STATISTICS 402 - Assignment 4
Solution
1. Chapter 4:
Exercise Set A, pg 109, # 2 and 8.
#2. Response: Percentage of the batch of eggs that hatch.
Conditions: Temperature of the water.
Material: Batches of fish eggs.
#8. Response: Number of gypsy moths killed.
Conditions: Chemical spray (BTU), tiny parasitic wasp.
Material: Gypsy moths.
Exercise Set B, pg 117, # 8 and 11.
#8. Response 2: Average milk yield, in pounds per week.
#11. Response 2: Percent of the exercises you get right.
Exercise Set C, pgs 124 – 125, # 2 and 3.
#2. Mothers in the first group (two children) tend to be younger than mothers in the
second group, and as you get older your blood pressure tends to go up. So for this
data set, number of children is confounded with age.
#3. a. If you looked at total number of deaths, the effect of climate would be
confounded with population. States that were more populous would tend to
have higher values.
b. Even death rate is not a good choice for a response, mainly because the effect
of climate would still be confounded with the ages of the people in the states.
Florida tends to have older people and Alaska younger people.
Exercise Set D, pgs 127 – 128, # 1, 2 and 3.
#1. a. A group of friends would tend to be more uniform.
b. A random sample would tend to be more representative.
c. A representative sample would be better because it would be more
indicative of attitudes of all students on campus.
#2. a. Chance errors are likely to be smaller i. using the first set of more uniform
subjects.
b. Bias is likely to be smaller ii. using the second set of less uniform subjects.
#3. a. Chance errors are likely to be smaller ii. using the second set of less
representative subjects.
b. Bias is likely to be smaller i. using the first set of more representative
subjects.
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2. The Biology Department at a large university that teaches introductory biology to
thousands of students each semester is interested in comparing students’
understanding of the frog’s anatomy based on students’ experiences with dissection.
The department decides to look at four methods to be used in labs accompanying the
lecture. One method has students dissecting real frogs. A second method has
students using a computer program that simulates the dissection of a frog. A third
method has students doing both real and virtual dissection. A fourth method has
students doing neither real nor virtual dissection. At the end of the unit on the frog’s
anatomy all students will take the same 100 point test on the anatomy of the frog. All
students will have instruction on the frog’s anatomy in lecture and through text
reading assignments. The dissection methods will be administered in lab sections that
accompany the lecture. Each lab section has 20 students. There are two different
ways to conduct the experiment. Experiment 1 – Randomly select 20 lab sections and
randomly assign methods to the lab sections so that there are five lab sections using
each method. The average test score for lab sections will be used as the response.
Experiment 2 – Select four lab sections at random and randomly assign one method to
each of the lab sections. Individual student test scores will be used as the response.
a) What are the conditions?
The conditions (treatments) are the different methods presented in lab:
dissecting real frog, virtual dissection, both real and virtual dissection, no
dissection.
b) What is the experimental material?
The experimental material consists of students in the biology labs.
c) Describe, in some detail, things that contribute to chance, random, variation in
Experiment 1.
Experiment 1: The experimental units are the 20 lab sections. Chance
variation comes from differences from one lab section to the next using the
same method. The lab sections are at different times, thus attracting
different types of students (off campus students may wish to take labs in the
middle of the day rather than early or late in the day). The labs could have
different instructors, thus effectiveness of instruction could vary from lab to
lab.
d) Describe, in some detail, things that contribute to chance, random, variation in
Experiment 2.
Experiment 2: The experimental units are the 4 lab sections. Chance
variation comes from differences from one student to the next within each
lab section. Students could have different study habits, have different test
taking abilities and general knowledge of biology.
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e) Suppose that the labs that used virtual dissection in Experiment 1 had a higher
mean test score than the other labs and that the difference in mean scores was
statistically significant. What generalization can be made about the effectiveness
of the methods?
Experiment 1:
Conclusion: Labs that used virtual dissection had a higher mean test score
than labs that used other methods. The difference in mean test scores was
statistically significant.
Generalization: Labs, made up of students similar to those in the experiment,
that use virtual dissection will have higher mean test scores, on average, than
labs that use the other methods.
f) Suppose that the lab that used both real and virtual dissection in Experiment 2 had
a higher mean test score than the other labs and that the difference in mean scores
was statistically significant. What generalization can be made about the methods?
Experiment 2:
Conclusion: For the four labs involved in the experiment, the lab that used
both real and virtual dissection had a statistically significant higher sample
mean test score than the other three labs that used different methods.
Generalization: Not much generalization can be made. Only one lab of
students used both real and virtual dissection. All we know is that the lab
that used both real and virtual dissection had a higher mean test score but
we don’t know if this is due to the method or to something else peculiar to
that one lab (e.g. maybe that lab had the best lab instructor or was made up
of honors biology students).
