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.