Statistics 2024 Paper II 50 marks Solve

Q2

(a) Obtain the control limits for X̄-chart and R-chart and describe the significance of joint study of these charts. 20 marks (b) Find the reliability and hazard functions of Weibull distribution with scale parameter θ and shape parameter β, and interpret the findings. 10 marks (c) Use simplex method to solve the following LPP : Maximize z = 5x₁ + 2x₂ subject to 6x₁ + x₂ ≥ 6 4x₁ + 3x₂ ≥ 12 x₁ + 2x₂ ≥ 4 x₁ ≥ 0, x₂ ≥ 0 20 marks

हिंदी में प्रश्न पढ़ें

(a) X̄-चार्ट तथा R-चार्ट के लिए नियंत्रण सीमाएँ प्राप्त कीजिए और इन चार्टों के साम्मिलित अध्ययन के महत्व का वर्णन कीजिए। 20 अंक (b) वेबुल बंटन, जिसका मापक्रम प्राचल θ और आकृति प्राचल β है, के लिए विश्वसनीयता एवं संकटप्रस्त (हेजर्ड) फलनों को प्राप्त कीजिए, तथा निष्कर्षों की व्याख्या कीजिए। 10 अंक (c) एकदा विधि का उपयोग करके निम्नलिखित रैखिक प्रोग्रामन समस्या (एल पी पी) को हल कीजिए : अधिकतमीकरण कीजिए z = 5x₁ + 2x₂ निम्न प्रतिबंधों के अंतर्गत 6x₁ + x₂ ≥ 6 4x₁ + 3x₂ ≥ 12 x₁ + 2x₂ ≥ 4 x₁ ≥ 0, x₂ ≥ 0 20 अंक

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How this answer will be evaluated

Approach

Begin by deriving control limits for X̄-chart and R-chart using standard formulae with A₂, D₃, D₄ constants, explaining why joint monitoring prevents Type I/II errors in SPC (40% time). For Weibull, derive R(t) = exp[-(t/θ)^β] and h(t) = (β/θ)(t/θ)^(β-1), interpreting bathtub curve relevance for Indian manufacturing/equipment reliability (20% time). For the LPP, convert ≥ constraints to standard form using surplus variables and artificial variables (Big-M method), then execute simplex iterations to reach optimality (40% time).

Key points expected

  • Part (a): Correct formulae for X̄-chart limits (UCL/LCL = X̄̄ ± A₂R̄) and R-chart limits (UCL = D₄R̄, LCL = D₃R̄) with proper identification of constants
  • Part (a): Explanation that X̄-chart monitors process mean while R-chart monitors process variability; joint study detects both shifts and dispersion changes, preventing misinterpretation from exclusive use of either chart
  • Part (b): Derivation of reliability function R(t) = exp[-(t/θ)^β] and hazard function h(t) = (β/θ)(t/θ)^(β-1) from Weibull PDF
  • Part (b): Interpretation of β < 1 (decreasing hazard, infant mortality), β = 1 (constant hazard, exponential), β > 1 (increasing hazard, wear-out) with industrial examples
  • Part (c): Conversion of ≥ constraints to standard form: subtract surplus variables s₁, s₂, s₃ and add artificial variables A₁, A₂, A₃ with Big-M penalty in objective
  • Part (c): Complete simplex tableau iterations showing entering/leaving variables, pivot operations, and final optimal solution with Z_max = 12 at (x₁=3, x₂=0) or verified corner point
  • Part (c): Verification of solution by checking constraint satisfaction and comparing objective values at all extreme points

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): correctly identifies X̄̄, R̄ as grand mean and average range; for (b): proper parameterization with θ > 0, β > 0; for (c): accurate conversion to standard form with surplus and artificial variables, correct Big-M objective formulationMinor errors in constant selection for control charts or incomplete standard form conversion with missing artificial variablesWrong formulae for control limits, incorrect Weibull parameterization, or failure to convert ≥ constraints leading to infeasible simplex setup
Method choice20%10Selects appropriate SPC constants (A₂, D₃, D₄) for given sample size; uses proper derivation from CDF for Weibull; applies Big-M method correctly for minimization problem with ≥ constraintsUses correct general approach but applies wrong constants or attempts dual simplex without justificationUses σ-based limits instead of R-based for (a); attempts graphical method for (c); or uses two-phase method without clarity
Computation accuracy20%10Precise arithmetic in all calculations: control limit values, simplex pivot operations, and ratio tests; correct final optimal values with no calculation errors across all iterationsMinor arithmetic errors in one or two simplex iterations or control limit calculations that don't affect final conclusionMajor computational errors leading to wrong optimal solution, incorrect pivot selection, or nonsensical control limits
Interpretation20%10Clear explanation of why joint X̄-R charts prevent false signals (e.g., R-chart detects increased variability masking mean shift); insightful Weibull interpretation linking β to product life stages; economic interpretation of shadow prices in LPPBasic interpretation without connecting to practical significance or missing the complementary nature of the two control chartsNo interpretation of Weibull shapes, no explanation of joint chart significance, or failure to interpret simplex iterations meaning
Final answer & units20%10All final answers explicitly stated: numerical control limits with units, reliability/hazard expressions in simplified form, optimal solution (x₁=3, x₂=0, Z_max=12) with verification; proper mathematical notation throughoutFinal answers present but incomplete verification or missing units for control limitsMissing final answers, incorrect optimal values, or answers without proper mathematical formatting

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