Math Options Be Bold! Inc. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 4, Number 3 May 2002 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This newsletter is best viewed with a Courier font, size 10. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note - The Road to DOE Excellence Schedule of Public Classes Math in Industry - Is a Transformation Necessary? Software Review - Java Plug-In for Windows XP Family Math - Summer Swoon and Gifts for the Gifted Guest Author - Richard Clark: Drafting - Math and Physics, NASCAR style Ask Statman - How to Determine Probabilities with the Normal Distribution ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If all of us would start to look for more troubles, and learn to handle them cheerfully and with good judgment, as opportunities rather than irritations, we would find ourselves getting ahead at a surprising rate. For it is a fact that there are plenty of big jobs waiting for men and women who aren't afraid of the trouble connected with them. Robert R. Updegraff When you attend "Performing Objective Experiments" you will gain the knowledge you need to tackle and solve problems too complicated for most people. William D. Kappele ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Reader, The road to DOE Excellence is now better paved! You can start your journey with the workshop, "Performing Objective Experiments," where you will learn not only the basics of DOE (Design of Experiments), but also how to use I-Optimal designs -- very efficient designs that have only been practical since the 1990's. The next step on the journey is "Objective Experiments for Mixtures and Discrete Factors," a workshop that will teach you techniques needed to study a wider variety of factors. You will also learn how to use every design in the I-Optimal Design Library. Your next step is "Creating Custom Experiment Designs," a workshop where you will learn to create experiment designs to your specifications. You will no longer have to rely on tabulated designs. The best news is that these classes teach you practical skills. You may not be able to sling the Statistical jargon with your colleagues, but you will be able to use DOE effectively in your work. Once you have taken these three steps, you will be ready for nearly any experimental situation that comes along. See you on your journey! Bill Kappele. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Sept. 17 - 19, 2002 Performing Objective Experiments Bellingham, WA Learn to use the power of DOE in your work. When you leave this workshop you will know how to identify the problem to be solved how interactions among factors affect your results how to optimize your product or process how to make contour plots to show to customers and your management how to measure and report precision in your results how to find the most important factors when experimenting on a tight budget how and when to use different types of designs for efficient, cost-effective, yet sufficiently thorough experiments You can learn more and register to attend at http://www.mathoptions.com/public.htm You can learn about hosting classes at your company at http://www.mathoptions.com/training1.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Industry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Is a Transformation Necessary? In the last issue you learned how to make a transformation of your data to insure that the standard deviation was the same for all of your experimental trials. This article will show you how to know if a data transformation is necessary. An assumption made by the authors of most DOE software is that the standard deviation is the same for every experimental trial. If this is not the case, you will need to make a transformation before analyzing your data. The key, then, is to test whether the standard deviation is the same for all trials. If you can do that, you will know if you need to make a transformation. Unfortunately, there is no perfect test that will tell you if the standard deviations for your trials are the same or different. Every test can give you a wrong answer. To compensate for this weakness, you will need to run three tests. You will then apply the 2-out-of-3 rule -- if two of the three tests say the standard deviations are the same, you will not make a transformation. If two of the three tests say the standard deviation is different, you will make a transformation. The first test is a residuals plot. A residual is the difference between an observed measurement for an experimental trial, and the predicted measurement for that same trial. If you calculate all of the residuals (one for each trial) and plot each against its corresponding observed measurement you should see a broad scattering of the points. A pattern in these points would indicate that the standard deviations are different for different trials. (Most DOE software will produce this "Residuals vs. Observations" plot for you.) The second test relies on the Normal Probability Plot. This plot is produced for you by most DOE software. (If you want to make one by hand, please see E-Math News .....) This plot should be a straight line. If you see large curving or points falling on more than one distinct line, your standard deviations may be different for different trials. The third test uses a Box-Cox Plot. A Box-Cox Plot plots the log of the average for replicate trials vs. the log of the standard deviation for the same replicate trials. You will need at least 5 such points -- at least 5 of your trials must be duplicated. If you see a straight line with no slope, your standard deviations are not likely to be different for different trials. If you see a rising or falling straight line, your standard deviations may be different. Remember, you will be looking for 2-out-of-3 of these tests to show you what to do. If two of these tests indicate that your standard deviation is the same for all trials, you don't need to make a transformation. If two of these tests indicate that the standard deviation is different for different trials, you will need to make a transformation. Now you know how to make a transformation and how to know if you need to make one. In the next issue you will learn how to know which transformation to make. