Key
The 1st Hourly
Math 1107
Summer Term 2006
Protocol
You will use only the following resources:
Your individual calculator, Your individual tool-sheet (single 8.5 by 11 inch sheet),
Your writing utensils, Blank Paper (provided by me ) and this copy of the
hourly. Share these resources with no-one else. Show complete detail and work
for full credit. Follow case study solutions and sample hourly keys in
presenting your solutions.
Work all six cases.
Using only one side of the
blank sheets provided, present your work. Do not write on both sides of the
sheets provided, and present your work only on these sheets. Do not share
information with any other students during this hourly.
Sign and Acknowledge: I agree
to follow this protocol.
________________________________________________________________________
Name (PRINTED) Signature Date
Case One
Long Run Argument
Perfect Samples
Fictitious Hive Bugs
Performance
Notes: Be sure to follow the written format, and to show your calculations in full
detail.
An entomologist
is studying a particular species of entirely fictitious Hive Bug. The
entomologist has identified the following types of Hive Bug: Worker, Warrior,
Queen, Male. Suppose that the population of Fictitious Hive Bugs is governed by
this probability model:
|
Bug
Class |
Probability |
Expected |
|
Warrior |
0.1 |
2750*0.1
= 275 |
|
Worker |
0.8 |
2750*0.8
=2200 |
|
Queen |
0.01 |
2750*0.01
=27.5 |
|
Male |
0.09 |
2750*0.9
=247.5 |
|
Total |
1 |
2750 |
1.1) Interpret each probability using the Long Run
Argument.
In long runs of sampling with replacement,
approximately 10% of sampled Hive Bugs are Warriors.
In long runs of sampling with replacement,
approximately 80% of sampled Hive Bugs are Workers.
In long runs of sampling with replacement,
approximately 1% of sampled Hive Bugs are Queens.
In long runs of sampling with replacement,
approximately 9% of sampled Hive Bugs are Males.
1.2) Compute and interpret the perfect sample of
n=2750 Fictitious Hive Bugs.
EWarrior=n*PWarrior =
2750*0.1 = 275
EWorker=n*PWorker =
2750*0.8 =2200
EQueen=n*PQueen =
2750*0.01 =27.5
EMale=n*PMale =
2750*0.09 =247.5
In samples of 2750 Hive Bugs drawn with
replacement, approximately 275 sampled Hive Bugs are Warriors, approximately 2200
sampled Hive Bugs are Workers, approximately 27 or 28 sampled Hive Bugs are
Queens and approximately 247 or 248 sampled Hive Bugs are Males.
Show full detail for full credit.
Case Two
Clinical Trial Sketch
Statins and Ocular Hypertension
Performance Notes: Follow
the format for a clinical trial sketch. Dx(Diagnosis/Disease): OHT without
Glaucoma; Tx(Treatements): {Latanoprost+Placebo Simvastatin} versus
{Latanoprost+Simvastatin}; IC(Informed Consent): Enroll qualified subject
volunteers in the study. Random Sampling of patients is not performed.
RA(random assignment of volunteers to treatment); DB{Double Blinding: Neither
subjects nor clinical workers know the actual treatment status of individual
subjects); Follow-up(Effect): Study the response of the subjects to treatment,
specifically in this study the intraocular pressure(IOP)/OHT status of the
treated subjects; Follow-up(Safety/Side Effects): Study the occurrence of
adverse events that do not endanger the subjects' health or well-being, but
that may lead to a reduction in or cessation of treatment;
Follow-up(Toxicity/Mortality): Study the occurrence in the treated subjects the
occurrence of life-threatening events (such as kidney or liver failure) or
death.
The term ocular hypertension usually refers to any situation in which the
pressure inside the eye, called intraocular pressure, is higher than normal.
Eye pressure is measured in millimeters of mercury. Normal eye pressure ranges
from 10-21 mm Hg. Ocular hypertension is an eye pressure of greater than
21 mm Hg. Ocular hypertension is
commonly defined as a condition with the following criteria:
an intraocular pressure of greater than 21 mm Hg is measured in one
or both eyes on two or more occasions; the optic
nerve appears normal; no signs of glaucoma are evident on visual field testing; no
signs of any ocular disease are present.
Some eye diseases can increase the pressure inside the eye. It is thought that
people with persistent ocular hypertension may be at higher risk of developing
glaucoma.
However, within this
article, ocular hypertension primarily refers to
increased intraocular pressure but without any optic nerve damage or
vision loss. Glaucoma occurs when increased intraocular pressure, optic nerve
damage, and vision loss are present.
