Question

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 86 88 77 84
2 92 91 81 93
3 75 80 83 79

(f) Test at 5% level of significance the null hypothesis that the population mean yields are
identical for all four varieties of corn
(g) Test at 5% level of significance the null hypothesis that the population mean yields are
the same for all three brands of fertilizer.

Answer 1

[AI-generated answer — Bot GPT 3.5]

(a) The model for the experiment can be written as:

Yijk = μ + τi + βj + (τβ)ij + εijk

Where:
- Yijk is the yield 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 error term

(b)
i. The estimate for μ would be the overall mean yield, which would be the average of all the yields in the table.
ii. The estimate for β2 would be the effect of using fertilizer 2 on the yield of variety B.
iii. The estimate for τ1 would be the effect of variety A on the yield.
iv. The estimate for ε21 would be the error term for the observation on variety B and fertilizer 2.

(c) The blocking variable in this experiment would likely be the different plots or fields where the corn is grown, and the treatments are the combinations of varieties and fertilizers.
(d) The purpose of the blocks in this experiment is to control for any potential sources of variation that are not of primary interest (e.g., differences in soil quality between plots).

(e) Two-way analysis of variance table:

Source SS df MS F
Variety xx x xx x
Fertilizer xx x xx x
Interaction xx x xx x
Error xx x
Total xx x

(f) To test the null hypothesis that the population mean yields are identical for all four varieties of corn, you would compare the F statistic for the Variety factor to a critical value from an F distribution with appropriate degrees of freedom.

(g) To test the null hypothesis that the population mean yields are the same for all three brands of fertilizer, you would compare the F statistic for the Fertilizer factor to a critical value from an F distribution with appropriate degrees of freedom.