The accompanying data resulted from a 23 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 (C) on the cleaning ability of a solution in washing tests (a larger number indicates better cleaning ability than a smaller number).

a. After obtaining cell totals xijk., compute estimates of β1, γ11AC and γ21AC.
b. Use the cell totals along with Yates's method to compute the effect contrasts and sums of squares. Then construct an ANOVA table and test all appropriate hypotheses using a 5 .05.
c. Suppose a low water temperature has been used to obtain the data. The entire experiment is then repeated with a higher water temperature to obtain the following data. Use Yates's algorithm on the entire set of 48 observations to obtain the sums of squares and ANOVA table, and then test appropriate hypotheses at level .05.
Condition ________Observations
d ..........................144, 154, 158
ad .........................239, 227, 244
bd .........................232, 242, 246
abd .......................364, 362, 346
cd .........................194, 162, 203
acd .......................284, 295, 291
bcd .......................291, 287, 297
abcd .....................411, 406, 395

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Factor Levels ABCCondition Observations 106, 93, 116 198, 200, 214 197, 202, 185 329, 331, 307 149, 169, 135 243, 247, 220 255, 230, 252 383, 360, 364 ab ac abc