lesson one - comap

570

77CHA P T E R

Testing 1, 2, 3LESSON ONE

Steroid Testing

LESSON TWO

Testing Models

LESSON THREE

Confirming the Model

LESSON FOUR

Solving the Model: Table and Graphs

LESSON FIVE

Solving the Model: Symbolic Methods

Chapter 7 Review

Examination Copy © COMAP Inc. Not for Resale


571

TESTING 1, 2, 3

Medical testing is common today. Sometimes such

tests are done because a patient is ill. Other times

they are done to screen members of a population.

In Chapter 6, Imperfect Testing, you saw that tests

used for screening are seldom perfect. With some tests, those who

test positive often do not have the condition the test is designed to

detect.

Tests that are used for screening should be inexpensive. However, a

cheap test is often less accurate than a costly one. Thus there can be a

tradeoff between accuracy and cost. Those who run testing programs

want to find reasonable ways to save money.

A common, but fairly expensive, test is used to detect steroids.

People from Olympic athletes to high school students undergo

steroid testing. In this chapter, you will use mathematical modeling

to look for less costly ways to test for steroids. You will learn about a

useful mathematical function called the quadratic. An understanding

of this function will improve your ability to use mathematics to

solve real-world problems.



PREPARATION READING:

Screening a Population

T esting for substance abuse is not limited to athletes.Some people must pass a test for substance abuse inorder to get a job. School-board members, employers,

and Olympic committees are among the groups who requiresuch screening tests. They want to do them fairly. But they alsowant to avoid high costs.

Sometimes those who mandate testing do not insist thateveryone be tested. Other times all members of the populationmust be tested. Regardless of the requirement, those in chargeof testing programs try to save money by making fewer tests.For example, if there are 100 people in the population, thetesters might look for a way to get by with 50 tests rather than100.

Mathematical modeling can be used to help people understandwhether a particular testing strategy will save money in thelong run.

As you know, the first step in mathematical modeling is toidentify a question to answer or a problem to solve. It is best tostart simply.

Once you have identified a problem that mathematics mighthelp solve, you must spend some time understanding thesituation. You must search for the important factors that affectthe problem.

In this lesson, you will consider the general issue of steroidtesting and one strategy for saving money on such tests. Youwill also consider factors that affect the number of tests thatcan be expected with this strategy.

DISCUSSION/REFLECTION

1. A testing program might try to save money by using areally cheap test. Why might this be a bad idea?

2. How might a testing program reduce the number of tests ifthere is not a requirement that everyone be tested?

3. How might a testing program reduce the number of tests ifthere is a requirement that everyone be tested?

4. When is a testing program less expensive than testing eachperson?

572 Preparation Reading

LESSON ONE

SteroidTesting

Key Concepts

Sample pooling

Mathematical model

Probability

Simulation

Expected value



573Preparation Reading

The Jerusalem Post July 2, 2000

By Judy Siegel-Itzkovich

Screening blood for dangerous viruses isexpensive. Medical school researchers atBen-Gurion University and from thePalestinian Authority are collaborating ona study aimed at reducing the cost andimproving the screening of blood andblood products.

According to professor Batia Sarov, thecost of detecting a positive blood samplefor a viral infection is $1352. That iswithout taking into account the expensiveverification procedures resulting fromfalse positives.

“The main idea of the proposed newscreening procedure,” asserts Prof. Sarov,“is based on the pooling of bloodsamples. This is different from thecurrent procedure that is based on testingeach individual blood sample.

“Thus, for example, if 16 blood sampleswere pooled together and determined tobe negative, this would be a great savingof resources. If a group of pooledsamples is found to be positive, there willbe a need to implement specialprocedures for identifying the individualsample which is infected.”

Because the majority of blood samples inIsrael are negative, the pooling of bloodsamples would make the process moreefficient.

Blood Brothers Join ForcesExamination Copy © COMAP Inc. Not for Resale


574

Activity 7.1: A Money-Saving Strategy

In this activity, you examine the strategy of pooling two samples in orderto save money in a steroid testing program.

