StdOrder RunOrder CenterPt Blocks Temp Time Conc Residue

$151112010263$

$261118010244$

$321112030250$

$411118030241$

$581112010658$

$641118010645$

$771112030652$

$831118030641$

$9151212010263$

$10131218010242$

$11161212030241$

$1291218030245$

$13141212010661$

$14101218010643$

$15121212030650$

$16111218030645$ In the restaurant business, the cleanliness of utensils $($forks$,$ knives, and spoons$)$ is critical for customer satisfaction. Temperature $(120\backslash$deg F$,180\backslash$deg F$),$ Time $(10$ seconds, $30$ seconds$),$ and Concentration of the cleaning solution $(2$ ppm$,6$ ppm$)$ have been identified as the key input variables that affect cleanliness.

After cleaning each utensil, the utensil is soaked in a solvent that removes any remaining dirt or grime. The solvent is then evaporated and the remaining residue is measured.

The problem is to find the factor settings that produce the cleanest utensils.

Objective:

You are looking for the treatment that produces the lowest residue. A meaningful improvement in cleanliness is a reduction in residue of five units. The standard deviation in residue is approximately two units. You must determine the number of replicates needed in a three$-$factor full factorial design to detect a difference of five units with at least $80\%$ power.

Data Collection:

Only eight runs can be performed in a single day. If you choose a design that requires more than eight runs, you need to add a blocking variable that accounts for the day on which the experiment is run.

Questions:

$(5$ points$)$ Using the information provided in the Problem and Objective statements above, you need to determine how many times you should replicate your design to achieve the power necessary to detect effects of size five units or larger at a significance level of $10\%.$ Show your work.

$(5$ points$)$ Identify $3$ ways in which you could increase the power of your design and provide a brief description of each approach.

$(5$ points$)$ Using the information provided in above and considering your answer to questions $1,$ create a Treatment design for this experiment using Minitab.

$(10$ points$)$ Despite your experimental design plans, the attached file shows the factorial design that was actually conducted. Analyze this data using the most complete model possible with Minitab. Provide a descriptive analysis of all figures and output.

$(10$ points$)$ The fact that concentration is not significant may seem counter$-$intuitive because you might expect that a higher concentration results in a lower residue. What practical reason might explain this outcome?

$(10$ points$)$ Analyze this data using the most appropriate model possible by removing insignificant terms from the model using a $10\%$ significance level. Provide a descriptive analysis of all figures and output.

$(10$ points$)$ Verify the model assumptions $($normality$,$ independence, and homogeneity of variance$).$ Which of the assumption$($s$),$ if any, are violated?

$(20$ points$)$ We have just learned that this dataset has a data point that was entered incorrectly. In fact, the observation in row $11($Temp$=120,$ Time$=30,$ Conc$=2)$ was coded as a $41,$ but should have been $51.$ The data must now be corrected and reanalyzed with the corrected data value before you can make any conclusions. Provide a descriptive analysis of all figures and output.

$(15$ points$)$ Create factorial plots for only significant $(\backslash$alpha $=0\mathrm{.}10)$ factors and provide a descriptive analysis for each figure.

$(10$ points$)$ It is beneficial to have an estimate of power for your tests to assess the validity of the results. Suppose your current process typically leaves an average residue of $55$ units. You want to improve your process so that you can detect a difference in the residue of $5$ units $($that is$,$ the residue is $50$ or Iess$).$ You used a $2-$level factorial design with $3$ factors, $8$ runs, $2$ replicates, $2$ blocks, and no center points $(\backslash$alpha $=0\mathrm{.}10).$ Using the adjusted mean square residual error as an estimate of process variance, find the power for this model. Provide a descriptive analysis of all figures and output.