Key | The Comprehensive Final Examination | Math
1107 | Summer 2010 | CJ Alverson
Protocol
You will use only the
following resources: Your
individual calculator; Your individual tool-sheets (two (2) 8.5 by 11 inch
sheets); Your writing utensils; Blank Paper (provided by me); This copy of the
hourly and
the tables provided by
me. 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.
Do not write on both sides of the sheets provided, and present your work only
on these sheets.
When youre done: Print
your name on a blank sheet of paper. Place your toolsheet,
test and work under this sheet, and turn it all in to me.
Do not share information
with any other students during this test.
Sign and
Acknowledge:
I agree to follow this
protocol. Initial: ______
______________________________________________________________________________________
Name
(PRINTED)
Signature
Date
Case One | Probability
Computation | Color Slot Machine
Here is our slot machine
on each trial, it produces a color sequence, using the table below:
|
Sequence* |
Probability |
|
RRYRRR |
.20 |
|
BBYYBR |
.15 |
|
GRRGGY |
.10 |
|
YGYRYG |
.25 |
|
YYRGBY |
.10 |
|
YYYYRG |
.20 |
|
Total |
1.00 |
*B-Blue, G-Green, R-Red,
Y-Yellow, Sequence is numbered from left to right. In each of the following, show your intermediate
steps and work.
a) Define the random
variable YC as the number of times that yellow shows in the color
sequence. List the values of YC, and compute the probabilities for those
values.
|
Sequence* |
YC |
Probability |
|
RRYRRR |
1 |
.20 |
|
BBYYBR |
2 |
.15 |
|
GRRGGY |
1 |
.10 |
|
YGYRYG |
3 |
.25 |
|
YYRGBY |
3 |
.10 |
|
YYYYRG |
4 |
.20 |
|
Total |
|
1.00 |
Pr{YC=1} = Pr{RRYRRR or GRRGGY shows} = Pr{RRYRRR} + Pr{GRRGGY} =
0.20 + 0.10 = 0.30
Pr{YC=2} = Pr{BBYYBR} = 0.15
Pr{YC=3} = Pr{YGYRYG or YYRGBY shows} = Pr{YGYRYG} + Pr{YYRGBY} =
0.25 + 0.10 = 0.35
Pr{YC=4} = Pr{YYYYRG} = 0.20
b) Pr{RY Shows}
|
Sequence* |
RY |
Probability |
|
RRYRRR |
Yes |
.20 |
|
BBYYBR |
No |
.15 |
|
GRRGGY |
No |
.10 |
|
YGYRYG |
Yes |
.25 |
|
YYRGBY |
No |
.10 |
|
YYYYRG |
No |
.20 |
|
Total |
|
1.00 |
Pr{One of RRYRRR or YGYRYG shows} = Pr{RRYRRR}
+ Pr{YGYRYG} = 0.20 + 0.25 = 0.45
c) Pr{Green Shows} Use the Complementary
Rule.
Other Event = Green Does Not Show
|
Sequence* |
Green Does Not Show |
Probability |
|
Yes |
.20 |
|
|
BBYYBR |
Yes |
.15 |
|
GRRGGY |
No |
.10 |
|
YGYRYG |
No |
.25 |
|
YYRGBY |
No |
.10 |
|
YYYYRG |
No |
.20 |
|
Total |
|
1.00 |
Pr{Green Does Not Show} = Pr{RRYRRR or BBYYBR shows} = Pr{RRYRRR} +
Pr{BBYYBR} = 0.20 + 0.15 = 0.35
Pr{Green Shows} = 1 Pr{Green Does Not Show} = 1 0.35
= 0.65
Case Two | Descriptive
Statistics |
Traumatic Brain Injury
Traumatic
Brain Injury (TBI) involves the
injury of the brain when it involves sudden or intense physical force resulting
in the presence of Concussion, Skull Fracture, or Bleeding and Tissue Damage
(Contusions, Lacerations, Hemorrhaging) involving the brain.
|
|
|
|
A random sample of TBI cases is acquired, and
the age at injury (in years) of the case is determined. The sample ages at
injury are listed below:
4,
5, 5, 6, 6, 7, 7, 8, 9, 12, 12, 13, 14, 15, 15, 16, 18, 18, 18, 19, 19, 20, 20,
20, 20, 21, 21, 22, 23, 25, 27, 27, 30, 30, 30, 31, 32, 32, 33, 35, 35, 36, 36,
37, 37, 38, 38, 39, 39, 40, 41, 41, 41, 42, 42, 45, 47, 50, 52, 60,63, 65, 70,
70, 71, 71, 71, 71, 72, 72, 72, 73, 73, 74, 74, 75, 75, 76, 76, 76, 77,
79, 80, 81, 89, 90, 91
Compute
and interpret the following statistics: sample size, p00, p25, p50, p75, p100,
(p100 p50),
(p75
p25), (p50 p00). Show complete detail and work
for full credit. Follow case study
solutions and sample hourly keys in presenting your solutions.
