ST 711. 9/24/2004 Midterm 1
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I have neither given nor received unauthorized aid on this
test.________________________________________
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the instructor) during this exam.

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If you have questions about terminology during the exam, ask the instructor.
Show your work. Partial credit is given.
1. A researcher is planning an experiment to study the relationship between X and
Y. He thinks there is a straight line relationship between X and Y, so that yi = β0
+ β1xi + εi . The researcher is considering the three designs using the following
values of X:
Design (i): X = 0, 0, 0, 10, 10, 10
Design (ii): X = 0, 2, 4, 6, 8, 10
Design (iii): X= 0,0, 5, 5, 10,10 .
The variance-covariance matrices of the parameter estimates are as follows for the
three designs:
(i)
Var(β) = | .3 -.03 | (ii) |.5 -.07 | (iii) | .4 -.05 |
| -.03 .007 |
| -.07 .01 |
| -.05 .01 |
(a) Which design gives the best (lowest variance) estimate of the slope?
(b) Which design, (i) or (iii), predicts Y at X=2 with the smaller variance?
(c) Would you recommend using Design (i)? List the advantages and
disadvantages of Design (i).
2. A forester is planning an experiment to compare the accuracy of four types of
GPS (Global Positioning System) units. Features in the landscape, such as
buildings and fences, can affect the accuracy of GPS measurements. The
proposed experiment is to have two factors:
A. Type of GPS unit (4 types)
B. Landscape features (2 landscapes: (1) open field, (2) near a fence).
(a) Use the following list of 16 random numbers to create a completely
randomized design with two replicates:
.14, .45, .10, .22, .32, .62, .56, .24, .13, .18, .09, .99, .65, .79, .97,.59
Show the steps in creating the design and draw a diagram of the resulting
design.
An experiment like this was conducted. Some SAS output from this experiment is
shown below.
Source
Sum of
Squares
DF
Model
Error
Corrected Total
7
8
15
Source
391.9375000
28.5000000
420.4375000
DF
A
B
A*B
Mean Square
3
1
3
Type III SS
55.9910714
3.5625000
Mean Square
297.1875000
60.0625000
34.6875000
A
B
y LSMEAN
1
1
2
2
3
3
4
4
1
2
1
2
1
2
1
2
11.0000000
12.5000000
5.5000000
6.0000000
14.0000000
21.5000000
10.5000000
16.5000000
99.0625000
60.0625000
11.5625000
Parameter
Intercept
a
a
a
a
b
b
a*b
a*b
a*b
a*b
Estimates
Estimate
16.50000000 B
1
-4.00000000 B
2
-10.50000000 B
3
5.00000000 B
4
0.00000000 B
1
-6.00000000 B
2
0.00000000 B
1 1
4.50000000 B
1 2
0.00000000 B
2 1
5.50000000 B
2 2
0.00000000 B
Standard
Error
1.33463478
1.88745861
1.88745861
1.88745861
.
1.88745861
.
2.66926956
.
2.66926956
.
a*b
a*b
a*b
a*b
3
3
4
4
2.66926956
.
.
.
.
1
2
1
2
-1.50000000
0.00000000
0.00000000
0.00000000
B
B
B
B
NOTE: The X'X matrix has been found to be singular, and a generalized
inverse was used to solve the normal equations. Terms whose estimates
are followed by the letter 'B' are not uniquely estimable.
(b) What is the standard error of the difference between the mean of GPS unit 2
and the mean of the other 3 GPS units?
(c) Landscape 2 is near a fence. It is known that fences can interfere with GPS
readings. What is the estimated mean difference between GPS units 2 and 4 if
the readings are taken in landscape 2?
(d) Show how to obtain this estimate from the section of the output labeled
“Parameter Estimates”.
(e) What is the standard error of the estimated mean difference computed in part
(c)?
3. A group making up ballots for an election has four candidates for office. They
want to make up 4 different ballots, with the candidate names in a different order
on each ballot. Propose a set of 4 ballots such that every candidate appears first on
one ballot, second on another ballot, third on another ballot and fourth on another
ballot.