Teaching Statistical Concepts with Activities, Data, and Technology Beth L. Chance and Allan J....

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 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

APSA Conference, Sept 2010 2

Schedule

Introductions Opening Activity Activity Sessions

Data Collection Data Analysis

<< lunch>> Randomness Statistical Inference

Resources and Assessment Q&A, Wrap-Up


Requests

Participate in activities 23 of them!

We’ll skip/highlight some Play role of student

Good student, not disruptive student!

Feel free to interject comments, questions


GAISE

Emphasize statistical literacy and develop statistical thinking

Use real data Stress conceptual understanding rather than mere

knowledge of procedures Foster active learning in the classroom Use technology for developing conceptual

understanding and analyzing data Use assessments to improve and evaluate student

learningwww.amstat.org/education/gaiseAPSA Conference, Sept 2010 5

Opening Activity

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


3S Strategy

Statistic Simulate

“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 Reject vs. plausible


Summary

Use real data/scientific studies Emphasize the process of statistical investigation

Stress conceptual understanding Idea of p-value on day 1/in one day!

Foster active learning You are a dot on the board

Use technology Could this have happened “by chance alone”? What if only 10 infants had picked the helper?


Data Collection Activities: Activity 2: Sampling Words 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…


Sampling Words

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?


Sampling Words

“Tactile” simulation Ask students to use computer or random number

table to take simple random samples Determine the sample mean in each sample Compare the distributions


Sampling Words

Java applet 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


Morals: Selecting a Sample

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.


Activity 3: Night Lights and Near-Sightedness 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 (near-sighted)?


Night Lights and Near-Sightedness

Darkness Night light Room light

Near-sighted

18 78 41

Normal refraction

114 115 22

Far-sighted

40 39 12


Night Lights and Near-Sightedness


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Darkness Night light Room light

Far-sighted

Normal refraction

Near-sighted

Morals: Confounding

Students can tell you that association is not the same as causation!

Need practice clearly describing how confounding variable Is linked to both explanatory and response

variables Provides an alternative explanation for observed

association


Activity 4: Have a Nice Trip

Can instruction in a recovery strategy improve an older person’s ability to recover from a loss of balance? 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?


Have a Nice Trip

Randomizing subjects applet How do the two groups compare?


Morals

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.


Activity 5: Cursive Writing

Does using cursive writing cause students to score better on the SAT essay?


Morals: Scope of Conclusions

Allocation of units to groups

By random assignment No random assignment

Selection of units

Random sampling

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 Inferences to

populations can be drawn

Not random sampling

A groups of study units is found; units are then randomly assigned to treatment groups

Collections of available units from distinct groups are examined

Cause and effect conclusions can be drawn

The Statistical Sleuth, Ramsey and Schafer


Activity 6: Memorizing Letters You will be asked to memorize as many

letters as you can in 20 seconds, in order, from a sequence of 30 letters Variables? Type of study? Comparison? Random assignment? Blindness? Random sampling?

More to come …APSA Conference, Sept 2010 23

Morals: Data Collection

Quick, simple experimental data collection Highlighting critical aspects of effective study

design Can return to the data several times in the

course


Data Analysis ActivitiesActivity 7: Matching Variables to Graphs Which dotplot belongs to which variable?

Justify your answer


Morals: Graph-sense

Learn to justify opinions Consistency, completeness

Appreciate variability Be able to find and explain patterns in the data


Activity 8: Rower Weights

2008 Men’s Olympic Rowing Team


Rower Weights


Mean Median

Full Data Set 197.96 205.00

Without Coxswain 201.17 207.00

Without Coxswain or 209.65 209.00

lightweight rowers

With heaviest at 249 210.65 209.00

With heaviest at 429 219.70 209.00

Resistance....

