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Sports Math 508 REAL-LIFE MATH years ϭ $250,000/year (bonus component); net salary ϭ $2,750,000/year. The player salary is treated as $2.75 mil- lion against the salary cap total of $76.8 million. This cal- culation will be made with respect to all 51 current contracts. Assume that the total player salaries are $71.2 million. The total available monies with which to sign other players to contracts is $5.6 million for the coming season, subject to releasing or otherwise terminating any existing contracts to create a greater cushion under the salary cap limit of $76.8 million. How does the salary cap work if a team wishes to acquire a player beyond their means? In the example, the available money for player acquisitions is $5.6 million. The team finished the previous season at eight wins and eight losses, and it did not qualify for the postseason play- offs. The head coach and the general manager believe that a certain wide receiver, who is not under contract to any team and is therefore a free agent, would be a player who might take the team that extra step needed to make the league playoffs in the coming year. This wide receiver is an elite player, and he is expected to command a salary of $10 million per season, and he will command a contract of four seasons. Can the sample team with only $5.6 million left in its salary cap sign this player? The capology options include: • No bid for this player: The current roster, subject to other contingencies such as injury, remains intact. (In a salary cap, where a player has been injured, they remain in receipt of their salary for the life of the contract, all counted in some fashion against the salary cap.) • Sign the elite player at $10 million per season for four seasons. To get “under” the salary cap in this exam- ple, the team would be required to cut other players whose salaries total $4.4 million for the coming season ($10 million in new salary, less the available $5.6 million). The team in this scenario would be required to assess whether the benefit to the team in terms of performance was worth the loss of other players; further, the variable of injury for the new player would be considered. • Sign the elite player, but structure the $10 million salary in year one of the four years as follows: Agree that the contract will be a $20 million bonus, and $20 million in salary over the following three years. The bonus is prorated over four years, meaning only $5 million would count against the salary cap this coming season. As $5.6 million is available as room under the team’s cap, the bonus/deferred salary structure works, at least for the first year. The team will have to assess how it deals with this contract in each successive year, as it will be required to count this player’s salary contract in year two as Bonus ϭ $5 million (25% calculated over four-year period) and Salary ϭ $20 million obligation now payable over three years. This math application in essence borrows from the team’s future to pay for the present needs of the team. In the realm of the salary cap, the best interests on the team on the field and the best financial interest of the team do not always exist in harmony. The more involved the mathematical equations deal- ing with salary cap, the less important are the players themselves. Further, it is a reasonable presumption that the greater the room available to a professional sports franchise in its salary cap, the greater potential profits to the ownership of the franchise. Some salary caps have a punitive component for those teams that breach the salary cap rule; these penalties are often referred to as a luxury tax. The premise behind these measures is that the richer franchises that exceed the salary cap limits will pay monies back into the general funds of the league, which are then distributed among the franchisees that abided by the salary cap rules. In the NBA, the tax on the individual player salary that broke the cap ceiling is 10%. The team is also obli- gated in general terms to pay a 10% team tax on its pay- roll that is in excess of the cap. There are a multitude of exemptions and qualifications; the bottom line for the owner is, are they prepared to exceed the salary cap and pay the penalties imposed if they get a team that might win a championship? MATH AND SPORTS WAGERING Team sports wagering has grown from its clandestine roots in taverns and clubs to a multi-billion dollar enter- prise that includes private bookmakers and state-run sport bets. All forms of sport gambling have a mathemat- ical basis, rooted in the concepts of probability and understanding the statistics relied upon by odds makers to establish betting systems. There are a number of dif- ferent types of wagers available, each generally involving a different math principle: • Straight bet: This is a wager placed on the final out- come of an event. For example, if