Statistics 2024 Paper II 50 marks Solve

Q8

(a) Four items are to be constructed so that they are equispaced on the difficulty scale. If the easiest item is passed by 85% of the group and the most difficult by 25%, find the percentage of individuals in the group passing the other two items. (Standard Normal tables are provided) 15 (b) Define Crude Birth Rate, General Fertility Rate and Age-specific Fertility Rate and indicate why each is considered an improvement on the preceding measure of fertility. Define Total Fertility Rate and its utility. 15 (c) (i) Discuss the problem of autocorrelation. What are the consequences of OLS estimators in estimating the parameters in the presence of autocorrelation ? (ii) Explain the Durbin-Watson test for testing the autocorrelation. 10+10=20

हिंदी में प्रश्न पढ़ें

(a) चार मदों का निर्माण किया जाना है ताकि वे कठिनाई पैमाने पर समान दूरी पर हों । यदि सबसे सरल मद समूह के 85% द्वारा पास किया जाता है तथा सबसे कठिन 25% द्वारा, तो समूह में अन्य दो मदों को पास करने वाले व्यक्तियों का प्रतिशत ज्ञात कीजिए । (मानक प्रसामान्य सारणी दी गई है) 15 (b) अशोधित जन्म दर, सामान्य प्रजनन दर और वय-विशिष्ट प्रजनन दर को परिभाषित कीजिए तथा बताइए कि प्रत्येक को प्रजनन के पूर्ववर्ती माप पर सुधार क्यों माना जाता है। संपूर्ण प्रजनन दर को परिभाषित कीजिए तथा इसकी उपयोगिता बताइए। 15 (c) (i) स्वसहसंबंध की समस्या की चर्चा कीजिए। स्वसहसंबंध की उपस्थिति में प्राचलों के आकलन में ओ एल एस आकलकों के क्या परिणाम होते हैं ? (ii) स्वसहसंबंध के परीक्षण के लिए डर्बिन-वॉटसन परीक्षण की व्याख्या कीजिए। 10+10=20

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How this answer will be evaluated

Approach

Begin with part (a) by converting percentages to z-scores using standard normal tables, then apply linear interpolation for equispaced difficulty; allocate ~30% time here. For part (b), define each fertility measure sequentially showing progressive refinement from crude to age-specific rates, then explain TFR's utility for population projection—spend ~30% time. For part (c), discuss autocorrelation consequences on OLS properties (BLUE violation), then detail Durbin-Watson test procedure with critical values—allocate ~40% time as this carries highest marks. Conclude with integrated insights on statistical applications in demographic and econometric analysis.

Key points expected

  • Part (a): Convert 85% and 25% to z-scores (-1.036 and 0.674), establish equidistant points on difficulty scale, calculate intermediate z-values, convert back to percentages (~58% and ~42%)
  • Part (b): CBR definition and limitation (ignores age-sex structure); GFR improvement (restricts to women 15-49); ASFR refinement (age-specific exposure); TFR as sum of ASFRs and utility for replacement-level fertility analysis (India's TFR ~2.0)
  • Part (c)(i): Autocorrelation causes (inertia, specification error, cobweb, data manipulation); consequences: OLS estimators remain unbiased but inefficient, standard errors biased, t/F tests invalid, R² misleading
  • Part (c)(ii): DW test assumptions (no lagged dependent variable, intercept, non-stochastic regressors); test statistic formula; decision zones (0 to 4 scale); inconclusive region problem; alternative tests (Durbin's h, Breusch-Godfrey)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Setup correctness20%10Correctly identifies z-scores for 85% and 25% in (a); properly sequences fertility measures showing logical progression in (b); accurately states autocorrelation assumptions and OLS framework in (c)Minor errors in z-score lookup or fertility measure sequencing; incomplete statement of assumptions for DW testWrong z-score identification, confused fertility measure hierarchy, or fundamental misunderstanding of OLS assumptions
Method choice20%10Uses linear interpolation on z-scale for equispaced items in (a); employs precise demographic formulas for fertility rates in (b); selects appropriate DW test over alternatives where suitable in (c)Correct method but imprecise execution; acceptable fertility formulas but weak comparative justification; standard DW presentation without nuanceArithmetic mean on percentages instead of z-scores in (a); incorrect rate formulas in (b); wrong test selection or procedure in (c)
Computation accuracy20%10Precise z-scores (-1.036, 0.674), correct intermediate calculations yielding ~58% and ~42%, accurate TFR summation formula, correct DW statistic boundsMinor rounding errors in z-scores or final percentages; acceptable but imprecise numerical presentationSignificant computational errors, wrong percentage conversions, or incorrect DW critical value references
Interpretation20%10Explains why z-scale interpolation preserves normality assumption in (a); clearly demonstrates how each fertility measure reduces denominator bias in (b); insightfully discusses BLUE violation consequences and DW test limitations in (c)Adequate interpretation without depth; standard explanations of measure improvements; basic consequence listingMissing interpretation of equispacing concept, no explanation of why measures improve, or failure to discuss DW inconclusive region
Final answer & units20%10Clear final percentages for two items in (a); complete fertility rate definitions with units (per 1000 women); explicit DW decision rule with conclusion; integrates Indian demographic context (e.g., NFHS-5 TFR data)Present but disorganized final answers; correct units but incomplete contextual applicationMissing final answers, wrong units (e.g., percentages for rates), or no conclusion on hypothesis tests

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