You must use a $2k$ factorial design and analysis of variance to

investigate a new direct energy deposition machine produced

by Optomec. Titanium powder is fused onto a surface using a

CO$2$ laser with variable power output. The gap between the

powder nozzle and substrate surface can be varied. The

powder flowrate can also be varied. The design factors are

therefore:

Factor A: the gap between the nozzle and surface $(cm)$

Factor B: the powder flowrate $(cm\frac{3}{m}in)$

Factor C: the laser power $($W$)$

Each factor is run at two levels $($low and high$),$ and the design is

replicated twice. The response variable is the metal fusion

depth $(m).$

The metal fusion depth data are shown in the table below, and the design is geometrically illustrated below

the table.

$1.$ Estimate all the factor effects $($A$,$ B$,$ C$,$ AB$,$ AC$,$ BC$,$ ABC$)$ and interpret the results $($i$.$e$.$ explain what these

results inform you about the relative importance of the various factors and factor combinations$).(10$

points for the correct values, $5$ points for the explanation of their relative importance.$)$

$2.$ You must now determine if these effects are statistically significant. Follow the procedure below and show

all your detailed calculations. Also state all your assumptions.

$2\mathrm{.}1.$ Calculate the variability $($S$2)$ and the standard error $($s$.$e$.)$ of the factor effects. $(5$ points$)$

$2\mathrm{.}2.$ Calculate the $95\%$ confidence limits for each factor effect $($A$,$ B$,$ C$,$ AB$,$ AC$,$ BC$,$ ABC$)$ and plot the factor

effects to help you interpret the results $(7$ points for calculation $3$ points for plot.$).$

$2\mathrm{.}3.$Describe which effects and interactions are statistically significant, and which are not. Do the results of

this analysis agree with the conclusions you made in Question $1?$ Explain. $(10$ points$)$

$2\mathrm{.}4.$ Write a linear regression equation that could be used to predict the metal fusion depth over the

region of powder flow rate, laser power, and gap used in this experiment. $(5$ points$)$

$2\mathrm{.}5.$ Use the regression analysis to generate a response surface contour plot for metal fusion depth.

Suppose it is critically important to obtain layer thickness of $812\backslash$mu m$.$ What settings of powder

flowrate, gap, and laser power would you recommend? $(10$ points$)$