Chapter B.2, Problem 46E

Silica Gel. Silica gel is a substance that absorbs moisture from the environment. Packets of silica gel are often placed inside packages that contain manufactured goods susceptible to damage by high hu-midity, such as delicate electronic equipment. In the paper "Improv-ing the Yield of Silica Gel in a Chemical Plant" (Quality Engineering, 9, pp. 355–361) K. N. Anand reported on the results of an experiment to increase the yield of silica gel. A key determinant of the yield is the density of the silica gel—the higher the density, the better the gel absorbs moisture. A number of variables are thought to affect the density. Silica gel is made by adding sodium silicate to a solution of hydrochloric acid (11C1) until a specified pH is reached. The mixture is then allowed to dry for a specified amount of time. Here we con-sider only two variables that are thought to affect density: (I) final pH of the sodium silicate/HC1 solution and (2) the amount of time the solution is allowed to dry, called the setting time. The data for this study am presented in the following table.

Output B.39 on page B-59 shows a scatterplot matrix for the three variables—density, pH, and settinte. The relationship between den-sity and pH exhibits some curvature, as possibly does the relationship

between density and settime. We will fit a second-order polynomial regression model and then assess the fit.

a. Output 8.40 on page B-60 provides the correlation coefficients between density and the first- and second-order terms in pH_c and settime_c, the centered predictor variables. Which predictor variables are most highly correlated with density? What are the

correlation coefficients between the first and second-order terms? Will these correlations adversely affect our ability to assess the effect of a term in the presence of the other terms? Explain your answer.

b. We obtained Output B.41 on page 8-60, showing a regression analysis of density using a second-ogler polynomial regression equation in the centered variables Mc and settime_c. Based on the t-tests for the individual utility of each term, which terms would you retain in the regression equation? Is it appropriate to use the t-tests here? Explain your answer.

c. Outputs B.42(a)—(d) on page B-61 provide plots of the residuals versus the fitted values, residuals versus pH_d, residuals versus set-timer, and a normal probability plot of the residuals. Perform a residual analysis to access the appropriateness of the regression equation, constancy of the conditional standard deviations, and normality of the conditional distributions. Check for outliers and influential observations.

d. Does your analysis in part (c) reveal any violations of the assump-tions for regression inferences? Explain.