Often the fitted regression model from a 2k factorial design is used to make predictions at points of interest in the design space. Assume that the model contains all main effects and two-factor interactions.

(a) Find the variance of the predicted response ̂y at a point x1, x2, . . . , xk in the design space. Hint: Remember that the x’s are coded variables and assume a 2k design with an equal number of replicates n at each design point so that the variance of a regression coefficient ????̂ is ????

2∕(n2k)

and that the covariance between any pair of regression coefficients is zero.

(b) Use the result in part

(a) to find an equation for a 100(1 − ????) percent confidence interval on the true mean response at the point x1, x2, . . . , xk in design space.