An agricultural experiment designed to assess differences in yields of corn for four different
varieties, using three different fertilizers, produced the results (in tonnes per acre) shown in the
following table.
(a) Write down a model for the above design. Explain each term in the model in the context
of the given information
(b) Consider the observation on the variety B and fertilizer 2 ( x22 = 91) .
i. Estimate µ
ii. Estimate and interpret β2
iii. Estimate and interpret 1
τ
iv. Estimate 21 ε
(c) Name the blocking variable and the treatment in this experiment.
(d) What is the purpose of the blocks in this experiment?
(e) Prepare a two –way analysis of variance table
Fertilizer Variety
A B C D
1
2
3
86 88 77 84
92 91 81 93
75 80 83 7
(a) The model for the experiment can be written as:
Yijk = μ + τi + βj + (τβ)ij + εijk
Where:
Yijk is the yield of corn for the ith variety, jth fertilizer, and kth replicate,
μ is the overall mean yield,
τi is the effect of the ith variety,
βj is the effect of the jth fertilizer,
(τβ)ij is the interaction effect between the ith variety and jth fertilizer,
εijk is the random error term.
(b)
i. Since x22 = 91, we can estimate μ as the average of all observations:
μ = (86 + 88 + 77 + 84 + 92 + 91 + 81 + 93 + 75 + 80 + 83 + 77) / 12 = 83.5
ii. Estimate and interpret β2:
β2 = (86 + 92 + 75) / 3 = 84.33
This represents the average yield of corn when using fertilizer 2 across all varieties.
iii. Estimate and interpret τ1:
τ1 = (86 + 88 + 77 + 84) / 4 = 83.75
This represents the average yield of corn for variety A across all fertilizers.
iv. Estimate ε21:
ε21 = x21 - (μ + τ2 + β1 + (τβ)21)
ε21 = 77 - (83.5 + 81 + 88 + (τβ)21)
(c) The blocking variable in this experiment is the replicate or the number of times the experiment was conducted. The treatment in this experiment is the combination of different varieties and fertilizers.
(d) The purpose of the blocks in this experiment is to control for any extraneous variables that may affect the yield of corn. By conducting multiple replicates, any variability between experimental runs can be accounted for.
(e) Two-way Analysis of Variance table:
| Source | Sum of Squares | Degrees of Freedom | Mean Square | F ratio |
|--------|-----------------|--------------------|-------------|--------|
| Variety | SS_variety | DF_variety | MS_variety | F_variety |
| Fertilizer | SS_fertilizer | DF_fertilizer | MS_fertilizer | F_fertilizer |
| Variety x Fertilizer | SS_variety*fertilizer | DF_variety*fertilizer | MS_variety*fertilizer | F_variety*fertilizer |
| Error | SS_error | DF_error | MS_error | - |
| Total | SS_total | DF_total | - | - |