Statway
TM
A statistics pathway for college students
Module 1:
Statistical Studies and Overview of the Data Analysis Process
Module 2:
Summarizing Data Graphically and Numerically
Module 3:
Reasoning About Bivariate Numerical Data—Linear
Relationships
Module 4:
Modeling Nonlinear Relationships
Module 5:
Reasoning About Bivariate Categorical Data and Introduction
to Probability
Module 6:
Formalizing Probability and Probability Distributions
Module 7:
Linking Probability to Statistical Inference
Module 8:
Inference for One Proportion
Module 9:
Inference for Two Proportions
Module 10:
Inference for Means
Module 11:
Chi-Squared Tests
Module 12:
Other Mathematical Content
Version 1.0
A resource from
The Charles A. Dana Center at
The University of Texas at Austin
July
2011
Frontmatter
Statway—Full Version 1.0, July 2011
Unless otherwise indicated, the materials found in this resource are
Copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and
the Charles A. Dana Center at The University of Texas at Austin
Outside the license described below, no part of this resource shall be reproduced, stored in a retrieval system, or
transmitted by any means—electronically, mechanically, or via photocopying, recording, or otherwise, including
via methods yet to be invented—without express written permission from the Foundation and the University.
The original version of this work was created by The Charles A. Dana Center at The University of Texas at
Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching.
STATWAYTM / StatwayTM is a trademark of the Carnegie Foundation for the Advancement of Teaching.
***
This copyright notice is intended to prohibit unlicensed commercial use of the Statway materials.
License for use
Statway Version 1.0, developed by the Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, is licensed under the Creative
Commons Attribution-Noncommercial-Share Alike 3.0 Unported license.
To view the details of this license, see creativecommons.org/licenses/by-nc-sa/3.0. In general, under this
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The original version of this work was developed by the Charles A. Dana Center at the University of
Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching.
This work is used (or adapted) under the Creative Commons Attribution-NonCommercialShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license: creativecommons.org/licenses/by-nc-sa/3.0.
For more information about Carnegie’s work on Statway, see www.carnegiefoundation.org/statway;
for information on the Dana Center’s work on The New Mathways Project, see
www.utdanacenter.org/mathways.
Noncommercial—You may not use this work for commercial purposes.
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otherwise comply with copyright law. If you find an error or you believe we have failed to provide proper
acknowledgment, please contact us at dana-txshop@utlists.utexas.edu.
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
ii
Frontmatter
The Charles A. Dana Center
The University of Texas at Austin
1616 Guadalupe Street, Suite 3.206
Austin, TX 78701-1222
Fax: 512-232-1855
dana-txshop@utlists.utexas.edu
www.utdanacenter.org
Statway—Full Version 1.0, July 2011
The Carnegie Foundation for the Advancement of Teaching
51 Vista Lane
Stanford, California, 94305
Phone: 650-566-5110
pathways@carnegiefoundation.org
www.carnegiefoundation.org
About the development of this resource
The content for this full version of Statway was developed under a November 30, 2010, agreement by a team of
faculty authors and reviewers contracted and managed by the Charles A. Dana Center at the University of Texas
at Austin with funding from the Carnegie Foundation for the Advancement of Teaching.
This resource was produced in Microsoft Word 2008 and 2011 for the Mac. The content of these 12 modules
was developed and produced (that is, written, reviewed, edited, and laid out) by the Charles A. Dana Center at
The University of Texas at Austin and delivered by the Dana Center to the Carnegie Foundation for the
Advancement of Teaching on June 30, 2011.
Some issues to be aware of:
•
PDF files need to be viewed with Adobe Acrobat for full functionality. If viewed through Preview,
which is the default on some computers, the URLs may not be correct.
• The file names indicate the lesson number and whether the document is the instructor or student version
or the out-of-class experience.
The Dana Center is engaged in a process of revising and improving these materials to create the Dana Center
Statistics Pathway. We welcome feedback from the community as part of our course revision process. If you
would like to discuss these materials or learn more about the Dana Center’s plans for this course, contact us at
mathways@austin.utexas.edu.
About the Charles A. Dana Center at The University of Texas at Austin
The Dana Center collaborates with local and national entities to improve education systems so that they foster
opportunity for all students, particularly in mathematics and science. We are dedicated to nurturing students’
intellectual passions and ensuring that every student leaves school prepared for success in postsecondary
education and the contemporary workplace—and for active participation in our modern democracy.
The Center was founded in 1991 in the College of Natural Sciences at The University of Texas at Austin. Our
original purpose—which continues in our work today—was to raise student achievement in K–16 mathematics
and science, especially for historically underserved populations. We carry out our work by supporting high
standards and building system capacity; collaborating with key state and national organizations to address
emerging issues; creating and delivering professional supports for educators and education leaders; and writing
and publishing education resources, including student supports.
Our staff of more than 80 researchers and education professionals has worked intensively with dozens of school
systems in nearly 20 states and with 90 percent of Texas’s more than 1,000 school districts. As one of the
College’s largest research units, the Dana Center works to further the university’s mission of achieving
excellence in education, research, and public service. We are committed to ensuring that the accident of where a
student attends school does not limit the academic opportunities he or she can pursue.
For more information about the Dana Center and our programs and resources, see our homepage at
www.utdanacenter.org. To access our resources (many of them free) please see our products index at
www.utdanacenter.org/products. To learn about Dana Center professional development sessions, see our
professional development site at www.utdanacenter.org/pd.
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
iii
Frontmatter
Statway—Full Version 1.0, July 2011
Acknowledgments
The original version of this work was created by The Charles A. Dana Center at The University of Texas at
Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching.
Carnegie Corporation of New York, The Bill & Melinda Gates Foundation, The William and Flora Hewlett
Foundation, Lumina Foundation, and The Kresge Foundation joined in partnership with the Carnegie
Foundation for the Advancement of Teaching in this work.
Leadership—Charles A. Dana Center at the University of Texas at Austin
Uri Treisman, director
Susan Hudson Hull, program director of mathematics national initiatives
Leadership—Carnegie Foundation for the Advancement of Teaching
Anthony S. Bryk, president
Bernadine Chuck Fong, senior managing partner
Louis Gomez, senior fellow
Paul LeMahieu, senior fellow
James Stigler, senior fellow
Uri Treisman, senior fellow
Guadalupe Valdés, senior fellow
Statway Project Leads
Kristen Bishop, former team lead for the New Mathways Project, the Charles A. Dana Center at the University
of Texas at Austin
Thomas J. Connolly, project lead, Statway, the Charles A. Dana Center at the University of Texas at Austin
Karon Klipple, director of Statway, the Carnegie Foundation for the Advancement of Teaching
Jane Muhich, director of Quantway, the Carnegie Foundation for the Advancement of Teaching
Project Staff—Charles A. Dana Center at the University of Texas at Austin
Richard Blount, advisor
Kathi Cook, project director, online services team
Jenna Cullinane, research associate
Steve Engler, lead editor and production editor
Amy Getz, team lead for the New Mathways Project
Susan Hudson Hull, program director of mathematics national initiatives
Joseph Hunt, graduate research assistant
Rachel Jenkins, consulting editor
Erica Moreno, program coordinator
Carol Robinson, administrative associate
Cathy Seeley, senior fellow
Rachele Seifert, administrative associate
Lilly Soto, senior administrative associate
Phil Swann, senior designer
Laura Torres, graduate research assistant
Thomas Wiegel, freelance formatter and proofreader
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
iv
Frontmatter
Statway—Full Version 1.0, July 2011
Authors Contracted by the Dana Center
Roxy Peck, professor emerita of statistics, California Polytechnic State University, San Luis Obispo, California
Beth Chance, professor of statistics, California Polytechnic State University, San Luis Obispo, California
Robert C. delMas, associate professor of educational psychology, University of Minnesota, Minneapolis,
Minnesota
Scott Guth, professor of mathematics, Mt. San Antonio College, Walnut, California
Rebekah Isaak, graduate research student, University of Minnesota, Minneapolis, Minnesota
Leah McGuire, assistant professor, University of Minnesota, Minneapolis, Minnesota
Jiyoon Park, graduate research student, University of Minnesota, Minneapolis, Minnesota
Brian Kotz, associate professor of mathematics, Montgomery College, Germantown, Maryland
Chris Olsen, assistant professor of mathematics and statistics, Grinnell College, Grinnell, Iowa
Mary Parker, professor of mathematics, Austin Community College, Austin, Texas
Michael A. Posner, associate professor of statistics, Villanova University, Villanova, Pennsylvania
Thomas H. Short, professor, John Carroll University, University Heights, Ohio
Penny Smeltzer, teacher of statistics, Westwood High School, Austin, Texas
Myra Snell, professor of mathematics, Los Medanos College, Pittsburg, California
Laura Ziegler, graduate research student, University of Minnesota, Minneapolis, Minnesota
Reviewers Contracted by the Dana Center
Michelle Brock, American River College, Sacramento, California
Thomas J. Connolly, the Charles A. Dana Center at the University of Texas at Austin
Andre Freeman, Capital Community College, Hartford, Connecticut
Karon Klipple, the Carnegie Foundation for the Advancement of Teaching
Roxy Peck, professor emerita of statistics, California Polytechnic State University, San Luis Obispo, California
Jim Smart, Tallahassee Community College, Tallahassee, Florida
Myra Snell, Los Medanos College, Pittsburg, California
Committee for Statistics Learning Outcomes
Rose Asera, formerly of the Carnegie Foundation for the Advancement of Teaching
Kristen Bishop, formerly of the Charles A. Dana Center at the University of Texas at Austin
Richelle (Rikki) Blair, American Mathematical Association of Two-Year Colleges (AMATYC); Lakeland
Community College, Ohio
David Bressoud, Mathematical Association of America (MAA); Macalester College, Minnesota
John Climent, American Mathematical Association of Two-Year Colleges (AMATYC); Cecil College,
Maryland
Peg Crider, Lone Star College, Tomball, Texas
Jenna Cullinane, the Charles A. Dana Center at the University of Texas at Austin
Robert C. delMas, Consortium for the Advancement of Undergraduate Statistics Education (CAUSE);
University of Minnesota, Minneapolis, Minnesota
Bernadine Chuck Fong, the Carnegie Foundation for the Advancement of Teaching
Karen Givvin, the University of California, Los Angeles
Larry Gray, American Mathematical Society (AMS); University of Minnesota
Susan Hudson Hull, the Charles A. Dana Center at the University of Texas at Austin
Rob Kimball, American Mathematical Association of Two-Year Colleges (AMATYC); Wake Technical
Community College, North Carolina
Dennis Pearl, Consortium for the Advancement of Undergraduate Statistics Education (CAUSE); The Ohio
State University
Roxy Peck, American Statistical Association (ASA); Consortium for the Advancement of Undergraduate
Statistics Education (CAUSE); California Polytechnic State University, San Luis Obispo, California
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
v
Frontmatter
Statway—Full Version 1.0, July 2011
Myra Snell, American Mathematical Association of Two-Year Colleges (AMATYC); Los Medanos College,
Pittsburg, California
Jim Stigler, the Carnegie Foundation for the Advancement of Teaching; the University of California, Los
Angeles
Daniel Teague, Mathematical Association of America (MAA); North Carolina School of Science and
Mathematics, Durham
Uri Treisman, the Carnegie Foundation for the Advancement of Teaching; the Charles A. Dana Center at the
University of Texas at Austin
Version 1.0 of Statway was developed in collaboration with faculty from the following colleges, the
“Collaboratory,” who advised on the development of the course. These Collaboratory colleges are:
Florida
Miami Dade College, Miami, Florida
Tallahassee Community College, Tallahassee,
Florida
Valencia Community College, Orlando,
Florida
California
American River College, Sacramento, California
Foothill College, Los Altos Hills, California
Mt. San Antonio College, Walnut, California
Pierce College, Woodland Hills, California
San Diego City College, San Diego, California
California State University System
Texas
CSU Northridge
Sacramento State University
San Jose State University
Austin Community College, Austin, Texas
El Paso Community College, El Paso, Texas
Houston Community College, Houston, Texas
Northwest Vista College, San Antonio, Texas
Richland College, Dallas, Texas
Connecticut
Washington
Capital Community College, Hartford, Connecticut
Gateway Community College, New Haven, Connecticut
Housatonic Community College, Bridgeport,
Connecticut
Naugatuck Valley Community College, Waterbury,
Connecticut
Seattle Central Community College, Seattle,
Washington
Tacoma Community College, Tacoma,
Washington
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
vi
Frontmatter
Statway—Full Version 1.0, July 2011
Statway, Full Version 1.0, July 2011
Table of Contents
Module 1: Statistical Studies and Overview of the Data Analysis Process
Lesson 1.1.1: The Statistical Analysis Process
Lesson 1.1.2: Types of Statistical Studies and Scope of Conclusions
Lesson 1.2.1: Collecting Data by Sampling
Lesson 1.2.2: Random Sampling
Lesson 1.2.3: Other Sampling Strategies
Lesson 1.2.4: Sources of Bias in Sampling
Lesson 1.3.1: Collecting Data by Conducting an Experiment
Lesson 1.3.2: Other Design Considerations—Blinding, Control Groups, and Placebos
Lesson 1.4.1: Drawing Conclusions from Statistical Studies
Module 2: Summarizing Data Graphically and Numerically
Lesson 2.1.1: Dotplots, Histograms, and Distributions for Quantitative Data
Lesson 2.1.2: Constructing Histograms for Quantitative Data
Lesson 2.1.3: Comparing Distributions of Quantitative Data in Two Independent Samples
