NaOH Silicate Test H2O2 Conc. Conc. Temp. Bright. No. (x1)

Chapter 13, Problem 13.54

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NaOH Silicate Test H2O2 Conc. Conc. Temp. Bright. No. (x1) (x2) (x3) (x4) (y) 11 1 1 1 1 82.6 12 1 1 1 1 85.1 13 1 1 1 1 84.5 14 1 1 1 1 86.0 15 1 1 1 1 84.0 16 1 1 1 1 85.4 17 2 0 0 0 82.9 18 2 0 0 0 85.5 19 0 2 0 0 85.2 20 0 2 0 0 84.5 21 0 0 2 0 84.7 22 0 0 2 0 85.0 23 0 0 0 2 84.9 24 0 0 0 2 84.0 25 0 0 0 0 84.5 26 0 0 0 0 84.7 27 0 0 0 0 84.6 28 0 0 0 0 84.9 29 0 0 0 0 84.9 30 0 0 0 0 84.5 31 0 0 0 0 84.6 Variables 2 1 0 1 2 x1 Hydrogen peroxide (100%), %wt .1 .2 .3 .4 .5 x2 NaOH, %wt .1 .2 .3 .4 .5 x3 Silicate (41B), %wt 5 1.5 2.5 3.5 4.5 x4 Process temp., F 130 145 160 175 190 a. When a (coded) model involving all linear terms, all quadratic terms, and all cross-product terms was fit, the estimated regression function was y 84.67 .650x1 .258x2 .133x3 .108x4 .135x2 1 .028x2 2 .028x2 3 .072x2 4 .038x1x2 .075x1x3 .213x1x4 .200x2 x3 .188x2 x4 .050x3 x4 Use this estimated model to predict brightness when H2O2 is .4%, NaOH is .4%, silicate is 3.5%, and temperature is 175. What are the values of the residuals for these values of the variables? b. Express the estimated regression function in uncoded form. c. SST 17.2567, and R2 for the model of part (a) is .885. When a model that includes only the four linear terms is fit, the resulting value of R2 is .721. State and test at level .05 the null hypothesis that specifies that the coefficients of all quadratic and cross-product terms in the regression function are zero. d. The estimated (coded) regression function when only linear terms are included is Yx1,x2,x3,x4 84.5548 .6500x1 .2583x2 .1333x3 .1083x4. When x1 x2 x3 x4 0, the estimated standard deviation of Y 0,0,0,0 is .0772. Suppose it had been believed that expected brightness for these values of the xis was at least 85.0. Does the given information contradict this belief? State and test the appropriate hypotheses. 550 CHAPTER 13 Nonlinear and Mult

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