Key
The 2nd Hourly
Math 1107
Fall Semester 2009
Protocol
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.
________________________________________________________________________
Name (PRINTED) Signature Date
Case One | Descriptive Statistics | Fictitious
City Half Marathon
A half marathon
is an event in which runners complete a course of approximately 13.1 miles.
Suppose that we have a random sample of finishers of the 2008 Fictitious City
Half Marathon, whose finishing times (in hours) are listed below:
1.03, 1.04,
1.08, 1.10, 1.22, 1.25, 1.30, 1.35, 1.40, 1.45, 1.50, 1.55, 1.60, 1.72, 1.75,
1.77, 1.80, 1.83, 1.85, 1.89, 1.90, 1.95, 2.00, 2.25, 2.30, 2.33, 2.40, 2.45,
2.50, 2.55, 2.57, 2.60, 2.63, 2.65, 2.70, 2.75, 2.78, 2.90, 2.95, 3.20, 3.22,
3.25, 3.27, 3.30, 3.35, 3.37, 3.39, 3.40, 3.45, 3.52, 3.62, 3.75, 3.90, 4.00,
4.08, 4.13
Compute
and interpret the following statistics: sample size, p00, p25, p50, p75, p100,
(p100 – p25), (p75 – p25), (p75 – p00).
n
p00 p25 p50
p75 p100 range41
range31 range30
56
1.03 1.735 2.475
3.26 4.13 2.395
1.525 2.23
range41 = p100 - p25 = 4.13 - 1.735 = 2.395
range31 = p75 - p25 = 3.26 - 1.735 = 1.525
range30 = p75 - p00 = 3.26 - 1.03 = 2.23
There
are 56 Fictitious City Half Marathon runners in the sample.
The
FCHM runner in the sample with the fastest finish ran the half marathon in 1.03
hours.
Approximately
25% of the FCHM runners in the sample ran the half marathon in 1.735 hours or less.
Approximately
50% of the FCHM runners in the sample ran the half marathon in 2.475 hours or less.
Approximately
75% of the FCHM runners in the sample ran the half marathon in 3.26 hours or less.
The
FCHM runner in the sample with the fastest finish ran the half marathon in 4.13
hours.
Approximately
75% of the FCHM runners in the sample ran the half marathon in between 1.735
and 4.13 hours. The largest difference in finish time between any pair of
runners in this upper ¾ sample is 2.395 hours.
Approximately
50% of the FCHM runners in the sample ran the half marathon in between 1.735
and 3.26 hours. The largest difference in finish time between any pair of
runners in this middle half sample is 1.525 hours.
Approximately
75% of the FCHM runners in the sample ran the half marathon in between 1.03 and
3.26 hours. The largest difference in finish time between any pair of runners
in this lower ¾ sample is 2.23 hours.
Case Two |
Summary Intervals | Fictitious Striped
Lizard
|
|
The Fictitious
Striped Lizard (FSL) is a native species of Lizard Island, and is
noteworthy for the both the quantity and quality of its stripes. Consider a
random sample of FSL, in which the number of stripes per lizard is
noted: |
29,
17, 13, 18, 21, 11, 12, 22, 10, 17, 18, 22, 22, 20, 19, 14, 18, 17, 8, 9, 17,
15, 5, 6, 10, 8, 11, 17, 15, 8,
14,
5, 6, 9, 16, 13, 6, 20, 20, 14, 5, 20, 7, 21, 17, 12, 14, 16, 12, 13, 16, 14,
10, 12, 14, 13, 15, 5, 1, 4, 13
20,
25, 25, 17, 27, 11, 12, 13, 22, 19, 22,
14, 20, 17, 17, 12, 14, 12, 15, 20, 19, 17, 23, 15, 25, 9, 22, 14
Let
m denote the sample mean finish time, 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.
n
m sd lower2sd upper2sd
lower3sd upper3sd
89
14.8764 5.70052 3.47537
26.2774 -2.22515 31.9780
lower2sd = m – 2*sd
= 14.8764 - 2*5.70052 =
3.47537[4]
upper2sd = m + 2*sd
= 14.8764 + 2*5.70052 =
26.2774[26]
lower3sd = m – 3*sd
= 14.8764 - 3*5.70052 =
-2.22515[0]
upper2sd = m + 3*sd
= 14.8764 + 3*5.70052 =
31.9780[31]
At least 75% of the Fictitious Striped
Lizards in the sample have between 4 and 26 stripes.
At least 89% of the Fictitious Striped
Lizards in the sample have between 0 and 31 stripes.
If the Fictitious Striped Lizards stripe counts cluster symmetrically around a central
value, becoming rare as the distance from the center increases, then:
Approximately 95% of the Fictitious Striped
Lizards in the sample have between 4 and 26 stripes.
Approximately 100% of the Fictitious
Striped Lizards in the sample have between 0 and 31 stripes.
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.
1. A clinical trial of a new Hepatitis C (HepC)
treatment is designed as follows: subjects are screened for HepC
infection. Untreated HepC
can lead to liver disease, liver failure, liver cancer and death. Those who
test positive for HepC infection are then told of
their status, and are offered treatment for HepC at
no cost, and are offered enrollment in a comparative clinical trial for the
treatment of HepC infection. Those who qualify and
who give informed consent are then stratified by risk of progression to
symptoms of hepatitis. Those judged to be at high risk are assigned to the new
treatment, those judged to be at moderate risk of progression to symptoms of
hepatitis are assigned to standard treatment, and those judged to be at minimal
risk of progression to symptoms of hepatitis symptoms are assigned to placebo
only.
