Designing a Better Paper Helicopter Using Response Surface Methodology Looking for A

Numb3rs, Sabermetrics, Joe Jackson, and Steroids

Larry Lesser Explains How Learning Stats Is FUN!

THE MAGAZINE FOR STUDENTS OF STATISTICS : : ISSUE 48 Designing a Better Paper Helicopter Using Response Surface Methodology Looking for a

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STATS “What nature does blindly, slowly, and ruthlessly, man may do providently, quickly, and kindly. As it lies within “his power, so it becomes his duty to work in that direction.” STATS Sir Francis Galton, 1905 contents guest writers Numb3rs, Sabermetrics, featuresSPECIAL TOPICS Joe Jackson, and Steroids Numb3rs, JIM ALBERT Sabermetrics,  ([email protected]) is Joe Jackson, and professor of mathematics Steroids and statistics at Bowling Green State University. Learning Stats Is His research interests are in Bayesian modeling, FUN... with the  statistical education, Right Mode and the application of statistics in sports (especially baseball). He is a former editor Designing a Better of Th e American Statistician, enjoys playing Paper Helicopter  tennis, and is an avid baseball fan. His dreams will come true when the Phillies Using Response next win the World Series. Surface Methodology Learning Stats Is FUN... with the page 7 Right Mode LAWRENCE (LARRY) LESSER is an associate professor at Th e University of Texas at El Paso and on the Research Advisory Board COLUMNScolumns of the Consortium for the Advancement EDITOR'S COLUMN 2 of Undergraduate Statistics Education

(CAUSE). His web site and contact ASK STATS information are at www.math.utep.edu/ What Is This Statistics Faculty/lesser/Mathemusician.html. Education Stuff? 12

AP STATISTICS Designing a Better Paper Vocabulary Is More Than Helicopter Using Response Just Knowing the Words 22 Surface Methodology ERIK BARRY ERHARDT REFERENCES and was introduced to response surface Additional methodology while Reading List 26 studying for a master’s of science degree in statistics at Worcester Polytechnic Institute. Currently, he is a PhD student at the University of New Mexico and a member of the Albuquerque Chapter of the ASA. He enjoys teaching and research, and he likes page 22 to continually improve things. EDITOR’S COLUMN

EDITOR Paul J. Fields Department of Statistics Brigham Young University, Provo, UT 84602 pjﬁ [email protected]

DIRECTOR OF PROGRAMS Martha Aliaga American Statistical Association 732 North Washington Street, Alexandria, VA 22314-1943 [email protected] n this issue, Jim Albert is our “lead-off batter,” with an article about statistics on television. Did you see the Numbrs episode, EDITORIAL BOARD I“Hardball,” last November about sabermetrics, Peter Flanagan-Hyde the statistical analysis of baseball? Albert is our Mathematics Department Phoenix Country Day School, Paradise Valley, AZ 85253 resident STATS sabermetrician. He poses some pfl [email protected] interesting questions based on that episode. You Schuyler W. Huck will be intrigued by his answers. By the way, do you Department of Educational Psychology and Counseling University of Tennessee know the origin of the word “sabermetrics”? “Saber” Knoxville, TN 37996 comes from the acronym SABR for the Society for [email protected]@cr.k12.ia.us American Baseball Research, and “metrics” refers to Jackie Miller measures of performance. Department of Statistics The Ohio State University, Columbus, OH 43210 Learning stats can be fun! Just ask Larry Lesser. [email protected] He has put together a huge collection of jokes, Chris Olsen Department of Mathematics songs, books, games, poems, and more to help George Washington High School, Cedar Rapids, IA 53403 us learn statistics and have fun at the same time. [email protected] Th ere is a long list of references to statistics fun in Bruce Trumbo References and Additional Reading starting on Page Department of Statistics California State University, East Bay, Hayward, CA 94542  and an even longer list on the STATS web site, [email protected] www.amstat.org/publications/stats. Th ink of some more ways to have fun learning stats and send them PRODUCTION/DESIGN to us to share with other STATS readers. Megan Murphy Jackie Miller, who writes for the column Ask Communications Manager American Statistical Association STATS, asked Joan Garfi eld, “What Is Statistics Val Snider Education?” STATS considers Garfi eld the foremost Publications Coordinator authority on statistics education in the country— American Statistical Association no, make that the world—and she has taken the Colby Johnson • Lidia Vigyázó Graphic Designers/Production Coordinators time to explain to us what statistics education is all American Statistical Association about. Read what a true pioneer has to say about this important topic. STATS: The Magazine for Students of Statistics (ISSN 1053-8607) is published three times a year, in the winter, What performance measure would make a paper spring, and fall, by the American Statistical Association, 732 helicopter “the best”? Maximum hang time, right? North Washington Street, Alexandria, VA 22314-1943 Statistics doctoral student Erik Erhardt explains USA; (703) 684-1221; www.amstat.org. STATS is published for beginning statisticians, including high school, undergraduate, in our cover feature how he and classmate Hantao and graduate students who have a special interest in statistics, Mai optimized the design of a paper helicopter to and is provided to all student members of the ASA at no get the best performance. At the right is a strobe additional cost. Subscription rates for others: $15.00 a year for ASA members; $20.00 a year for nonmembers; $25.00 for a photograph of his helicopters in action—a full-size Library subscription. model and a scaled-down “nanocopter.” Look at the Ideas for feature articles and materials for departments photo carefully. Which one has the longer hang time? should be sent to Editor Paul J. Fields at the address listed above. Material must be sent as a Microsoft Word After you read Erik’s article, try the experiment STROBE PHOTOGRAPH of a document. Accompanying artwork will be accepted in four yourself, and then let us know what you discover full-size model and scaled-down graphics formats only: EPS, TIFF, PDF, or JPG (minimum about paper helicopters. paper helicopters in flight 300 dpi). No articles in WordPerfect will be accepted. Requests for membership information, advertising rates In order to design good experiments to build a and deadlines, subscriptions, and general correspondence better paper helicopter, develop a new drug, or grow bigger shrimp, we need to should be addressed to the ASA offi ce. speak the language of experimentation. Peter Flanagan-Hyde says vocabulary is Copyright (c) 2007 American Statistical Association. more than just knowing the words, though. Read his AP Statistics article and fi nd out what he means by that. He gives us some lessons for learning statistics that will help us “take it to the next level.”

2 ISSUE 48 STATS Numb3rs, Sabermetrics, Joe Jackson, and Steroids

by Jim Albert

umbrs” is a prime time “ television show, currently in its Nfourth season, that follows FBI Special Agent Don Eppes and his brother, Charlie. Together, they fi ght crime and pursue criminals. Th ere are many shows of this type on television; what makes “Numbrs” notable is that Charlie is a mathematical genius, and his mathematical insights usually are crucial to solving the crimes. Th e show illustrates the use of mathematics in diff erent ways and throughout many walks of life. Generally supported by the mathematics community, “Numbrs” sends the positive message that “math is cool.” Th e “Numbrs” episode that aired November , , titled “Hardball,” focused on the use of steroids in baseball. Th is is a timely subject, as some professional baseball players recently admitted to using illegal human growth hormones to enhance their performance. In particular, many people have claimed that Barry Bonds, Mark McGwire, and Sammy Sosa—great home-run sluggers over the last  years—used steroids murder. When FBI agents explore the victim’s during their careers. Some think steroids may have recent email messages, they fi nd an interesting contributed to Bonds’ record of hitting  home attachment containing a long list of equations. Th ey runs during the  baseball season. discover the author of the equations is a young “sabermetrician,” an analyst of baseball statistics, Sabermetrics who claims to have discovered an equation that In the “Hardball” episode, a professional baseball uses batting statistics to detect players using player dies during a practice, and it is found that steroids. Charlie checks out the equation and he received a toxic dose of steroids, which suggests discovers it can reliably detect the players who

ISSUE 48 STATS 3 use illegal drugs. When Charlie is surprised at Joe Jackson this, “Numbrs” local physics professor Larry In , the Cincinnati Reds upset the Chicago Fleinhardt comments: White Sox in the World Series of baseball. It wouldn’t be the first time math revealed A year later, two key Chicago players confessed a surprising truth about the sport. In a to participating in a conspiracy to “throw,” 1993 article in The American Statistician, or deliberately lose, the  World Series. Jay Bennett, using sabermetrics, analyzed Eight Chicago players (labeled the “Black Sox”) ‘Shoeless’ Joe Jackson’s career. He was were tried and found innocent. However, the able to prove that Jackson played up commissioner of baseball banned all of the suspected players from Major League Baseball to his full potential in every one of the for life. 1919 ‘Black Sox’ series. …Math restored a man’s good name and reputation after 70 Th e most famous of the banned players was years. I find that rather beautiful. Jackson, who had the third-highest career batting average (.) in baseball history. Many fans have “Hardball” raises some interesting questions: raised questions about the  World Series. Is there evidence that the Black Sox did not play to How did Jay Bennett statistically show that their full potential during the series? Th e data Joe Jackson did not intentionally try to lose the show that Jackson actually performed well during  World Series? these games, obtaining  hits and a . batting average. But, even if he played well overall, Is it possible to use statistical methods to detect perhaps he did not perform well during “clutch” possible steroid use? situations in the game. Bennett, in his Th e American Statistician article, described a statistical method for determining the value of Jackson’s play during the  World Series. Here is how it works: At the beginning of the game, the teams have approximately the same ability, so the probability at the start that one team, say Chicago, will win the game is .. But every play in the game will change the game situation, such as the runners on base, number of outs, and runs scored, which will impact the probability that Chicago will win. One can measure a given player’s contribution to the game by looking at every batting and ﬁ elding play in which he was involved and the changes in the game-winning probabilities for all those plays. Using this measure of performance, Bennett found that Jackson was the third most valuable player on his team in contributing toward his team winning the games. Based on this probability of winning metric and using a resampling technique that redistributed Jackson’s at-bat outcomes with the situations in which they occurred, Bennett developed a distribution of possible batting performances for Jackson in the  World Series. If Jackson’s actual performance appeared at the extreme low end of the distribution, it would support Jackson’s detractors in their assertion that Jackson threw the series. However, Jackson’s performance was in the upper half of the distribution, which indicates he actually performed as well as could be expected with respect to the at-bat situations in which he appeared. In hypothesis-testing terms, the evidence does not support rejecting the null JOE JACKSON (with shoes) on the right and another legendary baseball great, Ty Cobb, on the left

