Math and Music Friends sing music together around a campfire. (credit: modification of work “Fire is hot! (2)” by Chetan Sarva/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Describe the basics of frequency related to sound. Describe the basics of pitch as it relates to music. Describe and evaluate musical notes, half-steps, whole steps, and octaves. Describe and find frequencies of octaves. “The world’s most famous and popular language is music.” Psy, South Korean singer, rapper, songwriter, and record producer Imagine a world without music and many of us would struggle to fill the void. Music uplifts, inspires, heals, and generally adds dimension to virtually every aspect of our lives. But what is music? For some it is a song; for others it may be the sounds of birds or the rhythmic sound of drumming or a myriad of other sounds. Whatever you consider music, it is all around us and is an integral part of our lives. “Music can raise someone’s mood, get them excited, or make them calm and relaxed. Music also—and this is important—allows us to feel nearly or possibly all emotions that we experience in our lives. The possibilities are endless” (Galindo, 2009). What music you listen to can impact your mood and emotions. In similar fashion, the music we choose can often tell those around us something about our current moods and emotions. Consider the music you may have been listening to as you today or even as you are reading this text. What cues to your mood do your music selections share? Albert Einstein is quoted as saying, “If I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.” What clues do your recent music choices say about your mood or how your day is going? Basics of Frequency as It Relates to Sound Every sound is created by an object vibrating and these vibrations travel in waves that are captured by our ears. Some vibrations we may be able to see, such as a plucked guitar string moving, whereas other vibrations we may not be able to see, such as the sound created when we hold our breath when accidentally dropping our cell phone on a hard floor. We don’t see the vibrations of our cell phone hitting the floor; however, any audible sound created in the fall is the result of vibrations in the form of sound waves, which can be pictured similarly to waves moving through the ocean. The waves of sounds each have a frequency , or rate of vibration of sound waves, that measures the number of waves completed in a single second and are measured in hertz (Hz; one Hz is one cycle per second). Louder sounds have stronger vibrations or are created closer to our ear. The further an ear is from the source of the sound, the quieter the sound will appear. Sounds range in frequency from 16 Hz to ultrasonic values, with humans able to hear sounds in a frequency range of about 20 Hz to 20,000 Hz. Adults lose the ability to hear the upper end of the range and typically top out in the ability to hear in a frequency of 15,000–17,000 Hz. Sounds with a frequency above 17,000 Hz are less likely to be heard by adults while still being audible to children. While frequency plays a key role in audible sounds, so too does the sound level, which can be measured in decibels (dB), which are the units of measure for the intensity of a sound or the degree of loudness. As a sound level increases, the decibel level increases. A person with average hearing can hear sounds down to 0 dB. Those with exceptionally good hearing can hear even quieter sounds, down to approximately –5 dB. The following table includes sample sounds with their related decibel values. Sound Decibels (dB) Firecrackers 140 Take-off of military jet from aircraft carrier 130 Clap of thunder 120 Auto horn standing next to the vehicle 110 Outboard motor 100 Motorcycle 90 Noise inside a car in city traffic 80 Typical washing machine 70 Public conversation, such as at a restaurant 60 Private conversation 50 Hum of a computer with fan blowing 40 Quiet whisper 30 Swishing leaves 20 Regular breathing 10 Lowest typical sound audible by teenagers 0 Selecting Decibel Value of Sounds Select the most representative decibel value for each of the following sounds: car wash: 25 dB, 55 dB, 85 dB vacuum cleaner: 15 dB, 70 dB, 90 dB ship’s engine room: 30 dB, 65 dB, 95 dB approaching subway car: 70 dB, 100 dB, 120 dB 85 dB 70 dB 95 dB 100 dB Basics of Pitch When considering the various sound levels the human ear can hear, the ear perceives sound both from the frequency level and the pitch of a sound. The quality of the sound is referred to as pitch , the tonal quality of a sound and how high or low the tone. Sounds with a high frequency have a high pitch, such as 900 Hz, and sounds with a low pitch have a low frequency, such as 50 Hz. Let’s take a look at frequency and pitch using a string instrument such as a guitar or piano. When a string is plucked on a guitar or a key is played on a piano, the related string vibrates at a frequency that is related to the length and thickness of the string. The frequency is measurable and has a singular value. The pitch of the note played is open for interpretation, as the pitch is a function of personal opinion. Graphing Calculators and Music It may be interesting to note that a TI-84 and TI-Nspire graphing calculators can be utilized to tune a musical instrument by measuring the frequency of a note using a small plug-in accessory that captures the sound waves from a note and displays the corresponding frequency. Using the displayed frequency, an instrumentalist can then make the needed adjustment to perfectly tune an instrument. This method of tuning an instrument can be helpful whether a novice player or a seasoned instrumentalist because the instrument can be tuned precisely to the correct frequency. Note Values, Half-Steps, Whole Steps, and Octaves “There are not more than five musical notes, yet the combinations of these five give rise to more melodies than can ever be heard.” Sun Tzu, Chinese strategist Piano Keys and Notes (credit: modification of work "Contemplate" by Walt Stoneburner/Flickr, CC BY 2.0) Moving our exploration to note values, the frequency of all notes is well defined by a specific and unique frequency for each note that is measurable. We will explore keys on a keyboard to discuss notes that have the same relationships with any instrument or musical piece. Let’s look at . The white keys are labeled with the letters A–G and the photo begins with middle C, which can be found in the middle of a keyboard. This labeling of the keys repeats across an entire keyboard and keys to the right have a higher pitch and frequency than keys to the left. Each of the keys correlates to