Statistics 2022 Paper II 50 marks Compulsory Distinguish

Q1

(a) Distinguish between process control and product control. Explain the various sources of variation encountered in a process control study. Suggest how they can be eliminated from the process. 10 marks (b) The management of ABC company is considering the question of marketing a new product. The fixed cost required in the project is ₹ 4,000. Three factors are uncertain, viz., selling price, variable cost and annual sales volume. The product has life of only one year. The management has the data on three factors as under: | Selling Price (₹) | Probability | Variable Cost (₹) | Probability | Sales Volume (units) | Probability | |---|---|---|---|---|---| | 3 | 0·2 | 1 | 0·3 | 2000 | 0·3 | | 4 | 0·5 | 2 | 0·6 | 3000 | 0·3 | | 5 | 0·3 | 3 | 0·1 | 5000 | 0·4 | Consider the sequence of thirty random numbers 81, 32, 60, 04, 46, 31, 67, 25, 24, 10, 40, 02, 39, 68, 08, 59, 66, 90, 12, 64, 79, 31, 86, 68, 82, 89, 25, 11, 98, 16 and using the sequence (first 3 random numbers for the first trial, etc.), simulate the average profit for the above project on the basis of 10 trials. 10 marks (c) If N(t) is a Poisson process and s < t, find P(N(s) = k | N(t) = n) and comment. 10 marks (d) What are the assumptions made in the theory of games? Describe the maximin principle and minimax principle. Explain the algebraic method for games without saddle point. 10 marks (e) What are the importances of censoring in life-testing experiments? Discuss the estimation of parameters involved in exponential distribution with mean θ, using type-2 censored sample. 10 marks

हिंदी में प्रश्न पढ़ें

(a) प्रक्रम नियंत्रण और उत्पाद नियंत्रण में विभेदन कीजिए। एक प्रक्रम नियंत्रण अध्ययन में आने वाले परिवर्तन के विभिन्न स्रोतों की व्याख्या कीजिए। सुझाइए कि इनको प्रक्रम से कैसे विलुप्त किया जा सकता है। 10 अंक (b) ABC कंपनी का प्रबंधन एक नये उत्पाद के विपणन के प्रश्न पर विचार कर रहा है। इस परियोजना में निश्चित लागत ₹ 4,000 की आवश्यकता है। तीन कारक अनिश्चित हैं, जैसे बिक्री मूल्य, परिवर्तनीय लागत और वार्षिक बिक्री मात्रा। इस उत्पाद का जीवन केवल एक वर्ष का है। प्रबंधन के पास तीन कारकों का डेटा निम्नलिखित है : | बिक्री मूल्य (₹) | प्रायिकता | परिवर्तनीय लागत (₹) | प्रायिकता | बिक्री मात्रा (इकाइयाँ) | प्रायिकता | |---|---|---|---|---|---| | 3 | 0·2 | 1 | 0·3 | 2000 | 0·3 | | 4 | 0·5 | 2 | 0·6 | 3000 | 0·3 | | 5 | 0·3 | 3 | 0·1 | 5000 | 0·4 | तीस यादृच्छिक संख्याओं के अनुक्रम 81, 32, 60, 04, 46, 31, 67, 25, 24, 10, 40, 02, 39, 68, 08, 59, 66, 90, 12, 64, 79, 31, 86, 68, 82, 89, 25, 11, 98, 16 पर विचार कीजिए और इस अनुक्रम का प्रयोग करते हुए (पहले अभिप्रयोग के लिए पहली 3 यादृच्छिक संख्याएँ आदि) 10 अभिप्रयोगों के आधार पर उपरोक्त परियोजना के लिए औसत लाभ का अनुकरण (सिमुलेट) कीजिए। 10 अंक (c) यदि N(t) एक प्वासों प्रक्रम है और s < t है, तो P(N(s) = k | N(t) = n) प्राप्त कीजिए और टिप्पणी दीजिए। 10 अंक (d) खेलों के सिद्धांत में कौन-सी कल्पनाएँ की जाती हैं? मैक्सिमिन नियम और मिनिमैक्स नियम का वर्णन कीजिए। पल्याण बिंदु रहित खेलों के लिए बीजीय विधि को समझाइए। 10 अंक (e) जीवन-परीक्षण प्रयोगों में खंड-वर्जन के क्या महत्व हैं? प्रकार-2 के खंड-वर्जित प्रतिदर्श का प्रयोग करके, चरघातांकी बंटन, जिसका माध्य θ है, के प्राचलों के आकलन का वर्णन कीजिए। 10 अंक