4 ISSUE 48 STATS Mickey Mantle Willie Mays HOME RUN RATE HOME RUN RATE 0.06 0.08 0.10 0.12 0.04 0.06 0.08 0.10

20 25 30 35 20 25 30 35 40 AGE AGE

Babe Ruth Mike Schmidt RATE RATE

RUN RUN

HOME HOME 0.04 0.06 0.08 0.10 0.00 0.04 0.08 0.12 20 25 30 35 40 25 30 35 AGE AGE

FIGURE 1. Home run rate by age for four all-time great sluggers

hypothesis (Jackson’s batting performance showed this mean Bonds was on steroids? Scott Berry, in a no adverse clutch eff ect) in favor of the alternate CHANCE article titled “A Juiced Analysis,” explains hypothesis (Jackson did not hit in the clutch). that there are many explanations for a jump in Bennett has analyzed all the World Series home run numbers for a particular player. It could games from  through  using this be due to steroids, but it also could be due to the probability of winning metric. Visit www.amstat. player doing a large amount of training during org/sections/sis to see the most valuable players the off -season or using a new hitting style. It also (MVPs) in these series according to Bennett’s could just be due to chance variation. approach. As you might expect, Bennett’s objective Although it is diffi cult to say any player is method of evaluating players’ performance to “juiced” based entirely on empirical evidence for determine the MVP sometimes confl icts with the a single season, we can look at patterns of hitting offi cial MVP chosen by sportswriters. over time and for players who exhibit unusual patterns. One useful measure of hitting strength Steroid Use is the home run rate defi ned as the proportion of Now let’s turn to steroid use by professional home runs hit among all the balls a player hits and baseball players. As statistical methodology was puts into play. Th e home run rate can be written as: useful in assessing Jackson’s performance during HR RATE = HR / (AB – SO), the  World Series, can it also be used to detect where HR is the number of home runs, AB is the steroid use? For example, suppose a particular number of offi cial at-bats, and SO is the number player—such as Barry Bonds—exhibits a large of strikeouts. If we look at the home run rate for increase in hitting home runs for one season. Does a particular player across all the seasons in his

ISSUE 48 STATS 5 Mark McGwire Barry Bonds HOME RUN RATE HOME RUN RATE 0.05 0.10 0.15 0.06 0.10 0.14 0.18 25 30 35 25 30 35 40 AGE AGE

Sammy Sosa

run rate shows a steady increase until his mid- s, when his home rate actually exceeded % at age . Bonds’ home run rate also shows an interesting pattern. When Bonds was in his early s, his home run rate started dropping oﬀ (as

HOME RUN RATE would be expected), before jumping back up and increasing until age . We also see that Bonds’

0.04 0.08 0.12 rate of %, when he hit  home runs at age , appears to be a very unusual event. In contrast, 20 25 30 35 Sosa shows a more typical career trajectory. AGE His home run rate peaked in his early s and dropped oﬀ toward the end of his career. FIGURE 2. Home run rate by age for three recent sluggers Have we proven that McGwire and Bonds used steroids when they were playing baseball? No. What these graphs show is that McGwire and Bonds have unusual career trajectories, in career, a typical pattern emerges. A player’s that they peak in home run hitting at atypically home run rate generally increases until the old ages. Few players in baseball history have middle of his career—usually when he is displayed similar career trajectories. What caused around  years old—and then diminishes these players to peak at such advanced ages? until retirement. For example, the graphs McGwire’s manager, Tony La Russa, responded to in Figure  plot the home run rates of four the steroid accusation as follows: historically great sluggers—Babe Ruth, Mickey It’s fabrication. The product of our good Mantle, Willie Mays, and Mike Schmidt—with play and strength of our players—Mark best-ﬁ tting quadratic curves placed on top. was a great example—what we saw was Th e career trajectories of most great home run a lot of hard work. And hard work will hitters in baseball history show a similar pattern. produce strength gains and size gains. Suppose we look at the home run rates in Figure  for recent home run leaders McGwire, As many other players also “work hard,” Bonds, and Sosa, who are suspected of using statistical analyses such as examining career steroids. Th e trajectories of McGwire and Bonds home run rates may suggest another reason is show a surprising pattern. McGwire’s home needed to explain these patterns of behavior.

6 ISSUE 48 STATS Learning Stats Is F N…with the Right Mode by Lawrence M. Lesser U

t one time or another, you may have felt “statistics anxiety” and A declared statistics to be part of the “inhumanities,” considering its focus on “mean” things. But maybe with the right “mode,” we can add some fun to learning statistics in order to make chi-square “chi-cool,” turn those factorial signs into exciting exclamation points, and not fall asleep while studying Zs.

Humor and Song As the word “median” is embedded in the word “comedian,” let’s start with some statistics jokes you can share with your statistics friends (the punch lines are on Page ):

❶ How many statisticians does it take to change a light bulb?

❷ Why did the statistician cross the road?

❸ What are the last available graves at a cemetery?

❹ Why did a statistician bring a bomb on the ﬂ ight? And, what if that jet-setting statistician was thrown in jail?

❺ What is a Western Union message sent by a snake?

❻ Speaking of snakes, what is the most statistical snake?

ISSUE 48 STATS 7 Books When you need a break from studying statistics Another way to make content in “mathematized” theoretical form, you statistics fun is through song. might enjoy reading books that use bits of humor For example, I have written to explain big ideas. Try Statistics with a Sense of fun songs ranging from Humor; Th e Cartoon Guide to Statistics; or Winning a parody about a not-yet- with Statistics: a Painless First Look at Numbers, informed lottery player sung Ratios, Percentages, Means, and Inference. to the tune of “Th e Gambler” Or, look at studies with realistic statistics to a break-up song I call delivered in an absurd or satirical context, such “Statistician’s BLUEs” (remember best as certain articles in Annals of Improbable Research linear unbiased estimators), stuff ed or Th e Journal of Irreproducible Results. Th is latter with two dozen statistics puns. Many journal recently sponsored a “funniest graph” people have been moved to write contest. What would your entry have been? fun songs about statistics, too, and I Perhaps a pie chart of how you and your friends helped assemble some of their songs shared some pie á la ‘mode’? into a searchable database on the CAUSE (Consortium for the Advancement of Games Undergraduate Statistics Education) Playing games with probabilistic structure provides web site (www.causeweb.org) for all a way to have fun while thinking about and to enjoy. You can look up the gambler discussing statistical ideas. It also can give you an and BLUEs songs there. Th e CAUSEweb edge over your less-statistical friends. For example, database also contains loads of cartoons, statistics can be used to determine good strategies jokes, and quotations that lend themselves to a for playing classic board games, such as Monopoly. lot of fun. Steve Abbott and Matt Richey, mathematics Besides being fun, rhyme and rhythm can be professors, used Markov chain models to helpful learning aids because they make it easier determine the spaces on which players most to recall statistical ideas. Psychology professor frequently landed, and then calculated expected Carmen Wilson VanVoorhis researched the value per roll and time to break-even by property eff ectiveness of using jingles as mnemonic devices. color groups with hotels. Th ey found, “Th e orange While teaching two sections of students who has far and away the lowest break-even time, had grade-point averages that were statistically owing to its high frequency and its relatively low equivalent, VanVoorhis had three statistics building costs. Enthusiasm for the green group’s defi nitions read aloud in one section and statistics high expected return per roll must be tempered by jingles sung for the same concepts in the other the knowledge that this monopoly may not yield section. On four short-answer test items, the profi ts for many rolls. [And] it is hardly worth the students in the jingles section did better trouble to purchase the feeble purples (Baltic and Mediterranean)…” (t = ., p-value < .) and had a statistically signifi cant correlation (r = ., p-value < .) Medical researcher Phil Woodward has shown between test scores and self-rated familiarity with that Yahtzee has more than  trillion outcomes, the jingles. which is why it is not trivial to fi nd the optimal You might enjoy writing your own jingles for choices of which dice to re-roll and which scoring statistical ideas. For example, I wrote “What P- group to use. As an example of a nonobvious Value Means,” to be sung to the tune of “Row, Row, scoring choice, he asks, “What should the dice Row Your Boat.” combinations ---- and ---- be scored against if obtained at the end of the fi rst turn?” It is key to know what p-value means. At the  Joint Statistical Meetings in It’s the chance, Seattle, statistics professor Paul Stephenson With the null, described GOLO, a “golf dice” game, with a You obtain probability-informed, multiple-rolls strategy. Data that’s at least that extreme. And at the  Joint Statistical Meetings, Mary Richardson, a statistics professor, and David Coff ey, a mathematics professor, asked, “What is the probability of pigging out?” Th ey looked at Editor’s Note: Th e References and Additional Reading List starting strategies for estimating the outcome probabilities on Page  is an annotated listing of all of the people mentioned in associated with rolling the nonstandard, pig- this article and a great resource for ideas about the fun ‘mode’ for shaped dice of Pass the Pigs. Th ey also looked learning statistics. at the implications on strategy for ending a turn, considering that one particular roll outcome can