Morals: Rower Weights

Think about the context

“Data are numbers with a context” -Moore Know what your numerical summary is

measuring Investigate causes for unusual observations

Anticipate shape


Activity 9: Cancer Pamphlets

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


Cancer Pamphlets

0

0.05

0.1

0.15

0.2

0.25

0.3

unde

r 3 3 4 5 6 7 8 9

10 11 12

abov

e 12

level

prop

orti

on

patientspamphlets


Morals: Importance of Graphs Look at the data

Think about the question

Numerical summaries don’t tell the whole story “median isn’t the message” - Gould


Activity 10: Draft Lottery

Draft numbers (1-366) were assigned to birthdates in the 1970 draft lottery

Find your draft number Any 225s?


Draft Lottery


month median

January 211.0

February 210.0

March 256.0

April 225.0

May 226.0

June 207.5

month median

July 188.0

August 145.0

September 168.0

October 201.0

November 131.5

December 100.0

Draft Lottery


Draft Lottery


Morals: Statistics matters!

Summaries can illuminate Randomization can be difficult


Activity 11:Televisions and Life Expectancy Is there an association between the two

variables?

So sending televisions to countries with lower life expectancies would cause their inhabitants to live longer?

r = .743


Morals: Confounding

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)


Activity 6 Revisited (Memorizing Letters) Produce, interpret graphical displays to

compare performance of two groups Does research hypothesis appear to be

supported? Any unusual features in distributions?


Lunch!

Questions? Write down and submit any questions you have

thus far on the statistical or pedagogical content…


Exploring RandomnessActivity 12: Random Babies Last Names First Names

Jones Jerry

Miller Marvin

Smith Sam

Williams Willy


Random Babies

Last Names First Names

Jones Marvin

Miller

Smith

Williams


Random Babies


Jones Marvin

Miller Willy

Smith

Williams


Random Babies


Jones Marvin

Miller Willy

Smith Sam

Williams



Jones Marvin

Miller Willy

Smith Sam

Williams Jerry

Random Babies



Jones Marvin

Miller Willy

Smith Sam 1 match

Williams Jerry

Random Babies


Random Babies

Long-run relative frequency Applet: www.rossmanchance.com/applets/ “Random Babies”


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


Random Babies

1234 1243 1324 1342 1423 1432

4 2 2 1 1 2

2134 2143 2314 2341 2413 2431

2 0 1 0 0 1

3124 3142 3214 3241 3412 3421

1 0 2 1 0 0

4123 4132 4213 4231 4312 4321

0 1 1 2 0 0


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

Random Babies


Goal: Interpretation in terms of long-run relative frequency, average value 30% chance of rain…

First simulate, then do theoretical analysis Able to list sample space Short cuts when are actually equally likely

Simple, fun applications of basic probability

Morals: Treatment of Probability


ELISA test used to screen blood for the AIDS virus Sensitivity: P(+|AIDS)=.977 Specificity: P(-|no AIDS)=.926 Base rate: P(AIDS)=.005

Find P(AIDS|+) Initial guess? Bayes’ theorem? Construct a two-way table for hypothetical population

Activity 13: AIDS Testing


Positive Negative TotalAIDSNo AIDSTotal 1,000,000

AIDS Testing


Positive Negative TotalAIDS 5,000No AIDS 995,000Total 1,000,000

AIDS Testing


Positive Negative TotalAIDS 4885 115 5,000No AIDS 995,000Total 1,000,000

AIDS Testing


Positive Negative TotalAIDS 4885 115 5,000No AIDS 73,630 921,370 995,000Total 1,000,000

AIDS Testing


Positive Negative TotalAIDS 4885 115 5,000No AIDS 73,630 921,370 995,000Total 78,515 921,485 1,000,000

AIDS Testing



P(AIDS|+) = 4885/78,515=.062

AIDS Testing



P(AIDS|+) = 4885/78,515=.062

P(No AIDS|-) = 921,370/921,485 =.999875

AIDS Testing


Intuition about conditional probability can be very faulty Confront misconception head-on

Conditional probability can be explored through two-way tables Treatment of formal probability can be minimized

Morals: Surprise Students!


