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.
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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.
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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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Estimated number of 50-­‐minute class sessions: 2 Learning Goals Students will begin to understand that •
•
•
•
a sample is only one of many possible samples that can be drawn from a population. the sample proportion summarizes information about a sample. samples vary. sample characteristics (e.g., sample proportions) vary. Students will begin to be able to •
•
•
identify the variable of interest. calculate a sample proportion. consider the role of variability in their statistical reasoning. Lesson Overview In an election for the mayor of a small city with two candidates, is it possible to predict which candidate will win before all votes are counted? Students work in small groups. They use a computer simulation to draw random samples from a population of voters. The data consist of counts of votes for each candidate from random samples taken from all polling stations in the city. The task is to use the data to predict whether a particular candidate will win the election. Each group first decides on the statistics it will use and a method for making a decision. Whole-­‐class discussion centers around the statistics students used to make their decisions and how decisions were made, with the aim of coming to a consensus on a common statistic (sample proportions). Each student then uses the spreadsheet to draw random samples and compute sample proportions. A dotplot of sample proportions is generated. Whole-­‐class discussion then centers on the variability of the sample proportions, why there is such variability, and how this affects predictions of who will win the election. If your students do not have access to computers with Excel, consider projecting the spreadsheet on a screen from an instructor workstation. Students can provide suggestions of values to use, and once values are settled on they can work in groups to answer questions about the results. Introduction to the Context of the Lesson [Student Handout] Your team is working for a candidate named Alicia Harper who is running in an election for the mayor of a small city. There is another candidate running for mayor named José Abasta. The candidate with a majority of the total votes wins the election. Your candidate wants to know as soon as possible on Election Day whether she is likely to win the election. Your team’s assignment is to conduct an opinion poll of voters as they exit the polling stations to predict whether Alicia will win the election. Opinion polls are often conducted by taking a random sample of several hundred to several thousand people. Your team will ask a random sample of people who finished voting at each polling station whom they voted for: Alicia Harper or José Abasta. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Task I: Creating a Sampling Plan [Student Handout] Discuss and devise a plan for how you would select people to ask whom they voted for. There are 25 polling stations in the small city holding the election. You want to collect information from 500 people. (Note: Once every group decides on a plan, your instructor will lead a whole-­‐class discussion of the different plans.) (1) How many people would you randomly select at each polling station? (2) How would you go about selecting each person so that the selection process was truly random? Wrap-­‐Up [Whole-­‐class discussion] Once each group has discussed the questions, have the groups report back to the class. You can expect that each group will plan to randomly select 20 people from each polling station (25 × 20 = 500). If a group has different plan (e.g., randomly generate a number between 10 and 30 to determine the sample size for each polling station), have the group state its rationale and discuss the advantages and disadvantages of the approach. Have the groups present how they will randomly select people. Discuss whether each method represents a random process or if method could introduce sources of selection bias. Transition to Task II In Task II, students use the simulation spreadsheet to observe how random sample proportions tend to be distributed and how care must be taken when making conclusions regarding sample proportions. In addition, students will begin to see that simulated sample proportions can look roughly normal in their distributions and that the mean of sample proportions appears to be approximately equal to the population proportion. The term sampling distribution is first used in this lesson, and immediately after a discussion regarding Sample Size versus Number of Samples is included. It might be worthwhile to discuss this issue in class, so students see the difference between observations in a sample and sample proportion observations in a sampling distribution. The task ends with a few basic questions regarding the likelihood of predicting the winner of the election incorrectly and a very informal confidence interval. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Task II: Estimating Chances of an Incorrect Prediction [Student Handout] Discuss the following questions in your small group: (3) If you use your sampling plan and find that Alicia Harper has a majority of voters in the sample of 500 people, does that mean that she will win the election? Why or why not? (4) In the total population of voters, if half of the people voted for each candidate, you would not want to declare that Alicia Harper would win the election. Could Alicia have a larger proportion than the other candidate in a random sample of 500 voters even if in the population there was an even split between the two candidates (in other words, each candidate wins exactly 1/2 of the votes)? To answer these questions, your team will use a spreadsheet that allows you to simulate many samples and see what might happen if you repeated the sampling process over and over again. This should help you see whether you can expect the same results every time you take a random sample for your poll. The spreadsheet is located at the following web address: http://math.mtsac.edu/statistics/MayoralRace.xls The Mayoral Race spreadsheet allows you to set two values: • The population proportion of all voters who will theoretically vote for each candidate. Enter •
1/2 (0.50) for Alicia Harper’s population proportion and the spreadsheet will use the rule of complements to assign a population proportion for José Abasta. The number of people you will survey in each sample. Enter 500 into this cell. With the proportions for each candidate and the sample size entered, the spreadsheet generates 1,000 simulated sample proportions for Alicia, distributed according to the binomial distribution. These sample proportions are then plotted on a dotplot. A sampling distribution is the distribution of all possible sample proportions from samples of a given size. The simulation spreadsheet generates 1,000 of these, but this is only a subset of the entire distribution. (5) Describe the shape of the distribution of Alicia Harper’s sample proportions in terms of shape and symmetry. (6) Visually estimate the mean of the simulated proportions for Alicia. (7) Is the estimated mean of Alicia’s sample proportions similar to the assumed population proportion that was entered in the spreadsheet for her? Sample Size Versus Number of Samples [For Lecture or Student Reading] When people learn about sampling distributions, many confuse the ideas of sample size and the number of samples. A sample consists of a collection of observations from population members. That sample has a certain size (e.g., the 500 randomly selected people for the election results samples). When you randomly select the 500 people, you have selected one sample. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability A randomly selected sample is just one possible sample from an entire population. While the many different samples you might gather have much in common, you should not expect each sample to be identical. This means that results, such as the proportion of votes for a particular candidate, will be different from one sample to the next. Some differences may be quite small and others quite large. To get a sense of how the proportions will vary, you can collect many samples, where each sample has the same size. How many samples do you need to get a good sense of the variation? Most statisticians agree that 1,000 samples can provide a good sense of the sampling variation. In summary, you are gathering 1,000 random samples, each containing 500 randomly selected people. The reason for gathering 1,000 samples is to provide a good picture of the variation present in a sampling distribution (how the proportions of voters who voted for Alicia Harper differ between random samples). There is one more feature of the Mayoral Race spreadsheet that can be changed called the Winning Proportion Criterion. For right now, set this to 0.50. When the values are entered, a total of 1,000 samples are automatically generated. Answer the following questions. (8) Is the proportion of people who voted for Alicia Harper the same for each of the 1,000 samples? How do you know this? (9) Estimate a typical range for the sample proportions of votes for Alicia. from _______________ to _______________ The spreadsheet can show you the percentage of sample proportions for Alicia Harper that are greater than a given sample proportion entered under Winning Proportion Criterion. This percentage is listed next to % of Samples Winning for Alicia. In the dotplot, a red vertical line is drawn at the winning proportion criterion value. Any dot to the right of this line represents a sample proportion that will lead to an incorrect decision regarding the winner of the election. (10) Across all of the 1,000 samples, what percent of samples have a proportion of votes for Alicia Harper greater than 50%? (11) Given that the theoretical situation is that both candidates will be tied in the election, what does your answer to Question 8 indicate about your chances of incorrectly predicting that Alicia will win the election if, in fact, the final vote shows a tie? Wrap-­‐Up [Whole-­‐class discussion] Once each group has discussed the questions, have them report to the class. Point to the similarities and differences in the estimates found across the different groups, but emphasize the similarities (e.g., similar values for the minimum and maximum and, therefore, about 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. 4 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability range). Ask why they think that the groups did not get exactly the same results, and why the results are similar. This can be related this to the large number of samples (1,000). Transition to Task III In Task III, students begin exploring the idea of choosing a (critical/criterion) value that corresponds to a chosen percentage of samples (the level of significance), 1% or 5%, whose sample proportions would yield incorrect decisions (Type I errors) regarding the winner of the election. These critical values are determined through trial and error, and students are led to see that choosing an appropriate criterion value can be helpful in preventing such errors. Task III: Reducing the Risk of an Incorrect Prediction [Student Handout] Statisticians are willing to take risks, but they usually do not take big risks. They know that they cannot be 100% certain when basing decisions on samples. Still, statisticians prefer to make correct decisions most of the time. They are willing to accept a small risk of being wrong, but not more frequently than around 1–5% of the time. You can never know the truth about an election until all votes are counted. Thus, when you collect a random sample before all votes are tallied, you cannot be 100% certain that the prediction is correct. However, you can be somewhat confident in the prediction of a winner if you set a decision criterion appropriately. Assuming that Alicia Harper actually ties with the other candidate when the full vote is counted, you would not want to predict that she is the winner based on a sample of 500 voters. One way to change your prediction is to only predict a victory if the sample proportion for Alicia is greater than or equal to a certain proportion. You might think of this as a winning proportion criterion, which is a type of cutoff (or critical) value. Use the cutoff value as follows: •
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If Alicia’s sample proportion is less than the criterion value, do not predict that she will win. If Alicia’s sample proportion is greater than or equal to the criterion value, predict that she will win. To determine this criterion value, use the Winning Proportion Criterion entry in the Mayoral Race spreadsheet to change the rule. It is initially set at 0.50, but you can change it as you like. The red vertical line, representing the criterion value, moves as the value changes. Remember that any dot to the right of the red line represents a proportion that will lead to an incorrect decision. The percent of Alicia’s sample proportions that are greater than or equal to the cutoff value is reported below it (initially it will be around 50%). Change the value to answer the following questions: (12) Use trial and error to determine the smallest cutoff value for Alicia's sample proportions that would lead you to incorrectly predict that she will win the election about 5% of the time? (13) What value would you use for winning proportion criterion if you wanted to be incorrect only about 1% of the time? Is this criterion higher or lower than the one you found for Question 9? Why do you think this is the case? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Wrap-­‐Up [Whole-­‐class discussion] Once each group has discussed the questions, have them report to the class. Did they come up with similar criteria for being incorrect 5% and 1% of the time? Why can they not be 100% certain (i.e., be incorrect 0% of the time)? You can ask for the highest percent difference from each group to illustrate that this varies. Transition to Task IV In Task III, students learned that introducing a criterion value that increases the requirement for a sample proportion before naming a winner can help to minimize the likelihood of incorrectly declaring a winner. In Task IV, they learn that sample size can be used in a way that can also diminish the number of incorrect predictions. Task IV: The Role of Sample Size [Student Handout] Discuss the following questions in your group. (14) If you increase the sample size to 1,000, what do you think will happen to the range of sample proportions for Alicia Harper? Do you expect it to be about the same, larger, or smaller as for a sample of 500? Why? (15) Test your ideas by changing the sample size to 1,000 and simulating the new samples. Does the distribution’s range change? How? (16) For the winning proportion criterion discovered in Task III, does the increased sample size increase or decrease the percentage of samples that would lead to an incorrect decision? Wrap-­‐Up Summarize what the class found out about sampling variability. •
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Why do you need to consider sampling variability when making a prediction for a population proportion or percentage? What is the shape of the sampling distribution of sample proportions for Alicia Harper? How is the shape of the distribution of sample proportions affected by the increasing of sample sizes? For a given winning proportion criterion, does increasing the sample size increase or decrease the percentage of samples that lead to incorrect decisions? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Homework [Student Handout] In the 2008 United States election for president, candidate Barack Obama received a population proportion of 53% voting for him. (1) Suppose you generate 1,000 samples, each containing 250 random votes. Given that Obama won a majority of the vote, do you expect that Obama won in each of the 1,000 samples? (2) Open the spreadsheet located at the following web address: http://math.mtsac.edu/statistics/PresidentialRace.xls In the second row of the blue table, the candidates’ names are entered: Obama, McCain (grouped with other votes not for Obama). Immediately under Obama’s name, enter his winning proportion: 0.53. The proportion for McCain & Other is entered automatically using the rule of complements. Set the winning proportion criterion to 0.50 (50%). In what proportion of the samples does Obama win a majority? (3) Suppose you require a 51% proportion for Obama before you will name him as the winner. Indicate this in the spreadsheet by entering 0.51 as the Winning Proportion Criterion. In what proportion of the samples are you led to conclude correctly that Obama won the election? In what proportion of the samples are you not led to conclude correctly that Obama won the election? (Use the rule of complements.) (4) Increase the sample size so that approximately 95% of the samples lead you to conclude that Obama won the election with a 51% winning proportion criterion. What sample size would lead you to this correct conclusion 95% of the time? With this sample size, what proportion of samples would not lead you to conclude correctly that Obama won the election? (5) What have you done here to improve your chances of a correct conclusion? (6) From the dotplot of Obama’s simulated sample proportions, estimate the mean and compare this to the population proportion entered under Obama’s name in the spreadsheet. How does the estimated mean of sample proportions compare to Obama’s population proportion? (7) Describe the dotplot of Obama’s sample proportions in terms of shape and symmetry. (8) Give the range of sample proportions observed in the dotplot. from _______________ to _______________ 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Your team is working for a candidate named Alicia Harper who is running in an election for the mayor of a small city. There is another candidate running for mayor named José Abasta. The candidate with a majority of the total votes wins the election. Your candidate wants to know as soon as possible on Election Day whether she is likely to win the election. Your team’s assignment is to conduct an opinion poll of voters as they exit the polling stations to predict whether Alicia will win the election. Opinion polls are often conducted by taking a random sample of several hundred to several thousand people. Your team will ask a random sample of people who finished voting at each polling station whom they voted for: Alicia Harper or José Abasta. Task I: Creating a Sampling Plan Discuss and devise a plan for how you would select people to ask whom they voted for. There are 25 polling stations in the small city holding the election. You want to collect information from 500 people. (Note: Once every group decides on a plan, your instructor will lead a whole-­‐class discussion of the different plans.) (1) How many people would you randomly select at each polling station? (2) How would you go about selecting each person so that the selection process was truly random? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Task II: Estimating Chances of an Incorrect Prediction Discuss the following questions in your small group: (3) If you use your sampling plan and find that Alicia Harper has a majority of voters in the sample of 500 people, does that mean that she will win the election? Why or why not? (4) In the total population of voters, if half of the people voted for each candidate, you would not want to declare that Alicia Harper would win the election. Could Alicia have a larger proportion than the other candidate in a random sample of 500 voters even if in the population there was an even split between the two candidates (in other words, each candidate wins exactly 1/2 of the votes)? To answer these questions, your team will use a spreadsheet that allows you to simulate many samples and see what might happen if you repeated the sampling process over and over again. This should help you see whether you can expect the same results every time you take a random sample for your poll. The spreadsheet is located at the following web address: http://math.mtsac.edu/statistics/MayoralRace.xls The Mayoral Race spreadsheet allows you to set two values: • The population proportion of all voters who will theoretically vote for each candidate. Enter •
