Often the fitted regression model from a 2^k factorial design is used to make predictions at points of interest in the design space.

(a) Find the variance of the predicted response ŷ at a point x₁, x₂, . x_k in the design space.  Remember that the x's are coded variables, and assume a 2^k design with an equal number of replicates n at each design point so that the variance of a regression coefficient β̂ is σ²/(n2^k) 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 - a) percent confidence interval on the true mean response at the point x₁, x₂ x_k in design space.