Engineers are interesting in finding the factor combinations that optimize the

workability of cement grout. The workability is a function of the water to cement

ratio $($WatCem$,{}_1),$ the percent of black liquor added $($BlackL$,{}_2),$ and the

percent of SNF added $($SNF$,{}_3).$ The following data were collected:

$($a$)$ What is the primary purpose of a response surface design?

$($b$)$ Using the above data, fit a first order model of the form

hat$(y{)}_i=$hat$({)}_0+$hat$({)}_1{x}_1+$hat$({)}_2{x}_2+$hat$({)}_3{x}_3$

$($c$)$ After fitting the first order model, the following output was produced in R:

Analysis of Variance Table

i$.$ Complete the ANOVA table above, clearly showing all calculations.

ii$.$ Comment on the adequacy of the first$-$order model. Is there evidence of

lack$-$of$-$fit for this model? Justify your answer.

iii. Using this output, how would the experimenter know they may be near

to the optimum?