Paper II — 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…
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)
Model answer
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(a) Percentage of picture tubes failing in the warranty period
Let T be the time to failure in hours, exponential with mean 2000, so λ = 1/2000 per hour. Warranty usage = 1.8 × 365 = 657 hours. P(T ≤ 657) = 1 − e^(−657/2000) = 1 − e^(−0.3285) = 1 − 0.7200 = 0.2800. About 28 per cent of the tubes fail during the warranty period, assuming average use of 1.8 hours a day and an exponential failure law.
(b) Control chart for defects per item
The number of defects per inspection unit is a count, so a c-chart is suitable. Total defects = 35, so c̄ = 35/20 = 1.75. UCL = c̄ + 3√c̄ = 1.75 + 3(1.3229) = 5.72; LCL = 1.75 − 3.97 < 0, so LCL = 0. Item 6 (8 defects) and item 11 (6 defects) lie above the UCL, so the process is not in statistical control. Remedial step: if assignable causes are found for items 6 and 11 and they are removed, the revised c̄ = 21/18 = 1.17 and UCL = 4.41; all remaining points (maximum 4) lie within the limits, so the chart can then be used for future monitoring.
(c) Producer's and consumer's risks
Producer's risk α is the probability of rejecting a lot of acceptable quality (p = AQL); consumer's risk β is the probability of accepting a lot of unacceptable quality (p = LTPD).
Here n = 100, accept if the number of defectives X ≤ 2, AQL = 0.01 and LTPD = 0.05. For a large lot, X ~ Bin(100, p).
- At p = 0.01: P(X ≤ 2) = 0.3660 + 0.3697 + 0.1849 = 0.9206, so α = 1 − 0.9206 = 0.0794 (about 7.9%).
- At p = 0.05: P(X ≤ 2) = 0.0059 + 0.0312 + 0.0812 = 0.1183, so β ≈ 0.118 (about 11.8%).
(With the Poisson approximation, np = 1 and 5, the values are α ≈ 0.080 and β ≈ 0.125.)
How this answer will be evaluated
Framework: Statistics, Paper 2
What earns full marks: Rigorous application of formulas with clear interpretation and correct control chart selection.
Evaluation rubric
Each sub-part is marked on its own, against the marks and word limit printed on the paper.
- (a) Find the percentage of picture tubes failing in the one-year warranty if MTTF is 2000 hours and usage is 1.8 hours a day, 15 marks 15 marks
calculate— given → formula → substitution → result with units → interpretation
Must cover
- Warranty usage 1.8 × 365 = 657 hours
- Exponential CDF 1 − exp(−657/2000)
- Result about 28%
Loses marks
- Using 365 × 24 hours
- Using the density instead of the CDF
Earns more
- Rate λ = 1/2000 per hour
- Stating the assumptions
Extra mark
- Comment on sensitivity to usage
- (b) Use a suitable control chart to decide whether the process producing the defect counts is in control, 15 marks 15 marks
analyse— intro → causes → effects → stakeholders/linkages → way forward
Must cover
- Choice of c-chart for counts per unit
- c̄ = 1.75, UCL = 5.72, LCL = 0
- Items 6 and 11 exceed UCL, so out of control
Loses marks
- Using a p-chart or X-bar chart
- Negative LCL not truncated
Earns more
- Total defects 35
- Revised limits after removing assignable causes
Extra mark
- Mention of Poisson basis of the c-chart
- (c) Explain producer's and consumer's risks and compute them for n = 100, acceptance number 2, AQL 1% and LTPD 5%, 20 marks 20 marks
explain— definition/context → points in order → small example → short close
Must cover
- Definitions of α and β in terms of AQL and LTPD
- Binomial calculation giving α = 0.0794
- β = 0.1183
Loses marks
- Swapping α and β
- Using P(X < 2)
Earns more
- Correct acceptance rule X ≤ 2
- Poisson approximation values 0.080 and 0.125
Extra mark
- Link to the OC curve
What "Solve" is asking you to do
Choose the method, then carry it through to a final answer. Identifying what kind of problem this is and why that method applies is the first thing marked; a correct figure arrived at invisibly earns almost nothing.
Structure that answers it
Given data and what is required → method chosen, with the reason it applies → set-up (equation, circuit, free body, trial balance) → working, step by step → answer with units and any condition of validity
Where marks are lost
Doing the middle steps mentally and writing only the result. In mathematics papers, a further loss comes from giving a decimal where the exact value in surds or fractions was wanted, or from skipping the justification a part explicitly asks for.
Practice this exact question
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