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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
PubHlth 640 Intermediate Biostatistics
Spring 2015
Examination 3
Unit 7 – Introduction to Analysis of Variance
Due: Wednesday May 6, 2015
Before you begin:
This is a “take-home” exam. You are welcome to use any reference materials you wish. You are
welcome to use the computer as you wish, too. However, you MUST work this exam by yourself and you
may not consult with anyone.
Instructions and Checklist:
__1. Start each problem on a new page.
__ 2. Write your name on every page.
__ 3. Make a photo-copy of your exam for safekeeping prior to submission
__ 4. Complete the signature page
__ 5. Please DO NOT submit a copy of the exam questions.
How to submit your exam (sorry – Faxed exams are NOT permitted):
(1) ONLINE Students
Please be sure your name is somewhere on your submission. Next, save it as a
SINGLE FILE pdf using the naming convention lastname_exam3.pdf. Email it to me
at: [email protected]
(2) Worcester Section.
If you’re all finished, you could bring your exam to
this last class. Otherwise, please send me your exam by U.S. post (see below for
address).
Our last class is Monday April 27, 2015.
(2) Amherst Section
You, too. If you are all finished, you could bring
your exam to this last class. Otherwise, please put your exam in my mailbox, located
in the mail room on the 4th floor of Arnold house.
Our last class is Wednesday April 29, 2015.
(3) Address for mailing by U.S. post – my home address, please!
UMass mail is sometimes slow … so …please mail with postmark no later than May 6,
2015 to:
Carol Bigelow
59 Butter Hill Road
Pelham, MA 01002-9760
Tel. 413-545-1319.
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
Signature
This is to confirm that in completing this exam, I worked independently and did not
consult with anyone.
Name: ___________________________________________________________
Date: ___________________________
Thank you!
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
1. (10 points, total)
1a. (2 points)
Multiple choice. Pick ONE. The purpose of analysis of variance is to compare
__a. the variances of several populations.
__b. the proportions of successes in several populations.
__c. the means of several populations.
1b (2 points)
Multiple choice. Pick ONE. A study of the effects of smoking classifies subjects as nonsmokers,
moderate smokers, or heavy smokers. The investigators interview a sample of 200 persons in each
group. Among the questions is “how many hours do you sleep on a typical night?” The degrees of
freedom for the analysis of variance F statistic for comparing mean hours of sleep are
__a. 2 and 197
__b. 2 and 597
__c. 3 and 597
The following setting pertains to questions 1c, 1d, and 1e.
A dental study evaluated the effect of tooth etch time on resin bonding strength. A total of 78
undamaged, recently extracted first molars (baby teeth) were randomly assigned to be etched with
phosphoric acid gel for either 15, 30, or 60 seconds. Composite resin cylinders of identical size were
then bonded to the tooth enamel. The researchers examined the bond strength after 24 hours by
finding the failure load (in megapascal) for each bond. Here are the summary data and ANOVA table
for this experiment.
Etch time
15 seconds
30 seconds
60 seconds
Source
Etch time
Error
Total, corrected
x
n
26
26
26
DF
2
75
77
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s
4.49
6.98
8.48
Sum of Squares
211.208
812.745
1023.953
Mean Square
105.604
10.837
2.28
3.15
4.17
F-statistic
9.745
p-value
0.0002
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
1c (2 points)
Multiple choice. Pick ONE. The most striking conclusion from the numerical summaries for the
three etch times is that
__a. there appears to be little difference among the etch times.
__b. on average, failure load increases with etch time
__c. on average, failure load decreases with etch time.
1d (2 points)
Multiple choice. Pick ONE. The conclusion of the analysis of variance test is that
__a. there is quite strong evidence (p=.0002) that the mean failure loads are not the same in all
three conditions.
__b. there is quite strong evidence (p=.0002)that the mean failure load is much lower for 15
seconds etch time than in any other two etch times.
__c. the data give no evidence (p=.0002) to suggest that mean failure load differ for the three etch
times.
1e (2 points)
Multiple choice. Pick ONE. If we used a series of two-sample t procedures to compare the three
conditions, we would have to give three 95% confidence intervals to compare all three pairs of etch
times. The weakness of doing this is that
__a. we won’t be 95% confident that all 3 intervals cover the true differences in population
means.
__b. the conclusions from the three intervals might not agree.
__c. the conditions for two-saomple t inference are not met for all 3 pairs of etch times.
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
2. (20 points total)
Each of the sets “A”, “B”, “C”, and “D” of observations is a simple random sample from a normal
distribution. Test the null hypothesis of equal population means using the appropriate analysis of
variance. In developing your answer, answer the questions below.
