UNIT 5—Testing 123

Teacher Materials

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TEKS Support
Teacher Notes
	Context Overview
	Mathematical Development
	Preparation Reading—"Anabolic Steroids: Use and Effect"
	Discussion—Modeling Process
	Activity 1—"It’s Only a Test"
	Supplemental Activity 1—"Middle of a Model" (Manipulative Version)
	Homework 1—"Let’s Rumble"
	Activity 2—"Middle of a Model"
	Homework 2—"Expect to Be Worth Something"
	Activity 3—Finally, A Model
	Homework 3—It’s a Good Fit
	Activity 4—Which Model Fits the Best?
	Activity 5—Verifying the Model as Quadratic
	Supplemental Activity 5—Introduction to the Parabola
	Homework 5—Quadratic Nature of Pairing Samples
	Activity 6—Solving the Problem
	Homework 6—Practice With Quadratics
	HANDOUT 1—Activity 1 Part 2
	HANDOUT 2—Activity 1 Part 4
	HANDOUT 3—Dracula Program
	HANDOUT 4—Linear Regression for Activity 3
	HANDOUT 5—Exponential Regression for Activity 3
	HANDOUT 6—Quadratic Regression for Activity 3
	HANDOUT 7—Graph Grid for Supplemental Activity 5
Annotated Student Materials
	Preparation Reading—Anabolic Steroids: Use and Effect
	Activity 1—It’s Only a Test
	Supplemental Activity 1—Middle of a Model (Manipulative Version)
	Homework 1—Let’s Rumble
	Activity 2—Middle of a Model
	Homework 2—Expect to Be Worth Something
	Activity 3—Finally, A Model
	HANDOUT 4—Answers
	HANDOUT 5—Answers
	HANDOUT 6—Answers
	Homework 3—It’s a Good Fit
	Activity 4—Which Model Fits Best?
	Activity 5—Verifying the Model as Quadratic
	Supplemental Activity 5—Introduction to the Parabola
	Homework 5—Quadratic Nature of Pairing Samples
	Activity 6—Solving the Problem
	Assessment—Working With Quadratics
	Unit Summary
	Mathematical Summary
	Key Concepts
Solution to Short Modeling Practice
	Mini Modeling Problem: Linear and Quadratic Equations
Solutions to Practice and Review Problems

TEKS Support

• Coordinate graphing
• Linear equations

• Probability
• Linear regression
• Use of residual patterns to judge "goodness of fit"
• Standard and parabolic forms of the quadratic equation
• Graphical solution of quadratic equations
• Solution of quadratic equations by completing the square
• Derivation and use of the quadratic formula

• Graphing calculator

Teacher Notes

"Simple Model"—Context Overview

The unit examines testing strategies for sampling pairs of individuals when cost is a consideration. Pooling of samples may require only one test and therefore save money, or may require three tests, losing money in the process. Whether pairs of samples should be initially pooled for testing depends on the probability that a single result is positive.

Students gather data by performing Monte Carlo experiments assuming different probabilities, and try to determine the mathematical model for this situation. They verify empirical results, deriving a theoretical relationship between the incidence of occurrence and the expected number of tests. Finally, they determine the "break-even" point, where it’s more cost-effective to test the samples individually than to pool the samples and test them in pairs.

Mathematical Development

Deciding when it is appropriate to pool samples is explored through discussion of the modeling process, concrete activity, and calculator simulation. The concept of expected value plays a major role in establishing the condition for the problem. Data analysis, using least squares regression, determines the constants that produce the best fit for linear, quadratic, and exponential functions. Examination of the residual patterns provides a basis for choosing which of these functions best describes the data. Probability area models are introduced to verify that the relationship between the probability of testing positive and the expected number of tests required is quadratic, and to determine the exact quadratic that models the situation. Finally, methods for solving the resulting quadratic equation are explored, and the problem of determining when it is cost-effective is solved.

Preparation Reading—"Anabolic Steroids: Use and Effect"

The reading provides some background into the problem of detecting steroid use and the physiological effects of taking anabolic steroids. The intent of the reading assignment is to generate a discussion of the use of steroids in the general population, and to build some understanding for the context of testing for steroid use. Prior to class, students should read the contents of this article, and be prepared to discuss some of the following questions:

• At major sporting events, athletes are routinely tested for steroid use. Why do you think these tests are run?
  (The purpose is to deter the misuse of steroids.)



• Why do some athletes take steroids?
  (Unfortunately, use of steroids provides a definite advantage to an athlete who uses them. If the use was widespread, many athletes who would not ordinarily use the drug would feel pressure to take them. This is the motivation for the ban.)



• What percent of athletes are thought to use steroids at the high-school level?
  (The article states that 6 to 7% of high school students use steroids. Some figures place steroid use at 11% for eleventh grade boys.)



• Is steroid use restricted to males?
  (No, some 1 to 2% of females at the high-school level are users and 3 to 5% of professional female athletes have admitted to using steroids.)



• What are the positive effects of steroids? Why were the drugs developed?
  (They are used in hormonal replacement therapy, to manage certain blood disorders and cancer, and to treat a relatively rare inherited skin condition.)



• What are the harmful side effects?
  (A great deal is still unknown, but there is concern that steroid use can contribute to hypertrophy, to cancer, to stunting bone growth, and to psychological and emotional problems in adolescents.)



• Does every athlete have to be tested, or would random sampling suffice?
  (Many sports events do rely on random testing as a deterrent, primarily because of the cost and logistics involved.)



• Suppose samples from two people are mixed and tested together in one test. What can be concluded from a test of a combined sample?
  (Make sure students understand that if a combined sample from two people tests negative, that would mean that each of the individual samples would also test negative. On the other hand, a positive test means that one or both of the individual samples would test positive.)

In addition, you might want to facilitate a discussion about current events that involve steroid testing or steroid use. Possibilities include: the Tour de France bicycling competition from the summer of 1998, Mark McGwire admitting that he takes Androä , Ben Johnson having his life ban from track and field upheld, or steroid testing of athletes at the local high school or college. Specific topics might include how the tests are administered, the stigma of a positive result, fairness issues when one sport bans the use of a substance and another allows it, "professional" wrestling and steroid use, or even talk-show spectacles about people who have to have a leg amputated from excessive steroid use.

Another option for you might be to have students research the topic of steroid use, using the sources cited below, the Internet, or other materials they find on public health. Have them bring in an interesting article on the subject to share with the class, or present a report on their research at various points in the unit.

The following resources were used to generate the Preparation Reading, and contain information you may find useful.

Wright, J.E., and V. Cowart, Anabolic Steroids: Altered States, Carmel, Indiana: Benchmark Press, 1990.

Donohue, T., and N. Johnson, Foul Play: Drug Abuse in Sports, New York: Basil Blackwell Ltd, 1986.

Meer, J., Drugs and Sports, New York: Chelsea House Publishers, 1987.

Discussion—Modeling Process

Having brought the issue of steroid testing to the students’ awareness, you need to shift their thinking to the mathematical nature of testing samples in pairs. The discussion of the modeling process has probably taken place in previous units, and begins with the students developing a well-defined question. While students may be considering many questions, cost is a critical factor in trying to determine when to pool samples.

• What is the "central" problem to try and solve in this situation?
  When is it cost-effective to pool samples and test them as a pair?

The second step of the modeling process is to identify key features or assumptions that will be part of the development of the model, and to introduce variables into the problem.

• To develop a mathematical model, what assumptions should we make about the problem?
  The tests will detect a steroid user 100% of the time (perfect blood test).



• What is the logic behind pooling pairs of samples?
  If the paired sample tests negative, it can be concluded that neither person is using steroids. If the paired sample tests positive, either one (or both) could be taking steroids.



