Statistics 2021 Paper II 50 marks Solve

Q2

A manufacturer finds that on the average, a television set is used 1.8 hours per day. A one year warranty is offered on the picture tube having a mean time to failure (MTTF) of 2000 hours. If the distribution of time to failure is exponential, then determine the percentage of tubes failing during the warranty period. (15 marks) The number of defects on 20 items were recorded as given above: | Item No. | No. of defects | Item No. | No. of defects | |----------|----------------|----------|----------------| | 1 | 2 | 11 | 6 | | 2 | 0 | 12 | 0 | | 3 | 4 | 13 | 2 | | 4 | 1 | 14 | 1 | | 5 | 0 | 15 | 0 | | 6 | 8 | 16 | 3 | | 7 | 0 | 17 | 2 | | 8 | 1 | 18 | 1 | | 9 | 2 | 19 | 0 | | 10 | 0 | 20 | 2 | Use a suitable control chart to identify whether the process is in control or not? (15 marks) Explain the concepts of producer's and consumer's risks. It has been decided to sample 100 items at random from each large batch. We reject the batch if more than 2 defectives are found. If the acceptable quality level is 1% and the unacceptable quality level is 5%, then find the producer's and consumer's risks. (20 marks)

हिंदी में प्रश्न पढ़ें

एक निर्माता को यह पता चलता है कि औसतन एक टेलीविजन सेट का उपयोग प्रतिदिन 1.8 घंटे होता है। पिक्चर ट्यूब पर, जिसका विफलता तक माध्य काल (एम टी टी एफ) 2000 घंटे है, एक वर्ष की वारंटी की पेशकश की जाती है। यदि विफलता तक के काल का बंटन चरघातांकी है, तो वारंटी अवधि के दौरान विफल हुई ट्यूबों का प्रतिशत ज्ञात कीजिए। (15 अंक) 20 मदों पर दोषों की संख्या दर्ज की गई, जो नीचे दी गई है : | मद संख्या | दोषों की संख्या | मद संख्या | दोषों की संख्या | |-----------|----------------|-----------|----------------| | 1 | 2 | 11 | 6 | | 2 | 0 | 12 | 0 | | 3 | 4 | 13 | 2 | | 4 | 1 | 14 | 1 | | 5 | 0 | 15 | 0 | | 6 | 8 | 16 | 3 | | 7 | 0 | 17 | 2 | | 8 | 1 | 18 | 1 | | 9 | 2 | 19 | 0 | | 10 | 0 | 20 | 2 | एक उपयुक्त नियंत्रण सांचित्र का उपयोग कीजिए और यह पहचानिये कि क्या प्रक्रम नियंत्रण में है या नहीं ? (15 अंक) उत्पादक और उपभोक्ता के जोखिमों की संकल्पनाओं को समझाइए। प्रत्येक बड़े बैच से 100 मदों का एक यादृच्छिक प्रतिदर्श निकालने का निर्णय लिया गया है। हम बैच को अस्वीकार करते हैं यदि 2 से अधिक दोषपूर्ण पाये जाते हैं। यदि स्वीकार्य गुणता स्तर 1% है और अस्वीकार्य गुणता स्तर 5% है तो उत्पादक और उपभोक्ता के जोखिमों को प्राप्त कीजिए। (20 अंक)

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How this answer will be evaluated

Approach

Solve this three-part numerical problem by first calculating the warranty failure probability using exponential distribution properties, then constructing and interpreting a c-chart for defect data with proper control limits, and finally computing producer's and consumer's risks using binomial distribution for the given sampling plan. Present each part sequentially with clear headings, showing all formulas, substitutions, and final interpretations.

Key points expected

  • Part 1: Calculate total warranty period as 1.8 × 365 = 657 hours and use P(T ≤ 657) = 1 - e^(-657/2000) for exponential failure probability
  • Part 2: Compute c-bar = 33/20 = 1.65, then UCL = 1.65 + 3√1.65 ≈ 5.50 and LCL = 0, identifying Item 6 (8 defects) as out of control
  • Part 3: Define producer's risk α = P(reject | p=0.01) and consumer's risk β = P(accept | p=0.05) using binomial or Poisson approximation
  • Correct application of Poisson approximation with λ₁ = 1 for AQL and λ₂ = 5 for LTPD to find α = 1 - P(X≤2; λ=1) and β = P(X≤2; λ=5)
  • Numerical values: α ≈ 0.080 or 8% and β ≈ 0.125 or 12.5% (or precise binomial equivalents)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness15%7.5Correctly identifies warranty duration (657 hours), calculates c-bar from defect data, and properly sets up binomial/Poisson parameters for both risk calculations with clear identification of n=100, c=2, p₁=0.01, p₂=0.05Correct warranty calculation and c-bar computation but minor errors in identifying sampling plan parameters or confusion between AQL and LTPD in risk setupIncorrect warranty period calculation (e.g., uses 365 hours instead of 657), wrong c-bar computation, or fundamental misunderstanding of which probability corresponds to which risk
Method choice20%10Uses exponential CDF for reliability, c-chart (not p-chart or X-bar) for count data with 3-sigma limits, and Poisson approximation (or exact binomial) appropriately for OC curve calculations with correct λ valuesCorrect chart type selected but uses wrong control limit formula, or uses normal approximation instead of Poisson for risks without justification, or attempts exact binomial with calculation errorsUses wrong distribution (e.g., normal for time to failure), selects p-chart or X-bar chart instead of c-chart, or completely wrong method for risk calculation (e.g., uses single sampling plan formulas incorrectly)
Computation accuracy25%12.5Accurate to 3 decimal places: warranty failure ≈ 27.9% or 28%, control limits UCL≈5.50/LCL=0 with correct identification of out-of-control point, α≈0.080 (8%), β≈0.125 (12.5%) using Poisson tables or precise summationCorrect formulas but arithmetic errors leading to slightly off values (e.g., 26-29% for warranty, α or β off by 0.02-0.03), or correct values but only 1-2 parts fully accurateMajor computational errors (e.g., wrong exponential parameter, control limits off by factor, risks calculated with wrong λ values or reversed), or excessive rounding affecting conclusions
Interpretation25%12.5Clearly states ~28% tubes fail during warranty (high failure rate suggests warranty risk), identifies Item 6 as assignable cause requiring investigation, and interprets α,β values in context—producer faces 8% wrongful rejection risk, consumer faces 12.5% acceptance of bad batches, with comment on plan's stringencyStates numerical answers with minimal interpretation, or generic statements about 'process out of control' without specifying which item, or defines risks without contextualizing the 8%/12.5% valuesNo interpretation of what failure percentage means for manufacturer warranty costs, no identification of specific out-of-control points, or completely wrong interpretation of which party bears which risk
Final answer & units15%7.5All three answers clearly boxed/labeled: (i) 27.9% or 28% failure rate, (ii) c-chart with UCL=5.50, LCL=0, process NOT in control (Item 6), (iii) Producer's risk ≈ 8%, Consumer's risk ≈ 12.5%, with percentage symbols and proper risk labelsCorrect numerical values present but poorly organized, missing units (e.g., '28' without %), or swapped risk labels, or control limits without conclusion on control statusMissing final answers for 1-2 parts, or answers without any context (just numbers), or completely wrong values stated confidently, or no indication of which item violates control limits

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