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
Variety
Fertilizer 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
(a) The model for the above design is:
Yij = μ + βi + τj + εij
where:
- Yij is the yield of corn for variety i and fertilizer j
- μ is the overall mean yield of corn
- βi is the effect of variety i on the yield of corn
- τj is the effect of fertilizer j on the yield of corn
- εij is the random error term
(b)
i. Estimate μ:
μ = (86 + 88 + 77 + 84 + 92 + 91 + 81 + 93 + 75 + 80 + 83 + 79) / 12
= 84.25 tonnes per acre
ii. Estimate and interpret β2:
β2 = (88 + 91 + 80) / 3
= 86.33 tonnes per acre
β2 represents the difference in yield when using variety B compared to the baseline variety A.
iii. Estimate and interpret τ:
τ = (86 + 92 + 75) / 3
= 84.33 tonnes per acre
τ represents the difference in yield when using fertilizer 2 compared to the baseline fertilizer 1.
iv. Estimate ε22:
ε22 = Y22 - μ - β2 - τ2
= 91 - 84.25 - 86.33 - 84.33
= -63.91 tonnes per acre
(c) The blocking variable in this experiment is the variety, and the treatment is the different fertilizers.
(d) The purpose of the blocks in this experiment is to control for the variability in yield that may be caused by differences in the varieties of corn being tested.
(e) Two-way analysis of variance table:
Source | Sum of Squares | Degrees of Freedom | Mean Square | F
Variety | xxx xxx | x | x | x
Fertilizer | xxx xxx | x | x | x
Interaction | xxx xxx | x | x | x
Error | xxx xxx | x | x
Total | xxx xxx | x
(f) To test the null hypothesis that the population mean yields are identical for all four varieties of corn, we would conduct an F-test using the data provided in the table. The results of the test would determine whether there is a statistically significant difference in yields between the four varieties.
(g) To test the null hypothesis that the population mean yields are the same for all three brands of fertilizer, we would again conduct an F-test using the data provided in the table. The results of the test would determine whether there is a statistically significant difference in yields between the three brands of fertilizer.