a musical note. Movement up or down between any two consecutive keys (black and white) or notes constitutes a half-step . Movement of one half-step sometimes involves a sharp (#) or a flat (♭) symbol. For example, D # is one half-step above D and D ♭ is one half-step below D. Note that this is not always true as one half-step above B is C, and one half-step below F is E. In similar fashion, a whole step is movement up or down between any two half-steps on a keyboard. Identifying Half-Steps Name which keys are one half-step up and one half-step down from the following: D E G# up D # , down D ♭ up F, down E ♭ up A, down G Identifying Whole Steps Name which keys are one whole step up and one whole step down from the following: F # E A ♭ Up G # , down E Up F # , down D Up B ♭ , down G ♭ You may have noticed that there are eight letters of the alphabet used to label notes. Selecting any one note and counting up 12 half-steps you will find that the numbering for notes begins at the same value as you started from. This collection of 12 consecutive half-notes is called an octave and is a basic foundational component in music theory. Listing All Notes in an Octave List the 12 notes forming an octave, beginning with the note C. C, C # , D, D # , E, F, F # , G, G # , A, A # , B Frequencies of Octaves Notes that are one octave apart have the same name and are related in frequency values. Given the frequency of any note, the frequency of same note one octave higher is doubled and this pattern continues as you move up and down the notes on a keyboard or any other musical instrument. Song writers and singers use this knowledge to change the pitch of a note up or down to align with a person’s vocal range. Regardless of which C is played or sung, the pitch is the same and the frequency is related by a power or two. Labeled keys on a keyboard are numbered for ease in identification. For example, middle C is labeled as C 4 on a full keyboard as it is the fourth C from the left in a set of eight notes. The frequency of C 4 is 262 Hz, rounded to the nearest whole number. Calculating the Frequency Values of Octaves Given that the frequency of C 4 is 262 Hz, find the approximate frequency of C 6 . The frequency of each consecutive higher octave doubles. Given that the frequency of C 4 is 262 Hz, the frequency of C 5 is found by doubling the frequency of C 4 , which is 524 Hz. In similar fashion, the frequency of C 6 is found by doubling the frequency of C 5 , which yields 1,048 Hz. David Cope David Cope has a somewhat eclectic list of job titles ranging from author and music professor to scientist and artificial intelligence researcher. Cope combined his interests when he developed software that can analyze a piece of music and create a new and unique musical piece in the same style as the original. Some of his well-known products have been based off of the classical music of Mozart, creating what has been called Mozart’s “42nd Symphony,” as well as other genres including opera and a range of current music styles. Cope has also composed original musical pieces in collaboration with a computer. We have explored some basics components of frequency, pitch, note relationships, and octaves, which are building blocks of music. It may be exciting to learn that the mathematical relationships found in music are vast and grow in complexity beyond the math commonly studied in high school. Spotify Royalty Payments Streaming services have grown exponentially in popularity thanks in large part to customized music listening through cell phone use and devices such as Google as well as Amazon Echo and Alexa devices for home and vehicles, adding to ways that artists are paid royalties. Spotify, which was launched in 2008, typically plays artists $0.06 per time a song is streamed, with some artists receiving up to $0.84 per play amounting to over $9 billion in revenue for Spotify in 2020. Since 2014, Spotify’s revenue has grown over a billion dollars a year, with roughly half of their revenue being paid out in royalties, which was good news for artists during the Covid-19 pandemic when in-person concerts and shopping were hindered. Check Your Understanding Key Terms frequency pitch Hertz decibel half-step whole step flat sharp octave Key Concepts As frequency of a sound increases, the pitch of the sound increases. Hertz is a unit of measurement for frequency. Decibel is a unit of measurement for the intensity of sound. Half-steps and whole-steps describe one type of movement between two notes on a keyboard. Octaves are a collection of any 12 consecutive notes, which is a foundation in music theory.

Math


Math and Sports Fans support their team by attending games and wearing team gear. (credit: “Cheering Touchdowns” by Steven Miller/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Describe why data analytics (statistics) is crucial to advance a team’s success. Describe single round-robin method of tournaments. Describe single-elimination method of tournaments. Explore math in baseball, fantasy football, hockey, and soccer (projects at the end of the section). Sports are big business and entertainment around the world. In the United States alone, the revenue from professional sports is projected to bring in over $77 billion, which includes admission ticket costs, merchandise, media coverage access rights, and advertising. So, whether or not you enjoy watching professional sports, you probably know someone who does. Some celebrities compete to be part of half-time shows and large companies vie for commercial spots that are costly but reach a staggering number of viewers, some who only watch the half-time shows and advertisements. Data Analytics (Statistics) Is Crucial to Advance a Team’s Success Analyzing the vast data that today’s world has amassed to find patterns and to make predictions for future results has created a degree field for data analytics at many colleges, which is in high demand in places that might surprise you. One such place is in sports, where being able to analyze the available data on your team’s players, potential recruits, opposing team strategies, and opposing players can be paramount to your team’s success. Hollywood turned the notion of using data analytics into a major motion picture back in 2011 with the release of Moneyball, starring Brad Pitt, which grossed over $110 million. The critically acclaimed movie, based on a true story as shared in a book by Michael Lewis, follows the story of a general manager for the Oakland Athletics who used data analytics to take a team comprised of relatively unheard of players to ultimately win the American League West title in a