Statistics 2024 Paper II 50 marks Explain

Q7

(a) Explain the problem of identification with a suitable example. Also discuss the conditions of identification. Check the identifiability of each equation of the following structural model : y₁ = 3y₂ – 2x₁ + x₂ + u₁ y₂ = y₃ + x₂ + u₂ y₃ = y₁ – y₂ – 2x₃ + u₃ 15 (b) Explain why mortality situations at two places cannot be compared on the basis of crude death rates. Describe the construction of standardised death rates for this purpose. What is a comparative mortality index and how is it used ? 20 (c) (i) Define reliability of a test. What is the effect of test length on the reliability of a test ? 5 (ii) Give different methods for estimating the reliability of a psychological test. 10

हिंदी में प्रश्न पढ़ें

(a) एक उपयुक्त उदाहरण के साथ अभिनिर्धारण की समस्या को समझाइए। अभिनिर्धारण की शर्तों की भी चर्चा कीजिए । निम्नलिखित संरचनात्मक मॉडल के प्रत्येक समीकरण की अभिज्ञेयता (अभिनिर्धारणीयता) की जाँच कीजिए : y₁ = 3y₂ – 2x₁ + x₂ + u₁ y₂ = y₃ + x₂ + u₂ y₃ = y₁ – y₂ – 2x₃ + u₃ 15 (b) स्पष्ट कीजिए कि दो स्थानों पर मृत्यु दर की स्थिति की तुलना अशोधित मृत्यु दरों के आधार पर क्यों नहीं की जा सकती । इस उद्देश्य के लिए, मानकीकृत मृत्यु दरों के निर्माण का वर्णन कीजिए । तुलनात्मक मृत्यु दर सूचकांक क्या है और इसका उपयोग कैसे किया जाता है ? 20 (c) (i) एक परीक्षण की विश्वसनीयता को परिभाषित कीजिए । एक परीक्षण की विश्वसनीयता पर परीक्षण की लंबाई का क्या प्रभाव होता है ? 5 (ii) एक मनोवैज्ञानिक परीक्षण की विश्वसनीयता के आकलन के लिए विभिन्न विधियों को बताइए । 10

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Directive word: Explain

This question asks you to explain. 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

Explain the identification problem with a concrete supply-demand example, then apply order and rank conditions to the given 3-equation system. For (b), explain why CDR fails using Indian state examples (e.g., Kerala vs. Uttar Pradesh age structures), then detail direct and indirect standardization methods. For (c), define reliability via Spearman-Brown prophecy, discuss test length effects, and enumerate split-half, KR-20, KR-21, and Cronbach's alpha methods. Allocate approximately 30% time to (a), 40% to (b), 20% to (c)(i), and 10% to (c)(ii) based on marks distribution.

Key points expected

  • (a) Clear explanation of identification problem using supply-demand simultaneous equations example; distinction between under-identified, just-identified, and over-identified equations
  • (a) Correct application of order condition (K - k ≥ m - 1) and rank condition to all three equations; accurate classification of each equation's identifiability status
  • (b) Explanation of CDR limitations due to varying age-sex compositions; use of Indian demographic examples (e.g., aging Kerala vs. younger Bihar populations)
  • (b) Construction of standardized death rates: direct method (standard population weights) and indirect method (standard mortality rates); formula for Comparative Mortality Index (CMI) and its interpretation
  • (c)(i) Formal definition of reliability as ratio of true score variance to observed score variance; statement and explanation of Spearman-Brown prophecy formula for test length effects
  • (c)(ii) Comprehensive coverage of reliability estimation methods: test-retest, parallel forms, split-half (including Spearman-Brown correction), Kuder-Richardson formulas (KR-20, KR-21), and Cronbach's coefficient alpha with appropriate formulas

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies endogenous variables (y₁, y₂, y₃) and exogenous variables (x₁, x₂, x₃) in the structural model; properly sets up identification conditions with clear matrix representation; correctly identifies the standard population concept for mortality standardization; provides psychometrically precise definition of reliabilityIdentifies most variables correctly but may confuse endogenous/exogenous classification; attempts order condition but with notation errors; describes standardization conceptually without clear population/rate distinction; defines reliability in loose intuitive termsFundamental errors in variable classification; fails to distinguish structural vs. reduced forms; confuses direct and indirect standardization purposes; provides circular or incorrect reliability definition
Method choice20%10Selects appropriate identification criteria (order and rank conditions) systematically; chooses suitable standardization method (direct/indirect) with justification for mortality comparison; selects appropriate reliability coefficients based on test properties (dichotomous vs. continuous items)Applies order condition correctly but may skip rank condition verification; describes one standardization method adequately but omits justification for method selection; lists reliability methods without matching to test characteristicsApplies incorrect identification rules (e.g., using OLS assumptions); confuses direct and indirect standardization formulas; selects inappropriate reliability methods (e.g., KR-20 for non-dichotomous items without justification)
Computation accuracy20%10Accurate count of excluded variables (K - k) and endogenous variables (m - 1) for each equation; correct determinant calculations for rank condition; precise CMI formula application; correct Spearman-Brown formula manipulation for reliability estimationCorrect arithmetic for order condition counts but possible errors in rank condition matrix setup; computational errors in CMI or standardization weights; correct formula statement but arithmetic errors in reliability calculationsMajor computational errors in identification counts; incorrect matrix rank determination; formula errors in standardization (e.g., reversing numerator/denominator); fundamental errors in reliability formulas
Interpretation20%10Interprets identification results to explain which equations can be estimated (just-identified vs. over-identified); explains why CDR comparisons mislead using concrete Indian demographic scenarios; interprets CMI values meaningfully; explains practical implications of reliability coefficients and test length decisionsStates identification conclusions without explaining estimation implications; notes CDR limitations without demographic context; states CMI formula without interpretation; mentions reliability improves with length without explaining diminishing returnsMisinterprets identification status (e.g., claiming under-identified equations estimable); fails to explain why standardization is necessary; provides no interpretation of CMI or reliability values; draws incorrect practical conclusions
Final answer & units20%10Clear final classification of all three equations (under-identified/just-identified/over-identified); explicit CMI value with correct interpretation; precise reliability coefficient values with confidence interpretation; all conclusions explicitly linked to original question requirementsPartial classification of equations with some correct conclusions; CMI mentioned without final computed value; reliability methods listed without final coefficient interpretation; conclusions present but not explicitly tied to questionMissing final identifiability verdict for equations; no CMI calculation or interpretation; no reliability coefficient computation or conclusion; fails to address all sub-parts in final answer

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