STATISTICS 402 - Assignment 4
Due March 3, 2006
1. Oleoresin 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 units 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
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
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.
(f) Come up with a randomization of the runs (assuming 3 units for each treatment
combination) for the experiment. That is, give me the randomized order that you
would use if you were the experimenter.
2. Most short-run supermarket strategies such as price reductions, media advertising, and
in-store 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-of-aisle display, 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 at a particular supermarket. Each application lasts a full week and the response variable of interest is unit sales for the 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 data, numbers of items sold, are given on the next page.
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Price
--------------------Regular Reduced
Cost
Display
Normal
949
1045
1051
1321
1327
1222
1557
1536
1638
Normal Plus
1031
1163
1151
1801
1940
1956
2502
2558
2461
Twice Normal
1201
1178
1080
1546
1521
1448
1772
1803
1912
(a) What is the response? What are the conditions? What are the units?
(b) Are there any differences among the nine treatments? Support your answer with
a statistical test of hypothesis.
(c) Are there any differences among the three different displays? Support your answer
with a statistical test of hypothesis.
(d) If there are any differences among the three different displays, where are those
differences? Support you answer with a multiple comparison procedure.
(e) Are there any differences among the three different price levels? Support your
answer with a statistical test of hypothesis.
(f) If there are any differences among the three different price levels, where are those
differences? Support you answer with a multiple comparison procedure.
(g) Is there interaction between the two factors? Support your answer with a statistical test of hypothesis.
(h) Construct an interaction plot. Comment on the plot and what it tells you about
interaction between the two factors.
(i) Construct a plot of residuals versus predicted values. Describe the plot and indicate what this tells you about the conditions necessary for the analysis of variance.
(j) Look at the distribution of residuals. Describe the distribution of residuals. Indicate what this tells you about the conditions necessary for the analysis of variance.
(k) What else does the distribution of residuals tell you about the experiment?
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