Q7
(a) Consider the following data given for a BIBD with v = b = 4, r = k = 3, λ = 2 and N = 12 : Analyse the design. [Given that : F₃,₅ (0·05) = 5·41] 15 (b) (i) The data matrix of a random sample of size n = 3 from a bivariate normal population BVN (μ₁, μ₂, σ₁², σ₂², ρ) is X = [6 10; 10 6; 8 2]. Test the null hypothesis H₀ : μ = μ₀ against H₁ : μ ≠ μ₀, where μ₀' = (8, 5), at 10% level of significance. [You are given : F₀.₁₀; ₂, ₁ = 49·5, F₀.₁₀; ₁, ₂ = 8·53] (ii) Suppose n₁ = 11 and n₂ = 12, observations are made on two random vectors X₁ and X₂ which are assumed to have bivariate normal distribution with a common covariance matrix Σ, but possibly different mean vectors μ₁ and μ₂. The sample mean vectors and pooled covariance matrix are X̄₁ = (-1, -1)', X̄₂ = (2, 1)', S_pooled = (7 -1; -1 5). Obtain Mahalanobis sample distance D² and Fisher's linear discriminant function. Assign the observation X₀ = (0, 1)' to either population Π₁ or Π₂. 10+10=20 (c) A sample of size n is drawn with equal probability and without replacement from a population with size N. Let Ŷ_N = Σᵣ₌₁ⁿ aᵣ yᵣ be any linear estimate of the population mean Ȳ_N, where aᵣ are constants and yᵣ denotes the value of the unit included in the sample at the rᵗʰ draw. (i) Show that Ŷ_N is an unbiased estimate of Ȳ_N if and only if Σᵣ₌₁ⁿ aᵣ = 1 (ii) Under above condition V(Ŷ_N) = (S²/N)[NΣᵣ₌₁ⁿ aᵣ² - 1] (iii) If aᵣ = 1/n, for what value of n may this variance of the sample mean in simple random sampling without replacement be exactly half the variance of the mean of a random sample of the same size taken with replacement ? 15
हिंदी में प्रश्न पढ़ें
(a) किसी बी.आई.बी.डी. (BIBD), जहाँ v = b = 4, r = k = 3, λ = 2 और N = 12, के लिए दिए गए निम्नलिखित आँकड़ों पर विचार कीजिए : अभिकल्पना का विश्लेषण कीजिए । [दिया गया है : F₃,₅ (0·05) = 5·41] 15 (b) (i) एक द्विचर प्रसामान्य समष्टि BVN (μ₁, μ₂, σ₁², σ₂², ρ) से लिए गए आमाप n = 3 के एक यादृच्छिक प्रतिदर्श का न्यास मैट्रिक्स X = [6 10; 10 6; 8 2] है। वैकल्पिक परिकल्पना H₁ : μ ≠ μ₀ के विरुद्ध निराकरणीय परिकल्पना H₀ : μ = μ₀, का परीक्षण 10% सार्थकता-स्तर पर कीजिए, जहाँ μ₀' = (8, 5) है। [आपको दिया गया है : F₀.₁₀; ₂, ₁ = 49·5, F₀.₁₀; ₁, ₂ = 8·53] (ii) मान लीजिए कि दो यादृच्छिक सदिशों X₁ और X₂, जो एक समान सहप्रसरण आव्यूह Σ, किन्तु सम्भवतः भिन्न माध्य सदिशों μ₁ और μ₂ के साथ द्विचर प्रसामान्य बंटन का अनुसरण करते माने जाते हैं, पर n₁ = 11 और n₂ = 12 प्रेक्षण बनाए जाते हैं। प्रतिदर्श माध्य सदिश और संयुक्त सहप्रसरण आव्यूह हैं : X̄₁ = (-1, -1)', X̄₂ = (2, 1)', Sसंयुक्त = (7 -1; -1 5)। महालनोबिस प्रतिदर्श दूरी D² और फिशर के रैखिक विभिक्तकर फलन को प्राप्त कीजिए। प्रेक्षण X₀ = (0, 1)' को या तो समष्टि Π₁ या Π₂ को निर्दिष्ट कीजिए। 10+10=20 (c) N आकार की समष्टि से n आकार का एक प्रतिदर्श समान प्रायिकता एवं प्रतिस्थापन रहित के साथ चुना गया । मान लीजिए कि Ŷ_N = Σᵣ₌₁ⁿ aᵣ yᵣ समष्टि माध्य Ȳ_N का कोई रैखिक आकल है, जहाँ aᵣ अचर हैं और yᵣ rवें ढंग पर प्रतिदर्श में सम्मिलित इकाई का मान है । (i) दर्शाइए कि Ŷ_N, Ȳ_N का एक अनभिनत आकल है यदि और केवल यदि Σᵣ₌₁ⁿ aᵣ = 1 (ii) उपर्युक्त प्रतिबंध के अंतर्गत V(Ŷ_N) = (S²/N)[NΣᵣ₌₁ⁿ aᵣ² - 1] (iii) यदि aᵣ = 1/n, तो n के किस मान के लिए प्रतिस्थापन रहित सरल यादृच्छिक प्रतिचयन में प्रतिदर्शी माध्य का यह प्रसरण उसी आकार के प्रतिस्थापन सहित लिए गए यादृच्छिक प्रतिदर्श के माध्य के प्रसरण का बिल्कुल आधा होगा ? 15
Directive word: Analyse
This question asks you to analyse. 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 'analyse' demands systematic examination with computational rigour across all sub-parts. Allocate approximately 30% time to part (a) BIBD analysis, 40% to part (b) multivariate tests and discriminant analysis, and 30% to part (c) sampling theory proofs. Structure as: brief identification of appropriate statistical methods for each sub-part → step-by-step computational working with formulae stated → interpretation of results in context → final conclusions with statistical significance statements.
