Statistics 2022 Paper II 50 marks Compulsory Calculate

Q5

(a) Apply the method of link relatives to the following data and calculate the seasonal indices : Price of Rice (in ₹ per 10 kg) | Quarter | 2001 | 2002 | 2003 | 2004 | |---------|------|------|------|------| | 1 | 75 | 86 | 90 | 100 | | 2 | 60 | 65 | 72 | 78 | | 3 | 54 | 63 | 66 | 72 | | 4 | 59 | 80 | 82 | 93 | 10 marks (b) Derive the means and variances of the sampling distributions of the OLS estimates of α and β in the two-variable linear model Y = α + βX + u. 10 marks (c) Consider, in the usual notations, the equation y = Y₁β + X₁γ + u, where y is an (n × 1) vector, Y₁ is an (n × (g-1)) matrix, X₁ is an (n × k) matrix. Derive the equations for the two-stage least square method of estimation. 10 marks (d) If the survivorship function l(x) in life table is linear between x and x+1, and complete expectations of life at ages 40 and 41 for a particular group of persons are 21·39 years and 20·91 years respectively and l(40) = 41176, find the number of persons that attain the age 41. 10 marks (e) Compute the T-scores corresponding to test score x for the following frequency distribution : | x | 1 | 2 | 3 | 4 | 5 | |-----|---|---|---|---|---| | f | 2 | 3 | 8 | 6 | 1 | (Cumulative Normal Distribution Table is given in Page No. 9) 10 marks

हिंदी में प्रश्न पढ़ें

(a) निम्नलिखित आँकड़ों पर शृंखलित आपेक्षिक विधि का प्रयोग कीजिए और ऋतुनिष्ठ सूचकांकों की गणना कीजिए : चावल का मूल्य (₹ में प्रति 10 किलोग्राम) | क्वार्टर | 2001 | 2002 | 2003 | 2004 | |---------|------|------|------|------| | 1 | 75 | 86 | 90 | 100 | | 2 | 60 | 65 | 72 | 78 | | 3 | 54 | 63 | 66 | 72 | | 4 | 59 | 80 | 82 | 93 | 10 (b) द्विचर रैखिक निदर्श Y = α + βX + u में α और β के ओ. एल. एस. आकलनों के प्रतिचयन बंटनों के मध्यों और प्रसरणों को व्युत्पन्न कीजिए। 10 (c) प्रचलित संकेतों में, समीकरण y = Y₁β + X₁γ + u पर विचार कीजिए, जहाँ y एक (n × 1) सदिश है, Y₁ एक (n × (g-1)) आव्यूह है, X₁ एक (n × k) आव्यूह है। आकलन की द्विचरण न्यूनतम वर्ग विधि के लिए समीकरणों को व्युत्पन्न कीजिए। 10 (d) यदि वय सारणी में उत्तरजीविता फलन l(x), x और x+1 के बीच रैखिक है तथा किसी विशेष व्यक्तियों के समूह के लिए आयु 40 और 41 पर जीवन की पूर्ण प्रत्याशा क्रमशः 21·39 वर्ष और 20·91 वर्ष हैं, और l(40) = 41176, तब आयु 41 तक पहुँचने वाले व्यक्तियों की संख्या ज्ञात कीजिए। 10 (e) निम्नलिखित बारंबारता बंटन के लिए परीक्षण स्कोर x के संगत (T-समांकी) का परिकलन कीजिए : | x | 1 | 2 | 3 | 4 | 5 | |-----|---|---|---|---|---| | f | 2 | 3 | 8 | 6 | 1 | (संचयी प्रसामान्य बंटन सारणी पृष्ठ सं० 9 में दी गई है) 10

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Directive word: Calculate

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Approach

This multi-part question requires precise calculation and derivation across five distinct statistical domains. Allocate approximately 20% time to each sub-part: (a) construct link relatives table and seasonal indices using chain relatives method; (b) derive OLS estimators' properties using Gauss-Markov assumptions; (c) set up 2SLS normal equations showing projection onto instruments; (d) apply linear survivorship assumption to solve for l(41); (e) compute percentile ranks then transform to T-scores using given normal table. Present each part clearly with step-by-step working.

Key points expected

  • For (a): Calculate chain relatives by expressing each quarter's value as percentage of preceding quarter, then obtain corrected relatives and seasonal indices normalized to 400
  • For (b): Derive E(α̂) = α and E(β̂) = β showing unbiasedness, then derive Var(α̂) = σ²ΣX²/(nΣx²) and Var(β̂) = σ²/Σx² using matrix or scalar algebra
  • For (c): State first stage projection Ŷ₁ = X(X'X)⁻¹X'Y₁, then second stage OLS of y on Ŷ₁ and X₁ to obtain 2SLS estimator β̂₂ₛₗₛ = (Z'PₓZ)⁻¹Z'Pₓy where Z = [Y₁|X₁]
  • For (d): Use linearity of l(x) to establish e°₄₀ = ½ + l(41)/l(40) × e°₄₁, then solve l(41) = l(40)(e°₄₀ - ½)/(e°₄₁ + ½) and compute numerical value
  • For (e): Compute cumulative frequencies, percentile ranks P = (100/N)(C - ½), find corresponding z-scores from normal table, then T = 50 + 10z for each score value

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly sets up all five sub-parts: proper link relatives table for (a), correct model specification with assumptions for (b), proper matrix dimensions and projection matrices for (c), correct linear interpolation formula for life table for (d), accurate cumulative frequency and percentile rank calculations for (e)Sets up most parts correctly with minor errors in one sub-part such as wrong matrix dimensions in (c) or incorrect percentile formula in (e)Major setup errors in multiple sub-parts, confuses methods across topics, or omits necessary initial calculations
Method choice20%10Selects appropriate methods throughout: link relatives method with correction for (a), classical linear regression theory for (b), instrumental variables/2SLS derivation for (c), linear survivorship assumption for (d), and normal deviate transformation for (e)Correct methods for most parts but uses simple average instead of link relatives for (a) or omits 2SLS first stage explicitly in (c)Wrong methods chosen such as ratio-to-trend for (a), simple correlation for (b), or direct OLS for (c) ignoring simultaneity
Computation accuracy20%10All calculations accurate: chain relatives, correction factors, seasonal indices summing to 400 for (a); correct algebraic derivation of variances for (b); correct matrix operations for 2SLS for (c); precise calculation yielding l(41) ≈ 40882 for (d); correct T-scores (e.g., T≈37, 43, 50, 57, 63) for (e)Minor computational slips in one sub-part such as arithmetic errors in seasonal indices or slight deviation in T-score valuesSignificant computational errors across multiple parts, wrong final values, or failure to complete calculations
Interpretation20%10Interprets seasonal pattern showing Q2-Q3 price dips for (a), explains BLUE properties and efficiency for (b), discusses identification and instrument relevance for (c), interprets mortality improvement for (d), and explains standardization meaning of T-scores for (e)Basic interpretation for some parts but misses economic significance of seasonal indices or statistical meaning of 2SLS consistencyNo interpretation provided, merely presents numbers without explaining what seasonal indices or T-scores represent
Final answer & units20%10All final answers clearly stated with appropriate units: seasonal indices as percentages for (a), variance expressions in terms of σ² for (b), explicit 2SLS estimator formula for (c), l(41) as count of persons for (d), T-scores with mean 50 SD 10 for (e)Most answers present but units missing or inconsistent (e.g., not specifying indices are percentages)Final answers incomplete, missing, or with wrong units; seasonal indices not normalized, T-scores without reference to mean 50

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