Question: Cooley, Franklin, and Elrod in a 1999 Quality Engineering article describe a 23 factorial experiment aimed at understanding the effects of A-hole type (in the fan "spider"), B-barrel type (to which the fan "spider" was attached), and C-assembly method on the torque, y (in ftlbs), required to break some industrial fans (large y is good). m = 8 fans of each of the 23 types were tested. Below are estimated mean responses for each of the 23 combinations. (For all except combination ac, these are sample means, but a peculiarity of the data collection for that combination necessitates the use of something more complicated for ac. (Here ignore this issue and just treat the values below as if they were sample means of m = 8 torques.)

(a) Use the Yates algorithm and compute the fitted effects for the "all high" treatment combination.

(b) Use 7.2 ftlbs as an estimate of the standard deviation of breaking torques for any fixed combination of hole type, barrel type and assembly method. (7.2 is not actually in this problem, but treat it as if it were.) Which of the 23 factorial effects do you judge to be "clearly more than noise"? Show appropriate calculations to support your conclusion. Use a 95 % confidence level.

(c) Suppose that one somehow judges that the only effects of practical importance in this study were A, B, and C main effects and the A × B interactions. What combination of hole type, barrel type, and assembly method has the largest predicted breaking torque, and what is this predicted value, y?