The Accompanying Data Resulted from A $2^3$ Experiment with Three Replications Per Combination of Treatments Designed to Replications

Essay

The accompanying data resulted from a $2^3$ experiment with three replications per combination of treatments designed to study the effects of concentration of detergent (A), concentration of sodium carbonate (B), and concentration of sodium carboxymethyl cellulose (C0 on cleaning ability of a solution in washing tests (a larger number indicates better cleaning ability than a smaller number). $$\begin{array} { c c c c c c } \hline \text { Factor Levels } \\A & B & C & \text { Condition } & \text { Observations } \\\hline 1 & 1 & 1 & ( 1 ) & \begin{array} { c } 106,93 , \\116\end{array} \\\hline 2 & 1 & 1 & a & 198,200,214 \\\hline 1 & 2 & 1 & b & 197,202 , \quad 185 \\\hline 2 & 2 & 1 & a b & 329,331,307 \\\hline 1 & 1 & 2 & c & 149 , \quad 169 , \quad 135 \\\hline 2 & 1 & 2 & a c & 243,247,220 \\\hline 1 & 2 & 2 & b c & 255,230,252 \\\hline 2 & 2 & 2 & a b c & 383,360,364 \\\hline\end{array}$$
a. After obtaining cell totals
compute estimates of $\beta _ { 1 } , \gamma _ { 11 } ^ { λ c } , \text {and } \gamma^ { λ c } _ { \mathbf {210 } }$
b. Use the cell totals along with Yate's method to compute the effect contrasts and sums of squares. Then construct an ANOVA table and test all appropriate hypotheses using $\alpha = .05$