Transcript Teaching Introductory Statistics with Activities and Data

Teaching Statistical Concepts with
Activities, Data, and Technology
Beth L. Chance and Allan J. Rossman
Dept of Statistics, Cal Poly – San Luis Obispo
Goals

Acquaint you with recent recommendations
and ideas for teaching introductory statistics
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Including some very “modern” approaches
On top of some issues we consider essential
Provide specific examples and activities that
you might plug into your courses
Point you toward online and print resources
that might be helpful
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Schedule
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Introductions
Opening Activity
Activity Sessions
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Data Collection
Data Analysis
<< lunch>>
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Randomness
Statistical Inference
Resources and Assessment
Q&A, Wrap-Up
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Requests
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Participate in activities
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23 of them!
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Play role of student
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We’ll skip/highlight some
Good student, not disruptive student!
Feel free to interject comments, questions
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GAISE
Emphasize statistical literacy and develop
statistical thinking
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Use real data
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Stress conceptual understanding rather than mere
knowledge of procedures
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Foster active learning in the classroom
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Use technology for developing conceptual
understanding and analyzing data
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Use assessments to improve and evaluate student
learning
www.amstat.org/education/gaise
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Opening Activity
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Naughty or nice? (Nature, 2007)
Videos:
http://www.yale.edu/infantlab/socialevaluation/
Helper-Hinderer.html
Flip 16 coins, one for each infant, to decide
which toy you want to play with (heads=helper)
Coin Tossing Applet:
http://www.rossmanchance.com/applets
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3S Strategy
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Statistic
Simulate
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“Could have been” distribution of data for each
repetition (under null model)
“What if” distribution of statistics across repetitions
(under null model)
Strength of evidence
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Reject vs. plausible
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Summary
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Use real data/scientific studies
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Stress conceptual understanding
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Idea of p-value on day 1/in one day!
Foster active learning
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Emphasize the process of statistical investigation
You are a dot on the board
Use technology
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Could this have happened “by chance alone”?
What if only 10 infants had picked the helper?
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Data Collection Activities:
Activity 2: Sampling Words
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Circle 10 representative words in the
passage
Record the number of letters in each word
Calculate the mean number of letters in your
sample
Dotplot of results…
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Sampling Words
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The population mean of all 268 words is
4.295 letters
How many sample means were too high?
Why do you think so many sample means are
too high?
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Sampling Words

