This problem is based on a real experiment conducted by the Ceramics Division in the Material Science and Engineering Laboratory at the National Institute of Standards and Technology $($NIST$).$ The response variable is Ceramic Strength $($higher the better$)$ and the factors are the following:

Factor $1=$ Table Speed with $2$ levels: $-1(0\mathrm{.}025$ m$/$s$)$ and $+1(0\mathrm{.}125$ m$/$s$)$

Factor $2=$ Down Feed Rate with $2$ levels: $-1(0\mathrm{.}05$ mm$)$ and $+1(0\mathrm{.}125$ mm$)$

Factor $3=$ Wheel Grit with $2$ levels: $-1($low$)$ and $+1($high$)$

Factor $4=$ Direction with $2$ levels: $-1($longitudinal$)$ and $+1($transverse$)$

Factor $5=$ Batch with $2$ levels: $-1($small$)$ and $+1($big$)$

Factors $1$ and $2$ are $$numeric$.$

Factors $3,4,$ and $5$ are $$text$.$

Data was collected for $2$ replicates of a full$-$factorial $(25)$ design, and the non$-$randomized data is provided as

Replicate $1$ Replicate $2$

$670675$

$730700$

$453672$

$667643$

$704675$

$642650$

$560702$

$669683$

$492480$

$476450$

$490510$

$568545$

$490420$

$410430$

$450412$

$590603$

$607580$

$500485$

$620700$

$638628$

$585575$

$586560$

$602612$

$608600$

$443434$

$434424$

$418408$

$511505$

$392435$

$360560$

$386392$

$447440$

Perform appropriate statistical analysis $($at $5\%$ level of significance$)$ and answer the following:

i$.$ If Table Speed $=0\mathrm{.}025$ m$/$s$,$ what should be the level of Wheel Grit?

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ii$.$ If Down Feed Rate $=0\mathrm{.}05$ mm$,$ what should be the level of Table Speed?

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iii. If Table Speed $=0\mathrm{.}125$ m$/$s$,$ what should be the level of Down Feed Rate?

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iv$.$ If Table Speed $=0\mathrm{.}025$ m$/$s$,$ what should be the level of Batch?

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v$.$ If Batch $=$ small, what should be the level of Direction?

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vi$.$ If Direction $=$ longitudinal, what should be the level of Batch?

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vii. What should be the level of Wheel Grit for a Big Batch?

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viii. For a Transverse Direction and Small Batch, recommend a combination of Table Speed and Down Feed Rate that helps achieve a Strength of at least $530.$

$($Answering in words like $$bottom$-$left corner$$ will be sufficient.$)$

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ix$.$ If you plot a Surface Plot for Strength, with Table Speed and Down Feed Rate as X and Y axes, respectively, why does the Surface Plot have a curvature?

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x$.$ Why doesn$$t Minitab let you choose Batch for X or Y$-$axis in the Contour Plot?

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