100%
Max 91
75%
Q3 71
50%
Median 37
25%
Q1 20
0% Min 4
n p00 p25
p50 p75 p100
Range42 Range31 Range20
87 4 20
37 71 91
54 51 33
range42 = p100 p50 = 91
37 = 54
range31 = p75 p25 = 71
20 = 51
range20 = p50 p00 = 37
4 = 33
There are 87 Traumatic
Brain Injury (TBI) cases in our sample.
The TBI case in our
sample with the earliest TBI injury was injured at 4 years of age.
Approximately 25% of the
TBI cases in our sample were injured at 20 years of age or younger.
Approximately 50% of the
TBI cases in our sample were injured at 37 years of age or younger.
Approximately 75% of the
TBI cases in our sample were injured at 71 years of age or younger.
The TBI case in our
sample with the latest TBI injury was injured at 91 years of age.
Approximately 50% of the
TBI cases in our sample were injured between 4 and 37 years of age. The largest
difference in age at injury between any pair of TBI cases in this lower half
sample is 33 years.
Approximately 50% of the
TBI cases in our sample were injured between 20 and 71 years of age. The
largest difference in age at injury between any pair of TBI cases in this
middle half sample is 51 years.
Approximately 50% of the
TBI cases in our sample were injured between 37 and 91 years of age. The
largest difference in age at injury between any pair of TBI cases in this upper
half sample is 54 years.
Case Three | Clinical
Trial Sketch | Traumatic Brain Injury, Prevention of Post-TBI Epilepsy
Epilepsy is a brain disorder in which clusters of nerve cells, or neurons,
in the brain sometimes signal abnormally. In epilepsy, the normal pattern of
neuronal activity becomes disturbed, causing strange sensations, emotions, and
behavior or sometimes convulsions, muscle spasms, and loss of consciousness.
Epilepsy is a disorder with many possible causes. Anything that disturbs the
normal pattern of neuron activity - from illness to brain damage to abnormal
brain development - can lead to seizures. Epilepsy may develop because of an
abnormality in brain wiring, an imbalance of nerve signaling chemicals called
neurotransmitters, or some combination of these factors. Having a seizure does
not necessarily mean that a person has epilepsy. Only when a person has had two
or more seizures is he or she considered to have epilepsy. EEGs and brain scans
are common diagnostic test for epilepsy. Traumatic brain injury (TBI)
causes epilepsy in up to 30% of civilian and 50% of military head injuries,
exacerbating chronic neurological disability. There is currently no method for
preventing epilepsy after TBI.
Topiramate, an
anticonvulsant, is used alone or with other medications to treat certain types
of seizures in people who have epilepsy. Phenytoin, an
anticonvulsant, is used to control certain type of
seizures, and to treat and prevent seizures that may begin during or after
surgery to the brain or nervous system. It works by decreasing abnormal
electrical activity in the brain.
Inclusion Criteria: 1)Moderate to
severe traumatic brain injury; 2)Time since TBI
less than 24 hours; 3)Age greater than or equal to 18 years and 4)Subject capable of giving informed
consent or have an acceptable surrogate capable of giving consent on the
subject's behalf.
Exclusion Criteria: 1) Prior history
of epilepsy or unprovoked seizures; 2)Irreversibly fatal TBI.
Endpoints:
Prevention of epilepsy; Reduction of epilepsy frequency, timing and severity
of seizures.
Treatment
Groups: Use three groups: Placebo Only versus Phenytoin
versus Phenytoin+Topiramate
Sketch
a basic clinical trial evaluating the effectiveness of Phenytoin
and Topiramate in the prevention and reduction of epilepsy
in subjects with recent TBI. 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 with
recent (acquired within 24 hours), moderate-to-severe traumatic brain injury,
excluding those with prior histories of epilepsy or unprovoked seizures, and
those with irreversibly fatal TBI. Those
giving informed consent (or proxy consent) are enrolled in the trial.
Enrolled subjects are randomly assigned to one
of three treatment groups: PlaceboPhenytoin
+ PlaceboTopiramate,
Phenytoin + PlaceboTopiramate
or Phenytoin + Topiramate.
Double blinding is employed, so that neither subjects nor clinical workers know
individual treatment status.
Treated subjects are tracked for the
occurrence of epilepsy, and in those who present epilepsy, timing, patterns,
frequency and severity of seizures. Safety and toxicity are also tracked in all
treated subjects
Case Four | Confidence
Interval, Population Proportion | Glioblastoma Multiforme
Glioblastoma multiforme (GBM) is the highest
grade glioma tumor and is the most malignant form of astrocytomas. These tumors
originate in the brain. GBM tumors grow rapidly, invade nearby tissue and
contain cells that are very malignant. GBM are among the most common and
devastating primary brain tumors in adults.
Suppose that we have a random sample of GBM patients,
with survival time (in months) listed below:
0,
1, 2, 2, 3 | 3, 3, 3, 4, 4 |4, 4, 4, 5, 5 | 5, 5, 5, 5, 6 | 6, 6, 6, 6, 7 | 7,
7, 8, 8, 8 | 9, 10, 10, 10, 10 | 11, 11, 11, 11, 12
12, 12, 12, 12, 13 | 13, 13, 13, 14, 14 | 14, 15,
15, 16, 16 | 16, 17, 17, 18, 18 | 18, 19, 19, 20, 21 | 22, 23, 24, 24, 25 27, 29,
30, 32, 36 | 38, 40, 58, 60, 61
Consider the proportion of GBM patients who survive 24
months or longer. Compute and interpret a 95% confidence interval for this
population proportion. Show
your work. Completely discuss and interpret your test results, as indicated in
class and case study summaries.