3. A study will be performed to evaluate the effectiveness of different hand cleaning
methods on reducing bacteria on hands. There are three hand cleaning methods:
washing with soap, washing with antibacterial soap, and spraying with an
antibacterial spray (65% Ethanol). The study will use volunteers with each volunteer
being assigned 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 is counted for each plate.
a) Explain briefly why this study is an experiment and not an observational study?
This study is an experiment because the method of hand cleaning is assigned
to the volunteers in the study. Hand cleaning method is manipulated to
create treatments not simply observed.
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b) What is the response variable?
The response is the number of bacterial after 2 days of incubation at 36o C.
c) What are the conditions of the experiment?
The conditions are the three methods of hand cleaning (Antibacterial Soap,
Antibacterial Spray, and Regular Soap).
d) What are the experimental units?
The experimental units are the volunteers, specifically their hands.
e) Give an example of an outside variable that is controlled in the study. How is the
variable controlled?
An outside variable that is controlled is what hand is used; only the right
hand.
Another outside variable that is controlled is the time and
temperature of incubation; all plates are incubated for 2 days at 36o C.
f) Give an example of an outside variable that is not controlled in the study? Explain
briefly. In your explanation you should indicate why this variable should be
controlled.
The amount of bacteria on a volunteer’s hand was not controlled. Depending
on what the volunteer was doing, there could be more or less bacteria to
begin with. Differences in the amount of bacteria could change the apparent
effectiveness of the hand cleaning method.
How long the volunteers cleaned their hands was not controlled. If some
volunteers clean for a longer period of time than others it could change how
much bacteria are left on the hands.
g) Is there a control group in this study? Explain briefly.
Maybe. The group of volunteers who use regular soap could be considered a
comparison group, if you believe washing with regular soap is the standard
method. Alternatively, a group that did not clean their hands at all could
also be considered a control group.
h) Is random selection used in this study? Explain briefly.
There is probably not random selection. The description indicates that
participants are “volunteers” and says nothing about randomly selecting the
participants from a large population of volunteers.
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i) Suppose the experimenter wanted to be able to detect a 0.6 standard deviation
difference in group means with Alpha = 0.05 and Beta = 0.10, how many
volunteers would be needed for the experiment?
According to the sample size tables, you would need 72 for each hand
cleaning method or a total of 216 volunteers.
j) Thirty three people volunteer for the experiment. Give two combinations of
Alpha, Beta and the size of the detectable difference in group means that
correspond to this number of volunteer.
Another combination is Alpha = 0.10, Beta = 0.05 and a difference in
treatment means of 1.6 standard deviations.
Another combination is Alpha = 0.05, Beta = 0.05 and a difference in
treatment means of 1.8 standard deviations.
Another combination is Alpha = 0.01, Beta = 0.10 and a difference in
treatment means of 2.0 standard deviations.
There are other possible combinations.
k) Describe in detail how you would randomly assign the hand cleaning methods to
volunteers so that there are an equal number (11) of volunteers using each
method. Once you have described what you will do, actually do the
randomization. Include your randomized assignment of the 3 methods to the 33
volunteers.
I would number the volunteers from 1 to 33. I would then use JMP. Have
one column labeled cleaning method and enter Anti Soap in the first 11 rows,
Anti Spray in the next 11 rows and Reg Soap in the next 11 rows. In a
second column labeled Volunteers, I would use Cols – Formula – Random –
Col Shuffle to shuffle the numbers from 1 to 33. The method on a row is
assigned to the volunteer whose number appears in the same row. The actual
assignment appears below.
Method
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Anti Soap
Volunteer
20
2
3
27
10
21
11
19
12
15
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Method
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Anti Spray
Volunteer
6
30
1
33
14
4
23
31
8
24
18
Method
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Reg Soap
Volunteer
32
13
28
9
25
5
16
22
29
26
17
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l) Is there replication within this experiment? Explain briefly.
Yes, there is replication within this experiment because there are 11
volunteers for every hand cleaning method.
4. A one factor, completely randomized experiment, as described in problem 3, is run.
The results appear on the next page.
a) Plot the data and compute summary statistics. If you use a computer program,
“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
variability within each method?
Means and Std Deviations
Method
Number
Antibacterial Soap
11
Antibacterial Spray
11
Regular Soap
11
Mean
95.0
37.0
101.0
Std Dev
6.49615
7.00000
5.74456
The antibacterial spray has the lowest average number of bacterial colonies
at 37.0. Both antibacterial soap and regular soap have about the same
average number of bacterial colonies, 95.0 and 101.0, respectively.
The standard deviations for the three methods are fairly similar. The
regular soap has the smallest standard deviation of 5.74 followed closely by
antibacterial soap (standard deviation = 6.50) and antibacterial spray has the
largest standard deviation of 7.00. The ratio of the largest to the smallest
standard deviation is about 1.2, well below the rule of thumb value of 2.
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b) Estimate the effect of each of the methods.
