1. 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 riderr@ecu.edu and Renea Baker, D. H. Conley High School bakerr.DHC@pitt.k12.nc.us Background image is a screenshot of Probability Explorer (Stohl, 2002) http://www.probexplorer.comhttp://www.probexplorer.com

2. 2 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)

3. 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

4. 4 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

5. 5 Schoolopoly Digital Dice Companies Pairs of students were assigned one company to investigate. Actual Probabilities (Unknown to Students) CompaniesP(1)P(2)P(3)P(4)P(5)P(6) Luckytown Dice Co.0.1500 0.2500 Dice R’ Us0.12500.1875 0.1250 High Rollers, Inc.0.13330.20000.13330.20000.13330.2000 Dice, Dice Baby!0.11110.16670.2222 0.16670.1111 Pips and Dots0.1667 Slice N’ Dice0.16000.2000 0.04000.2000

6. 6 Schoolopoly Physical Dice Dodgy Dice –Available from Highland Games –http://www.halfpast.demon.co.uk/http://www.halfpast.demon.co.uk/ –Either weighted toward a particular number or fair –Weights unknown

7. 7 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?

8. 8 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

9. 9 Sample Data Collection

10. 10 Sample Poster Artifacts

11. 11 Sample Poster

12. 12 Sample Poster

13. 13 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

14. 14 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

15. 15 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

16. 16 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?

17. 17 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.