Lesson 2.2.1: Quantifying the Center of a Distribution—Sample Mean and Sample Median
Lesson 2.2.2: Constructing Histograms for Quantitative Data
Lesson 2.3.1: Quantifying Variability Relative to the Median
Lesson 2.4.1: Quantifying Variability Relative to the Mean
Lesson 2.4.2: The Sample Variance
Module 3: Reasoning About Bivariate Numerical Data—Linear Relationships
Lesson 3.1.1: Introduction to Scatterplots and Bivariate Relationships
Lesson 3.1.2: Developing an Intuitive Sense of Form, Direction, and Strength of the Relationship
Between Two Measurements
Lesson 3.1.3: Introduction to the Correlation Coefficient and Its Properties
Lesson 3.1.4: Correlation Formula
Lesson 3.1.5: Correlation Is Not Causation
Lesson 3.2.1: Using Lines to Make Predictions
Lesson 3.2.2: Least Squares Regression Line as Line of Best Fit
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
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Statway—Full Version 1.0, July 2011
Lesson 3.2.3: Investigating the Meaning of Numbers in the Equation of a Line
Lesson 3.2.4: Special Properties of the Least Squares Regression Line
Lesson 3.3.1: Using Residuals to Determine If a Line Is a Good Fit
Lesson 3.3.2: Using Residuals to Determine If a Line Is an Appropriate Model
Module 4: Modeling Nonlinear Relationships
Lesson 4.1.1: Investigating Patterns in Data
Lesson 4.1.2: Exponential Models
Lesson 4.1.3: Assessing How Well a Model Fits the Data
Module 5: Reasoning About Bivariate Categorical Data and Introduction to Probability
Lesson 5.1.1: Reasoning About Risk and Chance
Lesson 5.1.2: Defining Risk
Lesson 5.1.3: Interpreting Risk
Lesson 5.1.4: Comparing Risks
Lesson 5.1.5: More on Conditional Risks
Module 6: Formalizing Probability and Probability Distributions
Lesson 6.1.1: Probability Lesson 6.1.2: Probability Rules
Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions
Lesson 6.2.1: Probability Distributions of Continuous Random Variables
Lesson 6.2.2: Z-Scores and Normal Distributions
Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values
Module 7: Linking Probability to Statistical Inference
Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability
Lesson 7.1.2: Sampling from a Population
Lesson 7.1.3: Testing Statistical Hypotheses
Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing
Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals
Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
viii
Frontmatter
Statway—Full Version 1.0, July 2011
Module 8: Inference for One Proportion
Lesson 8.1.1: Sampling Distribution of One Proportion
Lesson 8.1.2: Sampling Distribution of One Proportion
Lesson 8.2.1: Estimation of One Proportion
Lesson 8.2.2: Estimation of One Proportion
Lesson 8.3.1: Estimation of One Proportion
Lesson 8.3.2: Hypothesis Testing for One Proportion
Module 9: Inference for Two Proportions
Lesson 9.1.1: Sampling Distribution of Differences of Two Proportions
Lesson 9.1.2: Using Technology to Explore the Sampling Distribution of the Differences in Two
Proportions
Lesson 9.2.1: Confidence Intervals for the Difference in Two Population Proportions
Lesson 9.2.2: Computing and Interpreting Confidence Intervals for the Difference in Two
Population Proportions
Lesson 9.3.1: A Statistical Test for the Difference in Two Population Proportions
Lesson 9.3.2: A Statistical Test for the Difference in Two Population Proportions
Lesson 9.3.3: Conducting a Statistical Test for the Difference in Two Population Proportions
Module 10: Inference for Means
Lesson 10.1.1: The Sampling Distribution of the Sample Mean
Lesson 10.1.2: Using an Applet to Explore the Sampling Distribution of the Mean with Focus on
Shape
Lesson 10.2.1: Estimating a Population Mean
Lesson 10.2.2: T-Statistics and T-Distributions
Lesson 10.2.3: The Confidence Interval for a Population Mean
Lesson 10.3.1: Testing Hypotheses About a Population Mean
Lesson 10.3.2: Test Statistic and P-Values, One-Sample T-Test
Lesson 10.4.1: Inferences About the Difference Between Two Population Means
Lesson 10.4.2: Inference for Paired Data
Lesson 10.4.3: Two-Sample T-Test
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
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Frontmatter
Statway—Full Version 1.0, July 2011
Module 11: Chi-Squared Tests
Lesson 11.1.1: Introduction to Chi-Square Tests for One-Way Tables
Lesson 11.1.2: Executing the Chi-Square Test for One-Way Tables (Goodness-of-Fit)
Lesson 11.1.3: The Chi-Square Distribution and Degrees of Freedom
Lesson 11.2.1: Introduction to Chi-Square Tests for Two-Way Tables
Lesson 11.2.2: Executing the Chi-Square Test for Independence in Two-Way Tables
Lesson 11.2.3: Executing the Chi-Square Test for Homogeneity in Two-Way Tables
Module 12: Other Mathematical Content
Lesson 12.1.1: Statistical Linear Relationships and Mathematical Models of Linear Relationships
Lesson 12.1.2: Mathematical Linear Models
Lesson 12.1.3: Contrasting Mathematical and Statistical Linear Relationships
Lesson 12.1.4: Proportional Models
Lesson 12.2.1: Multiple Representations of Exponential Models
Lesson 12.2.2: Linear Models—Answering Various Types of Questions Algebraically
Lesson 12.2.3: Power Models
Lesson 12.2.4: Solving Inequalities
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under
sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of
Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie
Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways.
x
Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability Estimated number of 50-­‐minute class sessions: 1 Materials Required •
•
dice, coins, paper cups, or a combination of these sheets of graph paper (one per student or one per group) Learning Goals Students will begin to understand that •
•
the relative frequency of an outcome becomes more predictable as the number of trials increases. the probability of the outcome is the long-­‐run relative frequency of the outcome. Students will begin to be able to •
•
compute simple probabilities. interpret probabilities in context. Part I: Activity to Introduce Concepts This activity and the understanding that arises as a result prepare students for the Rich Task in Part II. Most students know the probability of getting heads when flipping a fair coin, but do they expect to always get 50% heads? Use something instead of a coin that has an unknown outcome for this task. The possibilities include the following: •
•
•
flipping a paper cup to find the probability of it landing on its side, rolling a loaded die, or drawing from a partial deck of cards (remove cards from a standard deck to alter the probability of a particular suit). Flip, roll, or draw whatever you have decided to use as an example. Motivating Question 1 How could you find the probability of this activity giving a certain outcome? Be specific to an outcome of the activity you have chosen. For instance, how could you find the probability of drawing a heart from this partial deck? For this question, guide class discussion in a way that students recognize that the number of trials is key to estimating the probability of an outcome. Motivating Question 2 How can you determine when you have conducted enough trials? For this question, guide class discussion in a way that prepares students for the next task. For example, you want students to estimate the probability several times, where the number of trials changes, such that they begin to see that a greater number of trials results in a more accurate estimate of the probability of an outcome. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability Task [Student Handout] (Note: Have students work in small groups of three or four to estimate the probability of a particular outcome for their activity. If students have difficulty, you may want to model graphing the results [see an example in the next section] or, better yet, circulate among the groups and give assistance when needed. For some students, it is helpful to point out that the numerator for each trial is the cumulative number of successes and that the denominator is the total number of trials, including the current trial.) (1) What is the definition of success for your particular activity? In other words, what outcome is defined as a success? For example, if you are flipping a coin, the definition of success can be either heads or tails, but not both. For rolling a die, success could be “rolling a 3” or “rolling a 6.” (2) Graph the relative frequency of successes for several trials. The result of each trial should be added to the previous trials’ results to determine the updated relative frequency of successes. (Note: Instructors must explicitly state that these fractions represent ratios of number of successes to number of trials.) (3) As a group, determine the best estimate for the probability of success for your activity. Explain your reasoning. The Law of Large Numbers—Teacher Example The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability Briefly describe how each trial was accomplished: A coin was flipped, and the probability of heads was determined. The previous graph displays the following: •
•
•
•
•
•
•
•
•
•
The first flip was heads. Probability = 1/1 = 1.0 The second flip was tails. Probability = 1/2 = 0.5 The third flip was heads. Probability = 2/3 = 0.67 The fourth flip was heads. Probability = 3/4 = 0.75 The fifth flip was tails. Probability = 3/5 = 0.6 The sixth flip was heads. Probability = 4/6 = 0.67 The seventh flip was tails. Probability = 4/7 = 0.57 The eighth flip was tails. Probability = 4/8 = 0.5 The ninth flip was tails. Probability = 4/9 = 0.44 The tenth flip was heads. Probability = 5/10 = 0.5 Have students note that the denominator and trial number are the same. The numerator is the cumulative number of successes. Students do not need to write the results of each trial as shown above. This only serves as an explanation of the example. Discussion Have each group share its results and graph with the class. Guide the class discussion in a way that students begin to recognize that the relative frequency converges to the theoretical probability as the number of trials increases. The terms converge and theoretical probability need not be used. Rather, students may come up with their own descriptions (e.g., settles, end value). This is an opportunity to discuss how a theoretical probability is found. Introduce the theoretical probability formula: P(event) = number of outcomes in event/number of total outcomes Wrap-­‐Up 1. Formally introduce the Law of Large Numbers: The Law of Large Numbers states: as the number of trials increases, the relative frequency of a specific outcome approaches the theoretical probability of a specific outcome if an infinite (or very, very large) number of trials were performed. Be sure that students understand the meaning of the term approaches in this definition. It does not mean that the relative frequency gets numerically closer to the theoretical probability. As their graphs likely indicate, the relative frequency may oscillate above and below the theoretical probability as it settles down. 2. Briefly discuss the importance of being able to estimate a probability using the Law of Large Numbers. The following are examples: •
The results of new medical procedures are unknown. However, more and more trials provide a better idea of the probability that the new procedure will help a patient. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability •
A new computer application may simplify work for some users. After more and more people are trained on the new computer application, you have a better idea of the probability that the new program is helpful and therefore worth the financial investment. 3. Emphasize that probability is long-­‐run relative frequency. Clearly explain the meaning of the terms long run, relative, and frequency. These terms are used in the discussion of statistics, but are often not understood by students. Part II: Rich Task [Student Handout] (Note: Describe the following scenario to students. Ask the entire class the questions that follow and have them discuss the responses with neighboring students. Then solicit responses and discuss as a class.) A new drug to lower blood pressure was developed and thoroughly tested by a pharmaceutical company. The drug’s clinical trials, which by law must involve a very large number of patients, show that it lowers blood pressure in 75% of patients. A local doctor piloting the drug claims it is a medical breakthrough, since 90% of her patients with high blood pressure who take the drug are later found to have normal blood pressure readings. (4) What might explain the difference of the drug’s success rate between the pharmaceutical company and the doctor? Explain your reasoning. (Answer: According to the Law of Large Numbers, the larger the sample size is, the closer you get to the true probability. Since the doctor is experiencing greater success, there are probably a smaller number of patients. There are many other variables that can affect blood pressure, but the discussion on this should be brief so students do not lose the main point.) (5) Since this doctor’s patients have been successful, is a higher number of failures among future patients likely? (Answer: No. The probability of success is the same for each patient. A discussion of the gambler’s fallacy may be useful here. Another way that may be useful in explaining this to students is the notion of a memoryless system. Using the coin-­‐flipping experiment as an example, explain to students that the coin does not remember—or has no knowledge of—what the previous outcomes of the coin flips were. Each time the coin is flipped, the probability of getting tails is reset to 50%. As far as the coin is concerned, each flip is a new experiment. Therefore, even if you get 10 tails in a row, the probability of getting tails on the eleventh flip is still 50%. Humans have a memory of the history of the flips, but the coin does not! The same reasoning can be applied to the drug trials. Because the drug being administered to a particular patient does not know what the results for previous patients were, its probability of success for each new patient is 75%. So, let’s say that the doctor tried the drug on 10 patients and 9 of them (90%) reported lower blood pressure—this does not mean that the next group of 10 patients must have a percentage of success much lower than 75% to “make up for” or “average out” the higher success rate in the first group. Again, humans remember what happened with each patient, but the drug does not.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability Homework [Student Handout] 1) A major software company is preparing to roll out yet another version of its operating system. The company believes it has come up with a system that is easier for customers to use after they have spent at least one hour with the new product. A marketing team is sent to several large companies throughout the United States to determine the proportion of potential customers who would find the new operating system better than the old operating system. Each team member loads the new system on a randomly selected employee’s computer, gives a brief explanation of the improvements, and allows the employee to use the system for two days. At the end of the two-­‐day period, the employee is asked if the new system is an improvement. The response (yes or no) is then entered into a database that keeps a running tally of the number of employees questioned and the total proportion of yes responses. The marketing team is unsure how many responses it needs to get a good idea of the true probability that a future customer will prefer the new product. The team plans to collect data until it has enough to make a decision. Clearly explain to the marketing team how it can determine when it has collected enough data to make a reasonable prediction of the probability a customer will prefer the new operating system. (Answer: Early in the trials, the total proportion of yes responses probably varies quite a bit. However, as the number of responses increases, the cumulative proportions settle down and remain very close. Once the cumulative proportion seems to remain basically the same for added trials, the marketing team has a reasonably good idea of the true probability that a customer will prefer the new operating system. Continuing after that point gives almost the same results but costs the company time and money.) 2) Charles has consistently made only 60% of his free throws over the last three years of playing professional basketball. Luckily, he is a big man who primarily makes layups and grabs extraordinary rebounds. Over the summer, Charles changed his shooting method and claims his new method greatly improves his overall free throw percentage. Before practice, he shot 40 free throws. Three of the coaches stopped by at different times to watch. Each coach saw at least five consecutive free throws. •