• What would be the next step, if the pooled sample tests positive?
  Test one of the samples individually.



• If the first individual sample tests negative, what can we conclude?
  We would know the first person is not taking steroids, but we can conclude that the second person must be taking steroids (the pooled test was positive).



• If the first individual sample tests positive, what can we conclude?
  We would know that the first person is taking steroids, but we won’t be able to conclude anything about whether the second person is also taking steroids. We will still need to test the second person individually to determine that.



• Under what circumstances is only one test required? Two tests? Three tests?
  One test will be required when the pooled sample tests negative. Two tests will be required when the pooled sample tests positive and the first individual sample tests negative. Three tests will be required when the pooled sample tests positive and the first individual sample tests positive.



• If the number of tests that are required varies (needs one, two, or three tests), how will we be able to determine whether to pool samples or test them individually?
  Work with the average number of tests over many situations, instead of the results of one individual trial.



• When does it save money to test the samples in pairs?
  When the average number of tests is below 2, since we would only need two tests if we were testing pairs of samples individually.



• What are the variables for this situation?
  p (probability of testing positive) and E (average number of tests required for a sample pair)

The third step of the modeling process is to begin to examine the mathematical nature of the problem and establish a relationship between the two variables.

• If the probability of testing positive for steroids is 0.0 among the population, how many tests are needed?
  Only 1 test, every time (pooled sample will always come up negative).



• If the probability of testing positive is 1.0, how many tests will be needed?
  3 tests, every time (pooled sample will always come up positive, and, no matter which individual you test, that will also come up positive, requiring a third test).



• If the probability of testing positive is 0.5, how many tests do you think will be needed?
  Each pair will require 1 to 3 tests. It is likely that the average will be near 2.



• What assumption were you making about the relationship between p and E?
  That kind of thinking assumes that there’s a linear relationship between p and E.

At this point, you can do one of two things:
1) announce to students that they will begin their exploration by trying to find out if the relationship between p and E is linear, and begin Activity 1, or
2) you can ask them how they might verify their conjecture to the number of tests needed when the probability of testing positive in the population is 0.5, and then begin Activity 1.

	Activity 1—"It’s Only a Test"

= 1, 2, 3, 4, 5, 7, 8

Part 1 of the activity is a review of the previous discussion. There are several ways in which you can proceed. You might let the students struggle with the questions from Part 1, and then have the discussion of the modeling process, with students recording answers for future reference. You might have the discussion first, and give students several minutes to answer these questions to make sure they understand what was discussed. Depending on the time in class, you could have the discussion in class and send the students home to formally answer the questions as part of a homework assignment.

Part 2 is where the students actually do the experiment. Model for the students how to perform the simulation; have two students playing the roles of the athletes, and you be the tester. One student should draw an object and put it back; then the other student should draw an object and put it back. Upon consulting with each other, they should announce whether the pooled test would be positive or negative. If positive, you select one of the students for an individual test, and he or she tells you the result. Finally, you record the results of the test on one row of the data table on Handout 1—the actual status determined by the color of the object drawn, the results of the test determined by the pooled result and which individual is tested first. Check to see if they understand what they are doing before allowing the students to begin the experiment.

• Why is the number of each color object (tile, bead, or chip) the same in the bag?
  We want to model the situation in which P (taking steroids) is 0.50.



• Which color represents the fact that I’m taking steroids?
  The choice is up to you, but be sensitive about using a color that might imply that one part of your student population is more susceptible to using steroids (drugs) than another.



• Why do I have to put my object back before the other athlete takes an object?
  That affects the probability that the second athlete is taking steroids, thereby changing the conditions for the simulation.



• How do the athletes determine whether the pooled test result is positive or negative?
  If both athletes pull the color that represents "not taking steroids," the pooled test will come up negative. Otherwise, it will come up positive.

In Part 3, students have the opportunity to answer the question, "How many tests should we expect when the probability of testing positive is 0.50?" Groups must communicate their results so the class data can be averaged together. The easiest way to do this is to have each group post its average number of tests on an overhead transparency or the board. Discuss the answers to Part 3 after the students have had a chance to think about them and formulate their own responses. Make sure that they understand that the results of the simulation indicate that it is possible to have the average number of tests be 2, but it’s not likely that we should expect the number of tests to average out to be 2. We need more data to minimize the fluctuations caused by individual trial results.

Part 4 has two versions to provide a little flexibility for individual teachers’ situations. The DRACULA simulation uses a calculator program to simulate the testing game with a larger number of trials; the commands that make up the program are included in the unit material as Handout 3. The other version, which uses coin tosses to simulate the testing game, is provided in case the calculators aren’t available, or if time is too limited to fully explore the activity. Handout 2 data table is for use with this version. In either case, students should discover two things about this situation: 1) increasing the number of trials to the simulation gives less individual fluctuation to the results, and 2) whatever the expected number of tests actually is, it’s slightly higher than 2 and not equal to 2.

	Supplemental Activity 1—"Middle of a Model" (Manipulative Version)

In case the graphing calculators are not available for Activity 2, an optional development is provided in which students repeat the testing game from Activity 1. Each group models one of the probabilities mentioned in Activity 1 by adjusting the relative numbers of objects from which they are drawing. They need to repeat the experiment 100 times, so that the data will be fairly patterned. Then they share their data as a class, and the rest of the activity is the same as Activity 1. Refer to those teacher notes for more details.

	Homework 1—"Let’s Rumble"

This is a collection of questions related to the testing game introduced in Activity 1. In discussing it with students afterward, ask students to connect Question 5 to the simulation done in class. Had they done 1000 trials, instead of 10 or 100, there would be even less fluctuation in the results and they would be even closer to the actual expected number of tests.

	Activity 2—"Middle of a Model"

This activity should begin with a continuation of the discussion of the modeling process. We’ve determined that a probability of 0.5 for testing positive doesn’t yield two tests as the expected number. But does that mean that the behavior is linear, and that particular point isn’t exactly on the line? Or is the behavior nonlinear? And, if it isn’t linear, the natural question that begs to be asked is: What is it?

• We’re trying to develop a mathematical relationship that connects the probability of testing positive for steroid use with our expectation of the average number of tests a sample pair will require. Let’s review what we have:



• When the probability of testing positive is 0.0, how many tests are needed?
  One.
• When the probability of testing positive is 1.0, how many tests are needed?
  Three.
• When the probability of testing positive is 0.5, how many tests are needed?
  Answers will vary, but it should be between 2.2 and 2.3 tests.



• We based our work in the previous activity on the idea that the relationship between the probability and the expected number of tests was linear. What have we found?
  We know that the relationship isn't a perfectly straight line. But we don't know if the situation is nonlinear, or if it is linear and the one value that we've investigated just doesn't lie on the line.



• To determine if the relationship truly is linear, what might be our next step in modeling the situation?
  Gather more data, find how the expected number of tests varies with the probability, and then analyze the data collected to see if they are linear.

At this point, you can distribute Activity 2 and the graphing calculators. Each group should be assigned a probability; 8 groups of students can run simulations with p = 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, 0.8, and 0.9. Remember that we already have data for p = 0.5, and the expected number of tests is determined when p = 0 and p = 1. The directions instruct the students to enter ‘1000’ as the number of trials; this will take several minutes. You could build in some short "sponge" activity while the students are waiting, or you could tell them to enter in a smaller number, like ‘100.’ The smaller number of trials will get an answer back much faster, but the data set won’t be as smoothly patterned. For the purposes of this activity and the rest of the unit, it doesn’t matter which way you go.