year’s time. The win caught the eye of other team managers and owners, which started an avalanche of other teams digging into the data of players and teams. In today’s world of sports, a team has multiple positions utilizing data analytics from road scouts who evaluate a potential recruit’s skills and potential to the ultimate position of general manager who is typically the highest-paid (non-player) employee with the exception of the coaches. Being able to understand and evaluate the available data is big business and is a highly sought after skill set. In college and professional sports, it is no longer sufficient to have a strong playbook and great players. The science to winning is in understanding the math of the data and using it to propel your team to excelling. Single Round-Robin Tournaments Single-Round Robin Tournament A common tournament style is single round-robin tournaments ( ), where each team or opponent plays every other team or opponent, and the champion is determined by the team that wins the most games. Ties are possible and are resolved based on league rules. An advantage of the round-robin tournament style is that no one team has the advantage of seeding , which eliminates some teams from playing against each other based on rank of their prior performance. Rather, each team plays every other team, providing equal opportunity to triumph over each team. In this sense, round-robin tournaments are deemed the fairest tournament style. One hindrance to employing a round-robin-style tournament is the potential for the number of games involved in tournament play to determine a winner. Determining the number of games can be found easily using a formula which, as we will see, can quickly grow in the number of games required for a single round-robin tournament. The number of games in a single round-robin tournament with n teams is n ( n - 1 ) / 2 . Calculating the Number of Games in Single Round-Robin Tournaments Find the number of games in a single round-robin tournament for each of the following numbers of teams: 4 teams 8 teams 20 teams Using the formula with 4 teams yields 4 ( 4 - 1 ) / 2 = 6 tournament games. Using the formula with 8 teams yields 8 ( 8 - 1 ) / 2 = 28 tournament games. Using the formula with 20 teams yields 20 ( 20 - 1 ) / 2 = 190 tournament games. As the examples show, single round-robin tournament play can quickly grow in the number of games required to determine a champion. As such, some tournaments elect to employ variations of single round-robin tournament play as well as other tournament styles such as elimination tournaments. Single-Elimination Tournament (credit: final TK) Single-Elimination Tournaments When desiring a more efficient tournament style to determine a champion, one option is single-elimination tournaments ( ), where teams are paired up and the winner advances to the next round of play. The losing team is defeated from tournament play and does not advance in the tournament, although some leagues offer consolation matches. The number of games in a single-elimination tournament with n teams is ( n - 1 ) . Calculating the Number of Games in Single-Elimination Tournaments Find the number of games in a single-elimination tournament for each of the following numbers of teams: 4 teams 8 teams 20 teams Using the formula with 4 teams yields ( 4 - 1 ) = 3 tournament games. Using the formula with 8 teams yields ( 8 - 1 ) = 7 tournament games. Using the formula with 20 teams yields ( 20 - 1 ) = 19 tournament games. A single-elimination tournament offers an advantage over single round-robin tournament style of play in the number of games needed to complete the tournament. As you can see, in comparing the number of games in a single round-robin tournament in with the number of games in single-elimination tournament as shown in , the number of games required for single round-robin can quickly become unmanageable to schedule. There are modifications to both the round-robin and elimination tournament styles such as double round-robin and double-elimination tournaments. Next time you observe a college or professional sporting event, see if you can determine the tournament style of play. Sports Popularity Shifts Sport has been a popular entertainment venue for hundreds of years and the popularity of various sports shifts over time as well as in different regions of the world today. In today’s world, the most popular sport is, overwhelmingly, soccer, with over 4 billion fans followed by cricket with 2.5 billion fans. It may surprise you to learn that American football doesn’t rank in the top 10 most popular sports in the world today, yet table tennis ranks in sixth place and golf fills the last spot in the top 10 world sports. In the early 1930s, baseball ranked a close second, with basketball virtually tying for third place. By the mid-1940s, hockey slightly led in first place over basketball. Ten years later, hockey remained in the number one sport, but cricket pushed basketball to third place. In the 1960s, soccer dominated in popularity. Over the next 30 years, basketball dropped in popularity and there was much movement in the popularity of sports. By the late 1990s, soccer swept to first place, where it has since continued to grow in popularity. Check Your Understanding Key Terms data analytics seeding Key Concepts Describe how a round-robin tournament is organized. Compute the number of games played in a round-robin tournament. Describe how a single-elimination tournament is organized. Compute the number of games played in a single-elimination tournament. Formulas Number of games in a single round-robin tournament with n teams is n ( n - 1 ) / 2 . Number of games in a single-elimination tournament with n teams is ( n - 1 ) . project Lucas Sequence and Fibonacci Sequence The Lucas numbers bear some similarity to the Fibonacci numbers and exhibit a stronger link to the golden ratio. Edouard Lucas is credited with naming the Fibonacci numbers and the Lucas numbers were so named in his honor. The Lucas numbers play a role in finding prime numbers that are utilized in encrypting data for actions such as using your debit card to obtain money at a cash machine or when making a credit card purchase for point of sale as well as when shopping online. Complete the following questions to explore numbers in the Lucas sequence as well as their relationships to the numbers in the Fibonacci sequence. Conduct an Internet search to find out what a Lucas number is and how the Lucas numbers are related to the Fibonacci numbers. What are the first two numbers in the Lucas sequence? Describe how the next number in the Lucas sequence is