•
•
Coach 1 claimed the new shot was not working and should be abandoned. Coach 2 claimed the new shot worked about the same as the old shot. Coach 3 claimed the new shot was a great improvement. Charles’s free throw outcomes are listed below in order from left to right (B = basket made; N = basket not made). Shot Number 1 2 3 4 5 6 7 8 9 10 Results of Shot B B B N N B B N B B Shot Number 11 12 13 14 15 16 17 18 19 20 Results of Shot B B N B N N B N B B The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability Shot Number 21 22 23 24 25 26 27 28 29 30 Results of Shot B B N B B B N B B B Shot Number 31 32 33 34 35 36 37 38 39 40 Results of Shot N N B B B N B N B B (Note: The goal here is to have students practice finding simple probabilities, look at short-­‐run relative frequencies versus long-­‐run relative frequencies, and make a decision based on the long run. Answers will vary.) (a) List the possible shots each coach may have watched Charles attempt. (Answer: Possible answers include the following: • Coach 1: Shots 12–16 or 4–6. • Coach 2: Shots 10–14 or 31–35. • Coach 3: Shots 8–12 or 24–28.) (b) Based on the data, do you think the new shooting method has improved Charles’s free throw shooting ability? Explain your reasoning. (Answer: Charles’s new method is improving his free throw percentage. Using all 40 shots, his overall percentage is 67.5%, which is better than his previous 60% shooting record.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability
Part I (1) What is the definition of success for your particular activity? In other words, what outcome is defined as a success? For example, if you are flipping a coin, the definition of success can be either heads or tails, but not both. For rolling a die, success could be “rolling a 3” or “rolling a 6.” (2) Graph the relative frequency of successes for several trials. The result of each trial should be added to the previous trials’ results to determine the updated relative frequency of successes. (3) As a group, determine the best estimate for the probability of success for your activity. Explain your reasoning. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability
Part II A new drug to lower blood pressure was developed and thoroughly tested by a pharmaceutical company. The drug’s clinical trials, which by law must involve a very large number of patients, show that it lowers blood pressure in 75% of patients. A local doctor piloting the drug claims it is a medical breakthrough, since 90% of her patients with high blood pressure who take the drug are later found to have normal blood pressure readings. (4) What might explain the difference of the drug’s success rate between the pharmaceutical company and the doctor? Explain your reasoning. (5) Since this doctor’s patients have been successful, is a higher number of failures among future patients likely? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability
Homework 1) A major software company is preparing to roll out yet another version of its operating system. The company believes it has come up with a system that is easier for customers to use after they have spent at least one hour with the new product. A marketing team is sent to several large companies throughout the United States to determine the proportion of potential customers who would find the new operating system better than the old operating system. Each team member loads the new system on a randomly selected employee’s computer, gives a brief explanation of the improvements, and allows the employee to use the system for two days. At the end of the two-­‐day period, the employee is asked if the new system is an improvement. The response (yes or no) is then entered into a database that keeps a running tally of the number of employees questioned and the total proportion of yes responses. The marketing team is unsure how many responses it needs to get a good idea of the true probability that a future customer will prefer the new product. The team plans to collect data until it has enough to make a decision. Clearly explain to the marketing team how it can determine when it has collected enough data to make a reasonable prediction of the probability a customer will prefer the new operating system. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.1.1: Probability
2) Charles has consistently made only 60% of his free throws over the last three years of playing professional basketball. Luckily, he is a big man who primarily makes layups and grabs extraordinary rebounds. Over the summer, Charles changed his shooting method and claims his new method greatly improves his overall free throw percentage. Before practice, he shot 40 free throws. Three of the coaches stopped by at different times to watch. Each coach saw at least five consecutive free throws. •
•
•
Coach 1 claimed the new shot was not working and should be abandoned. Coach 2 claimed the new shot worked about the same as the old shot. Coach 3 claimed the new shot was a great improvement. Charles’s free throw outcomes are listed below in order from left to right (B = basket made; N = basket not made). Shot Number 1 2 3 4 5 6 7 8 9 10 Results of Shot B B B N N B B N B B Shot Number 11 12 13 14 15 16 17 18 19 20 Results of Shot B B N B N N B N B B Shot Number 21 22 23 24 25 26 27 28 29 30 Results of Shot B B N B B B N B B B Shot Number 31 32 33 34 35 36 37 38 39 40 Results of Shot N N B B B N B N B B (a) List the possible shots each coach may have watched Charles attempt. (b) Based on the data, do you think the new shooting method has improved Charles’s free throw shooting ability? Explain your reasoning. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
Estimated number of 50-­‐minute class sessions: 1 Materials Required One bag (zipper-­‐lock bags are suitable) of 40 buttons is needed for each group of three to four students. Buttons must be in the quantities listed in the table below. Buttons of roughly the same size and style are preferable to ensure an approximately equal chance of selecting any button. White Black Red Two holes 8 9 3 Four holes 8 5 7 Learning Goals Students will understand •
•
the meaning of mutually exclusive events and independent events. basic probability rules and using addition, multiplication, and complements (including complements in cases requiring at least one). Students will be able to calculate basic probabilities, including conditional probabilities using tables and trees. Note: This lesson does not use formal notation for intersection and union or to state probability rules. Task Introduction [Student Handout] Now that you have a basic understanding of probability as the chance of success in the long run, you can work on more complex problems. To help you visualize the outcomes, you will use bags of buttons with different characteristics. Keep in mind that the buttons could represent similar classifications; for example, a shipment of new file cabinets that have two drawers or four drawers and have dents and scratches or no dents and scratches. (Note: Give each student a worksheet. Have students work in groups of three or four. Each group needs a bag of buttons. Since probability is not intuitive for many students, the goal is to provide students an opportunity to actually see the desired outcomes. Although moving buttons around may appear elementary, research confirms that physically doing an activity promotes long-­‐term memory better than simply working in the abstract.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
Buttoning Up Probability [Student Handout] (1) First, organize the data in the table. Carefully remove buttons from the bag and separate them into the listed categories. White Black Red Total Two holes 8 9 3 20 Four holes 8 5 7 20 Total 16 14 10 40 (Note: Answers are italicized in the table and do not appear in the Student Handout. Walk around and check the results of each group to ensure students filled the table in correctly before they get too far along with the problems.) (a) Is it possible to make two separate groups of buttons, one containing all the buttons with four holes and the other with all the red buttons? If no, why not? (Answer: No, some buttons are red and have four holes.) (b) Is it possible to make two separate groups of buttons, one containing all the red buttons and the other with all the white buttons? If no, why not? (Answer: Yes) (c) If you can make two separate groups, the two characteristics are mutually exclusive. Which of the following are mutually exclusive? Two holes and four holes Black and four holes (Answer: Two holes and four holes) (2) Sort the buttons as directed and perform the required calculation. (a) Separate out the red buttons. Find the probability that a randomly selected button is red. [Answer: P(red) = !"
= 0.25] #"
(b) Separate out the four-­‐holed buttons. Find the probability that a randomly selected button has four holes. [Answer: P(four holes) = !"
= 0.5] #"
(c) Separate out the buttons that are red or have four holes. Find the probability that a randomly selected button is red or has four holes. Note that some buttons have both characteristics, but they should only be counted once. [Answer: P(red or four holes) = !"
= 0.575] #$
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
(d) Find the probability that a randomly selected button is red and has four holes. [Answer: P(red and four holes) = !
= 0.175] "#
(e) Briefly explain why the probabilities in Questions 2c and 2d are different. (Answer: In Question 2c, students counted all buttons that are either red or had four holes. In Question 2d, they could only count buttons that have both characteristics: red and four holes.) (3) Sort the buttons as directed and perform the required calculation. (a) Separate out the white buttons. Find the probability that a randomly selected button is white. [Answer: P(white) = !"
= 0.4] #$
(b) Separate out the buttons with four holes. Find the probability that a randomly selected button is white, given that it has four holes. You may think “using the four-­‐holed buttons as the total, how many the four-­‐holed buttons are white.” Definition This is called a conditional probability. You want to find the probability that a button is white. However, there is an extra condition—the button must have four holes. So all other buttons—the ones with two holes—are irrelevant. You only choose from the group with four holes. That is the group you are given to choose from. The number of four-­‐holed buttons becomes the new total and is the denominator of the fraction. The probability can be written as P(white given four holes) to simplify your writing. [Answer: P(white given four holes) = !
= 0.4] "#
(c) Compare the two probabilities in Questions 3a and 3b. (Answer: They are the same.) (d) Separate out the buttons with four holes. Find the probability that a randomly selected button has four holes. [Answer: P(four holes) = !"
= 0.5] #"
(e) Find the probability that a button has four holes given it is white. [Answer: P(four holes given white) = !
= 0.5] "#
(f) Compare the two probabilities in Questions 3d and 3e. (Answer: They are the same.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
Definition When the probability of something happening is the same whether you are given another characteristic, the two characteristics or events are said to be independent. It does not matter which characteristic you use as the given characteristic. If two characteristics are independent, the probability of one characteristic alone will be equal to that same characteristic given the other one. So knowing that a button is white does not change the probability that it has four holes. (g) Are two holes and black independent events? Back up your answer with the appropriate probabilities. !"
= 0.5 #"
!
P(two holes given black) = = 0.643 "#
[Answers: P(two holes) = Since the two probabilities are not equal, having two holes and being black are not independent.] Questions 4 and 5 will help you understand why independence is such a big deal. (4) Sort the buttons as directed and perform the required calculation. (a) Separate out all the buttons that are both white and have four holes. Find the probability that a randomly selected button comes from this group when drawn from the entire group. [Answer: P(white and four holes) = !
= 0.2] "#
(b) Multiply the probability a button is white by the probability a button has four holes. Note: You have already found these probabilities in Question 3. [Answer: P(white) × P(four holes) = 0.4 × 0.5 = 0.2] (c) Compare your answers in Questions 4a and 4b. (Answer: They are the same.) (5) Sort the buttons as directed and perform the required calculation. (a) Separate out all the buttons that are both black and have two holes. Find the probability that a randomly selected button comes from this group when drawn from the entire group. [Answer: P(black and two holes) = !
= 0.225] "#
(b) Multiply the probability that a button is black with the probability that a button has two holes. [Answer: P(black) × P(two holes) = !"
!"
× = 0.35 × 0.5 = 0.175] "#
#"
(c) Compare the two probabilities in Questions 5a and 5b. (Answer: They are different.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
Independence—The Big Deal If events are independent, the probability of both events occurring is equal to multiplying the individual probabilities. Example: What is the probability of rolling a fair die three times and getting three sixes? Since each roll is independent, the probabilities remain the same. $
!
!
