Key
The 2nd Hourly
Math 1107
Spring Semester 2010
You will use only the
following resources: Your individual calculator; individual tool-sheet (one (1)
8.5 by 11 inch sheet), writing utensils, blank paper
(provided by me) and this copy of the hourly.
Do not share these resources
with anyone else. Show complete detail and work
for full credit. Follow case study
solutions and sample hourly keys in presenting your solutions.
Work all four
cases. Using only one
side of the blank sheets provided, present your work. Write on one side only of
the sheets provided, and present your work only on these sheets. Do not
share information with any other students during this hourly.
When you are
finished: Prepare a Cover Sheet: Print your name on an otherwise blank sheet of
paper. Then stack your stuff as follows: Cover Sheet (Top), Your Work Sheets, The Test Papers, Your Toolsheet. Then hand all of this in to me.
Sign and
Acknowledge: I agree to follow this protocol.
________________________________________________________________________
Case One | Descriptive Statistics | Framingham Heart Study
The objective of the
Framingham Heart Study was to identify the common factors or characteristics
that contribute to Cardiovascular disease (CVD) by following its development
over a long period of time (since 1948)
in a large group of participants who had not yet developed overt
symptoms of CVD or suffered a heart attack or stroke. Blood pressure is a
measurement of the force applied to the walls of the arteries as the heart
pumps blood through the body. Blood pressure readings are measured in
millimeters of mercury (mm Hg) and usually given as two numbers: the systolic
blood pressure (SBP) reading, representing the maximum pressure exerted when
the heart contracts and the diastolic blood pressure
(DBP) reading, representing the pressure in the arteries when the heart is at
rest.
Consider
the systolic to diastolic blood pressure ratio R = SBP/DBP.
A sample
of FHS adult subjects yields the following ratios:
1.86 1.71 1.43 1.82 1.62 1.57 1.53 1.42 1.55 1.75 1.95 1.58
2.33 1.69 1.56 1.85 1.51 1.50 1.72 1.50 2.06 1.36 1.78 1.56
1.62 1.75 1.78 1.52 1.83 1.60 1.54 1.80 2.13 1.67 1.86 1.60
1.48 1.47 1.45 1.44 1.50 2.05 1.50 1.79 1.82 1.59 1.74 1.86
1.64 1.54 2.03 1.91 1.92 1.97 1.54 1.67 2.00 1.63 1.82 1.49
Compute and interpret the following statistics: sample
size, p00, p25, p50, p75, p100, (p100 p25), (p75
p25), (p50 p25).
n=60
minimum = qo = p0 = 1.36
lower quartile = q1= p25 = 1.5375
median = q2 = p50 = 1.655
upper quartile = q3 = p75 = 1.8225
maximum = q4= p100 = 2.33
There are 60
Framingham Heart Study (FHS) subjects in our sample.
The FHS subject in our sample
with the lowest ratio has a ratio of 1.36 for systolic versus diastolic blood
pressure.
Approximately 25%
of the FHS subjects in our sample have systolic versus diastolic blood pressure
ratios (DtSBPRs) of 1.5375 or less.
Approximately 50%
of the FHS subjects in our sample have systolic versus diastolic blood pressure
ratios of 1.655 or less.
Approximately 75%
of the FHS subjects in our sample have systolic versus diastolic blood pressure
ratios of 1.8225 or less.
The FHS subject in our sample
with the highest ratio has a ratio of 2.33 for systolic versus diastolic blood pressure.
Range41 = Q4 Q1
= P100 P25 = 2.33 1.5375 = .7925
Approximately 75%
of the FHS subjects in our sample present systolic versus diastolic BP ratios (StDBPR) between 1.5375 and 2.33. The largest possible
difference in the StDBPR between any pair of FHS
subjects in this upper Ύ sample is 0.7925
Range31 = Q3 Q1
= P75 P25 = 1.8225 1.5375 = .285
Approximately 50%
of the FHS subjects in our sample present systolic versus diastolic BP ratios (StDBPR) between 1.5375 and 1.8225. The largest possible
difference in the StDBPR between any pair of FHS
subjects in this middle half sample is 0.285.
Range21 = Q2 Q1
= P50 P25 = 1.655 1.5375 = .1175
Approximately 25%
of the FHS subjects in our sample present systolic versus diastolic BP ratios (StDBPR) between 1.5375 and 1.655. The largest possible
difference in the StDBPR between any pair of FHS
subjects in this lower middle quarter sample is 0.1175.
Case Two |
Summary Intervals | Fictitious
Spotted Toad
The Fictitious Spotted Toad is a
native species of Toad Island, and is noteworthy for the both the quantity and
quality of its spots. Consider a random sample of toads, in which the number
of spots per toad is noted:
9, 10, 10, 10, 11, 12, 12, 12, 12, 13, 15,
16, 17, 18, 18, 18, 18, 19, 19, 20, 20, 21, 22, 22, 23
24, 24, 25, 25, 26, 29, 29, 29, 31, 31,
33, 33, 33, 33, 33, 34, 39, 40, 40, 40, 43, 43, 45, 47, 47
50, 56, 56, 56, 60, 63, 65, 66, 67, 98
Let
m denote the sample mean spot count, and sd
the sample standard deviation. Compute and interpret the
intervals m±2sd and m±3sd, using Tchebysheffs Inequalities and the Empirical Rule.