determined and compare this to how the next number is determined in the Fibonacci sequence. Complete the following table listing the first 10 terms in the Fibonacci and Lucas sequence: Term 0 1 2 3 4 5 6 7 8 9 Fibonacci Numbers 0 1 1 2 5 13 34 Lucas Numbers 3 7 18 47 Interestingly, many patterns can be found in looking at the relationships in the Fibonacci and Lucas numbers. Look closely at the chart in question 3 to discover one such pattern. Observe the Fibonacci numbers in the third and fifth terms and compare with the Lucas number in the fourth term of the sequence. Describe the pattern found. Does this pattern continue in the table? Research the Fibonacci or the Lucas numbers to find an application in our world distinct from what has been shared in this project and section. Write a paragraph sharing the findings of your research. Solar Array for a Residence One of the first steps in adding solar to a residence is determining the size of a system to achieve the desired output. In this project, we will explore the solar needs of a residence and estimate needs of a solar array to supply electrical output to meet various percentages of electrical need. Step 1: Obtain an electric bill from your apartment/home. Find the average monthly or yearly usage if listed or call the electric company to inquire. If an electric bill is not available, use the Internet to find an average monthly or yearly electric usage for your area. Step 2: Determine a daily and hourly usage. Divide the average monthly usage by 30 or the yearly average by 365. Divide again by 24 to calculate an average hourly electric usage, which will yield the average kilowatt-hours for how much electrical power your is being utilized in an hour. Step 3: Multiply your average hourly use (kilowatts) by 1,000 to convert to watts. Step 4: Use the Internet to determine the average daily peak hours of sunlight where you live. Step 5: Divide your average hourly watts (Step 3) by the average daily peak hours (Step 4) to calculate the average energy needed for a solar array to produce every hour. Step 6: Determine the average energy needed in a solar array per hour to meet each of the following: 100 % coverage of average energy (Step 5) 80 % coverage of average energy 50 % coverage of average energy Step 7: Using the values computed in Step 6, compute the residential savings based on an average cost of 12 cents per watt. Vaccine Validation Validation of vaccines is a topic that exploded in the news when the Covid-19 pandemic spread across the world. As governments and organizations looked for a vaccine to curb the spread and minimize the severity of infection, concern was expressed by some for what appeared to be a quick discovery for a Covid-19 vaccine. Conduct an Internet search to explore the following questions. Pay special attention to the sources you select to ensure that they are credible sources. Research the term efficacy rates . Express what efficacy means in your own words. Using a minimum of two sources, compose a well-developed paragraph describing what validation of a vaccine means. Using a minimum of two sources, compose a well-developed paragraph sharing the steps in validating a vaccine. Using a minimum of two sources, compose a well-developed paragraph describing how a vaccine is determined to be validated. Using your research for Questions 1–3, write a summary paragraph sharing your reflections on the validation of the Covid-19 vaccine or on validating a vaccine in general. What key components did you learn? What would you like to learn more about related to validating a vaccine? Frequency and Ultrasonic Sounds Ultrasonic sounds have been utilized for a variety of reasons, from purportedly repelling rodents and other animals as well as a variety of other applications. Using an Internet search, complete the following questions to explore some of these applications and examine the validity of various claims. Repelling Insects, Rodents, and Small Animals Some radio stations purport to play a high pitch sound dually with their music to aid in deterring insects and other annoying bugs to aid in providing a bug-reduced listening environment. To deter small rodents, some products claim to emit ultrasonic sounds that drive away mice and other similar pests. Research the science behind ultrasonic pest controls, paying attention to the source of the information that you find. Compare the information found on advertisements, reviews, and scientific articles. What frequency ranges do ultrasonic pest deterrent devices utilize and how do these frequencies compare to the audible range that humans can hear? Write a short paragraph comparing the claims in the advertisements with independent reviews and scientific articles. What do you conclude about ultrasonic pest controls and why? Disbursing Teenagers Some business owners and communities have turned to products such as the “mosquito” sonic deterrent device to discourage groups of teenagers from loitering around storefronts and community landmarks, citing a public nuisance issue and public safety concerns. Research the science behind ultrasonic deterrent devices as they apply to dispensing teenagers. Compare the information found on advertisements, reviews, and scientific articles. What frequency ranges do ultrasonic teenager deterrent devices utilize and how do these frequencies compare to the audible range that adults can hear? Write a short paragraph comparing the claims in advertisements with independent reviews and scientific articles. What are some of the ethical debates surrounding the use of ultrasonic teenager deterrent devices? What do you conclude about business owners or communities using ultrasonic teenager deterrent devices and why? Jewelry Cleaner Use of pastes and liquid chemicals to clean jewelry can be harsh on stones as well as metals. So how can we safely obtain the sparkling clean look at home that jewelry stores provide? Some would say the answer is to use an ultrasonic jewelry cleaner, but do these really work? Research the science behind ultrasonic jewelry cleaners, including the phrase “cavitation process” in your search. Compare the information found on advertisements, reviews, and scientific articles. Write a short paragraph detailing how ultrasonic jewelry cleaners work and what role the cavitation process plays in the claims for ultrasonic cleaning. Do ultrasonic jewelry cleaners utilize low or high frequencies? How do these frequencies compare to the audible range that humans can hear? Write a short paragraph comparing the claims in the advertisements with independent reviews and scientific articles. What do you conclude about