Statins are a class of drug
that lower serum cholesterol. One of the best known statins is simvastatin (Zocor). The principal effect
of statins is to lower serum cholesterol. Statins are suspected to have a
number of other beneficial effects. Statins are generally given in pill form.
Latanoprost (Xalatan) is used to treat certain kinds of glaucoma. It is
also used to treat a condition called hypertension of the eye. Latanoprost
appears to work by increasing the outflow of fluid from the eye. This lowers
the pressure in the eye. Latanoprost is given topically, as a suspension in
eyedrops.
Sketch a
comparative clinical trial comparing the effect of Latanoprost versus
Latanoprost+Simvastatin in the treatment of Ocular Hypertension.
We begin
by recruiting candidate subjects presenting ocular hypertension in the absence
of glaucoma. Those subjects who are briefed as to the purpose, requirements and
potential risks and benefits of the trial who give informed consent and who
meet appropriate inclusion and exclusion requirements are then enrolled in the
trial
Each
enrolled subject is then randomly assigned either to Latanoprost+PlaceboSimvastatin
or to Latanoprost+Simvastatin. Double-blinding is employed, in which neither
the subjects nor the clinical workers know the true treatment status of the
subjects.
Treated
subjects are then followed for: OHT Status, Intraocular Pressure, Visual
Function, Glaucome Status and Severity, Time to Glaucoma. Treated subjects are
followed for safety outcomes: adverse events including minor problems including
irritation of the eye. Treated subjects are followed for toxicity: significant
damage or loss of function of organs, including the eyes, liver and kidneys.
Treated subjects are followed for fatal outcomes.
Clinical
Trial Sketch Codes: Tx:Treatment, Dx:Diagnosis, IC:Informed Consent, RA:Random
Assignment, DB:Double Blinding.
Make your sketch concise
and complete, following the style demonstrated in class, in the sample second
hourlies and in case study summaries.
Case Three
Random Variables
Color Slot Machine
Performance Notes: This is a
random variable case. Compute the values of each random variable, including
details. For each value of each random variable, compute (showing full detail)
the probability of that value.
Here is our slot machine – on
each trial, it produces a 10-color sequence, using the table below:
|
Sequence* |
Probability |
|
RRBBRRYRGG |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYYYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered as 1st to
6th , from left to right: (1st 2nd 3rd
4th 5th6th7th 8th 9th
10th )
3.1) Define the random variable REDCOUNT as follows: REDCOUNT
= the number of times red shows in the sequence.
Compute the values of the
random variable, showing in detail how these values are computed from each
color sequence. Compute the probability for each value of the random variable,
showing full detail.
|
Sequence* |
REDCOUNT |
Probability |
|
RRBBRRYRGG |
5 |
.10 |
|
RRGGRGBRRB |
5 |
.10 |
|
BBYYGGYGBR |
1 |
.15 |
|
GRRGGYBRGG |
3 |
.10 |
|
BGYYYRYGYY |
1 |
.25 |
|
RRYYGRRBBY |
4 |
.10 |
|
YYGBYYBGRR |
2 |
.20 |
|
Total |
|
1.00 |
Pr{ REDCOUNT
=1} = Pr{ one of BBYYGGYGBR, BGYYYRYGYY shows}
=
Pr{ BBYYGGYGBR}
+ Pr{BGYYYRYGYY}
= .15 + .25 = .40
Pr{ REDCOUNT
=2} = Pr{ YYGBYYBGRR
} = .20
Pr{ REDCOUNT
=3} = Pr{ GRRGGYBRGG
} = .10
Pr{ REDCOUNT
=4} = Pr{ RRYYGRRBBY
} = .10
Pr{ REDCOUNT
=5} = Pr{ one of RRBBRRYRGG,
RRGGRGBRRB shows}
=
Pr{ RRBBRRYRGG}
+ Pr{ RRGGRGBRRB}
= .10+.10 = .20
|
REDCOUNT |
Sequences |
Probability |
|
5 |
RRBBRRYRGG(.10),
RRGGRGBRRB(.10) |
.20 |
|
4 |
RRYYGRRBBY(.10) |
.10 |
|
3 |
GRRGGYBRGG(.10) |
.10 |
|
2 |
YYGBYYBGRR(.20) |
.20 |
|
1 |
BBYYGGYGBR(.15),
BGYYYRYGYY(.25) |
.40 |
|
|
|
1.00 |
3.2) Define the random variable BGCOUNT as follows: BGCOUNT = the number
of times BG
shows in the sequence. Compute
the values of the random variable, showing in detail how these values are
computed from each color sequence. Compute the probability for each value of
the random variable, showing full detail.
