Statistics 2022 Paper II 50 marks Explain

Q7

(a) What are the various indices of mortality measure? Explain the purpose and procedure for standardizing them. (20 marks) (b) With usual notations, obtain logistic curve as given by P(t) = L / (1 + e^(r(β-t))) ; t > 0, β > 0, r > 0 for population growth model. Also discuss its any three properties. (15 marks) (c) In what way do total fertility rate (TFR), gross reproduction rate (GRR) and net reproduction rate (NRR) differ from one another as a measure of reproduction? (15 marks)

हिंदी में प्रश्न पढ़ें

(a) मृत्यता माप के विभिन्न सूचकांक क्या हैं? उनके मानकीकरण के लिए उद्देश्य एवं कार्यविधि की व्याख्या कीजिए। (20 अंक) (b) साधारण संकेतनों के साथ, जनसंख्या वृद्धि निदर्श के लिए बृद्धियत वक्र प्राप्त कीजिए, जो कि दिया जाता है P(t) = L / (1 + e^(r(β-t))) ; t > 0, β > 0, r > 0 के द्वारा। इसके किन्हीं तीन गुणों का भी वर्णन कीजिए। (15 अंक) (c) किस तरह से संपूर्ण जननक्षमता दर (टी० एफ० आर०), सकल जनन दर (जी० आर० आर०) और नेट जनन दर (एन० आर० आर०), प्रजनन के एक माप के रूप में, एक-दूसरे से भिन्न होते हैं? (15 अंक)

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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

The directive 'explain' demands conceptual clarity with logical exposition. For part (a) carrying 20 marks, allocate ~40% effort covering CDR, ASDR, IMR, MMR with direct and indirect standardization procedures; for (b) with 15 marks, spend ~30% on deriving the logistic curve from differential equation dP/dt = rP(1-P/L) and discussing properties like inflection point, asymptotic behavior, and symmetry; for (c) with 15 marks, devote remaining ~30% to contrasting TFR, GRR, NRR through formulas, assumptions, and replacement-level interpretations. Structure: brief intro → systematic part-wise treatment → integrated conclusion on demographic measurement evolution.

Key points expected

  • Part (a): Lists at least 5 mortality indices (CDR, ASDR, IMR, MMR, U5MR) with formulas; explains purpose of standardization (eliminating age-structure bias for inter-population/temporal comparison); describes direct standardization (applying standard population weights) and indirect standardization (applying standard rates to study population) with step-wise procedure
  • Part (a): Cites Indian context—SRS data, Sample Registration System mortality estimates, or NFHS standardized mortality ratios for interstate comparisons
  • Part (b): Derives logistic curve by solving dP/dt = rP(1-P/L) with initial condition P(0) = P₀; shows integration steps, substitution, and algebraic manipulation to reach given form with β = (1/r)ln[(L-P₀)/P₀]
  • Part (b): Discusses three properties—(i) sigmoid/S-shaped curve with inflection point at P=L/2, t=β; (ii) upper asymptote L (carrying capacity); (iii) growth rate parameter r determining steepness; may add symmetry or point of diminishing returns
  • Part (c): Distinguishes TFR (age-specific fertility rates summed, no mortality adjustment, both sexes), GRR (TFR × proportion female births, no mortality, female births only), NRR (GRR adjusted by survival probabilities lₓ to reproductive age, female generation replacement measure); notes NRR=1 indicates exact replacement, TFR≈2.1 replacement level for India
  • Part (c): Clarifies that TFR is period measure, GRR/NRR are generation measures; NRR most complete for population projection while TFR most commonly reported

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10For (a): Correctly defines all mortality indices with proper numerators/denominators and time references; for (b): Sets up differential equation correctly with all parameters defined; for (c): Accurately states formulas for TFR, GRR, NRR with correct summation limits and survival adjustmentsDefines most indices correctly but confuses numerator-denominator for one index; sets up differential equation with minor notational inconsistency; states TFR and GRR formulas but omits survival function in NRR or uses wrong subscriptsMisdefines key indices (e.g., CDR with wrong denominator); fails to set up differential equation or uses wrong form; confuses GRR with NRR or omits sex-specificity entirely
Method choice20%10For (a): Selects appropriate standardization method based on data availability—direct when age-specific rates known, indirect when only total deaths known; for (b): Chooses separation of variables method correctly with proper partial fractions; for (c): Selects cohort-component logic for reproduction rate hierarchyDescribes both standardization methods but unclear on when to use which; completes derivation with correct method but skips key substitution steps; explains differences descriptively without cohort perspectiveCannot distinguish direct from indirect standardization; attempts wrong derivation method (e.g., treating logistic as exponential); merely lists rates without methodological comparison
Computation accuracy20%10For (b): Executes integration flawlessly—∫dP/[P(L-P)] = (1/L)ln[P/(L-P)]; solves for constant using initial condition; algebraic manipulation to isolate P(t) is error-free with correct final exponent form; for (a) and (c): Any illustrative calculations (e.g., standardized rate computation) are arithmetically correctIntegration steps mostly correct but sign error or constant of integration mishandled; final form achieved but with minor algebraic slip (e.g., wrong sign in exponent); illustrative calculation has arithmetic error but method visibleIntegration fundamentally wrong; cannot solve for integration constant; final expression bears no resemblance to required form; no computational demonstration in (a) or (c)
Interpretation20%10For (a): Explains why crude rates mislead (e.g., Kerala vs. Bihar age structure) and how standardization enables valid policy comparison; for (b): Interprets β as time of maximum growth, r as intrinsic growth rate, L as carrying capacity with population biology relevance; for (c): Explains why NRR<1 implies eventual population decline even if TFR>2, linking to India's demographic transitionStates standardization removes age-structure effect without concrete example; describes curve shape without parameter interpretation; notes NRR accounts for mortality but weak on replacement-level implicationsNo interpretation of why standardization matters; describes logistic as 'S-curve' without explaining parameters; treats three rates as interchangeable without conceptual distinction
Final answer & units20%10For (b): Presents final derived equation exactly as specified with all parameters defined; for (a): Standardized rates expressed per 1000 with clear reference population; for (c): Rates correctly labeled (TFR/GRR/NRR as births per woman, dimensionless or per woman as appropriate); conclusion synthesizes that NRR is gold standard for replacement analysisFinal form correct but parameters poorly defined; units present but inconsistent (per 1000 vs. probability); conclusion merely summarizes without synthesisFinal equation garbled or missing; no units anywhere; abrupt ending without conclusion; fails to address all three sub-parts in final presentation

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