After the groups have run the simulation, a member from each group should record the group’s result on an overhead transparency or the board, and all students should record the class data on their activity pages in the data table. In analyzing the pattern, students are asked to "fit" the data with a line that starts at (0,1) and ends at (1,3). You may need to review how to find the equation of a line from two points. Make sure students understand the difference between observed value (numbers gotten from the calculator simulation) and expected value (numbers gotten by evaluating the line equation for the various values of p).

	Homework 2—"Expect to Be Worth Something"

The first part of the homework activity will require students to have a pair of dice. The activity is meant to provide them an opportunity to explore the problem’s various outcomes and payoffs before tackling the problem by mathematical analysis. An option for you is to do that part of the homework activity in class and let students do the analysis when they get home. Be sure that students understand the definition of probabilistic worth and expected value, or at least call to their attention the fact that the boldface words are actually definitions, and tell you how to proceed with the calculations.

	Activity 3—Finally, A Model

The purpose of this activity is to use regression analysis to determine if a linear, exponential, or quadratic function is the "best" type to use as a model. We’re continuing to use the data that have been collected on the expected number of tests and the probability of steroid use. Remind students that the process used in the previous activity (find an equation, calculate predicted values, calculate residuals, and examine the residual plot) is going to stay the same, but this time, the equations will be determined from regression analysis.

If students have never done regression, or were unsure of what they were doing, take the opportunity to show them how to do the linear regression (Part 1). Even if they are pretty clear of the idea, they may have forgotten how to use the graphing calculators to do that task, and reviewing with them the steps on the calculator isn’t a bad thing. Having done that, the students should proceed to do parts 2 and 3 on their own, so that they can discover the nature of the quadratic relationship themselves. Encourage them to explore other types of mathematical functions as well, even though that’s labeled an optional part.

Students should continue to Part 4 and answer the questions based upon the work that they’ve done. Have a brief discussion at the end, asking students to:

• Explain what you did in this activity.
• Why do you think the model is quadratic?
• If you ran the DRACULA simulation again and generated a new data set, would it still be quadratic? Would it be the same quadratic as the one you just determined?

	Homework 3—It’s a Good Fit

There are several options for the teacher in proceeding with this assignment. If students have access to graphing calculators, this assignment is a nice opportunity for them to practice developing linear models and determining if they truly are linear. If the students don’t have access to graphing calculators outside class, several options are still available.

1. Have each student make a hand drawing of the data set, draw a line on the graph that is a reasonable "best" fit line, then determine the equation of that line from two points on the line. They can use the equation to calculate the predicted values and the residuals, and finally make a residual plot to determine if the data set is linear. Depending on your students, doing both data sets in this fashion might be tedious; however, students need to see the contrasting results to reinforce whether data is linear. You could assign one problem to half the class and the other problem to the other half of the class.



2. Have students do all the data analysis in class with the calculators and record their table values; they would then go home to make the graphs and write their conclusions.



3. Have it be an optional reteaching activity to be assigned as in-class work if they don’t understand the process well enough from Activity 3.

	Activity 4—Which Model Fits the Best?

Students work in groups to analyze several data sets out of the context of steroid testing, and must determine the nature of the mathematical relationship among the data sets. There are a couple of ways to proceed with this activity:

• Since there are 5 data sets, organize the class into groups of five students. Each student is responsible for analyzing one data set, then the group compares the work done. Finally, students write summaries of their findings and logical arguments, including reasons they chose their models.



• Have smaller groups (maybe 3-4 students), and give each group only one or two data sets. Each group analyzes the data, comes up with a model, and then prepares an oral presentation to the class on its findings. Depending on how much time is available, students could spend some time in class discussing how to present their work, work on the presentation and their display materials that night, and have the group presentations the next day in class.

	Activity 5—Verifying the Model as Quadratic

 = 16, 17, 18

This activity illustrates the idea of probability as an area. The specific example, which is the focus for Part 1, is the original testing problem, in which the assumption is that 50% of the population is taking steroids. Depending on your students, you can either lead the class through a discussion of the questions that form Activity 1 (with them recording the answers as they go) or let them work on that part for about 10-15 minutes and have a discussion of the material at that time. It’s important that the students identify each area and its meaning with respect to the probability conditions. Be sure that they verbalize the condition and understand why the values are being multiplied. Then proceed to Part 2, in which the problem is generalized to a condition that the probability of steroid use in the population is p, instead of 0.5. Students may struggle with the algebraic simplifications needed for Questions 5 and 7, but that’s the "punch line" to verifying the model as being quadratic, and well worth the effort. If needed, use Transparency 1 to review the questions from Part 2 in the follow-up discussion.

Remind your class that the probability that each of two events happens is calculated as follows:

P(A and B) = P(A) · P(B½A).

That is, "the probability of event A and event B both happening is equal to the product of the probability of event A times the probability of event B knowing that event A has already happened. If the probabilities of event A and event B are not related (independent events), the probability of "B given that A has occurred" is simply the probability of event B. It is only because we have assumed that the two individual tests are independent that we can state:

P(A and B) = P(A) · P(B).

Note: As you saw in the unit Imperfect Testing, the area model of probability can also be represented using probability trees. Though they are not discussed on the student pages of this unit, you may want to encourage students to use both methods on some problems.

	Supplemental Activity 5—Introduction to the Parabola

This activity is designated as a supplemental activity only because it doesn’t specifically address the problem of steroid testing and the cost-effectiveness of pooling blood samples. However, it is strongly recommended that a day be spent working on the various parts of this activity if the students have never seen a quadratic function, or if they haven’t mastered the concepts of translations and stretches of quadratics and the distinction between standard-form and vertex-form equations for parabolas.

In part 1, students concentrate on the graphs of the parent function y = x² and translations on that function. Introduce them to the parent function by having them fill in the table of values and plot the points on the graph grid on Handout 7. Recall the concepts of domain and range, and reinforce the range limitation by asking the class to find the x-value that yields y = –1. Encourage discussion about "pairs" of points (x, y) and (–x, y), introducing the concept and terminology of line symmetry. Include several fractions and decimals in your table. Graph the function and label the vertex. (Use Transparency 2 to help you with the instruction, if needed). Use the graphing calculator to verify the drawing, and use the TRACE feature to verify the values recorded in the table.

Allow students to work in their groups to explore the transformations. There are four different transformations, so each student in the group can explore one of the shifts. Students should come together as a group to discuss their findings, and then answer Question 3. End the work for Part 1 by reviewing:
1) the shape of the parent parabola graph (y = x²),
2) the rules governing lateral translations of functions and
3) the domains and ranges of parabolas in the form y – k = (x – h)².

In Part 2, students concentrate on how the graph of the parent function y = x² is affected by stretch transformation and combinations of stretches and shifts. Once again, the work is divided into four parts, and each student in the group should explore two examples on his or her own. Then, the groups should collaborate to compare their results and draft responses to Questions 2 and 3. In the class discussion that follows the group explorations, ask if such stretch transformations change the domain or range of the parabola. (A horizontal stretch transformation would never change the domain or range, but a vertical stretch would change the range for parabolas whose vertex points are not on the x-axis.) Ask students if it matters whether one does the shift first and then the stretch, or the stretch first and then the shift. Close the discussion by reviewing how stretches in in both vertical and horizontal directions can be done.

In Part 3, students are asked to examine the two forms for the equation of a parabola: standard form and vertex form. They explore the parallel manipulation of the equation and its graph to discover how to change the equation from one form to another. Allow students to work through Questions 1-3, then discuss their findings before proceeding to Question 4, which gives students an additional opportunity to practice.

	Homework 5—Quadratic Nature of Pairing Samples

This assignment asks students to examine the expected value calculation again, first using a specific value for the probability of testing positive (0.3) and then examining the model developed. Students are led to examine how changes in the probability affect the model geometrically (how do the various shapes adjust?) and analytically (how do the various pieces of the equation behave?).

