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```					Using Simulations to Teach
Statistical Inference

Beth Chance, Allan Rossman (Cal Poly)

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Joint Work with

n   Soma Roy, Karen McGaughey (Cal Poly),
q   Alex Herrington (Cal Poly undergrad)
n   John Holcomb (Cleveland State),
n   George Cobb (Mt. Holyoke),
n   Nathan Tintle, Jill VanderStoep, Todd
Swanson (Hope College)
n   This project has been supported by the
National Science Foundation, DUE/CCLI
#0633349
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Outline

n   Motivation/Goals
n   Examples
q   Binomial process, randomized experiment- binary,
randomized experiment - quantitative response
q   Series of lab assignments
q   Discussion points
n   Student feedback, Evaluation results
n   Design principles & implementation
n   Observations, Open questions
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Motivation

n   Cobb (2007) – 12 reasons to teach
permutation tests…
q   Model is “simple and easily grasped”
q   Matches production process, links data production
and inference
q   Role for tactile and computer simulations
q   Easily extendible to other designs (e.g., blocking)
q   Fisherian logic
--”The Introductory Statistics Course:
A Ptolemaic Curriculum” (TISE)
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Goals

n   Develop an introductory curriculum that
focuses on randomization-based approach to
inference
q   vs. using simulation to teach traditional inference
q   From beginning of course, permeate all topics
n   Improve understanding of inference and
statistical process in general
q   More modern (computer intensive) and flexible
approach to inferential analysis

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Brief overview of labs

n   Case-study focus
n   Pre-lab
q   Background, Review questions submitted in advance
n   50-minute (computer) lab period
n   Online instructions
q   Directed questions following statistical process
q   Embedded applets or statistical software
n   Application/Extension
n   Lab report with partner

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Example 1: Friend or Foe
(Helper/Hinderer)
n   Videos
n   Research question
n   Pre-lab
n   Descriptive analysis
n   Introduction of null hypothesis,
p-value terminology
n   Plausible values
n   Conclusions

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Discussion Points

n   Can this be done on day one?
q   Yes if can motivate the simulation
n   Before reveal the data?

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<<After tactile simulation>> How many infants
would need to choose the helper toy for you to be
convinced the choice was not made “at random,”
but they actually prefer the helper toy?
n   Many students can reason inferentially
q   “If a choice is made at complete random, then
having 13 infants would be highly unlikely”
q   “Based on the coin flipping experiment, the results
stated that at/over 12 was extremely rare.
Therefore, at least 12 infants …
q   “Would be around 12-16 because it seems highly
unlikely that given a 50-50 option 12-16 would
choose the helper toy”

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<<After tactile simulation>> How many infants
would need to choose the helper toy for you to be
convinced the choice was not made “at random,”
but they actually prefer the helper toy?
n   But maybe not as well “distributionally”
q   Is it unusual? = “barely over half”
n   vs. unusual compared to distribution
n   Examine language carefully
q   “Unlikely that choice is random”
q   “Prove”
q   “Simulate”, “Repeated this study”
q   “At random” = 50/50, “model”
n   “Random” = anything is possible

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Discussion Points

n   Can this be done on day one?
q   Yes if can motivate the simulation
n   Before reveal the data?
n   Enough understanding of “chance model”?
n   Use of class data instead? (“observed” vs. research
study)
q   Yes, if return to and build on the ideas throughout
the course
n   So what comes next?

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Discussion Points

n   Tactile simulation
q   One coin 16 times vs. 16 coins
n   Population vs process
q   Defining the parameter
n   3Ss: statistic, simulate, strength of evidence
q   “could have been” distribution of data
q   “what if the null was true” distribution of statistic
n   Fill in the blank wording
n   Timing of final report
q   Follow-up in-class discussion

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Example 2: Two Proportions

n   Is Yawning Contagious?
q   Modelling entire process: data collection,
descriptive statistics, inferential analysis,
conclusions
q   Parallelisms to first example
q   Could random assignment alone produce a
difference in the group proportions at least this
extreme?
q   Card shuffling, recreate two-way table
q   Extend to own data
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Lab Instructions