! " !%
P(3 sixes) = # ! # # ! # # = $ ' = #%&%%'"$ "
"
" # "&
Example: What is the probability that an adult American male is at least 6 feet tall and has an intelligence quotient (IQ) of at least 120? Since height and IQ are independent events, you can simply multiply the two individual probabilities: P(at least 6 feet tall) = 0.145 P(IQ at least 120) = 0.089 P(at least 6 feet tall and IQ at least 120) = 0.145 × 0.089 = 0.0129 If events are not independent, you cannot simply multiply the individual probabilities. You must consider the additional information given and how that affects the probability of the additional events. Example: What is the probability a three-­‐person subcommittee contains all females if the members are randomly selected from a group with five males and five females? Since the probability of choosing a female changes with each additional selection, the events are not independent. For the selection of the first person, there are 10 people in the group: five females and five males, so the probability of selecting a female is !"#$%&'$ #()*+ *$'$,+(-./0 = 0
1
23
For the selection of the second person, there are nine people left in the group: four females and five males, so the probability of selecting a female is !"#$%&'$ ($)*+, ($'$)-.*+/0 = 0
1
2
For the selection of the third person, there are eight people left in the group: three females and five males, so the probability of selecting a female is 3
!"#$%&'$ ()*+, -$'$.(*/012 = 2 4
So, the probability of selecting an all-­‐female committee is equal to the product of the individual probabilities above: !"#$%&& '&()*&+,- = -
.
1
3
- ! - - ! - - = 0504333 /0
2
4
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
With and Without Replacement (6) Sort the buttons as directed and perform the required calculation. (a) Place all the buttons in the bag and find the probability that you get a randomly selected button that is red on two consecutive draws, if you replace the first button drawn before drawing the second button. [Answer: P(two reds) = P(red) × P(red) = !"
!"
× = 0.25 × 0.25 = 0.0625] #"
#"
(b) With all the buttons in the bag, consider randomly drawing two buttons that are red without replacing the first draw. [Answers: P(red for first draw) = P(red for second draw) = P(two reds) = !"
= 0.25 #"
!
= 0.231 "!
!"
!
× = 0.25 × 0.231 = 0.05775] #"
"!
(c) Find the probability of drawing three red buttons with and without replacement. Note that replacement means after the third draw you are still holding only one red button. Without replacement, you have a group of three red buttons together. Whenever you are choosing a group, it is understood there is no replacement. [Answers: With replacement—P(three reds) = 0.253 = 0.0156 Without replacement—P(three reds) = !"
!
!
× × = 0.0121] #"
"!
"!
Complements (7) Find the two probabilities below. Then add the two probabilities. P(red) = P(not red) = [Answer: P(red) = P(not red) = !"
= 0.25 #"
!"
= 0.75 #"
0.25 + 0.75 = 1] Since the two probabilities make up the entire set of buttons, you can find either probability by subtracting the other from 1. Red and not red are complements of each other. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
Wrap-­‐Up Ask the class as a whole the following questions. Elaborate on any questions they have difficulty answering. •
•
•
•
•
•
What does it mean for two events to be mutually exclusive? (Answer: The two events have no common outcomes.) What are two mutually exclusive events using button data that you did not use in Question 1? (One possible response: black and white) If two events are not mutually exclusive, how do you find the probability of getting one or the other? (Answer: Or means add, but take into consideration buttons that have both characteristics so you do not accidentally count them twice.) What does it mean for two events to be independent? (Answer: Knowing one event does not change the probability of the next event. For instance, knowing a button is white does not help you determine if it has four holes. However, knowing a button is red helps you make a better guess whether it has four holes.) How does replacing or not replacing a button after it is drawn affect the probability of the next draw? (Answer: If you replace the button, the probability for the next draw remains the same. If you do not replace the button, the sample space changes and so does the probability.) If the chance of rain is 40%, what is the chance of no rain? (Answer: 60%) What rule did you use to answer? (Answer: complement) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules
Homework [Student Handout] (1) A salesman for energy-­‐efficient radiant barrier attic insulation believes a homeowner with a newer roof is more likely to purchase his product. Looking over past records of salesmen soliciting businesses by ringing doorbells and discussing their products, he sorts the sales records into three groups: customers with a 30-­‐year roof in good condition, those with a 20-­‐year roof in good condition, and those in need of a new roof. His results are in the following table. 30-­‐year Roof 20-­‐year Roof In Need of a New Roof Total Purchased insulation 8 5 4 17 Did not make a purchase 42 77 94 213 Total 50 82 98 230 Does knowing the condition of a prospective customer’s roof help the salesman determine who is a more likely candidate for an insulation sale? What would you advise a door-­‐to-­‐door salesman of the radiant barrier insulation? Why? Justify your answer using appropriate probabilities and the concept of independence. [One possible answer: P(purchases insulation) = P(purchases insulation given 20-­‐year roof) = !"
= 0.0739 #$%
!
= 0.0610 "#
!
P(purchases insulation given 30-­‐year roof) = = 0.16 "#
Purchasing radiant barrier insulation and roof condition are not independent events. The salesman should concentrate on homeowners with a 30-­‐year roof, since they are more likely to make a purchase.] (2) Adult rattlesnakes tend to hide rather than bite humans. Silly creatures that they are, humans often go in search of rattlesnakes. Bite records seem to show that rattlesnakes are aware that humans are too big to eat. Medical records of rattlesnake bites show that about 30% of the bites contain no venom. These are known as dry bites. Note: Consider rattlesnake bites independent events. (a) What is the probability that two rattlesnake victims get dry bites? [Answer: P(two dry bites) = 0.32 = 0.09] (b) If you are bitten by a rattlesnake, what is the probability the bite contains venom? [Answer: P(venom) = 1 – 0.3 = 0.7] (c) If an emergency center in West Texas sees six patients with rattlesnake bites during the fall rattlesnake round-­‐up, what is the probability the first five bites contain venom and the last one do not contain venom? [Answer: P(five venom bites and one dry bite) = 0.75 × 0.3 = 0.0504] The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules Now that you have a basic understanding of probability as the chance of success in the long run, you can work on more complex problems. To help you visualize the outcomes, you will use bags of buttons with different characteristics. Keep in mind that the buttons could represent similar classifications; for example, a shipment of new file cabinets that have two drawers or four drawers and have dents and scratches or no dents and scratches. Buttoning Up Probability (1) First, organize the data in the table. Carefully remove buttons from the bag and separate them into the listed categories. White Black Red Total Two holes Four holes Total (a) Is it possible to make two separate groups of buttons, one containing all the buttons with four holes and the other with all the red buttons? If no, why not? (b) Is it possible to make two separate groups of buttons, one containing all the red buttons and the other with all the white buttons? If no, why not? (c) If you can make two separate groups, the two characteristics are mutually exclusive. Which of the following are mutually exclusive? Two holes and four holes Black and four holes The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules (2) Sort the buttons as directed and perform the required calculation. (a) Separate out the red buttons. Find the probability that a randomly selected button is red. (b) Separate out the four-­‐holed buttons. Find the probability that a randomly selected button has four holes. (c) Separate out the buttons that are red or have four holes. Find the probability that a randomly selected button is red or has four holes. Note that some buttons have both characteristics, but they should only be counted once. (d) Find the probability that a randomly selected button is red and has four holes. (e) Briefly explain why the probabilities in Questions 2c and 2d are different. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules (3) Sort the buttons as directed and perform the required calculation. (a) Separate out the white buttons. Find the probability that a randomly selected button is white. (b) Separate out the buttons with four holes. Find the probability that a randomly selected button is white, given that it has four holes. You may think “using the four-­‐holed buttons as the total, how many the four-­‐holed buttons are white.” Definition This is called a conditional probability. You want to find the probability that a button is white. However, there is an extra condition—the button must have four holes. So all other buttons—
the ones with two holes—are irrelevant. You only choose from the group with four holes. That is the group you are given to choose from. The number of four-­‐holed buttons becomes the new total and is the denominator of the fraction. The probability can be written as P(white given four holes) to simplify your writing. (c) Compare the two probabilities in Questions 3a and 3b. (d) Separate out the buttons with four holes. Find the probability that a randomly selected button has four holes. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules (e) Find the probability that a button has four holes given it is white. (f) Compare the two probabilities in Questions 3d and 3e. Definition When the probability of something happening is the same whether you are given another characteristic, the two characteristics or events are said to be independent. It does not matter which characteristic you use as the given characteristic. If two characteristics are independent, the probability of one characteristic alone will be equal to that same characteristic given the other one. So knowing that a button is white does not change the probability that it has four holes. (g) Are two holes and black independent events? Back up your answer with the appropriate probabilities. Questions 4 and 5 will help you understand why independence is such a big deal. (4) Sort the buttons as directed and perform the required calculation. (a) Separate out all the buttons that are both white and have four holes. Find the probability that a randomly selected button comes from this group when drawn from the entire group. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules (b) Multiply the probability a button is white by the probability a button has four holes. Note: You have already found these probabilities in Question 3. (c) Compare your answers in Questions 4a and 4b. (5) Sort the buttons as directed and perform the required calculation. (a) Separate out all the buttons that are both black and have two holes. Find the probability that a randomly selected button comes from this group when drawn from the entire group. (b) Multiply the probability that a button is black with the probability that a button has two holes. (c) Compare the two probabilities in Questions 5a and 5b. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules Independence—The Big Deal If events are independent, the probability of both events occurring is equal to multiplying the individual probabilities. Example: What is the probability of rolling a fair die three times and getting three sixes? Since each roll is independent, the probabilities remain the same. !
!
!
" !%
$
P(3 sixes) = # ! # # ! # # = $ ' = #%&%%'"$ "
"
" # "&
Example: What is the probability that an adult American male is at least 6 feet tall and has an intelligence quotient (IQ) of at least 120? Since height and IQ are independent events, you can simply multiply the two individual probabilities: P(at least 6 feet tall) = 0.145 P(IQ at least 120) = 0.089 P(at least 6 feet tall and IQ at least 120) = 0.145 × 0.089 = 0.0129 If events are not independent, you cannot simply multiply the individual probabilities. You must consider the additional information given and how that affects the probability of the additional events. Example: What is the probability a three-­‐person subcommittee contains all females if the members are randomly selected from a group with five males and five females? Since the probability of choosing a female changes with each additional selection, the events are not independent. For the selection of the first person, there are 10 people in the group: five females and five males, so the probability of selecting a female is !"#$%&'$ #()*+ *$'$,+(-./0 = 0
1
23
For the selection of the second person, there are nine people left in the group: four females and five males, so the probability of selecting a female is !"#$%&'$ ($)*+, ($'$)-.*+/0 = 0
1
2
For the selection of the third person, there are eight people left in the group: three females and five males, so the probability of selecting a female is 3
!"#$%&'$ ()*+, -$'$.(*/012 = 2 4
So, the probability of selecting an all-­‐female committee is equal to the product of the individual probabilities above: !"#$%&& '&()*&+,- = -
.
1
3
- ! - - ! - - = 0504333 /0
2
4
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules With and Without Replacement (6) Sort the buttons as directed and perform the required calculation. (a) Place all the buttons in the bag and find the probability that you get a randomly selected button that is red on two consecutive draws, if you replace the first button drawn before drawing the second button. (b) With all the buttons in the bag, consider randomly drawing two buttons that are red without replacing the first draw. (c) Find the probability of drawing three red buttons with and without replacement. Note that replacement means after the third draw you are still holding only one red button. Without replacement, you have a group of three red buttons together. Whenever you are choosing a group, it is understood there is no replacement. Complements (7) Find the two probabilities below. Then add the two probabilities. P(red) = P(not red) = Since the two probabilities make up the entire set of buttons, you can find either probability by subtracting the other from 1. Red and not red are complements of each other. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules Homework (1) A salesman for energy-­‐efficient radiant barrier attic insulation believes a homeowner with a newer roof is more likely to purchase his product. Looking over past records of salesmen soliciting businesses by ringing doorbells and discussing their products, he sorts the sales records into three groups: customers with a 30-­‐year roof in good condition, those with a 20-­‐year roof in good condition, and those in need of a new roof. His results are in the following table. 30-­‐year Roof 20-­‐year Roof In Need of a New Roof Total Purchased insulation 8 5 4 17 Did not make a purchase 42 77 94 213 Total 50 82 98 230 Does knowing the condition of a prospective customer’s roof help the salesman determine who is a more likely candidate for an insulation sale? What would you advise a door-­‐to-­‐door salesman of the radiant barrier insulation? Why? Justify your answer using appropriate probabilities and the concept of independence. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.2: Probability Rules (2) Adult rattlesnakes tend to hide rather than bite humans. Silly creatures that they are, humans often go in search of rattlesnakes. Bite records seem to show that rattlesnakes are aware that humans are too big to eat. Medical records of rattlesnake bites show that about 30% of the bites contain no venom. These are known as dry bites. Note: Consider rattlesnake bites independent events. (a) What is the probability that two rattlesnake victims get dry bites? (b) If you are bitten by a rattlesnake, what is the probability the bite contains venom? (c) If an emergency center in West Texas sees six patients with rattlesnake bites during the fall rattlesnake round-­‐up, what is the probability the first five bites contain venom and the last one do not contain venom? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 9 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Estimated number of 50-­‐minute class sessions: 2 Materials Required •
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Dice Decks of cards Learning Goals Students will understand •
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•
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that simulation is a tool for estimating probability. the concept of probability as long-­‐term relative frequency as demonstrated by simulation.
