Question IV: $($a$)$ Design a $2^{5-2}$ experiment using design generators $I=$ABD & $I=$ACE, complete the design of test conditions in the following table. $(4\%)$

$\backslash$table$[[$Run$,$A$,$B$,$C$,$D$,$E$],[1,+1,+1,+1,,],[2,-1,+1,+1,,],[3,+1,-1,+1,,],[4,-1,-1,+1,,],[5,+1,+1,-1,,],[6,-1,+1,-1,,],[7,+1,-1,-1,,],[8,-1,-1,-1,,]]$

$($b$)$ What is the complete defining relationship of this experiment? $(2\%)$

$($c$)$ Find the alias structure for main effects A & B and alias structure for two$-$factor interaction effects CD$.(3\%)$

$($d$)$ What is the resolution of this experiment? Explain your answer. $(3\%)$

$6$

$($e$)$ The experimenter finds that factors D&E actually has no impact on the test response and decide to drop them from further analysis. Tests using single replicate per combination of test conditions are given below. Using the data in the table below, draw the interaction plot of factors A and $C(5\%).$ Do factors A and $C$ have significant interaction effect? Explain. $(2\%)?$

$\backslash$table$[[$Run$,$Response$],[(1),56],[$a$,72],[$b$,68],[$ab$,89],[$c$.,43],[$ac$,62],[$bc$,59],[$abc$,75]]$

$($f$)$ Assume all interaction effects are not significant so one can use only main effect terms $($effects from A$,$ B and C$)$ to fit the response surface equation. Assume all variables are continuous and using the data above, fit a response model using the coded level of three factors: $(6\%)$

hat$(y)=65\mathrm{.}5+$

A$+$

B$+$

C$.$