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Exam Questions

n   Horizontal axis
n   Make up a research question

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Discussion Points

n   Starting with a significant result but when
n   How critical is authentic data?
n   Choice of statistic (count vs. difference in
proportion)
n   Role of traditional symbols and notation?
n   Visualization of bar graphs from trial to trial
n   Implementation of predict and test

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Example 3: Two means

n   Are there lingering effects to sleep
deprivation?
q   Randomized experiment
q   Quantitative data
q   Parallel inferential reasoning process
n   Index cards

n   Possible follow-up/extensions: what if -4.33?,
medians, plausible values

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Discussion Points

n   Role of tactile simulation
n   Scaffolding of lab report
q   Introductory sentences, labeling of graphs
q   Write conclusion to journal
n   When should “normal-based” methods be
introduced
q   Alternative approximation to simulation
q   Position, method for confidence intervals
n   Choice of technology
n   Applets, Minitab, R, Fathom

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Post-Lab Assessment (Fall 2010)

n   Following the lab comparing two groups on a
quantitative variable (65 responses)
q   Discuss the purpose of the simulation process
q   What information does the simulation process reveal
to help you answer the research question?
n   Essentially correct: 35.4% demonstrated
understanding of the big picture (looking at
repeated shuffles to assess whether the
observed results happened by chance)
n   Partially: 38.5% (one of null or comparison)
n   Incorrect: 26.1% (“better understand the data”)

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Post-Lab Assessment (Fall 2010)

n   Did students address the null hypothesis?
q   33.9% E/ 38.5% P/ 27.7% I
n   Did students reference the random assignment?
q   36.9% E/ 36.9% P/ 26.2% I
n   Did students focus on comparing the observed
result?
q   64.6% E/ 13.8% P/ 21.5% I
n   Did students explain how they would link the
pieces together and draw their conclusion?
q   24.6% E/ 60% P/ 15% I

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Student Surveys

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Student Surveys

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Student Surveys

n   Example 3 simulation

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Student Surveys

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Student Surveys

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Student Surveys

n   Helper/Hinderer (Winter 2011) – Did the lab
help you understand the overall process of a
statistical investigation?

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Student Surveys

n   Did subsequent labs increase understanding?

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Remainder of labs
n   Lab 4: Random babies
n   Lab 5: Reese’s Pieces (demo)
q   Normal approximation, CLT for binary
q   Transition to formal test of significance (6 steps)
n   Lab 6: Sleepless nights (finite population)
q   t approximation, CLT for quantitative, conf interval
n   Lab 7: Simulation of matched-pairs
n   Lab 8: Simulation of regression sampling
n   Chi-square, ANOVA
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Lab Report

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Student Feedback (Winter 2011)

n   Google docs survey during last week of
course
n   Two instructors

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Student end-of-course surveys (W 11)

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Student end-of-course surveys

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Top 2 most interesting labs

n   Instructor A
q   Is Yawning Contagious?
q   Heart Rates (matched pairs)

n   Instructor B
q   Friend or Foe
q   Is Yawning Contagious?
q   Reese’s Pieces

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q   Friend or Foe

q   Random babies
q   Melting away (intro two-sample t, paired)

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Exam 1

n   In a recent Gallup survey of 500 randomly
selected US adult Republicans, 390 said they
believe their congressional representative
should vote to repeal the Healthcare Law.
Suppose we wish to determine if significantly
more than three-quarters (75%) of US adult
Republicans favor repeal.
n   The coin tossing simulation applet was used to
generate the following two dotplots (A) and (B).
Which, if either, of the two plots (A) and (B) was
created using the correct procedure? Explain
how you know.

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Exam 1

n   35% picked B (usually citing null .75´500)
q   But some look at shape, or later p-value
n   29% picked A (observed result)
n   23% neither (wanted .5´500 = 250)
n   13% other responses: 0, .75, 50, can’t tell,
anything possible, label is wrong
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Exam 2

n   Heights of females are known to follow a normal
distribution with a mean of 64 inches and a
standard deviation of 3 inches. Consider the
behavior of sample means. Each of the graphs
below depicts the behavior of the sample mean
heights of females.
a. One graph shows the distribution of sample
means for many, many samples of size 10. The
other graph shows the distribution of sample
means for many, many samples of size 50.
Which graph goes with which sample size?

