Instructor Key with Performance Summary

The 1^st Hourly

Math 1107

Summer Term 2003

 

 

Please follow the protocol and directions…

 

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 );

            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 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

 

Cases were equally weighted at 20 points per case, for a maximum total of 120 points for all six cases.

 

Case One

Probability Computational Rules

Pair of Dice

 

In this experiment we have a weird pair of dice – they are telepathically linked so that they do not operate independently. In fact, the dice produce the following face-pairs with the following probabilities:

 

 

In this experiment we toss this weird pair of dice and note the resulting pair of faces. In each of the following, show your intermediate steps and work. If a rule is specified, you must use that rule for your computation.

 

Consider the SUM of the face values within each pair of face values.

 

List the possible SUMS of face values within the pairs,

 

 

 

 

and compute the probability for each value of SUM. Show full detail for full credit.

 

Pr{ SUM=2 } = Pr{ (1,1) } = .10

Pr{ SUM=3 } = Pr{ (1,2) } + Pr{ (2,1) } =  .10 + .10 = .20

Pr{ SUM=4 } = Pr{ (1,3) } + Pr{ (2,2) } =  .05 + .20 = .25

Pr{ SUM=5 } = Pr{ (1,4) } + Pr{ (2,3) } =  .15 + .20 = .35

Pr{ SUM=6 } = Pr{ (2,4) } =  .10

 

This one went very well, or not well at all. A few students ignored the model provided, and instead used a fair model. Others did not use the SUM random variable, as indicated.

 

Case Two

Clinical Trial Sketch                                                                                   

Crohn’s Disease

 

Crohn’s disease is a chronic inflammation of the intestinal wall.  Most often, lower areas of the small intestine (ileum) are involved, although any part of the digestive tract (from the mouth to the anus) may be involved. Most cases are noted between the ages of 14 and 25 years. In about 35% of cases, only the lower small intestine (ileum) is involved; in about 20% of cases, only the large intestine is involved and in about 45% of cases, both small and large intestines are involved.

 

Symptoms of Crohn’s disease include: chronic diarrhea, crampy abdominal pain, fever, suppressed appetite and weight loss.

Complications of Crohn’s disease include: intestinal obstructions, abnormal connections within the intestines, abnormal connections of the intestines and bladder, abnormal connections of the intestines and the skin surface, and intestinal infections. In children, symptoms may include joint inflammation, slow growth, fever and anemia.

 

Treatments of Crohn’s disease depend on the symptoms being treated. For infections, broad spectrum anti-biotics are employed. For inflammation, cortico-steriods are used, as well as other anti-inflammatories. Additionally, certain medications focus on the immune system for general symptom reduction, as well as keeping Crohn’s disease in remission. Cortico-steroids are used to provide relief for inflamed areas of the body. They lessen swelling, redness, itching, and allergic reactions.

 

Consider budesonide, which is an orally administered cortico-steroid that is released in the intestine, where it works locally and topically to decrease inflammation. Hopefully, patients taking budesonide experience fewer of the typical side effects associated with other cortico-steroids used to treat Crohn's Disease, such as prednisone or prednisolone tablets, because most of budesonide is not absorbed into the body. 

 

Possible side effects of cortico-steroids include: lowered resistance to infections, persistent infections, decreased or blurred vision; frequent urination; increased thirst, abdominal or stomach pain or burning, headache; irregular heartbeat; menstrual problems; muscle cramps or pain; muscle weakness; nausea.

 

Sketch a comparative clinical trial for budesonide versus standard cortico-steroids in the treatment of intestinal inflammation in patients with Crohn’s disease.

 

Population and Disease of Interest: Patients with Crohn’s Disease (CD)

 

Treatments: Standard CS – Standard suite of corticosteroids usually employed in the treatment of CD.  Budenoside – a hopefully kinder, gentler corticosteroid.

 

Recruitment: Begin with human candidate volunteers. They must meet all inclusion criteria, including having a diagnosis of CD, and they must have none of the exclusion criteria. After screening out those volunteers who do not have CD, or who meet one or more exclusion criteria, fully inform the remaining volunteers of the risks and possible benefits of trial study participation. Enroll those who give informed consent in the study. Those who give such consent are aware of how the trial will be conducted.

 

Assignment to Treatment: Randomly assign each volunteer to either standard CS or to Budesonide. Double blinding is employed, in that neither the study subjects nor the clinical workers know the actual treatment assignment of individual subjects.

 

Follow-up: Evaluate study subjects for intestinal inflammation, as well as for the known side effects of these treatments. Also track the occurrence of any other adverse events. Also track subjects for significant toxicity, such as kidney or liver damage.

 

A number of students did not present a clinical trial sketch as defined by course examples. Some details were insufficient on random assignment and double blinding. A few sharp students were very careful about informed consent and the possible ages of the subject/volunteers. Error codes used in the scoring: RA=Random Assignment; DB=Double Blinding; IC=Informed Consent.
Case Three

Probability Computational Rules

Color Slot Machine 

 

Here is our slot machine – on each trial, it produces a 4-color sequence, using the table below:

 

 

*          B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered as

            1^st to 4^th , from left to right: (1^st 2^nd 3^rd 4^th)

 

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.

