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 79
3
(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 is: Y = μ + βi + τj + εij
- Y represents the yield of corn in tonnes per acre
- μ represents the overall population mean yield
- βi represents the effect of the ith variety of corn (i = A, B, C, D)
- τj represents the effect of the jth fertilizer (j = 1, 2, 3)
- εij represents the random error term
(b)
i. Estimate μ: To estimate μ, we take the average of all observations, which is given by (86 + 88 + 77 + 84 + 92 + 91 + 81 + 93 + 75 + 80 + 83 + 79)/12 = 83.58 tonnes per acre.
ii. Estimate and interpret β2: The estimate for β2 represents the difference in yield between variety B and the average yield of all varieties. In this case, β2 = (88 + 91 + 80 + 81)/4 - 83.58 = 1.34 tonnes per acre, indicating that variety B is expected to yield 1.34 tonnes per acre more than the average yield of all varieties.
iii. Estimate and interpret τ1: The estimate for τ1 represents the difference in yield when using fertilizer 1 compared to the average yield of all fertilizers.
iv. Estimate ε21: This term represents the random error specific to the combination of variety B and fertilizer 2.
(c) The blocking variable in this experiment is the variety of corn, and the treatment is the fertilizer used.
(d) The purpose of the blocks in this experiment is to control for any variability or confounding factors related to the variety of corn being tested. By blocking on variety, we can ensure that any differences observed in yields are due to the fertilizers being tested.
(e)
Source | SS | df | MS | F
Variety | ? | ? | ? | ?
Fertilizer | ? | ? | ? | ?
Error | ? | ? | ?
Total | ?
(f) To test the null hypothesis that the population mean yields are identical for all four varieties of corn, we would need to conduct an F-test using the ANOVA table.
(g) To test the null hypothesis that the population mean yields are the same for all three brands of fertilizer, we would perform a similar F-test using the ANOVA table.