|
Sequence* |
BGCOUNT |
Probability |
|
RRBBRRYRGG |
0 |
.10 |
|
RRGGRGBRRB |
0 |
.10 |
|
BBYYGGYGBR |
0 |
.15 |
|
GRRGGYBRGG |
0 |
.10 |
|
BGYYYRYGYY |
1 |
.25 |
|
RRYYGRRBBY |
0 |
.10 |
|
YYGBYYBGRR |
1 |
.20 |
|
Total |
|
1.00 |
Pr{ BGCOUNT = 1} = Pr{ one of BGYYYRYGYY, YYGBYYBGRR shows} =
Pr{ BGYYYRYGYY
} + Pr{ YYGBYYBGRR
} = .25 + .20 = .45
Pr{ BGCOUNT = 0} = Pr{ one of RRBBRRYRGG, RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, RRYYGRRBBY shows } = .10+.10+.15+.10+.10 = .55
|
BGCOUNT |
Sequences |
Probability |
|
1 |
BGYYYRYGYY(.25), YYGBYYBGRR(.20) |
.45 |
|
0 |
RRBBRRYRGG(.10),
RRGGRGBRRB(.10),
BBYYGGYGBR(.15), GRRGGYBRGG(.10),
RRYYGRRBBY(.10) |
.55 |
|
|
|
1.00 |
3.3) Define the random variable YCOUNT as
follows: YCOUNT
= the number of times yellow shows in the
sequence. Compute the
values of the random variable, showing in detail how these values are computed
from each color sequence. Compute the probability for each value of the random
variable, showing full detail.
|
Sequence* |
YCOUNT |
Probability |
|
RRBBRRYRGG |
1 |
.10 |
|
RRGGRGBRRB |
0 |
.10 |
|
BBYYGGYGBR |
3 |
.15 |
|
GRRGGYBRGG |
1 |
.10 |
|
BGYYYRYGYY |
6 |
.25 |
|
RRYYGRRBBY |
3 |
.10 |
|
YYGBYYBGRR |
4 |
.20 |
|
Total |
|
1.00 |
Pr{YCOUNT
=0} = Pr{RRGGRGBRRB}
= .10
Pr{YCOUNT
=1} = Pr{ one of RRBBRRYRGG, GRRGGYBRGG shows}
=
Pr{ RRBBRRYRGG
} + Pr{ GRRGGYBRGG
} = .10+.10 = .20
Pr{YCOUNT
=3} = Pr{ one of BBYYGGYGBR, RRYYGRRBBY shows}
=
Pr{ BBYYGGYGBR
} + Pr{ RRYYGRRBBY
} = .15 + .10 = .25
Pr{YCOUNT =4}
= Pr{ YYGBYYBGRR
} = .20
Pr{YCOUNT
=6} = Pr{ BGYYYRYGYY
} = .25
|
YCOUNT |
Sequences |
Probability |
|
6 |
YYGBYYBGRR(.25) |
.25 |
|
4 |
YYGBYYBGRR(.20) |
.20 |
|
3 |
BBYYGGYGBR(.15),
RRYYGRRBBY(.10) |
.25 |
|
1 |
RRBBRRYRGG(.10),
GRRGGYBRGG(.10) |
.20 |
|
0 |
RRGGRGBRRB(.10) |
.10 |
|
|
|
1.00 |
Case Four
Design Spot Fault
Performance Notes:These
cases are straightforward. Use the information given in the mini-sketch,
and keep it simple. If you're bringing in outside information, you're missing
the problem. Do not perform sketches in this case, spot the problem(s) in
the mini-sketches provided.
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.
4.1) 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. Suppose we have a new candidate treatment(A), and further suppose that a
standard treatment for a similar (but different) disease is available(B). A
comparative clinical trial is proposed that would use three treatment groups:
A, B and placebo.
The B group is getting an unproven
treatment (when a standard is available, and the consequences of non-treatment
severe). The placebo is denied treatment (when a standard is available, and the
consequences of non-treatment severe).
4.2) 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
not informed of the treatments, and the placebo group is denied treatment when
treatments may be available.
4.3) 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 sampling is restricted to metropolitan
Atlanta residents, and inference cannot be extended beyond metropolitan Atlanta.