A really effective way of exploring Question 1 would be to construct an area model on the computer using a dynamic drawing program like Geometer’s Sketchpad. Display the areas of the three regions representing the three different tests, and the probabilities associated with those regions. Then drag the intersection point from the middle of the square (Point E) toward the upper left or lower right corner and watch what happens to the areas and the calculations!

	Activity 6—Solving the Problem

Closure on the problem of the cost-effectiveness of pooling blood samples is reached by examining the solution to the questions: "What is the break-even point? When is it cheaper to pool blood samples? When is it cheaper to test samples individually?" Various methods are explored for solving the quadratic equation describing the condition that our model should have a value equal to 2. Part 1 is pretty straightforward, using the calculator’s capability to determine intersection of two graphs. You might want to show your students the CALC menu feature that locates the intersection of two curves at this same time.

Part 2 examines the method of solving a quadratic equation called "completing the square." Students are led through this process and then asked to solve the modeling problem by applying their method. Let students work through Part 2; discuss individual aspects of the process as needed. After students solve the problem in Question 10, have them take a minute to compare their answers to the ones obtained in Part 1; they should verify the solution.

Finally, in Part 3, students are asked to generalize the work they did in Part 2 and to derive the quadratic formula. Let them work through Question 1, and have them tell you the steps involved. Make sure that there is no ambiguity or confusion in their process or thinking. Then let them write their answers to Question 2 before proceeding to the general equation. Discuss their work on Question 3 as needed, before letting the students go on to Question 4.

	Homework 6—Practice With Quadratics

This assignment should be an opportunity for students to practice what they have discovered in the previous work, and a chance for you to see which students really "got it" and which are still struggling with the algebra developed in this last activity.

HANDOUT 1—Activity 1 Part 2

HANDOUT 2—Activity 1 Part 4

HANDOUT 3—Dracula Program

(written for a TI-82 calculator)

PROGRAM:DRACULA

:0® C

:Disp "WHAT IS P?"

:Prompt P

:Disp "HOW MANY TRIALS?"

:Prompt N

:For(I,1,N,1)

:rand® A

:rand® B

:If A> P and B> P

:C+1® C

:If A< P

:C+3® C

:If A> P and B< P

:C+2® C

:End

:Disp "AVERAGE NUMBER"

:Disp "WAS"

:Disp C/N

:Stop

HANDOUT 4—Linear Regression for Activity 3

HANDOUT 5—Exponential Regression for Activity 3

HANDOUT 6—Quadratic Regression for Activity 3

HANDOUT 7—Graph Grid for Supplemental Activity 5

Annotated Student Materials

Preparation Reading—Anabolic Steroids: Use and Effect

Anabolic steroids, synthetic compounds created to act like the male hormone testosterone, remain in the forefront of discussions regarding athletes and sports. Many athletes such as bodybuilders, weight lifters, runners, swimmers, and football players believe that steroids will give them strength advantages.

Actually, the development of anabolic steroids was an important scientific breakthrough. There are, in fact, four legal uses for the drugs. They are effective in treating certain forms of anemia, some kinds of cancer, pituitary dwarfism, and serious hormone disturbances.

By virtue of their specific chemical structure (which is totally different from any vitamin or amino acid), anabolic steroids can stimulate the genetic apparatus (DNA) within the cell of the nucleus. The DNA reacts to this stimulus by directing the production of specific new proteins. It is very important to understand that the biological response that occurs is dependent on the location and number of receptors. If the receptor is muscle, there will be a tissue-building effect. If the receptor is in the brain, there may be noticeable psychological and even behavioral effects. There are, incidentally, more receptors in "sexual" tissue than in muscle, which raises concern about the prospect of hypertrophy or cancer of the prostate gland.

Although the development of bigger muscles is a result of the anabolic component of the steroid, there is also an androgenic associated with the steroid. Androgenics cause growth of facial hair, lengthening of the vocal cords that results in the voice "breaking", acne, and the premature closure of the space between parts of the body’s long bones. Anabolic steroids can have harmful effects on the liver and the cardiovascular system (causing a higher risk for developing high blood pressure and blood-clotting problems). There is growing concern that flooding a young adolescent with synthetic sex hormones may disrupt not only physical changes but also normal psychological and emotional maturation that occur during adolescence.

Several studies have placed the incidence of anabolic-androgenic steroid use among high-school students at 6 to 7%. Figures as high as 11 percent of 11th-grade boys using steroids have been reported, meaning that between 250,000 and 500,000 high-school students have used steroids. Even more disturbing are reports that approximately 40 percent who have used steroids have used five or more cycles, and approximately 40 percent began using steroids before the age of 16. As expected, the majority of teenage users are participants in school athletics, but as many as a third of the male users are not involved in sports and take these drugs in an attempt to enhance their appearance.

There have been only a few surveys among the male college population, all of which show an estimated 2 percent usage. However, among male college athletes, the estimated usage rises to between 5 and 17%. The problem is not restricted to males. As many as 1 to 2 percent of senior high school girls are users. Between 3 and 5 percent of Olympic and professional female athletes have admitted to using them at some time during their careers.

Source: Write, J. E., and V. Coward, Anabolic Steroids: Altered States, Carmel, Indiana: Benchmark Press, 1990.

	Activity 1—It’s Only a Test

 = 1, 2, 3, 4, 5, 6, 7, 8

Part 1: Introduction

1. What is the central problem that we’re trying to solve?

We’re trying to determine when it’s cheaper to "pool" blood samples in testing for steroid use.

2. What are the variables in this problem?

The likelihood, p, that someone is actually using steroids in the population, and the expected number of tests, E, that would be needed.

3. When one tests for steroid use by "pooling" two blood samples, how many tests do you expect to have to make?

The number of tests will always be one, two, or three tests.

4. In testing for anabolic steroid use by pooling two samples together:



1. If nobody is taking steroids, how many tests do you expect to have to make?

One test every time, because the pooled sample would come up negative (clean).

2. If very few people are taking steroids, how many tests do you expect to have to make?

Most of the time, only one test would be needed. Once in a while, a second (or third) test would be required. The average number would be close to one.

3. If almost all the people are taking steroids, how many tests do you expect to have to make?

Most of the time, three tests would be needed–the pooled sample test would come up positive, and the first individual test would come up positive. Some of the time, only two tests would be needed, since the first individual test might be negative. The average number would be close to three.

4. If everybody is taking steroids (and what a crazy world that would be!), how many tests do you expect to have to make?

Three tests every time, because the pooled sample would always come up positive, and the first individual test would also come up positive.

5. If exactly half the population were taking steroids, how many tests do you expect to need for each sample pair? Explain your reasoning.

I guess that two tests would be needed. Since p = 0.5 is halfway between none of the people and all of the people, I expect that halfway between one test and three tests should be my answer.

6. What modeling assumptions did you make in answering Question 5?

The nature of the relationship between the two variables is linear; the expected number of tests increases proportionally to the population occurrence.

Part 2: Simulation

In groups, you are going to explore the problem of what happens when half the population is taking steroids. Two people will play the role of Olympic athletes, who have to be tested for steroid use. The third person will be the lab worker performing the test. Go through the simulation 10 times, recording the actual status of each athlete, the result of the pooled test, and the result of any individual tests, in the data table provided for this activity. Use marks of ‘+’ for a positive result or ‘–’ for a negative result. Then, determine the number of tests needed for each trial, and finally the average number of tests needed for the 10 trials.

Part 3: Analyzing Class Results

In the table below, record each group’s results for the average number of tests needed for 10 trials. Answers will vary; sample results provided in the table.