Denial of informed consent, denial of treatment,
nonrandom assignment to treatment, inappropriate use of placebo
2. A sample
survey of US University and College undergraduates is used to study the
sexual habits, knowledge and attitudes of their parents and legal guardians.
Appropriate random sampling of students is used, and there are no problems with
the wording and delivery of the survey instrument.
Inappropriate
use of proxy respondents – ask the parents directly.
3. WidgetCorpsä is conducting an Employee Satisfaction Survey. They hire a third party to conduct the survey, and a random
sample of employees is employed in the survey.
The interviews are conducted after the annual performance and salary
reviews, and while the third party identifies the identities of the respondents
on the surveys, it tells the respondents that these will be removed before the
results are given to WidgetCorpsä.
Identification of respondents
may affect confidentiality. If the survey is too close to reviews, there may be
a distortion of responses.
4. Disease W is a disease caused by an infection. Left untreated, disease
W produces severe and occasionally fatal symptoms and complications. Suppose
that no effective, standard treatments are available. Suppose further that a
new treatment, ugorbitx is available for evaluation.
A clinical trial is proposed to
evaluate ugorbitx by giving all trial subjects ugorbitx. Their results would be compared to similar groups
of untreated patients who are not enrolled in the trial.
Use an internal
comparison group by randomly assigning subjects to either treatment or to
placebo.
Case Four | Clinical Trial Sketch | 2009 H1N1 Vaccine in
Pregnant Women Trial
|
|
This is a new influenza A(H1N1) virus
that has never before circulated among humans. This virus is not related to
previous or current human seasonal influenza viruses. Signs of influenza A(H1N1) including fever, cough,
headache, muscle and joint pain, sore throat and runny nose, and sometimes
vomiting and diarrhea. Influenza is a disease
characterized by fever, cough, headache, muscle and joint pain, sore throat,
congestion, and occasionally nausea and diarrhea. It is caused by a family of
viruses, all of which mutate from year to year. The 2009 H1N1 influenza virus
is a type A strain that is distinct from previous influenza viruses. It is
distinct in that it is more easily transmitted than the usual strains of
influenza viruses and in that it causes serious complications in
subpopulations not usually endangered by the usual strains of influenza
viruses. Serious complications of influenza include pneumonia and respiratory
failure. |
A vaccine reduces or prevents infection by exposing a person’s immune
system to selected components of a virus – this exposure leads to the
production of antibodies specifically designed to attack the original virus. An
effective vaccine induces the production of relevant antibodies, and these
antibodies then enhance the ability of a vaccinated person to resist infection
by the original virus.
The 2009 H1N1 Influenza virus is sufficiently distinct from other
strains of influenza viruses that a new vaccine is needed. Suppose that we have
developed a new vaccine, H1N12009PGXVax, based on a killed H1N1 influenza
virus. In particular, pregnant women are at higher than usual risk of
complications during H1N1-related influenza, as are children younger than two
years of age.
Inclusion Criteria: Pregnant female
between the ages of 18 and 39 years, Is from 14-34 weeks of gestation,
inclusive. Is in good health, as determined by vital signs (normal heart
function, normal blood pressure, no fever). Able to understand and comply with
planned study procedures. Provides written informed consent prior to initiation
of any study procedures.
Exclusion Criteria: have a known allergy to
eggs or other components of the vaccine, have suppressed immune system, have
cancer, use steroids or any immuno-suppressive drugs,
have been recently vaccinated, have a prior history of severe reactions
following previous immunization with influenza virus vaccines, had a recent fever,
had a prior H1N1 vaccine or infection within the past two years.
Outcome Measures: Safety: Occurrence of vaccine-associated serious adverse
events, including severe allergic reactions to the vaccine. Occurrence of
pregnancy-related adverse events. Immune:
Achieving a significant immune response, measured as an increase in H1N1
specific antibodies. Prevention:
Lower incidence of H1N1 influenza among vaccinated subjects.
Sketch a basic clinical trial evaluating
the safety and effectiveness of the new vaccine in the production of relevant
antibodies and in the prevention of H1N1 influenza. Make your sketch concise and complete, following the
style demonstrated in class, in the sample second hourlies
and in case study summaries.
We recruit H1N1-free pregnant females in good health, between the ages of 18 and 39
years, In the 14-34 weeks of gestation, inclusive. Who are able to understand
and comply with planned study procedures. Provides written informed consent
prior to initiation of any study procedures. Those who meet all inclusion/exclussion requirements, and who give informed consent are
enrolled in the trial.
Study subjects are
randomly assigned to either H1N12009PGXVax
or to a placebo version of H1N12009PGXVax. Double blinding is employed, so that
neither the subjects nor the study personnel know the actual treatment
assignments.
Treated subjects are then tracked for occurrence
of vaccine-associated serious adverse events, including severe allergic
reactions to the vaccine, occurrence of pregnancy-related adverse events,
significant immune response, measured as an increase in H1N1 specific
antibodies, as well as other side effects and toxicity.
Work all four (4) cases.