8 ISSUE 48 STATS wipe out all the points on a player’s current turn. A they know the puzzle’s solution. Who related dice game, HOG, was studied by mathematics Wants to Be a Millionaire? yields two professors Larry Feldman and Fred Morgan, who also types of strategy, explored by education were looking for the ‘best’ strategy. So, you can go professor Robert Quinn: () how much low, go hog-wild, or just pig out—all the while having guessing in the preliminary Fastest Finger “Learning and life are statistical fun. question may maximize the chance of maximal where play and Edward Packel, a mathematics professor who being the contestant with the correct work coincide.” studied backgammon, explains how probability four-item sequence in the fastest time and “ L. W. Gibbs can be used to support the following example of () the expected value of guessing when backgammon strategy: “If you must leave a blot you do not know an answer once you are [that] you do not want hit, leave it at least seven in the hot seat pursuing prize money. points from the threatening man, if possible, in Also, during the  Joint Statistical Meetings, mathematics professor Diane Evans explained the which case, the farther, the better (with  versus strategy for how to maximize the benefi t of that  as the lone exception). If you must leave the “– lifeline” based on the initial assignment blot within six points, move it as close as possible of probability estimates to the answer choices and to the threat man.” Th is would be diffi cult to fi gure whether the game show uses random or nonrandom out without using statistical analysis. elimination of choices. A multicultural example is the Jewish Hanukkah Of course, the decision of whether to keep one’s game of Dreidl, played with a four-sided top. Th is original choice after one or more options have been game can yield increasingly sophisticated analyses, eliminated brings to mind the Let’s Make a Deal game such as fi nding the expected payoff on the nth spin show’s “Monty Hall” problem (a.k.a the Th ree Doors with N players or what rule changes would make the problem or the Car and Goats problem). It has generated game ‘fair.’ Felicia Trachtenberg—mathematician, a huge number of articles. Statistics professor Federico statistician, and Dreidl enthusiast—and Robert O’Reilly’s article, “A Look at the Two-Envelope Paradox,” Feinerman, who is also a mathematics professor, published in the Spring  issue of STATS, is a looked at how the game can be fair and unfair. good example of another decision scenario involving In card games such as bridge, blackjack, and conditional probability. poker, there are many places for knowledge of Th e Price is Right game called Plinko involves trying probability to inform strategy. Th ere are particularly to choose the best place(s) to release a chip at the top of interesting situations, such as that attributed to a vertical board fi lled with rows of pegs so the chip has World Poker Champion Amarillo Slim, who allows an the greatest chance of fi nishing the maze in a favorable opponent to choose Hand  (♣, ♠), Hand  slot at the bottom of the board. Obviously, this game is (A♠, K♦), or Hand  (J♥, ♥). Woodward explains, similar to Sir Francis Galton’s quincunx device, in which “Slim was prepared to let you choose any of these beads dropped at the center form a bell-shaped curve. hands. He would then choose a diff erent one. …He Connections between Plinko and learning statistics have had the odds on his side because Hand  beats Hand been made by mathematicians Amy Biesterfeld, LaDawn  about % of the time, Hand  beats Hand  with Haws, Susie Lanier, and Sharon Barrs. Biesterfeld has probability around %, and Hand  will beat Hand looked at strategy for other Th e Price is Right games  on % of matchups.” He also notes that this involving probability, such as Range Game or Master phenomenon occurs in other two-player games, such as Key. Eric Wood, another mathematician, has used predicting what coin-tossing sequence of a given length probability to decide whether to take a second spin of will occur fi rst. the giant wheel (containing multiples of fi ve up to  Did you know there actually are magic tricks based in scrambled order) in the multiplayer competition. Th e on probability? For example, the famous “Kruskal goal is to get the highest score without exceeding  count” card trick is an application of linear algebra and earning a spot in the Showcase round. and Markov processes. Ivars Peterson, an author and A current popular game show is Deal or No Deal, editor, notes, “Th e underlying mechanics of the Kruskal in which players try to maximize the deal they get count also highlights how seemingly unrelated chains for their unopened briefcase, taking into account of events can lock together in sync after a while—a its expected value from a set of known monetary phenomenon worth watching for in other contexts.” amounts ranging from $. to $ million in light of  other unopened briefcases. Game Shows As one last example, Friend or Foe was analyzed to Televised game shows, such as Wheel of Fortune see whether age, gender, race, and the amount of prize or Who Wants to Be a Millionaire?, can provide money aff ect contestants’ strategies. another source of statistical fun. In Wheel of By learning statistics, you will be ready to win Fortune, contestants often must decide whether to big prizes if you are picked for a game show. Th e risk bankruptcy by spinning for more money when possibilities seem endless.

ISSUE 48 STATS 9 Internet More Fun Speaking of the seemingly endless, try searching Another diversion is coming up with your own the internet. Statistics professors Webster creative activities that can be subjected to West and Todd Ogden found and catalogued hypothesis testing. One of mine is to test the null online interactive demonstrations and games hypothesis that I cannot juggle three balls, using for learning statistics. You can fi nd applets to a value of fi ve seconds for the null mean amount make compositions from “Mozart’s Musical Dice of time before one of the balls hits the ground. Game.” A Google search will turn up musical Speaking of round things, next time you tear open performing groups with names such as “Statistics” a bag of m&ms, use a chi-square test to compare and “Random,” as well as web sites that randomly the observed color distribution to the offi cial generate songs or band names. Th ere are even distribution of that particular variety on the statistical “rap” videos on the internet. See Page company’s web site at http://us.mms.com/us/about/  for some sites I found. I am sure you can fi nd products. many more. Another of my favorites is to apply hypothesis testing to everyday events or subjects, such as Creative Works popular songs. Do hit songs have a mean length of Another way to have fun is to look for examples . minutes? Are they usually cowritten? Is there a of statistics in real-world creative works. Th ere correlation between a song’s tempo and the length are examples of statistics concepts and vocabulary of a song’s introduction? in popular song lyrics, such as in this one from It also can be fun fi nding and analyzing silly singer-songwriter David Wilcox: statements about correlation or causality, ranging from birth rates and stork counts to the Super Bowl and “So you come for a visit, but you lose your nerve Dow Jones. At the  Joint Statistical Meetings, out at the edges of the bell-shaped curve.” Harry Norton shared the following example: Th ere are also poems that use statistical images DAY ONE: Made loud noise behind frog. or words. One of these is Wislawa Szymborska’s “A Frog jumped  feet. Word on Statistics.” One of the best-known short stories ever is Shirley Jackson’s “Th e Lottery,” which DAY TWO: Immobilized one hind leg of fi rst appeared in  in Th e New Yorker and has frog, then made same loud noise as on since appeared in many compilations. Th e story’s day one. Frog jumped only  feet. vehicle is an annual village ritual that includes making sampling frames and conducting a sequence DAY THREE: Immobilized both hind legs of simple random samples. of frog, then made many loud noises— Th e classic novel Th e Phantom Tollbooth, by Norton louder than days one and two. Frog did Juster, has some entertaining passages concerning not jump at all. averages in Chapter , such as when lead character Milo encounters “one half of a small child who had CONCLUSION: When both hind legs of a been divided neatly from top to bottom”: frog are immobilized, it becomes deaf. What is the rest of your family like?” said Milo, this time a bit more sympathetically. Th e next time you are on a long road trip “Oh, we’re just the average family,” he said (perhaps driving to the Joint Statistical Meetings), thoughtfully, “mother, father, and 2.58 think of all the opportunities for statistical fun. children, and, as I explained, I’m the .58. Th e “license plate game” can be played with statistical variations (e.g., fi rst person to spot a It must be rather odd being only part of a license plate whose digits yield particular values person,” Milo remarked. “Not at all,” said for the mean, median, mode, or range…). You the child. “Every average family has 2.58 also might hunt for statistical terminology along children, so I always have someone to the roadside (e.g., Look, a park for ‘random play with. Besides, each family also has an variables’—RVs!). Another modifi ed car game is to average of 1.3 automobiles, and since I’m see how long you can take turns giving the name the only one who can drive three tenths of of a statistical term (or statistician) beginning with a car, I get to use it all the time. the letter of which the previous word ended. You also can play “free association” word games. Look around and I am sure you will fi nd many My graduate advisor, famous for his empirical more examples of statistics in popular songs, Bayes research, often would throw me and my stories, and movies. fellow members of S.O.S.—Students of Statistics— a pair of nonstatistical words to see if we could guess

10 ISSUE 48 STATS which word he was associating with “Bayesian” and since no one seems sure what day which one with “frequentist.” We correctly guessed in  Th omas Bayes was born in almost every time, to our advisor’s surprise. London, why not pick a date you With statistics, you even can have fun with the fi nd convenient for a “Bayes Day” clothes you wear (as long as you avoid “lack of fi t”). celebration? Check out the web sites Some of the current ASA T-shirt captions include: in the references section, where “I’m Statistically Signifi cant,” “Statisticians Are you can look up more birthdays of Spatial,” and “Approximately Normal.” Get some of famous people. your stats friends together and make some of your By the way, did you know the second own T-shirts with stats slogans. full week of February has been designated You also can explore paradoxes or statistical Random Acts of Kindness Week? Well, how situations with striking results. Eric Sowey has about picking your own week at random for “Kind shown how surprises are fun and help us learn. Test Acts of Randomness Week”? Th en, get some of your yourself with the following questions (Th e answers statistics friends together to celebrate by doing a are on Page .): service project for someone. Nothing is more fun than making the world a better place. Have some ➀ If the standard normal distribution is drawn to fun and pass it on. scale on a sheet of paper so its tails are  mm above the axis at z = ± , how high must the paper be at Learning Statistics Can Be Fun the mode to accommodate the drawing? I hope you are now inspired to fi nd your own ways ➁ How many people must be in the room for there of bringing fun into your statistical journeys, and to be at least a % chance of at least two people I hope you have a sense of the benefi ts and huge sharing a birthday? Th is problem was featured in variety of options for having fun with statistics— the Spring  issue of STATS. without unduly regressing or deviating! All we need is the right ‘mode.’ ➂ Th ere are three cards in a bag. Both sides of one card are green, both sides of one card are blue, and one side is green and one side is blue on the other card. You pull a card out of the bag at random and see that one side is blue. What is the probability that the other side is also blue?

