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
Show your work in answering the questions below.
(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 the experiment can be written as:
Yijkl = μ + τi + βj + (τβ)ij + εijkl
where:
- Yijkl represents the yield of corn for variety i, fertilizer j, and block k.
- μ is the overall mean yield.
- τi represents the effect of variety i.
- βj represents the effect of fertilizer j.
- (τβ)ij represents the interaction effect between variety i and fertilizer j.
- εijkl is the error term.
(b)
i. Estimate of μ: The estimate of μ can be calculated by taking the average of all the yield values, which is (86+88+77+84+92+91+81+93+75+80+83+79)/12 = 84.
ii. Estimate of β2: The estimate of β2 is the effect of fertilizer 2, which is calculated as the average yield for variety B with fertilizer 2 minus the overall mean (x22 - μ) = 91 - 84 = 7. Interpretation: On average, fertilizer 2 increases the yield of variety B by 7 tonnes per acre compared to the overall mean.
iii. Estimate of τ1: The estimate of τ1 is the effect of variety A, which is calculated as the average yield for variety A minus the overall mean (x11 + x12 + x13)/3 - μ = (86+92+75)/3 - 84 = 5. Interpretation: On average, variety A yields 5 tonnes per acre more than the overall mean.
iv. Estimate of ε21: The estimate of the error term for variety B with fertilizer 2 is (x22 - μ - β2 - τ2) = 91 - 84 - 7 - 5 = -5.
(c) The blocking variable in this experiment is not provided in the information given, but it could be a factor such as field location or weather conditions. The treatments in this experiment are the different combinations of corn varieties and fertilizers.
(d) The purpose of the blocks in this experiment is to control for any potential confounding variables that could affect the results, such as differences in soil quality or weather conditions in different locations.
(e) Two-way analysis of variance table:
- Source of Variation | SS | df | MS | F
- Variety | ? | 3 | ? | ?
- Fertilizer | ? | 2 | ? | ?
- Variety*Fertilizer | ? | 6 | ? | ?
- Error | ? | 0 | ? |
- Total | ? | 9
(f) To test the null hypothesis that the population mean yields are identical for all four varieties of corn, we can conduct an F-test using the variety factor in the analysis of variance table to compare the variation between varieties and the variation within varieties. If the calculated F-value is greater than the critical F-value, we reject the null hypothesis.
(g) To test the null hypothesis that the population mean yields are the same for all three brands of fertilizer, we can conduct an F-test using the fertilizer factor in the analysis of variance table to compare the variation between fertilizers and the variation within fertilizers. If the calculated F-value is greater than the critical F-value, we reject the null hypothesis.