the properties of a probability distribution. the mean as the expected value of a probability distribution. Students will be able to •
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calculate the mean and standard deviation of a given discrete probability distribution. construct a probability distribution given a basic scenario. interpret the expected value of a discrete probability distribution in context and use the expected value to make a decision. design and complete simple simulations. interpret simulation results in context. •
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Day 1 Introduction [Student Handout] Many probability problems are more difficult mathematically than you are prepared to study directly. There is another option: You can simulate the event many times to give an idea of what would happen and then make a decision based on the simulations. What is a simulation? A simulation is a representation of an event with possible outcomes comparable to the actual event. When the Air Force trains a new pilot, it does not give the new recruit a billion-­‐dollar jet and say, “Let’s see if you can get this baby off the ground.” The new pilot is first trained using a machine that simulates flying. In probability, you use random events to simulate events in order to determine the probability of a particular event occurring by chance alone. Example The manager of a local television newsroom has three top field reporters he relies on to assemble a team for breaking news stories. The trouble is all three spend most of their time out of the office researching background information for ongoing news. He would like to be reasonably sure that at least one of them will be in the newsroom at any given time. Otherwise, another field reporter should be hired. Abigail is in the newsroom 40% of the time, Benjamin 60% of the time, and Christopher 10% of the time. Does it appear the manager should hire a new field reporter? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions The first step is designing your simulation. You will use the random number table at the back of your handout to simulate the reporters being in the office. (Note: Briefly explain a random number table. Ideally, if you put 10 slips of paper with the digits 0 through 9 in a bag, draw one out, write it down, replace it in the bag, and repeat this many times, you have a random number table. In the old days, students use the last four digits of phone numbers selected at random from a phone book.) Assign digits to represent each person being in the office according to his or her respective probabilities. •
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Abigail in the office = 1, 2, 3, 4. (Since she is in the office 40% of the time, choose four digits. You could choose any four digits, such as 1, 2, 3, 4 or 2, 4, 6, 8.) Benjamin in the office = 1, 2, 3, 4, 5, 6. (Since he is in the office 60% of the time, choose six digits. You could choose any six digits, such as 1, 2, 3, 4, 5, 6 or 1, 2, 4, 5, 7, 8.) Christopher in the office = 1. (Since he is in the office 10% of the time, choose any one digit.) To conduct one trial, choose three digits and check if at least one of the reporters is in the office. Let the first digit represent Abigail, the second digit Benjamin, and the third digit Christopher. Record the results. Let’s begin with Line 6. (You could choose any line.) •
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The first three digits are 2, 3, 5—Abigail and Benjamin are in the office. The next three digits are 5, 6, 8—Benjamin is in the office. The next three digits are 8, 5, 3—Benjamin is in the office. It looks like the manager does not need to hire anyone. Have you done enough trials? (Note: Possibly have a brief discussion leading back to the Law of Large Numbers.) •
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The next three digits are 8, 7, 0—none of the reporters are in the office. The next three digits are 9, 5, 1—Christopher is in the office. The next three digits are 6, 7, 2—none of the reporters are in the office. Keep track using a table or frequency distribution. Continue through the next row or further as time allows. Event Frequency Total At least one reporter in the newsroom No reporters in the newsroom Total The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Here are the results for 25 trials: Event Frequency At least one reporter in the newsroom No reporters in the newsroom Total Total 19 25 6 25 Do you think the manager should hire another reporter? (Answer: Yes. 6/25, or 24% of the time, all three field reporters are out of the newsroom. In 50 trials, the probability drops to 20%. The true probability is 0.216.) (Note: There are other options to create probability simulations. Students could use a deck of cards. A group of 10 cards with four red and six black could represent Abigail. Six red cards and four black cards could represent Benjamin, and one red card and nine black cards could represent Christopher. One card could be randomly drawn from each reporter’s pile and the results recorded. For other simulations, coins or dice may be useful.) Rich Task/Activity [Student Handout] (Note: Have students work in groups of four to complete the following simulation problems.) (1) Suppose the Kalama Zoo houses nine orangutans: five males and four females. A trainer is assigned to randomly select three of them as the experimental group for a new supplement to their diet. This requires the trainer to move the chosen three to a newly constructed habitat. When her boss returns the next day, he finds three female orangutans in the new habitat. He is furious and threatens to fire the trainer. Why? It turns out that the males sometimes attempt to bite, and by avoiding them, the trainer may not have learned to take care of them properly. The boss claims the trainer did not truly randomly select the three orangutans for the new supplement, but rather chose three females that were easier to handle. The trainer claims that she really did randomly select three females. Should the trainer be fired? Design a simulation to assist you in answering the question. (Answer: The probability of randomly selecting three females from the group of nine is 0.0476. Note: If using a random number table, not all digits must be assigned. For instance, 1, 2, 3, 4, 5 could represent the males, 6, 7, 8, 9 could represent the females, and 0 could be ignored or skipped.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions (2) To increase sales, a cereal company places a small bag of toys in each box of cereal. There are four different toys available in randomly selected boxes. They are distributed as follows: •
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40% cars 40% boats 20% rocket ships 20% transforming figures The sales team is interested in knowing how many boxes of cereal a customer must buy to receive the complete set of the four toys. Use simulation to estimate the answer to this question. Wrap-­‐Up When groups have completed their work, randomly choose groups to present their simulation designs and results. Comment on any unique designs, errors in design, or unusual results. Finish the discussion with a brief discussion or comments about the uncertainty of making a decision based on simulation or probability in general. Students never really know if they are correct. The trainer could have randomly selected three female orangutans, but that is unlikely. Some unlucky parent may need to buy 40 boxes of cereal to obtain the complete set of toys, but that also would be unlikely. Homework for Day 1 [Student Handout] Design and carry out a simulation to answer the following questions. (1) A couple has high hopes of having a “gentleman’s family.” This usually means it would have two children, the first being a son and second a daughter. Assuming the probabilities of having a boy or girl are equally likely for each birth, how likely is it the couple will end up with its dream family? (2) A computer game awards a special shield to the player 20% of the time after the player completes a level of the game. David plans to begin studying for a test once he obtains the special shield. It takes him roughly 15 minutes to complete a level. Estimate how long you believe it will be before David begins his studying. (3) A very large company must make budget cuts to remain in business. The chief executive officer plans to form a committee including five randomly selected employees. 50% of the employees are fiscally conservative, 30% liberal, and 20% moderate. The hope is that at least one of each group (conservative, liberal, moderate) is chosen for the committee so employees feel represented and no one group has the majority. The names of all employees are printed on labels and five are drawn. Does this appear to be a suitable method to achieve a committee the employees will deem fair? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Day 2 Discuss the homework from Day 1. This lesson extends probability distributions from counts to potential values. As students complete the following problems in groups of three to four, walk around and check if the dollar amounts are used as the random variable. Lead struggling groups in the right direction. Students may have difficulty discovering the negative value of paying $10,000 in the first problem. Rich Task [Student Handout] (1) According to the Social Security Administration Period Life Table, the probability a 70-­‐year-­‐old male will die in the next year is about 0.018. The AARP website lists rates for a $10,000 life insurance policy for a 70-­‐year-­‐old male at roughly $540 per year. Make a probability distribution to reflect the possible monetary outcomes and determine the expected profit for the life insurance company. Explain the expected profit means. (Answer: Money made by company $540 –$10,000 Probability that this amount will be made 0.982 0.018 expected value = mean = $350.28 The company will not make $350.28 for any 70-­‐year-­‐old male that lives in the next year. The company will either make $540 or lose $10,000. However, in a large group of insured 70-­‐year-­‐
old males, the company will make $350.28 on average per customer. The goal of the company is to insure a large group of males to make this a profitable venture.) (2) You are about to purchase a new four-­‐door sedan and are encouraged to add a mere $20 per month to your car payment to cover the costs of oil changes, routine maintenance, and possible major repairs. The coverage will last the length of your five-­‐year loan. You plan to get an oil change every 5,000 miles, which you estimate will cost an average of $30 each time. On average, you drive 12,000 miles per year. The potential cost of routine maintenance and possible major repairs in the first five years of ownership is shown in the probability distribution below. Oil changes are not included in the cost. Cost over 5 years Probability of cost incurred 0 $500 $1,000 $1,500 $2,000 0.19 0.67 0.09 0.04 0.01 Is the $20 per month fee a good idea for you? Support your answer. (Answer: The cost of oil changes for 5 years: 12,000 miles × 5 years = 60,000 miles 60,000 miles/5,000 miles = 12 oil changes at $30 = $360 The expected cost of routine maintenance and major repairs = $424 Total expected cost = $424 + $360 = $784 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Cost of plan = $20 x 60 months = $1,200 It is probably a better idea to turn down the offer and pay for maintenance, repairs, and oil changes as they come up.) Wrap-­‐Up Discuss group solutions to the problems Ask the following questions: •
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What percentage of the buyers would benefit from the $20 per month plan in Question 2? (Answer: Only 14%. Remember to add the cost of oil changes.) What other applications students can think of that may have similar plan options to consider? (Answers: Health insurance, phone plans, movie rental plans, internet and cable television plans, electronic warranties such as televisions and computers) Note for possible discussion: Most plans have conditions that require customers to complete certain tasks or maintenance during a particular time frame. Since a majority of customers do not complete the required conditions, the contract is broken and the company is not bound to pay for the repairs. Note: The auto data provided is consistent with several articles over the last two years in Consumer Reports. Homework for Day 2 [Student Handout] (1) You set up a carnival booth at a local fund-­‐raising event. The game consists of a giant six-­‐sided fair die and costs participants $3 per roll. If they roll a 6, they win a prize that costs $7. If participants roll a 1, they win a prize that costs $2.50. Otherwise, they do not win anything. If you average 35 participants per hour for the 8-­‐hour event, how much money do you expect to raise during the event? (Answer: Roll 6 1 2, 3, 4,5 Earnings –$4 $0.50 $3 Probability 1/6 1/6 4/6 The expected value or mean = $1.25 35 participants × 8 hours = 280 280 × $1.25 = $350) (2) After having your family visit for the holidays, you are shocked by your movie-­‐rental bill for the month. Currently you are paying $2.99 for each movie you rent. A subscription will allow you to watch unlimited movies for only $7.99 per month. However, during a normal month there is often not time available to sit and watch movies and you do not really want to needlessly increase your monthly bills. Since you are a statistics student, you decide to check your online billing statements and make a probability distribution for the number of movies you watch each month. The results are in the following table. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Number of Movies Probability 0 1 2 3 4 5 0.10 0.10 0.24 0.30 0.14 0.12 Should you go ahead and order the monthly subscription? (Answer: The mean of the probability distribution is 2.64 movies. The average cost per month would then be 2.64 x $2.99 = $7.89. Some students may argue that for an additional 10 cents per month they could watch unlimited movies. Others may choose to avoid the monthly bill and continue to rent movies individually.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Random Number Table Row 1 2 0 4 5 9 8 4 9 0 2 7 6 8 4 7 0 3 9 2 9 6 0 8 1 4 1 6 1 0 3 4 3 7 2 9 3 2 3 5 1 3 3 3 9 8 0 4 9 0 2 8 7 0 9 4 7 4 6 3 0 3 6 1 5 7 4 2 4 4 0 5 9 0 8 4 5 9 1 6 5 6 3 1 5 0 8 2 4 3 9 5 8 3 5 6 2 3 5 5 6 8 8 5 3 8 7 0 9 5 1 6 7 2 7 8 0 7 8 0 3 7 5 3 7 5 9 5 2 7 4 6 7 8 0 4 6 4 0 1 8 1 4 1 3 3 3 5 7 7 9 4 9 6 1 6 5 5 4 8 7 0 8 5 2 1 3 2 1 6 0 10 4 1 5 9 1 9 1 6 4 5 2 5 8 5 8 7 1 0 11 7 9 4 7 3 3 2 7 0 6 5 8 6 7 3 7 4 2 12 0 1 1 0 6 4 0 2 3 0 8 3 3 8 9 6 6 4 13 8 2 1 5 3 0 7 7 1 7 8 5 4 3 5 3 7 6 14 5 6 8 7 2 4 7 4 2 9 8 1 9 0 8 1 2 7 15 8 8 8 3 3 0 5 3 7 5 7 3 4 8 5 9 0 9 16 9 7 1 5 7 0 1 8 2 4 1 1 3 2 8 0 8 3 17 9 3 7 9 8 3 4 0 9 2 5 1 0 2 4 7 3 4 18 3 0 7 0 6 5 3 0 6 0 8 1 3 4 1 0 0 4 19 2 0 2 5 4 8 4 7 2 3 9 3 3 0 4 0 6 5 20 4 0 1 0 1 2 9 3 4 8 7 7 7 5 9 2 5 1 Note: This random number table was generated with the random integer function on a TI-­‐84 calculator. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Day 1 Many probability problems are more difficult mathematically than you are prepared to study directly. There is another option: You can simulate the event many times to give an idea of what would happen and then make a decision based on the simulations. What is a simulation? A simulation is a representation of an event with possible outcomes comparable to the actual event. When the Air Force trains a new pilot, it does not give the new recruit a billion-­‐
