Statistics 2024 Paper II 50 marks Solve

Q4

(a) A manually handled toll-booth has two tellers, who are each capable of handling an average of 60 vehicles per hour, with the actual service times exponentially distributed. Vehicles arrive at the booth according to a Poisson process, at an average rate of 100 per hour. Determine the following : 15 marks (i) The probability that there are more than three vehicles in the booth at the same time (ii) The probability that a given teller is idle (iii) The probability that a vehicle spends more than 3 minutes in the booth (b) PQR Electronics produces 300 transistors per day, which go into the inventory. It supplies 150 transistors per day to XYZ Radios. The annual demand is 37,500 units. The inventory holding cost is $ 0·25 per transistor per year and the setup cost per production run is $ 200. Find the following : 15 marks (i) Economic Order Quantity (EOQ) (ii) Production run length (iii) Number of production runs per year (iv) Maximum Inventory Level (c) (i) Explain the terms 'chance causes' and 'assignable causes' of variation in quality control. Also provide some principal advantages of statistical quality control. 10 marks (ii) Describe the procedure of obtaining OC curve for single sampling plan. 10 marks

हिंदी में प्रश्न पढ़ें

(a) एक हस्तेन संभालने वाले टोल बूथ के पास दो टेलर हैं, जो प्रत्येक औसतन 60 वाहनों को प्रति घंटे की दर से संभालने के लिए सक्षम हैं, इनकी वास्तविक सेवा का समय चरघातांकीय बंटित है। वाहन बूथ पर प्वासों प्रक्रम के अनुसार औसतन 100 प्रति घंटे की दर से पहुँचते हैं। निम्नलिखित को निर्धारित कीजिए (ज्ञात कीजिए) : 15 अंक (i) एक ही समय पर बूथ में तीन से अधिक वाहनों के होने की प्रायिकता (ii) एक दिया गया टेलर खाली है, इसकी प्रायिकता (iii) एक वाहन बूथ में 3 मिनट से अधिक बिताता है, इसकी प्रायिकता (b) PQR इलेक्ट्रॉनिक्स प्रतिदिन 300 ट्रांजिस्टरों का उत्पादन करता है, जो तालिका (इन्वेंटरी) में चले जाते हैं। यह 150 ट्रांजिस्टर प्रतिदिन XYZ रेडियोज को आपूर्ति करता है। वार्षिक मांग 37,500 यूनिट्स है। तालिका धारण लागत $ 0·25 प्रति ट्रांजिस्टर प्रति वर्ष है और व्यवस्था लागत प्रति प्रोडक्शन रन $ 200 है। निम्नलिखित को ज्ञात कीजिए : 15 अंक (i) आर्थिक आदेश मात्रा (ई ओ क्यू) (ii) प्रोडक्शन रन की लंबाई (iii) प्रति वर्ष प्रोडक्शन रनों की संख्या (iv) अधिकतम तालिका (इन्वेंटरी) स्तर (c) (i) गुणवत्ता नियंत्रण में विचरण के पदों 'संयोग हेतु' और 'निर्देश्य कारणों' की व्याख्या कीजिए। सांख्यिकीय गुणवत्ता नियंत्रण के कुछ मुख्य लाभों को भी बताइए। 10 अंक (ii) एकल प्रतिचयन आयोजना के लिए संकारक अभिलक्षण (OC) वक्र को प्राप्त करने की प्रक्रिया का वर्णन कीजिए। 10 अंक

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Approach

Solve this multi-part numerical and theoretical question by allocating approximately 35% time to part (a) on M/M/2 queueing, 30% to part (b) on production inventory EOQ model, and 35% to part (c) on SQC theory. Begin with clear identification of model parameters for each part, show all formulas with standard notation, perform step-by-step calculations with proper unit conversions, and conclude with precise numerical answers and brief interpretations where asked.

