UPSC Statistics 2024 — Paper II

(a) calculate: given > formula > substitution > result with units > interpretation | (b) describe: define > structure or process in order > labelled diagram > significance | (c(i)) explain: definition/context > points in order > small example > short close | (c(ii)) explain: definition/context > points in order > small example > short close | (d) derive: given > assumptions > stepwise derivation > result > check | (e) explain: definition/context > points in order > small example > short close Full marks: All parts show correct method, clear notation, and contextual interpretation.

• States the parallel-system rule: the system fails only if all units fail
• Unit unreliability 1−0.90=0.10
• Uses independence: P(all fail)=(0.10)³=0.001
• R_system=1−(1−R)³=0.999
• Defines Sᵢ=Σ(x̄ⱼ−μ₀) plotted against sample number
• V-mask decision procedure with lead distance and angle
• Tabular CUSUM C⁺ᵢ, C⁻ᵢ with reference value k and decision interval h
• Applications: detecting small sustained shifts in industrial, pharmaceutical or continuous processes
• Probability α that a lot of acceptable quality (at AQL) is rejected
• Equivalent to a Type I error in the acceptance-sampling test
• Read from the OC curve as 1−Pₐ at p=AQL
• Commonly fixed at about 5%
• Context of rectifying inspection: rejected lots are 100% screened
• AOQ=p·Pₐ·(N−n)/N (≈p·Pₐ)
• AOQ rises then falls as p increases
• AOQL is the maximum AOQ over all incoming quality levels: the worst long-run average outgoing quality
• Starting basis is dual feasible (all cⱼ−zⱼ ≤ 0 for max) but primal infeasible (some bᵢ < 0)
• Leaving variable: basic variable with the most negative value
• Entering variable: minimum |(cⱼ−zⱼ)/a_rj| over a_rj < 0; none negative means infeasible
• Pivot and repeat until all basic variables are non-negative, giving the optimum
• Defines simulation using random numbers drawn from specified probability distributions
• Describes the steps: model, generate random inputs, evaluate, aggregate
• Uses: estimating integrals, probabilities and risk when analytic solutions are intractable
• Applications across queuing, inventory, reliability, finance, PERT or physics

Framework: Statistics, Paper 2. (a) describe: define > structure or process in order > labelled diagram > significance | (b) calculate: given > formula > substitution > result with units > interpretation | (c) calculate: given > formula > substitution > result with units > interpretation Full marks: Rigorous derivation, correct notation, and clear interpretation of statistical concepts.

• 3σ logic: X̄ ~ N(μ, σ²/n) with σ estimated by R̄/d₂
• X̄-chart: CL=X̿, UCL/LCL=X̿ ± A₂R̄ with A₂=3/(d₂√n)
• R-chart: CL=R̄, UCL=D₄R̄, LCL=D₃R̄ with D₄=1+3d₃/d₂, D₃=max(0,1−3d₃/d₂)
• Joint study: X̄ tracks location, R tracks dispersion; R-chart checked first because X̄ limits depend on R̄
• Density f(t)=(β/θ)(t/θ)^(β−1)exp[−(t/θ)^β]
• R(t)=exp[−(t/θ)^β]
• h(t)=f(t)/R(t)=βt^(β−1)/θ^β
• Interpretation: β<1 decreasing (infant mortality), β=1 constant (exponential), β>1 increasing (wear-out)
• Standard form with surplus and artificial variables (Big-M or two-phase)
• Phase I or Big-M iterations reaching a feasible basis
• Phase II iterations to the basis (x₁, s₁, s₂)=(4, 18, 4) with z=20
• s₃ has positive cⱼ−zⱼ but no positive column entry, so the solution is unbounded

(a) explain: definition/context > points in order > small example > short close | (b) define: precise definition > the distinguishing feature > one example | (c(i)) calculate: given > formula > substitution > result with units > interpretation | (c(ii)) comment: context > arguments both sides > judgment > close | (c(iii)) calculate: given > formula > substitution > result with units > interpretation Full marks: All parts fully answered with correct derivations, clear definitions, and proper interpretation.

• Feasible region is a convex polyhedron; an extreme point is not a strict convex combination of two other feasible points and corresponds to a basic feasible solution
• Fundamental theorem: if an optimum exists, it is attained at an extreme point, which justifies the simplex search
• Duality: weak duality inequality and strong duality (equal optimal values; one finite optimum implies the other)
• Complementary slackness: slackᵢ × yᵢ = 0 and surplusⱼ × xⱼ = 0 at optimality, with shadow-price interpretation
• TPM: non-negative entries, rows summing to 1
• Regular: some Pᵐ has all entries positive; ergodic: irreducible, aperiodic (positive recurrent)
• P has all positive entries, so it is regular and ergodic
• Solves πP=π: π=(5/9, 4/9), so lim Pⁿ has both rows (5/9, 4/9)
• Uses h(t)=f(t)/R(t)=−R′(t)/R(t)
• f(t)=(2/t₀)(1−t/t₀)
• h(t)=2/(t₀−t)
• States the range 0 ≤ t < t₀
• h′(t)=2/(t₀−t)² > 0
• Failure rate increases with time
• IFR distribution
• Interpretation: wear-out/ageing of the cutting assembly
• MTTF=∫₀^{t₀} R(t) dt
• Substitution u=1−t/t₀
• MTTF=t₀∫₀¹u² du
• MTTF=t₀/3