0, 1, 2, 2, 3 | 3, 3, 3, 4, 4 |4, 4, 4, 5, 5 | 5, 5, 5, 5, 6 | 6,
6, 6, 6, 7 | 7, 7, 8, 8, 8 | 9, 10, 10, 10, 10 | 11, 11, 11, 11, 12
12, 12, 12, 12, 13 | 13, 13,
13, 14, 14 | 14, 15, 15, 16, 16 | 16, 17, 17, 18, 18 | 18, 19, 19, 20, 21 | 22,
23, 24, 24, 25 27, 29, 30, 32,
36 | 38, 40, 58, 60, 61
|
n |
e |
p=e/n |
sdp=sqrt(p*(1-p)/n) |
Z |
lower=p-2*sdp |
upper=p+(2*sdp) |
|
80 |
13 |
0.1625 |
0.041245265 |
2 |
0.08000947 |
0.244991 |
n=80
event = GBM patient survives 24
months or longer
event count = e = 13
sample proportion for
event = e/n = 13/80 = 0.1625
sdp = sqrt(p*(1
p)/n) = sqrt( (13/80)*(67/80)/80 ) ≈
0.04124
from 2.00 0.02275
0.95450, Z=2.00
lower95 = p (2*sdp) = (13/80) 2*sqrt( (13/80)*(67/80)/80 ) ≈
0.0800
upper95 = p + (2*sdp) = (13/80) + 2*sqrt( (13/80)*(67/80)/80 ) ≈ 0.2449
Write the interval as
[0.0800, 0.2449].
We estimate the
population proportion of patients with Glioblastoma Multiforme surviving 24 months or more after diagnosis.
Each member of the
family of samples is a random sample of 80 GBM patients. The family of samples(FoS) consists of all possible samples of this type.
From each member of the FoS, compute the number e of GBM patients in the sample surviving 24 or more months
after diagnosis, then compute the sample proportion for event p = e//80, then compute
sdp = sqrt(p*(1
p)/n), then compute the interval [lower95, upper95] as
[ lower95 = p (2*sdp), upper95 = p + (2*sdp) ].
Doing this for each member of the FoS yields a family
of intervals (FoI), approximately 95% of which
contain the true population proportion of patients with Glioblastoma
Multiforme surviving 24 months or more after
diagnosis.
If our interval resides
in this 95% supermajority, then between 8.0% and 24.4% of Glioblastoma
multiforme patients survive 24 or more months after
diagnosis.
Table 1: Means and Proportions
|
Z(k) PROBRT
PROBCENT 0.05 0.48006 0.03988 0.10 0.46017 0.07966 0.15 0.44038 0.11924 0.20 0.42074 0.15852 0.25 0.40129 0.19741 0.30 0.38209 0.23582 0.35 0.36317 0.27366 0.40 0.34458 0.31084 0.45 0.32636 0.34729 0.50 0.30854 0.38292 0.55 0.29116 0.41768 0.60 0.27425 0.45149 0.65 0.25785 0.48431 0.70 0.24196 0.51607 0.75 0.22663 0.54675 0.80 0.21186 0.57629 0.85 0.19766 0.60467 0.90 0.18406 0.63188 0.95 0.17106 0.65789 1.00 0.15866 0.68269 |
Z(k) PROBRT PROBCENT 1.05 0.14686 0.70628 1.10 0.13567 0.72867 1.15 0.12507 0.74986 1.20 0.11507 0.76986 1.25 0.10565 0.78870 1.30 0.09680 0.80640 1.35 0.08850
0.82298 1.40 0.08075 0.83849 1.45 0.07352 0.85294 1.50 0.06680 0.86639 1.55 0.06057
0.87886 1.60 0.05479
0.89040 1.65 0.04947
0.90106 1.70 0.04456
0.91087 1.75 0.04005
0.91988 1.80 0.03593
0.92814 1.85 0.03215
0.93569 1.90 0.02871
0.94257 1.95 0.02558
0.94882 2.00 0.02275
0.95450 |
Z(k) PROBRT PROBCENT 2.05 0.020182 0.95964 2.10 0.017864 0.96427 2.15 0.015778 0.96844 2.20 0.013903 0.97219 2.25 0.012224 0.97555 2.30 0.010724 0.97855 2.35 0.009387 0.98123 2.40 0.008198 0.98360 2.45 0.007143 0.98571 2.50 0.006210 0.98758 2.55 0.005386 0.98923 2.60 0.004661 0.99068 2.65 0.004025 0.99195 2.70 .0034670 0.99307 2.75 .0029798 0.99404 2.80 .0025551 0.99489 2.85 .0021860 0.99563 2.90 .0018658 0.99627 2.95 .0015889 0.99682 3.00 .0013499 0.99730 |