Q7
(a) If c(x, t) denote observed proportion of females in the age group (x, x+t) and f(x, t) is the observed proportion of females giving birth to female children in the age group (x, x+t) at time t. Let us assume that X is uniformly distributed in (α, β). Then show that $$ \hat{B}_f(t)=\left[\hat{r}_{c,f|t} \hat{\sigma}_c \hat{\sigma}_f (\beta-\alpha) + \frac{[\hat{T}_f(t)]^2}{(\beta-\alpha)} \frac{1}{\hat{G}_f(t)}\right], $$ where $\hat{T}_f(t)$ is the estimated total fertility. $\hat{B}_f(t)$ is the estimated female birthrate at time t. $\hat{G}_f(t)$ is the estimated General Fertility rate. $\hat{r}_{c,f|t}$ represents product moment correlation coefficient between c and f given t. $\hat{\sigma}_c, \hat{\sigma}_f$ are observed standard deviations of c and f respectively. (15 marks) (b) What do you mean by Intelligence Quotient (I.Q.) ? Describe the procedure and test of measuring I.Q. How does an aptitude test differ from an Intelligence Test ? The reliability coefficient of a test of 60 items is 0·65. How much the test should be lengthened to raise the self correlation to 0·95 ? What effect will the doubling and tripling the test's length have upon the reliability coefficients ? What is the reliability of a test having 135 comparable items ? (15 marks) (c) Define instantaneous force of mortality (μₓ). Show that qₓ = (1/lₓ) ∫₀¹ μₓ₊ₜ lₓ₊ₜ dx where qₓ is the probability of dying within one year following the attainment of age x. Also prove that μₓ = (1/eₓ⁰) [1 + (deₓ⁰/dx)] where eₓ⁰ is the complete expectation of life. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) यदि c(x, t) आयु वर्ग (x, x+t) में महिलाओं का प्रेक्षित अनुपात है और f(x, t) आयु वर्ग (x, x+t) में उन महिलाओं का प्रेक्षित अनुपात है जो महिला बच्चों को जन्म देती हैं, समय t पर। हम यह मान लेते हैं कि X, (α, β) में एकसमान बंटित है। तब दर्शाइए कि $$ \hat{B}_f(t)=\left[\hat{r}_{c,f|t} \hat{\sigma}_c \hat{\sigma}_f (\beta-\alpha) + \frac{[\hat{T}_f(t)]^2}{(\beta-\alpha)} \frac{1}{\hat{G}_f(t)}\right], $$ जहाँ $\hat{T}_f(t)$ आकलित कुल उर्वरता है। $\hat{B}_f(t)$ समय t पर आकलित महिला जन्म दर है। $\hat{G}_f(t)$ आकलित सामान्य प्रजनन दर है। $\hat{r}_{c,f|t}$ निरूपित करता है c और f के बीच गुणन-आश्रित संबंध गुणांक को जब कि t दिया हुआ है। $\hat{\sigma}_c, \hat{\sigma}_f$ क्रमशः: c और f के प्रेक्षित मानक विचलन हैं। (15 अंक) (b) बौद्धिक स्तर (आई. क्यू.) से आप क्या समझते हैं ? आई. क्यू. को मापने की विधि और परीक्षण का वर्णन कीजिए। एक उपयुक्ता (एप्टीट्यूड) परीक्षण, एक बौद्धिक परीक्षण से किस प्रकार भिन्न है ? 60 मदों के एक परीक्षण का विश्वसनीयता गुणांक 0.65 है। परीक्षण को कितना लम्बा किया जाना चाहिए ताकि स्व-सहसंबंध (सेल्फ कोरिलेशन) बढ़ कर 0.95 हो जाए ? परीक्षण की लम्बाई को दो गुना और तीन गुना करने पर विश्वसनीयता गुणांकों पर क्या प्रभाव पड़ेगा ? 135 तुलनीय मदों वाले एक परीक्षण की विश्वसनीयता क्या है ? (15 अंक) (c) तत्क्षण मरता की तीव्रता (μₓ) को परिभाषित कीजिए । दर्शाइए कि qₓ = (1/lₓ) ∫₀¹ μₓ₊ₜ lₓ₊ₜ dx जहाँ qₓ आयु x प्राप्त करने के उपरान्त एक वर्ष के भीतर मरने की प्रायिकता है । यह भी सिद्ध कीजिए कि μₓ = (1/eₓ⁰) [1 + (deₓ⁰/dx)] जहाँ eₓ⁰ जीवन की पूर्ण प्रत्याशा है । (20 अंक)
Directive word: Derive
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How this answer will be evaluated
Approach
The directive 'derive' demands rigorous mathematical proofs and derivations. Allocate approximately 30% time to part (a) on female birthrate estimation using correlation structure, 30% to part (b) covering IQ definition, measurement procedures, aptitude-intelligence distinction, and reliability calculations using Spearman-Brown prophecy, and 40% to part (c) on force of mortality derivations and life table relationships. Structure with clear section headings, state assumptions explicitly, show step-by-step derivations, and conclude with precise final expressions.