The grand sample mean is 77.67. The estimated effect of each method is the
difference between the sample mean for that method and the grand sample
mean Yi   Y  .
Method
Anti Soap
Anti Spray
Reg Soap
Mean
95.0
37.0
101.0
Estimated Effect
17.33
–40.67
23.33
c) Construct an analysis of variance table giving sources of variation, degrees of
freedom, sums of squares, mean squares, appropriate F statistic and associated Pvalue. If you use a computer package, you can copy the results from the output
onto your answer sheet. I will not look at output simply attached at the end of
what you turn in.
Analysis of Variance
Source
Method
Error
C. Total
DF
2
33
35
Sum of Squares Mean Square
27485.3
13742.7
1242.0
41.4
28727.3
F Ratio Prob > F
331.95 <0.0001*
d) Give the value of R2 and an interpretation of this value.
R2 is 0.9568. Only 95.7% of the variation in the number of bacterial colonies
can be explained by the different hand cleaning methods.
e) Are there statistically significant differences amongst the three groups in terms of
sample mean number of bacterial colonies? Support your answer by referring to
the appropriate test of the null hypothesis: H 0 :  1   2   3  0 .
Yes, there is at least one statistically significant difference between sample
mean number of bacterial colonies. The value of the F Ratio is 331.95 with a
corresponding P-value less than 0.0001. Because the P-value is so small (<
α=0.05) we should reject the null hypothesis that all treatment effects are
zero.
f) Construct 95% confidence intervals for the differences in true mean change in
cholesterol.
 Antibacterial Spray compared to Antibacterial Soap
 Antibacterial Spray compared to Regular Soap
 Antibacterial Soap compared to Regular Soap
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The standard error for the difference in two sample means is
1 1
1 1
MS Error

 41.4
  2.7436 .
n1 n2
11 11
With 30 degrees of freedom and 95% confidence t* = 2.042.

Antibacterial Spray compared to Antibacterial Soap
95.0  37.0  2.0422.7436  58.0  5.60  52.4,63.6

Antibacterial Spray compared to Regular Soap
101.0  37.0  2.0422.7436  64.0  5.60  58.4, 69.6

Antibacterial Soap compared to Regular Soap
101.0  95.0  2.0422.7436  6.0  5.6  0.4,11.6
g) Based on the confidence intervals in f), what pairs of treatments have differences
in sample means that are statistically significant?
None of the confidence intervals contains zero, therefore the differences in
sample means for Antibacterial Spray compared to Antibacterial Soap,
Antibacterial Spray compared to Regular Soap and Antibacterial Soap
compared to Regular Soap are statistically significant. Antibacterial spray is
the most effective, having roughly 50 to 70 fewer bacterial colonies, on
average than either soap. Antibacterial soap is slightly more effective,
having more than zero but less than 12 fewer bacterial colonies, than regular
soap.
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 method of hand cleaning should be used.
From the analysis of the experiment’s data, no matter what method you use
to clean your hands, you will not get rid of all the bacteria. The antibacterial
spray does the best job of reducing the average number of bacterial colonies
that grow after two days of incubation. Antibacterial soap is slightly more
effective than regular soap in reducing the number of bacterial colonies, on
average. For individuals like the volunteers in this experiment, it is
recommended that they use antibacterial spray to reduce the average
number of bacterial colonies cultured from their hands.
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Data for Hand Cleaning Experiment
Number of Bacterial Colonies
Antibacterial
Soap
96
92
82
97
96
91
101
104
87
100
99
Antibacterial
Spray
47
48
42
33
30
35
30
29
43
32
38
Regular Soap
107
106
91
108
94
100
101
97
99
100
108
Oneway Analysis of Number Bacterial Colonies By Cleaning Method
Oneway Anova
Summary of Fit
Rsquare
Adj Rsquare
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.956766
0.953884
6.434283
77.66667
33
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Analysis of Variance
Source
DF Sum of Squares Mean Square F Ratio
Cleaning Method
2
27485.333
13742.7 331.9485
Error
30
1242.000
41.4
C. Total
32
28727.333
Prob > F
<.0001*
Means and Std Deviations
Level
Number Mean Std Dev Std Err Mean Lower 95% Upper 95%
Antibacterial Soap
11 95.000 6.49615
1.9587
90.636
99.36
Antibacterial Spray
11 37.000 7.00000
2.1106
32.297
41.70
Regular Soap
11 101.000 5.74456
1.7321
97.141
104.86
Means Comparisons
Comparisons for each pair using Student's t
Confidence Quantile
t
Alpha
2.04227
0.05
Ordered Differences Report
Level
- Level
Difference
Regular Soap
Antibacterial Spray 64.00000
Antibacterial Soap Antibacterial Spray 58.00000
Regular Soap
Antibacterial Soap
6.00000
Std Err Dif Lower CL Upper CL
2.743588 58.39685 69.60315
2.743588 52.39685 63.60315
2.743588
0.39685 11.60315
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