a team is chosen as the winner and does win, the successful bettor gets a return on their money 1:1. If $100 were wagered on the team, the winner recoups his initial bet, plus $100. • Odds: As with the straight bet, the wager is with respect to the final outcome, with the odds, or the Sports Math REAL-LIFE MATH 509 probability, of the event added to the wager. For example, as in the earlier example, if the team were not likely to beat the opponent, the odds of such an event occurring might be as remote as 10:1 against, meaning that it is stated to be 10 times more likely that the team will lose than win. If $100 were wagered on 10 to 1 odds, and the team were successful, the successful bettor would again recoup the initial $100 wagered, plus 10 ϫ 100, or $1,000. • Point spread (also referred to as the line and other terms): This variation in sports betting is very popu- lar in sports such as football and basketball. The nature of the point spread in any given game is typi- cally calculated by professional gambling organiza- tions, and published in major media. The bettor does not wager necessarily on the best team, but the wager is with respect to the difference in points between the team’s scores at the end of the game. For example, Team A and Team B are NFL football teams sched- uled to play on a Sunday afternoon. The professional gambling organization reviews the teams’ records, injury situation, home field advantage, and the play of each team to date, and determines that “Team A is a 5-point favorite,” which means that the gambling organization believes that Team A will beat Team B by 5 points or more. The organization will then take bets on the outcome of this game using that 5 points, referred to as the spread, as its betting standard for that game. The results in this type of bet for a bettor placing $100 on Team A are that Team A must win by 5 points or more. If Team A wins by 5 points exactly, the result is referred to as a “push”: the bettor gets his $100 back, less the fee charged by the gambling house, 10%. Another result is for a bettor who places $100 on Team B. Because Team A is favored by 5 points, this bet will succeed if either Team B wins altogether, or Team B loses by 5 points or less. As with the straight bets, these wagers pay on a 1:1 ratio, less the 10% customarily charged by the betting establishment. • Over/under: This bet and its variations are based upon the total number of points scored in a game, including any overtime played, by both teams; the win or loss of the game itself is not relevant. For example, in a basketball game, the wagering line would be established as 176 points, wagers invited as being over and under the mark. If a wager is successful in predicting whether the teams were a total over or under the line, the return is again a 1:1 ratio to the money wagered. • Parlay: This form of wagering permits the bettor to gamble on two or more games in one wager. The bet- ter must be correct in all of the individual wagers to claim the entire bet. The reward multiplies in parlay betting, as does the risk of missing out on one wager in the sequence:In three-game parlay, Game has 12.7:1 odds; Game 2 has 3.3:1 odds; Game 3 has 1.9:1 odds. On a $5.00 wager on this three-game parlay, the return if each team selected were successful would be 2.7 ϫ 3.3 ϫ 1.9 ϭ 16.93; $5 ϫ 16.93 ϭ $86.45. As is illustrated, a return of almost 17 times the initial $5 wager would be a successful gambler’s reward in this scenario; a loss of any of the three games would mean the bettor would lose the entire parlay. • Future event: It is common for both North American and world sporting events to be the subject of odds Key Terms Average: A number that expresses a set of numbers as a single quantity that is the sum of the numbers divided by the number of numbers in the set. Odds: A shorthand method for expressing probabilities of particular events. The probability of one particu- lar event occurring out of six possible events would be 1 in 6, also expressed as 1:6 or in fractional form as 1/6. Percentage: From the Latin term per centum meaning per hundred, a special type of ratio in which the sec- ond value is 100; used to represent the amount present with respect to the whole. Expressed as a percentage, the ratio times 100 (e.g., 78/100 ϭ .78 and so .78 ϫ 100 ϭ 78%). Statistics: Branch of mathematics devoted to the col- lection, compilation, display, and interpretation of numerical data. In general, the field can be divided into two major subgroups, descriptive statistics and inferential statistics. The former subject deals primarily with the accumulation and presenta- tion of numerical data, while the latter focuses on predictions. Sports Math 510 REAL-LIFE MATH posted by various professional gambling agencies. For example, in the lead-up to the World Cup of Soccer, every team will be the subject of odds of winning the quadrennial championship; a