The primary modeling goal of this chapter is to determine when thestrategy of pooling two samples can be expected to save money over thelong run. Sample pooling is the term used to describe the process ofcombing samples from two or more people.

The logic of the strategy of pooling two samples is simple.

• Mix two samples together. (Save part of each sample in case moretests have to be done.)

• Run one test on the pooled samples.

• If the test is negative, you do not have to test each sample.

• If the test is positive, test one individual sample. Test the otheronly if necessary.

1. Suppose you are testing people for steroid use. You takesamples from two people and you pool a portion of eachsample. If the pooled sample tests negative, then you do notneed to do more tests. You have saved half the cost of twoindividual tests.

a) If the pooled sample tests positive, what does this mean?

b) If the pooled sample is positive, how might youdetermine which people have used steroids?

c) How many more tests are needed to determine whichpeople have used steroids?

2. Imagine that you are a tester trying to save money bypooling two samples. To help you decide if this strategysaves money, you keep records of the number of tests youdo on each pair of samples. How might you organize yourdata in a concise way?

3. Suppose a tester uses the strategy of pooling two sampleswith a large number of people. She finds that the number ofsample pairs requiring one test and the number requiringthree tests are the same. Each test costs $80. Does thesample-pooling strategy save money?

4. In general, what does a tester have to see in the data toconclude that the strategy of pooling two samples has saved money?

There are a lot of waysto pool samples. Youcould pool two samples. You couldpool three samples. You could poolten. The simplest case is two samples.It’s best to start simply. Modelers often find that the mathematics theydevelop to handle the simplest casecan be adapted to more complexcases.

MODELINGNOTE

In Question 2, think about thevarious kinds of representationsyou have used in previouschapters (tables, graphs, andvarious kinds of diagrams) andselect at least one to try here.You may not have enoughinformation at this time tocomplete your representation, somake note of any missinginformation that would be helpful.

FYI

Chapter 7 Testing 1, 2, 3 Activity 7.1



575Activity 7.1 Lesson One

In this activity, you:

� determined the numbers of tests that might be required when thestrategy of pooling two samples is used.

� considered ways to represent data about the sample-pooling strategy.

� learned how to tell if the strategy of pooling two samples has savedmoney.

Activity SummaryExamination Copy © COMAP Inc. Not for Resale


576 Chapter 7 Testing 1, 2, 3 Research Activity

Research Activity: What Do You Know About Steroid Testing?

Your task in this activity is to learn as much as you can about steroidtesting.

Read one or more recent articles about steroid testing. Take notesand be prepared to share them with others in your class. Writedown anything you think is important. Here are some examplesof questions you might be able to answer.

1. Why do people abuse steroids?

2. Why do other people try to prevent steroid abuse?

3. What does a steroid test cost?

4. Are there legitimate medical reasons for using steroids?

5. What are some factors that might determine whether you can expectto save money by pooling steroid test samples? If you cannot findany information on this question, write down one or two conjectures.Give an explanation for each.


1. Suppose you are testing for steroid use in each of the followingsituations. You are using the strategy of pooling two samples. Howmany tests will you do on a pooled sample in each case?

a) All of the students are using steroids, and you do not know it.

b) None of the students are using steroids, and you do not knowit.

c) Almost all of the students are using steroids, and you do notknow it.

d) Very few of the students are using steroids, and you do notknow it.

2. Is the strategy of pooling two samples more likely to save moneyin a school where steroid use is common or one in which it isuncommon?

3. Review your answers to Questions 1 and 2. Name acharacteristic of a population that determines whether thesample-pooling strategy can be expected to save money in thelong run.

Mathematicalmodelers don’t knoweverything about everything.Often they do research tolearn more about a situationbefore they can build agood model.

MODELINGNOTE

No doubt there areseveral factors thataffect the cost of the sample-pooling strategy. In buildinga mathematical model, it’sbest to start with just asingle factor. Sometimesthat’s all that is necessaryfor an effective model. Thetrick is to identify the singlemost important factor.

