(a) Analyse and interpret the following data concerning output of wheat per field obtained as a result of experiment conducted to test four varieties of wheat A, B, C and D under a Latin square design at 5% level of significance.
[Given F(3, 6) = 4·7

Question

(a) Analyse and interpret the following data concerning output of wheat per field obtained as a result of experiment conducted to test four varieties of wheat A, B, C and D under a Latin square design at 5% level of significance.
[Given F(3, 6) = 4·76; F(4, 7) = 4·12] (20 marks)

(b)(i) Explain the need of factorial experiments with an example from pharmaceutical study. (6 marks)

(b)(ii) Divide the 16 treatments of 2⁴ factorial experiment into 4 blocks of 4 treatments each, confounding the interaction effect AB and CD completely with blocks. Which other interaction is automatically confounded in this design ? (9 marks)

(c) Define Horvitz-Thompson estimator for estimating the population total, and show that it is unbiased for probability proportional to size sampling without replacement. Also find its sampling variance. (15 marks)

UPSC Answer Check · Accepted Answer

Begin with the directive 'analyse' by breaking down the Latin square data in part (a) systematically—set up ANOVA table, compute F-statistic, and compare with critical value. Allocate approximately 40% of effort to part (a) (20 marks), 30% to part (b) combining theoretical explanation of factorial experiments with pharmaceutical example and confounding construction (15 marks), and 30% to part (c) for rigorous derivation of Horvith-Thompson estimator properties (15 marks). Structure as: (a) complete ANOVA with hypothesis testing, (b)(i) conceptual explanation with Indian pharmaceutical context like drug efficacy trials, (b)(ii) systematic block construction using confounding pattern, (c) formal definition followed by unbiasedness proof and variance derivation.
- For (a): Correct ANOVA setup for 4×4 Latin square with rows, columns, treatments; proper calculation of correction factor, total SS, row SS, column SS, treatment SS, error SS; correct F-test for varieties with df (3,6); comparison with given critical value F(3,6)=4.76; clear conclusion on significance
- For (b)(i): Explanation of factorial experiments need—simultaneous study of multiple factors, detection of interactions, efficiency over single-factor experiments; pharmaceutical example such as 2² factorial on drug dosage and administration timing effects on patient recovery in Indian clinical trials
- For (b)(ii): Construction of 2⁴ factorial in 4 blocks using AB and CD as confounded effects; identification of generalized interaction AB×CD = ABCD as automatically confounded; systematic block composition using even-odd rule or modulo 2 arithmetic on defining contrasts
- For (c): Formal definition of Horvitz-Thompson estimator as Σ(yᵢ/πᵢ) where πᵢ is inclusion probability; proof of unbiasedness showing E(Ŷ_HT) = Y using PPSWOR properties with πᵢ = npᵢ; derivation of variance formula involving πᵢ and πᵢⱼ using Yates-Grundy-Sen approach or alternative
- Cross-cutting: Appropriate use of statistical notation, clear statement of assumptions, and logical flow connecting theoretical derivations to practical experimental contexts

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	10	For (a): Correctly identifies Latin square structure with 4 treatments, 4 rows, 4 columns, and error df=6; for (b)(ii): Properly defines confounding pattern with generalized interaction; for (c): Correctly specifies PPSWOR framework with inclusion probabilities πᵢ=npᵢ	Identifies basic structure but makes minor errors in df calculation or confounding pattern; partial correct setup for PPSWOR	Wrong experimental design identification (confuses with RBD), incorrect df, or completely wrong sampling scheme specification
Method choice	20%	10	Selects appropriate ANOVA procedure for Latin square; uses correct method for constructing confounded blocks (defining contrasts/modulo arithmetic); applies proper indicator variable approach for HT estimator unbiasedness proof	Correct general method but inefficient or partially wrong technique; e.g., uses intuitive rather than systematic block construction	Wrong methodology entirely—uses CRD or RBD analysis for Latin square, or completely wrong variance derivation approach
Computation accuracy	20%	10	Accurate SS calculations with correct formulas; precise F-ratio computation; correct block composition for all 4 blocks; accurate variance expression with proper πᵢⱼ terms	Minor computational slips in one component (e.g., arithmetic error in one SS or block assignment) but overall structure correct	Major computational errors across multiple parts, wrong critical value usage, or completely garbled variance formula
Interpretation	20%	10	Clear hypothesis statement and conclusion for (a) with practical significance for wheat varieties; insightful explanation of why AB and CD chosen for confounding and implications; proper interpretation of HT estimator efficiency in PPSWOR context	Correct but superficial interpretation; states significance without context or misses practical implications of confounding	No interpretation of results, wrong conclusion from correct calculations, or completely missing practical significance
Final answer & units	20%	10	Explicit final answers: reject/fail to reject H₀ with reason for (a); complete block listing with treatments for (b)(ii); clean variance formula for (c); all notation defined; boxed or highlighted key results	Answers present but disorganized or missing one final component; some notation undefined	Missing final conclusions, incomplete block listings, or no variance expression; messy presentation with undefined symbols

Q7

Directive word: Analyse

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