ultrasonic jewelry cleaners and why? Specialized Ringtones As the use of cell phones has become commonplace and families grow towards each member having their own cell phone, specialized ringtones have become popular and can aid in identifying who is calling just by the ringtone. Ever hear of ringtones that can be heard by teens but often not their teachers? The banning of cell phone use by K–12 students during class time as been implemented across a wide array of schools and some students have purportedly found ways to get around teachers hearing a cell phone ring through the use of ultrasonic ringtones. Research the science behind ultrasonic ringtones. Compare the information found on advertisements, reviews, and scientific articles. Do ultrasonic ringtones utilize low or high frequencies? Write a short paragraph detailing how ultrasonic ringtones work. How do these frequencies compare to the audible range that adults can hear? Include ages that are purported to hear and not hear the ringtones as well as the frequency ranges utilized. Write a short paragraph comparing the claims in the advertisements with independent reviews and scientific articles. What do you conclude about ultrasonic ringtones and why? Streaming Services and Math With the ability to stream music virtually anywhere you are, it is not surprising that Google Play Music, Apple Music, and a slew of other companies such as Spotify, Amazon Music, YouTube Music, Sound Cloud, Pandora, Deezer Music, Tidal, Napster, and Bandcamp have invested heavily to bring streaming service to users worldwide. Streaming services have expanded to offer virtually every genre of music with vast libraries to meet diverse user requests. Considering all of the choices available for streaming music, there is a wide array of options for subscribing. Conduct an Internet research to review your current streaming choices, if any, and evaluate competitors’ products. In this project, you will explore options for music streaming service subscriptions. Select a minimum of five streaming services listed above that you are not currently utilizing and determine the below components. Format your findings in an easy-to-read format such as a table similar to the one shown below. Monthly subscription cost Available features Available Features Service Choice 1: Service Choice 2: Service Choice 3: Service Choice 4: Service Choice 5: Able to stream unlimited music Able to purchase individual songs Able to purchase individual whole albums Ability to create personalized play lists Ability to select specific songs to play Ability to listen Other = Premium cost per month, if available = ______ Feature(s) offered with premium monthly subscription = ______ What is (are) your current music streaming services, if any? What is your current monthly service charge(s)? What features does your current streaming service offer? Write a paragraph summarizing your findings and include if your current streaming service(s) meets your needs or if research for this project has you considering changing streaming services. Support your rationale. Math and Baseball Baseball is known to have one of the largest pools of statistics related to the game and its players. Managers, coaches, and pitchers study the statistics of the players on opposing teams to give their team an edge by knowing what pitches to throw for the best probability to be missed by a batter. In similar fashion, batters study pitchers’ statistics to learn a pitcher’s strength and how to predict what a pitcher will throw and how to best hit against a pitcher. The three primary baseball statistics are batting average, home runs, and runs batted in (RBIs), which are the components of the title of Triple Crown winner that is awarded to players who dominate in these three areas. However, there is a wealth of other statistics to evaluate when studying the performance of a player. Conduct an Internet search to research statistics and how they are calculated in the following categories: Batting Statistics There are about 30 batting statistics. Select a minimum of 10 batting statistics. Compose an organized list including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. As an example: AB/HR represents at bats per home run and is calculated by the number of times a player is at bat divided by home runs. Pitching Statistics There are about 40 pitching statistics. Select a minimum of 10 pitching statistics. Compose an organized list including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. As an example: K/9 represents strikeouts per nine innings and is calculated by the number of strikeouts times nine divided by the number of innings pitched. Fielding Statistics There are around 10 fielding statistics. Select a minimum of five fielding statistics. Compose an organized list including the name of the statistic, abbreviation, explanation of what it represents, as well as how it is calculated. As an example: FP represents fielding percentage is calculated by the number of total plays divided by the number of total chances. Overall Select a player from recent years to evaluate. The player can be one that you have followed, one from a favorite team, or any current player. Find the statistics shared in Questions 1–3 to use in evaluating the potential strengths and weaknesses of the player. Write a short paragraph analyzing your selected player, supported by the statistics from the answers to Questions 1–3. Math and Fantasy Football Fantasy football offers spectators an added dimension to football season with a competitive math-based game where the active components are real-life players in the current season. For clarity in this exercise, the actual fantasy football players will be denoted as FFP and actual professional team members will be denoted as players. While some fantasy football leagues have slightly different setups or scoring systems, most share some common elements. Often using a lottery system to determine who picks first, second, and so on, FFPs select 15 current players to comprise their personal fantasy football team. The players selected can be from any professional teams and a FFP can utilize any team recognized by their league. FFPs can elect to keep the same players on their team for the whole league play or trade for any player not selected by another FFP in their league. At the start of each week during football season, each fantasy football player selects their roster of actual players to comprise their roster of starting players. Typically, a starting roster consists of the following players: Number to Select Position Abbreviation Position Title 1 QB Quarterback 1 K Kicker 