Key points expected

  • For (a): Correct identification of M/M/2 queue parameters (λ=100/hr, μ=60/hr, c=2), calculation of traffic intensity ρ=λ/(cμ)=0.833, and use of multi-server queue formulas for P₀, Pₙ, and waiting time distribution
  • For (a)(i): Computation of P(n>3) = 1 - P₀ - P₁ - P₂ - P₃ using the formula Pₙ = (1/n!)(λ/μ)ⁿP₀ for n≤c and Pₙ = (1/c!cⁿ⁻ᶜ)(λ/μ)ⁿP₀ for n>c
  • For (a)(ii): Probability a given teller is idle = P₀ + ½P₁ (or equivalently 1 - ρ), recognizing that idle probability per server differs from system idle probability
  • For (b): Identification of production model parameters (p=300/day, d=150/day, D=37,500/yr, C₁=$0.25/yr, C₃=$200), and application of EPQ formulas rather than simple EOQ
  • For (b)(i)-(iv): Correct formulas for EOQ/EPQ = √[2DC₃p/(C₁(p-d))], production run length = Q/p, number of runs = D/Q, and maximum inventory = Q(p-d)/p with proper unit consistency
  • For (c)(i): Clear distinction between chance causes (random, inherent, unavoidable) and assignable causes (special, identifiable, removable), with 4-5 specific advantages of SQC such as early detection, reduced inspection costs, and customer satisfaction
  • For (c)(ii): Systematic description of OC curve construction: define p (lot fraction defective), calculate Pₐ using binomial/Poisson approximation, plot Pₐ vs p, and identify key points (AQL, LTPD, α, β)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies all model parameters for each sub-part: for (a) recognizes M/M/2 system with λ=100, μ=60, c=2 and computes ρ=5/6<1; for (b) distinguishes production model with p>d and converts all to consistent annual units; for (c) defines technical terms with precisionIdentifies most parameters correctly but makes minor errors like using simple EOQ instead of EPQ for (b), or confusing λ/μ values; some unit inconsistencies presentMajor parameter identification errors such as treating (a) as M/M/1, using wrong time units (minutes vs hours), or completely misidentifying the inventory model type
Method choice20%10Selects appropriate standard formulas: multi-server queue formulas for (a), production inventory model for (b), and describes OC curve construction using proper sampling theory for (c); cites relevant formulas by standard notation or nameUses generally correct methods but applies approximations where exact formulas needed, or mixes up similar-looking formulas between models; OC curve description lacks sampling distribution specificsApplies completely wrong methods such as using single-server formulas for two tellers, or simple EOQ for production model, or describes control charts instead of OC curves
Computation accuracy20%10All calculations accurate to 3-4 decimal places: P₀≈0.0909 for (a), P(n>3)≈0.512, idle probability≈0.167, P(W>3min) using correct exponential waiting time; EOQ≈10,954 units, run length≈36.5 days, runs/year≈3.42, max inventory≈5,477 for (b)Correct calculation approach with minor arithmetic errors or rounding issues; correct formulas but wrong substitution in 1-2 sub-parts; unit conversion errors (minutes to hours)Major computational errors, incorrect algebraic manipulation, or missing calculations for several sub-parts; answers without any working shown
Interpretation20%10Provides meaningful interpretation: for (a) notes 51% chance of queue buildup indicating need for third teller; for (b) interprets inventory policy implications; for (c)(i) gives practical examples of chance vs assignable causes from Indian manufacturing context; for (c)(ii) explains how OC curve balances producer and consumer riskBrief interpretations for some parts but missing for others; generic statements without connection to specific numerical results; theoretical descriptions without practical illustrationNo interpretation provided; or completely wrong interpretation of what results signify; confuses concepts (e.g., describing type I/II errors without linking to OC curve)
Final answer & units20%10All seven sub-parts answered with precise final values and correct units: probabilities dimensionless or as percentages, time in minutes/hours as specified, inventory quantities in units, costs in $; clearly labeled (i)-(iv) matching question structureMost answers present but units missing or inconsistent; some sub-parts skipped; final answers buried in working without clear highlightingMissing final answers for multiple sub-parts; serious unit errors (e.g., EOQ in dollars, probabilities >1); answers presented without any indication of which sub-part they address

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