Statistics 2025 Paper II 50 marks Solve

Q4

(a) Solve the game whose payoff matrix is $$ \begin{bmatrix} -1 & -2 & 8 \\ 7 & 5 & -1 \\ 6 & 0 & 12 \end{bmatrix} $$ (15 marks) (b) Use the penalty (Big M) method to solve the following linear programming problem : Minimize Z = 5x₁ + 3x₂ subject to the constraints 2x₁ + 4x₂ ≤ 12 2x₁ + 2x₂ = 10 5x₁ + 2x₂ ≥ 10 x₁, x₂ ≥ 0 (15 marks) (c) (i) Distinguish between a nonconforming unit and a nonconformity. State the appropriate conditions for constructing a control chart for nonconformities and derive the control limits for a control chart based on the average number of nonconformities per inspection unit. (2+8=10 marks) (ii) Describe the operating procedure of unit-by-unit sequential sampling plan by attributes. What is the unique feature of a sequential sampling plan? (5 marks) (iii) The time to failure for an electronic component used in a flat panel display unit is satisfactorily modelled by a Weibull distribution with the shape parameter β = ½ and the scale parameter θ = 5000 hours. Find the mean time to failure and the fraction of component that is expected to survive beyond 20000 hours. (2+3=5 marks)

हिंदी में प्रश्न पढ़ें

(a) उस खेल को हल कीजिए, जिसका भुगतान आव्यूह है $$ \begin{bmatrix} -1 & -2 & 8 \\ 7 & 5 & -1 \\ 6 & 0 & 12 \end{bmatrix} $$ (15 अंक) (b) पेनाल्टी (बिग M) विधि का उपयोग करके निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए : न्यूनतमीकरण कीजिए, Z = 5x₁ + 3x₂ निम्न प्रतिबंधों के अंतर्गत 2x₁ + 4x₂ ≤ 12 2x₁ + 2x₂ = 10 5x₁ + 2x₂ ≥ 10 x₁, x₂ ≥ 0 (15 अंक) (c) (i) गैर-अनुकूल इकाई तथा गैर-अनुकूलता के बीच अंतर बताइए। गैर-अनुकूलताओं के लिए एक नियंत्रण संचित्र (चार्ट) के निर्माण हेतु उपयुक्त शर्तों को बताइए तथा प्रति निरीक्षण इकाई में गैर-अनुकूलताओं की औसत संख्या पर आधारित नियंत्रण संचित्र के लिए नियंत्रण सीमाओं को न्यूनतम कीजिए। (2+8=10 अंक) (ii) गुणों के आधार पर इकाई-दर-इकाई अनुक्रमिक प्रतिचयन आयोजना की संचालन प्रक्रिया का वर्णन कीजिए। एक अनुक्रमिक प्रतिचयन आयोजना की अद्वितीय विशेषता क्या है? (5 अंक) (iii) एक फ्लैट पैनल डिस्प्ले यूनिट में उपयोग किए गए एक इलेक्ट्रॉनिक घटक की विफलता का समय संतोषजनक तरीके से एक वेबुल बंटन द्वारा मॉडल किया गया, जिसका आकृति प्राचल β = ½ और मापक्रम प्राचल θ = 5000 घंटे हैं। विफलता का माध्य समय तथा घटक का अंश, जो 20000 घंटों से अधिक जीवित रहने की आशा रखता है, प्राप्त कीजिए। (2+3=5 अंक)

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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.

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How this answer will be evaluated

Approach

The directive 'solve' demands complete working with optimal strategies and values for (a) and (b), while (c) requires theoretical exposition with derivations and calculations. Allocate approximately 35-40% time to part (a) given its 15 marks and computational complexity, 30% to part (b) for the Big M method iterations, and 30% to part (c) distributed as 10 marks for (c)(i), 5 marks for (c)(ii), and 5 marks for (c)(iii). Structure with clear part-wise headings, showing all matrix operations, simplex tableaus, and control limit derivations.

Key points expected

  • For (a): Identify the game has no saddle point, check for dominance, reduce using graphical method or solve 2×2 subgames, verify mixed strategy solution with value of game V = 17/5 ≈ 3.4
  • For (b): Convert to standard form by adding slack, surplus and artificial variables; use Big M penalty method with correct simplex iterations showing entering and leaving variables
  • For (c)(i): Define nonconforming unit as item with ≥1 nonconformity vs nonconformity as specific instance of non-fulfilment; state Poisson assumption for c-chart; derive UCL = c̄ + 3√c̄, LCL = max(0, c̄ - 3√c̄)
  • For (c)(ii): Describe sequential sampling with acceptance/rejection/continue regions; unique feature is ASN (average sample number) being smaller than fixed sampling for same protection
  • For (c)(iii): Calculate MTTF = θΓ(1+1/β) = 5000×Γ(3) = 10000 hours; survival probability S(20000) = exp[-(20000/5000)^0.5] = e^(-2) ≈ 0.1353 or 13.53%

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies no saddle point in (a) using maximin-minimax; properly converts all constraints in (b) with correct slack, surplus, artificial variables and Big M objective; accurately states Poisson conditions and Weibull parameters for (c)Identifies saddle point incorrectly or misses dominance; makes minor errors in constraint conversion or misses artificial variable for equality; partial correct conditions for c-chartFails to check for saddle point, wrong variable types in LP setup, or completely wrong distribution assumptions for quality control
Method choice20%10Selects appropriate 2×2 subgame solution or linear programming method for (a); correctly applies Big M simplex algorithm with proper penalty coefficients; uses correct formulas for c-chart limits and Weibull reliabilityUses dominance reduction but makes arithmetic errors; applies simplex but with incorrect M values; uses approximate formulas or wrong chart typeApplies wrong game theory method, uses regular simplex without Big M, or applies p-chart formulas instead of c-chart
Computation accuracy20%10Accurate mixed strategy probabilities (p₁=4/5, p₂=1/5, q₁=3/5, q₂=2/5) and game value 17/5; correct simplex iterations leading to x₁=4, x₂=1, Z=23; precise c-chart derivations and Weibull calculations with Γ(3)=2Minor arithmetic errors in probability calculations; one or two simplex tableau errors; approximate values for MTTF or survival probabilityMajor computational errors in game value, unbounded or infeasible solution in LP, or completely wrong reliability calculations
Interpretation20%10Interprets game value as gain to row player with optimal strategies; explains economic meaning of Big M and why artificial variables leave basis; relates c-chart to defect density monitoring; explains practical advantage of sequential sampling in reducing inspection costs for Indian manufacturingStates final answers without interpretation; minimal explanation of why methods work; generic statements about quality controlNo interpretation of results, fails to explain what optimal strategies mean, or no practical context for quality control applications
Final answer & units20%10All five sub-parts answered with boxed final answers: game value 17/5 or 3.4, optimal strategies as probability vectors; LP solution x₁=4, x₂=1, Z=23; c-chart formulas with ±3√c̄; sequential sampling ASN advantage stated; MTTF=10000 hours, survival fraction=0.1353 or 13.53%Most answers present but missing units or rounded excessively; incomplete final answers for one sub-partMissing final answers for multiple parts, no units where required (hours, probability), or answers without any working shown

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