perennial soccer power like Brazil might be listed at 3 to 1 odds, while a tra- ditionally less successful nation, such as Saudi Arabia or Japan, will be listed at more dramatic numbers such as 350 to 1. Wagers are typically binding at the odds quoted, no matter what might happen to the subject team in the period between the date of the wager and the date of the event. For example, if Brazil’s best scorer and best goaltender were injured, the actual odds quoted for Brazil might be quite higher at the start of the championships; the wager would remain payable at the initial 3 to 1 odds. Where to Learn More Books Adair, Robert K. The Physics of Baseball, 3rd ed. New York: Perennial, 2002. Holland, Bart K. What are the Chances? Voodoo Deaths, Office Gossip and Other Adventures in Probability. Baltimore, MD: Johns Hopkins University Press, 2002. James, Bill. Baseball Abstract, Revised ed. New York: The Free Press, 2001. Periodicals Klarneich, Erica. “Toss Out the Toss Up: Bias in Heads or Tails,” Science News, February 28, 2004. Postrel, Virginia. “Strategies on Fourth Down, From a Mathematical Point of View,”New York Times,September 9, 2002. REAL-LIFE MATH 511 Square and Cube Roots Overview Finding the square and cube roots of a number are amongst the oldest and most basic mathematical opera- tions. A number, when multiplied by itself, equals a num- ber called its square. For example, nine is the square of three. The square root of a number is the number that when multiplied by itself, equals the original number. For example, three is the square root of nine. The cube root is the same concept, but the cube root must be multiplied three times to yield the original number. These two con- cepts get their names from the relationship they have with the area of a square and the volume of a cube. In our three dimensional world, lines that have one dimension, squares that have two dimensions, and cubes that have three dimensions form the basic shapes that mankind uses to build models of the world. The square and cube of a number, and their inverses the square and cube roots, allow us to relate the length of a line to the area of a two-dimensional square or the volume of three- dimensional cube respectively. Examples of the square and cube roots will be found in any area of design where a model of an object will need to be conceptualized before the object can be built, for example in the architect’s plans for a new house or the maps for the construction of roads, or the blueprints of an aircraft. During the design phase, whenever areas and volumes need to be manipulated, the square and cube roots would be used to calculate these quantities. Fundamental Mathematical Concepts and Terms The definition of the square root is a number that when multiplied by itself, will yield the original number. As an example, again consider the value 9. It has a square root of 3, so 3 ϫ 3 ϭ 9. The value 9 is called the square of 3. The cube root is similar, but now the value that has to be multiplied is multiplied by itself three times, for exam- ple, the cube root of 8 is 2, so 2 ϫ 2 ϫ 2 ϭ 8 and the value 8 is called the cube of 2. The names square and cube root come from their relation with these shapes. Consider a square, where each side has an equal length; if you know the area of the square, the square root will give you the length of one side. Since all the sides are an equal length, you have found the length of them all. The area may be some square land where you want to know how much fencing is needed to mark the edge of your land. If the area is 100 square meters then the length of one edge is 10 meters. As Square and Cube Roots 512 REAL-LIFE MATH there are four edges to the square, you will need to buy 40 meters of fencing. The cubed root comes from the same idea. Imagine a wooden cube, where each edge is again exactly the same length. If we know the volume of this cube, the cube root will give us the length of one of the edges; since it is a cube, we know the length of all the edges. For example, an architect has calculated that his building will need a foun- dation with 1000 cubic meters of cement to hold the weight of the structure safely. The cube root of 1000 is 10, so the builders will know that by marking a 10 by 10 meter square out on the floor and digging down 10 meters this hole will be the right size for the cement. NAMES AND CONVENTIONS In mathematical text the radical symbol is used to indicate a root of a number. The square root is written as ͙ෆ9 ϭ 3. To indicate roots or higher than the square root, for example the cubed root, the number of the root is entered into the top left part of this symbol. For example the cubed root is written as 3 ͙ෆ8 ϭ2. This notation was developed over a period of about 100 years. The right hand slash and line above the num- bers first appeared in 1525 in the first German algebra book, Die Coss, by Christoff Rudolff (1499–1545). It is