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An Active Approach to Statistical Inference using

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An Active Approach to Statistical Inference using Powered By Docstoc
					An Active Approach to
Statistical Inference using
Randomization Methods
    Todd Swanson & Jill VanderStoep
    Hope College
    Holland, Michigan
Outline

   Background
   Content
   Pedagogy
   Example
   Assessment
   Future
Inspiration

  “Our curriculum is needlessly complicated
  because we put the normal distribution, as an
  approximate sampling distribution for the
  mean, at the center of the curriculum, instead
  of putting the core logic of inference at the
  center.”
           George Cobb (USCOTS 2005)


Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Previous Work
Concepts of Statistical Inference:
    A Randomization-Based Curriculum
   An NSF funded project in which modules
    were developed to teach inference through
    randomization techniques.
   Principle Investigators: Allan Rossman and
    Beth Chance (Cal Poly)
   Work done in 2007-08

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Development of Text
An Active Approach to Statistical Inference
   Along with Nathan Tintle, we
    developed first draft of a text
    in 2009
   Used the modules developed
    by Rossman and Chance as
    the base
   First used at Hope College in
    the Fall of 2009

 Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Development of Text

   Revisions were made during summer 2010.
   This fall we have joined up with Allan
    Rossman, Beth Chance and Soma Roy (all of
    Cal Poly) and George Cobb (Mt. Holyoke) to
    continue to make significant revisions to our
    materials.




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Content
   We begin with inference on the first day of
    the course and teach it throughout the entire
    semester
   First half of course is based on randomization
    methods and second half is based on
    traditional methods




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Table of Contents (Unit 1)
   Chapter 1: Introduction to Statistical Inference:
    One proportion
       Flipping coins and applets are used to model the null
        and their results are used to determine p-values.
   Chapter 2: Comparing Two Proportions:
    Randomization Method
       Explanatory and response variables are introduced
       Permutation tests are introduced
       First by using playing cards then with Fathom
        (perhaps applets in the future)
       Observational studies/experiments
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Table of Contents (Unit 1)
   Chapter 3: Comparing Two Means:
    Randomization Method
       Measurements of spread
       Permutation tests of means with cards and Fathom
       Type I and type II errors introduced
   Chapter 4: Correlation and Regression:
    Randomization Method
       Scatterplots, correlation, and regression are reviewed
       Permutation tests are used to test correlation


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Table of Contents (Unit 2)
    Chapter 5: Correlation and Regression: Revisited
        Sampling distributions are used to model scrambled
         distributions
        Confidence intervals (range of plausible values)
        Power is defined and students explore how it relates to
         sample size, significance level, and population
         correlation
    Chapter 6: Comparing Means: Revisited
        Standard deviation, normal distributions, and t-
         distributions
        The independent samples t test is introduced
        Confidence intervals and power
        Paired-data t test and ANOVA are also introduced
    Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Table of Contents (Unit 2)
    Chapter 7: Comparing Proportions: Revisited
        Power is explored in relationship to the difference in
         population proportions, sample size, significance level,
         and size of the two proportions
        The chi-square test for association is introduced
    Chapter 8: Tests of a Single Mean and Proportion
        Single proportion: binomial, normal distributions, and
         confidence intervals
        Single mean: t-test and confidence intervals
        Chi-squared goodness of fit test


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Main differences between our randomization
curriculum and traditional ones

   Traditional method of teaching introductory
    statistics:
       Descriptive statistics
       Probability and sampling distributions
       Inference


   Randomization method
       Inference on day one

Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Main differences between our randomization
curriculum and traditional ones
   Most of the time we visit and re-visit the core-
    logic of statistical inference as first
    demonstrated by randomization methods.
   We spend limited time teaching descriptive
    statistical methods and instead include time
    to review and reinforce the proper use of
    descriptive statistical methods through
    hands-on real data analysis experiences.
   We eliminate the explicit coverage of
    probability and sampling distributions.
Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Main differences between our randomization
curriculum and traditional ones
   We present an intuitive approach to power by
    looking at the relationships between power and
    sample size, standard deviation, difference in
    population proportions or means, etc. We think this
    helps students better understand the core logic of
    statistical inference.
   Confidence intervals are presented after tests. We
    demonstrate how tests of significance can be used
    to create ranges of plausible values for the
    population parameter.