“Tactile” simulation
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Ask students to use computer or random number
table to take simple random samples
Determine the sample mean in each sample
Compare the distributions
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Sampling Words
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Java applet
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www.rossmanchance.com/applets/
Select “Sampling words” applet
Select individual sample of 5 words
Repeat
Select 98 more samples of size 5
Explore the effect of sample size
Explore the effect of population size
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Morals: Selecting a Sample
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Random Sampling eliminates human
selection bias so the sample will be fair and
unbiased/representative of the population.
While increasing the sample size improves
precision, this does not decrease bias.
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Activity 3:
Night Lights and Near-Sightedness
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Quinn, Shin, Maguire, and Stone (1999)
479 children
Did your child use a night light (or room light
or neither) before age 2?
Eyesight: Hyperopia (far-sighted),
emmetropia (normal) or myopia (nearsighted)?
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Night Lights and Near-Sightedness
Darkness
Night light
Room light
Nearsighted
18
78
41
Normal
refraction
114
115
22
Far-sighted
40
39
12
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Night Lights and Near-Sightedness
100%
90%
80%
70%
60%
Far-sighted
50%
Normal refraction
Near-sighted
40%
30%
20%
10%
0%
Darkness
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Night light
Room light
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Morals: Confounding
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Students can tell you that association is not
the same as causation!
Need practice clearly describing how
confounding variable
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Is linked to both explanatory and response
variables
Provides an alternative explanation for observed
association
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Activity 4: Have a Nice Trip
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Can instruction in a recovery strategy
improve an older person’s ability to recover
from a loss of balance?
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12 subjects have agreed to participate in the study
Assign 6 people to use the lowering strategy and
6 people to use the elevating strategy
What does “random assignment” gain you?
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Have a Nice Trip
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Randomizing subjects applet
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How do the two groups compare?
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Morals
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Goal of random assignment is to be willing to
consider the treatment groups equivalent
prior to the imposition of the treatment(s).
This allows us to eliminate all potential
confounding variables as a plausible
explanation for any significant differences in
the response variable after the treatments are
imposed.
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Activity 5: Cursive Writing
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Does using cursive writing cause students to
score better on the SAT essay?
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Morals: Scope of Conclusions
The Statistical Sleuth, Ramsey and Schafer
Allocation of units to groups
Random sampling
By random assignment
No random
assignment
A random sample is
selected from one
population; units are then
randomly assigned to
different treatment groups
Random samples are
selected from existing
distinct populations
A groups of study units is
found; units are then
randomly assigned to
treatment groups
Collections of available
units from distinct
groups are examined
Inferences to
populations can
be drawn
Selection of units
Not random sampling
Cause and effect
conclusions can be drawn
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Activity 6: Memorizing Letters
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You will be asked to memorize as many
letters as you can in 20 seconds, in order,
from a sequence of 30 letters
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Variables?
Type of study?
Comparison?
Random assignment?
Blindness?
Random sampling?
More to come …
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Morals: Data Collection
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Quick, simple experimental data collection
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Highlighting critical aspects of effective study
design
Can return to the data several times in the
course
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Data Analysis Activities
Activity 7: Matching Variables to Graphs
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Which dotplot belongs to which variable?
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Justify your answer
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Morals: Graph-sense
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Learn to justify opinions
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Consistency, completeness
Appreciate variability
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Be able to find and explain patterns in the data
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Activity 8: Rower Weights
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2008 Men’s Olympic Rowing Team
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Rower Weights
Mean
197.96
201.17
209.65
Full Data Set
Without Coxswain
Without Coxswain or
lightweight rowers
With heaviest at 249
210.65
With heaviest at 429
219.70
Resistance....
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Median
205.00
207.00
209.00
209.00
209.00
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Morals: Rower Weights
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Think about the context
“Data are numbers with a context” -Moore
Know what your numerical summary is
measuring
Investigate causes for unusual observations
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Anticipate shape
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Activity 9: Cancer Pamphlets
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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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Cancer Pamphlets
0.3
0.25
pat ient s
pamphlets
proportion
0.2
0.15
0.1
0.05
above 12
12
11
10
9
8
7
6
5
4
3
under 3
0
level
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Morals: Importance of Graphs
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Look at the data
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Think about the question
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Numerical summaries don’t tell the whole
story
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“median isn’t the message” - Gould
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Activity 10: Draft Lottery
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Draft numbers (1-366) were assigned to
birthdates in the 1970 draft lottery
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Find your draft number
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Any 225s?
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Draft Lottery
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Draft Lottery
month
median
January
211.0
February 210.0
March
256.0
April
225.0
May
226.0
June
207.5
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month
median
July
188.0
August
145.0
September 168.0
October
201.0
November 131.5
December 100.0
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Draft Lottery
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Morals: Statistics matters!
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Summaries can illuminate
Randomization can be difficult
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Activity 11:
Televisions and Life Expectancy
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Is there an association between the two
variables?
r = .743
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So sending televisions to countries with lower
life expectancies would cause their
inhabitants to live longer?
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Morals: Confounding
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Don’t jump to conclusions from observational
studies
The association is real but consider carefully
the interpretation of graph and wording of
conclusions (and headlines)
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Activity 6 Revisited (Memorizing Letters)
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Produce, interpret graphical displays to
compare performance of two groups
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Does research hypothesis appear to be
supported?
Any unusual features in distributions?
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Lunch!

Questions?