Let’s Party For a couple of decades now, museums and schools throughout the country have celebrated Pi Day every March  (/). Maybe it’s time to create a statistics counterpart to this special day. Perhaps statistics students could celebrate “Happy Median Day” every July , enjoying a serving of pie á la ‘median’! Or maybe every January  and December  could be “% Conﬁ dence Days.” Suppose we celebrate the birthdays of one or more famous statisticians. Here is a list of  to get your started:

JANUARY 13 – GERTRUDE COX FEBRUARY 16 – SIR FRANCIS GALTON FEBRUARY 17 – RONALD FISHER MARCH 27 – KARL PEARSON MAY 12 – FLORENCE NIGHTINGALE JUNE 13 – WILLIAM GOSSET JUNE 16 – JOHN TUKEY JULY 15 – WILLIAM COCHRAN OCTOBER 14 – W. EDWARDS DEMING NOVEMBER 18 – GEORGE GALLUP

Gosset’s birthday could be celebrated with a ‘students’ tea,’ making sure the tea is served according to “students’ tea distribution.” And

ISSUE 48 STATS 11 ISSUE 48 STATS 11 ? ASK STATS What Is Th is Statistics Education STUFF JOAN GARFIELD is professor of educational psychology at the University of Minnesota, where she heads a graduate? program in statistics education. She serves as associate director of research for the Consortium for the Advancement of Undergraduate Statistics Education (CAUSE) and is involved in many national and international statistics education committees. She is coeditor of The Assessment Challenge in Statistics Education and The Challenge of Developing Statistical Literacy, Reasoning, and Thinking. Currently, she is coauthoring a book that connects research to practical teaching advice. [email protected]

by Jackie Miller

s someone with a one-of-a-kind PhD in Recently, there has been a growing interest statistics education from Th e Ohio State among mathematics educators in statistics education AUniversity, I am very interested in in K– settings, and on the preparation of K– statistics education. Th e statistics education teachers to teach statistics. However, while there are JACKIE MILLER is community has been growing and developing courses to prepare teachers to teach mathematics in a statistics education over the past two decades. Joan Garfi eld of the elementary and secondary schools, there is little, if specialist and auxiliary University of Minnesota is a mover and shaker in faculty member in any, formal preparation of teachers to teach statistics the Department of the statistics education community and someone at any level. Statistics at The Ohio I consider a mentor. In the hope of getting some State University. She of you interested in statistics education, I asked Is there a diff erent pedagogy that comes with the earned both a BA and teaching and learning of statistics? If so, why? BS in mathematics Garfi eld to answer the following questions. and statistics at Miami Statistics is a unique discipline, as people such University, along with How is statistics education diff erent from other as David Moore and George Cobb have so an MS in statistics disciplines, such as mathematics education? and a PhD in statistics eloquently written. Statistics is not mathematics, education from The I see most of the focus in mathematics education although mathematics is used in statistics, just as Ohio State University. being [put] on teaching and learning mathematics in mathematics is used in other disciplines, such as When not at school, kindergarten through high school (K–) classrooms, Miller enjoys a regular physics or economics. I believe there are unique life (despite what her or on the preparation and professional development methods for teaching statistics, and that it should students might think), of K– mathematics teachers. In contrast, most of not be taught as a mathematics course. Th e including keeping up the focus in statistics education is on the teaching recommended methods for teaching statistics include: with her many dogs! and learning of statistics in college-level classrooms. While most of the scholarship in mathematics Using real data education is by mathematics education professionals Focusing on concepts, rather than computations who do not actually teach mathematics, but rather Using statistical software and technological tools teach teachers to teach mathematics, most of the scholarship is by people who teach statistics and are Involving students in doing statistics, rather than concerned about improving the teaching and learning just learning about how to do statistics of statistics in statistics education.

12 ISSUE 48 STATS One of the pedagogical methods used by many in or university level. I expect he will statistics education is to utilize simulation tools to play an active role in the statistics “Statistics is help students visualize abstract statistical concepts education community and contribute not mathematics.” and procedures, such as sampling distributions and to the research and scholarship in the Central Limit Th eorem. this area. I have two other students in “ Currently, there is a set of guidelines on teaching the program, and a new one began in Fall  who I the introductory statistics course, as well as statistics hope will follow a similar path. in K– classes, that has been endorsed by the How many programs are there, and how many American Statistical Association: the Guidelines for current statistics education PhDs are there from Assessment and Instruction in Statistics Education these programs? (GAISE). We hope these guidelines will lead to high- quality statistics courses that will help students As far as I know, my program is the only stand- develop an understanding and appreciation of alone graduate program in statistics education. At statistical investigations. Ohio State, there are several students working on doctorates in mathematics education with a focus What is the purpose or objective in having a statistics on teaching statistics at the college level. Also, and education graduate program? new this year, there is a graduate education minor A graduate program in statistics education, such as in graduate interdisciplinary specialization in college the one I direct at the University of Minnesota, is to and university teaching. At the University of Georgia, both prepare excellent teachers of statistics at the graduate students in mathematics education can focus postsecondary level and to prepare graduate students on the area of statistics, but I think it is still directed to conduct research on the teaching and learning of to K– settings. statistics. To become an excellent teacher of statistics, What is the statistics education community like? graduate students need to take at least one course in statistics education. In such a course, they become I think the statistics education community is a familiar with the fi eld of statistics education: the wonderful, vibrant, exciting community. Th ere are current guidelines, important readings, resources, many bright, enthusiastic, energetic people who teaching methods, and technological tools. Graduate have dedicated their lives to teaching students and students also need to learn about good assessment to improving students’ experiences in statistics practices and resources. classes. Th ere are many wonderful texts, tools, and In addition, while statisticians are trained in other resources available today, so there is no excuse quantitative statistical methods, few of these for teaching an old-fashioned, traditional course methods apply to conducting research focused on that consists of a teacher lecturing to students the teaching and learning of statistics in college and demonstrating how to work out formulas and classrooms. Th is type of research requires an compute answers while students listen passively and understanding of current theories of learning so take notes. Th ere is no excuse for teaching a course that we can design and conduct studies with a that does not use real data, real examples, and real strong theoretical background. Qualitative methods, statistical tools. Th ese resources are all widely and educational measurement, and a solid background in freely accessible on the internet and elsewhere. foundational research in statistics education also are Today, we are fortunate to have CAUSEweb.org important and part of the coursework in a graduate —the web site for CAUSE—that provides a one-stop program in statistics education. site to locate valuable information about current Another purpose of a graduate program is activities, materials, and research. to provide careful and high-quality training and Is there anything else you would like to tell students of supervision in teaching as graduate students grow statistics about statistics education? and develop into teachers of statistics. I encourage all students interested in pursuing a career What results have we seen from graduates with in statistics education to become familiar with CAUSE degrees in statistics education? and CAUSEweb.org and to attend the United States In the past, a graduate student wanting to focus their Conference on Teaching Statistics (USCOTS) held at dissertation research on statistics education had to Th e Ohio State University in May . Th is conference do this on their own, as part of their course work in is held biannually and showcases some of the great existing programs, such as statistics, psychology, or thinkers and contributors to the fi eld and off ers an mathematics education. Th e graduate program in opportunity to network with graduate students and statistics education at the University of Minnesota statistics teachers at all levels. It is important to was launched four years ago, and the fi rst student become part of the statistics education community to complete a doctorate in the program graduated and not work in isolation. I welcome any readers with in May . He was hired for a one-year lecturer further questions to read about my graduate program position at the University of Minnesota and then at www.education.umn.edu/EdPsych/QME/stats-intro. planned to apply for academic jobs at the college html or contact me at [email protected].

ISSUE 48 STATS 13 Designing a Better Paper Helicopter USING RESPONSE SURFACE METHODOLOGY

by Erik uppose a group of your friends are having Barry a contest to design a paper helicopter that Sremains aloft the longest when dropped from a certain height. To be fair, everyone starts Erhardt with the helicopter pattern given in Figure , which is an easy pattern to modify and replicate. Armed with knowledge about response surface methodology and a desire to strive for excellence, you could have an advantage. Let us go through the steps together, and I will show you how my classmate, Hantao Mai, and I designed “a better paper helicopter” to become two-time paper helicopter champions at Worcester Polytechnic Institute. After you see what we did, you can try to do even better.

FIGURE 1. Initial helicopter pattern. Cut along the solid lines and fold along the dotted lines. The foot fold, D, paper weight, G, and fold direction, H, were not included as part of the initial pattern, but were added after brainstorming about potential factors that might influence flight time.

Deﬁ nitions of highlighted words can be found on FIGURE 2. Examples of response surfaces in three Page 20 dimensions: a maximum, a rising ridge, and a “saddle” with a maximum in one direction and a minimum in a transecting direction

14 ISSUE 48 STATS Response Surface Methodology SPOTLIGHT Response surface methodology (RSM) is a collection The concepts behind optimization through experimenta- of statistical and mathematical techniques to explore tion, now known as response surface methodology, were effi ciently the performance of a system to fi nd ways developed by George E. P. Box (a statistician) and K. B Wilson to improve it. A good reference on RSM is Response (a chemist). Box is emeritus professor at the University of Surface Methodology by Raymond Myers and Douglas Wisconsin-Madison, where he cofounded the Center for Montgomery. A response surface can be envisioned Quality and Productivity Improvement. His work has encom- as a curved surface representing how the system’s passed design of experiments, quality control, time series output performance (a dependent variable) is analysis, and Bayesian inference. It is his name in aff ected by specifi ed input factors (independent “Box-Jenkins time series models,” “Box-Cox transformation,” and “Box- variables). Examples of response surfaces in three- Behnken experimental designs.” dimensional space are shown in Figure . The idea for the paper helicopter experiment originated with C. B. “Kip” Rogers at Digital Equipment Corporation. It was popularized by Box as a George Box is the statistician credited with way to teach experimental design for quality improvement to engineers. fi rst proposing the ideas behind RSM more than Box is coauthor with J. Stuart Hunter and William G. Hunter of Statistics for  years ago. Th e major tools in RSM are design of Experimenters: Design, Innovation, and Discovery, which features the paper heli- experiments, multiple regression, and optimization. copter experiment in Chapter 12. In the paper helicopter project, our goal was to use these tools to fi nd the combination of design factors that would maximize fl ight time. We designed a set of experiments that allowed us to discover enough SPOTLIGHT about the shape of the response surface to design the Vilfredo Pareto (1848–1923) was an Italian economist, winning paper helicopter—twice! sociologist, and engineer. In statistics, his name is associated with the Pareto Chart and Pareto Distribution. He is best-known for observing that 20% of the population Design of Experiments owned 80% of the property in Italy. Experience has shown that a similar relationship is common in many other Brainstorming situations in the social, biological, and physical sciences. An important question to start with is, “What factors Consequently, Pareto’s observation has been generalized should we test?” A brainstorming session can reveal as the Pareto Principle. Also know as the “80–20 Rule,” many factors that might infl uence the response. it says that “80% of the consequences come from 20% of the causes.” The rule We brainstormed about helicopter design using our has many applications in quality control and design of experiments. For instance, knowledge of the aerodynamics of fl ight, and tried a Pareto Chart is a histogram showing the frequency of occurrence of events to think of everything about the design of a paper with the categories ordered from most frequent to least frequent. When study- helicopter that we could control. Th en, we made ing the performance of a system, a Pareto Chart can help identify the important a few helicopters, dropped them, and made some factors (the 20% or “vital few”) among the less important factors (the 80% modifi cations—a fun way to start a project. or “trivial many”). Based on this same idea, the Pareto Distribution has been used to describe the distribution of incomes, population sizes, particle sizes, Factorial Designs word frequencies, customer complaints, and many other empirical phenomena. In designing experiments, the workhorses of experimental designs are factorial designs. Th ese are effi cient designs for assessing the effects of several factors on a response. A k design is a factorial design for k factors, each tested at two levels, coded as + for “high” and – for “low.” Th e low and high levels are selected to span the range of possible values for each factor. A graphical representation of a  design is shown in Figure . Th e cube shows the orthogonal relationship of the experimental points. Center points are experimental runs with all factors at level  midway between – and +. Adding center points to the experiment is a logical way of checking curvature and obtaining an estimate of the error variance.