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Pedagogy

   Topics are introduced through a brief lecture
   Students work on activities to learn and
    reinforce the topics.
       Tactile learning (shuffling cards and flipping coins)
        to estimate p-values
       Computer based simulations
       Collecting data and running experiments




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All our classes meet in a computer classroom.

Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Real Data --- Real Research

   We try to avoid cute, but impractical
    illustrations of statistics. We include real data
    and research that matters.
   Homework problems and case-studies also
    involve real statistical data and research.
   Each chapter contains a research paper that
    students read and respond to questions.
   Students complete in-depth projects where
    they design a study, collect data, and present
    their results in both oral and written form.
Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Example: Bob or Tim?




Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Tim or Bob?

   A study in Psychonomic Bulletin and Review (Lea,
    Thomas, Lamkin, & Bell, 2007) presented evidence
    that “people use facial prototypes when they
    encounter different names.”
   Participants were given two faces and had to
    determine which one was Tim and which one was
    Bob. The researchers wrote that their participants
    “overwhelmingly agreed” on which face belonged to
    Tim and which face belonged to Bob.


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Hypotheses

   Alternative hypothesis: In the population, people
    have a tendency to associate certain facial features
    with a name. More specifically, the proportion of the
    population that correctly matches the names with
    the faces is greater than 0.5.
   Null hypothesis: In the population, people do not
    have a tendency to associate certain facial features
    with a name. More specifically, the proportion of the
    population that correctly matches the names with
    the faces is equal to 0.5.

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Did you get it correct?

Tim                               Bob




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Statistic---Simulate---Strength of Evidence

   Statistic: A recent class of statistics students
    (our sample) replicated this study and 23 of
    the 33 students (0.70) correctly identified the
    face that belonged to Tim.
   Simulate: To simulate the null hypothesis,
    we flip a coin 33 times and count the number
    of heads each time. (Repeat this 1000 times)
   Strength of Evidence: Just 17 out of 1000
    repetitions gave a result of 23 or more heads.
    Quite unlikely if the null was true.
Background ● Content ● Pedagogy ● Example ● Assessment ● Future
1000 repetitions of flipping a fair coin 33
times and counting the number of heads




                                              P-value = 0.017




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Conclusion

   We have evidence supporting that in the
    population of interest, the proportion of
    people that correctly identify which face
    belongs to Tim and which belongs to Bob is
    greater than 0.50.
   Thus based on our study we have evidence
    to support people have a tendency to
    associated certain facial features to a name.


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Assessment

   The Comprehensive Assessment of Outcomes in
    Statistics (CAOS)
   Students in our randomization course took this pre-
    and post-test in the Fall of 2009 (n = 202). These
    results were compared with students that took our
    traditional course in the Fall of 2007 (n = 198) and
    those from a national representative sample (n =
    768).
   Overall, learning gains were significantly higher for
    students that took the randomization course when
    compared to either those that took the traditional
    course at Hope or the national sample.

Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Questions where the new curriculum
faired significantly better
   Understanding that low p-values are desirable in
    research studies (Tests of significance)
   Understanding that no statistical significance
    does not guarantee that there is no effect (Tests
    of significance)
   Ability to recognize a correct interpretation of a
    p-value (Tests of significance)
   Ability to recognize an incorrect interpretation of
    a p-value. Specifically, probability that a
    treatment is not effective. (Tests of significance)
Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Questions where the new curriculum
faired significantly better
   Understanding of the purpose of randomization
    in an experiment (Data collection and design)
   Understanding of how to simulate data to find
    the probability of an observed value (Probability)




Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Questions where the new curriculum
faired significantly worse
   Ability to correctly estimate and compare
    standard deviations for different histograms.
    (Descriptive statistics)




Background ● Content ● Pedagogy ● Example ● Assessment ● Future
Moving Forward

   We welcome anyone that would like to field
    test the book.
   More information can be found at
              www.math.hope.edu/aasi
   Email
       Todd: swansont@hope.edu
       Jill: vanderstoepj@hope.edu



Background ● Content ● Pedagogy ● Example ● Assessment ● Future

				
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