Solutions to Spring 2002 Stat 402 Final Exam
1 a)
SST=6(42-39.25)^2+6(40-39.25)^2
+6(36-39.25)^2+6(39-39.25)^2=112.5
MSE=(8+6+5+9)/4=7
F=MST/MSE=5.36
Table value: F_0.05,3,20=3.10
Conclusion: Reject H_0. There were significant differences
among the means.
b) t=[(42-40)-(36-39)]/sqrt[(7/6)(1^2+(-1)^2+(-1)^2+1^2)]
=2.31
t_0.025,20=2.086
There was significant evidence of interaction.
c) For variety=V1, the confidence interval for F1-F2
is 2.8 to 9.2
For variety=V2, the confidence interval for F1-F2
is -2.2 to 4.2
When considering variety 1, fertilizer 1 is better than
fertilizer 2 by somewhere between 2.8 and 9.2 bushels per
acre on average.
When considering variety 2, there may be no difference
between the fertilizers. Fertilizer 1 could be better by
as much as 4.2 bushels per acre or it could be worse by as
much as 2.2 bushels per acre on average.
2. a)
F=5.69/11.51=0.49<5.59=F_0.05,1,7
No significant difference between the mean readings for
spectrometer A and B.
b) (12.69+1.26-1.55-11.51)/6=0.14833333
c) (1.55-1.26)/2=0.145
d) spectrometer*sample(plot)
3. a) b)
SOURCE DF
feed 1 pen(feed)
pen(feed) 10 random
drug 1 drug*pen(feed)
feed*drug 1 drug*pen(feed)
drug*pen(feed) 10 random
c.total 23
c) On each farm, I would randomly select one pen to get
the feed additive. The other pen would get the control.
I would assign the drug treatment exactly as before.
d)
farm 5 random
feed 1 farm*feed
farm*feed 5 random
drug 1 error
feed*drug 1 error
error 10 random
c.total 23
Note that error=farm*drug+farm*feed*drug.
Each contributes 5 of the 10 total d.f. for error.
4. a)
chamber a chamber b chamber c
run 1 12 24 8
run 2 24 8 12
run 3 8 12 24
Both run and chamber are factors that could affect the
response. We should treat both as blocks and use a
Latin square design.
b)
This is an example of a split-plot experiment where the whole-plot part of the experiment is designed as a Latin square. We talked about split-plot experiments where the whole-plot part was either a completely randomized design (CRD) or a randomized complete block design (RCBD), but any other design could be used for the whole-plot part of the experiment. If we just look at the whole-plot part of the experiment we would have the following table:
SOURCE DF
run t-1=2
chamber t-1=2
temp t-1=2
error (t-1)(t-2)=2
c.total t^2-1=8
You could also get the d.f. by noting that there are 9 whole-plot experimental units. Thus the c.total has to be 8. With 2 for each of run, chamber, and temp; that leaves 2 for error.
As for the split-plot part, we have four pots within each whole-plot experimental unit. These four pots are associated with the following split-plot "treatments":
1. inbred Mo17, chemical treatment U
2. inbred Mo17, chemical treatment V
3. inbred Co255, chemical treatment U
4. inbred Co255, chemical treatment V
It is natural to break down this split-plot "treatment" into two factors and their interaction:
inbred
chemical
inbred*chemical
We also consider the interaction between these split-plot fixed effects and the whole plot factor=temp.
The s.p. error d.f. are obtained by subtraction. We know that the c.total has to be 35 because there are 4*9=36 observations. Thus we need 18 d.f. for the s.p. error to get this to work out. The complete breakdown is below.
run 2
chamber 2
temp 2
w.p. error 2 <---could specify this in SAS as
inbred 1 run*chamber or run*temp or
chemical 1 temp*chamber or
inbred*chemical 1 run*chamber*temp
temp*inbred 2
temp*chemical 2
temp*inbred*chemical 2
s.p. error 18
c.total 35
Note that when writing SAS code, we need to specify the whole-plot error. We could put in any of the terms that I mentioned above to get SAS to compute the 2 d.f. error term for the whole-plot portion of the experiment. Regardless of which term you put (run*chamber, run*temp, chamber*temp, or the three-way interaction) the term will serve as the 2 d.f. whole-plot error. Note that any of these terms correspond to whole-plot experimental units. For example, if I pick a run and a chamber, that will correspond to exactly one of the 9 whole-plot experimental units. If I pick a run and a temperature, that will again correspond to one of the whole-plot experimental units. The same statement can be made for any of the interactions mentioned.
c) If the experiment had been conducted in a single run, the temperature factor would have been completely confounded with the chamber factor.