1/2 (0.50) for Alicia Harper’s population proportion and the spreadsheet will use the rule of complements to assign a population proportion for José Abasta. The number of people you will survey in each sample. Enter 500 into this cell. With the proportions for each candidate and the sample size entered, the spreadsheet generates 1,000 simulated sample proportions for Alicia, distributed according to the binomial distribution. These sample proportions are then plotted on a dotplot. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability A sampling distribution is the distribution of all possible sample proportions from samples of a given size. The simulation spreadsheet generates 1,000 of these, but this is only a subset of the entire distribution. (5) Describe the shape of the distribution of Alicia Harper’s sample proportions in terms of shape and symmetry. (6) Visually estimate the mean of the simulated proportions for Alicia. (7) Is the estimated mean of Alicia’s sample proportions similar to the assumed population proportion that was entered in the spreadsheet for her? Sample Size Versus Number of Samples When people learn about sampling distributions, many confuse the ideas of sample size and the number of samples. A sample consists of a collection of observations from population members. That sample has a certain size (e.g., the 500 randomly selected people for the election results samples). When you randomly select the 500 people, you have selected one sample. A randomly selected sample is just one possible sample from an entire population. While the many different samples you might gather have much in common, you should not expect each sample to be identical. This means that results, such as the proportion of votes for a particular candidate, will be different from one sample to the next. Some differences may be quite small and others quite large. To get a sense of how the proportions will vary, you can collect many samples, where each sample has the same size. How many samples do you need to get a good sense of the variation? Most statisticians agree that 1,000 samples can provide a good sense of the sampling variation. In summary, you are gathering 1,000 random samples, each containing 500 randomly selected people. The reason for gathering 1,000 samples is to provide a good picture of the variation present in a sampling distribution (how the proportions of voters who voted for Alicia Harper differ between random samples). 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability There is one more feature of the Mayoral Race spreadsheet that can be changed called the Winning Proportion Criterion. For right now, set this to 0.50. When the values are entered, a total of 1,000 samples are automatically generated. Answer the following questions. (8) Is the proportion of people who voted for Alicia Harper the same for each of the 1,000 samples? How do you know this? (9) Estimate a typical range for the sample proportions of votes for Alicia. from _______________ to _______________ The spreadsheet can show you the percentage of sample proportions for Alicia Harper that are greater than a given sample proportion entered under Winning Proportion Criterion. This percentage is listed next to % of Samples Winning for Alicia. In the dotplot, a red vertical line is drawn at the winning proportion criterion value. Any dot to the right of this line represents a sample proportion that will lead to an incorrect decision regarding the winner of the election. (10) Across all of the 1,000 samples, what percent of samples have a proportion of votes for Alicia Harper greater than 50%? (11) Given that the theoretical situation is that both candidates will be tied in the election, what does your answer to Question 8 indicate about your chances of incorrectly predicting that Alicia will win the election if, in fact, the final vote shows a tie? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Task III: Reducing the Risk of an Incorrect Prediction Statisticians are willing to take risks, but they usually do not take big risks. They know that they cannot be 100% certain when basing decisions on samples. Still, statisticians prefer to make correct decisions most of the time. They are willing to accept a small risk of being wrong, but not more frequently than around 1–5% of the time. You can never know the truth about an election until all votes are counted. Thus, when you collect a random sample before all votes are tallied, you cannot be 100% certain that the prediction is correct. However, you can be somewhat confident in the prediction of a winner if you set a decision criterion appropriately. Assuming that Alicia Harper actually ties with the other candidate when the full vote is counted, you would not want to predict that she is the winner based on a sample of 500 voters. One way to change your prediction is to only predict a victory if the sample proportion for Alicia is greater than or equal to a certain proportion. You might think of this as a winning proportion criterion, which is a type of cutoff (or critical) value. Use the cutoff value as follows: •
•
If Alicia’s sample proportion is less than the criterion value, do not predict that she will win. If Alicia’s sample proportion is greater than or equal to the criterion value, predict that she will win. To determine this criterion value, use the Winning Proportion Criterion entry in the Mayoral Race spreadsheet to change the rule. It is initially set at 0.50, but you can change it as you like. The red vertical line, representing the criterion value, moves as the value changes. Remember that any dot to the right of the red line represents a proportion that will lead to an incorrect decision. The percent of Alicia’s sample proportions that are greater than or equal to the cutoff value is reported below it (initially it will be around 50%). Change the value to answer the following questions: (12) Use trial and error to determine the smallest cutoff value for Alicia's sample proportions that would lead you to incorrectly predict that she will win the election about 5% of the time? (13) What value would you use for winning proportion criterion if you wanted to be incorrect only about 1% of the time? Is this criterion higher or lower than the one you found for Question 9? Why do you think this is the case? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Task IV: The Role of Sample Size Discuss the following questions in your group. (14) If you increase the sample size to 1,000, what do you think will happen to the range of sample proportions for Alicia Harper? Do you expect it to be about the same, larger, or smaller as for a sample of 500? Why? (15) Test your ideas by changing the sample size to 1,000 and simulating the new samples. Does the distribution’s range change? How? (16) For the winning proportion criterion discovered in Task III, does the increased sample size increase or decrease the percentage of samples that would lead to an incorrect decision? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability Homework In the 2008 United States election for president, candidate Barack Obama received a population proportion of 53% voting for him. (1) Suppose you generate 1,000 samples, each containing 250 random votes. Given that Obama won a majority of the vote, do you expect that Obama won in each of the 1,000 samples? (2) Open the spreadsheet located at the following web address: http://math.mtsac.edu/statistics/PresidentialRace.xls In the second row of the blue table, the candidates’ names are entered: Obama, McCain (grouped with other votes not for Obama). Immediately under Obama’s name, enter his winning proportion: 0.53. The proportion for McCain & Other is entered automatically using the rule of complements. Set the winning proportion criterion to 0.50 (50%). In what proportion of the samples does Obama win a majority? (3) Suppose you require a 51% proportion for Obama before you will name him as the winner. Indicate this in the spreadsheet by entering 0.51 as the Winning Proportion Criterion. In what proportion of the samples are you led to conclude correctly that Obama won the election? In what proportion of the samples are you not led to conclude correctly that Obama won the election? (Use the rule of complements.) 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.1.1: Predicting an Election—Statistics and Sampling Variability (4) Increase the sample size so that approximately 95% of the samples lead you to conclude that Obama won the election with a 51% winning proportion criterion. What sample size would lead you to this correct conclusion 95% of the time? With this sample size, what proportion of samples would not lead you to conclude correctly that Obama won the election? (5) What have you done here to improve your chances of a correct conclusion? (6) From the dotplot of Obama’s simulated sample proportions, estimate the mean and compare this to the population proportion entered under Obama’s name in the spreadsheet. How does the estimated mean of sample proportions compare to Obama’s population proportion? (7) Describe the dotplot of Obama’s sample proportions in terms of shape and symmetry. (8) Give the range of sample proportions observed in the dotplot. from _______________ to _______________ 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 Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Number of 50-­‐minute class sessions: 2 Materials/Resources Needed •
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One large bag of Reese’s Pieces candies and one paper cup per student. There are about 600 candies per pound (serving 24 students), so plan accordingly! Reese’s Pieces simulation process model figure—printed or projected on a screen Web applet at http://statweb.calpoly.edu/chance/applets/Reeses/ReesesPieces.html Learning Goals Students will understand •
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the difference between a population, a sample, and a sampling distribution. the difference between a sample proportion and a population proportion. how samples vary. that while statistics vary from sample to sample, the population parameter is fixed and unchanging. that statistics vary according to a pattern called the sampling distribution. the characteristics of a sampling distribution (center, shape, and variability). that the size of a sample affects the precision of an estimate from the sample, and that in particular, sample size affects the variability of the sampling distribution. that while large samples give better estimates, larger populations do not require larger samples for good estimates. Students will be able to •
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build and describe the characteristics of sampling distributions of sample proportions, in terms of shape, center, and spread. predict the effect of sample size on the shape and spread of a sampling distribution of sample proportions. understand why a prediction based on a single sample tends to be somewhat unreliable. Lesson Overview In this lesson, students make and test conjectures about sample proportions for orange-­‐colored candies in a population of Reese’s Pieces. They consider the population from which they are sampling to be the population of all Reese’s Pieces colors (orange or not orange) and compute sample proportions in an attempt to estimate the population proportion of Reese’s Pieces that are orange. Students take physical samples from a bag of Reese’s Pieces and construct distributions of sample proportions. Students then discuss an overview of the simulation process model to help them grasp the structure of a sampling distribution. Students then use a web applet to generate a larger number of candy samples, enabling them to examine the distribution of sample proportions for different sample sizes. Students map the simulation of sample proportions to the simulation process model, a visual scheme that distinguishes between the population, samples, and distribution of sample statistics. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Notes to Instructors In Part I of this activity, students gain first-­‐hand experience in constructing a sampling distribution. Each student measures one sample proportion of orange candies from a sample of 25 Reese’s Pieces candies. On your chalkboard or whiteboard, create an empty table in which students can record the number of orange candies in their sample, along with the sample proportion. Name Number of Orange Candies Sample Proportion From this table, students create a dotplot of all students’ sample proportions and then discuss the characteristics of the distribution. If there is time before class, it can speed things up if one cup of 25 Reese’s Pieces is prepared for each student for Part I of the activity. If this is not possible, it might be useful to have a container of the candy for each group from which students can count candies. Students should be encouraged to sample randomly from the container, avoiding selection bias (favoring any color over the others). Part II gives an overview of the simulation process, known as the simulation process model, in which students simulate a sampling distribution and then answer a collection of questions with their group. Part III extends the simulation activity by having students simulate many sample proportions, using a web-­‐based computer applet. If computers (and internet access) are not available for students to use in class, the applet can be projected onto a screen from the instructor’s computer, with the entire class working as a single group. Begin by discussing some of the following questions with students, depending on your view of their strengths and weaknesses in understanding. •
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What is the difference between a sample and a population? What is a proportion? Is a sample proportion a statistic or a parameter? Is a population proportion a statistic or a parameter? What is the distribution of statistics computed from all possible samples of a given size called? Can a small sample of Reese’s Pieces candies give a good estimate of the population proportion of all orange Reese’s Pieces candies produced by the Hershey Company? Take a moment to review proportions, reminding students that instead of representing proportions as percentages, they will record them as fractional values between 0 and 1. Also, make sure that students are as clear as possible regarding the meaning of a sampling distribution. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Introduction to the Context of the Lesson [Student Handout] In a previous lesson, you began thinking about the fact that from a given population, many samples can be gathered. If you are interested in measuring sample proportions, each sample can yield a sample proportion, but because samples are different, many sample proportions will arise in the process. When you use a dotplot to look at a collection of such sample proportions, you see that they tend to be distributed in a familiar pattern, and this familiarity helps you make inferences about the population from which the samples are drawn. In this lesson, you will continue thinking about •
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how samples are drawn from a population, how to estimate a feature of the population (a parameter) from a corresponding feature of a sample (a statistic), and how samples (and statistics) vary. Part I: Reese’s Pieces Party [Student Handout] Here, you will continue studying population proportions, but in the context of something that may seem less serious than elections or census-­‐taking: the proportions of different colors of Reese’s Pieces candy†. You may think that investigating something like this is unimportant, but this is not necessarily true. In addition to taste, other characteristics of food products—such as size, aroma, texture, “mouth feel,” shininess, and color—play an important role in consumer preferences. Companies care very much about such information. Analysts use statistical calculations to ensure that food products meet specified consumer preferences, since such preferences can greatly affect product sales. Image source: http://en.wikipedia.org/wiki/File:Reeses-­‐pieces-­‐
loose.JPG. Image released into the public domain by the copyright holder. In the following activities, you will focus on this question: What is the proportion of orange-­‐colored candies for the population of all Reese’s Pieces that are produced and sold nationwide? Such a proportion is called the population proportion, since it pertains to all Reese’s Pieces. It would be impossible or impractical to determine the exact value of the population proportion, since it would require us to count every single Reese’s Piece produced and to record whether it is orange or not. You can estimate the value of the population proportion by taking a sample of Reese’s pieces and calculating the proportion of orange-­‐colored candies in that sample. Such a proportion is called the sample proportion. Then, you can repeatedly take samples and record the sample proportions of orange-­‐colored candies for each sample by plotting them on a dotplot. As the number of samples—and thus the number of sample proportions that you calculate—increases, you will start to see how these sample proportions are distributed. From this distribution, you can determine an estimate of the population proportion. Before you do this, let’s consider a scenario that illustrates the concepts of a population, sampling, a sampling proportion, and a population proportion: † The Dana Center would like to acknowledge that the activities in this section were inspired by an exercise in Workshop Statistics, by Allan Rossman and Beth Chance. See the References section for a full citation. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Elliott loves Reese’s Pieces and has a very large bowl of them at his birthday party. What his guests do not know is that he is secretly doing a study for his Statistics class and wants them to help him with data collection. As his guests arrive, he hands each of them a small plastic cup and asks them to scoop up a cupful of Reese’s Pieces from the large bowl. Elliott asks them not to eat the candy until they each provide him with two numbers: the total number of Reese’s Pieces in the cup and the number of orange-­‐colored pieces. He then asks each of his guests to write his or her numbers on a large sheet of paper that is tacked to the wall. The results are shown in the following table. Total Number of Pieces Number of Orange Pieces Tom 35 18 Amy 36 20 Rachel 40 24 Susan 47 16 Erica 51 15 Jenna 47 21 Phillip 38 20 Kris 36 16 Steve 46 20 Joey 48 27 Name Proportion of Orange Pieces (p) (1) Work alone or with a partner to calculate the proportion of orange pieces for each of Elliott’s party guests. (Answer: Total Number of Pieces Number of Orange Pieces Proportion of orange pieces (p) Tom 35 18 0.51 Amy 36 20 0.56 Rachel 40 24 0.60 Susan 47 16 0.34 Erica 51 15 0.29 Jenna 47 21 0.45 Phillip 38 20 0.53 Kris 36 16 0.44 Steve 46 20 0.43 Joey 48 27 0.56 Name 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(Note: Facilitate class discussion by asking probing questions: What do you notice about the proportions you calculated? Did you expect different results? Most students will note that the sample proportions are in the vicinity of a single value, but that there is variability, due to the random nature of sampling, i.e., scooping out candies from the large bowl. Perhaps a few students expected all of the proportions of orange pieces to be the same. If nobody expected the proportions to be the same, ask the students why they would not expect the proportions to be the same. Ensure that students understand and are using the words variability and random in their proper contexts.) (2) The figure below represents the large bowl at Elliott’s party and the small cups of Reese’s Pieces that each guest scooped up. Match the terms below to their corresponding numbered labels in the figure. •
