Chapter 13.5, Problem 55E

The article “The Influence of Honing Process Parameters on Surface Quality, Productivity, Cutting Angle, and Coefficient of Friction” (Industrial Lubrication and Tribology, 2012: 77–83) included the following data on x₁ = cutting speed (m/s), x₂ = specific pressure of pre-honing process (N/mm²), x₃ = specific pressure of finishing honing process, and y = productivity in the honing process (mm³/s for a particular tool; productivity is the volume of the material cut in a second.

a.    The article proposed a multivariate power model $Y=\alpha x_1^{{\beta }_1}{x}_2^{{\beta }_2}{x}_3^{{\beta }_3}\in$. The implied linear regression model involves regressing ln(y) against the three predictors ln(x₁), ln(x₂), and ln(x₃). Partial Minitab output from fitting this latter model is as follows (the corresponding estimated power regression function appeared in the cited article).

Carry out the model utility test at significance level .05.

b.    The large P-value corresponding to the t ratio for ln(x₂) suggests that this predictor can be eliminated from the model. Doing so and refitting yields the following Minitab output.

S = 0.048848 R-Sq = 70.6% R-Sq(adj) = 59.5%

Given that ln(x₁) remains in the model, should ln(x₃) be retained?

c.    Fit the simple linear regression model implied by your conclusion in (b) to the transformed data, and carry out a test of model utility.

d.    The standardized residuals from the fit referred to in (c) are .03, .33. 1.69, .33, −.49, .96, .57, .33, −,25, −1.28, .29, −2.26. Plot these against ln(x₁). What does the pattern suggest?

e.    Fitting a quadratic regression model to relate ln(y) to ln(x₁) gave the following Minitab output. Carry out a test of model utility at significance level .05 (the pattern in residual plots is satisfactory). Then use the fact that $s_{\ln }{}_{\left({\hat{Y}}^{\prime }\right)}$ = .0178 [Y∙= ln(Y )] when x₁ = 1 to obtain a 95% prediction interval for productivity.