Statistics 2025 Paper II 50 marks Derive

Q2

(a) (i) What are control charts by variables and control charts by attributes? 5 marks (ii) Derive the control limits for the construction of control charts for the mean and variability based on sample standard deviation. 15 marks (b) (i) State the assumptions involved under sampling inspection plans by variables and describe the operating procedure of a single sampling plan by variables under the assumption of normal distribution for a quality characteristic. 5 marks (ii) Establish the relationship between the fraction defective and the acceptance probability under a single sampling plan by variables when the quality characteristic follows a normal distribution with mean μ and variance σ², where σ² is unknown, and when an upper specification limit is specified. Using the relationship, obtain the formula for finding the parameters of the sampling plan. 10 marks (c) (i) Given a system consisting of n components, define the state vector and the structure function of the system. What do they indicate? 5 marks (ii) Defining (1) a series system, (2) a parallel system and (3) a k-out-of-n system, obtain the associated expressions for the structure functions and the reliability functions. 10 marks

हिंदी में प्रश्न पढ़ें

(a) (i) चरों के लिए नियंत्रण संचित्र (चार्ट) तथा गुणों के लिए नियंत्रण संचित्र (चार्ट) क्या हैं? 5 अंक (ii) प्रतिदर्श मानक विचलन के आधार पर माध्य और परिवर्तनशीलता के लिए नियंत्रण संचित्रों के निर्माण के लिए नियंत्रण सीमाओं को व्युत्पन्न कीजिए। 15 अंक (b) (i) चरों द्वारा प्रतिदर्शी निरीक्षण आयोजनाओं के अंतर्गत मान्यताओं को बताइए तथा गुणता अभिलक्षण के लिए प्रसामान्य बंटन की कल्पना के अंतर्गत, चरों द्वारा एकल प्रतिचयन आयोजना की संचालन प्रक्रिया का वर्णन कीजिए। 5 अंक (ii) चरों द्वारा एकल प्रतिचयन आयोजना के अंतर्गत दुषितानुपात और स्वीकरण प्रायिकता के बीच संबंध स्थापित कीजिए, जबकि गुणता अभिलक्षण एक प्रसामान्य बंटन का अनुसरण करता है, जिसका माध्य μ और प्रसरण σ² है (σ² ज्ञात नहीं है), तथा जबकि ऊपरी विनिर्देश सीमा निर्दिष्ट है। संबंध का उपयोग करते हुए प्रतिचयन आयोजना के प्राचलों को ज्ञात करने के लिए सूत्र प्राप्त कीजिए। 10 अंक (c) (i) n घटकों की एक प्रणाली के लिए जाने पर, उसके अवस्था सदिश तथा संरचना फलन को परिभाषित कीजिए। ये क्या संकेत देते हैं? 5 अंक (ii) (1) एक श्रृंखला प्रणाली, (2) एक समांतर प्रणाली तथा (3) एक n-में-से-k प्रणाली को परिभाषित करते हुए संरचना फलनों और विश्वसनीयता फलनों के लिए संबंधित व्यंजकों को प्राप्त कीजिए। 10 अंक

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Directive word: Derive

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How this answer will be evaluated

Approach

Begin with clear definitions for (a)(i) distinguishing variables/attributes charts, then rigorously derive control limits for x̄ and s charts using sample standard deviation with proper statistical assumptions. For (b), state assumptions of normality and known/unknown variance, outline the operating procedure, then establish the OC function relationship showing how fraction defective links to acceptance probability via non-central t-distribution when σ² is unknown. For (c), define state vector and structure function mathematically, then derive expressions for series, parallel, and k-out-of-n systems using indicator functions and reliability theory. Allocate approximately 35% time to (a)(ii) derivation, 25% to (b)(ii) relationship establishment, 20% to (c)(ii) system derivations, and remaining 20% to definitional parts.

Key points expected

  • (a)(i) Clear distinction: variables charts for measurable characteristics (x̄, R, s charts) vs attributes charts for countable defects (p, np, c, u charts) with examples from Indian manufacturing
  • (a)(ii) Derivation of x̄ chart limits using s/c₄ as σ estimator: UCL/LCL = x̄̄ ± A₃s̄; s chart limits: UCL = B₄s̄, LCL = B₃s̄ with constants derived from χ² distribution
  • (b)(i) Assumptions: normality, single upper/lower specification limit, known or unknown σ; operating procedure: sample selection, computation of sample mean, comparison with acceptance criterion
  • (b)(ii) Relationship: p = P(X > U) = 1 - Φ((U-μ)/σ) for upper specification; acceptance probability Pa = P(accept|p) via non-central t when σ unknown; derivation of n and k parameters via producer/consumer risk points
  • (c)(i) State vector x = (x₁,...,xₙ) where xᵢ ∈ {0,1} indicates component state; structure function φ(x) ∈ {0,1} indicates system state; φ(x) = 1 iff system functions
  • (c)(ii) Series: φ(x) = Πxᵢ, Rₛ(t) = ΠRᵢ(t); Parallel: φ(x) = 1 - Π(1-xᵢ), Rₚ(t) = 1 - Π(1-Rᵢ(t)); k-out-of-n: φ(x) = 1 if Σxᵢ ≥ k, reliability via binomial/Beta or recursive formula

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies all statistical assumptions for each sub-part: normality and independence for control charts and sampling plans; binary state assumption for reliability; proper specification of null and alternative hypotheses for OC curvesStates basic assumptions but misses critical nuances like unbiasedness of s/c₄, distinction between known and unknown variance cases, or independence of component failuresMissing or incorrect assumptions; confuses variables with attributes; fails to specify distributional assumptions needed for derivations
Method choice20%10Selects appropriate derivation methods: uses χ² distribution properties for s chart constants; employs non-central t-distribution for unknown σ case in (b)(ii); uses indicator function algebra and combinatorial arguments for structure functionsUses standard formulas without showing derivation logic; applies correct methods but with gaps in justification; uses central t instead of non-central t where appropriateIncorrect methodological choices such as using population σ formulas for sample s charts, or incorrect combinatorial expressions for k-out-of-n systems
Computation accuracy20%10Flawless derivation of control limit formulas with correct constants (A₃, B₃, B₄ in terms of χ² percentiles); accurate establishment of p-Pa relationship with proper handling of unknown σ via t-distribution; correct reliability function derivations for all three system typesMinor algebraic errors in derivations; incorrect expressions for constants; partially correct handling of unknown variance case; correct final formulas but with gaps in derivationMajor computational errors; incorrect final formulas; confusion between known and unknown variance cases; incorrect structure function or reliability expressions
Interpretation20%10Interprets derived results practically: explains why s charts are preferred over R charts for large n or computer-based SPC; interprets OC curve shape and discriminatory power; explains physical meaning of structure functions and how system reliability exceeds component reliability in parallel systemsLimited interpretation beyond mathematical statements; generic explanations without connecting to quality control practice or system design implicationsNo interpretation provided; purely symbolic manipulation without explaining what results mean for quality assurance or system reliability engineering
Final answer & units20%10Presents all final formulas clearly: control limits in standard form, OC function explicitly stated, sampling plan parameters (n, k) formulas derived, all structure functions and reliability functions boxed and labeled; includes domain restrictions (e.g., k ≤ n)Final answers present but poorly organized; missing some formulas or presented without proper notation; incomplete specification of parameter formulasMissing final answers; incomplete derivations without conclusions; failure to present explicit formulas for sampling plan parameters or reliability functions

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