USE PYTHON ONLY$6\mathrm{.}9$ A router is used to cut locating notches on a printed circuit board. The vibration level at the surface of the board as it is cut is considered to be a major source of dimensional variation in the notches. Two factors are thought to influence vibration: bit size $(A)$ and cutting speed $(B).$ Two bit sizes $(\frac{1}{16}$ and $\frac{1}{8}$ in$.)$ and two speeds $(40$ and $90rpm)$ are selected, and four boards are cut at each set of conditions shown below. The response variable is vibration measured as the resultant vector of three accelerometers $,$ and $z$ on each test circuit board.

$\backslash$table$[[,,$Treatment,Replicate$],[A,B,$I,II$,$III,IV$,],[-,-,(1),18\mathrm{.}2,18\mathrm{.}9,12\mathrm{.}9,14\mathrm{.}4,],[+,-,a,27\mathrm{.}2,24\mathrm{.}0,22\mathrm{.}4,22\mathrm{.}5,],[-,+,b,15\mathrm{.}9,14\mathrm{.}5,15\mathrm{.}1,14\mathrm{.}2,],[+,+,ab,41\mathrm{.}0,43\mathrm{.}9,36\mathrm{.}3,39\mathrm{.}9,]]$

$($a$)$ Analyze the data from this experiment.

$($b$)$ Construct a normal probability plot of the residuals, and plot the residuals versus the predicted vibration level. Interpret these plots.

$($c$)$ Draw the $AB$ interaction plot. Interpret this plot. What levels of bit size and speed would you recommend for routine operation?