Suppose we have a oneway ANOVA with five treatments. Denote the treatment means by θ1,..., θ5, where θ1 is a control and θ2, ..., θ5 are alternative new treatments, and assume that an equal number of observations per treatment is taken. Consider the four contrasts ˆ‘aiθi defined by

(a) Argue that the results of the four t tests using these contrasts can lead to conclusions about the ordering of θ1,..., θ5. What conclusions might be made?
(b) Show that any two contrasts ˆ‘aii. formed from the four ais in part (a) are uncorrelated. (Recall that these are called orthogonal contrasts.)
(c) For the fertilizer experiment of Example 11.2.3, the following contrasts were planned:

Show that these contrasts are not orthogonal. Interpret these contrasts in the context of the fertilizer experiment, and argue that they are a sensible set of contrasts.

B2 = (0, 1,-3'-3'-3 = (0, 0, 1,-2,-2), a4 (0,0,0,1,-1) ai = (-1,1,0,0,0), as (0,0,1,-1,0), 84= (0,-1,0, 0, 1, ).