$(15$ points$)$ Exercise $6\mathrm{.}6$ of the Kuehl text describes a $3$ factor experiment to study the effects of

different treatments on the durability of fabric. The fabric is padded with a filler and the three factors

considered were the type of filler used to pad the fabric, the proportion of whichever filler was chosen,

and the type of surface treatment $($called S$1$ and S$2).$ The original experiment included $2$ types of fillers

with $3$ proportions, but I have modified the data by removing the middle proportion to construct a

$2\backslash$times $2\backslash$times $2$ factorial design. There are two observations for each treatment and the response is weight

loss in mg after an abrasion test. Less loss means that the fabric is more durable. The fabric

manufacturers would like to increase the proportion of filler to make a softer fabric, but they are

concerned that this may decrease the durability of the fabric. They would also like to know if there is a

difference in durability between the fabrics given the two surface treatments, since S$1$ is a much

cheaper treatment. Data for this question is available on Canvas in the file kuehl$\_6\_6\_$q$2.$csv$.$

a$.$ Write down the cell means model for the data and identify the components.

b$.$ Fit the ANOVA model and compute least squares estimates of the mean yield for each

treatment with Bonferroni corrected simultaneous $95\%$ confidence intervals.

c$.$ Assess the assumptions of the model and comment on any potential problems.