ calculations with clear formula application, and 25% to part (b) covering both assignment problem and dual formulation. Begin with clear problem identification for each part, show complete step-by-step working with proper tableaus for (a), cost matrix reduction for (b)(i), systematic dual conversion rules for (b)(ii), and standard EOQ model derivation followed by substitution for (c). Conclude each part with boxed final answers and appropriate units.

• Part (a): Convert LPP to standard form by introducing surplus, slack and artificial variables; set up Phase I objective to minimize artificial variable; execute simplex iterations till feasibility; proceed to Phase II with original objective; identify optimal solution at x₁ = 2, x₂ = 0, Z = 6
• Part (b)(i): Apply Hungarian algorithm to 4×4 cost matrix — row reduction, column reduction, minimum lines to cover zeros, adjust matrix, make optimal assignments; state final assignment with minimum total cost
• Part (b)(ii): Convert primal to dual by transforming maximization to minimization, reversing inequality signs for ≥ constraints, handling equality with unrestricted dual variables, and noting primal unrestricted variables become dual equality constraints
• Part (c): Define EOQ as optimal order quantity minimizing total inventory cost; list assumptions: constant demand, instantaneous replenishment, no stockouts, fixed ordering cost, constant carrying cost percentage
• Calculate EOQ = √(2×12000×100)/(0.20×50) = 490 units (or √480000 ≈ 693 if using 20% as 0.20 directly); number of orders = 12000/490 ≈ 24.5; time between orders = 365/24.5 ≈ 14.9 days

Solve this multi-part numerical problem by allocating approximately 40% time to part (a) transportation problem as it requires complete solution methodology, 35% to part (b) probability calculations involving normal and log-normal distributions, and 25% to part (c) queuing theory steady-state probabilities. Begin with clear problem setup for each part, show all computational steps with proper formulae, and conclude with interpreted final answers in correct units.

• For (a): Correctly set up the balanced transportation problem (check if supply equals demand, add dummy if needed), apply Vogel's Approximation Method or Least Cost Method to obtain degenerate/non-degenerate basic feasible solution with (m+n-1) allocations
• For (b)(i): Set up P(120 < X < 200) = 0.80, convert to standard normal Z-scores, use symmetry property to find z₀ such that P(-z₀ < Z < z₀) = 0.80, hence Φ(z₀) = 0.90, interpolate from table to find z₀ ≈ 1.28, then solve σ = 40/1.28 = 31.25 hours
• For (b)(ii): Transform log-normal to normal: P(X ≤ 10) = P(Y ≤ ln10) = P(Y ≤ 2.3026), calculate Z = (2.3026-3)/0.2 = -3.487, use table to find Φ(-3.49) ≈ 0.0002 or precise interpolation
• For (c): Identify this as M/M/3 queuing system with state-dependent service rates (μ, 2μ, 3μ for n=1,2,3+), use λ=10/hr, μ=5/hr, ρ=λ/3μ=2/3, apply birth-death process balance equations for steady-state probabilities p₀, p₁, p₂, and general formula for pₙ when n≥3
• Verify all calculations: check transportation cost arithmetic, confirm normal table reading with interpolation, validate queuing traffic intensity ρ < 1 for steady state existence, and ensure probability sum equals 1

This question requires explaining three distinct statistical concepts across psychometrics, demography, and epidemiology. Allocate approximately 35% time to part (c) covering reliability, validity definitions, their distinction, and the Spearman-Brown prophecy formula application; 35% to part (d) setting up and solving the differential equation for population growth; and 30% to part (e) comparing MIR and MPR with practical Indian public health examples. Begin with clear conceptual definitions, proceed to mathematical derivations where required, and conclude with contextual interpretations.

• Part (c): Define reliability (consistency/stability of test scores) and validity (extent test measures what it claims to measure); distinguish reliability as necessary but not sufficient for validity; apply Spearman-Brown prophecy formula to calculate new validity coefficient ≈ 0.67
• Part (c): Correctly identify that validity coefficient increases proportionally to square root of reliability ratio: r_new = r_old × √(0.90/0.80) = 0.60 × 1.0607 ≈ 0.636 or 0.64
• Part (d): Set up differential equation dP/dt = P×r(t) = P(0.01 + 0.0001t²); integrate ln(P) = ∫(0.01 + 0.0001t²)dt = 0.01t + 0.0001t³/3 + C
• Part (d): Apply initial condition P(0) = 1,000,000 to find C = ln(10⁶); compute P(30) = 10⁶ × exp[0.01(30) + 0.0001(27000)/3] = 10⁶ × e^1.2 ≈ 3,320,117
• Part (e): Recommend MIR (incidence rate) for medicine allocation as it measures new cases over time, directly indicating current disease burden and transmission dynamics requiring immediate intervention
• Part (e): Cite MPR usefulness for chronic disease planning like diabetes or hypertension prevalence studies in India where total existing cases matter for long-term healthcare infrastructure and resource allocation

Explain the theoretical foundations first, then demonstrate computational application. Allocate ~30% time to part (a) on least squares and trend fitting, ~30% to part (b) on stationarity concepts and testing the three given models, and ~40% to part (c) on heteroscedasticity detection with complete Goldfeld-Quandt test execution. Structure as: theoretical exposition → mathematical derivation → numerical computation → statistical inference.

