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Teaching Statistical Concepts with Activities, Data, and Technology

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### 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

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 learning

www.amstat.org/education/gaise

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)?

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

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 …

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

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?

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

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

- 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

Morals: Treatment of Probability

- 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

Activity 13: AIDS Testing

- 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

AIDS Testing

Positive Negative Total

AIDS 4885 115 5,000

No AIDS 73,630 921,370 995,000

Total 1,000,000

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

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

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

Morals: Surprise Students!

- 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

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

Morals: Sampling Distributions

- 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

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 p = proportion of all … who pick right front
- H0: p = .25
- Ha: p > .25
- Test statistic z =
- p-value = Pr(Z>z)
- How does this depend on n?
- Test of Significance Calculator

Which Tire?

nz-statistic p-value

50 1.14 .127

100 1.62 .053

150 1.98 .024

400 3.23 .001

1000 5.11 .000…

Morals: Formal Statistical Inference

- Fun simple data collection
- Effect of sample size
- hard to establish result with small samples
- Never “accept” null hypothesis

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 p, the probability a kissing couple turns right?

Coin Tossing applet

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

Kissing the Right Way

- Determine the plausible values for p, 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.

Kissing the Right Way

- How does this compare to estimate + margin of error?
- Or the even simpler approximation?

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 p from your sample proportion of orange
- Does everyone have same interval?
- Does every interval necessarily capture p?
- 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

Example 18: Dolphin Therapy

Subjects who suffer from mild to moderate depression were flown to Honduras, randomly assigned to a treatment

78

78

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)?

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

80

80

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

81

81

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)

83

83

Morals

- Re-emphasize meaning of significance and p-value
- Use of randomness in study
- Focus on statistical process, scope of conclusions

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)?

85

85

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)

86

86

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 p: (.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

Morals: Limitations of Inference

- Always consider sampling procedure
- Randomness is key assumption
- Garbage in, garbage out
- Inference is not always appropriate!
- Sample = population here

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

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

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 statistics

Thanks very much!

- Questions, comments?
- bchance@calpoly.edu
- arossman@calpoly.edu

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

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