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population sample sample proportion population proportion Image source: Dana Center Staff, created using OpenOffice.org software SPOILER ALERT!!! Do not read ahead until you have completed this activity! 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(Answers: 1: population 2: population proportion 3: sample 4: sample proportion) Before you continue to the next activity, you should become familiar with two important terms that are used to describe numerical information related to populations and samples. A sample proportion, such as the proportion of orange-­‐colored candies that you or one of your classmates calculated, is an example of a statistic—its value varies from sample to sample. A population proportion, such as the proportion of all orange-­‐colored Reese’s Pieces, is an example of a population parameter—its value does not change. (3) Now it’s time to have your own Reese’s Pieces party in class! This activity is a statistical simulation of the sampling process described in the scenario that you just read about. Your instructor will fill small cups with Reese’s Pieces from a large bag and distribute them to each student. Determine the sample proportion for your cup of Reese’s Pieces: the number of orange-­‐colored pieces divided by the total number of pieces. Report your sample proportion to the instructor. (Note: If you have a relatively small class, i.e., fewer than 15 students, it would be better to have each student take two or three samples to ensure that the plot to be generated will appear to be approximately normal. Either poll the students for their sample proportions or ask each student to write his/her sample proportion(s) on the board or on a large sheet of paper that the whole class can easily view from their seats.) (4) After the class has reported all their sample proportions, use this dataset to create a dotplot of the distribution of the sample proportions that you and your classmates calculated. This distribution is called the sampling distribution. Use the axes below to construct your dotplot. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(5) Based on the distribution created by the entire class, what would you estimate to be the population proportion of orange Reese’s Pieces candies? (Answer: Students should identify the estimate to be the center of the distribution they created.) (6) Did everyone in the class have the same number of orange candies? (Answer: No) (7) How do the actual sample values compare to the ones you estimated for the party guests? (Answer: They are similar.) (8) Did everyone have the same proportion of orange candies? (Answer: No) (9) Describe the variability of the distribution of sample proportions (drawn on the board) in terms of shape, center, and spread. (Answer: The distribution has one mode and is slightly symmetric, the center is around 0.45, and the range is around 0.40.) (10) Do you know the proportion of orange candies in the population of all Reese’s Pieces? (Answer: No) In the samples? (Answer: Yes) (11) Which value can you always calculate, the population proportion or the sample proportion? (Answer: The sample proportion) Which one do you have to estimate? (Answer: The population proportion) (12) Does the value of the parameter change each time you take a sample? (Answer: The parameter does not change.) Does the value of the statistic change each time you take a sample? (Answer: The statistic changes from sample to sample.) 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(13) How do the sample proportions compare to the population parameter (the proportion of all Reese’s Pieces candies produced by the Hershey Company that are orange)? (Answer: The estimates are likely to be spread on either side of the population proportion.) Wrap-­‐Up Discuss the following questions with the entire class: •
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What is the collection of all sample proportions computed from samples of a given size called? (Answer: The sampling distribution of sample proportions) Are most sample proportions in the sampling distribution (the statistics) equal to the population proportion (the parameter)? (Answer: No) Why are the statistics different from the parameter that they estimate? (Answer: They are measured from samples, and samples always yield sampling error.) What do you guess the shape is of an entire sampling distribution of all possible sample proportions from a given sample size? (Answer: Bell-­‐shaped) Lecture: The Levels of Data in Sampling Distributions You can describe the simulation of sample proportions using a simulation process model, which is simply an overview of the levels of data present in a sampling distribution of sample proportions. In the construction of a sampling distribution, three levels of data are considered: population, samples, and a sampling distribution of statistics. Discuss how these levels apply to the context of the Reese’s Pieces activity. The population consists of all Reese’s candy pieces made by the manufacturer—either orange or not orange. The unknown proportion of orange pieces in this population is π. Population Samples and the statistics computed from each sample The samples consist of the candy colors in cups of 25 candies, and the statistics are the sample proportions of orange candies in each cup. The distribution of sample statistics is the dotplot students drew on the board. It is only a part of the entire sampling distribution, as it depicts the distribution of all sample proportions from samples of a given size. Each sample proportion is denoted by p. Distribution of sample statistics for many samples 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 Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Simulation Process Model for Reese’s Pieces Activity [Student Handout] This diagram depicts the levels of data in the simulation process model. This particular process model shows that forty samples were taken from the population. Image source: Dana Center Staff, created using OpenOffice.org software, inspired by a Bob delMas’ image. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Part II: Simulating a Sampling Distribution [Student Handout] (Note: In Part II of this lesson, students use a computer to further simulate part of the sampling distribution of sample proportions of orange Reese’s Pieces. The concepts covered here are similar to the concepts covered in the students’ initial distribution, but the concepts are now revisited on a larger scale.) A larger scale investigation of the variability present in the sample proportions of orange Reese’s Pieces would be helpful to solidify your conjectures made in Part I. Counting out candies is time consuming and inefficient, so you will now use a computer applet to simulate additional sample proportions. The applet requires the population proportion of all Reese’s Pieces that are orange, which is denoted by π. Go to the course website and click on the Web Applet: Reese’s Pieces link, or browse to: http://statweb.calpoly.edu/chance/applets/Reeses/ReesesPieces.html In the applet, you will see a container of colored candies that represents the population of all Reese’s Pieces candies. (14) According to the applet, what is the population proportion (π) of candies that are orange? (15) How does 0.45 compare to the proportions of orange candies calculated by the party guests that you calculated in Question 1? (16) How does 0.45 compare to the center of the class distribution (check your dotplot from Question 4)? Does it seem like a plausible value for the population proportion of orange candies? Explain your reasoning. Simulation To create a single simulated proportion, click on the Draw Samples button in the Reese’s Pieces applet. One sample of 25 candies is taken, and the proportion of orange candies for this sample is plotted on the graph. Draw a second sample by repeating this process again. (17) Do you get the same or different values for each sample proportion? (18) How do these numbers compare to those generated by your class with real candy? (19) Are the sample statistics (proportion) similar to the population parameter? Further Simulation Next, proceed to the simulation of many sample proportions. First, uncheck the Animate box. Change the number of samples (num samples) to 500, click the Draw Samples button, and see the distribution of sample statistics (proportions). (20) Describe the shape, center, and spread of the distribution of sample statistics. 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. 10 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(21) How does this distribution compare to the one your class constructed on the board in terms of shape? Center? Spread? (22) Where does the value of 0.2 (i.e., 5 orange candies) fall in the distribution of sample proportions? Is it in a tail or near the middle? Does this seem like a rare or unusual result? Wrap-­‐Up In Part II, students learned that computers can simulate the sampling process to create a collection of sample proportions that are distributed similarly to those that are computed from real samples. The advantage to using computers, however, is that they are able to generate many sample proportions in a short period of time. This allows students to discover the nature of the distribution of sample proportions. In this simulation, they learned that sample proportions have a roughly symmetric distribution that appears bell-­‐shaped. Transition to Part III In this final part of this lesson, students explore the effect that sample size has on the shape and spread of the sampling distribution of sample proportions. Before beginning, ask students to guess what the effect of increased sample size will have on the shape and spread of the sampling distribution. Once students have thought about this question, they can return to the Reese’s Pieces applet to answer the questions. Part III: The Role of Sample Size [Student Handout] Consider what happens to the distribution of sample statistics if you change the number of candies in each sample (change the sample size). (23) What do you think will happen to the distribution of sample proportions if you change the sample size to 10? Explain your reasoning. (24) What do you think will happen if you change the sample size to 100? Explain your reasoning. To test your conjecture, change the Sample Size in the Reese’s Pieces applet to 10. Be sure the number of samples (num samples) is 500. Click the Draw Samples button. (25) How close are the sample statistics (proportions), in general, to the population parameter? Next, change the sample size in the Reese’s Pieces applet to 100, and draw 500 samples. Be sure the number of samples (num samples) is 500. Click the Draw Samples button. (26) How close are the sample statistics (proportions), in general, to the population parameter? 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. 11 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(27) As the sample size increases, what happens to the distance between the sample statistics and the population parameter? (28) Describe the effect of sample size on the distribution of sample statistics in terms of shape, center, and spread. Wrap-­‐Up Using computer simulations, students were able to learn the relationship between sample size and the shape and spread of a distribution. Larger samples tend to give better estimates of unknown parameters that therefore vary less. The distribution of estimates from larger sample sizes is, therefore, narrower. This principle is often summarized as the Law of Large Numbers. The Law of Large Numbers: Sample proportions p tend to become closer to the population proportion, π, as sample size increases. In this activity, students examined closely the construction of a sampling distribution of sample proportions through manual sampling and then through computer simulations. Knowing the nature of a distribution of statistics, such as sample proportions, can help in making inferences regarding the parameters that those statistics estimate. In the following lessons, students continue to develop the ideas of sampling distributions with the ultimate goal of developing a structure for making formal inferences. Reference Rossman, A., & Chance, B., Workshop Statistics, 3 ed., Wiley Publishing, 2008. Homework [Student Handout] For your homework, open the spreadsheet at: http://math.mtsac.edu/statistics/YellowReeses.xls This spreadsheet creates a simulation of sampling distributions for sample proportions of Reese’s Pieces that are yellow, corresponding to a particular sample size. Suppose you are interested in investigating if 1/3 of the population of all Reese’s Pieces is yellow. In the simulation spreadsheet, enter the assumed proportion (0.3333 [≈ 1/3]) of Reese’s Pieces that are yellow. The proportion of orange pieces is already set. (1) To gather evidence, suppose you sample of 25 Reese’s Pieces and observe that 9 are colored yellow. Compute the sample proportion corresponding to this sample. "
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# Enter this sample proportion into the spreadsheet next to Observed sample proportion of yellow. Enter the sample size into the spreadsheet, below the cell labeled n. 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. 12 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(2) The dotplot contains 1,000 simulated sample proportions of yellow Reese’s Pieces, generated under the assumption that 1/3 of all Reese’s Pieces are yellow. Read the spreadsheet to determine what percentage of the random samples had proportions greater than the one observed in Question 1. (3) Give a range of values within which nearly all (but not absolutely all) sample proportions lie. (4) Suppose in a sample of size n = 1,000, you observe the same proportion given in Question 1. Regenerate the simulated sampling distribution with this larger size, and give the proportion of samples that have a proportion of yellow candies greater than or equal to yours. (5) With this larger sample size, do you consider your observed proportion of yellow candies unusually large? Why? (6) You assumed without proof that 1/3 of all Reese’s Pieces are yellow. If your sample proportion was observed in a random sample of size 1,000, would you agree with the assumption that the proportion of all Reese’s Pieces that are yellow is 1/3? (7) Give a range of values within which nearly all (but not absolutely all) sample proportions lie with this larger sample size. (8) Enter 0.35 under Yellow, and use the percentage of sample proportions that are greater than yours to judge if yours is unusually large. Do you consider 0.35 to be a more likely population proportion for the yellow candies? 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. 13 Statway Student Handout April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Introduction to the Context of the Lesson In a previous lesson, you began thinking about the fact that from a given population, many samples can be gathered. If you are interested in measuring sample proportions, each sample can yield a sample proportion, but because samples are different, many sample proportions will arise in the process. When you use a dotplot to look at a collection of such sample proportions, you see that they tend to be distributed in a familiar pattern, and this familiarity helps you make inferences about the population from which the samples are drawn. In this lesson, you will continue thinking about •
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how samples are drawn from a population, how to estimate a feature of the population (a parameter) from a corresponding feature of a sample (a statistic), and how samples (and statistics) vary. Part I: Reese’s Pieces Party Here, you will continue studying population proportions, but in the context of something that may seem less serious than elections or census-­‐taking: the proportions of different colors of Reese’s Pieces candy†. You may think that investigating something like this is unimportant, but this is not necessarily true. In addition to taste, other characteristics of food products—such as size, aroma, texture, “mouth feel,” shininess, and color—play an important role in consumer preferences. Companies care very much about such information. Analysts use statistical calculations to ensure that food products meet specified consumer preferences, since such preferences can greatly affect product sales. Image source: http://en.wikipedia.org/wiki/File:Reeses-­‐pieces-­‐
loose.JPG. Image released into the public domain by the copyright holder. In the following activities, you will focus on this question: What is the proportion of orange-­‐colored candies for the population of all Reese’s Pieces that are produced and sold nationwide? Such a proportion is called the population proportion, since it pertains to all Reese’s Pieces. It would be impossible or impractical to determine the exact value of the population proportion, since it would require us to count every single Reese’s Piece produced and to record whether it is orange or not. You can estimate the value of the population proportion by taking a sample of Reese’s pieces and calculating the proportion of orange-­‐colored candies in that sample. Such a proportion is called the sample proportion. Then, you can repeatedly take samples and record the sample proportions of orange-­‐colored candies for each sample by plotting them on a dotplot. As the number of samples—and thus the number of sample proportions that you calculate—increases, you will start to see how these sample proportions are distributed. From this distribution, you can determine an estimate of the population proportion. Before you do this, let’s consider a scenario that illustrates the concepts of a population, sampling, a sampling proportion, and a population proportion: † The Dana Center would like to acknowledge that the activities in this section were inspired by an exercise in Workshop Statistics, by Allan Rossman and Beth Chance. See the References section for a full citation. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Elliott loves Reese’s Pieces and has a very large bowl of them at his birthday party. What his guests do not know is that he is secretly doing a study for his Statistics class and wants them to help him with data collection. As his guests arrive, he hands each of them a small plastic cup and asks them to scoop up a cupful of Reese’s Pieces from the large bowl. Elliott asks them not to eat the candy until they each provide him with two numbers: the total number of Reese’s Pieces in the cup and the number of orange-­‐colored pieces. He then asks each of his guests to write his or her numbers on a large sheet of paper that is tacked to the wall. The results are shown in the following table. Total Number of Pieces Number of Orange Pieces Tom 35 18 Amy 36 20 Rachel 40 24 Susan 47 16 Erica 51 15 Jenna 47 21 Phillip 38 20 Kris 36 16 Steve 46 20 Joey 48 27 Name Proportion of Orange Pieces (p) (1) Work alone or with a partner to calculate the proportion of orange pieces for each of Elliott’s party guests. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(2) The figure below represents the large bowl at Elliott’s party and the small cups of Reese’s Pieces that each guest scooped up. Match the terms below to their corresponding numbered labels in the figure. •
