A research team wishes to specify a manufacturing process so that Y, the area in a product affected by surface flaws is as small as possible. They have three levels of concentration of a chemical used to wash the product before the final manufacturing step and want to determine whether the concentration level changes E(Y). They run a balanced one-way layout with J=8 observations for each concentration with level 1 set at 10%, level 2 at 15%, and level 3 at 20%. They observe that y1=22.7 , y2=12.3, y3=18.4 where yi is the average of the ovservations taken on the ith level. They also observe that s1^2=421, s2^2=486, and s3^2= 454, where si^2 is the unbiased estimate of the variance for the observations taken on the ith level.
Q: Partition the sum of squres due to the concentrations into sum of squares due to the linear contrast and the quadratic contrast. The vector of coefficients for the linear contrast is (-1,0,1). The vector for the quadratic contrast is (1,-2,1). Evaluate the importance of each of these sums of squares. Give the 99% Scheffe confidence intervals for the linear and quadratic contrasts.