Write down and submit any questions you have
thus far on the statistical or pedagogical content…
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Exploring Randomness
Activity 12: Random Babies
Last Names
Jones
Miller
Smith
Williams
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First Names
Jerry
Marvin
Sam
Willy
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Random Babies
Last Names
Jones
Miller
Smith
Williams
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First Names
Marvin
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Random Babies
Last Names
Jones
Miller
Smith
Williams
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First Names
Marvin
Willy
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Random Babies
Last Names
Jones
Miller
Smith
Williams
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First Names
Marvin
Willy
Sam
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Random Babies
Last Names
Jones
Miller
Smith
Williams
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First Names
Marvin
Willy
Sam
Jerry
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Random Babies
Last Names
Jones
Miller
Smith
Williams
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First Names
Marvin
Willy
Sam 1 match
Jerry
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Random Babies
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Long-run relative frequency
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Applet: www.rossmanchance.com/applets/
“Random Babies”
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Random Babies: Mathematical Analysis
1234 1243
1324
1342
1423
1432
2134 2143 2314
2341
2413
2431
3124 3142 3214
3241
3412
3421
4123 4132 4213
4231
4312
4321
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Random Babies
1234
4
2134
2
3124
1
4123
0
1243 1324
2
2
2143 2314
0
1
3142 3214
0
2
4132 4213
1
1
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1342
1
2341
0
3241
1
4231
2
1423
1
2413
0
3412
0
4312
0
1432
2
2431
1
3421
0
4321
0
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Random Babies
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0 matches: 9/24=3/8
1 match: 8/24=1/3
2 matches: 6/24=1/4
3 matches: 0
4 matches: 1/24
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Morals: Treatment of Probability
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Goal: Interpretation in terms of long-run
relative frequency, average value
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First simulate, then do theoretical analysis
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30% chance of rain…
Able to list sample space
Short cuts when are actually equally likely
Simple, fun applications of basic probability
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Activity 13: AIDS Testing
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ELISA test used to screen blood for the
AIDS virus
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Sensitivity: P(+|AIDS)=.977
Specificity: P(-|no AIDS)=.926
Base rate: P(AIDS)=.005
Find P(AIDS|+)
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Initial guess?
Bayes’ theorem?
Construct a two-way table for hypothetical
population
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AIDS Testing
Positive Negative
AIDS
No AIDS
Total
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Total
1,000,000
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AIDS Testing
Positive Negative
AIDS
No AIDS
Total
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Total
5,000
995,000
1,000,000
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AIDS Testing
AIDS
No AIDS
Total
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Positive Negative
4885
115
Total
5,000
995,000
1,000,000
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AIDS Testing
Positive Negative
Total
AIDS
4885
115
5,000
No AIDS 73,630 921,370
995,000
Total
1,000,000
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AIDS Testing
Positive Negative
Total
AIDS
4885
115
5,000
No AIDS 73,630 921,370
995,000
Total
78,515 921,485 1,000,000
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AIDS Testing
Positive Negative
Total
AIDS
4885
115
5,000
No AIDS 73,630 921,370
995,000
Total
78,515 921,485 1,000,000
P(AIDS|+) = 4885/78,515=.062
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AIDS Testing
Positive Negative
Total
AIDS
4885
115
5,000
No AIDS 73,630 921,370
995,000
Total
78,515 921,485 1,000,000
P(AIDS|+) = 4885/78,515=.062
P(No AIDS|-) = 921,370/921,485
=.999875
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Morals: Surprise Students!
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Intuition about conditional probability can be
very faulty
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Confront misconception head-on
Conditional probability can be explored
through two-way tables
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Treatment of formal probability can be minimized
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Activity 14: Reese’s Pieces
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Reese’s Pieces
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Take sample of 25 candies
Sort by color
Calculate the proportion of orange candies in
your sample
Construct a dotplot of the distribution of
sample proportions
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Reese’s Pieces
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Turn over to technology
Reeses Pieces applet
(www.rossmanchance.com/applets/)
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Morals: Sampling Distributions
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Study randomness to develop intuition for
statistical ideas
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Not probability for its own sake
Always precede technology simulations with
physical ones
Apply more than derive formulas
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Activity 15: Which Tire?
Left Front
Right Front
Left Rear
Right Rear
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Which Tire?
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People tend to pick “right front” more than ¼
of the time
Variable = which tire pick
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Categorical (binary)
How often would we get data like this by
chance alone?
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Determine the probability of obtaining at least as
many “successes” as we did if there were nothing
special about this particular tire.
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Which Tire?
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Let p = proportion of all … who pick right front
H0: p = .25
Ha: p > .25
.32  .25
Test statistic z =
.25(.75) / n
p-value = Pr(Z>z)
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How does this depend on n?
Test of Significance Calculator
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Which Tire?
n
50
100
150
400
1000
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z-statistic
1.14
1.62
1.98
3.23
5.11
p-value
.127
.053
.024
.001
.000…
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Morals: Formal Statistical Inference
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Fun simple data collection
Effect of sample size
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hard to establish result with small samples
Never “accept” null hypothesis
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Activity 16: Kissing the Right Way
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Biopsychology observational study
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Güntürkün (2003) recorded the direction turned by
kissing couples to see if there was also a rightsided dominance.
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Kissing the Right Way
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Is 1/2 a plausible value for p, the probability a
kissing couple turns right?
Coin Tossing applet
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Is 2/3 a plausible value for p, the probability a
kissing couple turns right?
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Is the observed result in the tail of the “what if”
distribution?
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Kissing the Right Way
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Determine the plausible values for p, the
probability a kissing couple turns right…
The values that produce an approximate pvalue greater than .05 are not rejected and
are therefore considered plausible values of
the parameter. The interval of plausible
values is sometimes called a confidence
interval for the parameter.
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Kissing the Right Way
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How does this compare to estimate + margin
of error?
pˆ (1  pˆ )
pˆ  2
n
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Or the even simpler approximation?
1
pˆ 
n
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Morals: Kissing the Right Way
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Interval estimation as (more?) important as
significance
Confidence interval as set of plausible (not
rejected) values
Interpretation of margin-of-error
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Activity 17: Reese’s Pieces Revisited
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Calculate 95% confidence interval for p from
your sample proportion of orange
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Does everyone have same interval?
Does every interval necessarily capture p?
What proportion of class intervals would you
expect?
Simulating Confidence Intervals applet
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What percentage of intervals succeed?
Change confidence level, sample size
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Morals: Reese’s Pieces Revisited
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Interpretation of confidence level
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In terms of long-run results from taking many
samples
Effects of confidence level, sample size on
confidence interval
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Example 18: Dolphin Therapy
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Subjects who suffer from mild to moderate depression were
flown to Honduras, randomly assigned to a treatment
Subject improved
Subject did not
Total
Proportion
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Dolphin therapy
10
5
15
0.667
Control group
3
12
15
0.200
Total
13
17
30
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Dolphin Therapy
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Is dolphin therapy more effective than control?
Core question of inference:

Is such an extreme difference unlikely to occur by
chance (random assignment) alone (if there were no
treatment effect)?
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Some approaches
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Could calculate test statistic, p-value from
approximate sampling distribution (z, chi-square)
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But it’s approximate
But conditions might not hold
But how does this relate to what “significance” means?
Could conduct Fisher’s Exact Test
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But there’s a lot of mathematical start-up required
But that’s still not closely tied to what “significance” means

Even though this is a randomization test
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3S Approach
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Simulate random assignment process many times,
see how often such an extreme result occurs

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Assume no treatment effect (null model)
Re-randomize 30 subjects to two groups (using cards)


Determine number of improvers in dolphin group

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Assuming 13 improvers, 17 non-improvers regardless
Or, equivalently, difference in improvement proportions
Repeat large number of times (turn to computer)
Ask whether observed result is in tail of what if distribution

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Indicating saw a surprising result under null model
Providing evidence that dolphin therapy is more effective
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Analysis
http://www.rossmanchance.com/applets/
Dolphin Study applet
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Conclusion
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Experimental result is statistically significant

And what is the logic behind that?

Observed result very unlikely to occur by chance (random
assignment) alone (if dolphin therapy was not effective)
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Morals

Re-emphasize meaning of significance and
p-value
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Use of randomness in study
Focus on statistical process, scope of
conclusions
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Activity 19: Sleep Deprivation

Does sleep deprivation have harmful effects
on cognitive functioning three days later?
21 subjects; random assignment
sleep condition


deprived
unrestricted
-16
-8
0
8
16
24
improvement
32
40
Core question of inference:

Is such an extreme difference unlikely to occur by
chance (random assignment) alone (if there were
no treatment effect)?
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Sleep Deprivation


Simulate randomization process many times under null
model, see how often such an extreme result
(difference in group medians or means) occurs
Start with tactile simulation using index cards





Write each “score” on a card
Shuffle the cards
Randomly deal out 11 for deprived group, 10 for unrestricted
group
Calculate difference in group medians (or means)
Repeat many times (Randomization Tests applet)
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Sleep Deprivation

Conclusion: Fairly strong evidence that sleep
deprivation produces lower improvements, on
average, even three days later

Justification: Experimental results as extreme as
those in the actual study would be quite unlikely to
occur by chance alone, if there were no effect of
the sleep deprivation
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Exact randomization distribution

Exact p-value 2533/352716 = .0072 (for
difference in means)
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Morals: Randomizations Tests

Emphasizes core logic of inference
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
Takes advantage of modern computing power
Easy to generalize to other statistics
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Activity 6 Revisited (Memorizing Letters)

