1. (30%) For a 2k-P factorial design without any replicate, specify the following questions. (a) Write down the regression model with main effects and the two-factor interaction effects. (b) What is the relationship between the estimated effects and the estimated coefficients of the regression model? (c) Give the definitions of a fractional factorial design with resolution of III, IV, and V. (d) Specify what information of model parameters we can get more if we add each of the following runs. i. Add one more replicate for all the runs of the 2k design. ii. Add 5 central points to the 2k design. (e) In the screening design, there are three principles to screen potential important effects, hierarchy principle, sparsity principle, and heredity principle. What principles are used when i. using Lenth method to screen effects? ii. constructing the fractional factorial design? iii. an interaction term is significant and we consider at least one main factor is also significant