In their book Empirical Model Building and Response Surfaces (John Wiley, 1987), Box and Draper described an experiment with three factors. The data in the following table are a variation of the original experiment from their book. Suppose that these data were collected in a semiconductor manufacturing process.

(a) The response y₁ is the average of three readings on resistivity for a single wafer. Fit a quadratic model to this response.

(b) The response y₂ is the standard deviation of the three resistivity measurements. Fit a linear model to this response.

(c) Where would you recommend that we set x₁, x₂, and x₃ if the objective is to hold mean resistivity at 500 and minimize the standard deviation?

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-1 24.00 12.49 -1 120.33 8.39 -1 213.67 42.83 -1 86.00 3.46 136.63 80.41 -1 1 340.67 16.17 -1 1 112.33 27.57 1 -1 256.33 4.62 -1 271.67 23.63 0.00 81.00 -1 101.67 17.67 -1 357.00 32.91 -1 171.33 15.01 372.00 0.00 501.67 92.50 -1 264.00 63.50 427.00 88.61 1 730.67 21.08 1 220.67 133.82 239.67 1 23.46 1. -1 1 422.00 18.52 199.00 -1 29.44 1 485.33 44.67 158.21 1 673.67 -1 1 1 176.67 55.51 1 1 501.00 138.94 1 1 1010.00 142.45