Computer Lab Assignment 7 and 8
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Dec 6, 2023
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New York University Tandon School of Engineering
Biomedical Engineering
Applied Mathematics and Statistics for Biomedical Engineering
Fall 2023
Professor Mirella Altoe
Tuesday 6:00-8:30PM, Jacobs Academic Building, Rm 674
QUESTION 1:
The number of cases of Lyme disease, a tick-borne illness, more than
doubled between 2001 and 2015 in North America. A vaccine for Lyme disease was approved
by the U.S. Food and Drug Administration in 1998, but it was then removed from the
market just 3 years later (
Nigrovic and Thompson 2007
). Did it work? A clinical trial of the
efficacy of this vaccine compared 5469 people who received the vaccine to 5467 people who
received a placebo (
Steere et al. 1998
). Of these, 15 people with the vaccine had developed
Lyme disease after 2 years, and 65 people in the placebo control group had developed the
disease. In the same study, side effects were also tracked. The most severe possible side effect
was joint pain. Of the vaccinated group, 71 people had joint pain, while in the control group 6
people had joint pain.
A.
(0.2
5
pts.)
Calculate the relative risk of getting Lyme disease for patients who received
the vaccine compared to those who did not.
B.
(
0.2
5
pts.)
Based on that relative risk calculation, by what amount does the vaccine
reduce the rate of getting Lyme disease?
C.
(0.2
5
pts.)
Do these data provide evidence that the vaccine was effective in reducing the
rate of contracting Lyme disease? Carry out an appropriate hypothesis test.
D.
(0.2
5
pts.)
What is the relative risk of joint pain for vaccinated patients compared to
the controls?
E.
(0.2
5
pts.)
Is there evidence that the probability of joint pain is different between
vaccinated and control groups? Do a hypothesis test.
QUESTION 2:
Darwin suggested that plants pollinated by long-tongued insects would benefit
by having long flowers, because greater length would cause the insects to press themselves
farther into the flower to reach the nectar, increasing deposition and removal of pollen. Several
populations of the South African orchid,
Disa draconis
, evolved longer flowers after switching
pollinators to long-tongued tanglewing flies. To measure the advantage of the long flowers,
Johnson and Steiner (1997) experimentally shortened 59 of 118 flowers. The remaining 59
flowers were controls. One week later, 10 of the 59 shortened flowers had received pollen,
whereas 27 of the 59 control flowers had received pollen.
A.
(0.
7
5 pts.)
Illustrate these results in a mosaic plot.
B.
(
1.00
pts.)
What is the estimated odds ratio of not receiving pollen after experimental
shortening, as compared to control flowers? Provide a confidence interval for the
population odds ratio.
QUESTION 3:
Juvenile long-tailed tits (
Aegithalos caudatus
), a European relative of the
chickadee, “help” adult birds raise offspring, such as by feeding their nestlings. What is the
evolutionary advantage of helping behavior: practice for parenthood; increased changes of
inheriting the adults’ territory in future; or indirect genetic benefits via increased success of kin?
To investigate, Russell and Hatchwell (2001) monitored the behavior of 17 juveniles, each of
which lived equidistant from two nests of adult birds. In each case, one nest was parented by a
relative of the helper, and the other was parented by non-kin adults. Sixteen of the juveniles
helped at the nest of their kin, whereas one helped at the non-kin nest. Do these results provide
evidence for preferential helping at the nests of kin? Conduct the appropriate test.
A.
(0.25 pts.)
Clearly state an appropriate null hypothesis and an alternative hypothesis.
B.
(
1.00
pts.)
Do these results provide evidence for preferential helping at the nests of
kin? Conduct the appropriate test.
QUESTION 4:
Each member of a large genetics class grows 12 pea plants from an independent
pea family. Each family is expected to have 3/4 plants with smooth peas and 1/4 plants with
wrinkled peas.
A.
(0.5
0
pts.)
On average, how many wrinkled pea plants will a student see in her 12 plants?
B.
(0.5
0
pts.)
What is the standard deviation of the proportion of wrinkled pea plants
per student?
C.
(0.5
0
pts.)
What is the variance of the proportion of wrinkled pea plants per student?
D.
(0.5
0
pts.)
Predict what proportion of the students saw exactly two wrinkled pea
plants in their sample.
QUESTION 5:
In North America, between 100 million and 1 billion birds die each year by
crashing into windows on buildings, more than any other human-related cause. This figure
represents up to 5% of all birds in the area. One possible solution is to construct windows angled
downward slightly, so that they reflect the ground rather than an image of the sky to a flying
bird. An experiment by Klem et al. (2004) compared the number of birds that died as a result
of vertical windows, windows angled 20 degrees off vertical, and windows angled 40 degrees
off vertical. The angles were randomly assigned with equal probability to six windows and
changed daily; assume for this exercise that windows and window locations were identical in
every respect except angle. Over the course of the experiment, 30 birds were killed by windows
in the vertical orientation, 15 were killed by windows set at 20 degrees off vertical, and 8 were
killed by windows set at 40 degrees off vertical.
A.
(0.25 pts.)
Clearly state an appropriate null hypothesis and an alternative hypothesis.
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