Activities for Introducing Students to Statistics

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Activities for Introducing Students to Statistics

• Allan Rossman, Beth Chance, Nicole Walterman
• California Polytechnic State University
• San Luis Obispo, CA
• arossman_at_calpoly.edu
• http//statweb.calpoly.edu/rossman

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Outline

• Statistics Education Reform
• Calls, Features, Products
• Whats the Problem?
• Is the Problem Important?
• Proposed Alternative
• Principles, Content
• Sample Activities
• Testing, Dissemination

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Stat Ed Reform- Calls

• ASA/MAA Committee on Undergraduate Statistics
• www.maa.org/pubs/books/nte22.html
• NCTM K-12 Principles and Standards (2000)
• www.nctm.org/standards
• Advanced Placement Statistics curriculum
• www.collegeboard.com/ap/statistics/
• CBMS Mathematical Education of Teachers (2001)
• www.cbmsweb.org/MET_Document/index.htm

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Stat Ed Reform- Features

• Active Learning
• Conceptual Understanding
• Genuine Data
• Use of Technology
• Communication Skills
• Interpretation in Context
• Authentic Assessment

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Stat Ed Reform- Products

• Textbooks
• Activity Books
• Lab Manuals
• Books of Data Sources
• Books of Case Studies
• On-line Books
• Journals

• Dynamic Software
• Java Applets
• Web Sites
• Project Templates
• Assessment Instruments
• Workshops

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Whats the Problem?

• Vast majority of reform efforts have been
  directed at the Stat 101 service course
• Rarely reaches Mathematics or Statistics majors
• Option A Take Stat 101
• Option B Standard Prob and Math/Stat sequence

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Whats the Problem?

• Option A
• Does not challenge students mathematically
• Rarely counts toward major
• Option B
• Does not give balanced view of discipline
• Fails to recruit all who might be interested
• Leaves prospective K-12 teachers ill-prepared to
  teach statistics, implement reform methods
• Does not even prepare assistants for Stat 101!

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Is This Problem Important?

• The question of what to do about the standard
  two-course upperclass sequence in probability and
  statistics for mathematics majors is the most
  important unresolved issue in undergraduate
  statistics education.
• - David Moore, 1998 ASA President

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Is This Problem Important?

• The standard curriculum for mathematics majors
  allows little time for statistics until, at best,
  an upper division elective. At that point,
  students often find themselves thrust into a
  calculus-based mathematical statistics course,
  and they miss many basic statistical ideas and
  techniques that are at the heart of high school
  statistics courses.
• - CBMS MET report (ch. 5)

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Is This Problem Important?

• In most teacher preparation programs appropriate
  background in statistics and probability will not
  be provided by simply requiring a standard
  probability-statistics course for mathematics
  majors. It is essential to carefully consider the
  important goals of statistical education in
  designing courses that reflect new conceptions of
  the subject.
• - CBMS MET report (ch. 5)

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Proposed Alternative

• To develop and provide a
• Data-Oriented, Active Learning, Post-Calculus
• Introduction to Statistical
• Concepts, Applications, Theory
• Supported by the NSF DUE/CCLI 9950476, 0321973
• www.rossmanchance.com/iscat/

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Principles

• Motivate with real studies, data
• Foster active explorations
• Make use of mathematical competence to
  investigate underpinnings
• Use variety of computational tools
• Emphasize connections among study design,
  inference technique, scope of conclusion
• Use simulations (tactile, technology) often
• Introduce probability just in time

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Sample Activities

• Randomization Test (Friendly Observers)
• Sampling, Bias (Gettysburg Address)
• Confidence Procedures (Flat Tires)
• Matching Variables to Graphs
• Statistical Thinking (Cancer Pamphlets)
• Association vs. Causation (Televisions and Life
  Expectancy)

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Sample Activity Randomization Test

• Psychology experiment
• Butler and Baumeister (1998) studied effect of
  observer with vested interest on skilled
  performance
• Subjects played a video game ten times
• Established 70th percentile of performance score
  as threshold for each subject
• Played final game for prize, aiming to beat
  threshold

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Randomization Test (cont.)

• 24 subjects were randomly assigned to one of two
  groups
• Group A observer shared in prize
• Group B observer did not share
• Conjecture Those whose observer did not have a
  vested interest would perform better

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Randomization Test (cont.)
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Randomization Test (cont.)

• 3/12 lt 8/12
• Sample results support the conjecture, but by
  enough?
• How often would such an extreme sample occur by
  chance?

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Randomization Test (cont.)

• Simulate
• Let 11 black cards represent win and 13 red
  cards represent lose
• Shuffle the 24 cards and randomly deal 12 cards
  to represent Group A
• How many of the winners/black cards were assigned
  to Group A?
• How often do we find 3 or fewer winners in Group
  A if the assignment is purely random?

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Randomization Test (cont.)
empirical p-value 6/100 .06
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Randomization Test (cont.)

• Minitab macro
• put 11 1s, 13 0s in c1
• initialize counter k11
• sample 12 c1 c2
• let c3(k1)sum(c2)
• let k1k11

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Randomization Test (cont.)

