STATISTICS 402 - Assignment 4
Due March 4, 2011
1. Oleoresin, which is used to make turpentine and rosin, is obtained from pine trees by cutting
a hole in the bark and collecting the resin that seeps out. We want to design an experiment to
see if the shape of the hole (circle, triangle or rectangle) has an effect on the amount of resin
obtained. A second factor, the application of an acid (no acid or acid) to the hole is to be
investigated in a two factor completely randomized experiment.
a) What are the response, conditions (factors and levels the experimenter is going to
manipulate) and the experimental material used in this experiment?
b) The experimenter will use factorial crossing to create the treatment combinations. How
many treatment combinations will there be in the experiment? List all the treatment
combinations.
c) The experimenter would like to be able to detect a difference in treatment means of 2.5
standard deviations while keeping both chances of error at 5%. How many experimental
units will the experimenter need?
d) With this number of units, what size difference in factor level means can be detected
when Alpha=0.05 and Beta=0.10?
e) Because of budget constraints, only 3 units are available for each treatment combination.
How does this choice affect the size of the detectable difference in treatment means? in
factor level means? Use Alpha=0.05 and Beta=0.10.
Pine plantations are easily differentiated from natural pine forests due to the regular spacing
of pines in rows.
f) Explain how you would randomly select units for the experiment from a plantation
consisting of 20 rows with each row having 50 pine trees. Remember the budget
constraints in e).
g) Explain how you would randomly assign treatments to the units you selected in f).
Include a table that indicates the random assignment of treatments to units. Remember
the budget constraints in e).
2. Most short-run supermarket strategies such as price reductions, media advertising, and instore promotions and displays are designed to increase unit sales of particular products
temporarily. Factorial designs have been employed to evaluate the effectiveness of such
strategies. Two factors examined are Price Level (regular, reduced price and cost to the
supermarket) and Display Level (normal display space, normal display space plus end-ofaisle display, and twice the normal display space). A complete factorial experiment based on
these two factors involves nine treatments. Suppose each treatment is applied three times to a
particular product, 14 ounce boxes of Cheerios, at a particular supermarket. Each application
lasts a full week. To minimize treatment carryover effects, each treatment is preceded and
followed by a week in which the product is priced at its regular price and is displayed in its
normal manner. The experiment is run as a completely randomized design, that is, treatments
1
Regular Price
Normal Display
Normal Display
Plus End of Aisle
Twice Normal
Display
949
1045
1141
1031
1253
1151
1201
1178
1080
Reduced
Price
1021
1226
1122
1801
1940
1956
1486
1421
1308
Cost to
Supermarket
1557
1506
1668
2102
2188
2031
1772
1803
1912
a) What is the response? What are the conditions? What is the experimental material?
b) Why is random assignment of treatments to weeks during the experimental period
important for this experiment? Be specific and be sure your answer deals with this
experiment.
c) Give the full model for these data. Be sure to define all the terms in the model within the
context of the problem.
d) Are there any treatment effects different from zero? Support your answer with the
appropriate statistical test of hypothesis. Be sure to give the null and alternative
hypotheses, value of the appropriate test statistic, P-value, decision and reason for the
decision and a conclusion in the context of the problem.
e) Are there any display effects different from zero? Support your answer with a statistical
test of hypothesis. Be sure to give the null and alternative hypotheses, value of the
appropriate test statistic, P-value, decision and reason for the decision and a conclusion in
the context of the problem.
f) Where are the statistically significant differences in display sample means? Support you
answer with an appropriate multiple comparison procedure.
g) Are there any price level effects different from zero? Support your answer with a
statistical test of hypothesis. Be sure to give the null and alternative hypotheses, value of
the appropriate test statistic, P-value, decision and reason for the decision and a
conclusion in the context of the problem.
h) Where are the statistically significant differences in price level sample means? Support
you answer with an appropriate multiple comparison procedure.
i) Are there any interaction effects that are different from zero? Support your answer with a
statistical test of hypothesis. Be sure to give the null and alternative hypotheses, value of
the appropriate test statistic, P-value, decision and reason for the decision and a
conclusion in the context of the problem.
j) Construct an interaction plot. Comment on the plot and what it tells you about the
interaction between the two factors. Be specific and be sure your answer deals with this
experiment.
k) Construct a plot of residuals versus predicted values. Describe the plot and indicate what
this tells you about the Fisher conditions necessary for the analysis of variance.
l) Construct plots of the distribution of residuals. Describe each of the plots in the
distribution of residuals. Indicate what this tells you about the Fisher conditions
necessary for the analysis of variance.
m) What else does the distribution of residuals tell you about the experiment?
Turn in JMP output that you have used to answer the questions.
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