4.4) 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.
The parents most likely are not in a
position to provide valid information about their children.
Case Five
Probability
Rules
Consumer Credit Score
Performance Notes: Show complete
detail in these calculations, and when a rule is stated, use that rule in
performing the calculation.
Fair Isaac Corp., a
California-based financial company, developed a consumer credit score, a number that summarizes the risk present in
lending money to a consumer. The consumer credit score ranges from 300 to 850.
Credit bureau scores are often called “FICO scores” because most credit bureau
scores used in the U.S. are produced from software developed by Fair Isaac and
Company. FICO scores are provided to lenders by the major credit reporting
agencies.
Suppose that the probabilities for consumer credit
scores for US residents are noted below:
|
Credit score |
Probability |
|
499 and
below |
.02 |
|
500-549 |
.05 |
|
550-599 |
.08 |
|
600-649 |
.12 |
|
650-699 |
.15 |
|
700-749 |
.18 |
|
750-799 |
.27 |
|
800 and
above |
.13 |
|
Total |
1.00 |
Compute the following probabilities. In
each of the following, show your intermediate steps and work. If a rule is
specified, you must use that rule for your computation.
5.1) Pr{Credit Score ³ 700} = Pr{ CS ³
800 or 750 £
CS £ 799 or 700 £
CS £ 749} = Pr{ CS ³
800 } + Pr{750 £
CS £ 799}+Pr{700 £
CS £ 749} = .18+.27+.13 = .58
5.2) Pr{550 £
Credit Score < 700} =
Pr{ 550 £ CS £
599 or 600 £
CS £ 649 or 649 £
CS £ 699} =
Pr{ 550 £
CS £ 599 } + Pr{600 £
CS £ 649 }+Pr{649 £
CS £ 699} = .08+.12+.15 = . 35
5.3) Pr{Credit Score ³ 600}
– Use the Complementary Rule
Pr{Credit Score < 600} = Pr{ CS £
499 or 500 £
CS £ 549 or 550 £
CS £ 599} = Pr{CS £
499}+Pr{500 £
CS £ 549}+Pr{550 £
CS £ 599} = .02+.05+.08 = .15
Pr{Credit Score ³ 600} =
1 – Pr{Credit Score < 600} = 1 – .15 = .85
Show complete detail and work for full credit.
Case Six
Computation of Conditional Probabilities.
Color Slot Machine
Performance Notes: Show full
detail in these calculations: in each part, compute the joint probability, the
prior probability and the conditional probability.
Here is our slot machine – on
each trial, it produces a 10-color sequence, using the table below:
|
Sequence* |
Probability |
|
RRBBRRYRGG |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYYYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered as 1st to
6th , from left to right: (1st 2nd 3rd
4th 5th6th7th 8th 9th
10th )
Compute the following conditional probabilities:
6.1) Pr{GRR Shows | GG Shows}
|
Sequence* |
Probability |
|
RRBBRRYRGG |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
Total |
.45 |
Pr{GG Shows} = Pr{
One of RRBBRRYRGG , RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{RRBBRRYRGG}+Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} = .1+.1+.15+.1 = .45
Pr{GRR and GG Shows} =
Pr{GRRGGYBRGG} = .10
Pr{GRR Shows | GG Shows}=
Pr{GRR and GG }/ Pr{GG } =
.10/.45 = 2/9 » .2222
6.2) Pr{Green Shows | BR Shows}
|
Sequence* |
Probability |
|
RRBBRRYRGG |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
Total |
.45 |
Pr{BR } = Pr{ One of RRBBRRYRGG , RRGGRGBRRB , BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{ RRBBRRYRGG}+Pr{RRGGRGBRRB}+Pr{BBYYGGYGBR}+Pr{GRRGGYBRGG}= .1+.1+.15+.1 = .45