thought that the notation of adding the number 3 for a cube and numbers for higher roots as a symbol to the top left of the radical was first suggested by the Western philosopher, physicist, and mathematician René Descartes (1596–1650). The addition of the “vee” to the left side of the symbol is thought to have been developed in 1629 by Albert Girard (1595–1632), a French mathe- matician who had some of the first thoughts on the fun- damental theorem of algebra. The name root comes from a relationship with a fam- ily of equations called polynomials, these equations con- tain all the powers of a variable x in an infinite series and have the form, y ϭ a ϩ bx ϩ cx 2 ϩ dx 3 ϩ ex 4 . . . and so on, forever. All the letters on the right hand side of the equals sign, apart from the x, can have any values we want. Setting a value to zero will eliminate that term in the series. A Brief History of Discovery and Development In ancient times numbers held a deep religious and spiritual significance. Mathematics was heavily based on geometry, philosophy, and religion. Early thinkers about the nature of geometry saw lines and other geometrical shapes as the fundamental and logical building blocks of the heavens and Earth. The idea that nature could always be expressed with lines and shapes lead to the develop- ment of Pythagoras’ famous proof for triangles, a relation that uses the square root to calculate the final answer. Pythagoras of Samos (c. 500 B.C.), was an extremely important figure in the history of mathematics. Pythago- ras was an ancient Greek scholar who traveled extensively throughout his life. He founded a school of thought that had many followers. The society was extremely secretive but was based on philosophy and mathematics. The school admitted women as well as men to follow a strict lifestyle of thought and practice of mathematics. Pythagoras’ proof is for a triangle with one right angle and it relates the length of the longest side to the lengths of the other two sides. In the modern era, the proof is included in school textbooks and so it is hard for us to understand the deep impact on their way of life that this new method of logical thinking had on our ancestors. The proof—and knowledge of mathematics in general— were venerated as sacred secrets. Today, Pythagoras’ proof is learned as a formula with symbols, but this system of thinking would not have been known to its founder. Moreover, the proof that Pythago- ras found was based purely on geometry. Legend has it that a philosopher of Pythagoras’s society, called Hippa- sus, made the discovery at sea that if the two shorter sides of the triangle are set to 1 unit of length, then the result for the longer sided is an irrational number when the square root is taken. This special number could never be drawn with geometry and the legend goes that the other Pythagoreans were so shocked at this discovery that they threw him overboard to drown him and so keep his dis- covery a secret. There is another important property of taking roots of numbers that was not understood until English physi- cist and mathematician Sir Isaac Newton’s (1642–1727) time: the concept of taking the root of a negative number. If you try this on a calculator it will most likely give you an error. However, it was shown that it is possible to extend our number system to deal with taking the root of a negative number if we add a new number, given the symbol, i, in mathematics. This opened a whole new world of algebra that mathematicians call complex num- bers and allows solutions to be found for problems that had previously been thought impossible. From a practical viewpoint, this development affected almost every area of modern physics, which relies on complex numbers in some form or another. Some examples of their usage are found in electromagnetism, which gave us television, radio, and quantum mechanics, Square and Cube Roots REAL-LIFE MATH 513 which gave us, among many other things, computers and modern medical imaging techniques. PYTHAGOREAN THEOREM Using just pure geometry, Pythagoras is famous for proving that, for a right angled triangle, the square of the lengths of the longest side, called the hypotenuse, is equal to the sum of the squares of the other two sides. This rather long sentence is much easier to follow if it is writ- ten as an equation: h 2 ϭ a 2 ϩ b 2 . In this equation, the letter h is the length of the hypotenuse and a and b are the lengths of the other two sides. As this equation has only squared terms, we must take the square root if we want to find the actual length of h. For example, in a rectangular room, how long would a wire have to be if it was to be run in a straight line, across the floor, from the back, left hand corner, to the front, right hand corner? The room is full of furniture and it would be impossible to just measure the distance with a tape measure. However, we notice that the walls and