Refer to the Journal of Colloid and Interface Science study of water/oil mixtures, Exercise 11.27. Recall that

Question:

Refer to the Journal of Colloid and Interface Science study of water/oil mixtures, Exercise 11.27. Recall that three of the seven variables used to predict voltage (y) were volume (x₁), salinity (x₂), and surfactant concentration (x₅). The model the researchers fit is

E(y) = β₀ + β₁x₁ + β₂x₂ + β₃x₅ + + β₄x₁x₂ + β₅x₁x₅

a. Note that the model includes interaction between disperse phase volume (x₁) and salinity (x₂) as well as interaction between disperse phase volume (x₁) and surfactant concentration (x₅). Discuss how these interaction terms affect the hypothetical relationship between y and x₁. Draw a sketch to support your answer.

b. Fit the interaction model to the data. Does this model appear to fit the data better than the first-order model in Exercise 11.27? Explain.

c. Interpret the β estimates of the interaction model.

d. Conduct a test to determine whether the interaction terms contribute significantly to the prediction of voltage (y). Use α = .05.

Data from Exercise 11.27

In the oil industry, water that mixes with crude oil during production and transportation must be removed. Chemists have found that the oil can be extracted from the water/oil mix electrically. Researchers at the University of Bergen (Norway) conducted a series of experiments to study the factors that influence the voltage (y) required to separate the water from the oil. (Journal of Colloid and Interface Science, Aug. 1995.) The seven independent variables investigated in the study are listed in the table below. (Each variable was measured at two levels—a “low” level and a “high” level.) Sixteen water/oil mixtures were prepared using different combinations of the independent variables; then each emulsion was exposed to a high electric field. In addition, three mixtures were tested when all independent variables were set to 0. The data for all 19 experiments are saved in the WATEROIL file.