Suppose you have to perform the runs in two blocks. Propose a good block generator and indicate why it’s good.Write the full alias structure for this design.

1. Consider a fractional 25−2 design. The following runs were performed: de, ae, b, abd, cd, ac, bce, abcde.

a)Find the design generators.

b)Write the complete defining relation for this design.

c)Write the full alias structure for this design.

d)Suppose you have to perform the runs in two blocks. Propose a good block generator and indicate why it’s good.

e)Which runs will go in which block if you use the generator from (d)?

2. Using a response surface design, the fitted second order model of the form

for the response y was obtained.

Write down the matrix B for this model.
Find the stationary point.
Find the value ˆys of the predicted response at the stationary point.
Find the eigenvalues of B.

Write the model in the canonical form. What is the type of the stationary point? Can it be considered a ridge system?

3-A factorial experiment with four factors is conducted. Factors C and D is random, and all others are fixed. B is nested under A, C is nested under B, and D is nested under C. The number of levels is a, b, c, and d for A, B, C, and D, respectively. There are n replicates. Assume the restricted model.

Write down the full linear model for the response including all possible interactions.
Find the number of degrees of freedom for all sums of squares.
Write down the expressions for all expected mean squares.

Propose F-tests for the significance of all effects. In case an exact F-test is not possible, propose an approximate F-test.

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