Conduct randomization test to assess
strength of evidence in support of research
hypothesis



Enter data into applet
Summarize conclusion and reasoning
process behind it
Does non-significant result indicate that
grouping of letters has no effect?
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Activity 20: Cat Households


47,000 American households (2007)
32.4% owned a pet cat




or the other way around!
test statistic: z=-4.29
p-value: virtually zero
99% CI for p: (.31844, .32956)
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Morals: Limits of statistical significance

Statistical significance is not practical
significance


Especially with large sample sizes
Accompany significant tests with confidence
intervals whenever possible
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Activity 21: Female Senators

17 women, 83 men in 2010
95% CI for p:
= .170 + .074
= (.096, .244)
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Morals: Limitations of Inference

Always consider sampling procedure



Randomness is key assumption
Garbage in, garbage out
Inference is not always appropriate!

Sample = population here
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Activity 22: Game Show Prices






Sample of 208 prizes from The Price is Right
Examine a histogram
99% confidence interval for the mean
Technical conditions?
What percentage of the prizes fall in this
interval?
Why is this not close to 99%?
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Morals: Cautions/Limitations
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Prediction intervals vs. confidence
intervals
Constant attention to what the “it” is
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Activity 23: Government Spending
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2004 General Social Survey: Is there an
association between American adults’ opinion
on federal government spending on the
environment and political inclinations?
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Government Spending
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Descriptive analysis
Liberal
Moderate
Conservative
Total
Too Much
1
17
32
50
About Right
27
80
91
198
Too Little
127
158
113
398
Total
155
255
236
646
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Government Spending

Inferential analysis – 3S approach
1. Chi-square statistic
2. Simulate sampling distribution of chi-square test
statistic under null hypothesis of no association


Randomly mix up political inclinations, determine “could
have been” table
Repeat many times and examine “what if” distribution of
chi-square values under null hypothesis
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Government Spending
3. Strength of evidence


Is observed chi-square value in tail of distribution?
Summarize: What conclusions should be
drawn?
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Very statistically significant
Not cause and effect
Ok to generalize to adult Americans
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Government Spending

What about federal spending on the space
program?
More or less evidence of
association?
Larger or smaller p-value?
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General Advice
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Emphasize the process of statistical investigations, from posing
questions to collecting data to analyzing data to drawing
inferences to communicating findings
Use simulation, both tactile and technology-based, to explore
concepts of inference and randomness
Draw connections between how data are collected (e.g., random
assignment, random sampling) and scope of conclusions to be
drawn (e.g., causation, generalizability)
Use real data from genuine studies, as well as data collected on
students themselves
Present important studies (e.g., draft lottery) and frivolous ones
(e.g., flat tires) and especially studies of issues that are directly
relevant to students (e.g., sleep deprivation)
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General Advice (cont.)
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Lead students to “discover” and tell you important principles
(e.g., association does not imply causation)
Keep in mind the research question when analyzing data
Graphical displays can be very useful
Summary statistics (measures of center and spread) are helpful
but don’t tell whole story; consider entire distribution
Develop graph-sense, number-sense by always thinking about
context
Use technology to reduce the burden of rote calculations, both
for analyzing data and exploring concepts
Emphasize cautions and limitations with regard to inference
procedures
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Implementation Suggestions

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Take control of the course
Collect data from students
Encourage predictions from students
Allow students to discover/tell you findings
Precede technology simulations with tactile
Promote collaborative learning
Provide lots of feedback
Follow activities with related assessments
Intermix lectures with activities
Don’t underestimate ability of activities to teach materials
Have fun!
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Suggestion #1

Take control of the course



Not “control” in usual sense of standing at front
dispensing information
But still need to establish structure, inspire
confidence that activities, self-discovery will work
Be pro-active in approaching students



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Don’t wait for students to ask questions of you
Ask them to defend their answers
Be encouraging
Instructor as facilitator/manager
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Suggestion #2

Collect data from students



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Leads them to personally identify with data,
analysis; gives them ownership
Collect anonymously
Can do out-of-class
E.g., matching variables to graphs
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Suggestion #3

Encourage predictions from students


Fine (better…) to guess wrong, but important to
take stake in some position
Directly confront common misconceptions


Have to “convince” them they are wrong (e.g.,
Gettysburg address) before they will change their way of
thinking
E.g., AIDS Testing
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Suggestion #4