the wire form a triangle pattern. Each wall is at right angles and lengths of the walls form the shorter two sides of the right angle triangle. The wire, running across the room, forms that longer side, the hypotenuse. One wall is 3 meters long, and the other is 7 meters, so: h 2 ϭ 3 ϫ 3 ϩ 4 ϫ 4 ϭ 25. So the length of the wire is given by the square root of 25 as 5 meters long. Wall length “b” Wall length “a” Wire length “h” Finding the length of a wire h = h 2 a 2 b 2 =+ h 2 a 2 b 2 =+ HIPPASUS’ FATAL DISCOVERY How long is the wire in the previous example if we have a room where each wall is just 1 meter long? h 2 ϭ 1 ϫ 1 ϩ 1 ϫ 1 ϭ 2. Now take the square root of 2 to find 1.4142136. In fact the digits of this number go on forever. It is a member of the family of numbers called irrational num- bers. These numbers have the property that the fractional part of the digits continue forever and never repeat the same pattern. From the practical perspective of installing our wire, this is no problem as we would simply round up the length. However, in the exact world of mathematics the consequences are much more dramatic. Due to the fractional part having an infinite nature, it cannot be expressed as a ratio of integer values (a fraction). What is even stranger is that we have made this length in something that is a perfectly reasonable and real geometric shape, a square box with sides equal to 1 meter. In this case, what exactly does the length of the line from one corner to the opposite corner of the box “mean”? Something that at first glance would seem child’s play to measure is soon found to be impossible. No matter what we do, the length, given by the square root of 2, will always be wrong to some degree if we try to give it an exact value. In the legend of the death of Hippasus at the hands of his fellow Pythagoreans, it was the discovery of this anomaly that shattered the idea that the Heavens and Earth could be expressed totally and completely by lengths and their ratios. Real-life Applications ARCHITECTURE The knowledge that some lengths are related with squared ratios has been known since Egyptian times, even though they would not have known the proof. Examples of this include the lengths 3, 4, 5, which are related by Pythagoras’ theorem and are thought to be found in the construction of the Egyptian pyramids. Today, squared and cubed roots are used in con- struction and design. If you were to design a car you might wish to change the volume of the driver’s com- partment. A modern three-dimensional (3D) design would be stored, as a wire frame model, in the memory of a computer. A computer program will divide the 3D space into thousands of tiny cubes, a job that is easy for a computer to do. Next, a program is run that counts the number of cubes within the driver’s compartment and returns a value. The total volume is equal to the number of cubes found in the compartment, multiplied by the Square and Cube Roots 514 REAL-LIFE MATH volume of one cube. The one cube is called the unit cube and has real dimension; this allows us to make modifica- tions to the actual size of the 3D wire frame without alter- ing the wire frame itself. To change the volume of the compartment, you change the volume of the unit cube. The amount that you would need to scale the sides of the unit cube is found by taking the cubed root of the original volume NAVIGATION The use of Pythagoras’ theorem allows distance to be calculated on maps using coordinate systems. A coordi- nate system is a grid-like structure that is used as refer- ence for points on the map’s surface. Lines between one point and another form vectors and the calculation of lengths of vectors requires the use of square roots. Vectors can also be used to map velocity, a combination of speed and direction. These systems are used on land by the mil- itary, at sea by the navy and shipping firms, and in the air by aircraft, to plan and negotiate the terrain they are moving over. As an example, if two ships are moving per- pendicular to each other, i.e, at 90 degrees to each other, and one ship is traveling at 3 knots and the other at 4 knots, using Pythagoras’ relation, the navigators on the deck of each ship would measure the speed of the other as moving away from them at 5 knots. SPORT Football pitches, tennis courts, race tracks, and swim- ming pools are some examples of areas used by professional This paper, written in 1946, was written by Albert Einstein. He explains how he derived the formula E = Mc 2 , a consequence of his Special Theory of Relativity, first published in 1905. The formula specifies that c (the speed of light) is squared. AP/WIDE WORLD PHOTOS. REPRODUCED BY PERMISSION. Square and Cube Roots REAL-LIFE MATH 515 sports people that need to be accurately measured if the events are to be considered fair. The areas to be surveyed and locations of the various markings must be set down. The process of surveying