# 2004 Midterm 2

```Name
ST 711
Midterm 2
11/5/2004
You may not communicate with or receive information from any other person (except the
instructor) during this exam. Please sign the following honor pledge.
I have neither given nor received unauthorized aid on this test.
________________________________________


If you have questions about terminology during the exam, ask the instructor.
Show your work. Partial credit is given.
1. A consumer magazine wants to compare 10 different models of home charcoal grills for
grilling food outdoors. They contract with three chefs, who were selected randomly from a
large pool of volunteers. They give each chef one grill of each model to test, and instruct
each chef to cook one piece of salmon, one piece of trout, and a half dozen shrimp on each
grill. The response variable is a measure of moistness of the cooked seafood. Some SAS
output from Procs GLM and Mixed is given on page 3. The model, F-tests, and so forth
may or may not be correct for this problem, but you can compute all needed quantities from
the given output.
a. (10 pts) What type of experimental design is this?
b. (10 pts) Write down a statistical model matching the design. Define all terms
and subscripts. Write down the common assumptions for this model.
c. (10 pts) Compute the standard error for the mean difference between grill
models 1 and 4.
d. (10 pts) What proportion does variation among chefs contribute to the total
variance of y?
--------------------------------------------------------------------------------------------Problem 1. Compare Grill Models
1
The GLM Procedure
Class Level Information
Class
chef
grillmodel
fish
Levels
3
10
3
Values
1 2 3
1 2 3 4 5 6 7 8 9 10
salmon shrimp trout
Number of observations
90
------------------------------------------------------------------------------------------------Dependent Variable: y
Sum of
Source
DF
Squares
Mean Square
F Value
Pr &gt; F
Model
49
3712.291737
75.761056
58.92
&lt;.0001
Error
40
51.432448
1.285811
Corrected Total
89
3763.724185
R-Square
0.986335
Coeff Var
-51.14246
Root MSE
1.133936
y Mean
-2.217211
Source
chef
grillmodel
chef*grillmodel
fish
grillmodel*fish
DF
2
9
18
2
18
Type I SS
914.500692
125.823946
121.034928
2440.882800
110.049371
Mean Square
457.250346
13.980438
6.724163
1220.441400
6.113854
F Value
355.61
10.87
5.23
949.16
4.75
Pr &gt; F
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
Source
chef
grillmodel
chef*grillmodel
fish
grillmodel*fish
DF
2
9
18
2
18
Type III SS
914.500692
125.823946
121.034928
2440.882800
110.049371
Mean Square
457.250346
13.980438
6.724163
1220.441400
6.113854
F Value
355.61
10.87
5.23
949.16
4.75
Pr &gt; F
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
------------------------------------------------------------------------------------------------The GLM Procedure
Least Squares Means
chef
1
2
3
grillmodel
1
2
3
4
5
6
7
8
9
10
y LSMEAN
2.28119177
-4.21154419
-4.72128050
y LSMEAN
-3.23064228
-3.22636717
-1.97915027
-2.31673740
-3.34912816
-3.73283947
-1.82386441
-2.05503707
0.09054120
-0.54888472
Standard
Error
0.20702747
0.20702747
0.20702747
Standard
Error
0.37797872
0.37797872
0.37797872
0.37797872
0.37797872
0.37797872
0.37797872
0.37797872
0.37797872
0.37797872
Pr &gt; |t|
&lt;.0001
&lt;.0001
&lt;.0001
Pr &gt; |t|
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
&lt;.0001
0.8119
0.1543
Standard
fish
y LSMEAN
Error
Pr &gt; |t|
salmon
3.31376088
0.20702747
&lt;.0001
shrimp
-9.19430500
0.20702747
&lt;.0001
trout
-0.77108880
0.20702747
0.0006
------------------------------------------------------------------------------------------------The Mixed Procedure
Covariance Parameter
Estimates
Cov Parm
chef
chef*grillmodel
Residual
Estimate
15.0175
1.8128
1.2858
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
259.8
265.8
266.3
263.1
Type 3 Tests of Fixed Effects
Num
Den
Effect
DF
DF
F Value
grillmodel
9
18
2.08
fish
2
40
949.16
grillmodel*fish
18
40
4.75
Effect
grillmodel
grillmodel
grillmodel
grillmodel
grillmodel
grillmodel
grillmodel
grillmodel
grillmodel
grillmodel
fish
fish
fish
2.
fish
salmon
shrimp
trout
Least Squares Means
Standard
grillmodel
Estimate
Error
1
-3.2306
2.3985
2
-3.2264
2.3985
3
-1.9792
2.3985
4
-2.3167
2.3985
5
-3.3491
2.3985
6
-3.7328
2.3985
7
-1.8239
2.3985
8
-2.0550
2.3985
9
0.09054
2.3985
10
-0.5489
2.3985
3.3138
2.2603
-9.1943
2.2603
-0.7711
2.2603
Pr &gt; F
0.0890
&lt;.0001
&lt;.0001
DF
18
18
18
18
18
18
18
18
18
18
40
40
40
t Value
-1.35
-1.35
-0.83
-0.97
-1.40
-1.56
-0.76
-0.86
0.04
-0.23
1.47
-4.07
-0.34
Pr &gt; |t|
0.1947
0.1953
0.4201
0.3469
0.1796
0.1370
0.4569
0.4028
0.9703
0.8216
0.1505
0.0002
0.7348
A chemistry teacher wants his students to compare 7 methods of measuring volume. In
one class period there is time to complete just 3 of the measurement procedures. The
teacher divides the students into 7 groups and asks each group to measure a set volume of
water using 3 of the methods. He considers two different cyclic designs for assigning the
methods to the groups.
a. (10 pts) Construct a cyclic design for 7 treatments with starting block (0 1 3).
b. (10 pts) Construct a cyclic design for 7 treatments with starting block (0 2 4).
c. (10 pts) Which design do you recommend and why?
d. (10 pts) For the design of part (a), what is the variance of the mean difference
between method 1 and method 7?
e. (10 pts) Compute the efficiency factor for the design of part (a).
f. (10 pts) Relative efficiency compares two variances. For part (e) of this
problem what two variances would be compared? Be explicit about the type of
design and number of replicates in the designs that are compared.
```