dollar jet and say, “Let’s see if you can get this baby off the ground.” The new pilot is first trained using a machine that simulates flying. In probability, you use random events to simulate events in order to determine the probability of a particular event occurring by chance alone. Example The manager of a local television newsroom has three top field reporters he relies on to assemble a team for breaking news stories. The trouble is all three spend most of their time out of the office researching background information for ongoing news. He would like to be reasonably sure that at least one of them will be in the newsroom at any given time. Otherwise, another field reporter should be hired. Abigail is in the newsroom 40% of the time, Benjamin 60% of the time, and Christopher 10% of the time. Does it appear the manager should hire a new field reporter? The first step is designing your simulation. You will use the random number table at the back of your handout to simulate the reporters being in the office. Assign digits to represent each person being in the office according to his or her respective probabilities. •
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Abigail in the office = 1, 2, 3, 4. (Since she is in the office 40% of the time, choose four digits. You could choose any four digits, such as 1, 2, 3, 4 or 2, 4, 6, 8.) Benjamin in the office = 1, 2, 3, 4, 5, 6. (Since he is in the office 60% of the time, choose six digits. You could choose any six digits, such as 1, 2, 3, 4, 5, 6 or 1, 2, 4, 5, 7, 8.) Christopher in the office = 1. (Since he is in the office 10% of the time, choose any one digit.) To conduct one trial, choose three digits and check if at least one of the reporters is in the office. Let the first digit represent Abigail, the second digit Benjamin, and the third digit Christopher. Record the results. Let’s begin with Line 6. (You could choose any line.) •
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The first three digits are 2, 3, 5—Abigail and Benjamin are in the office. The next three digits are 5, 6, 8—Benjamin is in the office. The next three digits are 8, 5, 3—Benjamin is in the office. It looks like the manager does not need to hire anyone. Have you done enough trials? •
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The next three digits are 8, 7, 0—none of the reporters are in the office. The next three digits are 9, 5, 1—Christopher is in the office. The next three digits are 6, 7, 2—none of the reporters are in the office. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Keep track using a table or frequency distribution. Continue through the next row or further as time allows. Event Frequency Total At least one reporter in the newsroom No reporters in the newsroom Event Frequency Total At least one reporter in the newsroom No reporters in the newsroom Total Here are the results for 25 trials: Total 25 19 6 25 Do you think the manager should hire another reporter? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Task (1) Suppose the Kalama Zoo houses nine orangutans: five males and four females. A trainer is assigned to randomly select three of them as the experimental group for a new supplement to their diet. This requires the trainer to move the chosen three to a newly constructed habitat. When her boss returns the next day, he finds three female orangutans in the new habitat. He is furious and threatens to fire the trainer. Why? It turns out that the males sometimes attempt to bite, and by avoiding them, the trainer may not have learned to take care of them properly. The boss claims the trainer did not truly randomly select the three orangutans for the new supplement, but rather chose three females that were easier to handle. The trainer claims that she really did randomly select three females. Should the trainer be fired? Design a simulation to assist you in answering the question. (2) To increase sales, a cereal company places a small bag of toys in each box of cereal. There are four different toys available in randomly selected boxes. They are distributed as follows: •
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40% cars 40% boats 20% rocket ships 20% transforming figures The sales team is interested in knowing how many boxes of cereal a customer must buy to receive the complete set of the four toys. Use simulation to estimate the answer to this question. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Homework for Day 1 Design and carry out a simulation to answer the following questions. (1) A couple has high hopes of having a “gentleman’s family.” This usually means it would have two children, the first being a son and second a daughter. Assuming the probabilities of having a boy or girl are equally likely for each birth, how likely is it the couple will end up with its dream family? (2) A computer game awards a special shield to the player 20% of the time after the player completes a level of the game. David plans to begin studying for a test once he obtains the special shield. It takes him roughly 15 minutes to complete a level. Estimate how long you believe it will be before David begins his studying. (3) A very large company must make budget cuts to remain in business. The chief executive officer plans to form a committee including five randomly selected employees. 50% of the employees are fiscally conservative, 30% liberal, and 20% moderate. The hope is that at least one of each group (conservative, liberal, moderate) is chosen for the committee so employees feel represented and no one group has the majority. The names of all employees are printed on labels and five are drawn. Does this appear to be a suitable method to achieve a committee the employees will deem fair? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Day 2 (1) According to the Social Security Administration Period Life Table, the probability a 70-­‐year-­‐old male will die in the next year is about 0.018. The AARP website lists rates for a $10,000 life insurance policy for a 70-­‐year-­‐old male at roughly $540 per year. Make a probability distribution to reflect the possible monetary outcomes and determine the expected profit for the life insurance company. Explain the expected profit means. (2) You are about to purchase a new four-­‐door sedan and are encouraged to add a mere $20 per month to your car payment to cover the costs of oil changes, routine maintenance, and possible major repairs. The coverage will last the length of your five-­‐year loan. You plan to get an oil change every 5,000 miles, which you estimate will cost an average of $30 each time. On average, you drive 12,000 miles per year. The potential cost of routine maintenance and possible major repairs in the first five years of ownership is shown in the probability distribution below. Oil changes are not included in the cost. Cost over 5 years Probability of cost incurred 0 $500 $1,000 $1,500 $2,000 0.19 0.67 0.09 0.04 0.01 Is the $20 per month fee a good idea for you? Support your answer. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Homework for Day 2 (1) You set up a carnival booth at a local fund-­‐raising event. The game consists of a giant six-­‐sided fair die and costs participants $3 per roll. If they roll a 6, they win a prize that costs $7. If participants roll a 1, they win a prize that costs $2.50. Otherwise, they do not win anything. If you average 35 participants per hour for the 8-­‐hour event, how much money do you expect to raise during the event? (2) After having your family visit for the holidays, you are shocked by your movie-­‐rental bill for the month. Currently you are paying $2.99 for each movie you rent. A subscription will allow you to watch unlimited movies for only $7.99 per month. However, during a normal month there is often not time available to sit and watch movies and you do not really want to needlessly increase your monthly bills. Since you are a statistics student, you decide to check your online billing statements and make a probability distribution for the number of movies you watch each month. The results are in the following table. Number of Movies Probability 0 1 2 3 4 5 0.10 0.10 0.24 0.30 0.14 0.12 Should you go ahead and order the monthly subscription? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.1.3: Simulation, Discrete Random Variables, and Probability Distributions Random Number Table Row 1 2 0 4 5 9 8 4 9 0 2 7 6 8 4 7 0 3 9 2 9 6 0 8 1 4 1 6 1 0 3 4 3 7 2 9 3 2 3 5 1 3 3 3 9 8 0 4 9 0 2 8 7 0 9 4 7 4 6 3 0 3 6 1 5 7 4 2 4 4 0 5 9 0 8 4 5 9 1 6 5 6 3 1 5 0 8 2 4 3 9 5 8 3 5 6 2 3 5 5 6 8 8 5 3 8 7 0 9 5 1 6 7 2 7 8 0 7 8 0 3 7 5 3 7 5 9 5 2 7 4 6 7 8 0 4 6 4 0 1 8 1 4 1 3 3 3 5 7 7 9 4 9 6 1 6 5 5 4 8 7 0 8 5 2 1 3 2 1 6 0 10 4 1 5 9 1 9 1 6 4 5 2 5 8 5 8 7 1 0 11 7 9 4 7 3 3 2 7 0 6 5 8 6 7 3 7 4 2 12 0 1 1 0 6 4 0 2 3 0 8 3 3 8 9 6 6 4 13 8 2 1 5 3 0 7 7 1 7 8 5 4 3 5 3 7 6 14 5 6 8 7 2 4 7 4 2 9 8 1 9 0 8 1 2 7 15 8 8 8 3 3 0 5 3 7 5 7 3 4 8 5 9 0 9 16 9 7 1 5 7 0 1 8 2 4 1 1 3 2 8 0 8 3 17 9 3 7 9 8 3 4 0 9 2 5 1 0 2 4 7 3 4 18 3 0 7 0 6 5 3 0 6 0 8 1 3 4 1 0 0 4 19 2 0 2 5 4 8 4 7 2 3 9 3 3 0 4 0 6 5 20 4 0 1 0 1 2 9 3 4 8 7 7 7 5 9 2 5 1 Note: This random number table was generated with the random integer function on a TI-­‐84 calculator. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables Estimated number of 50-­‐minute class sessions: 0.5–1 Learning Goals Students will understand •
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the difference between discrete and continuous probability distributions. that area is a representation of probability. Students will be able to •
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interpret the mean and standard deviation of a continuous random variable. calculate probabilities for a continuous uniform distribution. Introduction [Student Handout] In previous lessons, you organized outcomes and probabilities in a probability distribution. This works well when the possible outcomes can actually be listed. The outcomes used were all discrete; this means that you can graph all outcomes as individual points on a number line. You could actually list all outcomes. What if that is not possible? Let’s say the data are a collection of the heights of freshmen at your college. Depending on the accuracy of the measurement tool, there are potentially an infinite number of possible heights. You could report the results to the nearest inch, half inch, quarter inch, eighth inch, and so on. These are not discrete data. You cannot possibly list all possible heights. These data are continuous. (Note: Ask students to determine whether the following random variables are continuous or discrete. Read each one and allow a quiet moment for students to consider their response. Then have them stand [or raise their hand or verbally respond] if the variables are continuous. •
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the time it takes to run a mile (continuous) the number of problems assigned for homework (discrete) the number of shoes in a closet (discrete) the lifetime of an automobile tire (continuous) the circumference of a hailstone (continuous) the length of a giraffe’s neck (continuous) the sum of the roll of two dice (discrete) Ask students how they can tell if a variable is discrete or continuous. Lead the discussion toward the idea of measuring versus counting.) You need a way to represent all possibilities for a continuous data set. •
