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 | 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
Compute and
interpret the following statistics: sample size, p00, p25, p50, p75, p100,
(p100 – p75), (p75 – p50), (p50 – p25), (p25 – p00).
n=60
minimum = q0 = p0 = 9
lower quartile = q1= p25 = 18
median = q2 = p50 = 27.5[27]
upper quartile = q3 = p75 = 40.75[40]
maximum = q4= p100 = 98
There are 60
Spotted Toads in our sample.
The toad in our
sample with the fewest spots has 9 spots.
Approximately 25%
of the toads in our sample have 18 or fewer spots.
Approximately 50%
of the toads in our sample have 27.5[27] or fewer spots.
Approximately725%
of the toads in our sample have 40.75[40] or fewer spots.
The toad in our
sample with the most spots has 98 spots.
Range43 = Q4 – Q3
= 98 – 40.75[40] =
57.25[57]
Approximately 25%
of the spotted toads in our sample have between 40.75[40] and 98 spots. The
largest possible difference in the spot count for any pair of spotted toads in
the upper quarter sample is 57.25[57] spots.
Range32 = Q3 – Q2
= 40.75[40] – 27.5[27] = 13.25[13]
Approximately 25%
of the spotted toads in our sample have between 27.5[27] and 40.75[40] spots.
The largest possible difference in the spot count for any pair of spotted toads
in the upper middle
quarter sample is 13.25[13] spots.
Range21 = Q2 – Q1
= 27.5[27] –
18 = 9.5[9]
Approximately 25%
of the spotted toads in our sample have between 18 and 27.5[27] spots. The
largest possible difference in the spot count for any pair of spotted toads in
the lower middle quarter sample is 9.5[9] spots.
Range10 = Q1 – Q0
= 18 – 9 = 9
Approximately 25%
of the spotted toads in our sample have between 9 and 18 spots. The largest possible
difference in the spot count for any pair of spotted toads in the lower quarter
sample is 9 spots.
Case Two |
Summary Intervals | 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.
Case Two |
Summary Intervals | Framingham Heart Study
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
Let
m denote the sample mean ratio, and sd
the sample standard deviation. Compute and interpret the
intervals m±2sd and m±3sd, using Tchebysheff’s Inequalities and the Empirical Rule.
Be specific and complete. Show your work, and discuss completely for full
credit.
lower2 = m – 2*sd =
1.696 – 2*0.205106 » 1.2857
upper2 = m + 2*sd =
1.696 + 2*0.205106 » 2.1062
At least 75% of
the Framingham Heart Study (FHS) subjects in the sample present diastolic to
systolic blood pressure
ratios between 1.29 and 2.10.
lower3 = m – 3*sd =
1.696 – 3*0.205106 » 1.0806
upper3 = m + 3*sd =
1.696 + 3*0.205106 » 2.3113
At least 89% of
the Framingham Heart Study (FHS) subjects in the sample present systolic to
diastolic blood pressure ratios (StDBPRs) between 1.08
and 2.31.
If the Framingham
Heart Study (FHS) subjects in the sample present systolic to diastolic blood
pressure ratios (StDBPRs) cluster symmetrically
around a central value, becoming rare
as the distance from the center increases, then:
Approximately 95% of
the Framingham Heart Study (FHS) subjects in the sample present systolic to
diastolic blood pressure ratios (StDBPRs) between
1.29 and 2.10.
Approximately 100%
of the Framingham Heart Study (FHS) subjects in the sample present systolic to
diastolic blood pressure ratios (StDBPRs) between
1.08 and 2.31.
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 left untreated leads to severe complications and possibly death. Suppose
we have a new candidate treatment, and further suppose that a standard
treatment for a similar (but different) disease is available. A comparative
clinical trial is proposed that would compare these treatments in patients with
condition Z.
The standard
treatment isn’t really a standard treatment for Z. The standard treatment group
requires a standard treatment for Z.
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 unaware of their participation in the
clinical trial – as far as they know, they’re getting treatment. If a standard
treatment is available, the use of a placebo group is inappropriate.
3.3) A
sample survey design targets a random sample of residents of metro Atlanta with
a well-designed questionnaire concerning driving/automotive safety practices.
The people running this survey sample design want to say that their results
will describe the driving/automotive safety practices of all Georgia drivers.
The survey
excludes non-metropolitan Atlanta residents of Georgia. Sample the entire
state, not just Atlanta.
3.4) A
random sample of parents of college/university first-year undergraduate
students is surveyed about the study practices of their children. The survey
questionnaire was properly written, and the sample of parents reasonably
selected. The parents responded to questions about their children's study
habits.
Directly survey
the students, not their parents.
Case Four | Clinical Trial Sketch | Age-related
Macular Degeneration
The retina is the light-sensitive portion
of the eye. Near the center of the
retina is the macula, the region of
the retina that provides central vision. In
wet macular degeneration, new blood vessels grow beneath the retina and leak
blood and fluid. This leakage causes permanent damage to light-sensitive
retinal cells, which die off and create blind spots in central vision. Choroidal neovascularization
(CNV), the underlying process causing wet AMD and abnormal blood vessel growth,
is the body's attempt to create a new network of blood vessels to supply more
nutrients and oxygen to the eye's retina. Instead, the process creates
scarring, leading to sometimes severe central vision loss.
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.
Avastin (bevacizumab) is derived from the same
monoclonal antibody as Lucentis (ranibizumab)
Visual
Acuity is measured by scoring the
patient against the ETDRS eye chart, and central vision by testing the patient
against the Amsler grid.
Preservation or improvement of
visual acuity can greatly affect quality of life.
Sketch a
comparative clinical trial comparing Lucentis versus Avastin in the treatment of age-related macular
degeneration. 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 Age-related Macular Degeneration. Those who give
informed consent are enrolled in the trial.
Enrolled subjects
are randomly assigned to either Lucentis with PlaceboAvastin or Avastin with PlaceboLucentis. 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. Central visual acuity as assessed against the Amsler
grid. They are also tracked for safety and toxicity, including kidney/liver,
and for changes in quality of life.
Work all four (4) cases.