Be specific and complete. Show your work, and discuss completely for full
credit.
lower2 = m 2*se = 31.5 2*18.39998 » - 5.2999 [0]
upper2 = m + 2*se = 31.5 + 2*18.39998 » 68.2999[68]
lower3 = m 3*se = 31.5 3*18.39998 » -23.6999 [0]
upper3 = m + 3*se = 31.5 + 3*18.39998 » 86.6999[86]
At least 75% of
the spotted toads in the sample have between 0 and 68 spots
At least 89% of
the spotted toads in the sample have between 0 and 86 spots
If the Fictitious
Spotted Toad spot counts cluster symmetrically around a central value, becoming
rare as the distance from the center increases, then:
Approximately 95% of
the spotted toads in the sample have between 0 and 68 spots
Approximately 100%
of the spotted toads in the sample have between 0 and 86 spots
Case Three | Design Fault
Spot
In each of the following a brief description of a
design is presented. Briefly identify faults present in the design. Use the
information provided. Be brief and complete.
3.1) In a comparative
clinical trial, treatment methods are compared in the treatment of Condition Z,
which when left untreated leads to severe complications and possibly death. A
standard treatment is available. Suppose we have a new candidate treatment, and
further suppose that a basic clinical trial is proposed to evaluate this new
candidate treatment.
The proposed
placebo group is inappropriate a standard treatment is available, and should
be use d as the basis for comparison.
3.2) A
clinical trial of a new Hepatitis C treatment is designed as follows: subjects
are screened for Hepatitis C infection. Those who test positive for Hepatitis C
infection are then told of their status, and are offered treatment for
Hepatitis C at no cost, and are given no further information. Those who accept
the free treatment offer are then randomly assigned to either a Placebo, or to
the New Treatment Plan.
The subjects are
denied informed consent they are not aware that they are participating in a
clinical trial.
If there is an
available standard of care, then the placebo group is not appropriate.
3.3) A survey of parents and
legal guardians of US high school students is used to study the sexual habits,
knowledge and attitudes of US high school students. An appropriate random
sample is used, and there are no problems with the wording and delivery of the
survey instrument.
Use of proxy
respondents in a survey may be misleading, especially in the study of sensitive
topics. Survey the students directly, not the parents.
3.4) The objective of a
sample survey is to study the attitudes of urban residents of the United States
regarding federal programs, taxation and spending. A random sample of urban
business owners is employed in this design. Assume that there are no problems
with the wording and delivery of the survey instrument.
The survey
excludes urban residents who do not own urban businesses, and includes urban
business owners who are not urban residents. Directly survey urban residents,
regardless of business ownership status.
Case Four | Clinical Trial Sketch | Radiation-related
Retinopathy
Melanin is a natural substance that gives color to hair, skin,
and the iris of the eye. It is produced by cells in the skin called melanocytes.
Melanoma is cancer based on malignancy of the melanocytes.
Melanoma is the rarest and most
serious of the skin cancers, but melanoma can arise wherever there are melanocytes, including the eye. Choroidal melanoma is the most common primary intraocular tumor in
adults. Initially appearing as a small freckle beneath the retina, choroidal melanoma can grow in height and diameter, and may
eventually spread to other organs of the body, causing death. Because choroidal melanoma is intraocular (occurring inside the
eye) and not usually visible to patients, patients with this disease may not
recognize its presence until the tumor grows to a size that impairs vision by
obstruction, retinal detachment, hemorrhage, or other complication. Periodic
dilated retinal examination is the best means of early detection.
Depending on the nature
of the cancer, primary treatment options include removal of the affected eye (enucleation) and radiation therapy (without enucleation). Primary
radiation therapy without enucleation preserves the
affected eye, however, it is common (30%-40% within five years
post-treatment) for patients to experience deterioration in visual acuity due
to radiation-related retinopathy (damage to the retina). The retina is the
light-sensitive layer of the eye.
Case Four | Clinical Trial Sketch | Radiation-related
Retinopathy
Lucentis (ranibizumab) binds to and inhibits
the action of vascular endothelial growth factor A (VEGF-A). VEGF may trigger the growth of
new vessels, which may leak blood and fluid into the eye. By blocking VEGF-A in
the eye, ranibizumab may prevent and reverse vision
loss.
Kenalog (triamcinolone acetonide) is an anti-inflammatory
medication. Post-radiation therapy related inflammation may contribute to loss
or degradation of visual acuity.
Visual
Acuity is measured by scoring the
patient against the ETDRS eye chart.
Preservation or improvement of
visual acuity can greatly affect quality of life.
Sketch a
basic clinical trial evaluating the effect of Lucentis+Kenalog
in the treatment of radiation-related retinopathy in patients treated for choroidal melanoma. Make your sketch concise and complete, following the
style demonstrated in class, in the sample second hourlies
and in case study summaries.
We recruit
subjects diagnosed with choroidal melanoma who were
treated with radiation therapy, but whose eyes were preserved. Those who give
informed consent are enrolled in the trial.
Enrolled subjects
are randomly assigned to either Lucentis and Kenalog or to PlaceboLucentis and PlaceboKenalog . Double blinding is employed, so that
neither the subjects nor their clinical workers know individual treatment
assignments.
Treated subjects
are the tracked for changes in visual acuity, as assessed against the ETDRS eye
chart. They are also tracked for safety and toxicity, including kidney/liver,
and for changes in quality of life.
Work all four (4) cases.