• Part (a): Principle of least squares (minimizing sum of squared residuals), its application in linear and non-linear trend fitting, and complete working for y=ab^tc^{t^2} using logarithmic transformation to linear form
• Part (b): Formal definition of weak/strong stationarity (constant mean, variance, autocovariance), Dickey-Fuller or graphical methods for testing, importance for valid inference, and classification of (i) random walk (non-stationary), (ii) random walk with drift (non-stationary), (iii) AR(1) process (stationary when |δ|<1)
• Part (c): Listing detection methods (graphical, Park test, Glejser test, White test, Goldfeld-Quandt test), complete Goldfeld-Quandt procedure with assumptions (normality, homoscedasticity under null, increasing/decreasing variance pattern)
• Correct computation of F-statistic = RSS2/RSS1 = 1536.8/377.17 = 4.075 with proper degrees of freedom (11,11)
• Proper hypothesis testing conclusion: F_calculated (4.075) > F_critical at 5% (2.82), reject null, heteroscedasticity present; also note significance at 1% level since 4.075 < 4.46 is false—actually 4.075 < 4.46, so not significant at 1%
• Recognition that RSS2 > RSS1 indicates increasing variance with income, confirming heteroscedasticity in consumption expenditure data

Begin with clear statement of assumptions for each derivation in 7(a), showing step-by-step integration for L_x formulas and summation manipulation for e_x^0. For 7(b), apply Z-score formula z = (X-μ)/σ systematically, then demonstrate linear transformation for standard scores with mean 50, SD 10. For 7(c), substitute given l_x = (100-x)/190 into survival probabilities, death probabilities, and life expectancy formulas, computing each numerical value with proper fraction handling. Allocate approximately 35% time to derivations in (a), 25% to standard score calculations in (b), and 40% to life table computations in (c) given its five sub-parts.

• 7(a)(i): Assume l_x linear in [x, x+1], integrate l(t)dt from x to x+1 to obtain (l_x + l_{x+1})/2
• 7(a)(ii): Assume l_x exponential (constant force of mortality), use l_t = l_x·e^{-μt} and integrate to derive harmonic mean form -d_x/log(p_x)
• 7(a)(iii): Express T_x = Σ L_{x+i} using linear assumption, substitute L_x = (l_x + l_{x+1})/2, rearrange to obtain e_x^0 = 1/2 + Σ i·d_{x+i}/l_x
• 7(b): Calculate Z-scores as (X-60)/12 for each raw score, then apply linear transformation 50 + 10·z for standard scores; for 1-10 scores, first find mean=5.5, SD=√8.25, then transform
• 7(c): Compute survival probabilities as l_60/l_5, l_80/l_30; death probability as 1-l_45/l_40; e_40^0 = T_40/l_40 with T_40 = Σ L_{40+i}; binomial probability for at least one survivor among three men

The question demands explanation across three distinct domains: agricultural data systems, fertility measures, and econometric identification. Allocate approximately 35% (15-18 marks) to part (a) covering data collection methods and official publications; 35% (15-18 marks) to part (b) distinguishing GFR/TFR with mathematical relationships and demographic interpretation; and 30% (12-15 marks) to part (c) applying order/rank conditions to the Keynesian model. Structure with clear sectional headings, begin each part with definitions, proceed to methodological details, and conclude with synthesis—ensuring the econometric section explicitly shows matrix calculations for identifiability.

• Part (a): Enumeration of agricultural data collection methods (census, sample surveys, administrative records) with specific Indian examples—Land Records, Agricultural Census, NSSO rounds; identification of official publications (Agricultural Statistics at a Glance, State Statistical Abstracts, FAO reports, DES publications)
• Part (a)(ii): Classification of statistics by agency—MoA&FW (crop area, yield, production), CSO (national income from agriculture), RBI (agricultural credit), NABARD (rural credit, WDRA data), State Directorates of Economics and Statistics
• Part (b): Precise distinction between GFR (age-specific births per 1000 women) and TFR (average births per woman completing reproductive span); interpretation of TFR=3.29 as replacement-level analysis; merits/demerits covering data requirements, period sensitivity, and cross-population comparability
• Part (b): Mathematical derivation TFR = 5 × GFR (assuming 5-year age groups) or TFR = Σ ASFR; GRR-TFR relationship via sex ratio at birth; GRR interpretation with population stability implications—GRR>1 (growing), GRR<1 (declining), GRR=1 (stationary, ignoring mortality)
• Part (c): Correct statement of order condition (K ≥ M-1, where K=excluded predetermined, M=endogenous in equation) and rank condition; systematic application to four-equation Keynesian model identifying endogenous (C,I,T,Y) and predetermined (G,Yt-1) variables; construction of coefficient matrix and explicit identifiability verdict for each equation