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population sample sample proportion population proportion Image source: Dana Center Staff, created using OpenOffice.org software SPOILER ALERT!!! Do not read ahead until you have completed this activity! 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Before you continue to the next activity, you should become familiar with two important terms that are used to describe numerical information related to populations and samples. A sample proportion, such as the proportion of orange-­‐colored candies that you or one of your classmates calculated, is an example of a statistic—its value varies from sample to sample. A population proportion, such as the proportion of all orange-­‐colored Reese’s Pieces, is an example of a population parameter—its value does not change. (3) Now it’s time to have your own Reese’s Pieces party in class! This activity is a statistical simulation of the sampling process described in the scenario that you just read about. Your instructor will fill small cups with Reese’s Pieces from a large bag and distribute them to each student. Determine the sample proportion for your cup of Reese’s Pieces: the number of orange-­‐colored pieces divided by the total number of pieces. Report your sample proportion to the instructor. (4) After the class has reported all their sample proportions, use this dataset to create a dotplot of the distribution of the sample proportions that you and your classmates calculated. This distribution is called the sampling distribution. Use the axes below to construct your dotplot. (5) Based on the distribution created by the entire class, what would you estimate to be the population proportion of orange Reese’s Pieces candies? (6) Did everyone in the class have the same number of orange candies? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(7) How do the actual sample values compare to the ones you estimated for the party guests? (8) Did everyone have the same proportion of orange candies? (9) Describe the variability of the distribution of sample proportions (drawn on the board) in terms of shape, center, and spread. (10) Do you know the proportion of orange candies in the population of all Reese’s Pieces? In the samples? (11) Which value can you always calculate, the population proportion or the sample proportion? Which one do you have to estimate? (12) Does the value of the parameter change each time you take a sample? Does the value of the statistic change each time you take a sample? (13) How do the sample proportions compare to the population parameter (the proportion of all Reese’s Pieces candies produced by the Hershey Company that are orange)? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
This diagram depicts the levels of data in the simulation process model. This particular process model shows that forty samples were taken from the population. Image source: Dana Center Staff, created using OpenOffice.org software, inspired by a Bob delMas’ image. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Part II: Simulating a Sampling Distribution A larger scale investigation of the variability present in the sample proportions of orange Reese’s Pieces would be helpful to solidify your conjectures made in Part I. Counting out candies is time consuming and inefficient, so you will now use a computer applet to simulate additional sample proportions. The applet requires the population proportion of all Reese’s Pieces that are orange, which is denoted by π. Go to the course website and click on the Web Applet: Reese’s Pieces link, or browse to: http://statweb.calpoly.edu/chance/applets/Reeses/ReesesPieces.html In the applet, you will see a container of colored candies that represents the population of all Reese’s Pieces candies. (14) According to the applet, what is the population proportion (π) of candies that are orange? (15) How does 0.45 compare to the proportions of orange candies calculated by the party guests that you calculated in Question 1? (16) How does 0.45 compare to the center of the class distribution (check your dotplot from Question 4)? Does it seem like a plausible value for the population proportion of orange candies? Explain your reasoning. Simulation To create a single simulated proportion, click on the Draw Samples button in the Reese’s Pieces applet. One sample of 25 candies is taken, and the proportion of orange candies for this sample is plotted on the graph. Draw a second sample by repeating this process again. (17) Do you get the same or different values for each sample proportion? (18) How do these numbers compare to those generated by your class with real candy? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(19) Are the sample statistics (proportion) similar to the population parameter? Further Simulation Next, proceed to the simulation of many sample proportions. First, uncheck the Animate box. Change the number of samples (num samples) to 500, click the Draw Samples button, and see the distribution of sample statistics (proportions). (20) Describe the shape, center, and spread of the distribution of sample statistics. (21) How does this distribution compare to the one your class constructed on the board in terms of shape? Center? Spread? (22) Where does the value of 0.2 (i.e., 5 orange candies) fall in the distribution of sample proportions? Is it in a tail or near the middle? Does this seem like a rare or unusual result? Part III: The Role of Sample Size Consider what happens to the distribution of sample statistics if you change the number of candies in each sample (change the sample size). (23) What do you think will happen to the distribution of sample proportions if you change the sample size to 10? Explain your reasoning. (24) What do you think will happen if you change the sample size to 100? Explain your reasoning. To test your conjecture, change the Sample Size in the Reese’s Pieces applet to 10. Be sure the number of samples (num samples) is 500. Click the Draw Samples button. (25) How close are the sample statistics (proportions), in general, to the population parameter? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Next, change the sample size in the Reese’s Pieces applet to 100, and draw 500 samples. Be sure the number of samples (num samples) is 500. Click the Draw Samples button. (26) How close are the sample statistics (proportions), in general, to the population parameter? (27) As the sample size increases, what happens to the distance between the sample statistics and the population parameter? (28) Describe the effect of sample size on the distribution of sample statistics in terms of shape, center, and spread. Reference Rossman, A., & Chance, B., Workshop Statistics, 3 ed., Wiley Publishing, 2008. 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 Student Handout April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
Homework For your homework, open the spreadsheet at: http://math.mtsac.edu/statistics/YellowReeses.xls This spreadsheet creates a simulation of sampling distributions for sample proportions of Reese’s Pieces that are yellow, corresponding to a particular sample size. Suppose you are interested in investigating if 1/3 of the population of all Reese’s Pieces is yellow. In the simulation spreadsheet, enter the assumed proportion (0.3333 [≈ 1/3]) of Reese’s Pieces that are yellow. The proportion of orange pieces is already set. (1) To gather evidence, suppose you sample of 25 Reese’s Pieces and observe that 9 are colored yellow. Compute the sample proportion corresponding to this sample. "
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# Enter this sample proportion into the spreadsheet next to Observed sample proportion of yellow. Enter the sample size into the spreadsheet, below the cell labeled n. (2) The dotplot contains 1,000 simulated sample proportions of yellow Reese’s Pieces, generated under the assumption that 1/3 of all Reese’s Pieces are yellow. Read the spreadsheet to determine what percentage of the random samples had proportions greater than the one observed in Question 1. (3) Give a range of values within which nearly all (but not absolutely all) sample proportions lie. (4) Suppose in a sample of size n = 1,000, you observe the same proportion given in Question 1. Regenerate the simulated sampling distribution with this larger size, and give the proportion of samples that have a proportion of yellow candies greater than or equal to yours. (5) With this larger sample size, do you consider your observed proportion of yellow candies unusually large? Why? 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. 10 Statway Student Handout April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.2: Sampling from a Population
(6) You assumed without proof that 1/3 of all Reese’s Pieces are yellow. If your sample proportion was observed in a random sample of size 1,000, would you agree with the assumption that the proportion of all Reese’s Pieces that are yellow is 1/3? (7) Give a range of values within which nearly all (but not absolutely all) sample proportions lie with this larger sample size. (8) Enter 0.35 under Yellow, and use the percentage of sample proportions that are greater than yours to judge if yours is unusually large. Do you consider 0.35 to be a more likely population proportion for the yellow candies? 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. 11 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses Estimated number of 50-­‐minute class sessions: 1 Materials Required At least 40 pennies—each student should receive at least 1 penny, but more than 1 if you have fewer than 40 students. Learning Goals Students will understand •
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formal ideas of statistical inference. the process and language of significance tests at an informal level. the use of simulation to conduct an informal test of significance. the use of a P-­‐value in a test of significance. Students will be able to •
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connect informal to formal ideas of statistical inference. use a simulated sampling distribution to estimate a P-­‐value. use an estimated P-­‐value to test if a result is statistically significant. Lesson Structure This lesson uses the context of tossing a coin to introduce formal ideas of hypothesis testing. Students gather data for tossed coins to count the proportion of times their penny falls landing heads up. This proportion is used to test a null distribution based on equally likely outcomes. All ideas of tests of significance are introduced without any formulas or computations. The idea of the P-­‐value is examined visually and conceptually. The argumentation metaphor is used to explain the logic of testing hypothesis. Notes to Instructors In the data-­‐gathering activity of this experiment, students toss a coin (or balance it on its edge) 10 times to consider whether it is likely that the coin is fair. After 10 tosses are performed, students compute a sample proportion and add this proportion to a group dotplot on the classroom whiteboard or chalkboard. In a perfect scenario, about 10% of the samples have sample proportions that are unusual (p ≥ 0.80 or p ≤ 0.20). It is nice if a few unusual values occur. Obviously, this is more likely to occur as more sample proportions are accumulated. If your class is small, consider having each student test two pennies (10 tosses per penny), generating two corresponding sample proportions to add to the class dotplot. If at least 40 pennies are tested (10 times each), there should be several cases where unusually high or low proportions of heads occur. The activity depends on observing an unusual value for the number of heads. In the end, if no student observes 0–2 (unusually small values) or 8–10 (unusually large values), consider having everyone test another penny. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses In the activity, the authors suggest students toss their coin, or balance it on its edge over a level surface, with the presumption that it quickly falls. The advantage of balancing a coin is that it minimizes the number of coins flying around the room and the accompanying confusion (as fun as that might be). Coin balancing requires a level surface, so if students have sloped desks they may choose to gently toss their coins (or use the floor for balancing) instead. Opening Discussion/Questions Begin by holding up a penny, asking students: •
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If I toss this penny and catch it, what are the possible outcomes? If I toss the penny 10 times, what type of probability experiment am I conducting? Is either outcome more likely to occur than the other? What does it mean for a penny to be fair? What does it mean for a penny to be biased? How do you determine if a penny is fair? Explain to students that this lesson is a first look into formal statistical inference. The goal is to develop a process by which they can use statistical data to support or refute claims regarding a population parameter. This lesson uses a distribution of sample proportions generated by the class to get an idea of the range of values that can be expected from a collection of sample proportions. This understanding will help them understand why some results can be classified as unusual and lead them to question the validity of any assumptions made about the population of sample proportions. Introduction to the Context of the Lesson [Student Handout] This lesson introduces the formal process of statistical inference known as hypothesis testing. A hypothesis test is a process that statisticians use to determine whether an assumption made about a population is reasonable. The process for testing a hypothesis is quite similar to the process of making a formal argument. The steps to a formal argument typically include the following: •
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Hypothesis: What hypotheses (assumptions) can you agree on at the outset of the argument? Evidence: What data can be presented to support or refute the hypothesis? Conclusion: Does the evidence provided give sufficient cause to warrant rejection of the initial hypothesis? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses Balancing Coins [Student Handout, estimated time: 30 minutes plus wrap-­‐up] In this lesson, join your classmates in tossing a collection of pennies to generate a distribution sample of proportions of tosses that land heads. Then use this distribution to examine a particular coin to conclude whether it is fair—that it is just as likely to land heads as it is to land tails. Heads (Source: U.S. Treasury) Tails (Source: U.S. Treasury) Make a Hypothesis—Your Penny Is Fair [Student Handout] (1) When tossing or balancing a penny on its edge, it is bound to eventually land on one side or the other—heads or tails. Assuming your penny is fair, make a hypothesis regarding the probability that your coin lands heads. P(heads) = In statistical testing, always begin tests with a hypothesis you assume to be true. This assumption is known as the null hypothesis. Your statement regarding the proportion of tosses for a given coin that land heads is the null hypothesis for this experiment. (2) If you assume the coin has a 50% chance of landing heads, you are saying you have a fair, unbiased coin. Discuss with your group and record a possible method for testing such a hypothesis. Consider Evidence for Statistical Inference [Student Handout] (3) Toss your coin n = 10 times, carefully tallying the number of heads (x) that occur. x = (4) Compute the sample proportion for the n = 10 tosses. "
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Add your sample proportion to the dotplot of all students’ sample proportions on the classroom whiteboard or chalkboard. If your class needs additional data, run the test on a different penny by repeating the previous steps and add another data point to your class’s dotplot. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses (5) Recreate the class dotplot on the graph below. 22
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(6) Estimate the likelihood of any unusual outcomes. On your dotplot, look for extreme values, either on the right side of the distribution or the left. Circle those values, on one side only, that you consider to be extreme. (a) What is the value of the least extreme of the circled values? p = (b) Now focus on the coin that yielded this extreme sample proportion to consider whether it is fair. Estimate the probability of observing a value at least as extreme as the proportion in Question 6a by computing the proportion of all dots on the dotplot that are circled. The probability you are estimating is known as a P-­‐value. !!"#$%&'('
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Conclusion: Make a Decision [Student Handout] In general, the P-­‐value for a statistical observation is the probability of observing a sample proportion that is at least as extreme as the one given in Question 6a. When the P-­‐value of an outcome is very small, consider the outcome statistically significant. This means that under the assumption made in the null hypothesis (that the coin is fair), the event is quite unusual. In this experiment, the P-­‐value is the probability that the observed number of heads for the chosen coin occurred by chance (under the assumption that the coin is fair). If it is quite unlikely that the observed outcome occurred by chance, consider rejecting your null hypothesis—the assumption that the coin is fair. When you conduct a statistical test, you want to have a rule for how small a P-­‐value should be before you reject the null hypothesis. This rule is a value, denoted by the Greek letter α (alpha), 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses called the level of significance, which represents the probability of an event that you consider to be statistically significant. Typically, the level of significance ranges from around α = 0.01 to α = 0.05. For this experiment, use the level of significance α = 0.05. You have chosen a sample proportion that you consider unusual, corresponding to a single penny. If the estimated P-­‐value computed in Question 6b is less than the chosen level of significance, reject the null hypothesis (the coin is fair) in favor of the alternative (the coin is biased). Otherwise, you fail to reject the null hypothesis and may continue to assume that the coin is fair (though you have not proven it). (7) Do you reject the null hypothesis that the coin is fair? (8) Write a sentence describing your conclusion regarding the outcome of the experiment. Consider the Possibility of an Incorrect Conclusion [Student Handout] (9) Assume that you rejected the assumption that the coin is fair. Is there any possibility that your conclusion is incorrect? (10) Write a sentence to describe the error that might be made with such a conclusion. In general, whenever you decide against an assumption you made based on sample evidence, there is always a chance you made the wrong decision. This error is known as a Type I error. In the context of the coin-­‐tossing experiment, the Type I error is rejecting the assumption that the coin is fair when in fact it actually is fair. (11) Assume you did not conclude that the chosen coin is not fair. Write a sentence to describe any error that may have occurred in this conclusion. Whenever you fail to disprove an assumption that is actually false, you make an error as well. An error of this type is known as a Type II error. For the coin-­‐tossing experiment, a Type II error is concluding that the coin is fair when in fact it is actually biased (the probability of heads is not 50%). Wrap-­‐Up [Whole-­‐Class Discussion] One important flaw in this experiment that must be established with the class is the fact that the coin chosen as a possibly biased coin was chosen through what is often referred to as data mining. Given a large enough collection of randomly generated sample proportions from a sampling distribution, extreme values always occur eventually. Data mining occurs when researchers ignore sample after sample of unremarkable data and focus only on data sets that exhibit extreme outcomes. With a large collection of coin toss experiments, you are bound to eventually discover a sample proportion that is extreme—but this does not necessarily mean that the coin represented by that extreme is biased. Further testing of that coin (replication) would probably indicate that the coin is 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. 5 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses Discuss the following questions with your class. •
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How many groups rejected the null hypothesis? What does this imply about the process? Did groups fail to reject the null hypothesis? What does this imply about the process? What can you do to make a better, more-­‐informed decision? What could be done as a follow-­‐up on the coin chosen as a possibly biased coin to further investigate the truth of the matter? What is a null hypothesis? What is a P-­‐value? What is a Type I error? What is a Type II error What is data mining? Review the main ideas behind hypothesis testing (hypothesis, evidence, decision rule) and the analogy to making a convincing argument. References Scheaffer, R.L., Watkins, A. Witmer, J., & Gnanadesikan, M., (2004a). Activity-­‐based statistics: Instructor resources (2nd edition, Revised by Tim Erickson). Key College Publishing. Scheaffer, R.L., Watkins, A., Witmer, J., & Gnanadesikan, M., (2004b). Activity-­‐based statistics: Student guide (2nd edition, Revised by Tim Erickson). Key College Publishing. Images of United States Penny are from the U.S. Treasury Department, and are in the Public Domain. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses This lesson introduces the formal process of statistical inference known as hypothesis testing. A hypothesis test is a process that statisticians use to determine whether an assumption made about a population is reasonable. The process for testing a hypothesis is quite similar to the process of making a formal argument. The steps to a formal argument typically include the following: •