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can learn Design of Experiments in a practical, hands-on workshop at your company. Let Bill Kappele show you how to USE DOE in your work - not just talk about it. Please visit http://www.mathoptions.com/training1.htm for details. Have you taken "Performing Objective Experiments" but are feeling pretty rusty? You can repeat the workshop for $495. Please call Bill Kappele for details - (888) 764-3958. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Software Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Java Plug-In for Windows XP Java is one of the most useful technologies on the Internet. Java can make sites interactive, power online calculators of various types, and simplify your use of the Internet. Math Options provides a simulated laboratory where you can run experiments to test different techniques. This lab is written in Java and relies on the Java Plug-In to function. (You can find a version for your browser at http://www.mathoptions.com/browser.htm ) Unfortunately, Microsoft has chosen not to support Java in Windows XP. If you are using Windows XP, you may not be able to use many of the great Java applets on the Internet. Fortunately, Sun Microsystems is making the Java Plug-In easily available to you. They even have an automated installation process for XP users. You can find the Plug-In at http://java.sun.com/getjava/download.html If you are using another operating system and need the plug-in, you can get it from http://java.sun.com/j2se/1.4/download.html Happy surfing! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Summer Swoon and Gifts for the Gifted by Beth Heffernan For the majority of children, summer vacation is a welcome holiday from all school thoughts, especially those about math. Unfortunately for them, I am here to tell you "Don't let 'em slide!" Teachers spend much of the first quarter of each school year repeating math skills that have been lost over the summer for lack of use. At least three times a week your child needs to work some problems at his skill level. You can be direct (worksheets) or subtle- "How many cups of lemonade must you sell at $0.25 per cup to make the three dollars you need for the movies?" You can tell them that you are actually saving them future grief when school starts if they keep the math gears oiled all summer. Well, some of them might buy it! This also applies to reading skills. For the gifted child or those keenly interested in math and/or computer science, John Hopkins University offers correspondence courses through their Center for Talented Youth. These are not remedial programs. They are for home-schooled children or those with real talent in these areas. Programs include: An accelerated K-6 sequence Problem Solving Courses, developed by Riverdeep software designers Middle School Mathematics sequence, including pre-algebra Secondary School sequence, including algebra, geometry and pre- calculus with trigonometry. Introduction to C programming, Java programming and Physics. Perhaps the school district in your area may pick up some of the cost for these courses through its gifted minor programs. Further, CTY offers Advanced Placement Level Courses in Calculus A,B, and C written by the College Board which prepare your child for the Advanced Placement exams. This could save college tuition money later. CTY's courses are $490 per course, with a $40 non-refundable application fee. See their website www.jhu.edu/gifted/math for further information. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailto:Beth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Guest Author: Richard Clark ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Drafting - Math and Physics, NASCAR style The fastest growing spectator sport in America is Stock Car racing. Even if you are not a fanatic like I am you probably watch the Daytona 500 every year in February. If you do, you hear the term "drafting." At various times during the race the networks will even display 3D diagrams of the aerodynamics involved within the phenomenon but they never really explain how two or more cars travel faster than only one car. It begins with understanding the where and why -- only then can you understanding the how. Stock car racing takes place 2 times per year at the Daytona International Speedway and twice per year at the Talladega Superspeedway in Alabama. These tracks are different than the other tracks around the country. Daytona is 2.5 miles long with all of the turns banked at 31°. The Talladega Superspeedway is an amazing 2.66 miles long with bankings of 33° in the turns. These track specifications allow drivers to run full speed the entire lap, every lap. When the cars are up to top speed the accelerator stays on the floor until