Hints and/or comments about Problem 16
For part (a), you will have to derive the proper df formula,
using the method of Satterthwaite, to use with a simple approximate
chi-square pivot. My guess is that you should look over some pages in the
two-way mixed effects model part of Miller's book in order to figure
out how to modify what I did in the notes for the one-way random effects
model to handle the situation at hand. But since we have n = 1,
you'll have to look back in the two-way fixed effects model part of
Miller to see the proper ANOVA table for the n = 1 case, and use
some of what you find there along with some of what you get from the
mixed efects model section of Miller's book. (The main thing that is
different for a mixed effects model when n = 1 is the error term
part --- ones needs to use (I - 1)(J - 1) degrees of freedom.)
To derive the df formula, you should first make sure that you
understand the derivation I did on p. E31 of the class notes. Then all
you need to do is replace the estimator of the random effects term's variance
for the one-way model with the corresponding estimator for the two-way
mixed effects model (for the n = 1 case), and go through a
derivation of the df using similar steps.
Once you got the df, it should be fairly easy to finish things
off and obtain the desired confidence interval --- you'll just need to
use a modification (since the variance estimator is different) of the
approximate pivot result given near the bottom of p. E31. If you got
part (b) of Problem 14 wrong, it may be a good idea to make sure that
you understand how to do it correctly. In particular, given that one
should use 5 df, make sure that you can then obtain the lower and
upper confidence bounds. I've posted the correct answers on the
homework solutions web page, but
if you need more details then you may want to try to track me down tonight,
or tomorrow before class.