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Hypothesis: What hypotheses (assumptions) can you agree on at the outset of the argument? Evidence: What data can be presented to support or refute the hypothesis? Conclusion: Does the evidence provided give sufficient cause to warrant rejection of the initial hypothesis? Balancing Coins In this lesson, join your classmates in tossing a collection of pennies to generate a distribution sample of proportions of tosses that land heads. Then use this distribution to examine a particular coin to conclude whether it is fair—that it is just as likely to land heads as it is to land tails. Heads (Source: U.S. Treasury) Tails (Source: U.S. Treasury) Make a Hypothesis—Your Penny Is Fair (1) When tossing or balancing a penny on its edge, it is bound to eventually land on one side or the other—heads or tails. Assuming your penny is fair, make a hypothesis regarding the probability that your coin lands heads. P(heads) = In statistical testing, always begin tests with a hypothesis you assume to be true. This assumption is known as the null hypothesis. Your statement regarding the proportion of tosses for a given coin that land heads is the null hypothesis for this experiment. (2) If you assume the coin has a 50% chance of landing heads, you are saying you have a fair, unbiased coin. Discuss with your group and record a possible method for testing such a hypothesis. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses Consider Evidence for Statistical Inference (3) Toss your coin n = 10 times, carefully tallying the number of heads (x) that occur. x = (4) Compute the sample proportion for the n = 10 tosses. "
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Add your sample proportion to the dotplot of all students’ sample proportions on the classroom whiteboard or chalkboard. If your class needs additional data, run the test on a different penny by repeating the previous steps and add another data point to your class’s dotplot. (5) Recreate the class dotplot on the graph below. 22
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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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses (6) Estimate the likelihood of any unusual outcomes. On your dotplot, look for extreme values, either on the right side of the distribution or the left. Circle those values, on one side only, that you consider to be extreme. (a) What is the value of the least extreme of the circled values? p = (b) Now focus on the coin that yielded this extreme sample proportion to consider whether it is fair. Estimate the probability of observing a value at least as extreme as the proportion in Question 6a by computing the proportion of all dots on the dotplot that are circled. The probability you are estimating is known as a P-­‐value. !!"#$%&'('
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Make a Decision In general, the P-­‐value for a statistical observation is the probability of observing a sample proportion that is at least as extreme as the one given in Question 6a. When the P-­‐value of an outcome is very small, consider the outcome statistically significant. This means that under the assumption made in the null hypothesis (that the coin is fair), the event is quite unusual. In this experiment, the P-­‐value is the probability that the observed number of heads for the chosen coin occurred by chance (under the assumption that the coin is fair). If it is quite unlikely that the observed outcome occurred by chance, consider rejecting your null hypothesis—the assumption that the coin is fair. When you conduct a statistical test, you want to have a rule for how small a P-­‐value should be before you reject the null hypothesis. This rule is a value, denoted by the Greek letter α (alpha), called the level of significance, which represents the probability of an event that you consider to be statistically significant. Typically, the level of significance ranges from around α = 0.01 to α = 0.05. For this experiment, use the level of significance α = 0.05. You have chosen a sample proportion that you consider unusual, corresponding to a single penny. If the estimated P-­‐value computed in Question 6b is less than the chosen level of significance, reject the null hypothesis (the coin is fair) in favor of the alternative (the coin is biased). Otherwise, you fail to reject the null hypothesis and may continue to assume that the coin is fair (though you have not proven it). (7) Do you reject the null hypothesis that the coin is fair? (8) Write a sentence describing your conclusion regarding the outcome of the experiment. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.1.3: Testing Statistical Hypotheses Consider the Possibility of an Incorrect Conclusion (9) Assume that you rejected the assumption that the coin is fair. Is there any possibility that your conclusion is incorrect? (10) Write a sentence to describe the error that might be made with such a conclusion. In general, whenever you decide against an assumption you made based on sample evidence, there is always a chance you made the wrong decision. This error is known as a Type I error. In the context of the coin-­‐tossing experiment, the Type I error is rejecting the assumption that the coin is fair when in fact it actually is fair. (11) Assume you did not conclude that the chosen coin is not fair. Write a sentence to describe any error that may have occurred in this conclusion. Whenever you fail to disprove an assumption that is actually false, you make an error as well. An error of this type is known as a Type II error. For the coin-­‐tossing experiment, a Type II error is concluding that the coin is fair when in fact it is actually biased (the probability of heads is not 50%). 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Estimated number of 50-­‐minute class sessions: 1 Materials Required This lesson is developed using a statistical simulation spreadsheet. If possible, each student group should have internet access and a computer running Excel. If this is not possible, the entire class may view the simulation on the instructor’s computer (with internet access) and use the simulation to answer the questions. Learning Goals Students will understand •
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how to apply the simulation process in a new context. the process of hypothesis testing. the idea of a confidence interval. Students will be able to •
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informally carry out the steps of hypothesis testing. state why the null hypothesis can or cannot be rejected. use simulation to estimate a confidence interval. provide an informal interpretation of a confidence interval. Lesson Overview This lesson builds on Lesson 7.1.3, using the context of flipping coins with a different type of coin (a Euro) to test hypotheses and learn the language of tests of significance. This lesson also introduces the idea of a confidence interval in a new context (an opinion survey), helping students see the two parts of the interval (sample statistic and margin of error) and different ways of reporting confidence intervals. Students also begin to learn how to interpret a confidence interval. Part I: Testing Statistical Hypotheses [Student Handout] The euro, represented by the symbol €, is the unit of currency for the European Union. The reverse side of all 1€ coins shows a map of Europe, along with the text: 1 EURO. The obverse sides of 1€ coins feature a design from one of the member countries. An Italian 1€ coin, whose obverse side features Leonardo DaVinci’s famous drawing Vitruvian Man, is shown in the photo. For the purposes of this lesson, let’s consider the obverse side to be “heads” and the reverse side to be “tails.” Suppose that you flip a 1€ coin one hundred times and get 33 heads and 67 tails. Does the number of heads seem like a surprisingly small outcome? Is the number of heads observed low enough for you to consider the coin unfair in a statistical sense? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing In this lesson, you will formalize some ideas introduced in Lesson 7.1.3 for hypothesis tests. The question is the same, but now you focus on a single coin that appears to be giving strange results. You will use a coin-­‐flipping simulation to test the hypothesis that the proportion of heads when flipping a 1€ coin is 0.50. As previously discussed, a hypothesis test is a method of reasoning that involves making a decision about an assumed value of a population parameter. In this case, the parameter is a population proportion, denoted as π. Make a Hypothesis [Student Handout] As before, assume the 1€ coin is fair. The parameter π represents the proportion of all possible flips by this coin that would be heads. The null hypothesis (HO) is based on the idea that the coin is fair and uses this to assume a value for π. (1) Complete the null hypothesis. HO: π = __________ (2) While assuming that the null hypothesis is true, you will decide on an alternative that you would like to prove. Begin by examining the sample data. In the experiment described in this problem, the 1€ coin was flipped n = 100 times, and only x = 33 of these flips were heads. Compute the corresponding sample proportion in decimal form. "
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(3) Based on the value of p (the sample proportion computed in Question 2), what do you claim regarding the population proportion (π) as an alternative to the null hypothesis? Circle one of the following: π < 0.50 or π > 0.50 (4) The circled inequality in Question 3 is called the alternative hypothesis, which you denote as Ha. Rewrite this circled inequality in the blank space below. Ha: _______________ While you always assume that the null hypothesis is true, the process of the hypothesis test involves examining sample data to determine whether there is sufficient evidence that leads you to reject the null hypothesis and support the alternative. Consider the Evidence for Statistical Inference [Student Handout] You have already examined the sample proportion (p) of flips that resulted in heads. The question you must address is whether the observed value of p is sufficiently far from the assumed value of the population proportion (HO: π = 0.5) to warrant rejection of this assumption. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing One way to answer this question is to examine the sampling distribution of sample proportion, under the assumption that the 1€ coin is fair (so the population proportion of heads is π = 0.50). Open the coin flipping spreadsheet located at http://math.mtsac.edu/statistics/EuroFliping.xls. In the spreadsheet, below the cell labeled n, enter the given sample size. Directly below this cell, enter the observed sample proportion computed in Question 2. With these values entered, a simulated distribution of 1,000 sample proportions is generated, and a red vertical line is plotted at the value of the observed sample proportion. (5) How many simulated proportions are less than or equal to the observed sample proportion (p)? (6) Recall that under the assumption of the null hypothesis, the P-­‐value for a hypothesis test is the likelihood of randomly observing a sample proportion that is at least as extreme as the one provided by the sample data. This probability can be estimated by the percentage of simulated sample proportions that are less than or equal to the observed sample proportion (p). Read the simulation spreadsheet carefully to determine this percentage, and convert it to a decimal proportion. P-­‐value ≈ _________________ (7) When a P-­‐value indicates an unlikely statistical observation, consider the event statistically significant. Given the P-­‐value estimated in Question 6, do you consider the value observed for p in Question 2 to be statistically significant? Conclusion: Make a Decision [Student Handout] As stated previously, you make your decision about the null hypothesis by considering whether the P-­‐value is small enough to indicate that the observed statistic is statistically significant. It was also stated that the rule for how small the P-­‐value needs to be is called the level of significance, denoted by the Greek letter α (alpha). For this hypothesis test, use a 1% level of significance (α ≈ 0.01). If the P-­‐value is less than or equal to α, consider the sample proportion (p) significantly different from the assumed population proportion (π) and reject the null hypothesis in favor of the alternative. This decision rule is summarized below. Decision Rule: Reject HO and support Ha whenever P-­‐value ≤ α. If the P-­‐value is not less than or equal to α, then fail to reject the null hypothesis. (8) Do you reject or fail to reject the null hypothesis? Express this decision in the context of the fairness of the 1€ coin of this problem. (9) Given your decision regarding the null hypothesis, do you support the alternative hypothesis? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing (10) Does the alternative hypothesis indicate that your coin favors heads, tails, or neither? (11) Recall the definitions of the Type I and II errors given previously. •
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Type I error—Rejecting a true null hypothesis Type II error—Failing to reject a false null hypothesis Is it possible that your decision involved one of these errors? Which error might have been made? Do you think that it is likely that a randomly selected 1€ coin is biased? Wrap-­‐Up: The Hypothesis Test in Detail In general, there are several items that must be addressed in every hypothesis test. Discuss the following questions with your class: •
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Identify the research question. What question were the students trying to answer? Identify a parameter related to the research question. Recall that a parameter is a (generally unknown) numerical summary of an entire population. What is the parameter addressed in this lesson? Write the statistical hypotheses in terms of the parameter. In the example, the statistical hypotheses were the following: HO: π = 0.50 (The population proportion of heads when flipping a 1€ coin repeatedly is 0.50.) Ha: π < 0.50 (The population proportion of heads when flipping a 1€ coin is less than 0.50.) •
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Collect data and calculate a sample statistic. What sample proportion was observed? Find the P-­‐value. Under the assumption of the null hypothesis, the P-­‐value is the likelihood of observing a sample proportion at least as extreme as yours. What P-­‐value did you observe? Make a decision. If the P-­‐value is less than the probability of a significant event (α, the level of significance), then you reject the null hypothesis in favor of the alternative. Otherwise, you fail to reject the null hypothesis. Did you reject or fail to reject the null hypothesis? Did you support the alternative? What does this mean regarding the 1€ coin in question? The following are other questions to consider: •
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How certain are you regarding your conclusion? Will your decisions be correct every time you do a significance test? What are the main components of any hypothesis 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. 4 Statway Instructor’s Notes April 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Part II: President Obama’s Approval Rating and Confidence Intervals [Student Handout] (Note: In Part II, students extend the ideas of statistical inference to the confidence interval. Again, a simulation is used to generate a sampling distribution, and through trial and error, students approximate the margin of errors corresponding to 95% and 99% confidence.) Gallup, Inc., conducts phone surveys to gather the opinions of American adults on a variety of political issues. One such question is whether American adults approve of the current president’s job performance. A phone survey conducted by Gallup at the end of 2010 asked a random sample of about 1,500 United States adults whether they approved of the job President Barack Obama was doing. At that time, 47% of the respondents indicated that they approved of President Obama’s job performance. In your small group, discuss the following questions and write a group response. (12) Based on the given information, what do you think was the true proportion of all U.S. adults who approved of President Obama’s approval rating during this time period at the end of 2010? Explain your reasoning. (13) Do you think the true proportion of the population that approved of President Obama’s job performance at the end of 2010 could have been as high as 55%? As high as 60%? Explain your reasoning. Do you think it could have been as low as 45%? As low as 40%? Explain your reasoning. (14) The sample proportion (p) mentioned in this problem is a single value that is often referred to as a point estimate. Give the sample proportion (p) as a point estimate below. p = ________________ Estimating the Population Proportion [Student Handout] Because you have only a sample of American adults, you know that the proportion of all American adults who approved of the president’s job performance could be lower or higher than the sample proportion. The question is how much lower or higher? A common way to provide an estimate of a population parameter is to construct an interval estimate called a confidence interval. To construct a confidence interval you need two things: •
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an estimate of a population parameter (a statistic), and a margin of error. The margin of error is subtracted from the statistic to give a lower limit for the confidence interval and added to the statistic to give an upper limit for the confidence interval. A simple formula for the lower and upper limits of the confidence interval is statistic ± margin of error For example, Gallup reported that the confidence interval for the true percentage of all American adults who approved of President Obama’s job performance at the end of 2010 was 47% ± 3%, or somewhere between 44% and 50% of all American adults. The 3% is the margin of error. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Typically, the estimate for the margin of error is chosen so you can be 95% to 99% confident the true population percentage is between the lower and upper limits. There are formal procedures for estimating the margin of error that you will learn in later lessons. For the moment, use a simulation to learn how the margin of error and confidence interval are determined. Open the spreadsheet simulation at the following address: http://math.mtsac.edu/statistics/ObamaApproval.xls Use this spreadsheet to create an interval estimate for the true percentage of American adults who approved of President Obama’s job performance at the end of 2010. The spreadsheet requires input in the cells shaded in darker blue. •
