Statistics 402 B & C
May 7, 2008
Final Exam
Name: __________________________
INSTRUCTIONS: Read the questions carefully and completely. Answer the questions
and show work in the space provided. This is the only work that I will look at. Partial
credit will not be given if work is not shown. Be sure to answer all questions within the
context of the problem. Refer to the computer printout and graphs provided when
appropriate. If the computer printout has the answer use it, you do not have to calculate
by hand. Pace yourself. Do not spend too much time on any one problem. Point values
for each problem are given.
1. [14 pts] For each of the following situations give the response, conditions of
interest and experimental units. Explain what design you would use to
accomplish the stated purpose. Be sure to support your choice of design.
a. [7] The investigator wishes to see if the time it takes a baseball player to
run from home plate to 2nd base is affected by the way the player rounds
1st base. There are three different ways to round first base; “round out,”
“narrow angle,” and “wide angle”. The three ways are illustrated below.
2nd
narrow angle
round out
1st
wide angle
Home
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b.
[7] Do students from different colleges score differently on multiplechoice and problem solving statistics tests? Students, randomly selected,
from Engineering, Business, Agriculture and Liberal Arts and Sciences
Colleges will participate. Each student will take both a multiple-choice
test and a problem-solving test over the same material in an introductory
statistics course. The order of the test, multiple choice or problem solving
first, will be randomized for each student.
2. [16 pts] For each of the following situations give the design that was used
(completely randomized, randomized complete block, Latin square, split
plot/repeated measures). If there is factorial crossing, indicate what the factors
are. Give a partial ANOVA table listing all sources of variation and degrees of
freedom.
a. [8] A fishing magazine hires a statistician to conduct an experiment on the
effects of A: reel type (Zebco and Generic), B: line type (Berkeley and
Generic), and C: sinker weight (25 g and 50 g) on the distance the line
casts in meters. Forty fishing rods, all of the same type, are purchased.
Five rods are randomly assigned to each of the eight combinations of A:
reel, B: line and C: sinker weight. The order in which the 40 rods are used
is also randomized.
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b. [8] An experiment is performed to study the effect of depth of planting (½
inch, 1 inch) and date of planting (April 30, May 7, May 14) on corn
yields. There are 18 fields available for the experiment. Six of the fields
are assigned at random to each of the planting dates. On half of each field
corn seed is planted at a depth of ½ inch, on the other half of each field
corn seed is planted at a depth of 1 inch. Which half is planted at each
depth is determined by the flip of a coin for each field. All the fields are
harvested on the same day in late October and the yield of corn (bushels
per acre) is recorded.
3. [31 pts] A friend of yours needs some help analyzing some data he has collected
for a psychology experiment. He is interested in memory and his response is the
number of errors on a memory trial. The data are listed below. He also has
output from several different models he has run in JMP and SPSS. You should
refer to this output when answering the following questions.
Type 1
Type 2
Type 3
Type 4
18
14
12
6
19
12
8
4
14
10
6
2
16
12
10
4
12
8
6
2
18
10
5
1
16
10
8
4
18
8
4
1
19
12
6
2
19
16
10
8
16
14
10
9
16
12
8
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a. [5] When asked to describe the purpose for the experiment, he says that he
is interested in seeing if there is a difference among the four types of
memory trials. Given only this information:
i. [2] Are there statistically significant differences amongst the four
types of memory trials? Support your answer with an appropriate
F-test.
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ii. [3] Specifically which pairs of types are significantly different? Be
sure to report the value of smallest significant difference used in
the appropriate multiple comparison procedure.
b. [5] When you show your friend the results from (a) he likes what he sees
but then mentions that he only used 12 subjects and that each subject
experienced all four types of memory trials in a random order. Given this
new information.
Subject
Type 1
Type 2
Type 3
Type 4
1
18
14
12
6
2
19
12
8
4
3
14
10
6
2
4
16
12
10
4
5
12
8
6
2
6
18
10
5
1
7
16
10
8
4
8
18
8
4
1
9
19
12
6
2
10
19
16
10
8
11
16
14
10
9
12
16
12
8
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i. [2] Are there statistically significant differences amongst the four
types of memory trials? Support your answer with an appropriate
F-test.
ii. [3] Specifically which pairs of types are significantly different? Be
sure to report the value of smallest significant difference used in
the appropriate multiple comparison procedure.