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Exam 2

n   85% matched n=10 and n = 50

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Exam 2

n Suppose we wish to test the following
hypotheses about the population of Cal Poly

n For which graph (A or B) would you expect
the p-value to be smaller? Explain using the
appropriate statistical reasoning.

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Exam 2

n   77% picked B
q   Mixture of appealing to smaller SD/outliers, larger
sample size means smaller p-value, and thinking
in terms of test statistic
q   A few choices not internally consistent
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Student understanding of p-value

n   CAOS questions (final exam)
q   Statistically significant results correspond to small
p-values
n   Randomization (Hope/CP): 95%/95%
q   Recognize valid p-value interpretation
n   Randomization (Hope/CP): 60/72%
q   p-value as probability of Ho - Invalid
n   Randomization (Hope/CP): 80%/89%

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Student understanding of p-value

n   CAOS questions (final exam)
q   p-value as probability of Ha – Invalid
n   Randomization (Hope/CP): 45/67%
q   Recognize a simulation approach to evaluate
significance (simulate with no preference vs.
repeating the experiment)
n   Randomization (Hope/CP): 32%/40%

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Student understanding of p-value

n   p-value interpretation in regression (final
exam)

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Student understanding of process

n   Video game question (Final exam: NCSU, Hope,
Cal Poly, UCLA, Rhodes College)
q   What is the explanation for the process the
student followed?
q   Which of the following was used as a basis for
simulating the data 1000 times?
q   What does the histogram tell you about whether
\$5 incentives are effective in improving
performance on the video game?
q   Which of the following could be the approximate p
-value in this situation?

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Student understanding of process

n   Simulation process
q   Fall: over 40% chose “This process allows her to
determine how many times she needs to replicate
the experiment for valid results.”
q   About 70% pick “The \$5 incentive and verbal
encouragement are equally effective at improving
performance.” as underlying assumption
q   Still evidence some look at center at zero or
shape as evidence of no treatment effect
q   1/3 to ½ could estimate p-value from graph

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Example – 2009 AP Statistics Exam

n   A consumer organization would like a method
for measuring the skewness of the data. One
possible statistic for measuring skewness is
the ratio mean/median….
q   Calculate statistic for sample data…
q   Draw conclusion from simulated data …

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Design Principles

n   Tactile simulation
n   Visual, contextual animation of tactile simulation
n   Intermediate animation capability
n   Level of student construction
q   Ease of changing inputs
q   Connect elements between graphs
n   Carefully designed, spiraling activities
q   “Stop!”
q   Thought questions
n   Allow for student exploration

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Implementation
n   Early in course
n   Repetition through course, connections
n   Normal approximations
n   Lab assignments
q   Focus on entire statistical process
q   Motivating research question
q   Follow-up application
q   Thought questions
q   Screen captures
q   Pre-lab questions
n   Exam questions

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Observations

n   Students quickly get sense of trying to
determine whether a result could be “just due
to chance”
n   Still struggle with more technical
understanding
q   Under the null hypothesis
q   Observed vs. hypothesized value
n   Students may fail to see connections
between scenarios
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Suggestions/Open Questions

n   Begin with class discussion/brain-storming on
how to evaluate data before show class
results
q   Loaded dice, biased coin tossing
q   Thought questions
n   Student data vs. genuine research article
q   “the result” vs. “your result”
n   Choice of first exposure
q   Significant?
q   Random sampling or random assignment
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Suggestions/Open Questions

n   Scaffolding
q   Observational units, variable
n   How would you add one more dot to graph?
q   At some point, require students to enter the
correct “observed result” (e.g., Captivate)
q   At some point, ask students to design the
simulation?
q   Start with fill in the blank interpretation?

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Suggestions/Open Questions

n   One crank or more?
n   When connect to normal approximations?
q   How make sure traditional methods don’t overtake
once they are introduced?
q   How much discuss exact methods?
n   Confidence intervals

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Summary

n   Very promising but also need to be very
careful, and need a strong cycle of repetition
closely tied to rest of course…

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