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Enter the sample size: n = 1,500. Enter the sample proportion: p = 0.47. The spreadsheet immediately generates 1,000 sample proportions, with the assumption that the population proportion (π) is equal to your best estimate, 0.47 (much like a null hypothesis). This assumption is explained later. There is an additional input cell for the margin of error, which represents an upper limit for the expected error—with a certain level of confidence—that a random sample proportion (p) is from the population proportion (π). Using trial and error, experiment with the value of the margin of error until about 95% of the sample proportions are contained in the red box. This red box graphically depicts a confidence interval. (15) What margin of error that captures 95% of the sample proportions? E = ______________ (16) Record the values of the lower and upper limits of the confidence interval. Lower limit: ________________________ Upper limit: ________________________ Although you have plotted a sampling distribution centered at an assumed value of the population proportion (π), recognize that this proportion is unknown. Fortunately, you do not need π to estimate the margin of error, since error depends only on the spread of the distribution. The true sampling distribution of sample proportions is centered at π. If you assume the sampling distribution has similar spread to the distribution plotted on the spreadsheet, then about 95% of sample proportions will be within the margin of error (E) of the true unknown proportion. This fact tells you that the distance from 95% of sample proportions to the population proportion is less than the maximum error. |π – p| < E 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing If the value of p was randomly selected, then it is likely that it is one of the 95% of all p values that satisfy this inequality. In fact, you are 95% confident that the inequality is true. The inequality above is equivalent to the compound inequality, –E < π – p < E Adding p to each part of this inequality gives the confidence interval, p – E < π < p + E (17) Using the values of p and E in the spreadsheet, give the 95% confidence interval for the population proportion (π). __________ < π < __________ When interpreting a 95% confidence interval, you are 95% confident that the interval contains the population proportion. This means that 95% of all such intervals actually contain π as they indicate. Of course, this means that 5% of such intervals do not contain π as they indicate. (18) Returning to the spreadsheet and using trial and error, change the margin of error so that about 99% of sample proportions are contained within the red box. Give the margin of error. E = ________________ (19) Compute the 99% confidence interval using p and the new value of E. p – E < π < p + E __________ < π < __________ (20) Interpret this confidence interval. Use wording similar to the interpretation provided at the end of Question 17. (21) Compare the margins of error given in Questions 15 and 18. What is the effect of increasing the confidence level on the margin of error? Wrap-­‐Up Spend some time discussing the meaning of the confidence interval and how to interpret it. Pay special attention to students who misinterpret the confidence interval, gently steering them away from statements regarding probability that the population is between the endpoints and toward statements regarding how confident they are that the interval’s endpoints contain the population proportion. Remind students that while there are many sample proportions—each of which may be used to construct a confidence interval—there is only one population proportion. The level of confidence is the proportion of confidence intervals that actually contain the population proportion. 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Below are a few questions you can talk over with students. •
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Is it true that President Obama’s true approval rate is equal to 47%? Why or why not? Are you absolutely sure that President Obama’s true approval rate is in the confidence interval? What is the difference between a point estimate and an interval estimate? What is a margin of error? Do you know the true percentage of all American adults who approved of President Obama’s performance at the end of 2010? How do you interpret the confidence interval? References Seier, E., & Robe, C. (2002). Ducks and green—An introduction to the ideas of hypothesis testing. Teaching Statistics, 24(3), 82–86. Garfield, J.B. & Ben-­‐Zvi, D. (2008). Developing Students’ Statistical Reasoning: Connecting Research and Teaching Practice, Springer Publishing, New York, NY. Homework [Student Handout] A Hypothesis Test Using the Coin Flip Simulation (1) Suppose another 1€ coin is flipped 100 times, and 57 heads are observed. Do you think this evidence is strong enough to conclude that the coin is biased? (2) What proportion of flips were heads? "
!!"! !" _______ = # (3) What null hypothesis regarding the value of the population proportion (π) of all flips for this coin that are heads would assume that the coin is fair? HO: π = __________ (4) What alternative hypothesis regarding π would assume that the coin favors tails? Ha: π = ___________ Open the Euro flipping applet at http://math.mtsac.edu/statistics/EuroFliping.xls, and enter the sample size and sample proportion in the appropriate cells. (5) Estimate the P-­‐value for this test by giving the percentage (in a nonpercentage decimal form) of simulated values greater than the sample proportion observed above. P-­‐value ≈ _________________ 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 Instructor’s Notes April 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing (6) With a 5% level of significance (α = 0.05), reject the null hypothesis whenever the P-­‐value is less than α. Do you reject the null hypothesis? (7) Do you support the alternative hypothesis? (8) Use a complete sentence to explain your conclusion in the context of the fairness of the coin. A Confidence Interval Using the Obama Approval Rates Simulation (9) In Spring 2011, a sample of 1,000 voters included 527 people who approved of President Obama’s job in office. What is the sample proportion of approvals? "
!!"! !" _______ = # Open the spreadsheet simulation at http://math.mtsac.edu/statistics/ObamaApproval.xls, and enter the appropriate sample size and sample proportion computed above. (10) Using trial and error, experiment with the margin of error until the spreadsheet’s confidence interval contains 95% of the simulated sample proportions. What is the margin of error? E = ______________ (11) Compute the 95% confidence interval below for the population proportion (π) of voters who approved of President Obama’s work in Spring 2011. p – E < π < p + E __________ < π < __________ (12) Write a brief interpretation of the confidence interval. 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 Student Handout April 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Part I: Testing Statistical Hypotheses The euro, represented by the symbol €, is the unit of currency for the European Union. The reverse side of all 1€ coins shows a map of Europe, along with the text: 1 EURO. The obverse sides of 1€ coins feature a design from one of the member countries. An Italian 1€ coin, whose obverse side features Leonardo DaVinci’s famous drawing Vitruvian Man, is shown in the photo. For the purposes of this lesson, let’s consider the obverse side to be “heads” and the reverse side to be “tails.” Suppose that you flip a 1€ coin one hundred times and get 33 heads and 67 tails. Does the number of heads seem like a surprisingly small outcome? Is the number of heads observed low enough for you to consider the coin unfair in a statistical sense? In this lesson, you will formalize some ideas introduced in Lesson 7.1.3 for hypothesis tests. The question is the same, but now you focus on a single coin that appears to be giving strange results. You will use a coin-­‐flipping simulation to test the hypothesis that the proportion of heads when flipping a 1€ coin is 0.50. As previously discussed, a hypothesis test is a method of reasoning that involves making a decision about an assumed value of a population parameter. In this case, the parameter is a population proportion, denoted as π. Make a Hypothesis As before, assume the 1€ coin is fair. The parameter π represents the proportion of all possible flips by this coin that would be heads. The null hypothesis (HO) is based on the idea that the coin is fair and uses this to assume a value for π. (1) Complete the null hypothesis. HO: π = __________ (2) While assuming that the null hypothesis is true, you will decide on an alternative that you would like to prove. Begin by examining the sample data. In the experiment described in this problem, the 1€ coin was flipped n = 100 times, and only x = 33 of these flips were heads. Compute the corresponding sample proportion in decimal form. "
!!"! !" _________ = #
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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing (3) Based on the value of p (the sample proportion computed in Question 2), what do you claim regarding the population proportion (π) as an alternative to the null hypothesis? Circle one of the following: π < 0.50 or π > 0.50 (4) The circled inequality in Question 3 is called the alternative hypothesis, which you denote as Ha. Rewrite this circled inequality in the blank space below. Ha: _______________ While you always assume that the null hypothesis is true, the process of the hypothesis test involves examining sample data to determine whether there is sufficient evidence that leads you to reject the null hypothesis and support the alternative. Consider the Evidence for Statistical Inference You have already examined the sample proportion (p) of flips that resulted in heads. The question you must address is whether the observed value of p is sufficiently far from the assumed value of the population proportion (HO: π = 0.5) to warrant rejection of this assumption. One way to answer this question is to examine the sampling distribution of sample proportion, under the assumption that the 1€ coin is fair (so the population proportion of heads is π = 0.50). Open the coin flipping spreadsheet located at http://math.mtsac.edu/statistics/EuroFliping.xls. In the spreadsheet, below the cell labeled n, enter the given sample size. Directly below this cell, enter the observed sample proportion computed in Question 2. With these values entered, a simulated distribution of 1,000 sample proportions is generated, and a red vertical line is plotted at the value of the observed sample proportion. (5) How many simulated proportions are less than or equal to the observed sample proportion (p)? (6) Recall that under the assumption of the null hypothesis, the P-­‐value for a hypothesis test is the likelihood of randomly observing a sample proportion that is at least as extreme as the one provided by the sample data. This probability can be estimated by the percentage of simulated sample proportions that are less than or equal to the observed sample proportion (p). Read the simulation spreadsheet carefully to determine this percentage, and convert it to a decimal proportion. P-­‐value ≈ _________________ (7) When a P-­‐value indicates an unlikely statistical observation, consider the event statistically significant. Given the P-­‐value estimated in Question 6, do you consider the value observed for p in Question 2 to be statistically significant? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Make a Decision As stated previously, you make your decision about the null hypothesis by considering whether the P-­‐
value is small enough to indicate that the observed statistic is statistically significant. It was also stated that the rule for how small the P-­‐value needs to be is called the level of significance, denoted by the Greek letter α (alpha). For this hypothesis test, use a 1% level of significance (α ≈ 0.01). If the P-­‐value is less than or equal to α, consider the sample proportion (p) significantly different from the assumed population proportion (π) and reject the null hypothesis in favor of the alternative. This decision rule is summarized below. Decision Rule: Reject HO and support Ha whenever P-­‐value ≤ α. If the P-­‐value is not less than or equal to α, then fail to reject the null hypothesis. (8) Do you reject or fail to reject the null hypothesis? Express this decision in the context of the fairness of the 1€ coin of this problem. (9) Given your decision regarding the null hypothesis, do you support the alternative hypothesis? (10) Does the alternative hypothesis indicate that your coin favors heads, tails, or neither? (11) Recall the definitions of the Type I and II errors given previously. •
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Type I error—Rejecting a true null hypothesis Type II error—Failing to reject a false null hypothesis Is it possible that your decision involved one of these errors? Which error might have been made? Do you think that it is likely that a randomly selected 1€ coin is biased? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Part II: President Obama’s Approval Rating and Confidence Intervals Gallup, Inc., conducts phone surveys to gather the opinions of American adults on a variety of political issues. One such question is whether American adults approve of the current president’s job performance. A phone survey conducted by Gallup at the end of 2010 asked a random sample of about 1,500 United States adults whether they approved of the job President Barack Obama was doing. At that time, 47% of the respondents indicated that they approved of President Obama’s job performance. In your small group, discuss the following questions and write a group response. (12) Based on the given information, what do you think was the true proportion of all U.S. adults who approved of President Obama’s approval rating during this time period at the end of 2010? Explain your reasoning. (13) Do you think the true proportion of the population that approved of President Obama’s job performance at the end of 2010 could have been as high as 55%? As high as 60%? Explain your reasoning. Do you think it could have been as low as 45%? As low as 40%? Explain your reasoning. (14) The sample proportion (p) mentioned in this problem is a single value that is often referred to as a point estimate. Give the sample proportion (p) as a point estimate below. p = ________________ 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Estimating the Population Proportion Because you have only a sample of American adults, you know that the proportion of all American adults who approved of the president’s job performance could be lower or higher than the sample proportion. The question is how much lower or higher? A common way to provide an estimate of a population parameter is to construct an interval estimate called a confidence interval. To construct a confidence interval you need two things: •
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an estimate of a population parameter (a statistic), and a margin of error. The margin of error is subtracted from the statistic to give a lower limit for the confidence interval and added to the statistic to give an upper limit for the confidence interval. A simple formula for the lower and upper limits of the confidence interval is statistic ± margin of error For example, Gallup reported that the confidence interval for the true percentage of all American adults who approved of President Obama’s job performance at the end of 2010 was 47% ± 3%, or somewhere between 44% and 50% of all American adults. The 3% is the margin of error. Typically, the estimate for the margin of error is chosen so you can be 95% to 99% confident the true population percentage is between the lower and upper limits. There are formal procedures for estimating the margin of error that you will learn in later lessons. For the moment, use a simulation to learn how the margin of error and confidence interval are determined. Open the spreadsheet simulation at the following address: http://math.mtsac.edu/statistics/ObamaApproval.xls Use this spreadsheet to create an interval estimate for the true percentage of American adults who approved of President Obama’s job performance at the end of 2010. The spreadsheet requires input in the cells shaded in darker blue. •
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Enter the sample size: n = 1,500. Enter the sample proportion: p = 0.47. The spreadsheet immediately generates 1,000 sample proportions, with the assumption that the population proportion (π) is equal to your best estimate, 0.47 (much like a null hypothesis). This assumption is explained later. There is an additional input cell for the margin of error, which represents an upper limit for the expected error—with a certain level of confidence—that a random sample proportion (p) is from the population proportion (π). Using trial and error, experiment with the value of the margin of error until about 95% of the sample proportions are contained in the red box. This red box graphically depicts a confidence interval. (15) What margin of error that captures 95% of the sample proportions? E = ______________ 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing (16) Record the values of the lower and upper limits of the confidence interval. Lower limit: ________________________ Upper limit: ________________________ Although you have plotted a sampling distribution centered at an assumed value of the population proportion (π), recognize that this proportion is unknown. Fortunately, you do not need π to estimate the margin of error, since error depends only on the spread of the distribution. The true sampling distribution of sample proportions is centered at π. If you assume the sampling distribution has similar spread to the distribution plotted on the spreadsheet, then about 95% of sample proportions will be within the margin of error (E) of the true unknown proportion. This fact tells you that the distance from 95% of sample proportions to the population proportion is less than the maximum error. |π – p| < E If the value of p was randomly selected, then it is likely that it is one of the 95% of all p values that satisfy this inequality. In fact, you are 95% confident that the inequality is true. The inequality above is equivalent to the compound inequality, –E < π – p < E Adding p to each part of this inequality gives the confidence interval, p – E < π < p + E (17) Using the values of p and E in the spreadsheet, give the 95% confidence interval for the population proportion (π). __________ < π < __________ When interpreting a 95% confidence interval, you are 95% confident that the interval contains the population proportion. This means that 95% of all such intervals actually contain π as they indicate. Of course, this means that 5% of such intervals do not contain π as they indicate. (18) Returning to the spreadsheet and using trial and error, change the margin of error so that about 99% of sample proportions are contained within the red box. Give the margin of error. E = ________________ (19) Compute the 99% confidence interval using p and the new value of E. p – E < π < p + E __________ < π < __________ 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing (20) Interpret this confidence interval. Use wording similar to the interpretation provided at the end of Question 17. (21) Compare the margins of error given in Questions 15 and 18. What is the effect of increasing the confidence level on the margin of error? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing Homework A Hypothesis Test Using the Coin Flip Simulation (1) Suppose another 1€ coin is flipped 100 times, and 57 heads are observed. Do you think this evidence is strong enough to conclude that the coin is biased? (2) What proportion of flips were heads? "
!!"! !" _______ = # (3) What null hypothesis regarding the value of the population proportion (π) of all flips for this coin that are heads would assume that the coin is fair? HO: π = __________ (4) What alternative hypothesis regarding π would assume that the coin favors tails? Ha: π = ___________ Open the Euro flipping applet at http://math.mtsac.edu/statistics/EuroFliping.xls, and enter the sample size and sample proportion in the appropriate cells. (5) Estimate the P-­‐value for this test by giving the percentage (in a nonpercentage decimal form) of simulated values greater than the sample proportion observed above. P-­‐value ≈ _________________ (6) With a 5% level of significance (α = 0.05), reject the null hypothesis whenever the P-­‐value is less than α. Do you reject the null hypothesis? 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 24, 2012 (Full Version 1.0) Initiating Lesson 7.2.1: Two Types of Inferential Procedures—Estimation and Hypothesis Testing (7) Do you support the alternative hypothesis? (8) Use a complete sentence to explain your conclusion in the context of the fairness of the coin. A Confidence Interval Using the Obama Approval Rates Simulation (9) In Spring 2011, a sample of 1,000 voters included 527 people who approved of President Obama’s job in office. What is the sample proportion of approvals? "