Evaluate your answer to this question → See all 2022 Statistics questions

Directive word: Distinguish

This question asks you to distinguish. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Distinguish requires clear differentiation followed by explanation. Allocate approximately 20% time to part (a) on process vs product control with variation sources, 25% to part (b) simulation with proper random number mapping, 15% to part (c) conditional probability derivation for Poisson, 20% to part (d) game theory assumptions and algebraic method, and 20% to part (e) censoring importance and MLE for exponential Type-2. Begin with definitions, proceed to analytical derivations or computational steps, and conclude with interpretations.

Key points expected

  • Part (a): Clear distinction between process control (monitoring during production) and product control (acceptance sampling); identification of chance causes (random, inherent) and assignable causes (special, identifiable) of variation; remedial measures for each source
  • Part (b): Correct probability interval mapping for selling price (00-19, 20-69, 70-99), variable cost (00-29, 30-89, 90-99), and sales volume (00-29, 30-59, 60-99); proper profit calculation as (Price - Variable Cost) × Volume - Fixed Cost; accurate simulation table with 10 trials and average profit computation
  • Part (c): Derivation of P(N(s)=k|N(t)=n) using independent increments property resulting in Binomial(n, s/t) distribution; recognition that conditional distribution depends only on ratio s/t not on rate parameter λ
  • Part (d): Assumptions: two players, finite strategies, zero-sum, simultaneous moves, complete information; maximin (player A's security level) and minimax (player B's security level) principles; algebraic method using mixed strategies with probability variables p and q, solving simultaneous equations for value of game
  • Part (e): Importance of censoring: time/cost efficiency, ethical considerations, handling heavy-tailed distributions; Type-2 censoring with r failures out of n items; MLE derivation for θ with estimator T/r where T is total time on test, showing unbiasedness and variance properties

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a) correctly classifies control types with industrial examples; for (b) establishes proper random number intervals matching given probabilities; for (c) states Poisson process properties correctly; for (d) lists all game theory assumptions; for (e) defines Type-2 censoring precisely with r and n specifiedPartial setup with minor errors in probability intervals or missing 1-2 assumptions; vague on censoring typesConfused process/product control, wrong probability mappings, missing critical assumptions, or undefined censoring parameters
Method choice20%10For (a) uses Shewhart's classification framework; for (b) applies Monte Carlo simulation systematically; for (c) employs conditional probability with independent increments; for (d) selects algebraic method appropriately for no-saddle-point games; for (e) uses MLE with correct likelihood for censored dataCorrect methods chosen but with suboptimal presentation or missing justification for method selectionWrong methods (e.g., analytical instead of simulation for b), confused maximin/minimax, or incorrect estimation approach for censored data
Computation accuracy20%10For (b) all 10 trials computed correctly with proper random number usage and profit values; for (c) exact binomial derivation with correct parameters; for (d) accurate algebraic solution with correct value of game; for (e) correct MLE formula T/r with proper T calculationMinor arithmetic errors in 1-2 trials or algebraic steps, but overall structure correctMajor computational errors, wrong profit formula, incorrect conditional probability result, or invalid game value/estimator
Interpretation20%10For (a) explains why assignable causes must be eliminated; for (b) interprets average profit with decision recommendation; for (c) comments on time-homogeneity and lack of λ dependence; for (d) explains why algebraic method works when pure strategies fail; for (e) discusses efficiency loss due to censoringBasic interpretation present but lacking depth or missing commentary on key insightsNo interpretation of results, purely mechanical answers, or incorrect conclusions from computations
Final answer & units20%10Clear final answers: (a) summarized distinction table; (b) explicit average profit in ₹ with simulation table; (c) final conditional probability formula with binomial identification; (d) game value and optimal strategies; (e) MLE formula with properties; all units (₹, units, time) specifiedFinal answers present but poorly organized or missing some unitsMissing final answers, no units, or incorrect presentation of results across multiple parts

Practice this exact question

Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.

Evaluate my answer →

More from Statistics 2022 Paper II