it's time for a pit-stop. This is where the cars become equal and Physics takes over. Picture an object, such as a car, traveling at a constant speed of 185 Mph. The car is fighting two forms of resistance. First, the car is pushing a wall of air equal to the winds created by a Hurricane or Tornado. As the car pushes the air out of it's way a vacuum (area void of matter) is created behind the car. The air flowing over the car fills the vacuum behind the car (just as the wake left from a large boat is quickly filled by the water). The effect of the air filling the vacuum directly behind the car creates drag, which attempts to slow the car from propelling forward through the air. After only a few minutes the car reaches terminal velocity. The laws of Physics will not allow the car to move any faster because the action (power created by a 450 horsepower racing engine) is equal to the reaction (the resistance of the 2 forces of nature working against the car). Isaac Newton couldn't have demonstrated it any better. Now let's add a second car. When two or more cars are traveling together around the high-banks of Daytona or Talladega they "hook up" together and begin to increase velocity because of the momentum of "the draft." Imagine two cars traveling nose-to-tail, or bumper-to-bumper, around these tracks. These cars will also reach terminal velocity but they work together to cheat the laws of Physics. Here's how it works. As two cars travel through the air at 185 Mph, they are fighting the same two forms of resistance, the push of the wall of air and the drag from the vacuum being filled. The difference is a lack of vacuum behind the lead car. Why? Because the trail car is filling the void. The air cannot move in behind the lead car because the trail car is in the space. Now the air flows over the trail car, which has a vacuum behind it. As the air fills the vacuum, the drag resistance tries to slow the second car. The cars move faster because they are traveling through the air as only one object. Instead of 450 Horsepower of action working against the reaction, the two cars have 900 Horsepower, thus they can travel faster. That's drafting. The most incredible Math and Science Fair project ever, on display "Live" in our living rooms, 4 times a year. When you understand this phenomenon you will never be able not to watch it. The best part is when you learn the language of the sport and begin to understand the scientific "play-by-play" the race announcers are giving you as it unfolds. They'll say: "the number 99 car needs a push if he's gonna get by the 24." Translation: If another car pulls up behind the 99, the effect of the draft will allow both cars to travel faster, and they'll pass the 24 car. Another classic is: "the 48 car just got hung out to dry?" This means the 48 was the trail car in a draft of cars making a pass together, when the car ahead of the 48, his "drafting partner", completed his pass, the driver moved over before enough room opened up for both his car and the 48 to fit. That left the 48 all alone with no drafting help beside a line of 5 or 6 cars (the pack) drafting together. The 48 has no choice but to let the line of cars "freight train by him" and fall in at the back of the pack. Sometimes a car behind the race leader tries to "make a move" on the leader. As the cars hook up and draft by leader (gang up and get by him), the leader realizes that he just got "shuffled out." It all becomes a Math and Physics "Chess match" at 190 miles-per-hour and once you understand it you'll wonder how you ever overlooked it. So tune in on July 6th and see for yourself. Or as the announcers say? "roll the dice and see what happens !!!" ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Richard Clark works in Measurement Science and Measurement Systems Analysis in Portland, Indiana. E-mail feedback to mailto:rcmetrology@yahoo.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Statman: I am trying to calculate the probability of our power supplies producing voltages out of specification. The voltages follow a bell curve, but I don't know the formula for finding the probabilities. The average voltage is 8.85 volts with a standard deviation of 0.23 V. The specification is 8.4 to 9.2 V. Can you help? Thanks, Wanting to Know Editor's Note: This article includes graphics that are not easily represented in text. You can find a Microsoft Word document at http://www.MathOptions.com/statman.normal.doc with the full text and graphics. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you have a question for Statman, please send it to mailto:Statman@MathOptions.com. Statman will answer questions about basic statistics that are of general interest to people working in industry. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright 2002 by William D. Kappele, Beth Heffernan, Richard Clark, and Charles S. Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you like E-Math News, please forward it to a friend. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A free newsletter published every other month by Math Options Inc. http://www.MathOptions.com 814 Lakeway Drive #179 FAX (503) 218-6587 Bellingham, WA, 98226 Toll Free (888) 764-3958 William D. Kappele, Editor Bill@MathOptions.com To subscribe to or unsubscribe from E-Math News please visit http://www.mathoptions.com/e-math.htm. If you don't have access to the World Wide Web, please send E-Mail to mailto:EMathNews-request@listdelivery.com with either "subscribe" or "unsubscribe" in the message body.