!!"! !" _______ = # Open the spreadsheet simulation at http://math.mtsac.edu/statistics/ObamaApproval.xls, and enter the appropriate sample size and sample proportion computed above. (10) Using trial and error, experiment with the margin of error until the spreadsheet’s confidence interval contains 95% of the simulated sample proportions. What is the margin of error? E = ______________ (11) Compute the 95% confidence interval below for the population proportion (π) of voters who approved of President Obama’s work in Spring 2011. p – E < π < p + E __________ < π < __________ (12) Write a brief interpretation of the confidence interval. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals Estimated number of 50-­‐minute class sessions: 1 Materials Required Once again, this lesson uses a computer simulation spreadsheet. If students do not have workstations with Excel, the authors suggest that the spreadsheet be projected from an instructor workstation. The instructor can enter any required parameters into the spreadsheet, and students can answer questions individually regarding the resulting simulation. Learning Goals Students will understand •
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what the level of confidence actually means. how sample size and confidence levels affect the width of the confidence interval. how to use statistical software to find a confidence interval. how changing the sample size and confidence level will affect the confidence interval. connections between confidence intervals and hypothesis tests. Students will be able to •
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reason about confidence intervals. provide a correct interpretation of what a 95% confidence interval represents. find a confidence interval using statistical software. use a confidence interval to test a hypothesis. Lesson Overview This lesson features simulation exercises that enable students to develop a deeper understanding of confidence intervals by manipulating the parameters that affect confidence intervals —that is, the sample size and the confidence level. Further, students will be able to investigate — in context— and dispel commonly held misconceptions about the information that confidence intervals provide. Students use a spreadsheet to generate 100 random samples of 25 Reese’s Pieces candies and, for each sample, to calculate a 95% confidence interval for the proportion of orange candies in the population of all Reese’s Pieces. They investigate the percentage of confidence intervals that include the population proportion. After this step, students use the simulation to determine the effects of varying confidence levels and sample sizes on the width of the confidence interval. Class discussion is focused on identifying the relationships between sample size, confidence level, and confidence interval width. Opening Discussion Remind students of the activity that looked at the characteristics of the sampling distribution for the proportion of orange candies in 25-­‐piece samples taken from a population of Reese’s Pieces candies (that is, the sampling distribution was roughly symmetric and bell-­‐shaped). The center of the distribution was the population proportion for orange candies, which is equal to 0.45. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals In reality, you do not know the population proportion of orange Reese’s Pieces. To estimate it, you could draw a random sample, compute a sample proportion of orange candies, and produce a confidence interval. Suppose you gathered many different random samples (as done previously). Do you think that each confidence interval would include the true population proportion of π = 0.45? Introduction to the Context of the Lesson [Student Handout] In Lesson 7.1.2, you explored the distribution of proportions of orange Reese’s Pieces candies for samples of size 25, randomly selected from a population of Reese’s Pieces candies. In this activity, you will use that same context to learn about different characteristics of confidence intervals. Most importantly, you will develop a better understanding of what the 95% means in the phrase 95% confidence interval. Part I [Student Handout, estimated time: 25 minutes plus wrap-­‐up] Recall the Reese’s Pieces activity. In that lesson, you assumed (based on prior information) that the Hershey Company manufactures 45% of its Reese’s Pieces to be orange. Also previously, you learned the basic ideas behind a confidence interval for a sample proportion. In this activity, you further investigate the meaning of confidence interval and how such intervals can vary depending on several factors. Open the simulation spreadsheet located at http://math.mtsac.edu/statistics/ManyConfInts.xls. In the spreadsheet, next to population proportion, enter the value assumed to be the proportion of all Reese’s Pieces that are orange: π = 0.45. Enter 25 for the sample size and 0.95 (= 95%) for the level of confidence. Once the values are entered, a graph is generated using 100 randomly simulated sample proportions. On the graph, a red vertical line is plotted at the value of the population proportion (π = 0.45). One hundred orange horizontal lines are also plotted. Each horizontal line represents a confidence interval corresponding to 1 of the 100 sample proportions. The left endpoint of each line is plotted at the lower limit of its respective confidence interval, and the right endpoint is at the upper limit. It is actually quite easy to see when the population proportion (π = 0.45) is not contained in a confidence interval. If you scan down the length of the red line, wherever the dark background touches the red line is a confidence interval that does not contain the population proportion. (1) Scan the graph visually and count the number of confidence intervals that do not contain the population proportion. How many are there? (2) Of the 100 intervals represented, how many contain the population proportion? Does this agree with the value reported on the spreadsheet? (3) What percent of the confidence intervals contain the population proportion? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals (4) If you randomly selected 1 of the 100 sample proportions and its corresponding interval, how confident are you that the population proportion would be in that interval (give your answer as a percent from 0% to 100%). (5) Is your answer to Question 4 similar to the level of confidence entered in the spreadsheet? (6) Suppose your answer to Question 4 is not the same as the level of confidence, why might this be different? (7) Suppose you gathered 1,000 randomly generated sample proportions and computed a 95% confidence interval for each. About what number of these confidence intervals would include the population proportion of π = 0.45? (8) Discuss these ideas in your group and then write a statement for what the 95% means when you say you have a 95% confidence interval. Wrap-­‐Up Lead the discussion toward a common understanding that one interpretation of the confidence level is the expected percent of all confidence intervals that include the true population proportion. When you draw a single random sample, you cannot know if its corresponding confidence interval includes the true population proportion. However, if you construct a 95% confidence interval, the interval contains the population proportion 95% of the time. Part II [Student Handout] (Note: In Part II, students experiment with sample size and confidence level to determine the effect they have on the width of a confidence interval. In this activity, students continue to use the simulation at http://math.mtsac.edu/statistics/ManyConfInts.xls.) (9) In the simulation spreadsheet, continue to work with the population proportion of orange Reese’s Pieces (π = 0.45) and a confidence level of 0.95 (= 95%). Watch the widths of the confidence intervals as you increase the sample size to n = 1,000. Do the intervals get wider or narrower? (10) With the larger sample size, has the margin of error increased or decreased in each interval? (11) Change the sample size to n = 500. Are the intervals wider or narrower? (12) With the smaller sample size, has the margin of error increased or decreased in each interval? (13) Lower the level of confidence to 0.90 (= 90%). Does this cause the intervals to become wider or narrower? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals (14) With the lower confidence level, is the margin of error larger or smaller? Why do you suppose that is? (15) What proportion of the 90% confidence intervals contains the population proportion (π = 0.45)? Is this similar to the level of confidence? (16) Raise the confidence level to 0.99 (= 99%). Does this cause the intervals to become wider or narrower? (17) With the higher level of confidence, is the margin of error larger or smaller? Why do you suppose that is? (18) What proportion of the 99% confidence intervals contains the population proportion (π = 0.45)? Is this similar to the level of confidence? Wrap-­‐Up Make sure the class comes to a general understanding that a larger sample produces a smaller margin of error, and smaller sample size produces a larger margin of error. In addition, spend some time discussing the idea that larger error accompanies larger levels of confidence, while smaller error accompanies smaller levels of confidence. This is strange, but it helps to use the analogy that you have more confidence when you shoot at larger targets, allowing for more error. Continue to steer students away from the idea that a 95% interval has a 95% probability of being correct and toward the idea that they are 95% confident that a given interval includes the population proportion. Discuss the following questions with students: •
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When and why do you use a confidence interval? Can you ever be 100% certain that the confidence interval includes the true population proportion? Name two ways to reduce the margin of error. Name two ways to increase the margin of error. What is the best way to lower the margin for error—by reducing confidence levels or increasing sample size? Why? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals Homework [Student Handout] At this point, you are intimately familiar with the assumption that 45% of all Reese’s Pieces are orange. Let’s assume that 35% of all Reese’s Pieces are yellow. Open the simulation used in this lesson at http://math.mtsac.edu/statistics/ManyConfInts.xls. For population proportion, enter the assumed proportion of Reese’s Pieces that are yellow (π = 0.35). Enter 100 for the sample size and 0.99 (= 99%) for the level of confidence. (1) Recall that the vertical red line represents the population proportion (π = 0.35) and that each horizontal orange line is a confidence interval. How many of the 100 confidence intervals contain the population proportion? (2) What percent of the confidence intervals contain the population proportion? Is this similar to the level of confidence? (3) What can be done so that a higher percentage of the intervals contain the population proportion? (4) Try your suggestion in Question 3. Does this yield the desired result? (5) For a level of confidence to 0.99 (= 99%), what percentage of the intervals should contain the population proportion? (6) Test your answer to Question 5 on the spreadsheet. Does this yield the desired result? (7) Name two things you can do to reduce the margin of error. Which of these increases the quality of your results? Which decreases the quality? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals In Lesson 7.1.2, you explored the distribution of proportions of orange Reese’s Pieces candies for samples of size 25, randomly selected from a population of Reese’s Pieces candies. In this activity, you will use that same context to learn about different characteristics of confidence intervals. Most importantly, you will develop a better understanding of what the 95% means in the phrase 95% confidence interval. Part I Recall the Reese’s Pieces activity. In that lesson, you assumed (based on prior information) that the Hershey Company manufactures 45% of its Reese’s Pieces to be orange. Also previously, you learned the basic ideas behind a confidence interval for a sample proportion. In this activity, you further investigate the meaning of confidence interval and how such intervals can vary depending on several factors. Open the simulation spreadsheet located at http://math.mtsac.edu/statistics/ManyConfInts.xls. In the spreadsheet, next to population proportion, enter the value assumed to be the proportion of all Reese’s Pieces that are orange: π = 0.45. Enter 25 for the sample size and 0.95 (= 95%) for the level of confidence. Once the values are entered, a graph is generated using 100 randomly simulated sample proportions. On the graph, a red vertical line is plotted at the value of the population proportion (π = 0.45). One hundred orange horizontal lines are also plotted. Each horizontal line represents a confidence interval corresponding to 1 of the 100 sample proportions. The left endpoint of each line is plotted at the lower limit of its respective confidence interval, and the right endpoint is at the upper limit. It is actually quite easy to see when the population proportion (π = 0.45) is not contained in a confidence interval. If you scan down the length of the red line, wherever the dark background touches the red line is a confidence interval that does not contain the population proportion. (1) Scan the graph visually and count the number of confidence intervals that do not contain the population proportion. How many are there? (2) Of the 100 intervals represented, how many contain the population proportion? Does this agree with the value reported on the spreadsheet? (3) What percent of the confidence intervals contain the population proportion? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals (4) If you randomly selected 1 of the 100 sample proportions and its corresponding interval, how confident are you that the population proportion would be in that interval (give your answer as a percent from 0% to 100%). (5) Is your answer to Question 4 similar to the level of confidence entered in the spreadsheet? (6) Suppose your answer to Question 4 is not the same as the level of confidence, why might this be different? (7) Suppose you gathered 1,000 randomly generated sample proportions and computed a 95% confidence interval for each. About what number of these confidence intervals would include the population proportion of π = 0.45? (8) Discuss these ideas in your group and then write a statement for what the 95% means when you say you have a 95% confidence interval. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals Part II (9) In the simulation spreadsheet, continue to work with the population proportion of orange Reese’s Pieces (π = 0.45) and a confidence level of 0.95 (= 95%). Watch the widths of the confidence intervals as you increase the sample size to n = 1,000. Do the intervals get wider or narrower? (10) With the larger sample size, has the margin of error increased or decreased in each interval? (11) Change the sample size to n = 500. Are the intervals wider or narrower? (12) With the smaller sample size, has the margin of error increased or decreased in each interval? (13) Lower the level of confidence to 0.90 (= 90%). Does this cause the intervals to become wider or narrower? (14) With the lower confidence level, is the margin of error larger or smaller? Why do you suppose that is? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals (15) What proportion of the 90% confidence intervals contains the population proportion (π = 0.45)? Is this similar to the level of confidence? (16) Raise the confidence level to 0.99 (= 99%). Does this cause the intervals to become wider or narrower? (17) With the higher level of confidence, is the margin of error larger or smaller? Why do you suppose that is? (18) What proportion of the 99% confidence intervals contains the population proportion (π = 0.45)? Is this similar to the level of confidence? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals Homework At this point, you are intimately familiar with the assumption that 45% of all Reese’s Pieces are orange. Let’s assume that 35% of all Reese’s Pieces are yellow. Open the simulation used in this lesson at http://math.mtsac.edu/statistics/ManyConfInts.xls. For population proportion, enter the assumed proportion of Reese’s Pieces that are yellow (π = 0.35). Enter 100 for the sample size and 0.99 (= 99%) for the level of confidence. (1) Recall that the vertical red line represents the population proportion (π = 0.35) and that each horizontal orange line is a confidence interval. How many of the 100 confidence intervals contain the population proportion? (2) What percent of the confidence intervals contain the population proportion? Is this similar to the level of confidence? (3) What can be done so that a higher percentage of the intervals contain the population proportion? (4) Try your suggestion in Question 3. Does this yield the desired result? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.2: Connecting Sampling Distributions and Confidence Intervals (5) For a level of confidence to 0.99 (= 99%), what percentage of the intervals should contain the population proportion? (6) Test your answer to Question 5 on the spreadsheet. Does this yield the desired result? (7) Name two things you can do to reduce the margin of error. Which of these increases the quality of your results? Which decreases the quality? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Estimated number of 50-­‐minute class sessions: 1 Materials Required Once again, students interact with a simulation spreadsheet for this lesson. A computer with Excel is required. If students cannot use a computer, project the simulation from an instructor workstation. Students can break into groups to answer questions once the simulation has been run. Learning Goals Students will understand •
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•
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the process of hypothesis testing. the two types of errors that may occur when conducting hypothesis tests. the relationships between the probabilities of Type I and Type II errors. the tradeoffs involved when choosing a level of significance. Students will be able to •