1. As the tester, when would you be saving money in pooling samples?

When the average number of tests was under 2 tests, since testing the two athletes individually would always require 2 tests.

2. When half the population is using steroids, is it possible to save money by pooling samples in testing for steroid use?

Yes. There were specific trials that required only 1 test, and, in certain groups, the average number < 2 tests.

3. In this situation, is it likely that you would save money by pooling samples?

Probably not. There were more group results that were > 2 tests, and the average of those results was » 2.2 tests. However, given the range of results, it’s hard to be sure. Some of the groups may have been very unlucky in choosing which of the individual samples to test first when the pooled sample came up positive.

4. Think back to the simulation. Are there problems or limitations in the way in which we collected our data that might affect the results that we got?

Possible answers include: not replacing the first person’s chip before drawing the second, having the chips stick to each other, nonuniform weights or shapes for the objects drawn, not mixing or shaking up the objects well enough, or looking at the objects as they are being drawn.

5. How could we modify our simulation, to better understand what’s going on in the situation where half the population is taking steroids?

Do more trials. Specifically, if each group did the test 100 times, the range of results might narrow down a bit, and the average of those results would get close to the "true" results.

Part 4: DRACULA Simulation of Situation

Each group will run the DRACULA program. When asked the first question, "What is P? P = ?", enter in the value ‘0.5’ (remember, we’re studying what happens when half the population is taking steroids). When asked the second question, "How many trials? N = ?", enter in the value ‘100’. After a few seconds, the calculator will provide the average number of tests needed.

In the table below, record each group’s results for the average number of tests needed for 100 trials.

1. Do you think it’s cheaper to pool samples in testing for steroid use when half the population is taking steroids? Explain.

No. The range of possible results when only 10 trials were taken was [1.8, 2.5]. When the simulation was repeated with 100 trials, the range became [2.04, 2.37]. In every situation, the average number of tests was greater than 2, and, when you consider the total number of tests (n = 1000), the average » 2.23. Incidentally, the average of the total number of tests before was » 2.2 as well. I thought that the results were supposed to be an average of 2 tests. I wasn’t sure if the way the simulation was being run, or a streak of bad luck, might be causing me to get a different answer. But now, I’m convinced that it’s not supposed to average to 2 tests, and, if I needed even more convincing, I would simply repeat the simulation again, doing an even larger number of trials.

2. In modeling the situation, two questions come up as a result of this activity. Review the problem and the original assumptions, and see if you can come up with those questions.



1. The problem of "when is it cheaper to pool samples?" is now narrowed to a range of possible probabilities between [0, 0.5]. So a natural question to ask would be: "What is the probability of steroid use in the general population where the break-even point occurs?" or "At what probability would the number of tests be expected to be exactly two tests?"



2. The assumption is that the relationship between p and E is not linear, or else we would have found that the number of tests would average to be 2 tests. So a natural question to ask at this point would be: "What is the exact nature of the relationship between p (the incidence of steroid use in the population) and E (the expected number of tests required in pooling two samples)?"

Part 5: Simulation by Tossing Pennies

You will repeat the simulation from today’s class, only this time using pennies. We’ll agree that a penny that comes up ‘tails’ will be a negative test result, and a penny that comes up ‘heads’ will be a positive test result. You will need someone else to play the role of the tester, but you might have figured out there is a shortcut to doing this:

• If both pennies come up tails, you need 1 test (both people are negative, right?).
• If both pennies come up heads, you need 3 tests.
• If one penny is heads, while the other is tails, have the "tester" randomly select one of the pennies for an individual test. That will determine whether you need only two tests or if a third test would be required.

Record the results of the coin-toss simulation in the data table provided. Calculate the average number of tests needed for your simulation, and come to class tomorrow ready to share your results.

In the table below, record 10 students’ results for the average number of tests needed for 100 trials.

1. Do you think it’s cheaper to pool samples in testing for steroid use when half the population is taking steroids? Explain.

No, the range of possible results when only 10 trials were taken was [1.8, 2.5]. When the simulation was repeated with 100 trials, the range became [2.04, 2.37]. In every situation, the average number of tests was greater than 2, and, when you consider the total number of tests (n = 1000), the average » 2.23. Incidentally, the average of the total number of tests before was » 2.2 as well. I thought that the results were supposed to be an average of 2 tests. I wasn’t sure if the way the simulation was being run, or a streak of bad luck, might be causing me to get a different answer. But now, I’m convinced that it’s not supposed to average to 2 tests, and, if I needed even more convincing, I would simply repeat the simulation again, doing an even larger number of trials.

2. In modeling the situation, two questions come up as a result of this activity. Review the problem and the original assumptions, and see if you can come up with those questions.



1. The problem to solve becomes "What is the probability of steroid use in the general population where the break-even point occurs?" or "At what probability would the number of tests be expected to be exactly two tests?"
2. The modeling question becomes "What is the exact nature of the relationship between p (the incidence of steroid use in the population) and E (the expected number of tests required in pooling two samples)?"

	Supplemental Activity 1—Middle of a Model (Manipulative Version)

In Activity 1, we assumed that the probability was 0.50 for a person selected at random to be taking steroids. This activity will explore how different probability rates affect the number of tests required.

Your group will be assigned a particular probability to use, and then you will do the Testing Game again. Modify the number of each kind of marker used, so that it models the probability your group is working with. Then do the simulation 100 times and determine the average number of tests. When done, record your results in a table and share them with other groups according to directions given.

Record the class results for the average number of tests needed for 100 trials in the space below the appropriate probability of occurrence. Use a table similar to the data table in Handout 2, Activity 1.

1. Using the data collected, make a graph of Expected Value (E) vs. probability (p).



2. In creating a model, the first clue is the pattern showing up in the scatterplot. Describe the graph; what kind of pattern seems to relate these quantities?

It appears to be fairly linear, starting at (0,1) and going to (1,3).

3. You may have answered the previous question by saying that the pattern looks linear. Let’s assume that it is. The next step is to describe the pattern in mathematical terms.



1. Draw a line that starts at the first point (0,1), and goes to the last point (1,3). Find the slope of that line.

Slope = (3 – 1) / (1 – 0) = 2

2. Write the equation for the line in slope-intercept form; this will be our first model for these data.

E = 2p + 1 or y = 2x + 1

4. Now, we have to see how good a job our model does in describing the observed data. Fill in the table below; the actual values are the ones found from the simulation and recorded on the previous page. The equation above tries to "predict" those data, and so predicted values are found by taking each value for the probability and using it in the equation you just derived. Finally, errors (residuals) are how far the predictions are from the observed data, so they are found by subtraction:

Error = Observed – Expected

Probability (p)	Observed Value (E)	Expected Value	Error (residual)
0.0	1.00	1.00	0.00
0.1	1.25	1.20	+ 0.05
0.2	1.56	1.40	+ 0.16
0.3	1.85	1.60	+ 0.25
0.4	2.06	1.80	+ 0.26
0.5	2.24	2.00	+ 0.24
0.6	2.41	2.20	+ 0.21
0.7	2.69	2.40	+ 0.29
0.8	2.79	2.60	+ 0.19
0.9	2.86	2.80	+ 0.06
1.0	3.00	3.00	0.00



5. Finally, in the grid provided above, make a graph of errors vs. probability (p).



6. A good model will show a random pattern of residuals, with a balance of points above and below the zero line. Does the model you’ve developed have those properties?

No. The residual plot has no points below the zero line and is not showing a random pattern.

	Homework 1—Let’s Rumble

1. Dante Takum is trying to arrange tag-team matches for his wrestlers, half of whom are currently taking anabolic steroids. The event organizers insist on having the wrestlers tested for steroid use. Each test costs $160.



