Using Computer Simulations to Teach Basic Probability

Transcription

1 Using Computer Simulations to Teach Basic Probability CSU Symposium on University Teaching Pomona, California April 2, 2005 Bruce E. Trumbo Department of Statistics, CSU East Bay (formerly CSU Hayward) (Now) (Soon) 1

2 Pedagogy of Probability Euclid formulated axioms of geometry 2300 years ago Kolmogorov's axioms of probability: 1935 Why the time gap? Probability is harder. Practical experience precedes formalism.! You can draw a triangle with a stick in the sand. Pyramid builders used geometry.! Probability revealed in large numbers. Labs : Gambling casinos began in 1600s. 2

3 Role of Simulation in Class Small-scale random experiments using real objects can be used for introduction. (Coins, dice, drawing beads from a bag.) But time consuming, yield small-scale results. Computer simulation provides large-scale results quickly. High-quality software necessary. Minitab, SPSS, SAS, S-Plus, R are OK (not Excel). R excellent for simulations and free, but not point-and-click: 3

4 Illustration: Margin of Error in a Public Opinion Poll * Many polls quote a margin of sampling error based on the number n of people sampled:! ± 2% for n = 2500! ± 3% for n = 1100! ± 4% for n = 625 For a candidate's lead : Double the margin of error. * In part, similar to a paper to be presented at JSM 2005 (Aug 05) jointly with Eric A Suess and (CSUEB BS Student) Shuhei Okumura. 4

5 What is the basis for these numbers? Assume a random sample. Ignore administrative / subjective issues (nonresponse, phrasing of questions, deceptive answers, etc.). Margin of error based on probability rules: hard to prove at an elementary level. Just say Because I said so? Simulation illustrates these rules. Experimental, exploratory approach. 5

6 Simulating an Election Poll To simplify, suppose:! No undecided voters in population.! 53% of all voters favor Candidate A.! 47% of all voters favor Candidate B.! Randomly sample 25 voters. Can we detect from data on 25 subjects that Candidate A is in the lead? 6

7 One Simulation Using R: > sample(0:1, 25, rep=t, prob=c(.47,.53)) [1] [14] Interpret 0s & 1s as candidate preferences A B A A B B A B B A B B B B A B A B A A A B A A B Only 12 / 25 = 48% for A. Misleading result. What can we conclude from this small poll? 7

8 Now plot the remaining 20 subjects. 8

9 Endpoint is 48%. What would other polls say? 9

10 Note: Here and below, vertical axis from 25% to 75% to help show detail. 10

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13 Results Differ Widely. Next: On one graph, overlay 20 traces. 13

14 All 20 traces end between 40% and 70%, most above 50%. 14

15 Vast majority of traces end between 30% and 75%. Misleading: 377 of the 1000 traces end below 50%. 15

16 Results fit a predictable, but useless pattern. A 25-subject poll can t show if A is winning. 16

17 Polls with n = 2500 Subjects Major polling organizations report such polls as having a ± 2% margin of sampling error. Intended Meaning: If the true proportion for A is 53%, then most 2500-subject polls (95%) will have endpoints between 51% and 55%, thus detecting that A is the favorite. 17

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22 Red marks at right indicate 53% ± 2%. Green line at 50%. 22

23 On average run: 19 out of 20 traces (95%) within margin of error. 23

24 95% of endpoints within margin of error; only 3 in 1000 below 50%. 24

25 n=2500 n=25 In the interval from 51% to 55% for Candidate A : Black curve has 95% probability. Blue curve only about 12%. 25

26 More Complex Simulations Random processes can have a wide variety of limiting behaviors: Process that does not have a limit. (Brownian motion process on a circle.) Process with "absorbing" states. (Brother-Sister mating. Relax: it's botany.) Results of using a bad random number generator. Non-normal processes and limiting distributions. Just for fun: The limit is a fractal object. 26

27 From an example in a forthcoming book by E. Suess and B. Trumbo (Springer). 27

28 From an example in a forthcoming book by E. Suess and B. Trumbo (Springer). 28

29 x Index A bad pseudorandom number generator (black sawtooth ) causes rapid convergence to an incorrect limit. A good generator (irregular green) shows slower convergence to the correct limit. 29

30 5000 Experiments Each With Tosses of a Fair Coin Proportion of Time Head Count Exceeds Tail Count Based on W. Feller (1957): Intro. to Probability Theory and Its Applications, Vol. I [The "bathtub shaped" distribution is BETA(1/2, 1/2).] Over 10,000 tosses either Heads or Tails is likely to remain consistently in the lead. 30

31 From: B. Trumbo, E. Suess, C. Fraser: Using Computer Simulation to Investigate Relationships Between the Sample Mean and Standard Deviation, STATS Sample mean and standard deviation are dependent for non-normal distribution. 31

32 Based on M. Barnsley: Fractals Everywhere, Academic Press (1988) 32

33 Simulation Papers Presented by CSU East Bay Students at Joint Statistics Meetings, Toronto, Canada, August Papers illustrate pedagogical uses of R. Our sessions were extraordinarily well attended because of the current interest in using simulation to teach statistics. 33

34 B. Trumbo, E. Suess, C. Schupp: Using R to Compute Probabilities of Matching Birthdays. (Also to appear in STATS Magazine, Spring 2005) There are n people in a room. What is the probability two (or more) have the same birthday? If birthdays are uniform throughout the year, then the following graph (based on a simple formula) gives the answer. 34

35 There is better than a chance of a match when more than 23 people are in the room. Almost sure when there are more than 50. But US birthdays aren t uniform (36 months shown): 104% of average in Summer; 96% in Winter except for a spike of tax deduction babies in December. Empirical Daily Birth Proportions: By Month Jan '97 - Dec '99 Percent (Percent of Uniform = 1/365 Per Day) Month Source: Nat'l Ctr. for Health Statistics 35

36 Simulation shows actual US nonuniformity doesn t much change the probability of getting no match, but extreme nonuniformity (left side of each graph) would. Assume half of annual births spread uniformly in the number of Weeks shown, and the other half uniformly in the rest of the year. Weeks=26 is uniform over the whole year. Each point above is based on 50,000 simulated rooms, with 25 people in each room. 36

37 B. Trumbo, E. Suess, R. Brafman: Classroom Simulation: Are Variance- Stabilizing Transformations Really Useful?" If data are exponentially distributed (instead of normally), some statistical procedures, such as an analysis of variance, may not work as they should. Simulation shows when log and rank transformation of such data are helpful. 37

38 Power Against Various Alternatives When M = 1, H0 is true; when M = 2, the group means are 1, 2, 4; and when M = 4, the group means are 1, 4, 16; etc. Solid = Original Dotted = Log Transf Dashed = Rank Transf. Simulations for this graph required generating about 3 million ANOVAs, each with 15 observations. These transformations are helpful when samples are small. 38