Framework: Operations Research (Queuing Theory & Inventory Management). (a) calculate: given > formula > substitution > result with units > interpretation | (b) calculate: given > formula > substitution > result with units > interpretation | (c(i)) explain: definition/context > points in order > small example > short close | (c(ii)) describe: define > structure or process in order > labelled diagram > significance Full marks: Correct formulas, accurate calculations, clear interpretation, and proper context for all parts.

• Identifies M/M/2 with ρ=λ/(2μ)=5/6 < 1 and P₀=1/11
• P₁=5/33, P₂=25/198, P₃=125/1188, so P(n>3)=625/1188≈0.526
• P(given teller idle)=P₀+P₁/2=1/6
• P(W>0.05 h)=(25/22)e⁻¹−(3/22)e⁻³≈0.411 using the waiting-time distribution
• Recognises finite production rate: Q*=√[2DS/(H(1−d/p))]
• Q*=√(1.2×10⁸)≈10,954 units
• Run length Q*/p≈36.5 days and runs per year D/Q*≈3.42
• Maximum inventory Q*(1−d/p)≈5,477 units
• Chance causes: many small, random, inherent sources; process stable and in statistical control
• Assignable causes: identifiable, non-random special causes (tool wear, wrong setting, bad material, operator error) that put the process out of control
• Control charts separate the two and trigger corrective action
• Advantages: early detection, reduced scrap/rework and inspection cost, rational acceptance decisions, standards and continuous improvement
• Plan (N, n, c): accept if defectives d ≤ c
• Pₐ(p) via hypergeometric, binomial or Poisson as appropriate
• Compute Pₐ for a range of p and plot Pₐ against p
• Read producer's risk at AQL and consumer's risk at LTPD from the curve

Framework: UPSC Statistics Paper 2. (a) discuss: intro > 3-4 dimensions > example > balanced close | (b) calculate: given > formula > substitution > result with units > interpretation | (c) explain: definition/context > points in order > small example > short close | (d) calculate: given > formula > substitution > result with units > interpretation | (e) define: precise definition > the distinguishing feature > one example Full marks: Precise definitions, correct formulas, clear derivations, and accurate calculations with proper interpretation.

• Decentralised system coordinated by MoSPI, with central and state agencies (DES in states)
• Names key bodies: NSO, NSC (2005, Rangarajan Commission), RGI/Census, RBI, ISI
• NSO formed in 2019 by merging CSO and NSSO; its divisions for national accounts, economic and social statistics, survey design, field operations and data processing
• NSO functions: large-scale surveys (PLFS, HCES), national accounts, CPI/IIP, standards and dissemination
• Model Xₜ−μ=φ(Xₜ₋₁−μ)+εₜ
• X̂ₙ₊₁=μ+φ(xₙ−μ)=70.148
• X̂ₙ₊ₕ=μ+φʰ(xₙ−μ) with φ⁵≈0.0604
• X̂ₙ₊₅≈73.886, approaching the mean
• Structure versus model: structural equations with endogenous and predetermined variables; reduced form
• Endogenous regressors are correlated with the disturbances
• OLS is biased and inconsistent (simultaneous-equation bias)
• Remedies: identification, ILS, IV, 2SLS, LIML/3SLS
• Average age = x + ∫₀ⁿ t l(x+t)μ(x+t)dt/(lₓ−lₓ₊ₙ)
• Simplifies to x+(ₙLₓ−n lₓ₊ₙ)/(lₓ−lₓ₊ₙ)=x+(Tₓ−Tₓ₊ₙ−n lₓ₊ₙ)/dₓ
• μ(x)=−d ln l(x)/dx = 1/[2(100−x)]
• μ(84)=1/32=0.03125
• Intelligence tests are standardised measures of cognitive abilities (verbal/performance, individual/group)
• Use: norm-referenced comparison for selection, diagnosis, guidance
• Mental age: age level whose average performance matches the examinee's
• IQ=(MA/CA)×100 (ratio IQ) and the deviation IQ (mean 100, SD 15)

Framework: UPSC Statistics Paper 2. (a) compare: paired headings or point-wise contrast > key differences > significance > conclusion | (b) explain: definition/context > points in order > small example > short close | (c) compare: paired headings or point-wise contrast > key differences > significance > conclusion Full marks: Rigorous derivations, correct calculations, clear distinctions, and proper interpretation of results.