Activity 14: Reese’s Pieces


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

Reese’s Pieces


Turn over to technology Reeses Pieces applet

(www.rossmanchance.com/applets/)

Reese’s Pieces


Study randomness to develop intuition for statistical ideas Not probability for its own sake

Always precede technology simulations with physical ones

Apply more than derive formulas

Morals: Sampling Distributions


Left Front Right Front

Left Rear Right Rear

Activity 15: Which Tire?


People tend to pick “right front” more than ¼ of the time

Variable = which tire pick Categorical (binary)

How often would we get data like this by chance alone? Determine the probability of obtaining at least as

many “successes” as we did if there were nothing special about this particular tire.

Which Tire?


Let = proportion of all … who pick right front H0: = .25

Ha: > .25 Test statistic z = p-value = Pr(Z>z)

How does this depend on n? Test of Significance Calculator

n/)75(.25.

25.32.

Which Tire?


n z-statistic p-value

50 1.14 .127

100 1.62 .053

150 1.98 .024

400 3.23 .001

1000 5.11 .000…

Which Tire?


Fun simple data collection Effect of sample size

hard to establish result with small samples Never “accept” null hypothesis

Morals: Formal Statistical Inference


Activity 16: Kissing the Right Way Biopsychology observational study

Güntürkün (2003) recorded the direction turned by kissing couples to see if there was also a right-sided dominance.


Kissing the Right Way

Is 1/2 a plausible value for the probability a kissing couple turns right?Coin Tossing applet

Is 2/3 a plausible value for the probability a kissing couple turns right? Is the observed result in the tail of the “what if”

distribution?



Determine the plausible values for the probability a kissing couple turns right…

The values that produce an approximate p-value 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.



How does this compare to estimate + margin of error?

Or the even simpler approximation?


n

ppp

)ˆ1(ˆ2ˆ

np

1ˆ

Morals: Kissing the Right Way Interval estimation as (more?) important as

significance Confidence interval as set of plausible (not

rejected) values Interpretation of margin-of-error


Activity 17: Reese’s Pieces Revisited Calculate 95% confidence interval for from

your sample proportion of orange Does everyone have same interval? Does every interval necessarily capture ? What proportion of class intervals would you

expect? Simulating Confidence Intervals applet

What percentage of intervals succeed? Change confidence level, sample size


Morals: Reese’s Pieces Revisited Interpretation of confidence level

In terms of long-run results from taking many samples

Effects of confidence level, sample size on confidence interval


7878

Example 18: Dolphin Therapy Subjects who suffer from mild to moderate depression were

flown to Honduras, randomly assigned to a treatment

Dolphin therapy Control group TotalSubject improved 10 3 13Subject did not 5 12 17

Total 15 15 30Proportion 0.667 0.200


Dolphin Therapy

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)?


8080

Some approaches

Could calculate test statistic, p-value from approximate sampling distribution (z, chi-square) But it’s approximate But conditions might not hold But how does this relate to what “significance” means?

Could conduct Fisher’s Exact Test 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


8181

3S Approach

Simulate random assignment process many times, see how often such an extreme result occurs Assume no treatment effect (null model) Re-randomize 30 subjects to two groups (using cards)

Assuming 13 improvers, 17 non-improvers regardless Determine number of improvers in dolphin group

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

Indicating saw a surprising result under null model Providing evidence that dolphin therapy is more effective


8282

Analysis

http://www.rossmanchance.com/applets/

Dolphin Study applet


8383

Conclusion

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)


Morals

Re-emphasize meaning of significance and p-value Use of randomness in study

Focus on statistical process, scope of conclusions


8585

Activity 19: Sleep Deprivation Does sleep deprivation have harmful effects

on cognitive functioning three days later? 21 subjects; random assignment

Core question of inference: Is such an extreme difference unlikely to occur by

chance (random assignment) alone (if there were no treatment effect)?

improvement

sleep c

onditio

n

4032241680-8-16

deprived

unrestricted


8686

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)


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


Exact randomization distribution

Exact p-value 2533/352716 = .0072 (for difference in means)


Morals: Randomizations Tests Emphasizes core logic of inference

Takes advantage of modern computing power Easy to generalize to other statistics


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?