1. If you wanted to test each wrestler individually, what would the total cost be?

(2)($160) = $320

2. Based on the work done in Activity 1, how much would it cost (on average) to test the pooled sample from the two wrestlers on the tag team first, and then test individuals if needed?

Based on an average of 2.23 tests per pair, it would cost (2.23)($160) = $356.80

2. Carefully consider the following table that represents results from repeating the testing simulation from Activity 1. Was the game properly played? How do you know?

Trial Number	1	2	3	4	5	6	7	8	9	10
Number of Tests	1	3	3	1	1	1	3	1	1	3

Probably not. The situation in which two tests were required never came up.

3. In Activity 1, we modeled a situation in which half the population was taking steroids; we used two types of markers and had the same number of each color. If green means "steroid user," determine the smallest number of blue and green markers needed to model a population in which the probability of steroid use is:



1. 75%

3 green, 1 blue

2. 15%

3 green, 17 blue

3. 23%

23 green, 77 blue

4. Suppose you want to model a situation in which the probability of an individual taking steroids is 25%. Use the same color scheme as in the previous question.



1. What is the smallest number of chips that you would need?

1 green and 3 blue

2. If you drew a blue marker on the first draw, and did not replace the chip, what would be the probability of drawing a blue on the second draw? A green on the second draw?

2/3 or 0.67; 1/3 or 0.33

3. If you drew a green on the first draw and did not replace the chip, what is the probability of drawing a green on the second draw? A blue on the second draw?

0%; 100%

5. Suppose that you use a coin to simulate the situation in which the probability is 0.5 that a person in the general population is taking anabolic steroids, and that you use a "fair coin."



1. Would that mean that, if you flipped the coin twice, you would always get one head and one tail?

No, but that would be the most likely outcome.

2. If you flipped the coin a million times, would you always get 500,000 heads and 500,000 tails?

No, but that (very unlikely) event would still be the most likely outcome.

3. Which do you think is more likely: The number of heads in 10 tosses would be between 4 and 6, or the number of heads in 100 tosses would be between 40 and 60?

The second option is more likely.

4. Suppose a fair coin was flipped ten times and happened to land heads eight of those times. If you flipped the coin another 100 times, how many heads would you expect to get?

Approximately 50. This would naturally draw the overall percentage closer to 50% (i.e. 58 out of 110, if exactly 50 of the next 100 tosses are heads). Be careful not to assume that the data will automatically compensate for the initial results.

6. In your simulation in Activity 1, it was important to replace the drawn chip before drawing another. Can you explain why? Can you think of a different situation where it would be important not to replace the chip?

Not replacing the drawn chip changes the probability of drawing a green chip on the second draw. We want to model the situation in which both athletes have the same probability of testing positive.

Many states have lotteries in which a collection of six numbers is chosen from 44 to 51 different possible numbers. In that situation, the numbers shouldn’t be replaced.

	Activity 2—Middle of a Model

 = 6, 9, 10, 11

In Activity 1, we assumed that the probability was 0.50 for a person selected at random to be taking steroids. This activity will explore how different probability rates affect the expected number of tests required.

The teacher will assign a particular probability to each group, which will then run the DRACULA program. When prompted by the calculator program, give it the probability (decimal, please!), and set the number of trials to be 1000. Settle in for a couple of minutes; after the calculator has done all those trials, record your results in the table below and share them with other groups according to directions given.

In the tables below, record the class results for the average number of tests needed for 1000 trials in the space below the appropriate probability of occurrence:

1. Using the data collected, make a graph of Expected Value (E) vs. probability (p).



2. In creating a model, the first clue is the pattern showing up in the scatterplot. Describe the graph; what kind of pattern seems to relate these quantities?

It appears to be fairly linear, starting at (0,1) and going to (1,3).

3. You may have answered the previous question by saying that the pattern looks linear. Let’s assume that it is. The next step is to describe the pattern in mathematical terms.



1. Draw a line that starts at the first point (0,1), and goes to the last point (1,3). Find the slope of that line.




Slope = (3 – 1) / (1 – 0) = 2

2. Write the equation for the line in slope-intercept form; this will be our first model for these data.

E = 2p + 1 or y = 2x + 1

4. Now, we have to see how good a job our model does in describing the observed data. Fill in the table below; the actual values are the ones found from the simulation and recorded on the previous page. The equation above tries to "predict" those data, and so predicted values are found by taking each value for the probability and using it in the equation you just derived. Finally, errors (residuals) are how far the predictions are from the observed data, so they are found by subtraction:

Errors = Observed – Expected

Probability (p)	Observed Value (E)	Expected Value	Error (residual)
0.0	1.000	1.000	0.000
0.1	1.300	1.200	+ 0.100
0.2	1.554	1.400	+ 0.154
0.3	1.879	1.600	+ 0.279
0.4	2.042	1.800	+ 0.242
0.5	2.237	2.000	+ 0.237
0.6	2.418	2.200	+ 0.218
0.7	2.632	2.400	+ 0.232
0.8	2.727	2.600	+ 0.127
0.9	2.873	2.800	+ 0.073
1.0	3.000	3.000	0.000



5. Finally, in the grid provided above, make a graph of errors vs. probability (p).



6. A good model will show a random pattern of residuals, with a balance of points above and below the zero line. Does the model you’ve developed have those properties?

No. The residual plot has no points below the zero line and is not showing a random pattern.

	Homework 2—Expect to Be Worth Something

1. A dice game is defined by two rules:



• Each player will roll a pair of dice two times.
• The score is the product of the two numbers rolled.

Find the expected value.

1. Play the game several times with a friend or a family member. Describe the strategy that you think would be best to apply to this game.

Answers will vary. Possibilities include "12," since it comes up in a lot of different ways; "36," since it is the biggest outcome.

2. Now, let’s apply some mathematics. First, we need to determine what the possible outcomes are when you multiply the two dice together, and how often they come up. Fill in the chart provided below with the winning answers:




Outcome for Dice 2
	1	2	3	4	5	6
1	1	2	3	4	5	6
2	2	4	6	8	10	12
3	3	6	9	12	15	18
4	4	8	12	16	20	24
5	5	10	15	20	25	30
6	6	12	18	24	30	36




3. Some outcomes come up more than once, and some outcomes pay off more than others. Here we consider that.




Value (n)	Prob. (p)	Worth (W)
1	1/36	1/36 » 0.028
2	2/36	4/36 » 0.111
3	2/36	6/36 » 0.167
4	3/36	12/36 » 0.333
5	2/36	10/36 » 0.278
6	4/36	24/36 » 0.667
8	2/36	16/36 » 0.444
9	1/36	9/36 » 0.250
10	2/36	20/36 » 0.556
12	4/36	48/36 » 1.33
15	2/36	30/36 » 0.833
16	1/36	16/36 » 0.444
18	2/36	36/36 » 1.00
20	2/36	40/36 » 1.11
24	2/36	48/36 » 1.33
25	1/36	25/36 » 0.833
30	2/36	60/36 » 1.67
36	1/36	36/36 » 1.00




• In the first column, list all possible outcomes. Those are also the values for the outcomes.



• In the second column, record the probability the outcome will happen. There are 36 possible dice rolls, so the probability is the (number of times the outcome takes place) ¸ 36.



• The third column is for what’s called the probabilistic worth. It’s found by multiplying the value of an outcome by the probability that it happens.



2. Now what should your strategy be for playing the dice game?

I would predict "30" every time, since it provides the greatest worth, and there is no penalty for incorrect guesses.