• OLS from minimising (Y−Xβ)′(Y−Xβ): β̂=(X′X)⁻¹X′Y
• GLS with Var(u)=σ²Ω via transformation P′P=Ω⁻¹: β̃=(X′Ω⁻¹X)⁻¹X′Ω⁻¹Y
• Both unbiased; Var(β̃)=σ²(X′Ω⁻¹X)⁻¹ and Var(β̂)=σ²(X′X)⁻¹X′ΩX(X′X)⁻¹
• Gauss–Markov (OLS BLUE when Ω=I) vs Aitken (GLS BLUE); OLS inefficient and its usual variance estimate biased when Ω≠I
• TRT: P₀₁×P₁₀=1; FRT: P₀₁×Q₀₁=Σp₁q₁/Σp₀q₀; circular test P₀₁×P₁₂×P₂₀=1
• Σp₀q₀=3530, Σp₁q₀=3365, so P₀₁=0.9533
• Σp₁q₁=3400, Σp₀q₁=3600, so P₁₀=1.0588
• Product ≈1.0093 ≠ 1: Laspeyres fails TRT
• Estimates refer to past/present dates (post-censal, intercensal) using censuses and registration data
• Projections are conditional future computations based on assumptions about fertility, mortality and migration
• Component method: base population by age and sex, survival ratios from life tables, births from ASFRs, net migration
• Cohort advancement repeated for each projection interval; balancing equation P₁=P₀+B−D+M

Framework: UPSC Statistics Paper 2. (a) explain: definition/context > points in order > small example > short close | (b) explain: definition/context > points in order > small example > short close | (c(i)) define: precise definition > the distinguishing feature > one example | (c(ii)) enumerate: list the items in order > one line each > no commentary Full marks: Precise definitions, correct application of order/rank conditions, clear distinction of standardisation methods, and accurate formulas for reliability.

• Explains identification as recovering structural parameters from the reduced form, with a demand–supply example
• Order condition K−k ≥ m−1 (excluded exogenous ≥ included endogenous − 1) and rank condition (rank M−1 of the excluded-variable coefficient matrix)
• Equation 1: order met exactly, rank 2, exactly identified; Equation 2: K−k=2 > 1, rank 2, over-identified
• Equation 3: K−k=2=m−1, rank 2, exactly identified
• CDR is confounded by differing age–sex composition of populations
• Direct standardisation: apply each area's age-specific rates to a common standard population
• Indirect standardisation: apply standard rates to the local population to get expected deaths and SMR; adjusted rate = SMR × standard CDR
• CMI = standardised rate of area ÷ that of the standard × 100 (or observed/expected × 100), used to rank areas
• Reliability = consistency of scores; r=σ_T²/σ_X² in classical test theory
• X=T+E with error uncorrelated with true score
• Spearman–Brown: r_n = nr/[1+(n−1)r]
• Longer tests (with parallel items) increase reliability with diminishing returns
• Test–retest (stability) and its limitations
• Parallel/equivalent forms
• Split-half with Spearman–Brown correction
• Internal consistency: KR-20, KR-21 and Cronbach's alpha formulas

Framework: UPSC Statistics Paper 2. (a) calculate: given > formula > substitution > result with units > interpretation | (b) define: precise definition > the distinguishing feature > one example | (c(i)) discuss: intro > 3-4 dimensions > example > balanced close | (c(ii)) explain: definition/context > points in order > small example > short close Full marks: Precise definitions, correct calculations, clear logical flow, and accurate statistical terminology.

• Difficulty measured on the normal scale: z=Φ⁻¹(1−p)
• z₁=−1.036 (p=0.85) and z₄=0.674 (p=0.25) from the table
• Equal spacing h=(z₄−z₁)/3=0.570, so z₂=−0.466 and z₃=0.104
• Pass percentages ≈67.9% and ≈45.9%
• CBR=B/P×1000: denominator includes people not exposed to childbearing
• GFR=B/W₁₅₋₄₉×1000: restricts to women of reproductive age
• ASFR=Bₐ/Wₐ×1000: removes the effect of age distribution within 15–49
• TFR=5ΣASFR/1000: children per woman under current rates; age-structure free, compared with replacement level 2.1
• Definition: Cov(uₜ,uₛ)≠0 for t≠s, typically AR(1) uₜ=ρuₜ₋₁+εₜ
• Causes: inertia, omitted variables, misspecification, data smoothing
• OLS stays unbiased and consistent (exogenous regressors) but is not BLUE; true variance σ²(X′X)⁻¹X′ΩX(X′X)⁻¹
• Usual standard errors, t and F tests invalid (often understated with positive ρ); with a lagged dependent variable OLS becomes inconsistent
• d=Σ(eₜ−eₜ₋₁)²/Σeₜ² from OLS residuals
• d≈2(1−r), range 0 to 4, d≈2 no autocorrelation
• Decision rule with d_L, d_U: reject if d<d_L, do not reject if d>d_U, inconclusive between; 4−d for negative ρ
• Assumptions and limits: intercept, non-stochastic X, no lagged y (use Durbin h), first-order only