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
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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.