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.
Fertilizer Variety
A B C D
1 86 88 77 84
2 92 91 81 93
3 75 80 83 79
(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 τ1
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
(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 this experiment can be written as:
Yijkm = μ + τi + βj + (τβ)ij + εijkm
Where:
- Yijkm is the yield of corn for the kth replicate of the jth variety under the ith fertilizer treatment.
- μ is the overall mean yield.
- τi represents the effect of the ith fertilizer treatment.
- βj represents the effect of the jth variety.
- (τβ)ij represents the interaction effect between the ith fertilizer treatment and the jth variety.
- εijkm is the random error term.
(b)
i. Estimate μ: To estimate the overall mean yield, we take the average of all the yield values: (86+88+77+84+92+91+81+93+75+80+83+79)/12 = 84.0833
ii. Estimate β2: To estimate the effect of variety B, we need to take the average yield for variety B across all fertilizer treatments: (88+91+80)/3 = 86.33. Therefore, β2 = 86.33.
iii. Estimate τ1: To estimate the effect of fertilizer 1, we need to take the average yield for fertilizer 1 across all varieties: (86+92+75)/3 = 84.33. Therefore, τ1 = 84.33.
iv. Estimate ε22: To estimate the random error term for the observation on variety B and fertilizer 2, we subtract the estimated values of μ, β2, and τ1 from the observed value of x22: ε22 = 91 - 84.0833 - 86.33 - 84.33 = -63.74
(c) The blocking variable in this experiment could be the plot or field where the corn was grown. The treatments are the combination of different varieties of corn and different fertilizers.
(d) The purpose of blocking in this experiment is to control for any external factors that could influence the yield of corn, such as soil differences, sunlight exposure, or other environmental conditions. By blocking, we can ensure that any differences in yield are due to the treatments being studied.
(e) Two-way ANOVA table:
Varieties Fertilizers Observed yield Sum of squares Degrees of freedom Mean squares F-ratio
A 1 86 XX XX XX XX
A 2 88 XX XX XX XX
A 3 77 XX XX XX XX
B 1 84 XX XX XX XX
B 2 91 XX XX XX XX
B 3 80 XX XX XX XX
C 1 77 XX XX XX XX
C 2 81 XX XX XX XX
C 3 83 XX XX XX XX
D 1 84 XX XX XX XX
D 2 93 XX XX XX XX
D 3 79 XX XX XX XX
(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 to compare the mean squares for varieties and the error term. The test would involve comparing the F-ratio for varieties to the critical value at the 5% significance level.
(g) To test the null hypothesis that the population mean yields are the same for all three brands of fertilizer, we would conduct a similar F-test comparing the mean squares for fertilizers to the error term in the ANOVA table. Again, we would compare the F-ratio to the critical value at the 5% significance level.