Key points expected
- Part (a): Verify BIBD parameters satisfy λ(v-1) = r(k-1), construct ANOVA table with SST, SSB, SStr, SSE, compute F-ratio and compare with critical value 5.41 for treatment significance
- Part (b)(i): Compute sample mean vector, sample covariance matrix S, Hotelling's T² statistic, convert to F-statistic using F = (n-p)/((n-1)p) × T² with p=2, compare with given critical value
- Part (b)(ii): Calculate Mahalanobis D² = (X̄₁-X̄₂)'S_pooled⁻¹(X̄₁-X̄₂), derive Fisher's linear discriminant function Z = a'X where a = S_pooled⁻¹(X̄₁-X̄₂), compute discriminant scores and classify X₀
- Part (c)(i): Prove unbiasedness by showing E(Ŷ_N) = Ȳ_N requires Σaᵣ = 1 using linearity of expectation and equal probability sampling properties
- Part (c)(ii): Derive variance expression using V(yᵣ) = σ² and Cov(yᵣ, yₛ) = -σ²/(N-1) for r≠s, expand V(Σaᵣyᵣ) and simplify
- Part (c)(iii): Set V(SRSWOR) = ½ V(SRSWR), i.e., (N-n)/(Nn) × S² = ½ × S²/n, solve to get n = N/2
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies all BIBD parameters for (a), specifies bivariate normal assumptions and degrees of freedom for (b)(i)-(ii), and states sampling scheme properties for (c); writes correct null hypotheses and alternative hypotheses with proper notation | Identifies most parameters correctly but has minor notational errors or misses one structural assumption (e.g., confuses pooled vs unpooled covariance) | Major setup errors: wrong parameter identification, incorrect distributional assumptions, or failure to specify hypotheses and conditions for validity |
| Method choice | 20% | 10 | Selects ANOVA for BIBD with proper decomposition, Hotelling's T² for multivariate test, Mahalanobis distance and Fisher's discriminant for classification, and uses indicator variable approach for sampling proofs; justifies each method choice | Correct methods chosen but lacks justification or has minor formula selection errors (e.g., uses wrong degrees of freedom adjustment) | Wrong methods selected (e.g., univariate t-tests instead of Hotelling's T², ignores BIBD structure, or uses population formulas instead of sample) |
| Computation accuracy | 20% | 10 | Accurate arithmetic throughout: correct matrix operations (inverses, determinants), precise F-statistic calculations, correct D² and discriminant coefficients, exact algebraic simplification in proofs; shows intermediate steps | Generally correct with minor calculation slips (e.g., sign error in covariance, arithmetic mistake in final decimal place, algebraic oversight in variance expansion) | Major computational errors: incorrect matrix inversion, wrong F-conversion formula, miscalculated discriminant scores, or algebraic errors preventing proof completion |
| Interpretation | 20% | 10 | Clear statistical interpretation: states whether treatments differ significantly in (a), interprets multivariate test result and classification decision with probability context, explains sampling efficiency implications; relates results to given critical values explicitly | Basic interpretation provided but lacks depth: states significance without context, or gives classification without explaining decision rule, or proves results without discussing practical meaning | Missing or incorrect interpretation: fails to compare with critical values, no conclusion on hypothesis tests, or no explanation of classification outcome |
| Final answer & units | 20% | 10 | All six sub-parts answered with precise final values: F-calculated vs F-critical with conclusion for (a), T²/F value with decision for (b)(i), D² value, discriminant function equation, and assigned population for (b)(ii), completed proofs and n = N/2 for (c); proper statistical notation throughout | Most answers present but incomplete: missing one final value, or correct calculation without final boxed/concluded answer, or proofs completed but final simplification omitted | Multiple missing answers, or answers without units/context (e.g., 'significant' without α level), or numerical answers without supporting calculations |
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