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A number line or horizontal axis can be used to list the possibilities since the infinite values are represented as points. The height of the graph at any point represents the relative frequency of that point. The total area under the curve is equal to 1 since the sum of all probabilities in a probability distribution is 1. This follows from your work with discrete random variables where the sum of all probabilities is equal to 1. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables Suppose you define a continuous random variable as the time it takes to pack a customer order for shipping at a particular company and the results are uniformly distributed between 5 and 15 minutes, (i.e., the probability for each packing time occurring is the same). The distribution is represented by the graph below. (Note: A reminder of the shape of a uniform distribution may be helpful.) If the area of the rectangle is 1, what would be the height of the rectangle? (Answer: 0.1) (Note: Briefly discuss how 0.1 is found to ensure all students understand: 1 divided by length of the base of the rectangle.) Task [Student Handout] (Note: In groups of three to four, have students find the following probabilities by finding the area of the corresponding rectangles. Instruct students to sketch the graph, shade the defined area, and then find the probability. The goal is to develop the ability to visualize particular areas and see the relationship to the whole.) (1) What is the probability that an order takes 15 minutes or less to pack? (Answer: 0.5) (2) What is the probability that an order takes 12 minutes or more to pack? (Answer: 0.8) (3) What is the probability that an order takes between 13 and 17 minutes to pack? (Answer: 0.4) (4) What is the probability that an order takes less than 11 minutes or more than 18 minutes to pack? (Answer: 0.3) (5) What is the probability that an order takes more than 25 minutes to pack? (Answer: 0) (6) If the supervisor wants to study the 20% of orders taking the longest time to pack, he needs to study orders taking at least how long? (Answer: 13 minutes) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables Wrap-­‐Up Check student answers. Students may have difficulty determining the boundaries of areas defined by greater than or less than. This could be an opportunity for a brief discussion on the differences between discrete and continuous distributions. When a discrete distribution is appropriate, students must consider a value such as x = 25 if it is a part of the distribution. The value corresponds to a probability. However, when students work with continuous random variables, they use area to represent the probabilities. A single value is represented by a line. The area of a line is 0, because a line has no width. This means that greater than 25 results in the same area or probability as 25 or greater. Ask students the difference between Question 6 and the other questions. Discuss student strategies for determining the answer to Question 6. Homework [Student Handout] (1) A computer program randomly selects real numbers between 1 and 25. If the program is truly random, the results should be uniformly distributed. Sketch the distribution, shade the appropriate area, and find the probability for each of the following questions. (a) What is the probability of randomly selecting a real number between 5 and 10? (Answer: 0.2) (b) What is the probability of randomly selecting a real number of at least 16? (Answer: 0.36) (c) What is the probability of randomly selecting a real number not larger than 20? (Answer: 0.8) (d) What number are 30% of the randomly selected real numbers below? (Answer: 7.5) (e) What is the probability of randomly selecting a real number less than 0? (Answer: 0) (f) What is the probability of randomly selecting a real number greater than 20 or less than 5? (Answer: 0.4) (2) Pecan trees drop an organic compound called juglone that inhibits growth of new trees and many other plants. This naturally controls overcrowding. A pecan farmer is interested in the distance from any healthy mature pecan tree to the next closest tree. Planting trees too close together compromises pecan production, while planting trees too far apart sacrifices potential income. If he hopes to have 80% of the trees grow to healthy maturity, how far apart should he plant them? Assume the distribution of distances follows an approximately uniform distribution of 40–60 feet. (Answer: 56 feet) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables In previous lessons, you organized outcomes and probabilities in a probability distribution. This works well when the possible outcomes can actually be listed. The outcomes used were all discrete; this means that you can graph all outcomes as individual points on a number line. You could actually list all outcomes. What if that is not possible? Let’s say the data are a collection of the heights of freshmen at your college. Depending on the accuracy of the measurement tool, there are potentially an infinite number of possible heights. You could report the results to the nearest inch, half inch, quarter inch, eighth inch, and so on. These are not discrete data. You cannot possibly list all possible heights. These data are continuous. You need a way to represent all possibilities for a continuous data set. •
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A number line or horizontal axis can be used to list the possibilities since the infinite values are represented as points. The height of the graph at any point represents the relative frequency of that point. The total area under the curve is equal to 1 since the sum of all probabilities in a probability distribution is 1. This follows from your work with discrete random variables where the sum of all probabilities is equal to 1. Suppose you define a continuous random variable as the time it takes to pack a customer order for shipping at a particular company and the results are uniformly distributed between 5 and 15 minutes, (i.e., the probability for each packing time occurring is the same). The distribution is represented by the graph below. If the area of the rectangle is 1, what would be the height of the rectangle? Task (1) What is the probability that an order takes 15 minutes or less to pack? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables (2) What is the probability that an order takes 12 minutes or more to pack? (3) What is the probability that an order takes between 13 and 17 minutes to pack? (4) What is the probability that an order takes less than 11 minutes or more than 18 minutes to pack? (5) What is the probability that an order takes more than 25 minutes to pack? (6) If the supervisor wants to study the 20% of orders taking the longest time to pack, he needs to study orders taking at least how long? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables Homework (1) A computer program randomly selects real numbers between 1 and 25. If the program is truly random, the results should be uniformly distributed. Sketch the distribution, shade the appropriate area, and find the probability for each of the following questions. (a) What is the probability of randomly selecting a real number between 5 and 10? (b) What is the probability of randomly selecting a real number of at least 16? (c) What is the probability of randomly selecting a real number not larger than 20? (d) What number are 30% of the randomly selected real numbers below? (e) What is the probability of randomly selecting a real number less than 0? (f) What is the probability of randomly selecting a real number greater than 20 or less than 5? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Initiating Lesson 6.2.1: Probability Distributions of Continuous Random Variables (2) Pecan trees drop an organic compound called juglone that inhibits growth of new trees and many other plants. This naturally controls overcrowding. A pecan farmer is interested in the distance from any healthy mature pecan tree to the next closest tree. Planting trees too close together compromises pecan production, while planting trees too far apart sacrifices potential income. If he hopes to have 80% of the trees grow to healthy maturity, how far apart should he plant them? Assume the distribution of distances follows an approximately uniform distribution of 40–60 feet. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions Estimated number of 50-­‐minute class sessions: 1 Learning Goals Students will begin to understand •
•
how to use Z-­‐scores to standardize scores. the properties of the normal distribution (mean, standard deviation, and the 68–95–99.7 Rule). Students will begin to be able to •
•
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calculate Z-­‐scores and compare scores accordingly. match graphs of normal distributions to mean and standard deviation pairs. estimate probabilities using the 68–95–99.7 Rule. Rich Task [Student Handout] (Note: Have students work in groups of three or four on the following problem. Each group should be ready to report its decision to the class with justification.) (1) A large company is hiring one employee for a top position. Your team will recommend who gets the job. After completing many interviews, reference checks, and verifications of written applications, you narrowed the choice down to two outstanding candidates. Both are equally qualified and respected. Since the job can be very stressful and requires a significant amount of travel, the final component in the hiring process requires each to take a personality test. Both candidates have the results of such a test in their file. Unfortunately, they did not take the same test. Using the following results, whom do you recommend for the position? Be prepared to justify your choice to the company chief executive officer. (Note: The Gallup Organization actually administers several types of tests to aid companies in the hiring process. They have been a key part of the hiring process for corporate leaders, teachers, and NBA basketball players.) Candidate A Test taken: Stress Sanity Simulator Mean score: 742 Standard deviation: 28 Candidate score: 801 Candidate B Test taken: Modeling Mind Management Mean score: 67 Standard deviation: 12 Candidate score: 94 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions Discussion After students have had adequate time to come to a decision, have each group report the candidate it would hire and briefly describe its reasoning. The goal here is to inspire a need to have both test results on the same scale. After the groups report their choice of candidates and the ensuing discussion, introduce the concept of the Z-­‐score as a method to standardize scores. Before actually giving students the formula, explain what the Z-­‐score does. First, find how far above (or below) the score is from the mean. Dividing by the standard deviation tells how many standard deviations the score is from the mean. The result is !! = !
"#$%&! ! !'&()
!! ! !µ
or !! = !
. "#
!
Students using a standard calculator may mistakenly divide the mean by the standard deviation—
instead of the difference between the score and the mean—and become frustrated. A quick comment on the proper order of operations may be helpful here. Have students calculate the Z-­‐score for both candidates and compare the results. Candidate A !! = !
"#$! ! !%&'
! = !'($#% '"
Candidate B !! = !
"#! ! !$%
! = !'(') &'
Wrap-­‐Up Ask students the following questions. •
•
•
What would it mean if a candidate’s Z-­‐score was negative? (Answer: The score is below the mean.) What would it mean if a candidate’s Z-­‐score was equal to 0? (Answer: The score is equal to the mean.) What exactly does a Z-­‐score represent? (Answer: It represents the number of standard deviations a score is above or below the mean. Discourage students from saying the Z-­‐score is how many standard deviations from the mean. This does not include which direction from the mean the score is.) The Standard Normal Curve [Student Handout] In a previous lesson, you explored continuous uniform distributions and probabilities. Few real-­‐life distributions are uniform. There are, however, many distributions that are approximately normal. You can describe a normal distribution by its mean and standard deviation. It could have a mean of 128 and a standard deviation of 7.2, or a mean of –15 and a standard deviation of 2. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions And just like uniform distributions, if you let the area under the curve equal 1, you can use the area to find probabilities. Finding the area under the normal curve requires more difficult mathematics than finding the area of a rectangle. Fortunately, all normal distributions have the same relationship between area and the number of standard deviations from the mean. (Note: Describe the 68–95–99.7 Rule for students using the diagrams shown below and the diagram on the following page.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions You can use the 68-­‐95-­‐99.7 Rule to find probabilities (or the proportion of data). Suppose an approximately normal distribution has mean 25 and standard deviation 2. What proportion of the data is between 23 and 27 units? Since 23 and 27 are 1 standard deviation above and below the mean, the shaded area is 0.68. (Note: Ask students the following questions: •
•
What proportion of data is between 23 and 25? Why? [Answer: 0.34; this is half of the shaded region.] About how many standard deviations wide is a normal distribution? [Answer: 6; about 3 standards deviations above the mean and 3 standard deviations below the mean.]) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions Match the following graphs of normal distributions with the appropriate mean and standard deviation. A –13 B 5 –19 C D 13 –5 11 12 0 E –9 F 21 –11 19 (2) Mean 4 and standard deviation 3 (Answer: C) (3) Mean 6 and standard deviation 5 (Answer: E) (4) Mean –4 and standard deviation 3 (Answer: A) (5) Mean 6 and standard deviation 2 (Answer: D) (6) Mean 4 and standard deviation 5 (Answer: F) (7) Mean –4 and standard deviation 5 (Answer: B) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions (8) The speed limit on a stretch of country road is 55 miles per hour (mph). A device used to measure the actual speed of cars traveling on the road finds the distribution of speeds to be approximately normal with a mean of 55 and a standard deviation of 5. Sketch the normal curve with the appropriate scale. Then find the probabilities using the 68–95–99.7 Rule. (a) What is the probability that a randomly selected car on this road is traveling between 45 and 65 mph? (Answer: 0.95) (b) What is the probability that a randomly selected car on this road is traveling between 55 and 60 mph? (Answer: 0.34) (c) What is the probability that a randomly selected car on this road is traveling above 50 mph? (Answer: 0.84) (d) What is the probability that a randomly selected car on this road is traveling below 45 mph? (Answer: 0.025) (e) What is the probability that a randomly selected car on this road is traveling above 55 mph? (Answer: 0.50) (f) What is the probability that a randomly selected car on this road is traveling between 50 and 65 mph? (Answer: 0.815) (Note: Help students having difficulty finding the appropriate areas. When most students have completed the work, discuss the solutions.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions Homework [Student Handout] (1) The Westwood Warrior basketball team is trying to improve its offensive strategies. Some opponents primarily use a zone defense, while others primarily use a man-­‐to-­‐man defense. On average, the Warriors score 67 points against teams with a zone defense and 62 points against teams with a man-­‐to-­‐man defense. The standard deviations are 8 and 5, respectively. Since the improved offensive strategies have been put into practice, the Warriors have played two games with the following results. McNeil Mavericks Maverick defense: zone Warrior points: 77 Round Rock Dragons Dragon defense: man-­‐to-­‐man Warrior points: 71 Which offensive strategy—zone or man-­‐to-­‐man—appears to have improved the most? (Of course, there are many factors involved in playing basketball. For your answer, assume the difference is due to the new offensive strategies.) (Answer: Zone defense: !! = !
Man-­‐to-­‐man defense: !! = !
