The Brazilian Journal of Chemical Engineering (Vol. 22, 2005) published a study to compare two foaming agents in the flotation of sulfured copper materials process. The two agents were surface-active bio oil (SABO) and pine oil (PO). A 2 × 2 × 2 × 2 factorial design was used to investigate the effect of four factors on the percentage of copper in the flotation concentrate. The four factors are: foaming agent (SABO or PO), agent-to-mineral mass ratio (low or high), collector-to-mineral mass ratio (low or high), and liquid-to solid ratio (low or high). Percentage copper measurements (y) were obtained for each of the 2 × 2 × 2 × 2 = 16 treatments. The data are listed in the table.

a. Write the complete model appropriate for the 2 × 2 × 2 × 2 factorial design.

b. Note that there is no replication in the experiment. (That is, there is only one observation for each of the 16 treatments.) How will this impact the analysis of the model, part a?

c. Write a model for E(y) that includes only main effects and two-way interaction terms.

d. Fit the model, part c, to the data. Give the least-squares prediction equation.

e. Conduct tests (at α = .05) for the interaction terms. Interpret the results.

f. Is it advisable to conduct any main effect tests? If so, perform the analysis. If not, explain why.

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FOAM Collector-to- Liquid-to- Solid Ratio % Copper Agent-to- Mineral Mass Ratio Mineral Mass Ratio SABO PO L 6.11 6.96 H L L 6.17 7.31 L H L 6.60 7.37 H. H. L 7.15 7.52 L L H. 6.24 7.17 H L H. 6.98 7.48 L H H. 7.19 7.57 H. 7.59 7.78