2. (15 points) Consider the smallest central composite design with k = 2 factors, nf = 2 cube points and no center points. The factorial portion is defined by I = AB. Theorem 10.1 shows that this is a second-order design as long as ne > 0 and a > 0. We verify this by doing the following (a) Write down the design matrix for ne = 1 center point. (b) Write down the second-order model and the model matrix X. (c) Find XTx. (d) Verify that the determinant of XT X is not zero for a = 1 and 2. (Note that det(XTX) + O implies that this is a second-order design.) (e) Find (XTX)-1 when a = 1. (f) Assume the experimental error e has variance o2. Find the variances of the least squares estimates for all parameters in the second-order model when a = 1. (g) Redo (f) for ne = 4. What are changed when we increase nc? Instruction: Answer the questions with necessary graphs and computer outputs. 2. (15 points) Consider the smallest central composite design with k = 2 factors, nf = 2 cube points and no center points. The factorial portion is defined by I = AB. Theorem 10.1 shows that this is a second-order design as long as ne > 0 and a > 0. We verify this by doing the following (a) Write down the design matrix for ne = 1 center point. (b) Write down the second-order model and the model matrix X. (c) Find XTx. (d) Verify that the determinant of XT X is not zero for a = 1 and 2. (Note that det(XTX) + O implies that this is a second-order design.) (e) Find (XTX)-1 when a = 1. (f) Assume the experimental error e has variance o2. Find the variances of the least squares estimates for all parameters in the second-order model when a = 1. (g) Redo (f) for ne = 4. What are changed when we increase nc? Instruction: Answer the questions with necessary graphs and computer outputs