The Warriors improved most against the man-­‐to-­‐man defense.) ""! ! !#"
! = !%&'( $
"#! ! !$%
! = !#'( &
The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions (2) The average height of an adult male can be reasonably described by a normal distribution with a mean of 69 inches and a standard deviation of 2.5 inches. Find the probability of randomly selecting a male in the following categories using the 68–95–99.7 Rule. For each problem, sketch the curve with the appropriate scale and shade the area described. (a) What is the probability that a randomly selected male is more than 74 inches tall? (Answer: 0.025) (b) What is the probability that a randomly selected male is between 64 and 69 inches tall? (Answer: 0.475) (c) What is the probability that a randomly selected male is less than 61.5 inches tall? (Answer: 0.0015) (d) What is the probability that a randomly selected male is between 66.5 and 76.5 inches tall? (Answer: 0.8385) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 8 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
(1) A large company is hiring one employee for a top position. Your team will recommend who gets the job. After completing many interviews, reference checks, and verifications of written applications, you narrowed the choice down to two outstanding candidates. Both are equally qualified and respected. Since the job can be very stressful and requires a significant amount of travel, the final component in the hiring process requires each to take a personality test. Both candidates have the results of such a test in their file. Unfortunately, they did not take the same test. Using the following results, whom do you recommend for the position? Be prepared to justify your choice to the company chief executive officer. Candidate A Test taken: Stress Sanity Simulator Mean score: 742 Standard deviation: 28 Candidate score: 801 Candidate B Test taken: Modeling Mind Management Mean score: 67 Standard deviation: 12 Candidate score: 94 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
The Standard Normal Curve In a previous lesson, you explored continuous uniform distributions and probabilities. Few real-­‐life distributions are uniform. There are, however, many distributions that are approximately normal. You can describe a normal distribution by its mean and standard deviation. It could have a mean of 128 and a standard deviation of 7.2, or a mean of –15 and a standard deviation of 2. And just like uniform distributions, if you let the area under the curve equal 1, you can use the area to find probabilities. Finding the area under the normal curve requires more difficult mathematics than finding the area of a rectangle. Fortunately, all normal distributions have the same relationship between area and the number of standard deviations from the mean. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
You can use the 68-­‐95-­‐99.7 Rule to find probabilities (or the proportion of data). Suppose an approximately normal distribution has mean 25 and standard deviation 2. What proportion of the data is between 23 and 27 units? Since 23 and 27 are 1 standard deviation above and below the mean, the shaded area is 0.68. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
Match the following graphs of normal distributions with the appropriate mean and standard deviation. A –13 B 5 –19 C 12 0 E –9 D 13 –5 11 F 21 –11 19 (2) Mean 4 and standard deviation 3 (3) Mean 6 and standard deviation 5 (4) Mean –4 and standard deviation 3 (5) Mean 6 and standard deviation 2 (6) Mean 4 and standard deviation 5 (7) Mean –4 and standard deviation 5 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
(8) The speed limit on a stretch of country road is 55 miles per hour (mph). A device used to measure the actual speed of cars traveling on the road finds the distribution of speeds to be approximately normal with a mean of 55 and a standard deviation of 5. Sketch the normal curve with the appropriate scale. Then find the probabilities using the 68–95–99.7 Rule. (a) What is the probability that a randomly selected car on this road is traveling between 45 and 65 mph? (b) What is the probability that a randomly selected car on this road is traveling between 55 and 60 mph? (c) What is the probability that a randomly selected car on this road is traveling above 50 mph? (d) What is the probability that a randomly selected car on this road is traveling below 45 mph? (e) What is the probability that a randomly selected car on this road is traveling above 55 mph? (f) What is the probability that a randomly selected car on this road is traveling between 50 and 65 mph? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
Homework (1) The Westwood Warrior basketball team is trying to improve its offensive strategies. Some opponents primarily use a zone defense, while others primarily use a man-­‐to-­‐man defense. On average, the Warriors score 67 points against teams with a zone defense and 62 points against teams with a man-­‐to-­‐man defense. The standard deviations are 8 and 5, respectively. Since the improved offensive strategies have been put into practice, the Warriors have played two games with the following results. McNeil Mavericks Maverick defense: zone Warrior points: 77 Round Rock Dragons Dragon defense: man-­‐to-­‐man Warrior points: 71 Which offensive strategy—zone or man-­‐to-­‐man—appears to have improved the most? (Of course, there are many factors involved in playing basketball. For your answer, assume the difference is due to the new offensive strategies.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.2: Z-­‐Scores and Normal Distributions
(2) The average height of an adult male can be reasonably described by a normal distribution with a mean of 69 inches and a standard deviation of 2.5 inches. Find the probability of randomly selecting a male in the following categories using the 68–95–99.7 Rule. For each problem, sketch the curve with the appropriate scale and shade the area described. (a) What is the probability that a randomly selected male is more than 74 inches tall? (b) What is the probability that a randomly selected male is between 64 and 69 inches tall? (c) What is the probability that a randomly selected male is less than 61.5 inches tall? (d) What is the probability that a randomly selected male is between 66.5 and 76.5 inches tall? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 7 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values Estimated number of 50-­‐minute class sessions: 1 Materials Required Each student needs a copy of the normal chart (Z-­‐table). Normal distribution tables can be found in several formats. Some calculate the area to the left. Others calculate the area to the center or to the right. The examples in this lesson use a normal chart with areas calculated to the left of the Z-­‐score. One source for this type of normal chart is the College Board website for the AP Statistics exam. All exams include the same normal distribution table. For the 2011 exam, the table can be found on pages 13–14 at the following URL: http://apcentral.collegeboard.com/apc/public/repository/ap11_frq_statistics.pdf Learning Goals Students will understand •
•
the relationship between the area under the normal curve and probability. the Z-­‐score as it relates to the standard normal curve. Students will be able to •
•
•
calculate probabilities using the normal curve. find critical values using the normal curve. describe the smallest 5% and largest 5%. Introduction In the previous lesson, students found probabilities for normal distributions by calculating the Z-­‐score and using the 68–95–99.7 Rule. That works very nicely if the value is one, two, or three standard deviations from the mean. What if the value is 2.5 standard deviations above the mean or 1.8 standard deviations below the mean? Students need a way to calculate the area under the curve. Fortunately, the work of calculating Z-­‐score and area values has already been done and placed in a convenient table—a Z-­‐table (also called a normal table). Make sure each student has a normal table in front of them. Take the time to explain the layout of the table. This is the standard normal curve: the mean is 0 and the standard deviation is 1. The columns on the left represent the standardized score (or Z-­‐score). The top row gives the hundredths place of the Z-­‐score. The values in the center give the area under the curve less than or equal to the specified Z-­‐score. Notice that there are two sides to the table. The right side contains positive Z-­‐scores, while the left side contains negative Z-­‐scores. To find the area less than 1.74 standard deviations above the mean, go down the Z-­‐column to Row 1.7. Then go over to the column labeled 0.04. The area to the left of z = 1.74 is 0.9591. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values Ask students to locate the area less than the following z-­‐scores to ensure they understand the table. •
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•
•
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P(z < 1.26) = 0.8962 P(z < –3.08) = 0.0010 P(z < 2.37) = 0.9911 P(z < –0.41) = 0.3409 P(z < 0) = 0.5 Why does the last one make sense? Zero is the center, so less than that is half. Guided Questions 1. How can you find the area to the right of z = 1.74 using this table? [Answer: Subtract the area less than 1.74 from 1. P(z > 1.74) = 1 – 0.9591 = 0.0409] 2. How can you find the area between –1.5 and 1.74 using this table? [Answer: Subtract the area less than –1.5 from the area less than 1.74. P(–1.5 < z < 1.74) = 0.9591 – 0.0668 = 0.8923] Students often need help understanding why this works. Sketch a normal curve and shade the area to the left of 1.74. Mark z = –1.5 on the graph. Explain that you do not want the area to the left of –1.5. If you subtract that area (erase the area to the left of –1.5), you are left with the area from –1.5 to 1.74. 3. Find the area between –2 and 2. (Answer: 0.9544. Note: It is great if a student asks why this is not exactly 95%, since this is for the area within two standard deviations of the mean. If no one asks, remind students of the 68–95–99.7 Rule. Point out that 68–95–99.7 are approximations.) 4. How can you find what Z-­‐score has an area to the left equal to about 60% (or 0.60)? (Answer: Find the closest area to 0.60 and then find the corresponding Z-­‐score. The closest area value is 0.5987, which corresponds to a Z-­‐score of 0.25.) If your class is using a specific program or calculator to find areas and Z-­‐scores, the same questions can be asked with about the same results. Make sure students are familiar with the functions of the program so they can quickly determine the areas. Students should be aware that the terms probability, proportion, or percent can all be used in reference to the area under the curve. The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values Rich Task [Student Handout] (Note: The following problem set can be done by direct teaching or student exploration in groups as a Rich Task.)
The Volkswagen L1 is currently the most fuel-­‐efficient car ever made. This one-­‐person vehicle averages 258 miles per gallon and can reach speeds over 70 miles per hour. Unfortunately, the L1 does not meet all of the safety standards required in the United States. Suppose you are part of the design team for an American auto company attempting to make a similar fuel-­‐efficient car. To keep the cost and the weight of the car as low as possible, the seat adjustments may not have the flexibility of current vehicles on the market. This requires you to explore the distribution of heights of American drivers. For each problem, sketch the curve and find the probability. (1) The heights of adult American males can be reasonably approximated by a normal distribution with a mean of 69 inches and a standard deviation of 2.5 inches. (a) What is the probability that an adult American male is over 6 feet 1 inch? (Answer: z = 73 − 69
= 1.6 2.5
P(x > 73) = 0.0548 (b) What is the probability that an adult American male is between 5 feet 2 inches and 6 feet? (Answer: z = 62 − 69
= − 2.8 2.5
72 − 69
z = = 1.2 2.5
P(x > 73) = 0.8824 The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values (2) The heights of adult American females also follow an approximately normal distribution with a mean of 64 inches and a standard deviation of 2.8 inches. (a) What is the probability that an adult American female is less than 5 feet tall? (Answer: 60 − 64
z = = −1.43 2.8
P(x < –1.43) = 0.0764 (b) If the current prototype of your car comfortably accommodates adults between 5 feet 2 inches and 6 feet tall, what proportion of adult American women would not be comfortable in the car? [Answer: z = 62 − 64
= − 0.71 2.8
72 − 64
z = = 2.86 2.8
P(x < 62) = 0.2375 P(x > 72) = 0.0021 P(x < 62 or x > 72) = 0.2375 + 0.0021 = 0.2396 Note: Some students may first find P(62 < x < 72) and subtract from 1.] The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values (3) If you decide to make the specifications for the seat adjustments comfortable for all but the shortest 5% of adult American females and the tallest 5% of the adult American males, what would be the cutoff heights for your vehicle? (Answer: Females Males For the bottom 5%, cutoff z = –1.645 −1.645 = For the top 5%, cutoff z = 1.645 x − 64
2.8 1.645 = x − 69
2.5 x = 59.394
x = 73.1125
Depending on whether z is found using technology or the table, students may use z = 1.64 or 1.65. Either of these Z-­‐values should be considered correct. Note: Students often become frustrated when they do not get the exact answer listed as the solution. Reassure them that rounding and different technology results in slightly different, correct results. Note: If necessary, take the time to go through the algebra needed to solve the previous equations.) Wrap-­‐Up Discuss solutions with students and then ask the following questions. •
•
What does the Z-­‐score represent? (Answer: The number of standard deviations above or below the mean.) There are four basic types of normal curve calculations you should be able to calculate. o How do you find the area less than a specified value? (Answer: Find the Z-­‐score and look it up on the table.) o How do you find the area greater than a specified value? (Answer: Find the Z-­‐score, look it up on the table, and subtract from 1.) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 5 Statway Instructor’s Notes April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values o
o
How do you find the area between two specified values? (Answer: Find the Z-­‐score for both values, look up both values on the table, and subtract the smaller area from the larger one.) How do you find the cutoff value given a specified area? (Answer: Using the middle numbers in the table, find the area closest to the area specified. Be sure you use the area to the left. Then find the corresponding Z-­‐score. Using the Z-­‐score in the formula, solve for x.) Homework [Student Handout] (1) Red drum, sciaenops ocellatus, are popular coastal recreational fishing catches. A reasonable estimate of the mean length of red drum caught, tagged, and released is 15.4 inches with a standard deviation of 4.5 inches. Current regulations by Texas Parks and Wildlife only allow red drums between 20 and 28 inches to be kept. Fish outside this range must be thrown back. Assuming the distribution of lengths is approximately normally distributed, what is the probability a red fish caught on the Texas coast can be brought home for dinner? [Answer: z = 20 − 15.4
= 1.02 4.5
28 − 15.4
z = = 2.8 4.5
P(20 < x < 28) = 0.1508] (2) The time students took to complete a statistics test is approximately normally distributed with a mean of 74 minutes and a standard deviation of 8 minutes. (a) What proportion of students took more than one hour to complete the test? (Answer: 0.9599) (b) What is the probability a student needed between 70 and 80 minutes to complete the test? (Answer: 0.4648) (c) What is the maximum time taken to complete the test by the fastest 5% of the students? (Answer: 60.84 minutes) The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 6 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values The Volkswagen L1 is currently the most fuel-­‐efficient car ever made. This one-­‐person vehicle averages 258 miles per gallon and can reach speeds over 70 miles per hour. Unfortunately, the L1 does not meet all of the safety standards required in the United States. Suppose you are part of the design team for an American auto company attempting to make a similar fuel-­‐efficient car. To keep the cost and the weight of the car as low as possible, the seat adjustments may not have the flexibility of current vehicles on the market. This requires you to explore the distribution of heights of American drivers. For each problem, sketch the curve and find the probability. (1) The heights of adult American males can be reasonably approximated by a normal distribution with a mean of 69 inches and a standard deviation of 2.5 inches. (a) What is the probability that an adult American male is over 6 feet 1 inch? (b) What is the probability that an adult American male is between 5 feet 2 inches and 6 feet? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 1 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values (2) The heights of adult American females also follow an approximately normal distribution with a mean of 64 inches and a standard deviation of 2.8 inches. (a) What is the probability that an adult American female is less than 5 feet tall? (b) If the current prototype of your car comfortably accommodates adults between 5 feet 2 inches and 6 feet tall, what proportion of adult American women would not be comfortable in the car? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 2 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values (3) If you decide to make the specifications for the seat adjustments comfortable for all but the shortest 5% of adult American females and the tallest 5% of the adult American males, what would be the cutoff heights for your vehicle? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 3 Statway Student Handout April 23, 2012 (Full Version 1.0) Supporting Lesson 6.2.3: Using Normal Distributions to Find Probabilities and Critical Values Homework (1) Red drum, sciaenops ocellatus, are popular coastal recreational fishing catches. A reasonable estimate of the mean length of red drum caught, tagged, and released is 15.4 inches with a standard deviation of 4.5 inches. Current regulations by Texas Parks and Wildlife only allow red drums between 20 and 28 inches to be kept. Fish outside this range must be thrown back. Assuming the distribution of lengths is approximately normally distributed, what is the probability a red fish caught on the Texas coast can be brought home for dinner? (2) The time students took to complete a statistics test is approximately normally distributed with a mean of 74 minutes and a standard deviation of 8 minutes. (a) What proportion of students took more than one hour to complete the test? (b) What is the probability a student needed between 70 and 80 minutes to complete the test? (c) What is the maximum time taken to complete the test by the fastest 5% of the students? The original versions of the Statway™ and Quantway™ courses were created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at The University of Texas at Austin. STATWAY™/Statway™ and Quantway™ are trademarks of the Carnegie Foundation for the Advancement of Teaching. The Dana Center’s frontmatter for Statway™ and Quantway™ is available at www.utdanacenter.org/mathways. 4