1 TE Tight End 2 RB Running Back 1 D/ST Defense 2 WR Wide receiver 1 RB or WR Flex As actual professional games are played, points are tallied based on your league’s scoring system. The points the team members on your starting roster make during the week are computed and whichever FFP has the highest score for the week wins that week. The FFPs with the best records of wins versus losses enters fantasy football playoffs to determine the ultimate league champion and collects the pot. The above overview of fantasy football describes the basic game play. The fun comes in understanding and analyzing the math behind the scoring. Talk to people you know who have played fantasy football in the past and interview them. Inquire about how they select players and how league(s) they have played in have computed the weekly scoring. Was a league manager recruited to manage the record keeping of scores? Was an online scoring system utilized or did the fantasy football use a streamlined or specialized scoring system? What strategies were used to select the team of 15 players and who would be on the weekly starting roster? Were any online resources utilized to aid in choosing which players to select? How was math utilized to select team members or who to place as a starter each week? Be as specific as possible, citing examples when available. Conduct an Internet research to find two different resources that a FFP could utilize when selecting players for their team or for their starting roster. Describe the math involved in the resources. Conduct an Internet research to learn more about the scoring utilized by fantasy football leagues. Write a short paragraph sharing how the use of math and knowledge of football statistics is used and how they are calculated to aid a FFP in selecting their team and starting roster. Math and Hockey Hockey is full of math from obvious components such as scoring and statistics to the shape of the rink and the angles involved in puck movement. Collegiate and professional hockey games are 60 minutes long and are divided into three periods of 20 (60/3) minutes each. At any one time, there are five players and one goalie on the ice for each team. If a player is called on a penalty and is placed in the penalty box, that team now has four players, which is 20% less players on the ice competing against five opponents. In some instances, a team may have two players in the penalty box at one time, resulting in three players, or 40% less players on the ice compared to a full team. Being one player down for a 2-minute penalty or potentially 5 minutes for a major penalty leads to an imbalance on the ice and calls for a quick change of offense and defense strategy. Rink Composition—North American An ice rink is comprised of various geometrical shapes, each with precise dimensions. Research the dimensions of a hockey rink and draw an accurate scale model on graph paper. Be sure to include the scale for your model and indicate all units. List the shapes and the numbers of each kind of hockey rink. You should find four different shapes, some with multiple dimensions. Describe the shape and dimensions of a hockey puck using geometric vocabulary. Statistics Using an Internet search, select two top hockey players from the same league to answer the following questions: Write a paragraph sharing a minimum of five statistics for each player you have selected. Describe how each of the statistics are calculated and what each statistic means. Write a paragraph comparing the two players and determine who you believe is the better player. Support your choice. Scoring The basics of scoring in ice hockey is simple, the team with the most goals is the winner. But, how to score the most goals involves much math! Research one of the following components involved in hockey puck movement on the ice and write a paragraph summarizing your findings. Be specific and detailed in your summary. Angles Velocity vectors Angle of incident and angle of return Speed and acceleration (player as well as puck movement) Math and Soccer As the world’s most popular sport, you’ll be excited to confirm that soccer is full of mathematics ranging from scoring and statistics to footwork, angles, and field shape. Soccer requires understanding of mathematical concepts and equations as well as skill, fitness, and game knowledge. One such example is angles, which you all will remember from your geometry class. While players are not carrying protractors and measuring angles during play, mental calculation of angles is a constant in any successful player’s thinking. A goalie is not physically able to cover the entire open net region and a player must calculate an angle to kick the ball consistent with the net opening while predicting the ability of the goalie to stop the ball from entering the net. Research angles as they apply to soccer play. Provide two examples of different plays indicating angle of a player’s body, angle of foot striking the ball, and angle to the net. Include relevant dimensions. Adding a diagram that may aid in clarity is an option. Draw a scale model of a soccer field including dimensions with labels. Using an Internet search, select two top soccer players from the same league to answer the following questions: Write a paragraph sharing a minimum of five statistics for each player you have selected. Describe how each of the statistics are calculated and what each statistic means. Write a paragraph comparing the two players and determine who you believe is the better player. Support your choice. Chapter Review Section 13.1 Math and Art Section 13.2 Math and the Environment Section 13.3 Math and Medicine Section 13.4 Math and Music Section 13.5 Math and Sports Chapter Test

Math


