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1 Apply What You Know to a Problem You Have Never Seen: How AP Students Apply Statistical Reasoning to Solve a Task Using Empirical Data Robin Rider, East Carolina University [email protected] and Renea Baker, D. H. Conley High School [email protected] Background image is a screenshot of Probability Explorer (Stohl, 2002) http:// www.probexplorer.com

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3 Students’ Schoolopoly Assignment Investigate whether the die sent to you by the company is, in fact, fair. That is, are all six outcomes equally likely to occur? You will need to create a poster to present to the School Board. The following three questions should be answered on your poster: Would you recommend that dice be purchased from the company you investigated? What evidence do you have that the die you tested is “fair” or “unfair”? Use your experimental results to estimate the theoretical probability of each outcome, 1-6, of the die you tested. Stohl & Tarr, 2002

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Apply What You Know to a Problem You Have Never Seen: How AP Students

Apply Statistical Reasoning to Solve a Task Using

Empirical Data Robin Rider, East Carolina University

[email protected] and

Renea Baker, D. H. Conley High [email protected]

Background image is a screenshot of Probability Explorer (Stohl, 2002) http://www.probexplorer.com

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Classical vs. Frequentist• Classical - probability is a quantity determined a

priori trials being conducted and derived theoretically from the sample space

VERSUS• Frequentist - probability is a quantity determined

a posteriori trials being conducted and derived experimentally from data,.

(Borovcnik, Bentz, & Kapadia, 1991)

 

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Students’ Schoolopoly AssignmentInvestigate whether the die sent to you by the company is, in fact, fair. That is, are all six outcomes equally likely to occur? You will need to create a poster to present to the School Board. The following three questions should be answered on your poster:

• Would you recommend that dice be purchased from the company you investigated?

• What evidence do you have that the die you tested is “fair” or “unfair”?

• Use your experimental results to estimate the theoretical probability of each outcome, 1-6, of the die you tested.

Stohl & Tarr, 2002

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Simulation vs. Physical Dice• Each pair of students investigated two

situations• Computer Simulated Dice

– Probability Explorer (Stohl, 2002)– Assigned company could produce weighted or

unweighted dice• Physical Dice

– Assigned dice could be weighted or unweighted

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Schoolopoly Digital Dice CompaniesPairs of students were assigned one company to investigate.

Actual Probabilities (Unknown to Students)Companies P(1) P(2) P(3) P(4) P(5) P(6)Luckytown Dice Co. 0.1500 0.1500 0.1500 0.1500 0.1500 0.2500

Dice R’ Us 0.1250 0.1875 0.1875 0.1875 0.1875 0.1250

High Rollers, Inc. 0.1333 0.2000 0.1333 0.2000 0.1333 0.2000

Dice, Dice Baby! 0.1111 0.1667 0.2222 0.2222 0.1667 0.1111

Pips and Dots 0.1667 0.1667 0.1667 0.1667 0.1667 0.1667

Slice N’ Dice 0.1600 0.2000 0.2000 0.2000 0.0400 0.2000

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Schoolopoly Physical Dice• Dodgy Dice

– Available from Highland Games– http://www.halfpast.demon.co.uk/– Either weighted toward a particular number or fair– Weights unknown

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Research Questions• How are students reasoning about fairness,

sample sizes, and the probability of each outcome on the die?

• What do students consider to be evidence to support their claim about the fairness of the die?

• What are the difference in reasoning with physical dice vs computer simulated dice?

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Sources of Data

• Video of computer actions and conversation• Video of students at desk• Transcripts• Students’ final report to the school board• Students’ poster and video of presentation to

class

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Sample Data Collection

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Sample Poster Artifacts

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Sample Poster

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Sample Poster

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Preliminary Results• All students used an

appropriate statistical test (Chi Square) for the data collected

• Students who collected multiple samples were attentive to variation between trials and used variation to support their claims of fairness

• All students had some incorrect application of the CLT– Collected ONE sample and

used the fact that their sample size was greater than or equal to 30 to justify that it was large enough to make conclusions about probability

– After collecting a sample students immediately applied a hypothesis testing procedure

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Preliminary Results• With both the physical

dice and the simulation students made comments as to a “race” between outcomes, cheering outcomes which were “behind”– “Two’s are catching up”,

“Three’s are taking a strong lead”, “No clear cut winner”

– Seems to indicate a desire for the dice to be fair

– This “racing” phenomena has been noted in middle school students also

• Lee, Rider, & Tarr (under review)

• Used similar reasoning for both physical and computer simulated dice– Students who tested the

physical dice first tended to use the same sample size for the computer simulation, students who did the simulation first used much larger samples for the simulation than for the physical dice

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Not surprising• Students were very

procedural in their approach to the problem– Immediately applied a HT

to the data – Collected few, relatively

small samples based on lack of understanding of the CLT

• Students were unfamiliar with the process of having to collect their own data to make conclusions and had difficulty in how to approach the problem– Students particularly had

difficulty in determining how to estimate the theoretical probability

Surprising• Because of the small

samples (even though it met the “rule of thumb” for Chi Square) a surprising number of pairs made Type II errors (not rejecting a false null hypothesis)– Limit power of the Chi

Square Test for Goodness of Fit

• Only one group of students took multiple (3) samples and averaged the proportions for each outcome to get a better estimate of the theoretical probability

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Where Do We Go from Here?• Curricular focus on

procedures in AP Statistics– Develop more

opportunities to integrate empirical data collection activities to reason about theoretical probabilities

– Better understanding of CLT and Law of Large numbers

• Replication of study after integrating more empirical data collection activities– Will students have a

deeper conceptual and practical understanding of CLT and Law of Large Numbers?

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References• Borovcnik, M., Bentz, H.-J., & Kapadia, R. (1991). A

probabilistic perspective. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education (pp. 27-71). Boston: Kluwer Academic Publishers.

• Lee, H. S., Rider, R. L., & Tarr, J. E. (2005). Making connections between empirical data and theoretical probability: Students’ generation and analysis of data in a technological environment. Manuscript currently under review.

• Stohl, H. & Tarr, J. E. (2002). Developing notions of inference with probability simulation tools. Journal of Mathematical Behavior 21(3), 319-337.