The article "The Equilibrium Partitioning of Titanium Between Ti3+ and Ti4+ Valency States in CaO-SiO2-TiOx Slags" (G. Tranell, O. Ostrovski, and S. Jahanshahi, Metallurgical and Materials Transactions B, 2002:61-66) discusses the relationship between the redox ratio Ti3+/Ti4+ and oxygen partial pressure pO2 in CaO-SiO2-TiOx melts. Several independent measurements of the redox ratio were made at each of five different partial pressures: 10ˆ’7, 10ˆ’8, 10ˆ’9, 10ˆ’10, and 10ˆ’12 atmospheres. The results for the runs at 14 mass percent TiOx are presented in the following table.

a. Denoting the redox ratio by y and the partial pressure by x, theory states that y should be proportional to xβ for some β. Express this theoretical relationship as a linear model.
b. Compute the least-squares line for this linear model. Plot the residuals versus the fitted values. Does the linear model hold?
c. Further theoretical considerations suggest that under the conditions of this experiment, y should be proportional to xˆ’1/4. Are the data in the preceding table consistent with this theory? Explain.

Oxygen Partial Pressure 10-7 10-8 10-9 10-10 10-12 Redox Ratio Measurements 0.011, 0.017, 0.034, 0.039 0.018, 0.01, 0.026, 0.050, 0.034, 0.068, 0.061 0.027, 0.038, 0.076, 0.088 0.047, 0.069, 0.123,0.162 0.160, 0.220,0.399,0.469