(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…

(a)(i) Analysis of the 2² Experiment

The three non-blank entries are the estimated factorial effects A, B and AB; the (–,–) row is the reference. Here r = 2, N = 8, G = 50.00, ∑y² = 510.5. Grand mean = 50/8 = 6.25 yield units. Total SS = ∑y² – G²/N = 510.5 – 50²/8 = 510.5 – 312.5 = 198.0 (yield units)². For a 2² design with r replicates, SS_effect = r(effect)².

Thus SS_A = 2(8)² = 128, SS_B = 2(–5)² = 50, SS_AB = 2(2)² = 8. Treatment SS = 186. Error SS = 198.0 – 186 = 12.0. df: A 1, B 1, AB 1, Error 4, Total 7. MS_A = 128, MS_B = 50, MS_AB = 8, MS_E = 12/4 = 3. F_A = 128/3 = 42.67, F_B = 50/3 = 16.67, F_AB = 8/3 = 2.67.

With F(1,4,0.05) = 7.71, A and B are significant; AB is not. The F tests require independent normal errors with common variance.

(a)(ii) Regression Model and Predicted Yield at Low Levels

With coded variables x_A, x_B = ±1, the regression coefficients of a 2² design are half the factorial effects and the intercept is the grand mean: β₀ = 6.25, β₁ = 8/2 = 4, β₂ = −5/2 = −2.5, β₃ = 2/2 = 1.

Since only A and B are significant, the fitted regression model retains the main effects:

ŷ = 6.25 + 4x_A − 2.5x_B.

At x_A = x_B = −1: ŷ = 6.25 − 4 + 2.5 = 4.75 yield units.

If the non-significant interaction is retained (full model ŷ = 6.25 + 4x_A − 2.5x_B + x_Ax_B), the prediction is 6.25 − 4 + 2.5 + 1 = 5.75, which equals the fitted (−, −) cell mean. Dropping AB is preferred because its F = 2.67 < 7.71.

(b)(i) Difference, Ratio and Regression Estimates of Ȳ

For SRS without replacement, n = 1000, N = 1,000,000, f = 0.001.

• Difference estimator, valid if X and Y are on the same scale: Ȳ_D = ȳ + (X̄ – x̄) = 10 + (15 – 14) = 11 Y units.

• Ratio estimator: Ȳ_R = (ȳ/x̄)X̄ = (10/14)(15) = 75/7 ≈ 10.7143 Y units.

• Regression estimator: b = sₓᵧ/s²ₓ = 15/25 = 0.6 Y units per X unit; Ȳ_reg = ȳ + b(X̄ – x̄) = 10 + 0.6(1) = 10.6 Y units.

(b)(ii) MSEs and Choice of Estimator

The finite-population correction is (1 – f)/n = 0.999/1000 = 0.000999.

• Difference: MSE_D = 0.000999[s²ᵧ + s²ₓ – 2sₓᵧ] = 0.000999(20 + 25 – 30) = 0.014985 (Y units)².

• Ratio: R̂ = ȳ/x̄ = 5/7. MSE_R ≈ 0.000999[s²ᵧ + R̂²s²ₓ – 2R̂sₓᵧ] = 0.000999[20 + (25/49)(25) – 2(5/7)(15)] = 0.000999(555/49) = 0.011315 (Y units)². This is the usual large-sample approximation; X is positive, so the ratio is valid.

• Regression: MSE_reg = 0.000999[s²ᵧ – (sₓᵧ)²/s²ₓ] = 0.000999(20 – 225/25) = 0.010989 (Y units)².

The regression estimator has the smallest estimated MSE, so choose the regression estimator.

(c) Two-Way Classification with Interaction: Model, Hypotheses and ANOVA

The fixed-effects two-way model with interaction is Y_ijk = μ + α_i + β_j + (αβ)_ij + ε_ijk, i = 1,...,a, j = 1,...,b, k = 1,...,n. Assumptions: ε_ijk are independent N(0,σ²); effects are fixed with ∑α_i = 0, ∑β_j = 0, ∑_i(αβ)_ij = 0, ∑_j(αβ)_ij = 0; n ≥ 2.

Hypotheses:

• H₀: α_i = 0 for all i, against some α_i ≠ 0.

• H₀: β_j = 0 for all j, against some β_j ≠ 0.

• H₀: (αβ)_ij = 0 for all i,j, against some interaction ≠ 0.

Let Y_i.. = ∑_j∑_kY_ijk, Y_.j. = ∑_i∑_kY_ijk, Y_ij. = ∑_kY_ijk, Y_.... = ∑_i∑_j∑_kY_ijk, N = abn, CF = Y_....²/N.

• SS_Total = ∑Y_ijk² – CF.

• SS_A = (1/bn)∑Y_i..² – CF.

• SS_B = (1/an)∑Y_.j.² – CF.

• SS_AB = (1/n)∑∑Y_ij.² – (1/bn)∑Y_i..² – (1/an)∑Y_.j.² + CF.

• SS_Error = ∑Y_ijk² – (1/n)∑∑Y_ij.².

ANOVA:

• A: df a–1, MS = SS_A/(a–1), F = MS_A/MS_E.

• B: df b–1, MS = SS_B/(b–1), F = MS_B/MS_E.

• AB: df (a–1)(b–1), MS = SS_AB/[(a–1)(b–1)], F = MS_AB/MS_E.

• Error: df ab(n–1), MS = SS_Error/[ab(n–1)].

• Total: df N–1.

Expected mean squares: E(MS_E) = σ²; E(MS_A) = σ² + bn∑α_i²/(a–1); E(MS_B) = σ² + an∑β_j²/(b–1); E(MS_AB) = σ² + n∑∑(αβ)_ij²/[(a–1)(b–1)].