STATISTICS 402 - Assignment 6
Due March 30, 2012
1. For each of the following problem statements, identify the response, conditions and
experimental material. Indicate the design (completely randomized, randomized
complete block, Latin square) that is used to collect the data and indicate if factorial
crossing is used. Construct a partial ANOVA table by identifying all sources of
variability and associated degrees of freedom.
a. Industrial psychologists wish to investigate the effect of music in the factory on
the productivity of workers. Four distinct music programs (Country, Soft Rock,
Hard Rock, and Classical) and no music (a control) make up the five treatments.
In order to account for variability from day to day and week to week, a different
program is used on each day (M, T, W, Th, and F). The programs are rotated
from week to week, so that over 5 randomly chosen weeks, each program appears
once on any specific working day of the week.
b. An experiment is conducted to see the effect of fishing rod length and sinker
weight on the length of the cast. There are three fishing rod lengths: 6 feet, 6.5
feet and 7 feet. There are four sinker weights: 2 ounces, 3 ounces, 4 ounces, and
5 ounces. Each of the twelve combinations of fishing rod length and sinker
weight will be replicated four times. The order in which the 48 casts will be made
is completely randomized.
c. A sports physiologist is conducting an experiment on eye focus time. She is
interested in the distance of the object from the eye on the focus time. Five
different distances are of interest. She chooses 10 players at random from a
college softball team. Each player will have her focus time measured at each of
the five distances. The order of the distances will be randomized for each player.
d. An experiment is performed to see whether different operators obtain different
results in the routine analysis to determine the amount of nitrogen in soil. 50 soil
samples are chosen at random and divided at random into 5 groups of 10. Each
operator is assigned a group of 10 soil samples at random and asked to determine
the amount of nitrogen in each sample.
2. A grain refiner is a chemical added to a molten metal or alloy to restrict grain growth
that would otherwise lead to softening of the material at high temperatures. A master
alloy manufacturer produces grain refiners in four furnaces. Each furnace has unique
operating characteristics and so must be considered a nuisance factor for any
experiment run in the foundry. Process engineers want to investigate the effect of
stirring rate on the grain size of the final product. Each furnace can be run at four
different stirring rates; 5 rpm, 10 rpm, 15 rpm and 20 rpm. All four stirring rates are
used with each furnace and the order in which the stirring rates are used is
randomized for each furnace. The data on grain size appear on the next page.
a. What is the response?
b. What are the conditions?
c. Why is this block design? How are blocks made?
1
Furnace 1
Furnace 2
Furnace 3
Furnace 4
5 rpm
8
4
4
6
Stirring Rate
10 rpm
15 rpm
14
14
5
6
6
8
9
8
20 rpm
18
9
12
11
d. Compute the sample means for the four stirring rates and an overall grand sample
mean.
e. Compute the estimated effect for each stirring rate. Which stirring rate(s) appear
to produce the smallest grain size? Explain briefly.
f. Construct a complete ANOVA table including all appropriate sources of
variation.
g. Are there statistically significant differences among the stirring rates in terms of
average grain size? Report the appropriate F- and P-values and explain why these
support your answer.
h. If there are statistically significant differences among the stirring rates, indicate
where those differences lie. Give the value of HSD. Clearly state which of the
positions for the feet have statistically significant differences in sample means and
which do not.
i. If one wishes to have the smallest grain size, what statistically valid
recommendations can you make for the choice of stirring rate? Explain your
reasoning.
j. Is there a statistically significant linear relationship between stirring rate and grain
size? Support your answer with an appropriate test of hypothesis.
k. Construct appropriate plots of the residuals and comment on whether the equal
variance condition and the normally distributed errors condition are satisfied.
3. Traffic engineers attempt to design traffic flow systems so that the maximum number
of vehicles can go through the system safely in the least amount of time with the
fewest stops at red lights. A study was done to evaluate five different traffic-light
signal sequences (A, B, C, D and E). The traffic engineers were interested in the
unused red light time, which is the amount of waiting time for traffic facing a red
light when there is no traffic in the direction(s) of the green light(s). The experiment
was conducted with five randomly selected intersections and five time periods during
the day. A lower unused red light time indicates a better outcome in terms of traffic
flow. The data appear on the next page.
2
Intersection
1
Intersection
2
Intersection
3
Intersection
4
Intersection
5
a.
b.
c.
d.
e.
f.
g.
h.
i.
7 to 9 am
C
12.1
B
16.5
A
15.2
E
14.6
D
10.7
10 am to 12
D
31.4
C
26.5
B
33.8
A
31.7
E
34.2
1 to 3 pm
B
30.2
A
22.7
E
29.1
D
23.8
C
21.6
4 to 6 pm
E
17.0
D
19.2
C
13.5
B
16.7
A
19.5
7 to 9 pm
A
31.5
E
25.8
D
27.4
C
26.3
B
27.2
What is the response?
What are the conditions?
Why is this experiment a Latin square design?
Compute the sample means for the five traffic-light signal sequences and an
overall grand sample mean.
Compute the estimated effect for the five traffic-light signal sequences. Which
signal sequence appears to produce lowest average unused red light time?
Construct a complete ANOVA table including all appropriate sources of
variation.
Are there statistically significant differences among the traffic-light signal
sequences in terms of average unused red light time? Report the appropriate Fand P-values and explain why these support your answer.
If there are statistically significant differences among the signal sequences,
indicate where those differences lie. Give the value of HSD. Clearly state which
of the signal sequences have statistically significant differences in sample means
and which do not.
What statistically valid recommendations can you make for minimizing the
unused red light time? Is there a best or worst traffic-light signal sequence?
Explain your reasoning.
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