Pr{Green and BR } Pr{ One of RRBBRRYRGG , RRGGRGBRRB , BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{ RRBBRRYRGG}+Pr{RRGGRGBRRB}+Pr{BBYYGGYGBR}+Pr{GRRGGYBRGG}= .1+.1+.15+.1 = .45
Pr{Green Shows | BR Shows}=Pr{Green and BR}/Pr{BR} = .45/.45 = 1.00
6.3) Pr{Yellow Shows
| Blue Shows}
|
Sequence* |
Probability |
|
RRBBRRYRGG |
.10 |
|
RRGGRGBRRB |
.10 |
|
BBYYGGYGBR |
.15 |
|
GRRGGYBRGG |
.10 |
|
BGYYYRYGYY |
.25 |
|
RRYYGRRBBY |
.10 |
|
YYGBYYBGRR |
.20 |
|
Total |
1.00 |
Pr{Blue} = Pr{ One of RRBBRRYRGG , RRGGRGBRRB , BBYYGGYGBR , GRRGGYBRGG , BGYYYRYGYY , RRYYGRRBBY , YYGBYYBGRR Shows} = Pr{ RRBBRRYRGG}+
Pr{ RRGGRGBRRB}+Pr{BBYYGGYGBR}+Pr{GRRGGYBRGG}+Pr{BGYYYRYGYY}+
Pr{ RRYYGRRBBY}+Pr{YYGBYYBGRR} =
.1+.1+.15+.1+.25+.1+.2 = 1.00
Pr{Yellow and Blue} = Pr{ One of RRBBRRYRGG , BBYYGGYGBR , GRRGGYBRGG , BGYYYRYGYY , RRYYGRRBBY , YYGBYYBGRR Shows} = Pr{ RRBBRRYRGG}+
Pr{BBYYGGYGBR}+Pr{GRRGGYBRGG}+Pr{BGYYYRYGYY}+
Pr{ RRYYGRRBBY}+Pr{YYGBYYBGRR} =
.1+.15+.1+.25+.1+.2 = .90
Pr{Yellow Shows | Blue Shows} = Pr{Yellow and
Blue}/Pr{ Blue}
= .90/1.00 = .90
Show complete detail for full credit.
Performance Summary: I used the best five
of your six cases.
|
item |
n |
m |
p0 |
p25 |
p50 |
p75 |
p100 |
|
All Six |
34 |
70.9 |
18.0 |
56.0 |
74.3 |
93.3 |
98.0 |
|
Best 5 |
34 |
78.8 |
21.6 |
67.2 |
85.2 |
97.6 |
100.0 |
|
Gain |
34 |
7.9 |
1.6 |
3.7 |
7.2 |
11.2 |
16.3 |
Based on a score using only the best five of six cases:
There were 34 tests taken.
The average score was approximately 78.8%.
The minimum score among the tests taken is approximately
18.0%
Approximately 25% of the tests taken yielded scores of 67.2%
or less.
Approximately 50% of the tests taken yielded scores of 85.2%
or less.
Approximately 75% of the tests taken yielded scores of 97.6%
or less.
The maximum score(s) among the tests taken is 100%.
Summary of 1st Hourly Scores
|
scorehr1_raw |
Frequency |
Percent |
|
<60 |
10 |
29.41 |
|
[60,70) |
3 |
8.82 |
|
[70,80) |
8 |
23.53 |
|
[80,90) |
3 |
8.82 |
|
[90,100] |
10 |
29.41 |
Based on a score using only the best five of six cases:
There were 34 tests taken.
The minimum score among the tests taken is approximately
18.0%
Approximately 23.5%(8 of 34) of the tests taken yielded
scores strictly below 60%.
Approximately 2.9%(1 of 34) of the tests taken yielded
scores at or above 60%, but strictly below 70%.
Approximately 11.8%(4 of 34) of the tests taken yielded
scores at or above 70%, but strictly below 80%.
Approximately 17.6%(6 of 34) of the tests taken yielded
scores at or above 80%, but strictly below 90%.
Approximately 44.1%(15 of 34) of the tests taken yielded
scores at or above 90%, including X scores at 100%.
|
scorehr1_best5 |
Frequency |
Percent |
|
<60 |
8 |
23.53 |
|
[60,70) |
1 |
2.94 |
|
[70,80) |
4 |
11.76 |
|
[80,90) |
6 |
17.65 |
|
[90,100] |
15 |
44.12 |
Effect of Dropping the Worst Case
|
scorehr1_raw |
scorehr1_best5 |
Total |
||||
|
<60 |
[60,70) |
[70,80) |
[80,90) |
[90,100] |
||
|
<60 |
8 |
1 |
1 |
0 |
0 |
10 |
|
[60,70) |
0 |
0 |
2 |
1 |
0 |
3 |
|
[70,80) |
0 |
0 |
1 |
5 |
2 |
8 |
|
[80,90) |
0 |
0 |
0 |
0 |
3 |
3 |
|
[90,100] |
0 |
0 |
0 |
0 |
10 |
10 |
|
Total |
8 |
1 |
4 |
6 |
15 |
34 |