• Fishers exact test p-value

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Sample Activity Sampling

• Select 10 representative words from the
  population of 268 words in the Gettysburg Address
• Calculate average length ( letters) per word
• Do you expect this sampling process to produce
  samples that are representative of the
  population?

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Sample Activity Sampling

• Students analyze results for evidence of bias
• Population mean is 4.295 letters per word

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Sample Activity Sampling

• Students then take a random sample of 5 words
• Discover that this method is unbiased
• Even with smaller sample
• May or may not be more precise

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Sample Activity Sampling

• Then students turn to technology (applet) to
  investigate long-run behavior of sample mean
• Effect of sample size
• Non-effect of population size

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Sample Activity Confidence

• Folk story Two students miss an exam and claim
  to have had a flat tire. Teacher agrees to give
  them a make-up exam
• Which tire was it?
• In a recent class, 8 of 20 students chose the
  right front tire
• Is this convincing evidence that the right front
  is chosen more often than expected?

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Confidence (cont.)

• Is p.25 a plausible value for Pr(right front)
  since . 4 with n20?
• Yes, since Pr(Xgt8) .1018
• Is p.5 a plausible value?
• Yes (at 95 level), since Pr(Xlt8) .2517
• What are the plausible values for Pr(RF)?
• Those values for which Pr(more extreme than
  observed 8) is not too small

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Confidence (cont.)

• Exact binomial 95 CI for p (.191, .639)
• Approximate 95 CI for p based on normal
  distribution (.185, .615)
• What does it mean to be 95 confident?
• Simulate!

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Confidence (cont.)

• Agresti and Coull (1998)
• (.219, .614)

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Confidence (cont.)

• Coverage probabilities of 10,000 repetitions
• top conventional, bottom shrinkage

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Sample Activities Summary

• Content introduced through student explorations
• Experiments, randomization, variability,
  significance, hypergeometric distribution
• Sampling, bias, random sampling, precision,
  effect of sample size
• Binomial distribution, confidence intervals,
  confidence levels, duality,

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Sample Activities Summary

• Students construct their own knowledge
• Hands-on simulations build intuition
• Focus on concepts, interpretation
• Motivation through real studies, data
• Statistical applications of mathematical models
• Context plays important role

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Sample Activity Minimization Criteria

• Total points in NBA games played on December 10,
  1999
• 140, 163, 184, 190, 196,
• 198, 204, 205, 206, 224
• Criterion for measuring center?
• Sum of absolute deviations
• Sum of squared deviations
• Sum of deviations to other power
• Maximum of absolute deviations
• Median of absolute deviations

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Minimization Criteria (cont.)
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Minimization Criteria (cont.)
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Minimization Criteria (cont.)
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Minimization Criteria (cont.)
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Minimization Criteria (cont.)
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Minimization Criterion (cont.)
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Sample Activity Matching Variables to Graphs

• (a) Annual snowfall in U.S. cities
• (b) Margins of victory in MLB
• (c) Weights of Cal Poly football players
• (d) Jersey s of Cal Poly football players
• (e) Ages at which mothers had first child
• (f) Monopoly property prices
• (g) Weights of 1999 cars

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Sample Activity Matching Variables to Graphs

• (a) Annual snowfall in U.S. cities F
• (b) Margins of victory in MLB E
• (c) Weights of Cal Poly football players G
• (d) Jersey s of Cal Poly football players C
• (e) Ages at which mothers had first child A
• (f) Monopoly property prices D
• (g) Weights of 1999 cars B

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Sample Activity Matching Variables to Graphs

• Learn to justify opinions
• Develop graph-sense
• Appreciate variability
• Context matters

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Sample Activity Statistical Thinking

• Researchers in Philadelphia investigated whether
  pamphlets containing information for cancer
  patients are written at a level that the cancer
  patients can comprehend

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Sample Activity Statistical Thinking
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Sample Activity Statistical Thinking

• Look at the data
• Think about the question
• Simple techniques can answer important questions
• Dont use high-powered techniques if theyre not
  needed, dont apply

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Sample Activity Association vs. Causation

• Is the number of people per television set in a
  country associated with the countrys life
  expectancy?

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Sample Activity Association vs. Causation
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Sample Activity Association vs. Causation

• Buy another television

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Sample Activity Association vs. Causation

• Buy another television
• Association is not causation
• Students can learn this without being told

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Testing and Dissemination

• National advisory group providing guidance on
  development
• Sample activities have been tested at a variety
  of institutions
• Opportunities for more extensive class testing
• Materials to be published by Duxbury
• Summer faculty development workshops
• June 21-25, 2004 in San Luis Obispo, CA
• July 14-16, 2004 in Huntsville, AL

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Summary

• Need to rethink introductory course for
  mathematically inclined students
• Provide balanced view of discipline
• Recruit more statisticians
• Prepare better teachers
• Recommendations
• Use real data, studies
• Foster active learning
• Utilize mathematical abilities

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Questions, Comments?

• More information about this project
• www.rossmanchance.com/iscat/
• Java applets
• www.rossmanchance.com/applets/
• Please contact us
• arossman_at_calpoly.edu
• bchance_at_calpoly.edu
• nickimw11_at_hotmail.com