Spring 2016
STATISTICS 402B
Sample Exam II Questions
1. The effect of five different catalysts (A, B, C, D, E) on the reaction time of a chemical process is being
studied. A catalyst is added to a mixture of chemicals, the process initiated, and timed until the reaction
is complete.
(a) What is the response?
(b) What are the treatments?
(c) What are the experimental units/runs?
(d) If the experiment is to be conducted using a completely randomized design, how many runs would you
need to detect a difference between any pair of catalyst means of 2 standard deviations with α = 0.05
and β = .1?
(e) Suppose that each prepared batch of the chemical mixture is large enough to do five runs of the
process and that six batches were prepared. Describe how you would perform the experiment using a
randomized complete block design (RCBD), stating your blocking factor clearly.
(f) Describe in detail how you would do the randomization in the RCBD experiment you described above.
(g) Complete the following skeleton ANOVA table for the RCBD you described.
Source of Variation d.f.
Catalyst
Batch
Error
Total
(h) Give a reason why conducting the experiment as a block design with batches as a blocking variable is
better than a completely randomized design using mixtures from different batches completely randomly
for performing the runs.
(i) Suppose each run of the process requires approximately 1.5 hours, so only five runs can be completed
in one day. The experimenter decides to run the experiment using a Latin square design. Describe
how you would perform the experiment using a latin square design (LSD) with batches and days as
blocking factors. Show the experimental plan.
(j) Given below is the anova table constructed from the results of latin square experiment:
Source of Variation d.f.
SS
MS
F p − value
Catalyst
4 141.44 35.36 11.309
.0005
Batch
4 15.44
3.86
Day
4 12.24
3.06
Error
12 37.52 3.1267
Total
24 206.64
Test the hypothesis that the catalyst means are equal against the alternative that some means are
different. Give the F-statistic, it’s degrees of freedom, the p-value, and your decision using α = .05.
(k) Compute the LSD (α = .05)for comparing the catalyst means.
(l) The following are the catalyst means:
Catalyst A
B
C
D
E
Means
8.4 5.6 8.8 3.4 3.2
Use the LSD to find pairs of means that are different.
1
2. An experiment is conducted to explore the relationship between height of step (5.75 in or 11.5 in) and rate
of stepping (14 steps/min, 21 steps/min or 28 steps/min) on the change of heart rate of college students.
Six college students each are randomly allocated to each step height and stepping rate combination. There
are 6 combinations of step height and stepping rate. Thus there were 36 students participating in the
experiment. Each student experiences each combination. The order is randomized for each student and
enough time separates the trials so that students heart rates return to a resting rate. The resting heart
rate for each student is taken before each trial and the heart rate at the end of 3 minutes of the stepping
combination is also measured. The change in heart rate is calculated by subtracting the resting heart rate
from the heart rate after stepping. Refer to the JMP output for the Stepping Experiment. Note this output
has been edited so that there are several blank spots.
(a) What are the response, conditions of interest and experimental material?
(b) What design was used to collect the data? Explain how you determined what design was used.
(c) Write down the complete ANOVA table for this experiment.
(d) Comment on the interaction plot. Describe what you see in the plot and what it indicates about the
possible interaction between step height and stepping frequency.
(e) Do the results from the ANOVA analysis confirm your above conclusion about interaction? Explain
why?
(f) Are there statistically significant differences among the sample means for the step heights? Report
the appropriate F-statistic, P-value, decision, reason for the decision and conclusion.
(g) Are there statistically significant differences among the sample means for the stepping frequencies?
Report the appropriate F-statistic, P-value, decision, reason for the decision and conclusion.
(h) For comparing treatment (combination of height and frequency) means the value of q is 3.08179.
Compute the value of HSD.
(i) What would your recommendation be if you wanted the largest average increase in heart rate? Support
your answer statistically.
3. (a) Give two advantages of using a factorial experiment instead of single factor experiments to study
effects of several factors.
(b) Explain how to define the main effect A of a 22 factorial with factors A and B. Recall that the
treatment means are identified using the notation (1), a, b, ab.
(c) Explain how to define the interaction effect AB of a 22 factorial with factors A and B. Recall that the
treatment means are identified using the notation (1), a, b, ab.
(d) What is a 23 factorial experiment? How many runs does a 23 factorial with each treatment combination
replicated twice have?
(e) What are the numbers of degrees of freedom for Treatment and Error in the 23 factorial experiment
in part (b)?
(f) Give the defining contrast for any two-factor interaction in a 23 factorial experiment.
(g) Suppose the 3-fi contrast and the
Catalyst (1) a
b
ab
Contrast
+
+
Means
6.4 5.6 6.8 3.4
Calculate the 3-fi effect.
treatment
c
ac
+
3.2 7.2
totals for the experiment in part (d) are:
bc abc
+
4.5 5.0
2
JMP Analysis of Stepping Experiment Data
Analysis of Variance
Source
DF
Model
Error
C. Total
Sum of
Squares
6126.0000
3631.0000
9757.0000
35
Mean Square
F Ratio
1225.20
121.03
10.1228
Prob > F
<.0001*
Effect Tests
Source
Nparm
DF
Frequency
Height
Frequency*Height
Sum of
Squares
3578.1667
2466.7778
81.0556
F Ratio
Std Error
Mean
<.0001*
<.0001*
0.7181
Effect Details
Frequency
Least Squares Means Table
Level
14
21
28
Least Sq
Mean
16.416667
28.250000
40.833333
16.4167
28.2500
40.8333
Height
Least Squares Means Table
Level
05.75
11.5
Least Sq
Mean
20.222222
36.777778
Std Error
Level
14,05.75
14,11.5
21,05.75
21,11.5
28,05.75
28,11.5
Least Sq
Mean
10.166667
22.666667
19.500000
37.000000
31.000000
50.666667
Interaction Plot
Mean
20.2222
36.7778
Frequency*Height
Least Squares Means Table
Std Error
4.4913497
4.4913497
4.4913497
4.4913497
4.4913497
4.4913497
Prob > F