3. The expected value of an event is defined as the sum of the probabilistic worths of all the possible outcomes. Suppose you buy a $1 ticket for a raffle. One grand prize valued at $1,000 will be awarded. Two second prizes valued at $300 each will be awarded, and fifty consolation prizes valued at $20 each will be awarded. There are 10,000 tickets sold in the raffle. What is the expected value of one ticket? What is the meaning of the answer?

$1000 (0.0001) + $300 (0.0002) + $20 (0.005) – $1 (1.000) = – $0.74

Each time you "play" this raffle, you can expect to lose about 74 cents.

4. What is the expected value of the dice game played on the previous page? What is the meaning of that answer?

(1/36) + (4/36) + (6/36) + (12/36) + (10/36) + (24/36) + (16/36) + (9/36) + (20/36) + (48/36) + (30/36) + (16/36) + (36/36) + (40/36) + (48/36) + (25/36) + (60/36) + (36/36) = 49/4

As a long-term behavior, the average payoff for the dice game is around 12.25 points.

5. The table below shows data from a high school that was testing for steroid use among its athletes. Did the school save money by pooling blood samples?




Number of Tests	1	2	3
Proportion of samples	0.28	0.31	0.41

The average number of blood tests needed would be calculated in this way:

(1) (0.28) + (2) (0.31) + (3) (0.41) = 2.13.

So, the school would have been better off testing the athletes individually.

6. What does the mathematics of expected value have to do with the unit problem of steroid testing?

The number of tests in each logical category is the "value," and the fraction of times those categories come up is the "probability." The expected value for this situation is the average number of tests required, which indicates whether it is cost-effective (below 2.00) or not.

	Activity 3—Finally, A Model

 = 12, 14, 15

In modeling whether to pool samples when testing for drug use, we need to determine how the variables are related. We speculated that increases in the probability of steroid use (p) produce proportional increases in the expected number of tests required (E). Even though the data look linear, we found that the line connecting the first and last data points was a poor model.

In this activity, you are going to use the calculator’s statistical regression ability to fit the data in the "best" way possible. The menu that you will use allows you to "fit" data with functions that are linear, exponential, quadratic, or any other kind you might want to try. Regression equations will balance the number of data points on either side of the line or curve, and will minimize the sum of the squares of the residuals. Modelers must also pay attention to which kind of equation to use. To determine that, you must look for a random pattern in the residual plot (errors vs. probability).

Part 1: Linear Regression (uses Activity 3 Handout 4)

1. In the 2nd column, record the data for the expected values (E) obtained from the work done in Activity 2.
2. Enter the data from the table into your calculator. On the grid provided, draw a copy of the scatterplot of your data.
3. Do a linear regression on the data. Record the equation obtained in the appropriate place on the handout. Sketch the graph of the line into your scatterplot.
4. Calculate the predicted values, using the regression equation. Record the values in the 3rd column of the data table.
5. Calculate the errors in using that equation to "model" the data. Record those values in the 4th column of the data table.
6. On the other grid provided, draw the residual plot for this regression analysis.

Part 2: Exponential Regression (uses Activity 3 Handout 5)

Repeat the steps from the previous part of the activity, using exponential regression.

HANDOUT 4—Answers

Linear Regression for Activity 3

HANDOUT 5—Answers

Exponential Regression for Activity 3

Equation:   E = 1.228 (2.767)^p

Part 3: Quadratic Regression (uses Activity 3 Handout 6)

Repeat the steps from the previous parts of the activity, using quadratic regression.

HANDOUT 6—Answers

Quadratic Regression for Activity 3

Equation:   E = –1.028p² + 3.0p + 1.01

Optional Part: There are more types of regression built into the calculator you are using. Try some of the other methods, but be sure to follow the steps as you did earlier. And remember to record everything as part of this activity work.

Part 4: Finally, A Model!

1. What do you think is the connection between the probability (p) and the expected value (E) in testing for steroid use?

It’s quadratic. The model for these data is described by the equation:

E = –1.028p² + 3p + 1.01

2. How did you arrive at that conclusion?

When the data were "fit" with a quadratic equation, the residual pattern was random. The actual equation was found by doing quadratic regression on the calculator, so it is the type of regression that minimizes the sum of the squares of the errors.

3. Think back over the process used to arrive at this model. In simulating the drug testing, you either used a calculator program with 1000 trials or drew markers 100 times. If you had analyzed data that represented only 10 trials instead:



1. Would it have changed the kind of model you chose (linear, exponential, quadratic, or so on.)?

No, the nature of pooling samples is quadratic; the number of trials won’t change that.

2. Would it have changed the actual equation you recorded?

Yes, the regression equation was based on the data that were generated by the calculator program having done 1000 trials. Unless we got the exact same numbers when doing only 10 trials, we would get a slightly different equation.

3. Would it be a "better" model? Explain what that means.

It is likely that the experimental results will more closely approach the expected value as the number of trials increases.

	Homework 3—It’s a Good Fit

Below are two data sets. Graph each separately; be sure to indicate the scales used. For each graph, find the equation for the line that you think models the data most closely. Plot the line, then plot the residuals. Does there seem to be a pattern to the residuals? Do you think a line is the best kind of equation to fit these data?

1. 
   

   x	–4	10	22	32	36	51	62	68	78	89	96
   y	39	66	102	145	192	233	285	355	414	481	533

   

   

   The line of best fit is roughly y = 5.1x + 8.2 The residuals seem to "curve upward" (line is not the best kind of equation).

2. 
   

   x	4	6	20	30	40	47	60	70	79	92	99
   y	65	58	52	46	44	32	31	22	18	14	13

   

   

   The line of best fit is approximately y = –0.54x + 63. The residuals seem to be scattered around the zero line, so a line is the best kind of equation to use.

	Activity 4—Which Model Fits Best?

We’ve seen how the kinds of models that you use to describe the patterns in data are varied—linear, exponential, quadratic, and so on. Each can be used to fit the data, but one will work best by producing a random residual plot.

Work on the data set(s) assigned to your group. Prepare an analysis of your data, make the graphs of your data and the residuals, and state the model that best fits your data. Write a summary of your results and a logical argument defending your choice of model for your group’s presentation to the class. If the data are nonlinear, explain what might cause the rate of change to not be constant.

Data Set 1

Volume of a Drop of Water From a Beral^TM Pipet

(A Beral pipet is similar to a medicine dropper and is used in chemistry experiments.)

This is strongly linear: y = –0.029 + 0.046x

Data Set 2

Oil Leaking from a Container

Data Set 3

U.S. Electricity Consumption

E is billions of kilowatt-hr/year.

(Years are set up so that 1900 is year 0)

Data Set 4

A Falling Stone

Data Set 5

One Swing of a Pendulum

	Activity 5—Verifying the Model as Quadratic

 = 16, 17, 18

We’ve seen how the data collected on the expected number of tests are best described by a quadratic function. To be completely satisfied with this result, it would be nice to understand why it is quadratic in nature. Think back to the original simulation for this problem (Testing Game). We assumed that the probability of a positive test occurring would be 0.50. Let’s see how the problem relates to the square below, which is divided into regions. The sides of the square all have a length of 1. The total area represents a total probability of 1. The rows represent the first person’s probabilities on the test, and the columns represent the second person’s probabilities.

Part 1: Specific example

1. What outcome does the region EFIH represent?

Both people testing negative

2. What outcome does the region EFIH represent?

(0.5)(0.5) = 0.25

1. Under what conditions are exactly two tests needed?

When the combined sample was positive and first of the two athletes tested negative, the second athlete had to be positive. Then exactly two tests were needed. Thus, since the first person gets the first individual test, two tests are needed when only the second person tests positive.

