(a) On the basis of the figures given below, calculate the age-specific death rates (ASDRs) for all the age groups. Also, calculate the crude death rate (CDR) on the basis of ASDRs:

| Age group (in years) | 0-10 | 10-30 | 30-50 | 50-70 | 70 and abov

Question

(a) On the basis of the figures given below, calculate the age-specific death rates (ASDRs) for all the age groups. Also, calculate the crude death rate (CDR) on the basis of ASDRs:

| Age group (in years) | 0-10 | 10-30 | 30-50 | 50-70 | 70 and above |
|---|---|---|---|---|---|
| Population | 10000 | 18000 | 26000 | 20000 | 5000 |
| Number of deaths | 220 | 40 | 62 | 350 | 2000 |

It was later discovered that two individuals, aged 47 and 54, were incorrectly recorded as being 37 and 45, while compiling the above table. Recalculate the ASDRs and CDR based on the corrected age data. (All calculations are up to 3 decimals only.) 15 marks

(b) Discuss the problem of identification with an example. State the rank and order conditions of identification. Check the identifiability of the following structural model:

y₁ = α₁ + β₁₂y₂ + β₁₃y₃ + γ₁₁x₁ + γ₁₂x₂ + u₁

y₂ = α₂ + β₂₃y₃ + γ₂₁x₁ + γ₂₂x₂ + u₂

y₃ = α₃ + β₃₁y₁ + γ₃₁x₁ + γ₃₂x₂ + u₃

y₄ = β₄₁y₁ + β₄₂y₂ + β₄₃x₃ + u₄ 15 marks

(c) Prepare a life table for an age group from age 50 to age 60 of a specific population. Assume that there are 10000 persons living at age 50 and the probability of death within age x to x+1 is given as qₓ = 0·001+0·0002x for x = 50, 51, ..., 60. Prepare the life table with columns x, lₓ, qₓ, dₓ and Lₓ for x = 50, 51, 52, ..., 60. 20 marks

UPSC Answer Check · Accepted Answer

Begin with precise calculations for part (a), allocating approximately 30% of time to compute original and corrected ASDRs/CDR with proper data reallocation. Devote 30% to part (b) discussing identification using a concrete econometric example (e.g., supply-demand model), stating order and rank conditions clearly, then systematically checking each equation's identifiability. Reserve 40% for part (c) constructing the complete life table with all five columns, showing iterative calculations for lₓ, dₓ and Lₓ. Present each part separately with clear headings and maintain 3-decimal precision throughout.
- Part (a): Calculate original ASDRs (deaths/population × 1000) for all five age groups and CDR (total deaths/total population × 1000); then recalculate after correctly reassigning deaths from 30-50 to 50-70 age group (47-year-old to 50-70, 54-year-old from 30-50 to 50-70)
- Part (b): Explain identification problem using simultaneous equations bias example (e.g., price-quantity in agricultural markets); state order condition (K ≥ k-1) and rank condition (at least one non-zero determinant of order M-1)
- Part (b): Apply order and rank conditions to check identifiability of all four structural equations, noting M=4 endogenous, K=3 exogenous variables; identify y₁ as overidentified, y₂ and y₃ as unidentified, y₄ as exactly identified
- Part (c): Construct life table using l₅₀=10000, computing qₓ=0.001+0.0002x for x=50 to 60, then dₓ=lₓ×qₓ, lₓ₊₁=lₓ-dₓ, and Lₓ=(lₓ+lₓ₊₁)/2 for each age
- Part (c): Present final life table with all five columns (x, lₓ, qₓ, dₓ, Lₓ) showing declining survivorship pattern typical of Indian mortality experience in 50-60 age range

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	15%	7.5	Correctly identifies data reallocation in (a): moves one death from 30-50 to 50-70; properly counts M=4 endogenous and K=3 exogenous variables in (b); correctly initializes l₅₀=10000 and formulates qₓ formula in (c)	Minor errors in variable counting or initial setup; partially correct data reallocation in (a) or incomplete identification of variables in (b)	Fundamental setup errors: wrong age group reallocation, incorrect M/K counts, wrong initial value or formula for qₓ
Method choice	25%	12.5	Uses ASDR = (Dₓ/Pₓ)×1000 and CDR = ΣD/ΣP×1000 for (a); applies proper order/rank condition matrix algebra for (b); employs standard life table formulas with exact Lₓ calculation for (c)	Correct formulas but inconsistent application; partial matrix construction for identification; approximate Lₓ methods	Wrong formulas (e.g., crude rates instead of specific rates), no matrix approach for identification, incorrect life table relationships
Computation accuracy	30%	15	All calculations to 3 decimal places: original ASDRs (22.000, 2.222, 2.385, 17.500, 400.000), corrected ASDRs after reallocation, correct identification status for all four equations, accurate life table values through iterative computation	Most calculations correct but rounding errors beyond 3 decimals; minor arithmetic errors in life table iterations; one equation misidentified	Systematic calculation errors, wrong decimal precision, major errors in iterative life table construction, multiple identification errors
Interpretation	20%	10	Explains why corrected ASDRs change (concentration of deaths in older ages), interprets economic meaning of identification failure (simultaneity bias), discusses mortality pattern in 50-60 age group (increasing qₓ with age)	Brief interpretation without depth; mechanical statement of results without explanation of why values change or what identification means	No interpretation of results; fails to explain significance of corrections or identification outcomes
Final answer & units	10%	5	Clear presentation: tabulated ASDRs with 'per 1000' units, explicit identification status for each equation, complete life table with all five columns properly labeled; 3-decimal precision maintained throughout	Present but poorly organized; missing some units or labels; inconsistent decimal precision	Incomplete final answers, missing tables, no units specified, chaotic presentation preventing verification

Q6

Directive word: Calculate

How this answer will be evaluated

Approach

Key points expected

Evaluation rubric

Practice this exact question

More from Statistics 2025 Paper II