Allow students to discover, tell you findings


E.g., Televisions and life expectancy
“I hear, I forget. I see, I remember. I do, I
understand.” -- Chinese proverb
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Suggestion #5

Precede technology simulations with tactile/
concrete/hands-on simulations



Enables students to understand process being
simulated
Prevents technology from coming across as
mysterious “black box”
E.g., Gettysburg Address (actual before applet)
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Suggestion #6

Promote collaborative learning

Students can learn from each other


Better yet from “arguing” with each other
Students bring different background knowledge

E.g., Matching variables to graphs
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Suggestion #7

Provide lots of feedback


Danger of “discovering” wrong things
Provide access to “model” answers after the fact

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Could write “answers” on board
Could lead discussion/debriefing afterward
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Suggestion #8

Follow activities with related assessments

Or could be perceived as “fun and games” only



Assessments encourage students to grasp
concept


Require summary paragraphs in their own words
Clarify early (e.g., quizzes) that they will be responsible
for the knowledge
Can also help them to understand concept
E.g., fill in the blank p-value interpretation
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Suggestion #9

Inter-mix lectures with activities

One approach: Lecture on a topic after students
have performed activity


Another approach: Engage in activities toward
end of class period


Students better able to process, learn from lecture
having grappled with issues themselves first
Often hard to re-capture students’ attention afterward
Need frequent variety
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Suggestion #10

Do not under-estimate ability of activities to
“teach” material


No dichotomy between “content” and “activities”
Some activities address many ideas

E.g. “Gettysburg Address” activity




Population vs. sample, parameter vs. statistic
Bias, variability, precision
Random sampling, effect of sample/population size
Sampling variability, sampling distribution, Central Limit
Theorem (consequences and applicability)
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Suggestion #11

Have fun!
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Assessment Advice

Two sample final exams


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
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

Carefully match the course goals
Be cognizant of any review materials you have given the students
Use real data and genuine studies
Provide students with guidance for how long they should spend per
problem
Use multiple parts to one context but aim for independent parts (if a
student cannot answer part (a) they may still be able to answer part (b))
Use open-ended questions requiring written explanation
Aim for at least 50% conceptual questions rather than pure calculation
questions
(Occasionally) Expect students to think, integrate, apply beyond what
they have learned.
Sample guidelines for student projects
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Promoting Student Progress



Document and enhance student learning
Element of instruction
Interactive feedback loop





Diagnostic with indicators for change
Throughout the course
To student and instructor
Encourage self-evaluation
Multiple indicators
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Student Projects




Best way to demonstrate to students the
practice of statistics
Experience the fine points of research
Experience the “messiness” of data
From beginning to end



Formulation and Explanation
Constant Reference
statweb.calpoly.edu/bchance/stat217/projects.html
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Resources

www.causeweb.org
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Resources

GAISE reports
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Resources

TeachingWithData.org
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Resources

Inter-University Consortium for Political and
Social Research (ICPSR)
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Resources
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
www.rossmanchance.com/applets/
http://statweb.calpoly.edu/csi/
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Resources

https://app.gen.umn.edu/artist/
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Resources



http://lib.stat.cmu.edu/DASL/
www.amstat.org/publications/jse/
/jse_data_archive.html
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Background Readings





Guidelines for teaching introductory statistics
Reflections on what distinguishes statistical
content and statistical thinking
Educational research findings and
suggestions related to teaching statistics
Collections of resources and ideas for
teaching statistics
Suggestions and resources for assessing
student learning in statistics
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Thanks very much!

Questions, comments?

[email protected]
[email protected]

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My Syllabus Briefly








W1: Collecting Data
W2: Graphical/Numerical
W3: Normal
Project 1
W4: Exam 1
Project 2
W5: Probability
W6: Sampling Distributions
W7: Inference
W8: Inference
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My Syllabus Briefly







W9: Two Samples
W10: Exam II
Project 3
W11: Two variables
W12: Inference for Regression
W13: Two-way Tables Project 4
W14: ANOVA
W15: Presentations
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Non-simulation approach

Exact randomization distribution



Hypergeometric distribution
Fisher’s Exact Test
p-value = 1317  1317  1317  1317
= .0127
10 5  11 4  12 3  13 2 
           
 30
 
Distribution Plot
 15 
Hypergeometric, N=30, M=13, n=15
0.30
0.25
Probability
0.20
0.15
0.10
0.05
0.0127
0.00
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X
10
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