Co-Req Appendix: Integer Powers of 10 Nonnegative Integer Powers of 10 The phrase nonnegative integers refers to the set containing 0, 1, 2, 3, … and so on. In the expression 10 5 , 10 is called the base , and 5 is called the exponent , or power . The exponent 5 is telling us to multiply the base 10 by itself 5 times. So, 10 5 = 10 × 10 × 10 × 10 × 10 = 100,000 . By definition, any number raised to the 0 power is 1. So, 10 0 = 1 . In the following table, there are several nonnegative integer powers of 10 that have been written as a product. Notice that higher exponents result in larger products. What do you notice about the number of zeros in the resulting product? Exponential Form Product Number of Zeros in Product 10 0 1 0 10 1 10 1 10 2 10 × 10 = 100 2 10 3 10 × 10 × 10 = 1,000 3 10 4 10 × 10 × 10 × 10 = 10,000 4 10 5 10 × 10 × 10 × 10 × 10 = 100,000 5 That’s right! The number of zeros is the same as the power each time! Negative Integer Powers of 10 The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 10 is 1 10 . We use negative exponents to indicate a reciprocal. For example, 10 − 1 = 1 10 1 = 1 10 . Similarly, any expression with a negative exponent can be written with a positive exponent by taking the reciprocal. Several negative powers of 10 have been simplified in the table that follows. What do you notice about the number of zeros in the denominator (bottom) of each fraction? Exponential Form Equivalent Simplified Expression Number of Zeros in Denominator 10 − 1 1 10 1 = 1 10 1 10 − 2 1 10 2 = 1 10 × 10 = 1 100 2 10 − 3 1 10 3 = 1 10 × 10 × 10 = 1 1,000 3 10 − 4 1 10 4 = 1 10 × 10 × 10 × 10 = 1 10,000 4 That’s right! The number of zeros is the same as the positive version of the power each time. In the following table, we will write the same powers of 10 as decimals. Count the number of decimal places to the right of the decimal point. What do you notice? Exponential Form Equivalent Simplified Expression Number of Decimal Places to Right of Decimal 10 − 1 1 10 1 = 1 ÷ 10 = 0.1 1 10 − 2 1 10 2 = 1 ÷ 100 = 0.01 2 10 − 3 1 10 3 = 1 ÷ 1,000 = 0.001 3 10 − 4 1 10 4 = 1 ÷ 10,000 = 0.0001 4 That’s right! The number of decimal places to the right of the decimal point is the same as the positive version of the power each time. Multiplying Integers by Positive Powers of 10 Did you know that the distance from the sun to Earth is over 90 million miles? This value can be represented as 90,000,000, or we can write it as a product: 9 × 10,000,000 = 9 × 10 7 , which is actually a more compact way of writing 90 million. Notice that the power of 7 reflects the number of zeros in 90 million. Several products of positive integers and powers of 10 are given in the table that follows. Notice that the number of zeros is the same as the exponent except in one case. Exponential Form Product Number of Zeros in Product 5 × 10 1 5 × 10 = 50 1 13 × 10 2 13 × 100 = 1,300 2 8 × 10 3 8 × 1,000 = 8,000 3 15 × 10 4 15 × 10,000 = 150,000 4 70 × 10 5 70 × 100,000 = 7,000,000 6 The only case in which the number of zeros didn’t equal the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70 which also had a zero. So, the product had a zero from the 70 and 5 zeros from 10 5 for a total of 6 zeros in 7,000,000. Multiplying by Negative Powers of 10 As we have seen, negative powers of 10 are decimals. Several products of positive integers and powers of 10 are given in the table below. Notice that multiplying an integer by 10 raised to a negative integer power results in a smaller number than you started with. Also, the number of decimal places to the right of the decimal point is the same as the exponent except in one case. Exponential Form Product Number of Decimal Places to Right of Decimal 3 × 10 − 1 3 × 0.1 = 0.3 1 13 × 10 − 2 13 × 0.01 = 0.13 2 9 × 10 − 3 9 × 0.001 = 0.009 3 15 × 10 − 4 15 × 0.0001 = 0.0015 4 70 × 10 − 5 70 × 0.00001 = 0.00070 or 0.0007 5 ( 6 if we leave on the extra 0 ) The only case in which the number of decimal places to the right of the decimal point didn’t equal the positive version of the exponent was the last case. Why do you think that happened? That’s right! We multiplied by 70, which ended in zero. Moving the Decimal Place A helpful shortcut when multiplying a number by a power of 10 is to “move the decimal point.” The following table shows several powers of 10, both positive and negative. Compare the location of the decimal point in the original number to the location of the decimal point in the product. How has it changed? Exponential Form Product How the Position of the Decimal Point Changed 5 × 10 1 5. × 10 = 5 ∧ 0 ⌣ . = 50 1 place to the right 13 × 10 2 13. × 100 = 13 0 ⌣ ∧ 0 ⌣ . = 1,300 2 places to the right 8 × 10 3 8. × 1,000 = 8 ∧ 0 ⌣ 0 ⌣ 0 ⌣ . = 8,000 3 places to the right 15 × 10 4 15. × 10000 = 15 ∧ 0 ⌣ 0 ⌣ 0 ⌣ 0 ⌣ . = 150,000 4 places to the right 70 × 10 5 70. × 100,000 = 70 ∧ 0 ⌣ 0 ⌣ 0 ⌣ 0 ⌣ 0 ⌣ . = 7,000,000 5 places to the right 3 × 10 − 1 3. × 0.1 = . 3 ⌣ ∧ = 0.3 1 place to the left 13 × 10 − 2 13. × 0.01 = . 1 ⌣ 3 ⌣ ∧ = 0.13 2 places to the left 9 × 10 − 3 9. × 0.001 = . 0 ⌣ 0 ⌣ 9 ⌣ ∧ = 0.009 3 places to the left 15 × 10 − 4 15. × 0.0001 = . 0 ⌣ 0 ⌣ 1 ⌣ 5 ⌣ ∧ = 0.0015 4 places to the left 70 × 10 − 5 70 × 0.00001 = . 0 ⌣ 0 ⌣ 0 ⌣ 7 ⌣ 0 ⌣ ∧ = 0.0007 5 places to the left Notice that multiplying by a positive power of 10 moves the decimal point to the right, making the value larger, while multiplying by a negative power of 10 moves the decimal point to the left, making the value smaller. Also, the number of decimal places that the decimal point moves is exactly the positive version of the exponent.

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Preface Welcome to Introductory Business Statistics 2e , an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost. About OpenStax OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. Our mission is to make an amazing education accessible for all. Through our partnerships with philanthropic organizations and our alliance with other educational resource companies, we’re breaking down the most common barriers to learning. Because we believe that everyone should and can have access to knowledge. 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Coverage and scope Introductory Business Statistics 2e began as a customized version of OpenStax Introductory Statistics by Barbara Illowsky and Susan Dean. Statistics faculty at The University of Oklahoma have used the business statistics adaptation for several years, and the author has continually refined it based on student success and faculty feedback. The book is structured in a similar manner to most traditional statistics textbooks. The most significant topical changes occur in the latter chapters on regression analysis. Discrete probability density functions have been reordered to provide a logical progression from simple counting formulas to more complex continuous distributions. Many additional homework assignments have been added, as well as new, more mathematical examples. 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Introduction We encounter statistics in our daily lives more often than we probably realize and from many different sources, like the news. (credit: modification of work “New office” by Phil Whitehouse/ Flickr, CC BY 2.0) You are probably asking yourself the question, "When and where will I use statistics?" If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess." Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics. Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data can be distinguished from "bad."