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c. [12] When you show your friend the results from (b) your friend is pleased
but then informs you there was a second factor of interest, a treatment that
has four levels. Three subjects were randomly assigned to each treatment
level. Each subject experienced all four of the memory trials in a random
order. Given this new information.
Treatment
Subject
Type 1
Type 2
Type 3
Type 4
1
18
14
12
6
1
2
19
12
8
4
3
14
10
6
2
4
16
12
10
4
2
5
12
8
6
2
6
18
10
5
1
7
16
10
8
4
3
8
18
8
4
1
9
19
12
6
2
10
19
16
10
8
4
11
16
14
10
9
12
16
12
8
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i. [2] Are there statistically significant differences amongst the four
types of memory trials? Support your answer with an appropriate
F-test.
ii. [3] Specifically which pairs of types are significantly different? Be
sure to report the value of smallest significant difference used in
the appropriate multiple comparison procedure.
iii. [2] Are there statistically significant differences amongst the four
treatments? Support your answer with an appropriate F-test.
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iv. [3] Specifically which pairs of treatments are significantly
different? Be sure to report the value of smallest significant
difference used in the appropriate multiple comparison procedure.
v.
[2] Is there a statistically significant interaction between type of
memory trial and treatment? Support your answer with an
appropriate F-test.
d. [6] For comparing the types of memory trials, how has the mean square
error and degrees of freedom for error differed for the three analyses? For
each analysis give the type of design, model #, MSError and dfError.
Analysis
a.
b.
c.
Design
Model #
MSError
dfError
e. [3] Suppose we wanted to do a Randomized Complete Block Design with
the 4 Types of memory trials crossed with the 4 Treatments with 12
Subjects acting as blocks. We will reuse subjects so that each subject will
experience all type by treatment combinations in a random order. Give
the partial ANOVA table showing all sources of variation and the
associated degrees of freedom.
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4. [9 pts] The three fundamental principles of a well-designed experiment are
control of outside variables, randomization and replication.
a. [3] Why is control of outside variables important?
b. [3] If some outside variables are not controlled how can randomization
help?
c. [3] Why is replication within an experiment important?
5. [30] An experiment on the fuel economy (mpg) of a particular make and model of
an automobile was designed using five factors, each at 2 levels.
Factor
A: Engine Size (L)
B: Driving Speed (mph)
C: Tire Pressure (psi)
D: Fuel Octane
E: Air Conditioning
Low Level (–1)
4.0
55
22
87
Off
High Level (+1)
4.5
65
28
93
On
The experiment was run as a completely randomized design with one observation
at each of the 32 treatment combinations. Refer to the JMP output “Fuel
Economy Experiment”.
a) [1] What is the overall average fuel economy for the 32 treatment
combinations?
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b) [2] What is the average fuel economy when driving with the Air Conditioning
On?
3
.99
2
.95
.90
1
.75
0
.50
.25
Normal Quantile Plot
c) [4] Using the normal plot of full effects given below, identify those factors
and interactions that appear to have a large effect on fuel economy.
-1
.10
.05
-2
.01
-3
-5
-4
-3
-2
-1
0
1
2
3
d) [4] A statistician suggests fitting a reduced model using a full factorial in XA,
XB, XC, and XE. Why is this a reasonable suggestion?
e) [5] Based on the full factorial model in XA, XB, XC, and XE what factors and
interactions are statistically significant? Report values of test statistics and Pvalues to support your answer. Remember that you are doing 15 simultaneous
tests.
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f) [3] Give the prediction equation for fuel economy. Include only those terms
that you found to be statistically significant in e).
g) [5] Give a recommendation for factor settings (give me the actual settings for
the factors not the +1/–1 codes) to obtain the maximum predicted mpg.
h) [2] Using your recommendation in g), what is the predicted mpg?
i) [4] If we used 4-way and 5-way interactions to estimate error variation what
would MSError be? How does this compare to the MSError for the full
factorial model in XA, XB, XC, and XE?
I will probably not have the final exams graded until Monday, May 12. You
can pick up your graded final exam the week of May 12 at my office in
Wilson Hall (1407). Be sure to set up a time with me to pick up your exam so
that I will be there to give it to you. You can also give me a self-addressed
stamped envelope ($1.14 postage should be enough) and I will mail you your
graded final exam. If you have a campus mail address, write that below and I
will mail your graded final exam there for free.
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