2. What region(s) of the figure represent(s) the probability of needing exactly two tests?

DEHG

3. What is the probability of needing exactly two tests?

(0.5)(0.5) = 0.25

1. Under what conditions are three tests needed?

When the combined sample was positive and the first athlete was positive, then three tests are needed. Whenever the first person tests positive, three tests will be needed.

2. What region(s) represent(s) the probability of needing exactly three tests?

ABED and BCFE, or ACFD

3. What is the probability of needing exactly three tests?

(0.5)(0.5) + (0.5)(0.5) = 0.50 or (1.00)(0.5) = 0.50

4. What is the sum of the probability answers to questions 1, 2, and 3? Explain the meaning of your result.

One. Since you would always need 1, 2, or 3 tests, it would happen 100% of the time. Therefore, the probability should be 1.

5. Write an expression for the expected number of tests, and then calculate it. Compare your answer to the value obtained in Activity 1.

1(0.25) + 2(0.25) + 3(0.50) = 2.25.

In Activity 1, the expected value for a probability p = 0.5 was around 2.23. That was an experimental result based on the performance of the simulation; the answer of 2.25 is a theoretical result based on probability.

6. Explain how the area model represents probabilities. How could you go about generalizing this problem?

The area of the region actually corresponds to the value of the probability. Each region corresponds to two conditions (test result for first and second persons), and, in finding the overall probability of both events taking place, one must multiply the individual probabilities together. The dimensions of the regions correspond to the actual probabilities for those events, and, in calculating area, one must multiply the dimensions together.

To generalize the problem, introduce variables to represent the probability that one of the people tests positive for steroid use.

7. If p is the probability of testing positive, how can you represent the probability of not testing positive?

Use the expression 1 – p

Part 2: Generalization

Let’s go through the same analysis as before, but this time we will use the variable p to represent the incidence of steroid use in the population rather than a specific number like 0.5. The diagram provided at right will help you answer the following questions.

1. What probability is represented by the area of EFIH, the region requiring only 1 test?

(1 – p)²

2. What region(s) represent(s) the probability of needing exactly two tests? What probability is represented by the area of that region?

DEHG, because this area represents a positive pooled sample and a negative first individual test. The probability for that case is represented by p (1 – p).

3. What region(s) represent(s) the probability of needing exactly three tests? What probability is represented by the area of that region?

ACFD; students may also say there are two—BCFE and ABED. The probability for that situation is represented by p.

4. Write an algebraic expression that represents the total area for the model as a sum of its parts.

(1 – p)² + (1 – p)p + p(1 – p) + p²

5. Simplify the algebraic expression you wrote for Question 4 and show that the total has a value of 1.

(1 – p)² + (1 – p)p + p(1 – p) + p² = 1 – 2p + p² + p – p² + p – p² + p² = 1

6. Write an algebraic expression for the expected number of tests needed when pooling samples.

1(1 – p)² + 2(1 – p)p + 3(p(1 – p) + p²)

7. Use the distributive property to simplify the algebraic expression you wrote for Question 6.

1(1 – p)² + 2(1 – p)p + 3(p(1 – p) + p²) = 1(1 – 2p + p²) + 2(p – p²) + 3(p – p² + p²) = 1 – 2p + p² + 2p – 2p² + 3p = 1 + 3p – p²

8. Compare the equation you just got in Question 7 with the model developed from doing quadratic regression from Activity 3.

The model developed by using quadratic regression was given by the equation E = –1.028p² + 3p + 1.01. You can see that they are both quadratic, they have exactly the same linear coefficient, and the other two coefficients are almost exactly the same.

	Supplemental Activity 5—Introduction to the Parabola

Part 1

1. To begin, let’s look at the simplest parabola, y = x². Complete the table below, and then, using your graphing calculator, plot some points and sketch the curve.




x	0	1	3	–1	2	–2	can’t	–1.414	2/3	1/2	–2.24	2.24
y	0	1	9	1	4	4	–4	2	4/9	1/4	5	5




2. The following four tables represent parabolas that have been changed. Use the information that you just found to complete the table, plot the points, and sketch the graph. Each person in the group should tackle one of the problems, then share their work with the group before going on to answer Question 3.

Person 1: y = (x + 3)²

x

–3

–2

0

–4

–1

–5

can’t

»–1.59

–2 1/3

–2 1/2

»–0.76

»–5.3

y

0

1

9

1

4

4

–4

2

4/9

1/4

5

5


Person 2: y = x² – 3

x	0	1	–1	2	–2	3	–3		–	1/2	3/2	–3/2
y	–3	–2	–2	1	1	6	6	2	2	–1 1/4	–3/4	–3/4


Person 3: y = (x – 2)²

x	2	1	–1	3	0	4	can’t	2+	8/3	5/2	2+	6
y	0	1	9	1	4	4	–4	2	4/9	1/4	5	16


Person 4: y = x² + 2

x	0	1	–1	2	–2	3	–3		–	1/2	3/2	–3/2
y	2	3	3	6	6	11	11	7	7	9/4	17/4	17/4




1. How would you describe the shift for the new graph you drew, from where you drew the graph of y = x²?

   Person 1: Shifts 3 units to the left	Person 3: Shifts 2 units to the right
   Person 2: Shifts down 3 units	Person 4: Shifts up 2 units

   
   
2. The lowest point (vertex) for the graph of y = x² is (0,0). Where is this point located for your graph?

Person 1: (–3, 0)	Person 3: (2, 0)
Person 2: (0, –3)	Person 4: (0, 2)



3. What would be a rule that would explain how to draw the graph of a parabola that has been changed by adding or subtracting a value in the equation for the parabola?

Adding/subtracting before the squaring operation shifts to the left/right.

Adding/subtracting after the squaring operation shifts up/down.

Part 1 Graphs

Part 2

You can stretch functions by multiplying or dividing by a number in particular places in equations. Recall what the basic parabola, y = x², looks like; a table of values has been listed again to help you get started.

1. Work in groups; use your graphing calculators to complete the tables, plot some points, and sketch the curves.

y = x²

x	0	1	–1	2	–2	3	–3
y	0	1	1	4	4	9	9





Person 1: y/2 = x2									Person 2: y = (x – 2)2 – 1
x	0	1	–1	2	–2	3	–3		x	2	1	3	0	4	5	–1
y	0	2	2	8	8	18	18		y	–1	0	0	3	3	8	8





y = (x/2)2									y = (x/2 – 2)2 – 1
x	0	2	–2	4	–4	6	–6		x	4	2	6	0	8	10	–2
y	0	1	1	4	4	9	9		y	–1	0	0	3	3	8	8





Person 3: y = 3x2										Person 4: y = (x/4)2
x	0	1	–1	2	–2	3	–3			x	0	4	–4	8	–8	12	–12
y	0	3	3	12	12	27	27			y	0	1	1	4	4	9	9





y = (x/2 – 1)2 – 2									y = (x – 1)2 – 2
x	4	2	6	0	8	10	–2		x		2	1	4	0	–1	3	–2
y	–1	–2	2	–1	7	14	2		y		–1	–2	7	–1	2	2	7




2. How would you describe the way the new graph looks, compared to y = x²?




Person 1:	First one is thinner; second one is wider.		Person 2:	First one is shifted to the right by 2 and down 1; second one is shifted the same, but opens wider.
Person 2:	First one is narrower; second one is shifted to the right 1 and down 2, and also wider.		Person 4:	First one opens wider; second one is shifted to right 1 and down 2.




3. What rule would explain how to draw the graph of a parabola that has been changed by multiplying or dividing a value in the equation for a parabola?

Multiplying the x-variable by a number greater than 1 (or dividing the y-variable) will "stretch" the graph, making it steeper and the U-shape narrower. Dividing the x-variable by a number greater than 1 (or multiplying the y-variable) will "shrink" the graph, making the U-shape wider.