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Definitions of Statistics, Probability, and Key Terms The science of statistics deals with the collection, analysis, interpretation, and presentation of data . We see and use data in our everyday lives. In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics . Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics . Statistical inference uses probability to determine how confident we can be that our conclusions are correct. Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life. Probability Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 1 2 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996 2000 is equal to 0.498 which is very close to 0.5, the expected probability. The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data. Key Terms In statistics, we generally want to study a population . You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample . The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population. Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16-ounce can contains 16 ounces of carbonated drink. From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter, in this case the mean. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter. One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample . We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter. A variable , or random variable, usually notated by capital letters such as X and Y , is a characteristic or measurement that can be determined for each member of a population. Variables may be numerical or categorical . Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense). Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value. Two words that come up often in statistics are mean and proportion . If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is 22 40 and the proportion of women students is 18 40 . Mean and proportion are discussed in more detail in later chapters. NOTE The words " mean " and " average " are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean," and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean." Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively. The population is all first year students attending ABC College this term. The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population). The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term: the population mean. The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample. The variable could be the amount of money spent (excluding books) by one first year student. Let X = the amount of money spent (excluding books) by one first year student attending ABC College. The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225. Try It Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively. Try It Solutions The population is all families with children attending Knoll Academy. The sample is a random selection of 100 families with children attending Knoll Academy. The parameter is the average (mean) amount of money spent on school uniforms by families with children at Knoll Academy. The statistic is the average (mean) amount of money spent on school uniforms by families in the sample. The variable is the amount of money spent by one family. Let X = the amount of money spent on school uniforms by one family with children attending Knoll Academy. The data are the dollar amounts spent by the families. Examples of the data are $65, $75, and $95. Determine what the key terms refer to in the following study. A study was conducted at a local college to analyze the average cumulative GPA’s of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below. 1. Population ____ 2. Statistic ____ 3. Parameter ____ 4. Sample ____ 5. Variable ____ 6. Data ____ all students who attended the college last year the cumulative GPA of one student who graduated from the college last year 3.65, 2.80, 1.50, 3.90 a group of students who graduated from the college last year, randomly selected the average cumulative GPA of students who graduated from the college last year all students who graduated from the college last year the average cumulative GPA of students in the study who graduated from the college last year 1. f 2. g 3. e 4. d 5. b 6. c Try It Determine what key terms refer to in the following study. A survey of athletes in a university was conducted to study the heights of athletes, in meters. Fill in the letter of the phrase that best describes each of the items below. Population ____ Statistics ____ Parameter ____ Sample ____ Variable ____ Data _____ the average height of athletes in the university the average height of athletes in the survey all athletes in the university all students in the university the height of one athlete a group of athletes randomly selected 1.82, 1.76, 1.69, 1.93 c b a f e g Determine what the key terms refer to in the following study. As part of a study designed to test the safety of electric automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies. Here is the criterion they used: Speed at which cars crashed Location of “drivers” (i.e. dummies) 35 miles/hour Front Seat Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars. The population is all cars containing dummies in the front seat. The sample is the 75 cars, selected by a simple random sample. The parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population. The statistic is proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample. The variable X = whether a dummy (if it had been a real person) would have suffered head injuries. The data are either: yes, had head injury, or no, did not. Try It Determine what the key terms refer to in the following study. A survey is conducted to check the time taken by a mobile for charging of battery from 50% to 100%. The criteria used to collect the data are: Wattage of charger used Type of mobile used 30 W Android We want to know the proportion of Android mobiles that are charged to 100% within 30 minutes. We start with a simple random sample of 200 mobiles. The population is all Android mobiles. The sample is the 200 mobiles, selected by a simple random sample. The parameter is the proportion of Android mobiles that are charged to 100% within 30 minutes in the population. The statistic is the proportion of Android mobiles that are charged to 100% within 30 minutes in the survey. The variable X = the number of Android mobiles that are charged to 100% within 30 minutes. The data are either: yes, charged to 100% within 30 minutes; or no, charged to 100% taking more than 30 minutes. Determine what the key terms refer to in the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit. The population is all medical doctors listed in the professional directory. The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population. The sample is the 500 doctors selected at random from the professional directory. The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample. The variable X = whether an individual doctor has been involved in a malpractice suit. The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not. Try It Determine what the key terms refer to in the following study. A study is to be conducted by a news agency to find the proportion of all truck drivers that have no points on their license. The agency selects 1000 truck drivers randomly from the directory of truck drivers and determines the number of truck drivers in the sample who have no points on their license. The population is all truck drivers present in the directory. The parameter is the proportion of truck drivers that have no points on their license in the population. The sample is the 1000 truck drivers selected randomly from the directory. The statistics is the proportion of truck drivers that have no points on their license in the sample. The variable X = the number of truck drivers that have no points on their license. The data are either: yes, they have no points on their license; or no, they have points on their license. Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study: You want to determine the average (mean) number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were 1, 0, 1, 3, and 4 glasses of milk. References The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013). Chapter Review The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. HOMEWORK For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. A fitness center is interested in the mean amount of time a client exercises in the center each week. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally. all children who take ski or snowboard lessons a group of these children the population mean age of children who take their first snowboard lesson the sample mean age of children who take their first snowboard lesson X = the age of one child who takes their first ski or snowboard lesson values for X , such as 3, 7, and so on A cardiologist is interested in the mean recovery period of their patients who have had heart attacks. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance. the clients of the insurance companies a group of the clients the mean health costs of the clients the mean health costs of the sample X = the health costs of one client values for X , such as 34, 9, 82, and so on A politician is interested in the proportion of voters in their district who think the politician is doing a good job. A marriage counselor is interested in the proportion of clients she counsels who stay married. all the clients of this counselor a group of clients of this marriage counselor the proportion of all their clients who stay married the proportion of the sample of the counselor’s clients who stay married X = whether a client stayed married yes, no Political pollsters may be interested in the proportion of people who will vote for a particular cause. A marketing company is interested in the proportion of people who will buy a particular product. all people (maybe in a certain geographic area, such as the United States) a group of the people the proportion of all people who will buy the product the proportion of the sample who will buy the product X = whether a person bought the product buy, not buy Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter. What is the population she is interested in? all Lake Tahoe Community College students all Lake Tahoe Community College English students all Lake Tahoe Community College students in the instructor's classes all Lake Tahoe Community College math students Consider the following: X = number of days a Lake Tahoe Community College math student is absent In this case, X is an example of a: variable. population. statistic. data. a The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a: parameter. data. statistic. variable. Average also called mean or arithmetic mean; a number that describes the central tendency of the data Categorical Variable variables that take on values that are names or labels Data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous . Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage) Mathematical Models a description of a phenomenon using mathematical concepts, such as equations, inequalities, distributions, etc. Numerical Variable variables that take on values that are indicated by numbers Parameter a number that is used to represent a population characteristic and that generally cannot be determined easily Population all individuals, objects, or measurements whose properties are being studied Probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur Proportion the number of successes divided by the total number in the sample Representative Sample a subset of the population that has the same characteristics as the population Sample a subset of the population studied Statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter. Survey a study in which data is collected as reported by individuals. Variable a characteristic of interest for each person or object in a population

Math