these areas requires the use of roots in the calculations of various lengths for the markings STOCK MARKETS Many of the transactions used in stock markets use statistics to estimate the market trends and the best times to buy and sell stocks and shares. These calculations will often use something called the standard deviation, a measure of the spread of random events, and will give the traders some idea of the accuracy of their estimates. This calculation will require the use of roots. Another occurrence of the root comes when the errors of predictive models are calculated. Models used to predict the stock market or anything else will have some sort of error depending on the accuracy of the data fed into it. If the error is much smaller than the size of the result, then the result can be trusted. For example, if your model suggests that you buy gold next Wednesday, within an error of one hour, this is fine, but if the error is ten years then the it would be fool- ish to trust the result. As there may be many sources of error they will all have to be accounted for they need to be combined to give a final overall error. This technique is well defined in statistics, which requires the use of the square root. Potential Applications GLOBAL ECONOMICS As global finance becomes more sophisticated, math- ematicians and economists investigate the patterns of these transactions and look for relationships that will indicate the growth and decline of large groups of com- panies or even countries. It has only been recently, with the large scale computing and the application of a num- ber of areas of science to economics that such models have come into use. Successful interpretation of these trends, and new ideas and concepts in understanding the trends, are vital to the future development and stability of corporations and governments. This science, macroeconomics, is sta- tistical in nature and allows predictions of important eco- nomic indicators such as inflation, interest rates, and the prices of materials. The use of squared and cubed roots in making these judgments incorporates fundamental for- mulas of probability and statistics that rely on square and cube roots. Where to Learn More Web sites Wolfram. MathWorld. Ͻhttp://mathworld.wolfram.com/Ͼ (February 1, 2005). Key Terms Cubed root: The relation of the volume of a cube to one of its edges. Root: The solutions of a polynomial equation, of which the square and cube root are special cases. 516 REAL-LIFE MATH Overview Statistics is the branch of applied mathematics con- cerned with characterization of populations by the collec- tion and analysis of data. Its applications are broad and diverse. Politicians rely on statistical polls to learn how their constituents feel about issues; medical researchers analyze the statistics of clinical trials to decide if new med- icines will be safe for the general public; and insurance companies collect statistics about automobile accidents and natural disasters to help them set rates. Baseball fans immerse themselves in statistics that range from slugging percentages to earned run averages. Nervous travelers comfort themselves by reminding themselves that, statisti- cally speaking, it is safer to travel in a commercial airliner than in an automobile. Students preparing for college fret over grade point averages and standardized test score per- centiles. In short, almost every facet of daily life involves statistics to one degree or another. Fundamental Mathematical Concepts and Terms POPULATIONS AND SAMPLES A statistic is a numerical measure that characterizes some aspect of a population or group of values known as random variables. They are random variables because the outcome of any single measurement, trial, or experiment involving them cannot be known ahead of time. The weight of men and women, for example, is a random variable because it is impossible to pick a person at ran- dom and know his or her weight before he or she steps on a scale. Random variables are discrete if they can take on only a finite number of values (for example, the result of a coin toss or the number of floods occurring in a cen- tury) and continuous if they can take on an infinite num- ber of values (for example, length or height). In some cases the populations are finite, for example the students in a classroom or the citizens of a country. While it may be impractical to do so if the population is large, a statistician can in theory measure each member of a finite population. For example, it is possible to measure the height of every student attending a particular school because the population is finite. In other cases, especially those related to the outcome of scientific experiments or measurements, the populations are infinite and it is impossible to measure every possible value. An oceanog- rapher who wants to determine the salt content of sea water using an electronic probe is faced with an infinite population because there are an infinite number of places where he or she could place the probe. Statistics [...]