MODELINGNOTE



Individual Work 7.1: Saving Money on Steroid Tests

577

In this individual work, you reconsider the strategy of pooling twosamples and its economics. You also develop a first model to determinewhen this strategy can be expected to save money over the long run.

1. A high school testing program uses the strategy of pooling twosamples and has kept a record of the results. They are shown in a bargraph in Figure 7.1. Note that the graph shows the number of pairs ofstudents tested.

a) How many students were tested altogether?

b) Did the school save money by using the sample-pooling strategy?Explain.

c) What was the average number of tests required to test a pooledsample?

d) How can the average number of tests per pooled sample be usedto tell whether the school saved money?

e) Estimate the percentage of students who tested positive forsteroids. Explain your answer.

2. Figure 7.2 is a bar graph that describes steroid testing results inanother school. This school also used the strategy of pooling pairs ofsamples.

Individual Work 7.1 Lesson One

Testing results

Number of tests

Num

ber

of p

airs

of s

ampl

es

0

10

20

30

40

50

60

70

80

90

1 2 3

Figure 7.1.



578 Chapter 7 Testing 1, 2, 3 Individual Work 7.1

a) Did this school save money by using the sample-poolingstrategy? Explain.

b) Estimate the percentage of students who tested positive forsteroids. Explain your answer.

You have made quite a bit of progress toward this chapter’s modelinggoal. A summary might be helpful before you do the rest of thequestions in this Individual Work.

• The modeling question is, when can the strategy of pooling twosamples be expected to save money over the long run in a steroid testingprogram?

• The most important factor in determining the answer to thisquestion is the rate of steroid use in the population.

The rate of steroid use in a population is called theincidence. Incidence can be expressed as a percentage ordecimal. It is sometimes stated as a verbal fraction such as7 out of 1000.

The next step is to use mathematics to describe a relationship betweenthe incidence of steroid use and the number of tests per pair of samples.

3. In Questions 1 and 2, you saw that relating the number of tests to thepercentage of people who use steroids is tricky. However, it is fairlyeasy if all the people in a group use steroids or if none of the peopledo. Although these extreme cases are not likely to occur, they can beuseful in creating a model.

Testing results

Number of testsPe

rcen

tage

of p

airs

of s

ampl

es0

5

10

15

20

25

30

35

40

45

50

1 2 3

Figure 7.2.

The bargraph inFigure 7.2

is labeled differently from theone in Figure 7.1. Onedisplays the number of pairsof students tested. The otherdisplays percentages of allpairs of students tested.

TAKE NOTE



579Individual Work 7.1 Lesson One

a) Three tests are necessary if 100% of the people use steroids.One test is necessary if 0% of the people use steroids. Usethese two facts to create a model for a relationship betweenincidence and average number of tests per pair of samples.Use at least one of the representations from previouschapters (that is, tables, graphs, equations) to describe yourmodel.

b) Use your model to describe incidence values for whichpooling two samples can be expected to save money. Explain your reasoning.

4. In the remainder of this lesson, you will put the model you created inQuestion 3 to a test. Thus, it will be important to keep in mind anyassumptions you have made. If your model does not work well, youcan revisit these assumptions.

a) What assumption did you make about the accuracy of the steroidtest used?

b) What assumption did you make about the nature of therelationship between incidence and the number of tests?

Sinceincidenceexplains

the number of tests you canexpect to perform, anymathematical function youuse should have incidence asthe explanatory variable.

TAKE NOTE



580 Chapter 7 Testing 1, 2, 3 Activity 7.2

Activity 7.2: The Testing Game

In this activity, you test the model that you built in Individual Work 7.1.

Your first model may have led you to conclude that poolingpairs of samples saves money if less than half the people usesteroids. Your task in this activity is to design and perform asimulation to determine if an average of two tests can beexpected when 50% of the people use steroids.