Key points expected
- Part (a): Derivation of female birthrate formula using uniform distribution assumption, correlation structure between c(x,t) and f(x,t), and proper substitution of T_f(t) and G_f(t) with algebraic manipulation of (β-α) terms
- Part (b): Precise definition of IQ (Mental Age/Chronological Age × 100 or deviation IQ), Stanford-Binet and Wechsler procedures, distinction between aptitude (specific potential) and intelligence (general ability) tests
- Part (b): Application of Spearman-Brown prophecy formula n = r₂(1-r₁)/r₁(1-r₂) to find required test length for reliability 0.95, and calculation of new reliabilities for doubled/tripled lengths and 135 items
- Part (c): Definition of μₓ as instantaneous death rate and derivation of qₓ = (1/lₓ)∫₀¹ μₓ₊ₜ lₓ₊ₜ dt using relationship between force of mortality and survival function
- Part (c): Proof of μₓ = (1/eₓ⁰)[1 + (deₓ⁰/dx)] using complete expectation of life definition eₓ⁰ = Tₓ/lₓ and differentiation with respect to age
- Correct handling of Indian demographic context: mention of SRS (Sample Registration System) data for fertility estimation and applicability to Indian population studies
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies uniform distribution of X in (α,β), defines all terms (c, f, T_f, G_f) with demographic meaning; For (b): accurately defines IQ variants (ratio vs deviation), states Spearman-Brown assumptions; For (c): properly defines μₓ as limit of deaths/exposure, sets correct integral bounds and survival relationships | Basic definitions present but some confusion between T_f(t) and G_f(t) in (a), mixes MA/CA formula with deviation IQ in (b), or has incorrect integral limits in (c) | Missing key definitions, confuses fertility rates with mortality rates, applies wrong reliability formula, or fundamental misunderstanding of force of mortality concept |
| Method choice | 20% | 10 | For (a): uses covariance/correlation structure and algebraic expansion of product terms; For (b): selects Spearman-Brown prophecy formula appropriately, distinguishes group vs individual tests; For (c): applies integration by parts or direct survival analysis, uses Tₓ = ∫ₓ^∞ lᵧ dy relationship correctly | Correct general approach but skips crucial steps like justifying uniform distribution simplification, or uses crude approximation for reliability extension without showing formula derivation | Wrong methodological approach such as using Cronbach's alpha instead of Spearman-Brown, or attempting discrete approximations where continuous derivations are required |
| Computation accuracy | 20% | 10 | For (a): precise algebraic manipulation showing r_c,f σ_c σ_f (β-α) + [T_f]²/[(β-α)G_f] structure; For (b): correct calculation n = 0.95×0.35/(0.65×0.05) = 10.23, so extend to ~614 items; doubled reliability = 0.79, tripled = 0.86, 135 items gives r = 0.90; For (c): exact integration yielding qₓ = 1 - pₓ and correct differentiation of eₓ⁰ | Correct final answers but arithmetic errors in intermediate steps, or correct formula application with calculation mistakes in reliability extension factors | Major computational errors such as wrong reliability formula application (r₂ = nr₁), incorrect integration limits giving wrong probability expressions, or algebraic mistakes in covariance expansion |
| Interpretation | 20% | 10 | For (a): explains demographic significance of correlation term capturing age-fertility interaction; For (b): interprets reliability 0.95 as high precision for individual decisions, discusses cultural bias in Indian IQ testing context; For (c): interprets μₓ as age-specific death intensity, explains why eₓ⁰ derivative appears in relationship | Limited interpretation beyond mathematical results, or generic statements about reliability importance without specific demographic/psychometric context | No interpretation of results, fails to explain what derived formulas represent in practical demographic or psychological measurement terms |
| Final answer & units | 20% | 10 | All three parts: boxed/clearly stated final expressions—(a) complete B̂_f(t) formula with all terms defined; (b) specific items needed (10.23× or ~614 items), reliabilities 0.79, 0.86, 0.90; (c) both proved identities with proper notation; dimensional consistency checked (rates per year, dimensionless probabilities) | Final answers present but poorly organized, missing units for rates, or incomplete presentation of one sub-part's conclusion | Missing final answers, wrong units (e.g., giving reliability >1 or <0), or fundamental notation errors like confusing lₓ with Lₓ in life table formulas |
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