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correctly identify Type I and Type II errors. correctly interpret the probability of a Type I error. choose a level of significance based on the nature of the possible errors. Notes to Instructors In this lesson, students run a coin toss simulation to generate a distribution of sample proportions under the null hypothesis that the true population proportion of heads is 0.50. Students identify the events that represent a Type I error, and the probability of a Type I error is defined as the proportion of samples where the null hypothesis is rejected when the null is true. A second distribution of sample proportions is generated under the alternative hypothesis that the true proportion of heads is 0.67. Students identify the sample proportions that represent a Type II error, and the probability of a Type II error is defined as the proportion of samples where the null is not rejected when the specific alternative hypothesis is true. The whole-­‐class discussion focuses on the consequences of each type of error, the fact that even if you estimate that a particular set of outcomes is rare it is still possible, and therefore you are never 100% certain that your decision is correct. Review of Hypothesis Testing (Note: It may be wise to remind students of the process applied in a hypothesis test. Below an outline of such a review is provided.) In an earlier lesson, you learned how to test claims regarding the value of an unknown population parameter. You learned that certain steps were required to conduct such tests. These steps are reviewed in the following sections. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing A Hypothesis to Test Each statistical test begins with an assumed value of the population parameter. This assumed value is the null hypothesis, denoted symbolically as HO. The null hypothesis is countered by a contrary claim—the parameter in question is either less than, greater than, or not equal to the value assumed in the null hypothesis. This contrary claim is Ha, the alternative hypothesis. Evidence for Statistical Inference The evidence required for inference is a statistic from a sample, along with some idea of the nature of the sampling distribution of such statistics. The sampling distribution is constructed based on the assumption that the null hypothesis is true. When you compare the statistic to simulated sampling distribution values, you can see how unusual or surprising the observed statistic is. If the statistic lies within one of the tails of the distribution (a low-­‐probability event), it becomes unlikely that the null hypothesis is true. Conclusion: Making a Decision Using simulation or rules of probability, you compute a P-­‐value, which is the likelihood of randomly gathering a statistic that is at least as extreme as the one you observed. The P-­‐value is a measure of how significant the observed statistic is; it is compared to the level of significance (α). Whenever the P-­‐value is less than the required level for significance, you reject the null hypothesis and support the alternative. In this lesson, you revisit the process of testing a hypothesis and learn a few new things about testing hypotheses. You revisit the idea of making a wrong conclusion and explore the idea that you can never be 100% certain when you draw a conclusion from a statistical test. Introduction to the Context of the Lesson [Student Handout] Imagine you have a coin that you know is fair when tossed. In other words, in the long run, the coin will come up heads half of the time when it is flipped and allowed to land on the ground. However, imagine some friends of yours are skeptical and want to test whether the coin really is fair. They toss the coin 100 times and record the proportion of times it lands heads up. Without going into the results of the coin tossing, let’s explore the possible erroneous decisions that they might make. Assuming the coin is fair, is it possible your friends end up with a result that causes them to claim the coin is not fair? In this activity, the assumption is the coin is fair, the population proportion of all possible tosses for the coin is π = 0.50, and this is the null hypothesis. The alternative hypothesis is that the coin is not fair—it is biased. HO: The coin is fair, π = 0.50. Ha: The coin is biased, π ≠ 0.50. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Part I [Student Handout, estimated 50-­‐minute periods: 0.5] Your friends toss your coin 100 times, and only conduct this test once. This is the sample that they will compare to a sampling distribution created under the assumption of the null hypothesis. You have seen repeatedly that sample proportions from random samples of a given size and population can vary. Discuss the following question in your small group. (1) Is it possible that even though the coin is fair, your friends could get a sample proportion so large or so small that they reject the null hypothesis and claim your coin is not fair? Next, use a simulation to address the fairness of the coin in question. Open the simulation spreadsheet located at the following address. http://math.mtsac.edu/statistics/CriticalRegions.xls In this spreadsheet, enter the following values: •
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Set the assumed population proportion to 0.50. Set the sample size to 100 (the number of coin tosses). The spreadsheet immediately simulates 1,000 sample proportions from samples of size 100, under the assumption of the null hypothesis that the population proportion is 50%. You will use the simulation to estimate the proportion of samples that would cause you to reject the null hypothesis (the coin is fair). To make a decision regarding the null hypothesis, a P-­‐value must be computed. The P-­‐value represents the likelihood of observing a random sample proportion of heads that is at least as extreme as the proportion observed in your friends’ experiment. The rule for rejection of the null hypothesis will be the following: If the P-­‐value is less than a 5% level of significance (α = 0.05), reject the null hypothesis. The spreadsheet displays two rejection values: a lower rejection value and an upper rejection value. These are used as limits for the range of values that you should consider not unusual and that will not cause a rejection of the null hypothesis. If the observed proportion of heads tossed is outside of these values, consider the observation unusual and reject the null hypothesis. Determine the rejection values that would lead you to wrongly reject the null hypothesis 5% of the time. Using trial and error, adjust the value of the allowed deviation from π (π = 0.50) until the percentage of simulated sample proportions outside of this range is approximately equal to the level of significance (α = 0.05 = 5%). On the graph, a red box is plotted. The vertical lines on the box separate the usual values from the unusual values. All dots that are not in the box are colored red. The red dots represent sample proportions that would cause you to reject the null hypothesis and declare the coin to be biased. (2) What value did you discover for the allowed deviation from π that causes about 5% of sample proportions to be outside of the box (in the rejection region)? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing (3) What range of values would not cause a rejection of the null hypothesis? from __________________ to __________________ In your small group, discuss the following questions. (4) Describe what is meant when you say that your friends may make a correct decision. (5) Based on the values entered into the spreadsheet, what are the chances that your friends will make a correct decision? (6) Describe what is meant when you say that your friends may make an incorrect decision. (7) What are the chances that your friends will make an incorrect decision? (8) Why do you think that it is possible for your friends to decide incorrectly? (9) Would you agree that the level of significance (α = 0.05) is equal to the probability that your friends conclude incorrectly that the coin is biased? Remember the definitions of Type I and Type II errors: •
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A Type I error occurs when you reject a null hypothesis that is true. A Type II error occurs when you fail to reject a null hypothesis that is false. (10) If your friends incorrectly decide that your coin is biased, which type of error have they made? (11) The level of significance (α) is the probability of which type of error—Type I or II? Whole-­‐Class Discussion Once each group has discussed the questions, have them report to the class. Check that students are focusing on the correct areas of the dotplot when referring to the probabilities of making a correct and incorrect decision. Spend a moment to emphasize what has been discovered to this point. The level of significance is now being seen in two ways: •
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The level of significance is a rejection criterion for the rejection of the null hypothesis. You reject the null hypothesis whenever the P-­‐value is less than the null hypothesis. The level of significance is the probability of a Type I error. Whenever the null hypothesis is true, there is always a chance that you will reject it, depending on the nature of the random sample. The probability of this Type I error is α, the level of significance. Also, take a moment to discuss the idea that you have some control over the likelihood of a Type I error. If a Type I error is worse than a Type II Error, you can minimize its likelihood by picking a smaller level of significance. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Part II: Managing Type I and Type II Errors [Student Handout, estimated 50-­‐minute periods: 0.5] For Part II of this lesson, you will no longer continue to be absolutely sure that your coin is fair. You are very sure, but must admit to some uncertainty. You would like to minimize the probability of a Type I error, but still do not want to commit a Type II error. (12) How can you choose the level of significance to minimize the probability of a Type I error? (13) Returning to the simulated sampling distribution spreadsheet, make sure that the sample size is set at n = 100 and that the proportion of heads for your coin is π = 0.50. Using trial and error, increase the allowed deviation from the assumed value of π until the simulated value of α is around 1% (α ≈ 1%). (14) What range of values would not cause a rejection of the null hypothesis? from __________________ to __________________ (15) Is the range of values that would not cause a rejection of the null hypothesis wider or narrower than when α = 0.05 (see Question 3). (16) By lowering the value of α to approximately 1%, have you made it less or more probable that you will reject the null hypothesis? (17) Suppose the null hypothesis (the coin is fair) is false. While using the smaller value of α, are you more or less likely to commit a Type II error? (18) Do smaller levels of significance (α) make the probability of a Type II error larger or smaller? In hypothesis tests in general, sometimes a Type I or Type II error can be serious and have grave consequences (in medical or engineering settings, for example). When you choose a level of significance, consider the consequences of the Type I and Type II errors. (19) If a Type I error is worse than a Type II error, should you pick a smaller or larger level of significance? (20) If a Type II error is worse than a Type I error, should you pick a smaller or larger level of significance? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Wrap-­‐Up The whole-­‐class discussion can focus on the following items: •
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The meaning of each type of error. Even when a particular error is unlikely, it is still possible. The probability of a Type I error is equal to the level of significance (α). Reducing α decreases the probability of a Type I error, but increases the probability of a Type II error. Raising α increases the probability of a Type I error, but reduces the probability of a Type II error. Homework [Student Handout] Suppose a pharmaceutical company manufactures a medication that it advertises as having serious side effects on less than 1% of the general population of people using it. To investigate this issue further, you gather 1,000 users of the medication and measure the proportion who experienced serious side effects. With these data, you will test the hypothesis that the proportion (π) of all users who experience serious side effects from this medication is less than 1% (= 0.01) against the null hypothesis that the proportion is equal to 1%. If the sample data support the hypothesis that the proportion of users who experience serious side effects is less than 1%, no action will be taken. If however, this hypothesis cannot be supported, the medication will be removed from the market. (1) What is the null hypothesis for this hypothesis test? HO: π ______________ (2) What is the alternative hypothesis? Ha: π ______________ (3) Suppose the data lead you to reject the null hypothesis and support the alternative. What type of error could possibly be made in this case? (4) Describe the error in the context of the medical study. (5) Suppose the data do not lead you to reject the null hypothesis. What type of error might be made in this case? (6) Which error is worse in this situation—the Type I or Type II error? (7) Given your decision regarding which error is worse in the context of this problem, which value for the level of significance might be more appropriate: α = 0.01 or α = 0.05? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Imagine you have a coin that you know is fair when tossed. In other words, in the long run, the coin will come up heads half of the time when it is flipped and allowed to land on the ground. However, imagine some friends of yours are skeptical and want to test whether the coin really is fair. They toss the coin 100 times and record the proportion of times it lands heads up. Without going into the results of the coin tossing, let’s explore the possible erroneous decisions that they might make. Assuming the coin is fair, is it possible your friends end up with a result that causes them to claim the coin is not fair? In this activity, the assumption is the coin is fair, the population proportion of all possible tosses for the coin is π = 0.50, and this is the null hypothesis. The alternative hypothesis is that the coin is not fair—it is biased. HO: The coin is fair, π = 0.50. Ha: The coin is biased, π ≠ 0.50. Part I Your friends toss your coin 100 times, and only conduct this test once. This is the sample that they will compare to a sampling distribution created under the assumption of the null hypothesis. You have seen repeatedly that sample proportions from random samples of a given size and population can vary. Discuss the following question in your small group. (1) Is it possible that even though the coin is fair, your friends could get a sample proportion so large or so small that they reject the null hypothesis and claim your coin is not fair? Next, use a simulation to address the fairness of the coin in question. Open the simulation spreadsheet located at the following address. http://math.mtsac.edu/statistics/CriticalRegions.xls In this spreadsheet, enter the following values: •
•
Set the assumed population proportion to 0.50. Set the sample size to 100 (the number of coin tosses). The spreadsheet immediately simulates 1,000 sample proportions from samples of size 100, under the assumption of the null hypothesis that the population proportion is 50%. You will use the simulation to estimate the proportion of samples that would cause you to reject the null hypothesis (the coin is 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. 1 Statway Student Handout April 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing To make a decision regarding the null hypothesis, a P-­‐value must be computed. The P-­‐value represents the likelihood of observing a random sample proportion of heads that is at least as extreme as the proportion observed in your friends’ experiment. The rule for rejection of the null hypothesis will be the following: If the P-­‐value is less than a 5% level of significance (α = 0.05), reject the null hypothesis. The spreadsheet displays two rejection values: a lower rejection value and an upper rejection value. These are used as limits for the range of values that you should consider not unusual and that will not cause a rejection of the null hypothesis. If the observed proportion of heads tossed is outside of these values, consider the observation unusual and reject the null hypothesis. Determine the rejection values that would lead you to wrongly reject the null hypothesis 5% of the time. Using trial and error, adjust the value of the allowed deviation from π (π = 0.50) until the percentage of simulated sample proportions outside of this range is approximately equal to the level of significance (α = 0.05 = 5%). On the graph, a red box is plotted. The vertical lines on the box separate the usual values from the unusual values. All dots that are not in the box are colored red. The red dots represent sample proportions that would cause you to reject the null hypothesis and declare the coin to be biased. (2) What value did you discover for the allowed deviation from π that causes about 5% of sample proportions to be outside of the box (in the rejection region)? (3) What range of values would not cause a rejection of the null hypothesis? from __________________ to __________________ In your small group, discuss the following questions. (4) Describe what is meant when you say that your friends may make a correct decision. (5) Based on the values entered into the spreadsheet, what are the chances that your friends will make a correct decision? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing (6) Describe what is meant when you say that your friends may make an incorrect decision. (7) What are the chances that your friends will make an incorrect decision? (8) Why do you think that it is possible for your friends to decide incorrectly? (9) Would you agree that the level of significance (α = 0.05) is equal to the probability that your friends conclude incorrectly that the coin is biased? Remember the definitions of Type I and Type II errors: •
•
A Type I error occurs when you reject a null hypothesis that is true. A Type II error occurs when you fail to reject a null hypothesis that is false. (10) If your friends incorrectly decide that your coin is biased, which type of error have they made? (11) The level of significance (α) is the probability of which type of error—Type I or Type II? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Part II: Managing Type I and Type II Errors For Part II of this lesson, you will no longer continue to be absolutely sure that your coin is fair. You are very sure, but must admit to some uncertainty. You would like to minimize the probability of a Type I error, but still do not want to commit a Type II error. (12) How can you choose the level of significance to minimize the probability of a Type I error? (13) Returning to the simulated sampling distribution spreadsheet, make sure that the sample size is set at n = 100 and that the proportion of heads for your coin is π = 0.50. Using trial and error, increase the allowed deviation from the assumed value of π until the simulated value of α is around 1% (α ≈ 1%). (14) What range of values would not cause a rejection of the null hypothesis? from __________________ to __________________ (15) Is the range of values that would not cause a rejection of the null hypothesis wider or narrower than when α = 0.05 (see Question 3). (16) By lowering the value of α to approximately 1%, have you made it less or more probable that you will reject the null hypothesis? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing (17) Suppose the null hypothesis (the coin is fair) is false. While using the smaller value of α, are you more or less likely to commit a Type II error? (18) Do smaller levels of significance (α) make the probability of a Type II error larger or smaller? In hypothesis tests in general, sometimes a Type I or Type II error can be serious and have grave consequences (in medical or engineering settings, for example). When you choose a level of significance, consider the consequences of the Type I and Type II errors. (19) If a Type I error is worse than a Type II error, should you pick a smaller or larger level of significance? (20) If a Type II error is worse than a Type I error, should you pick a smaller or larger level of significance? 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing Homework Suppose a pharmaceutical company manufactures a medication that it advertises as having serious side effects on less than 1% of the general population of people using it. To investigate this issue further, you gather 1,000 users of the medication and measure the proportion who experienced serious side effects. With these data, you will test the hypothesis that the proportion (π) of all users who experience serious side effects from this medication is less than 1% (= 0.01) against the null hypothesis that the proportion is equal to 1%. If the sample data support the hypothesis that the proportion of users who experience serious side effects is less than 1%, no action will be taken. If however, this hypothesis cannot be supported, the medication will be removed from the market. (1) What is the null hypothesis for this hypothesis test? HO: π ______________ (2) What is the alternative hypothesis? Ha: π ______________ (3) Suppose the data lead you to reject the null hypothesis and support the alternative. What type of error could possibly be made in this case? (4) Describe the error in the context of the medical study. 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 24, 2012 (Full Version 1.0) Supporting Lesson 7.2.3: Connecting Sampling Distributions and Hypothesis Testing (5) Suppose the data do not lead you to reject the null hypothesis. What type of error might be made in this case? (6) Which error is worse in this situation—the Type I or Type II error? (7) Given your decision regarding which error is worse in the context of this problem, which value for the level of significance might be more appropriate: α = 0.01 or α = 0.05? 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