... discovered until the early twentieth century by notables such as Karl Pearson (1857– 193 6), A.N Kolmogorov ( 190 3– 198 7), R.A Fisher (1 890 – 196 2), and Harold Hotelling (1 895 – 197 3), for whom numerous statistical methods and tests are named One of the most unusual statisticians of the early twentieth century was William S Gosset (1876– 193 7), who wrote under the pseudonym Student He is best known for the t-test and... being incorrect if the null hypothesis is rejected S TAT I S T I C A L H Y P O T H E S I S T E S T I N G In a previous example it was shown that the arithmetic mean of the numbers 8 .95 , 6 .93 , 11.07, 10.21, and 10.31 is 9. 49 Could the numbers have been drawn at random from a normal distribution with a mean of 9 or less, even though the calculated sample mean is greater than 9? Possibilities such as this... 8 .95 and 6 .93 The second value, 6 .93 , is smaller than the first value, 8 .95 , so the positions of the two values are switched Next, the third value, 11 .93 , is compared to the first two Because 11 .93 is greater than both of the first two values, none of their positions in the list are switched The fourth value, 10.21, is then compared It is greater than the first two R E A L - L I F E M A T H values, 9. 93... can be evaluated using statistical hypothesis tests, which are formulated in terms of a null hypothesis (commonly denoted as H0) that can be rejected with a specified level of certainty Statistical hypothesis tests can never prove that a hypothesis is true They can only allow statisticians to reject null hypotheses with a specified level of confidence One common hypothesis test, the t-test, is used to... to the normal distribution To determine if the numbers 8 .95 , 6 .93 , 11.07, 10.21, and 10.31 are likely to have been drawn from a population with an arithmetic mean of 9 or less, first define a null hypothesis In this case, the null hypothesis is that the arithmetic mean of the population from which the sample is drawn is less than or equal to 9 The result of the ttest, which can be performed by many... the null hypothesis rejected each time Scientists often use a threshold (also known as a level of significance) of 0.05, so in this case the null hypothesis cannot be rejected because it is greater than either of those commonly used values It can be tempting to interpret the failure to reject a null hypothesis at an 0.05 level of significance as a 0 .95 , or 95 %, probability that the null hypothesis is... the arithmetic mean is (8 .95 ϩ 6 .93 ϩ 11.07 ϩ 10.21 ϩ 10.31)/5 ϭ 9. 49 Another kind of mean, the geometric mean, is calculated using the logarithms of the values The geometric mean is calculated as follows: First, find the logarithm of each number in the sample or population For the example list of five values used above, the natural (base e ϭ 2.7183) logarithms are: 2. 19, 1 .94 , 2.40, 2.32, and 2.33... Ͻhttp://www.fas.org/nuke/control/start1Ͼ (March 19, 2005) Internal Revenue Service Form 1040 Ͻhttp://www.irs.gov/ pub/irs-pdf/f1040.pdfϾ (March 17,2005) The Math Lab Subtraction in your head! An algebraic method for eliminating borrowing Ͻhttp://www.themathlab.com/ Pre-Algebra/basics/subtract.htmϾ (March 19, 2005) National Air and Space Museum Oral History Project Interviewee: James E Webb, November 4, 198 5 Ͻhttp://www nasm.si.edu/research/dsh/TRANSCPT/WEBB9.HTMϾ... 198 5 Ͻhttp://www nasm.si.edu/research/dsh/TRANSCPT/WEBB9.HTMϾ (March 19, 2005) National Sleep Foundation Myths and Facts About Sleep Ͻhttp://www.sleepfoundation.org/NSAW/pk_myths cfmϾ (March 19, 2005) Spaceref.com Press Release: NASA Evolutionary Software Automatically Designs Antenna Ͻhttp://www.spaceref.com/ news/viewpr.html?pid=14 394 Ͼ (March 19, 2005) U.S Department of Energy; EV America General... following list of values as an example: 8 .95 , 6 .93 , 11.07, 10.21, and 10.31 In order to calculate the range, first identify the minimum and maximum values in the list In this case, as in most real life applications, the minimum and maximum values are not the first and last values The minimum and maximum values in this example are 6 .93 and 11.07, so the range is 11.07 Ϫ 6 .93 ϭ 4.14 R E A L - L I F E M A T H . twentieth century by notables such as Karl Pearson (1857– 193 6), A.N. Kolmogorov ( 190 3– 198 7), R.A. Fisher (1 890 – 196 2), and Harold Hotelling (1 895 – 197 3), for whom numerous statistical methods and tests. are: 2. 19, 1 .94 , 2.40, 2.32, and 2.33. Second, calculate the mean of the loga- rithms, which is (2. 19 ϩ 1 .94 ϩ 2.40 ϩ 2.32 ϩ 2.33)/5 ϭ 2.24. Finally, raise e to that power, or e 2.24 ϭ 9. 37. Any base. compared. It is greater than the first two values, 9. 93 and 8 .95 , but smaller than the third value, 11 .93 . Therefore, the positions of 10.21 and 11 .93 are switched. The same procedure is repeated