The 1st Hourly

Math 1107

Summer Term 2004

 

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

 

Case One

Probability Computational Rules and Random Variable

Pair of Dice 

 

We have a pair of dice – a fair d2 {faces 0, 8} and a d4 {faces 1,3,5,7} – note the probability model for the d4 below. 

 

Face

Probability

1

0.15

3

0.35

5

0.25

7

0.25

 

We assume that the dice operate separately and independently of each other. Suppose that our experiment consists of tossing the dice, and noting the resulting face-pair.

 

1.a)      List the possible pairs of face values, and compute a probability for each pair of face values.

 

1.b)      Define THING as follows. When the higher face value in the pair is less than or equal to 4, define THING as the product of the face values in the pair. Otherwise, define THING as the sum of the face values in the pair. Compute and list the possible values for the variable THING, and compute a probability for each value of THING.

 

Show full detail for full credit.

 

Case Two

Clinical Trial Sketch                                                                                    

Post-Lumpectomy Radiotherapy – Breast Cancer

 

Lumpectomy: A lumpectomy is one type of surgery for breast cancer. The malignant tumor and a surrounding margin of normal breast tissue are removed. Lymph nodes in the armpit may also be removed.

 

Early Breast Cancer Detection: Lumpectomy is a surgical treatment option typically available to patients whose breast cancer is detected  in an early state. The procedure is advantageous in that only the cancer tissue and a minimum of adjacent tissue is removed. This minimal surgical approach can minimize post-surgical complications, as well as minimize the need for breast reconstruction.

 

The Need for Post-Surgical Radiation: Post-operative radiation therapy is required after lumpectomy. A full course of post-lumpectomy radiotherapy can require daily radiation doses for a full six weeks.

 

A Rapid Alternative – Single Dose Radiotherapy: The extended nature of required post-surgical radiotherapy deters many women from selecting lumpectomy. Consider an experimental technique called intra-operative radiotherapy. Intra-operative radiotherapy uses a miniature radiation probe right after a lumpectomy. The probe is inserted inside the cavity created by the removal of the tumor, and radiation equivalent to six weeks of doses is emitted for about 25 minutes.

Sketch a clinical trial for intra-operative radiotherapy versus standard radiotherapy in post-lumpectomy breast cancer patients.

For full credit, discuss completely, per examples discussed in class and on the web page. 

 

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:

 

Sequence*

Probability

GBRB

.25

BBGG

.10

GBBR

.25

RGYB

.10

BBYY

.10

RRYY

.10

YBGR

.10

Total

1.00

 

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

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

 

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 1st or 3rd }

 

 

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

 

 

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

  

Show full detail for full credit.

 

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.

 

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

 

 

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

 

 

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

 

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.

 

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.

 

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.

           

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.

           

Case Six

Perfect Samples

Landsteiner's Human Blood Types

 

In the early 20th century, an Austrian scientist named Karl Landsteiner classified blood according to chemical molecular differences – surface antigen proteins. Landsteiner observed two distinct chemical molecules present on the surface of the red blood cells. He labeled one molecule "A" and the other molecule "B." If the red blood cell had only "A" molecules on it, that blood was called type A. If the red blood cell had only "B" molecules on it, that blood was called type B. If the red blood cell had a mixture of both molecules, that blood was called type AB. If the red blood cell had neither molecule, that blood was called type O. So the blood types are O, A, B, AB.

 

Suppose that the probabilities for blood types for US residents are noted below:

 

Blood Type

Probability

O

.40

A

.30

B

.10

AB

.20

Total

1.00

 

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

 

6.b)      Compute the perfect sample of n=150 US residents, and describe the relationship of this perfect sample to real random samples of US residents.

 

Show all work and detail for full credit.

 

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