(a) A 2²-factorial design was used to develop the yield of a crop. Two factors A and B were used at two levels: low (–1) and high (+1). The experiment was replicated two times with completely randomized way. The data obtained are as follows:

| Facto

Question

(a) A 2²-factorial design was used to develop the yield of a crop. Two factors A and B were used at two levels: low (–1) and high (+1). The experiment was replicated two times with completely randomized way. The data obtained are as follows:

| Factor A | Factor B | Estimated Average Effect |
| – | – | |
| + | – | 8 |
| – | + | –5 |
| + | + | 2 |

The sum of squares of all the yields = 510.5
The grand total of all the yields = 50.00

(i) Analyze the data and identify the significant factors. (12 marks)
(ii) Develop the regression model and predict the yield when A and B both are at low level (–1). (8 marks)
[Given, F₍₁, ₄, ₀.₀₅₎ = 7.71]

(b) To estimate the population mean Ȳ of a characteristic Y, a simple random sample of size 1000 was selected from a population of size 1000000 by without replacement. The population mean of an auxiliary character X is X̄ = 15. The other results are given below: s²ᵧ = 20, s²ₓ = 25, sₓᵧ = 15, x̄ = 14, ȳ = 10.
(i) Estimate Ȳ using difference, ratio and regression estimators. (6 marks)
(ii) Estimate the MSE of these estimators. Which estimator should we choose to estimate Ȳ? (9 marks)

(c) Write down the model used in the analysis of a two-way classification with interactions, stating the assumptions. What are the hypotheses tested in this scenario? Obtain the expression for the sum of squares and complete the ANOVA. (15 marks)

UPSC Answer Check · Accepted Answer

Solve this multi-part numerical problem by allocating approximately 35% time to part (a) [20 marks], 25% to part (b) [15 marks], and 40% to part (c) [25 marks]. Begin with clear model specification and ANOVA table construction for the 2² factorial in (a), followed by systematic calculation of difference, ratio and regression estimators in (b), and complete theoretical derivation of two-way ANOVA with interaction in (c). Present all computational steps in tabular format with explicit F-test conclusions and MSE comparisons.
- For (a)(i): Calculate main effects A and B, interaction effect AB, construct complete ANOVA table with 3 d.f. for treatments and 4 d.f. for error, compare F-calculated with F-critical=7.71 to identify significant factors
- For (a)(ii): Develop regression equation Y = β₀ + β₁A + β₂B + β₁₂AB with coded variables, substitute A=-1, B=-1 to predict yield at low-low combination
- For (b)(i): Compute difference estimator ȳ_D = ȳ + (X̄ - x̄), ratio estimator ȳ_R = ȳ(X̄/x̄), and regression estimator ȳ_lr = ȳ + b(X̄ - x̄) where b = s_xy/s_x²
- For (b)(ii): Calculate MSE for each estimator using appropriate formulas (MSE(ȳ_D), approximate MSE for ratio, and MSE(ȳ_lr) = (1-f)(s_y²(1-ρ²))/n), select estimator with minimum MSE
- For (c): State model y_ijk = μ + α_i + β_j + (αβ)_ij + ε_ijk with assumptions (normality, independence, homoscedasticity, Σα_i=Σβ_j=Σ(αβ)_ij=0), hypotheses H₀: all α_i=0, all β_j=0, all (αβ)_ij=0, derive SSA, SSB, SSAB, SSE with degrees of freedom and complete ANOVA table

Dimension	Weight	Max marks	Excellent	Average	Poor
Setup correctness	20%	12	Correctly identifies 2² factorial with r=2 replications giving 8 total observations; properly specifies sampling design in (b) as SRSWOR with n=1000, N=1000000, f=0.001; accurately writes two-way ANOVA model with interaction including all constraints and assumptions	Identifies basic design structure but makes minor errors in replication count or degrees of freedom; model specification in (c) misses one assumption or constraint	Confuses factorial design structure, incorrect sample size or sampling fraction, omits interaction term or constraints in model specification
Method choice	20%	12	Selects Yates' algorithm or contrast method for factorial effects; applies correct finite population correction; uses optimal regression coefficient for regression estimator; chooses standard ANOVA decomposition for two-way classification	Uses correct general methods but with minor formula variations; acceptable alternative approaches with some inefficiency	Applies incorrect method (e.g., treats as 2³ design, uses with-replacement variance formulas, omits interaction in two-way analysis)
Computation accuracy	20%	12	Precise calculation of effects (A=5, B=-1.5, AB=-1.5 from given data), correct SS values leading to F_A≈33.3, F_B≈3.0, F_AB≈3.0; accurate estimator values ȳ_D=11, ȳ_R≈10.71, ȳ_lr=13; exact ANOVA expressions with correct df breakdown	Minor arithmetic errors in one component (e.g., effect calculation or variance estimation) with mostly correct methodology	Major computational errors in multiple parts, incorrect sum of squares decomposition, wrong F-values, or invalid estimator calculations
Interpretation	20%	12	Correctly concludes only Factor A is significant (F_A>7.71, others<F-critical); interprets negative B effect as yield decrease; justifies regression estimator choice via minimum MSE; explains practical meaning of interaction test in agricultural context	Correct significance conclusions with weak practical interpretation; acceptable estimator comparison without clear recommendation	Wrong significance conclusions, fails to interpret effect directions, no comparison or justification for estimator selection
Final answer & units	20%	12	Predicted yield at (-1,-1) as 6.5 (or 3.5 if using grand mean approach with clear justification); all estimators with explicit values; complete ANOVA table with all SS, df, MS, F values; proper units (yield units, squared units for MSE)	Final answers present with minor rounding issues or missing one component; incomplete ANOVA table	Missing final answers, no ANOVA table, incorrect prediction values, or complete absence of units throughout

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