STAT 411/616
Homework 3 – Design of Experiments
Rules
The report for the assignment should be in a single .pdf file, uploaded to Canvas with hard copy to the TA
at or before the due date and time. Your report must include:
A cover page with name, class, assignment number, and date;
All of the code you used to solve each problem, along with RELEVANT output. You can include this
as part of the body of the report, or (preferably) you can put all of the code in an appendix of the report. Do
not include incorrect code or error messages, unless they are related to some point you are raising in your
discussion. All code/output must be in a monospace font (not proportional);
Handwritten or typed mathematic derivations or proofs if needed;
Any graphs that you used in answering the question. See R's help facility or go to the site
https://statistics.berkeley.edu/computing/saving-plots-r for information on saving graphs from R in a form
that can be incorporated into your report;
A written description of your conclusion for each question asked in the assignment.
Assignment overview
This assignment has 12 problems with 1 additional STAT 616-only problem.
STAT 411 Students choose 6 of these problems and solve.
STAT 616 Student choose 8 of these problems plus the 616 only problem.
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1. A series of runs were performed to determine how the wash water temperature and the detergent
concentration affect the bacterial count on the palms of subjects in a hand washing experiment.
a. Identify the experimental unit.
b. Identify the factors.
c. Identify the response.
2. Explain the difference between an experimental unit and a sub-sample or sub-unit in relation to the
experiments described in l.
3. Explain the difference between a sub-sample and a duplicate in relation to the experiment described in
1.
4. Describe a situation within your realm of experience (your work, your hobby, or school) where you
might like to predict the result of some future action. Explain how an experimental design, rather than
an observational st udy, might enhance your ability to make this prediction.
5. The effect of plant growth regulators and spear bud scales on spear elongation in asparagus was
investigated by Yang-Gyu and Woolley (2006). Elongation rate of spears is an important factor
determining final yield of asparagus in many temperate climatic conditions. Spears were harvested
from 6-year-old J ersey Giant asparagus plants grown in a commercial planting at Bulls (latitude 40.2S,
longitude 175.4E), New Zealand. Spears were harvested randomly and transported from field to lab for
investigation. After trimming to 80mm length, spears were immersed completely for 1h in aqueous
solutions of 10 mg 1-1 concentration of indole-3-acetic acid (IAA), abscisic acid (ABA), GA3, or CPPU
(Sitofex EC 2.0%; SKW, Trostberg, Germany) in test tubes. Control spears were submerged in distilled
water for 1h.
The experiment was a completely randomized design with five replications (spears) per treatment. The
resulting data (final spear length in mm) is shown below.
a. Perform the analysis of variance to test the hypothesis of no treatment effect.
b. Use the Tukey method to test all pairwise comparisons of treatment means.
c. STAT 616. Use the Dunnett procedure to compare all treatment group means to the control
mean
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6. Paint used for marking lanes on a highway must be very durable. In one trial paint from four
differrence suppliers, labeled GS, FD, L and ZK, were tested on six different highway sites, denoted
1,2,3,4,5,6. After considerable lenght of time, which included different levels of traffic and weather, the
average wear for the samples at the six sites was as follows:
The objective was to compare the wear of the paints from the different suppliers.
a. What kind of an experimental design is this.
b. Make an ANOVA.
c. Obtain confidence limits for the supplier averages.
d. Make checks that might indicate departures from assumptions.
e. Do you think these data contain bad values?
f. What can you say about the relative resistance to wear of the four paints~
g. Do you think this experimental arrangement was helpful?
7. Consider planning an experiment to determine how the flavor duration of gum is affected by the
characteristics of the gum such as Factor A (type of gum stick or tablet), Factor B (flavor of gum), and
Factor C (regular or sugar-free), see Rogness and Richardson (2003).
a. If several subjects were recruited to participate in the study and each was asked to chew and
record the flavor duration (in minutes) of one or more types of gum, what would the
experimental unit be?
b. How could experimental units be blocked to reduce the variance of experimental error and
increase the power for detecting differences?
c. Choose at least two of the factors listed above and provide a randomized list for an RCB design
8. Lew (2007) presents the data from an experiment to determine whether cultured cells respond to two
drugs. The experiment was conducted using a stable cell line plated onto Petri dishes, with each
experimental run involving assays of responses in three Petri dishes: one treated with drug 1, one
treated with drug 2, and one untreated serving as a control. The data are shown in the table below:
STAT 411/616 Homework 3
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a. Analyze the data as if it came from a completely randomized design using the model
ij i ijy μ τ ε= + + . Is there a significant difference between the treatment groups?
b. Analyze the data as an RCB design, where experiment number represents a blocking factor.
c. Is there any difference in the results you obtain in (a) and (b)? If so explain what may be the
cause of the difference in the results and which method would you recommend?
9. Consider a 28-4 fractional factorial design. Answer the following questions:
a. How many factors does this design have.
b. How many runs are involved in this design.
c. How many levels for each factor.
d. How many independent generators are there for this design.
10. Repeat problem 9. for a 26-1 design
11. Melo et al. (2007) used a 24-1 factional factorial design with generator D = ABC to study the factors that
influence the production of levan by aerobic fermentation using the yeast Zymomonas mobilis. Levan
is a sugar polymer of the fructan group, which has been shown to have anti-tumor activity against
sarcoma and Ehrlich carcinoma in Swiss albino mice. The factors varied in the fermentation medium
and their levels are shown in the table below.
a. What is the defining relation and complete alias structure for this design?
b. What is the resolution of this design?
c. The fermentations were carried out batchwise in Pyrex flasks. After 72 hours of fermentation,
the levan produced was extracted from the fermented medium and its dry weight was
determined. The results (in g.L-1) for the eight experiments (in standard order) were: 4.70, 14.67,
1.71, 3.73, 9.47, 7.61, 0.15, 4.78. From this data, calculate the seven effects and make a normal
probability plot to determine what is significant.
d. Delete the smallest three effects and fit the model again. Are the four effects left in the model
significant?
e. Based on the effect heredity principle, what do you think the significant string of aliased two-
factor interactions could represent?
f. Can the design be collapsed to a full factorial by ignoring insignificant factors?
g. Based on what you said in (e) write an interpretation of the significant effects and interaction.
Based on your model determine the factor levels that will produce the maximum dry weight of
levan.
12. Create a central composite design (CCD) for two factors using R's rsm package.
a. Create the uniform precision CCD and store the design along with random numbers (simulated
response) in a data frame.
b. Use the Vdgraph package to make a variance dispersion graph or a fraction of design space
plot of the design you created.
c. Repeat (a) through (b) for a face-centered cube design (i.e., CCD with axial points at ±1).
d. Based on the graphs you made, which design do you prefer? State the reason.