Hocking (1985) described an example for which the nested factorial model is appropriate. Observations are taken at laundromats using different types of washing machines and different detergents. For the study we have seven laundromats using three types of machines and four brands of detergent. Each laundromat uses, exclusively, one type of machine, and the laundromats generally have different numbers of machines. The detergents are allocated at random to the machines within a laundromat. The response is the average amount of phosphorous in grams per liter from daily one-hour samples over a seven-day period. Assume the model in (12.79) with machines being factor one, detergents factor two, and laundromats factor three. The sample cell means and cell frequencies are shown in the following table. The residual sum of squares is 40.5.

a. Develop the reparameterized model that arises by removing the redundancies in the over-parameterized model by imposing the standard conditions (e.g.,

(a) = 0) with the restrictions suggested in Section 12.5. Identify the hypotheses associated with the parameter definitions.

1 Data for Exercise 12.17 Machine type 2 Laundromats within machine types 3 12 1 2 3 1 2 D 1 6.0 5.5 2.5 3.5 4.0 4.0 4.5 T (3) (2) (2) (2) (3) (2) (2) e 2 4.5 4.0 3.0 4.0 4.0 3.0 3.0 . (2) (3) (2) (2) (2) (3) (2) 8 3 4.5 4.0 3.7 4.5 4.0 1.5 2.0 (4) (2) 4 2.0 4.0 (3) (3) (3) (2) (2) (2) (2) 2.3 2.0 3.0 (3) (2) (3) 1.0 1.3 (2) (3)

b. Develop the analysis of variance table for the laundromat data by formulating the appropriate regression model for this choice of parameters and using R(6, 2), where, and contain all parameters in the model.

e. Use any available computer software to analyze the data, and compare with the analysis in part

b. d. Use a sequential procedure for including the effects for the model in part

a, and compare the AOV with parts b and c.