___________________________________________________________________
A
B
C
D
__________________________________________________________________
49
49
44
58
42
44
57
54
47
50
34
64
76
58
48
60
69
70
50
53
58
64
52
42
___________________________________________________________________
2a. (5 points)
State the appropriate analysis of variance model using notation µ and τ i and σ e2 as appropriate.
Please don’t forget to define your terms.
2b. (5 points)
Complete the analysis of variance table below. I have provided you with the column headings.
It’s up to you to identify the row names and to provide the numeric entries to the table.
Source
DF
Sum of Squares
Mean Square
F-statistic
p-value
2c. (10 points)
Carry out the appropriate test of the null hypothesis that the population means are equal,
providing:
i.
ii.
iii.
iv.
v.
The null and alternative hypothesis (2 points)
The name of the test statistic and its formula (2 points)
The value of the test statistic (2 points)
The achieved significance (p-value) (2 points)
Interpretation (2 points).
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
3. (20 points)
A population of 10,000 households is sub-divided geographically into three regions (“A”, “B”, and “C”)
containing 1,000, 4,000, and 5,000 households, respectively. Random samples of size 200 each were
taken from each region, and the distance to work was measured. The means were
X A =10 miles, X B = 20miles, and XC = 4 miles and the pooled variance was s 2p = 5.76 . Test the null
hypothesis of equal mean distances to work using the appropriate analysis of variance. In developing
your answer, answer the questions below.
3a. (5 points)
State the appropriate analysis of variance model using notation µ and τ i and σ e2 as appropriate.
Please don’t forget to define your terms.
3b. (10 points)
Complete the analysis of variance table below. I have provided you with the column headings.
It’s up to you to identify the row names and to provide the numeric entries to the table.
Source
DF
Sum of Squares
Mean Square
F-statistic
p-value
3c. (5 points)
Carry out the appropriate test of the null hypothesis that the mean distances to work among the
region means are equal, providing:
i.
ii.
iii.
iv.
v.
The null and alternative hypothesis (1 point)
The name of the test statistic and its formula (1 point)
The value of the test statistic (1 point)
The achieved significance (p-value) (1 point)
Interpretation (1 point).
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
4. (15 points total)
Consider a two-way analysis of variance, fully factorial, that is investigating seed type (Factor I) at 2
levels and fertilizer level (Factor II) at 3 levels. The outcome is yield (bushels/acre). Suppose that the
true population means µij are as indicated in the following table.
Factor I
Seed type
Factor II (Fertilizer)
j=2
18
10
j=1
14
12
i=1
i=2
j=3
19
17
Recall the definition of the model.
Xijk = µ + α i + β j + (αβ )ij + ε ijk
4a. (1 point)
What is the value of
µ
?
4b. (2 points)
What are the values of each of the α i ?
4c. (3 points)
What are the values of each of the
4d. (6 points)
What are the values of each of the
βj ?
(αβ )ij ?
4e. (3 points)
In 1-3 sentences at most, what is your conclusion about the influences of seed type and fertilizer
level on yield. Is there an interaction? Explain. Tip – There is no data here. Just evaluate the means.
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
5. (35 points total)
In a study of intelligence of children with heart disease that is either acyanotic or cyanotic and who have
undergone surgery or not, the following changes (change = last – first) in IQ were obtained.
Factor I (Surgery)
No
Yes
Factor II (Disease Type)
Acyanotic
Cyanotic
9
2
-1
1
-10
-4
3
-5
-2
0
-7
5
-7
10
-12
9
-13
2
-12
15
5a. 5 points
State the analysis of variance model. Use deviation from means parameterization. Define all
terms and constraints on the parameters.
5b. 5 points
Now state the appropriate model, formulated as a regression using reference cell coding. Define
all terms and constraints on the parameters.
5c. 10 points
Complete the following analysis of variance table.
Source
DF
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Sum of Squares
Mean Square
F-statistic
p-value
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PubHlth 640 Exam 3 – Spring 2015
Name __________________________________________________
5. (35 points total) - continued
5d. 5 points
Test the null hypothesis that changes in IQ are not affected by an interaction between surgery and
disease type. In 1-2 sentences, what do you conclude?
5e. 5 points
Drawing upon your conclusion in problem 5d, in words, what null hypothesis is being tested in the
test of “main effect” of disease (factor II)?
5f. 5 points
In 2-3 sentences at most, report the findings and your interpretation of this analysis of variance.
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