STATISTICS 402 - Assignment 3
Due February 15, 2008
1. An experiment is performed to see the effectiveness of different hand cleaning
methods on eliminating bacteria from hands. There are three hand cleaning
methods: washing with soap, washing with antibacterial soap, and spraying with
an antibacterial spray (65% Ethanol). Volunteers are assigned to groups with
each group using one of the hand cleaning methods. After hands are cleaned the
right hand of each volunteer is placed on a sterile media plate. Bacteria still on
the hand will be transferred to the media that is designed to facilitate bacterial
growth. The plates are incubated for 2 days at 36o C. After incubation the number
of bacterial colonies are counted.
a) Why is this an experiment and not an observational study?
b) Identify the response, conditions and experimental units.
c) Give an example of an outside variable that is controlled in the study. How is
the variable controlled?
d) Give an example of an outside variable that is not controlled in the study?
Explain briefly.
e) If the experimenter wishes to detect a difference in mean bacterial counts as
small as 1 standard deviation with Alpha=0.05 and Beta=0.10, how many
volunteers are needed?
f) The experimenter is only able to recruit 36 volunteers. Give two combinations
Δ
that correspond to this number of volunteers. Briefly
of Alpha, Beta and
σ
explain the difference between your two choices.
g) Describe in detail how you would randomly assign the volunteers to the three
groups so that there are an equal number of volunteers in each group. Once
you have described what you will do actually do the randomization. Include
your randomized assignment of the 36 volunteers to the 3 groups.
2. A completely randomized experiment, as described in problem 1, is run with the
following results.
Mean, y
Variance, s 2
Antibacterial Spray
47
48
42
33
30
35
37
30
29
43
32
38
37.0
44.54
Antibacterial Soap
96
92
82
97
96
91
101
104
87
100
99
101
95.5
41.36
Regular Soap
107
106
89
91
108
94
100
101
97
99
100
108
100.0
42.00
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a) Plot the data. If you use a computer program to plot the data, cut and paste the
output on your answer sheet. Based on the plot what can you say about the
effectiveness of the various methods? What can you say about the consistency of
the methods?
b) Estimate the effect of each of the hand cleaning methods.
c) Construct an analysis of variance table giving sources of variation, degrees of
freedom, sums of squares, mean squares, the appropriate F statistic and associated
P-value. If you use a computer package, you can copy the results from the output
onto your answer sheet.
d) Give the value of R2 and an interpretation of this value.
e) Are there statistically significant differences amongst the three groups in terms of
mean counts of bacterial colonies? Support your answer by referring to the
appropriate test of the null hypothesis: H 0 : τ 1 = τ 2 = τ 3 = 0 .
f) If there are statistically significant differences, which groups are different?
Support your answer with a multiple comparisons method.
g) Look at the distribution of residuals. What does this tell you about the conditions
necessary for the analysis of variance? Be sure to tell me what you see in the plots
and how that relates to the specific conditions.
h) Write a brief summary (one or two sentences is enough) of the findings of the
experiment. In this summary make a recommendation based on your analysis as
to what hand cleaning method should be used.
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