11. Steiner et al. (2007) describe a mixture-process variable (MPV) experiment conducted by students to nd an optimal homemade bubble solution for use in a battery operated bubble-blowing toy. The response was the average number of bubbles produced by the bubble-blowing toy. The homemade bubble solution was made as a mixture of x1: dish soap, x2: water, and x3:

glycerin. The following constraints were placed on the mixture components, based on recipes found in children's books and cost, 0:04  x1  0:35 0:60  x2  0:98 0:00  x3  0:15:

Two process variables were also included in the study, which were z1: brand of soap (-1=Joy, +1=Ivory) and z2: type of water (-1=spring water, +1=tap water).

(a) Create a design for this experiment by crossing a design appropriate for tting a Schee quadratic model in the mixture components and a 22 factorial in the process variables.

(b) Write the model that can be t from the design you created.

(c) Given the actual data from Steiner et al. (2007) experiments shown in Table 11.14, that were run in a completely random order, t the model you propose in

(b) to the data.

(d) Is the process water signicant? Can it be safely removed from the model?

(e) Make a contour plot over the constrained mixture space for each level of the process variable soap type.

(f) Make a recommendation for the optimal bubble solution.