1. Design a role-playing simulation for testing a population inwhich 50% of the people use steroids. Use three people.

• Two people should play the roles of the people being tested.

• The third person should play the role of the tester.

Since you are simulating a situation in which 50% of the people usesteroids, you need to use a random event in which two outcomeshave the same chance of occurring. For example, you can flip a coinor you can put the same number of two colors of chips in a containerand draw one.

• Each tested person should use a random event to determinewhether he or she is a steroid user.

• The two tested people should then tell the tester if their pooledsample is positive or negative. (Recall that a pooled sample ispositive if at least one person tests positive.)

• If the pooled sample is positive, the tester should use a randomevent to pick one of the two to test first.

Perform the simulation a total of ten times. Keep a record of thenumber of times one, two, and three tests are required. When youhave finished, share the results with your class. Combine the resultsfrom all groups in a table showing the number of times one, two, orthree tests occurred.

a) Find the average number of tests per trial.

b) Compare your results to the solution you obtained from yourmodel in Individual Work 7.1. Does this simulation confirm thesolution?

Computer or calculator simulations are less time consuming thansimulations performed with coins. In addition, they use random numbergenerators that are free of biases that can occur with other methods.

Testing a model isthe final step of themodeling process. Anexcellent way to test amodel is with real-worlddata. When reliable dataare not available,modelers turn tosimulations. They often usecomputers or calculators toperform the simulations.

MODELINGNOTE



581Activity 7.2 Lesson One

2. Perform another simulation in which 50% of the peopleuse steroids. This time use a computer or calculatorsimulation such as the Dracula program thataccompanies this book. Do a much larger number oftrials than your class did in Question 1.

a) Does this simulation confirm the solution youobtained from your model?

b) Are you more confident in the result you obtainedfrom the simulation done with a program than fromthe simulation done by hand?

In this activity, you:

� performed a role-playing simulation to test the model you built inIndividual Work 7.1.

� performed a computer or calculator simulation for the same purpose.


1. Based on the simulations in Activity 7.2, how do you feel about themodel you built in Individual Work 7.1?

2. If the simulations cast doubts on your model, then return to themodeling assumptions you listed in Question 4 of Individual Work7.1 and discuss which assumption(s) you think is faulty.

3. Again, if you have doubts about your model, how might you adaptthe simulations you did in Activity 7.2 to help you revise the model?

Most computer andcalculator programswork with the portion

of people using steroids rather than thepercentage. So you need to enter 0.5rather than 50. Also, the programprobably uses the letter p to stand forthe portion using steroids. Portions willbe used frequently in the rest of thischapter.

TAKE NOTE

Activity Summary




Individual Work 7.2: Probability and Expected Value

Whether you can expect to save money by pooling two test samplesdepends on the average number of tests per pooled sample. In thisIndividual Work, you take a closer look at the average number of tests.You also examine simulations like the ones in Activity 7.2.

1. A group of three students played the testing game. When they werefinished, they organized their results into the table in Figure 7.3.What do you think of these results? Explain.

2. To simulate a situation in which 50% of the people use steroids, youcan flip a coin. If you want to simulate a situation in which thepercentage is not 50, coin flipping doesn’t work. Drawing chips outof a container is a better option. For example, suppose that you wantto simulate a situation in which 30% of the people use steroids. Youdecide to use ten chips, some red and some blue. You decide that redrepresents a steroid user.

a) How many red chips and how many blue chips should be in thecontainer?

b) The person tested first should put the chip back before the secondperson draws a chip. Why is this necessary?

c) Suppose you use 100 chips instead of 10. Would putting the firstchip back before drawing the second be as important?

3. Calculator and computer programs simulate steroid testing bygenerating a random decimal number. You can think of generating arandom number as throwing a dart at a one-dimensional dartboard(see Figure 7.4).

1 2 3 4 5 6 7 8 9 10

1 3 3 1 1 1 3 1 1 3

Total number

Number of tests required

Figure 7.3. Testing game results.

If the dart landson this side of 0.3,the test is positive.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

+ positive + – negative –

If the dart landson this side of 0.3,the test is negative.

