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.