When the x1's, x2's, ... , and/or the xk's are equally spaced, the calculation of the 
's can be simplified by using the coding suggested in Exercise 15. Rework Exercise 80 coding the x1-values -1, 0, and 1 and the x2-values -1 and 1. (Note that for the coded x1's and x2's, call them z1's and z2's, we have not only 
Exercise 15
When the x's are equally spaced, the calculation of 
and can be simplified by coding the x's by assigning them the values ... , -3, -2, -1, 0, 1, 2, 3, ... when n is odd, or the values ... , -5, -3, -1, 1, 3, 5, ... when n is even. Show that with this coding the formulas for and become
Exercise 80
The following data were collected to determine the relationship between two processing variables and the hardness of a certain kind of steel:
Fit a plane by the method of least squares, and use it to estimate the average hardness of this kind of steel when the copper content is 0.14 percent and the annealing temperature is 1,100?F.