Figure 7.4. A one-dimensional dartboard.




For example, suppose 30% of the people use steroids.

• If the random decimal number is below 0.3, the program countsthe person as testing positive.

• If the random number is above 0.3, the program counts the personas testing negative.

In each of the following, state whether the program counts the testpositive or negative.

a) The percentage using steroids is 20, and the random number is0.7873.

b) The percentage using steroids is 30, and the random number is0.2955.

c) The percentage using steroids is 14, and the random number is0.1433.

4. Figure 7.5 shows two tables that display the results of the sample-pooling strategy in two schools.

a) Use the average number of tests to determine whether School 1saved money by pooling pairs of samples.

b) Use the average number of tests to determine whether School 2saved money by pooling pairs of samples.

c) Explain how you calculated the average number of tests in parts(a) and (b).

Questions 5 and 6 use the term “expected value,” which is used often inthe rest of this chapter.

The expected value of a variable is the average that canbe expected over the long run.

1 2 3

84 6 14

Number of tests

Number of pairs of samples

School 1

1 2 3

28 31 41

Number of tests

Number of pairs of samples

School 2Figure 7.5. Testing results in two schools.




In the context of this chapter, expected value is just the average numberof tests per pooled sample that you can expect over many trials. You canestimate the expected value by calculating the average number of tests ina simulation. Be aware that the average that you calculate from asimulation doesn’t always match the expected value exactly. Forexample, when you flip a coin, you expect 50% heads. But you are notsurprised if the actual average is a little over 50% or a little under 50%.

The term “expected value” can be misleading; it does not tell you thenumber of tests on any sample. For example, it isn’t correct to say thatyou expect to do 2.3 tests on a pair of samples. In fact, you never do 2.3tests on a pair of samples—you do either 1, 2, or 3 tests.

5. Here is another example of expected value. Suppose that you are oneof 10,000 people who pay $1 each for a raffle ticket. There is onegrand prize of $1000. There are two second prizes of $300 each and50 consolation prizes of $20 each. The amounts of money you mightwin (or lose) and the probability you will win each amount can beorganized in a table (see Figure 7.6).

a) Explain the numbers in the “Amount won or lost” row.

b) Supply the missing probabilities.

c) Find the expected value.

d) What does this expected value mean to the people whobought tickets?

6. Some carnivals feature a dice game in which you bet a dollaron number 1, 2, 3, 4, 5, or 6. Three dice are rolled. You win $1if your number comes up once, $2 if it comes up twice, or $3if it comes up three times. If you win, you also get to keepthe dollar you bet. Of course, if your number doesn’t comeup, you lose your dollar.

a) Organize the amounts you might win or lose in a table. Whenthree dice are rolled, the probabilities of a number coming uponce, twice, or three times are approximately 0.347, 0.0694, and0.00463.

b) Find the expected value.

c) What does the expected value mean to the players?

7. A school decides to pool three tests when testing students forsteroids. On average, how many tests per sample should schoolofficials expect if they want to save money?

$–1 $19 $299 $999Amount won or lost

Probability

Figure 7.6.Amount won orlost in a raffle.

Since theprobabilitiesin Figure 7.6

are like decimal portions, youcan calculate expected valueusing the same method youused to find the averagenumber of tests. Just as youmultiplied the number of testsby the number of sample pairs(the portions), you multiply theamount won or lost by thedecimal portion.

TAKE NOTE




8. A tester is using a strategy of pooling three samples. If the pooledsample tests positive, the tester selects one of the three samples totest. Then the tester continues testing individual samples until nomore tests are necessary. The results are shown in Figure 7.7.

a) Use the average number of tests to determine whether the testersaved money with this strategy.

b) Explain why two tests are never needed.

c) Is there a way the tester could change the strategy so that twotests are needed? Explain.

1 2 3 4

58 0 18 12

Number of tests

Number of pooled samples

Figure 7.7. Testresults when threesamples are pooled.



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