             

3.a)      Pr{ Red Shows 3^rd or 4^th }

 

Pr{ Red Shows 3^rd or 4^th } = Pr{ GBRB }+ Pr{ GBBR } + Pr{ YBGR } = .25 + .25 + .10 = .60

 

3.b)      Pr{ Red Shows and Green Does Not Show }

 

Pr{ Red Shows and Green Does Not Show } = Pr{ RRYY } = .10

 

3.c)      Pr{ Blue Shows} - Use the Complementary Rule.       

  

Pr{ Blue Does Not Show} = Pr{ RRYY } = .10

Pr{ Blue Shows } 1- Pr{ Blue Does Not Show } = 1 - .10 = .90

 

A number of students did not use the complementary rule in 3c, or got it backwards somehow. Code used in 3.c: CR=Complementary Rule.

 

Case Four

Conditional Probability

Color Bowl/Draws without Replacement

 

We have a bowl containing the following colors and counts of balls (color@count):

 

White @ 1, Black @ 2, Blue @ 4, Green @ 4, Red @ 3, Yellow @ 1

 

Each trial of our experiment consists of five draws without replacement from the bowl. Compute the following conditional probabilities.

 

Compute these directly.

 

This one went very well or very badly. The idea was to do these directly, as in the example in class.

 

4.a)      Pr{ Yellow shows 2^nd  | Yellow shows 1^st}

 

Pr{ Yellow shows 2^nd  | Yellow shows 1^st} =

0 (Yellow available for 2^nd draw) / 14 (Total For 2^nd Draw) = 0

 

4.b)      Pr{ Black shows 5^th | Black shows 1^st, Red shows 2^nd, Blue shows 3^rd, and Green shows 4^th }

 

 

Pr{ Black shows 5^th | Black shows 1^st, Red shows 2^nd, Blue shows 3^rd, and Green shows 4^th } = 1/11

 

4.c)      Pr{ Blue shows 3^rd | Blue shows 1^st, Blue shows 2^nd }

 

 

Pr{ Blue shows 3^rd | Blue shows 1^st, Blue shows 2^nd } = 2/13

 

Case Five

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.

5.a)      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, 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 proposed trial would compare treatments intended for two different diseases.  The comparative design would use an inappropriate treatment – the lack of a standard treatment for CZ  requires a placebo.

5.b)      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.

 

Informed consent is not employed – the patients need to be fully informed of their Hepatitis C Status, and then properly recruited into the study.

 

5.c)      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.

           

            A sample that systematically excludes rural Georgia drivers cannot claim to apply to all           Georgia drivers. Moreover, the difference between reported driving practices and actual            street practices may be large. The subject of those drivers between ages 15 and 17 may   warrant a separate study. In principle, we might want to narrow the design to adult drivers, though in practice this may be difficult to do in advance.

5.d)      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.

            Parents aren’t necessarily capable of giving accurate information of this type – direct   responses are generally preferred to proxy responses.

As usual, this case went pretty well.

 

Case Six

Perfect Samples

PuzKertin’s Syndrome

Severity of cases of PuzKertin’s Syndrome (PKS) is noted as:  (F)atal, (S)evere, (Mo)derate, or (Mi)ld. Suppose that severity probabilities  for the population of PKS patients are: 15% Severe, 15% Moderate, 20% Mild and  50% Fatal.

6.a)      Interpret each probability using the Long Run Argument. Be specific and complete for full credit.

In long runs of draws with replacement from this population, approximately 20% of sampled patients with PKS have mild cases.

In long runs of draws with replacement from this population, approximately 15% of sampled patients with PKS have moderate cases.

In long runs of draws with replacement from this population, approximately 15% of sampled patients with PKS have severe cases.

In long runs of draws with replacement from this population, approximately 50% of sampled patients with PKS have fatal cases.

6.b)      Compute the perfect sample of n=150 PKS cases, and describe the relationship of this perfect sample to real samples of PKS cases. Show all work and detail for full credit.

 

Random samples of cases of PuzKertin’s Syndrome (PKS) yield approximately 30 mild cases, 22 or23 moderate cases, 22 or 23 severe cases and 75 fatal cases.

 

For some reason, people got parts 6.a. and 6.b. blurred together. The long run argument is a general statement of how sample runs are governed by the model probabilities, while the probabilities themselves apply directly and exactly to the entire population as a whole. The perfect sample is a specific blueprint for a specific sample size under a specific model, which real samples follow approximately.

 

Be certain that you have worked all six (6) cases.

 

 

 

 

The maximum absolute score for the first hourly was 120 points. To get your percentage score, divide your total by 120.

 

There were 30 scores.

The minimum score was 0%

Approximately 10% of scores were at 52.5% or lower.

Approximately 20% of scores were at 66.7% or lower.

Approximately 25% of scores were at 71.7% or lower.

Approximately 30% of scores were at 73.3% or lower.

Approximately 40% of scores were at 81.3% or lower.

Approximately 50% of scores were at 84.6%  or lower.

The average score was approximately 78.1%.

Approximately 60% of scores were at 87.9% or lower.

Approximately 70% of scores were at 89.2% or lower.

Approximately 75% of scores were at 89.2% or lower.

Approximately 80% of scores were at 92.9% or lower.

There were four scores at 97.5%, these were the highest scores.

 

 

 

 

 

 

Do not interpret your individual score as a grade. In this course, there is only one letter grade, based on a weighted average of the hourlies and the final. If this is your best in-class hourly, it will account for up to 40 points of your course score. Otherwise, it will count for up to 20 points of your course score.