Solutions to the Spring 2001 Stat 402 Final Exam

1. a)

SOURCE         DF        
garden          1
rate            2    <--- These three lines would add to
garden*rate     2         model d.f. of 5


The error would have 18-6=12 d.f.

b)  yhat=-1.11+3.69+-.58*rate+1.33*rate=2.58+0.75*rate

c)  yhat=-1.11+1.33*rate

d) t=-2.38  or F=5.69 p-value=0.0318

e) garden 1: 2.58+0.75*7=7.83
   garden 2: -1.11+1.33*7=8.2

Garden 2 would have the higher expected yield.

2 a) F=11.08   p-value=0.0076
There was a significant difference between soil types.

b)

Soil 1:   (8+3.5+10)/3=7.1667
Soil 2:   (6+2+7.5)/3=5.1667

c) Potting soil 2 because it had the smallest lsmean and
that mean was significantly less than the lsmean for
potting soil 1 (p-value=0.0076).
d)  (8+6)/2 - (3.5+2)/2
            + or - 
2.228*sqrt[(0.5^2)/2+(0.5^2)/4+(0.5^2)/4+(0.5^2)/1]
2.67 to 5.83

e) Variety 1 took significantly longer to germinate
than variety 2.

3. a) Latin square

12345
23451
34512
45123
41234

b)

Because of the breaks over the weekend, Mondays are likely 
to be more similar than days in general.  The same goes for 
Tuesdays, Wednesdays, etc.  Thus it is important to 
construct blocks according to day of the week.  (For 
example, the 5 Mondays are one block, the 5 Tuesdays are 
another block, etc.)

Performance could also vary with week.  For example, the 
professor could get a cold that would reduce performance 
for a whole week.  The professor might get tired by week 5 
or perhaps stronger.  Thus it is also important to consider 
the days in one week a block.

4. a)

Use a randomized complete block design (RCBD) with the
feed lots as blocks.  For example...

FEED       PEN
LOT   1  2  3  4  5
1     A  C  D  E  B
2     D  B  A  E  C
3     A  D  E  C  B
4     E  D  C  B  A

b)  pens

c) steers

d)

SOURCE           DF
lot               3
diet              4
lot*diet         12
steer(lot diet) 100
c.total         119

e)  y_ijk=mu+L_i+d_j+(LD)_ij+e_ijk

y_ijk is ribeye area of the kth steer at feed lot i that
      received diet j.

mu is overall mean

L_i are random effects for feed lots  (i=1 to 4)
 
d_j are fixed effects for diets (j=1 to 5)

(LD)_ij are random effects for lot*diet interaction
        they correspond to pens

e_ijk are random effects for steers in pens (k=1 to 6)

All random effects are normally distributed and independent
with mean 0.  The L_i have one common variance component.
The (LD)_ij have one common variance component.  The
e_ijk have one common variance component.

proc glm;
  class lot diet;
  model y=lot diet lot*diet;
  random lot lot*diet;
run;

proc mixed method=type3;
  class lot diet;
  model y=diet;
  random lot lot*diet;
run;


5.  a)                           b)

SOURCE             DF     
time                1          bank(time)
bank(time)          8          random
grass               3          grass*bank(time)
time*grass          3          grass*bank(time)
grass*bank(time)   24          random
c.total            39

c) (i)  numerator=3, denominator=24
   (ii) There was no significant difference among grasses
        at one month, but there are significant differences
        among grasses at three months.  This makes sense as
        it probably takes awhile for the grasses to appear.