Pareto Principle and Factor Screening FIGURE 3. A 24 factorial design consists of all combinations of four factors, each Th e Pareto Principle says there are a “vital few” taking two levels (+1, –1), represented here with a letter for +1 and without a letter important factors and a “trivial many” less for –1. For example, point abd in the figure corresponds to factors A, B, and D set to high level +1 and C to low level –1. important factors. A screening experiment can distinguish the trivial variables from the vital few important factors infl uencing the response. Th e ISSUE 48 STATS 15 dimensionality of the experimental region often is We produced  helicopters at the design points greatly reduced, exponentially decreasing the number listed in Table . Th en, we dropped them in random of experimental runs required. order from  feet and recorded the fl ight time of each. Fractional Factorial Designs All fl ights were conducted indoors in still air. A full factorial design contains all possible combinations We used the SAS macro EFFECTS from Applied of the k factors at the tested levels. When the number Statistics for Engineers and Scientists by J. D. of factors is large, a k design will have a large number Petruccelli, B. Nandram, and M. Chen to compute the of runs. For example, k =  requires  =  runs. factor eff ect estimates, as shown in Table . Fractional factorial designs can be used for screening to fi nd the factors that contribute most to the response, if a full factorial design requires a prohibitive number of runs. TABLE 2. Screening experiment with coded factor levels and Fractional factorial designs are designated as k–p, with center points p =  for a design with one-half the full factorial runs, Factors and Coded Levels p =  for one-fourth the full factorial runs, etc. Flight Th e price paid for the effi ciency of a k–p design is Run Order ABCDEFGH Time that ambiguity is introduced through confounding of the eff ects. In a useful screening design, the main eff ects will 1 12 –1–1–1–1–1–1–1–1 11.8 0 not be confounded with each other, although there can 2 7 –1–1–11111–1 8.29 be confounding with the interaction eff ects. Because the main eff ects are not confounded with each other, their 3 11–1–11–111–11 9.00 relative importance can be distinguished among each 4 15–1–111–1–111 7.21 other, but not from certain interactions. 5 1 –11–1–11–111 6.65 Our Screening Experiment Th e goal during a screening experiment is to identify 6 4 –1 1 –1 1 –1 1 –1 1 10.26 those factors with the most infl uence on the 7 16–111–1–111–1 7.98 response, and then eliminate the remaining factors from further analysis. To determine the vital few 8 8 –11111–1–1–1 8.06 factors that contribute to fl ight time, we performed a 9 3 1–1–1–1–1111 9.20 screening experiment using a – fractional factorial design with the initial helicopter pattern as our 10 10 1 –1 –1 1 1 –1 –1 1 19.35 center point. We selected eight factors possibly infl uencing the 11 9 1 –1 1 –1 1 –1 1 –1 12.08 fl ight time of the helicopter: the original fi ve factors plus three that we added (i.e., foot length, paper 12 5 1–11 1–11–1–1 20.50 weight, and fold direction). Paper weight levels were 13 6 1 1–1–11 1–1–1 13.58 phone book white pages paper light (–) and standard copy paper heavy (+). Fold direction levels indicated 14 13 1 1–11–1–11–1 7.47 the fold direction was against (opposite) or with the 15 14 1 1 1 –1 –1 –1 –1 1 9.79 direction of rotation. Table  shows the coded and uncoded values for the eight factors we investigated. 162 11111111 9.20

TABLE 1. Factors potentially influencing paper helicopter flight time and the factor levels for the screening experiment in actual units for each coded value. Length and width were measured in centimeters. With only a single measurement for each combination of factor levels in the screening design, Coded Values we could not estimate the error, as all the degrees- Factors –1 0 +1 of-freedom would be used for estimating the factor A = Rotor Length 5.5 8.5 11.5 eff ects. A Pareto Chart of the absolute value of the B = Rotor Width 3.0 4.0 5.0 eff ects (see Figure ) shows that the main eff ects A, B, and G should be considered the vital few, as they C = Body Length 1.5 3.5 5.5 are more than twice as large as the next largest main D = Foot Length 0.0 1.25 2.5 eff ect (i.e., D). (Th e references on Page  by Cuthbert E = Fold Length 5.0 8.0 11.0 Daniel and Russell V. Lenth discuss other ways of F = Fold Width 1.5 2.0 2.5 analyzing data from unreplicated factorial designs.) G = Paper Weight light (none) heavy We were left with three main eff ects, but by H = Fold Direction against (none) with inspection, we could see that G = – (the light paper)

16 ISSUE 48 STATS TABLE 3. Factor effects estimated with SAS macro EFFECTS. Multiple Regression Factors A, B, and G are indicated as the vital few. Regression Modeling In building a regression model, we assume the Factor Estimate response of fl ight time, y, depends upon k controllable factors, such as rotor length and width, A 3.9900 * ξ ,…,ξ , in yf=+()ξξ,…, ε , where f is an 1 k 1k B –3.0550 * unknown function and ε is random error with expected value E=0()ε and common variance C –0.3475 Var()εσ = 2 . Th e response surface is the expected D 1.2825 fl ight time for varying factor values defi ned by E 0.2500 Ey=f() (ξξ , …, ). For convenience, the uncoded 1 k variables, ξ ,…,ξ , in their original units, are F 0.7000 1 k transformed to the coded variables, x,…x,, G –4.2825 * μ 1k with zero mean i =0. The response surface is in H –1.1375 terms of the coded variables E(y)= f(x ,…,x ). 1k AB –2.2175 Because we do not know the form of f, we need AC 0.8400 to approximate it. For this purpose, second-order polynomials are useful. A second-order polynomial AD 1.6850 has the form: BC –0.3850 k k-1 k k 2 BD –2.0350 E(y)=+ββ∑ x+ ∑∑ β x x + ∑β x 0 ii ij i j ii ii i=1 i=1j=i+ 1 i=1 CD 0.2475 Th e fi rst-order terms describe a plane, while the ABCD 1.5625 cross-product terms (also called interaction terms) and the second-order terms describe curvature. With only two factors, it is practical to use a  always gave a higher response. Also, we fi xed H = – replicated design. New measurements were taken (fold against), as we observed that this setting gave for all helicopters, as shown in Table . more stable helicopters. Further testing suggested favorable levels of body length C = , foot length TABLE 4. The results of a 22 design with replication to D = , fold length E = , and fold width F =  in determine the initial estimate of the response surface their uncoded units of centimeters. So, we had Obs ABTime only two factors to consider (A, rotor length, and B, rotor width) to start our model-building process. 1–1–110.24 2–1–1 9.11

4.00 3 1–116.52 3.50 4 1–116.99 3.00 5–1 110.20 2.50

2.00 6–1 1 9.26 Abs (Effect) Abs 1.50 7 1 110.02 1.00

0.50 8 1 1 9.94

0.00 9–1–111.31 G A B AB BD AD ABCD D H AC F BC C E CD Factors 10 –1 –1 10.94

FIGURE 4. Pareto Chart of the absolute value of the effects, in 11 1 –1 12 . 58 order from largest to smallest. The paper weight, rotor length, and rotor width stand out as the important factors. 12 1 –1 13.86 13 –1 1 8.20 14 –1 1 9.92 15 1 1 9.95 16 1 1 9.93 ISSUE 48 STATS 17 Our Initial Model A (rotor length) and correspondingly –. cm We applied multiple regression to the results in for factor B (rotor width). Five steps along the path Table  to determine our initial model to estimate of steepest ascent, starting at the base, are shown the response surface as: in Table . An approximate maximum was found at Step . ˆyxx=+bb + b 01122 =+11.. 1163 1 2881xx − 1 . 5081 Central Composite Designs 12 Th e basic experimental designs to estimate Optimization second-order response surface models are central Steepest Ascent composite designs. A two-dimensional central What is the shortest path to the top of a hill? composite design is shown in Figure . Th ese Always take the steepest possible step. Th e goal is designs consist of a factorial design for estimating ﬁ rst-order and two-factor interactions, k axial (or to move the process variables, x, to a new region of “star”) points at improved response by following the path of steepest ascent. Th e procedure is to compute the path of (± a, , … , ), (, ± a, , … , ), … , (, … , , ± a) steepest ascent using the ﬁ rst-order model and for estimating the second-order terms, and conduct experiments along this path until a local replicated center points to estimate the second- maximum is reached. Th e direction and magnitude order terms and to provide an estimate of error. of steepest ascent is the gradient. For the response Th e value of a is set at k for rotatability, so surface Ef(y)=… (x,, x ), the gradient is 1 k the accuracy of prediction with a quadratic equation will not depend on direction.

