Consider given Example concerning the efficiency of synthetic fuel for which the factors are an additive and a catalyst. The original data are given in Table. The experiment is conducted using only five randomly chosen automobiles (A, B, C, D, and E). Each treatment is used on each automobile. The reason for choosing only five automobiles and replicating the experiment is to eliminate any variation in the fuel efficiencies that might arise due to differences in the automobiles.

The data in Table is interpreted as follows. For additive level I and catalyst level 1, the response value is 75 for automobile A, 72 for automobile B, and so forth. Similar interpretations hold for the other treatments.

(a) What is your conclusion regarding the significance of the factors or their interactions? Test at the 5% level of significance.

(b) Find a 95% confidence interval for the mean efficiency when the additive is at level I and the catalyst is at level 2.

(c) Find a 95% confidence interval for the difference in the treatment means when the additive is at level I and the catalyst is at level 3 versus when the additive is at level II and the catalyst is at level 3. What is your conclusion?

(d) What levels of additive and catalyst would you prefer to choose?

(e) Was it beneficial to select only five automobiles and replicate the treatments on only those automobiles? Test at the 5% level of significance.

Example
In a study on the effectiveness of synthetic automobile fuels, two factors are of importance. Factor A is an additive that is to be tested at three levels of use in successively increasing amounts. Factor B is a catalyst, for which three levels of use (1, 2, and 3, representing successively increasing amounts) are also to be tested.