Name_________________________________
ST 711. 12/6/2004 Final Exam
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.
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If you have questions about terminology during the exam, ask the instructor.
Show your work. Partial credit is given.
You may use one page (front and back) of notes.
1. A researcher proposes to test the effects of three factors (A, B, C) on the reliability of
a diagnostic test. Each of the three factors has two levels (a 23 factorial). Only four
treatment combinations can be run at each laboratory. The AB and AC interactions
are of particular interest to the researcher, but the BC and ABC interactions are of less
interest.
a. (10 points) Suppose that two laboratories are available. Which of the
following two assignments of treatments to laboratories would you
recommend, Design I or Design II? Explain why.
Design I
lab 1: abc ac bc c
lab 2: ab a b (1)
Design II
lab 1: ab ac bc (1)
lab 2: a b c abc
b. (10 points) Give a skeleton analysis of variance table for design I. Include
sources of variation and degrees of freedom.
c. (5 points) Suppose the researcher started with design II and then two more
laboratories become available. Propose a treatment assignment for these two
additional labs. Justify your recommendation.
2. A Christmas tree grower wants to compare 25 different families of Sitka Spruce,
which is a species of Christmas tree. There are 30 sites available, and each site has
space to grow 5 of the families.
a. (5 points) What type of design would you recommend? Choose one of the
following: Latin square, strip plot, balanced lattice, or cyclic incomplete block
design.
b. (5 points) List all of the important features of the design you recommended in
part (a) (e.g., the number of replicates, etc.).
c. (5 points) Show the first three replicates of the recommended design.
3. An airport plans to test four baggage handling procedures, labeled 0, 1, 2, and 3. The
airport has four terminals and plans to use the following experimental design.
Terminal
1 2 3 4
Day
0 1 3 2
Monday
3 0 2 1
Tuesday
Wednesday 1 2 0 3
2 3 1 0
Thursday
a. (5 points) What kind of design is this (give the name)?
b. (5 points) List all the important features of this design (e.g., the number of
replicates, etc.).
c. (5 points) Write an appropriate model for this experiment. Define all terms and
subscripts.
d. (5 points) State the usual assumptions. If you think that any of these assumptions
might be violated, give a one sentence explanation for each of how you think it
might be violated.
4. A psychologist is interested in the effects of chocolate on brain activity in men and
women. He conducts a study with three volunteers of each sex. The volunteers each
eat a standardized lunch. Two hours after eating lunch, each volunteer eats a piece of
dark chocolate. The brain activity of each volunteer is measured at two time points:
(i) just before eating the chocolate, and (ii) while the chocolate is in the mouth.
The model for this experiment is
Yijk = μ+ Si + ε(1)j(i) + Ck +(SxC)ik + ε(2)ijk,
where i= 1,2 sexes, j=1,…,3 subjects, and k=1,2 times (before or during eating
chocolate). There are several independent random effects:
Si ~ iid N(0,σs2), ε(1)j(i)~ iid N(0,σ12), and ε(2)ijk~ iid N(0,σ22).
The sample means of the four treatments are given in the following table.
Before Eating Chocolate While Eating Chocolate
Men
20
30
Women
26
36
MSE(1)=100
MSE(2)=50
a. (5 points) What does ε(1) stand for? Give a one-sentence explanation in nonstatistical terms.
b. (5 points) What does ε(2) stand for? Give a one-sentence explanation in nonstatistical terms.
c. (5 points) Compute the standard error for the mean difference between men
and women.
d. (5 points) Compute the standard error for the difference between brain activity
while eating chocolate and before eating chocolate for women.
e. (10 points) This model can be written in the form
y = Xτ + Zβ + ε.
Putting all fixed effects and the intercept in X and τ and random effects in Z and
β, write out the matrices y, X, and τ using effect coding.
f. (5 points) Write the hypotheses that the effect of chocolate is the same for
men and women in matrix notation, H0: Lτ = 0. Write out the elements of the
L matrix.
g. (5 points) Write out the test statistic, compute its value, and give the rejection
criterion for testing the hypothesis of part (f). Specify the distribution and
degrees of freedom.