⎛ ⎞ ′ ∂f ∂f ∂f ∇f = ⎜ , ,,… ⎟ , a vector of slopes. ⎜ ∂x ∂x ∂x ⎟ ⎝ 12 k ⎠ In particular, for the ﬁ rst-order model

y =+bbxbx + +…+ bx, the resulting 0kk11 22 estimated direction of steepest ascent is ′ (bb,,,… b) . 12 k Th erefore, a step change proportional to the regression coeﬃ cient for each factor will be the path to follow in subsequent experiments to “climb the hill” to the optimum. For our climb, we decided each step would be . cm for factor FIGURE 5. A two-dimensional central composite design

TABLE 5. Steepest ascent coordinates and results in uncoded units (centimeters). Time measured in seconds. Although the fold width (F) was not a factor, it is shown in the table because it had to vary with rotor width (B). F had to decrease as B decreased. For our central composite design, a= k = 2 , as shown in Table  in coded and uncoded units Factors centered at the maximum steepest ascent point Step ABF Time (Step , in Table ). Th e results are shown in Table . Base 8.50 4.00 2.0 12.99 1 9.503.612.015.22 TABLE 6. Central composite design factor levels in coded and 210.503.222.016.34uncoded units (centimeters) 3 11.50 2.83 1.5 18.78 Factor− 2 –1 0 1 2 4 12.50 2.44 1.5 17.39 A10.0810.5011. 50 12.50 12.91 513.502.051.2 7.24B2.282.442.83 3.22 3.38

18 ISSUE 48 STATS TABLE 7. Experimental results from the central composite Our Final Model design. Time in seconds. The order of runs was randomized to minimize the risk of the sequence of experimentation Using regression analysis, the estimated second- affecting the response. order model is =+ + + +2 + 2 Run Order ABTime ˆy bbxbxbxxbxbx 011223124152 1 7 –1 –1 13.65 = 166.. 713−+− 0 702xxxxx... 0 . 735 0 . 510 −− 1 . 3112 1 554x2 23 1–113.74 12121 2 311–1 115.48 Model Validation 45 1 113.53 To validate our model, we checked the signifi cance 59 0 017.38 of the model terms, performed a lack-of-fi t test, and checked our statistical assumptions. 62 0 016.35 Using t-tests for the regression coeffi cients, 71 0 016.41 Table  indicates that all factors except the interaction term were signifi cant at the . level. 810+1.414 012.51 To test lack-of-fi t, the pure error in the process can be estimated using the center points. Th e 94–1.414 015.17 lack-of-fi t test suggests the model adequately describes the data because the p-value = . in 10 6 0 +1.414 14.86 Table  is large; we do not reject the null hypothesis 11 8 0 –1.414 11.85 (H) of fi t (i.e., no lack of fi t).

TABLE 8. Second-order response surface parameter estimates are all significant, Finding an Optimum except for the interaction term, indicating the interaction term could be dropped from the model. A near-optimum location was found and a second- T order model was used to describe the curvature Parameter df Estimate Std Err p-value near the optimum to determine optimal conditions. Intercept 1 16.713 0.408 40.98 0.0001

Near the optimum, the response surface is curved x = a 1–0.7020.250–2.810.0374 and can be approximated by a second-order model. x = b 1 0.735 0.250 2.94 0.0322 A maximum or minimum will occur at a stationary x = a x b 1–0.5100.353–1.440.2083 point, where the partial derivatives are all ,  x  = a x a 1–1.3110.297–4.410.0070 ∂ˆy  = 01, forik =,,… . Determining the shape of x  = b x b 1 –1.554 0.297 –5.23 0.0034 ∂  xi the surface is easy if k is  or  by plotting the surface or looking at contour plots. If there are many x variables, then visualizing the surface is TABLE 9. Lack-of-fit test indicates the model is adequate (i.e., it does not “lack fit”). diffi cult, but information about curvature can be Residual df SS MS Fp-value understood analytically. Canonical analysis takes Lack of Fit 3 1.826 0.609 1.82 0.3737 a square matrix of the βij coeffi cients and Pure Error 2 0.668 0.334 calculates eigenvalues λλ,,… and eigenvectors. 1k Total Error 5 2.495 0.499 Th e eigenvalues tell us the direction and magnitude of curvature, with the sign indicating downward (– = concave) or upward (+ = convex) curvature and magnitude indicating fairly fl at (near ) to steeply We checked that the statistical requirements of peaked (far from ). Wanting a maximum in independent and identically distributed residuals our experiment, we hoped all the λs would were satisfi ed. Th e residuals plot indicated the be negative. residuals were random with constant variance. Additionally, we tested the residuals for normality. Th e test had a large p-value (p = .), indicating the normality assumption should not be rejected. Th erefore, our second-order model appeared adequate.

ISSUE 48 STATS 19 We used SAS’s RSREG (Response Surface Design of Experiments Regression) procedure to predict the stationary point at (a= –., b=.) with the predicted response λλ =− − y=.. Th e eigenvalues ()(ab, 115.,. 171 ) VOCABULARY indicated a maximum—yes!

Effect: The mean difference in Our Optimum Design response due to changing a factor’s level, such as the mean Th e estimated response surface is plotted in Figure , difference in flight time of with the % confi dence region for the location of the paper helicopters when rotor maximum emphasized at the top. (See the reference length is changed from short on Page  by Enrique Del Castillo and Suntara Cahya to long. about computing confi dence regions on a response surface stationary point.) For our model, R = ., which indicates the model should be able to explain Experimental Design: A systematic about % of the variation in the response over the procedure for changing factor range of factor values we tested. Th e fi nal optimal levels to measure and compare helicopter design is shown in Figure . the responses, such as changing rotor length and rotor width on paper helicopters.

Factor: An independent variable deliberately changed in an experiment to study its effect on the response, such as the length of a paper helicopter’s rotors.

Factorial Design: An experiment that measures the responses from all possible combinations of two or more factors at fixed levels, such as the four combinations of two rotor lengths and two rotor widths for paper helicopters.

FIGURE 6. Estimated response surface with 90% confidence Interaction: When the response region for the coordinates of the maximum response due to one factor is influenced by another factor, such as the influence of rotor width Conﬁ rmatory Experiment on rotor length for paper Last, we conducted a confirmatory experiment at helicopters. the optimum location to verify the second-order prediction. The results of our confirmatory Response: Th e performance of an experiment consisting of six new helicopters at the experimental unit, such as the predicted optimal setting are shown in Table . The ﬂ ight time of a paper helicopter. mean response is . seconds with standard deviation . seconds, with a % confidence Response Surface: A curved surface representing how a system’s output performance is affected TABLE 10. Results from the confirmatory experiment by specified input factors, such corroborate the model’s predicted response. as how paper helicopter flight Flight time is affected by rotor length Time (sec) 15.5 16.4 19.7 19.4 18.6 17.3 and rotor width.

20 ISSUE 48 STATS interval on the mean response of (., .). Our model’s predicted response, ˆy=. seconds, falls within this interval, thus confirming the usefulness of our response model for the factor levels we tested and under the environmental conditions in our experiments.

Answers from Learning Stats Is FUN

Page 7 ❶ One…plus or minus . ❷ To get data from the other side of the median ❸ Residual plots FIGURE 7. The optimal helicopter design is longer and narrower ❹ Just think how unlikely it is for there to be two and has a shorter base than the initial helicopter design. bombs on board. She’d lose her degrees of freedom. There You Have It ❺ A hisss-togram So, there you have it. Th at is what we did. Now, see ❻ if you can do even better. Good luck, and please let Th e South-American ANOCOVA us know what you ﬁ nd out about using response Page 11 surface methodology to design a better paper ① . km (about  miles) helicopter. Have fun. ②  people ③ ⅔

ISSUE 48 STATS 21 AP STATISTICS

Vocabulary Is More Than Just Knowing the WORDS

by Peter Flanagan-Hyde

have participated in grading the Advanced Many of the students who managed to explain Placement (AP) statistics examinations for the design of the experiment correctly had large gaps Ia number of years, and it always amazes me in how they expressed these fundamental terms, how reading many students’ papers and seeing often confusing and mis-stating their analyses where their misunderstandings cluster help me of the design. As I struggled to fi gure out where reach a deeper understanding of some of the they were going off track, it helped me think more most basic ideas of the AP statistics course. deeply about the fundamental ideas of experimental The  exam was no exception, as students had design. It powerfully struck me, while reading this consistent diffi culty on its Question , about the year’s exams, that there were many students who design of an experiment. In particular, students could quote the defi nitions of the terms and quote had trouble eff ectively using the vocabulary of statements of interpretations without having experimental design on this question, especially the anything but the most superfi cial understanding of meaning of the words factor, treatment, variation, and confounding. the concepts involved.

Shrimp Growth Experiment PETER FLANAGAN- HYDE has been a math teacher for 27 years and has taught AP Statistics since its inception in the 1996-1997 school year. With a BA from Williams College and an MA from Teachers College, Columbia University, he has pursued a variety of professional interests, including geometry, calculus, physics, and the use of technology in education.

FIGURE 1. Shrimp growth experiment factors, levels, and treatments

22 ISSUE 48 STATS Th e scenario for the question was a two-factor experiment on the growth of shrimp. Th e experiment was to be conducted on a speciﬁ c species of shrimp— tiger shrimp—with two factors: salinity and growth- enhancing nutrient. Th ere were two levels of salinity Confounding is a word for which many students seem to (high and low) and three levels of supplemental know the definition, but not the meaning of the term.” nutrients arbitrarily labeled A, B, and C. Students needed to state the treatments for the experiment, “ design a completely randomized experiment, and comment on both the advantages and disadvantages of restricting the experiment to one species of shrimp. Factor is an explanatory or predictor Factor and Treatment variable to be studied in an Many students revealed in their answers that their investigation. understanding of the terms factor and treatment did not really distinguish one from the other, as The possible values of a factor are they tended to use the terms interchangeably. In called the levels of that factor this two-factor design, there were six treatments, one for each of the possible combinations of the factors. Many students, however, listed the factors Treatment is a specific combination and their levels when asked about the treatments, of the levels of the factors. just stating high salinity, low salinity, nutrient A, nutrient B, and nutrient C. Th ey completely missed the idea that a treatment consisted of a particular Confounding variable is a combination of the factors (salinity and nutrient) at variable whose effect on the particular levels (high salinity with nutrient A, etc.). response variable cannot be separated from the effect of the However, for many students, the confusion explanatory variable (factor) about the vocabulary did not prevent them from on the response variable. It correctly designing a completely randomized is a variable that offers an experiment in which there were, indeed, six alternative explanation for the treatments. Th ey just did not know exactly what result of the study. the term “treatment” referred to in the ﬁ rst part of the question. Part of the problem may have been that students were much less familiar with the two- Variation refers to observed factor setting than with the more typical classroom changes or inconsistencies in a problem of a single-factor experiment. With only characteristic, attribute, or action. one factor, the distinction between treatment and factor is less clear in students’ minds, as they can essentially interchange factor level with treatment.