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 (.31844, .32956)


Morals: Limits of statistical significance Statistical significance is not practical

significance Especially with large sample sizes

Accompany significant tests with confidence intervals whenever possible


Activity 21: Female Senators

17 women, 83 men in 2010

95% CI for :

= .170 + .074

= (.096, .244)


Always consider sampling procedure Randomness is key assumption Garbage in, garbage out

Inference is not always appropriate! Sample = population here

Morals: Limitations of Inference


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%?


Morals: Cautions/Limitations

Prediction intervals vs. confidence intervals

Constant attention to what the “it” is


Activity 23: Government Spending 2004 General Social Survey: Is there an

association between American adults’ opinion on federal government spending on the environment and political inclinations?


Government Spending

Descriptive analysis


Liberal Moderate Conservative

Total

Too Much 1 17 32 50About Right

27 80 91 198

Too Little 127 158 113 398Total 155 255 236 646

Government Spending

Inferential analysis – 3S approach1. 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


Government Spending

3. Strength of evidence Is observed chi-square value in tail of distribution?

Summarize: What conclusions should be drawn? Very statistically significant Not cause and effect Ok to generalize to adult Americans


Government Spending

What about federal spending on the space program?


More or less evidence of association?Larger or smaller p-value?

General Advice

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)


General Advice (cont.)

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


Implementation Suggestions 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!


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

Don’t wait for students to ask questions of you Ask them to defend their answers Be encouraging

Instructor as facilitator/manager


Suggestion #2

Collect data from students Leads them to personally identify with data,

analysis; gives them ownership Collect anonymously Can do out-of-class E.g., matching variables to graphs


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


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


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)


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


Suggestion #7

Provide lots of feedback Danger of “discovering” wrong things Provide access to “model” answers after the fact

Could write “answers” on board Could lead discussion/debriefing afterward


Suggestion #8

Follow activities with related assessments Or could be perceived as “fun and games” only

Require summary paragraphs in their own words Clarify early (e.g., quizzes) that they will be responsible

for the knowledge Assessments encourage students to grasp

concept Can also help them to understand concept

E.g., fill in the blank p-value interpretation


Suggestion #9

Inter-mix lectures with activities One approach: Lecture on a topic after students

have performed activity Students better able to process, learn from lecture

having grappled with issues themselves first Another approach: Engage in activities toward

end of class period Often hard to re-capture students’ attention afterward

Need frequent variety


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)


Suggestion #11

Have fun!


Assessment Advice

Two sample final exams 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


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


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


Resources

www.causeweb.org


Resources

GAISE reports


Resources

TeachingWithData.org


Resources

Inter-University Consortium for Political and Social Research (ICPSR)


Resources

www.rossmanchance.com/applets/ http://statweb.calpoly.edu/csi/


Resources

https://app.gen.umn.edu/artist/


Resources

http://lib.stat.cmu.edu/DASL/ www.amstat.org/publications/jse/ /jse_data_archive.html


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 statisticsAPSA Conference, Sept 2010 126

Thanks very much!

Questions, comments?

[email protected] [email protected]


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


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


Non-simulation approach

Exact randomization distribution Hypergeometric distribution Fisher’s Exact Test p-value =

= .0127 0.30

0.25

0.20

0.15

0.10

0.05

0.00

X

Pro

bability

10

0.0127

3

Distribution PlotHypergeometric, N=30, M=13, n=15

15

30

2

17

13

13

3

17

12

13

4

17

11

13

5

17

10

13


Teaching Statistical Concepts with Activities, Data, and Technology Beth L. Chance and Allan J....

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