Variation and Confounding Th e second half of Question  asked about the statistical advantage and disadvantage of restricting the experiment to one species of Variation due to diff erences in treatments is shrimp. Students were better at articulating the the ‘good’ variation in an experiment. In this disadvantage—limited generalizability of the experiment, the good variation is diff erences in results, because other shrimp species might react weight gain due to diff erences in salinity levels diff erently to the treatments—than they were and added nutrients. We are trying to quantify this about the advantage. To correctly answer this variation as specifi cally as possible, but this is made part of the question, students needed to have a more diffi cult by ‘bad’ variation that arises due to secure understanding of variation, its sources in factors other than the treatments. an experimental setting, and the strategies used to Th ese other factors are extraneous variables— manage unwanted variation. other characteristics of the conditions and subjects

ISSUE 48 STATS 23 in the experiment that cause diff erences in able to identify as the advantage of having only the response. In this experiment, there can be tiger shrimp in the experiment. diff erences in the conditions in the tanks in which However, student responses about the the experiment takes place, as well as innate statistical advantage revealed commonly held diff erences among the shrimp in the experiment. misconceptions about an important term in Th e design of the experiment should attempt to experimental design: confounding. Perhaps the minimize the eff ects of these sources of variation. most commonly stated incorrect advantage was For the tanks, this means eff orts on the part that having only one type of shrimp “eliminates of the experimenters to make the conditions in confounding.” Confounding is a word for which the tanks as identical as possible. Despite these many students seem to know the defi nition, but eff orts, there can be no guarantee that there are not the meaning of the term. Th ey are typically not diff erences that remain between the tanks. able to chant “confounding is when you cannot tell Randomly assigning the treatments to the tanks if the response is due to the treatment or another is how experimenters can make sure there is no variable,” and they know it is ‘bad.’ Th ey also know systematic way in which diff erences in the tanks it prevents you from easily seeing the true eff ects are associated with particular treatments. of a variable on a response. Having more than one species of shrimp potentially increases the diffi culty of seeing the true eff ects of salinity and nutrients, but this does not make it confounding. As stated in the AP Scoring Guidelines for this question, “In this completely randomized design, confounding is not possible.” Th is is worded bluntly and strongly, I think, to reinforce the point that confounding is widely misunderstood by students and perhaps their teachers. For confounding to be present, there needs to be a systematic way in which the treatments are tied together or to an extraneous variable. Th is occurs often in observational studies, but should not in an experiment, when the treatments are randomly assigned and all combinations of factors are tested. It is possible to imagine situations in which confounding would be present in trying to understand the eff ects of salinity and nutrients. For example, if the experiment were conducted in such a way that the higher level of salinity was applied inadvertently to only tiger shrimp, while the lower level to only some other type of shrimp, confounding may well be present. However, species could be included in the experiment as a third factor without causing confounding by testing all factor combinations and by randomizing. Th en, if a diff erent species of shrimp responds (grows) diff erently than tiger shrimp in the various treatments, that eff ect could be determined. Some students used the term “confounding” in an almost casual way, making statements such as “Having one species of shrimp reduces variability For the shrimp, this is partly addressed by and so eliminates potential confounding variables.” randomly assigning the shrimp to the tanks. Th is use conveys a sense that the student’s concept Minimizing the diff erences between shrimp, of a confounding variable is anything that makes however, is a second step that helps reduce the the situation murkier in determining an eff ect. variation in weight gain due to diff erence from If so, the student has not separated the ideas of shrimp to shrimp. Th is is the point of limiting the variability and confounding in his or her mind, as shrimp to one particular species: tiger shrimp. Th is both can contribute to making it harder to see if reduction in variation is what students should be an eff ect is present.

24 ISSUE 48 STATS In this problem, with a well-designed, completely phrased just so. Furthermore, they will not be able randomized experiment, confounding should not be to use what they ‘know’ on any test or challenge present. Removing the possibility of confounding is that frames the same question diff erently.” really the point of doing an experiment, allowing one Th e idea that we should be teaching and to make a conclusion about cause and eff ect. learning for an understanding that is fl exible, adaptable, and able to be transferred to new Lessons for Learning Statistics situations is not new. It dates back to at least In the students’ incorrect responses, there are the early s when Benjamin Bloom developed two lessons for learning statistics. Th e fi rst is to his taxonomy of educational objectives. His words be exposed to a variety of experimental situations about how to design useful assessment tasks could and to make sure some have multiple factors. have come from the current writers of the AP exam: Th e second is that it is often diffi cult to judge “Ideally, we are seeking a problem which will test the from the fi nal product—in this case, a correctly extent to which an individual has learned to apply the designed experiment—whether all the constituent abstractions in a practical way.” concepts that support the fi nal product are Th e capability to apply concepts in new situations correctly confi gured. It takes some probing with is the appropriate measure of understanding. As careful assessment along the way to make sure that we move through the curriculum, there is a natural when we have learned to reproduce a particular tendency to reduce the concepts to a phrase or experimental design, we also are able to apply each two. At times, we need to rely on memorized of the concepts in somewhat diff erent situations. formulations to help us reach a beginning Th e depth of our understanding of a particular understanding of an idea, but we need genuine concept can be measured by the distance we carry practice in using the concepts in a variety of it to reach a new situation, and that takes situations to develop a deeper understanding. repeated practice. It is certainly correct to say, “Confounding As teachers and students, we need to fi nd is when you cannot tell if the response is due eff ective ways to learn the vocabulary of statistics to the treatment or another factor,” but if this that produce something more than recognizing is the extent of our knowledge of the term, we the terms and being able to replay each defi nition. will not have the kind of fl exible and adaptable Th is is at the heart of teaching and learning for understanding we can transfer to new situations understanding, an idea that has been promoted by as they arise, either on tests or—much more Grant Wiggins and Jay McTighe, among others. importantly—when we use the ideas we have Th ese authors contend that students “cannot be learned in new settings and new situations in said to understand their own answer, even though future coursework or when reading the paper some it is correct, if they can only answer a question Saturday morning.

CALLING ALL STUDENTS OF STATISTICS

STATS: The Magazine for Students of Statistics is interested in publishing articles that illustrate the many uses of statistics to enhance our understanding of the world around us. We are looking for engaging topics that inform, enlighten, and motivate readers, such as:

STATISTICS IN EVERYTHING from STATISTICIANS IN HISTORY and the STUDENT PROJECTS using statistics to sports to medicine to engineering classic problems they studied answer interesting research questions in creative ways “STATISTICS IN THE NEWS,” discussing HOW TO USE particular probability current events that involve statistics and distributions in statistical analyses Send a description statistical analyses EXAMINING SURPRISING EVENTS and of your concepts for STATISTICS ON THE INTERNET, asking, “What are the chances?” and then covering new web sites with statistical providing the answers feature articles to resources such as data sets, programs, and examples REVIEWS OF BOOKS about statistics that Paul J. Fields, are not textbooks INTERVIEWS WITH PRACTICING [email protected]. STATISTICIANS working on intriguing problems

ISSUE 48 STATS 25 References and Additional Reading List Th e references for each article in this issue of STATS are included in the listing below, along with suggestions for additional reading on related topics. Th e page numbers are the numbers in blue.

Numb3rs, Sabermetrics, Joe Ranking of  All-Time Greats, Several humor collections are in 3Jackson, and Steroids. McFarland & Company, Inc., Hershey Friedman, Linda Friedman, You can see the “Hardball” Publishers, . and Taiwo Amoo’s “Using Humor in episode of Numbrs at www.cbs.com/ Th e web site for the Society for the Introductory Statistics Course,” primetime/numbrs. American Baseball Research (SABR) Journal of Statistics Education, (): Jim Albert’s work appeared in can be found at www.sabr.org. , accessible at www.amstat.org/ STATS when he wrote, “Does a For the stories behind the stats, publications/jse/vn/friedman.html. Baseball Hitter’s Batting Average see www.baseballlibrary.com; baseball General connections between Measure Ability or Luck?”, STATS, stats are available at www.baseball- music and statistics are in Jan Beran’s Issue , . reference.com. Statistics in Musicology, Chapman and Th e details of Jay Bennett’s Hall/CRC, . methodology appear in “Did Shoeless Books and Journals Joe Jackson Th row the  World Learning Stats Is Fun…with the Series?”, Th e American Statistician, (): 7 Right Mode. For a statistics book with a –, . It was republished in Humor and Song humorous style, read Fred Pyrczak’s Anthology of Statistics in Sports, edited Examples of Larry Lesser’s Statistics with a Sense of Humor, by J. Albert, J. Bennett, and J. Cochran, statistical music appear in “Average Pyrczak Publishing, . American Statistical Association and Love Songs,” Amstat News, (): Another statistics book with a light Society of Industrial and Applied ; “Stat Song Sing-Along,” STATS, touch is Larry Gonick and Woollcott Mathematics, . : –, ; and “Musical Means: Smith’s Th e Cartoon Guide to Statistics, Scott Berry’s study of steroid Using Songs in Teaching Statistics” Harper Collins, . use appears in, “A Juiced Analysis,” Teaching Statistics, (): –, . Your library might have a copy CHANCE, (): . Larry Lesser presented “Making of Richard Runyon’s Winning with Th e American Statistical Statistics Learning Fun” in a Statistics: A Painless First Look Association’s Section on Statistics in Consortium for the Advancement of at Numbers, Ratios, Percentages, Sports web site is at www.amstat.org/ Undergraduate Statistics Education Means, and Inference, Addison- sections/sis. (CAUSE) webinar, accessible at www. Wesley, . causeweb.org/webinar/-. Joe Jackson’s biography by David Sources of irreverent uses of L. Fleitz is Shoeless: Th e Life and Times For information about CAUSEweb, statistics in ‘research’ include The of Joe Jackson, McFarland & Company, see www.causeweb.org and www. Journal of Irreproducible Results Inc., Publishers, . causeweb.org/resources/fun. at www.jir.com and the Annals of Th e web site for the Shoeless Joe Carmen Wilson VanVoorhis’ Improbable Research at www.improb.com. Jackson Society is at www.blackbetsy. research using songs to learn com/society.htm. statistics, “Stat Jingles: To Sing or Games Not To Sing,” appears in Teaching of J. Albert and J. Bennett’s book about Steve Abbott and Matt Richey Psychology, (): –, . baseball statistics is Curve Ball: Baseball, “Take a Walk on the Boardwalk” in Statistics, and the Role of Chance in the Randall Garner describes his College Mathematics Journal, (): Game, Copernicus Press, . research about using humor to learn –, . statistics in “Humor in Pedagogy: W. F. McNeil and P. Palmer’s Phil Woodward solves the popular How Ha-Ha Can Lead to Aha!”, College sabermetrics book is Backstop: A dice game Yahtzee in “Yahtzee: the Teaching, (): –, . History of the Catcher and Sabermetric Solution,” CHANCE, (): –,

26 ISSUE 48 STATS . He discusses poker in “Call My Millionaire?’ ” in Teaching Statistics, Internet Bluff ,” Signifi cance, (): –, . (): –, . Webster West and Todd Ogden, Paul Stephenson, Mary Richardson, Diane Evans published “Conditional “Interactive Demonstrations for and John Gabrosek analyze the dice Probability and the : Lifeline on Statistics Education on the World game GOLO in “How LO Can You GO? ‘Who Wants to Be a Millionaire?’ ” in Wide Web,” Journal of Statistics Analyzing Probabilities for the Dice- the  JSM Proceedings [CD-ROM], Education, (): , accessible at Based Golf Game GOLO,” a paper in –. www.amstat.org/publications/jse/vn/ the  JSM Proceedings [CD-ROM], Ed Barbeau presents the three- applets/LetsMakeaDeal.html. –. You can fi nd GOLO at door problem in “Fallacies, Flaws, One place to fi nd Mozart’s Musical http://igolo.com. and Flimfl am,” College Mathematics Dice Game is http://sunsite.univie. Mary Richardson and David Coff ey Journal, (): –, . ac.at/Mozart/dice. analyze the game Pass the Pigs in Matthew Carlton also looks at the A web site that randomly generates “What Is the Probability of ‘Pigging three-door scenario in “Pedigrees, songs is www.amiright.com/cgi-bin/ Out’?” on the  ASA Proceedings Prizes, and Prisoners: Th e Misuse songrandom.cgi; one that makes band [CD-ROM], –. of Conditional Probability,” Journal names is www.bandnamemaker.com. John Kern provides a Bayesian of Statistics Education, (): , Statistics ‘rap’ lyrics can be found view of Pass the Pigs in his paper, accessible at www.amstat.org/ at www.lawrence.edu/fast/jordanj/rap. “Pig Data and Bayesian Inference on publications/jse/vn/carlton.html. html and statistics ‘rap’ videos can be Multinomial Probabilities,” Journal Federico O’Reilly took “A Look at found at http://video.google.com/ of Statistics Education, (): , the Two-Envelope Paradox” in STATS, videoplay?docid=. accessible at www.amstat.org/ : –, Spring . publications/jse/vn/datasets.kern. LaDawn Haws discusses Creative Works html. “Plinko, Probability, and Pascal” in An example of published song lyrics Larry Feldman and Fred Morgan Mathematics Teacher, (): –, with statistical language is David go ‘HOG wild’ in “The Pedagogy and . Wilcox’s “Down Here,” Underneath, Probability of the Dice Game HOG,” Amy Biesterfeld shows that “Th e Vanguard –, . Journal of Statistics Education, (): Price (or Probability) Is Right” in Wislawa Szymborska’s poem, “A , accessible at www.amstat.org/ Journal of Statistics Education, (): Word on Statistics,” appears in Th e publications/jse/vn/feldman.html. , accessible at www.amstat.org/ Atlantic Monthly, (): , . Edward Packel examines publications/jse/vn/biesterfeld.html. Shirley Jackson’s “Th e Lottery” is in Backgammon and other games Susie Lanier and Sharon Barrs X. J. Kennedy, Dorothy Kennedy, and in The Mathematics of Games and say, “Let’s Play Plinko: A Lesson Jane Aaron’s Th e Bedford Reader, th Gambling, Mathematical Association in Simulations and Experimental ed., Bedford Books, , accessible at of America, . Probabilities” in Mathematics Teacher, http://fi sheaters.com/thelottery.doc. Felicia Trachtenberg explains (): –, , with resources Norton Juster’s novel, Th e Phantom “The Game of Dreidel Made Fair” available at http://mathdemos.gcsu. Tollbooth, published by Random in College Mathematics Journal, : edu/mathdemos/plinko/index.html. House, Inc., in . –, . Eric Wood explains “Probability, Robert Feinerman gives a statistical Problem Solving, and ‘Th e Price Is More Fun ‘spin’ to “An Ancient Unfair Game” in Right’ ” in Mathematics Teacher, (): You can have fun testing American Mathematical Monthly, : –, . hypotheses about packaged candy, –, . You can track your own success such as m&m’s, using resources such Ivars Peterson explains the in maximizing the off er by playing as the Mars, Inc., web site at http:// probability of “Guessing Cards” on a simulation of “Deal or No Deal” at us.mms.com/us/about/products. MAA Online, , accessible at www. www.nbc.com/Deal_or_No_Deal/game. Th e frog ‘causality’ example and many maa.org/mathland/mathtrek___ David Kalist looks at “Data from other fun ideas appear in Harry Norton’s .html. the Television Game Show ‘Friend paper, “Keeping an Introductory or Foe?’ ” in Journal of Statistics Statistics Course Interesting: Use of Game Shows Education, (): , accessible Demonstrations, Examples, Rewards, Robert Quinn goes “Exploring the at www.amstat.org/publications/jse/ and a Little Humor,” presented at the Probabilities of ‘Who Wants to Be a vn/datasets.kalist.html. Joint Statistical Meetings, Seattle, Washington, .

ISSUE 48 STATS 27 Eric Sowey catalogues examples and the International Statistical University of Wisconsin– of intriguing statistical situations in Institute, . Madison. “Striking Demonstrations in Teaching For information about the David Annis, “Rethinking the Statistics,” Journal of Statistics graduate education minor in graduate Paper Helicopter: Combining Education, (): , accessible at interdisciplinary specialization in Statistical and Engineering www.amstat.org/publications/jse/vn/ college and university teaching at Th e Knowledge,” Th e American sowey.html. Ohio State University, contact Jackie Statistician, (): –. Buy humorous T-shirts from the Miller at [email protected]. Terry Siorek and Raphael American Statistical Association at Haftka, “Paper Helicopter— www.amstat.org/asastore. Experimental Optimum Designing a Better Paper Explore the birthday problem Engineering Design Classroom 14Helicopter Using Response with Larry Lesser in “Exploring Surface Methodology. Problem,” American Institute of the Birthday Problem with Raymond H. Myers and Douglas Aeronautics and Astronautics, Spreadsheets,” Mathematics Teacher, C. Montgomery, Response Surface . (): -, . Methodology: Process and Product J. Adam Johnson, Scott Bruce Trumbo, Eric Suess, and Optimization Using Designed Widener, Howard Gitlow, Clayton Schupp use R for “Simulation: Experiments, Wiley, . and Edward Popovich, “A ‘Six Computing the Probabilities of SAS and Minitab macros Sigma’ Black Belt Case Study: Matching Birthdays,” STATS, : accompany the text by Joseph G. E. P. Box’s Paper Helicopter –, . Petruccelli, Balgobin Nandram, and Experiment Part A and Part Minghui Chen, Applied Statistics B,” Quality Engineering, (): Let’s Party for Engineers and Scientists, –. For “Pi Day” ideas, see “Slices of Prentice Hall, . Available Pi: Rounding Up Ideas for Celebrating at http://users.wpi.edu/~jdp/ Vocabulary Is More Than Pi Day,” Texas Mathematics Teacher, downloads/book_macros/main.html. 22Just Knowing the Words. (): –, , or visit www.math. Cuthbert Daniel explains the “Use Blooms’ Taxonomy is utep.edu/Faculty/lesser/piday.html. of Half-Normal Plots in Interpreting presented by Benjamin Find famous statisticians’ birthdays Factorial Two-Level Experiments” in Bloom (editor) in Taxonomy at www.york.ac.uk/depts/maths/ Technometrics, : –, . of Educational Objectives: histstat/people or the math history Russell V. Lenth gives a “Quick Classifi cation of Educational Goals, site, www-history.mcs.st-andrews. and Easy Analysis of Unreplicated Longman, Green, & Co., . ac.uk/BiogIndex.html. Factorials” in Technometrics, ; – Six facets of understanding , . are discussed by Grant What Is This Statistics Enrique Del Castillo and Suntara Wiggins and Jay McTighe 12 Education Stuff? Cahya show “A Tool for Computing in Understanding by Design, Joan Garfi eld’s web site is located Confi dence Regions on the Stationary Association for Supervision and at http://education.umn.edu/EdPsych/ Point of a Response Surface” in Th e Curriculum Development, . faculty/Garfi eld.html. American Statistician, : –,  AP Statistics Exam Th e Guidelines for Assessment , available at http://lib.stat.cmu. questions can be found at www. and Instruction in Statistics edu/TAS/BH. collegeboard.com/prod_downloads/ Education (GAISE) can be accessed To learn about design of student/testing/ap/statistics/ at www.amstat.org/Education/gaise/ experiments and the paper helicopter, ap_frq_statistics.pdf. GAISECollege.htm. see Chapter  of Statistics for Scoring guidelines for the Th e United States Conference on Experimenters: Design, Innovation, AP Statistics exam are available Teaching Statistics (USCOTS) web site and Discovery, by George E. P. Box, at http://apcentral.collegeboard. is www.causeweb.org/uscots. J. Stuart Hunter, and William G. com/apc/public/repository/_ap_ For more about statistics Hunter, nd ed., John Wiley & Sons, statistics_sg.pdf. education, see D. Ben-Zvi and J. Inc., . Garfi eld’s (editors) Th e Challenge of Developing Statistical Literacy, See also the following: Reasoning, and Thinking, Kluwer George Box, “Teaching Engineers Academic Publishers, , and I. Experimental Design with a Paper Gal and J.B. Garfield’s (editors) Helicopeter,” Report No. , The Assessment Challenge in December , Center for Quality Statistics Education, IOS Press and Productivity Improvement,

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