Probability and Statistics

Transcription

1 CHAPTER 9 Probability and Statistics THEME: Sports When the Cubs send their right-handed power hitter to the plate in the ninth inning, the Mets counter with a left-handed pitcher. Why? The manager of the Mets is simply playing the odds. Since its inception, baseball has kept meticulous records. By studying the data, managers, players, announcers, and fans use the concepts of probability and chance to make predictions. Today, players and managers use computers and calculators to record and analyze data. They know that the more effectively they use statistics and probability, the better they will do their jobs. Team Dietitians (page 39) plan meals and nutritional plans for athletes. They use their knowledge of nutrition to help team members maintain overall health, muscle health, and bone strength. Physical Therapists (page 4) determine exercises to strengthen muscles after injuries. Through specially designed exercise programs, they improve mobility, relieve pain, and reduce injuries. 380 mathmatters3.com/chapter_theme

2 Home Run Greats Then and Now Player Year Home Games At bats Batting Runs runs played average batted in Jim Thome Barry Bonds Alex Rodriguez Roger Maris Babe Ruth Home Runs Month by Month Player Year Apr May Jun Jul Aug Sept Oct Jim Thome Barry Bonds Alex Rodriguez Roger Maris Babe Ruth Data Activity: Home Run Greats 3. Thome: 3.64; Bonds: 3.; Rodriguez: 3.85; Maris: 3.66; Ruth: 3.58; Rodriguez s average is the greatest. Use the tables for Questions 4.. For each player, divide the at bats by the total of home runs and round to the nearest hundredth. Compare the unit rates. Which player had the best unit rate? Thome:.30; Rodriguez: 0.95; Bonds: 6.5; Maris: 9.67; Ruth: Bonds s rate is best.. What percent of Barry Bond s home runs were hit in August and September? about 33% 3. Find the average number of at bats per game for each player. Round to the nearest hundredth. Which player had the greatest average? 4. In 00, Bonds hit 49 singles. For what percent of his at bats did he reach first base by hitting a single? About 0% CHAPTER INVESTIGATION Baseball has been called America s Pastime. In recent years, the game s appeal has become international. Using statistics and probability, fans at home can predict what will happen when the bases are loaded in the bottom of the ninth inning. Working Together Gather baseball statistics and design your baseball simulation game using dice and percentile cards. Make a lineup and play a game. Decide whether the game s outcome matches your predictions. Use the Chapter Investigation icons to guide your group. Chapter 9 Probability and Statistics 38

3 CHAPTER 9 Are You Ready? Refresh Your Math Skills for Chapter 9 The skills on these two pages are ones you have already learned. Stretch your memory and complete the exercises. For additional practice on these and more prerequisite skills, see pages PERCENTS, DECIMALS, AND FRACTIONS Convert each fraction or decimal to a percent. Round to the nearest tenth if necessary % % % % % % 37.5% 4% 43.8% % % 64.6% 8.6% % % 46 Convert each percent to a decimal. Round to the nearest thousandth if necessary % % % % % % % % % % 0.7 Convert each percent to a fraction in lowest terms % % 8. 68% 9. 9% % % % % % % MEASURES OF CENTRAL TENDENCY Find the mean, median, mode and range of each set of data. Round to the nearest tenth if necessary. mean 75.; median mean 34.8; median ; mode 74; ; mode 34; range range mean 48.6; mean 6.5; median median 48; range 6.5; no mode; ; mode range mean 88.8; median ; mode 87, 9; mean 64.6; median 65; mode range 3 65; range 9 REDUCING FRACTIONS Determine the greatest common factor of the numerator and denominator. Divide both the numerator and denominator by the factor to write the fraction in lowest terms ; ; 3 4; ; 4; 7; ; ; 3 5; ; ; 5; Chapter 9 Probability and Statistics

4 HISTOGRAMS Draw a histogram for each frequency chart. 54. Interval Tally Frequency For 54 59, see additional answers. Interval Tally Frequency Interval Tally Frequency Interval Tally Frequency Interval Tally Frequency Interval Tally Frequency AREA Find the area of the shaded region of each figure. Use 3.4 for. Round to the nearest hundredth if necessary cm cm 7 cm 8 4 in. 4 in. 4. m 6 in. 48 in cm m Chapter 9 Are You Ready? 383

5 9- Review Percents and Probability Goals Find experimental and theoretical probabilities. Applications Sports, Card Games, Test Taking Results will vary. Probability theory shows that a group of 3 people has a Work with a partner. 50% chance that two members share a birthday; for a group of 40, the probability is about 90%.. Discuss: How large must a group of people be for there to be a 50% chance that two members of the group will share a birthday? Make a guess.. Take a survey of the students in one of your classes to look for common birthdays. What did you learn? BUILD UNDERSTANDING Recall that probability theory is the mathematics of chance. Probability is used to describe how likely it is that an event will occur. Probabilities are reported using fractions, decimals, and percents. The greater the probability of an event, the more likely the event is to occur. One way to find the likelihood of an event occurring in the real world is to conduct an experiment. In an experiment, you either take a measurement or make an observation. A probability determined by observation or measurement is called experimental probability. An outcome is the result of each trial of an experiment. The experimental probability of an event E is defined as follows. P(E) number of favorable observations of E total number of observations Example RECREATION Lions fans attending a recent 3-game series were asked whether the team should have a mascot. The table shows how many fans thought it should. According to these results, what is the probability that a Lions fan wants the team to have a mascot? Game Attendance In favor of mascot Friday Saturday Sunday Solution Use the experimental probability formula. number of favorable observations of E P(E) total number of observations P(fan favoring mascot) The probability that a fan interviewed at a Lions game will favor having a team mascot is Chapter 9 Probability and Statistics

6 The set of all possible outcomes of an experiment is called the sample space. Problem Solving Tip Example Solution SPORTS A baseball team has 8 pitchers and 3 catchers. The manager is choosing a pitcher-catcher combination. How many are possible? One way to show all possible outcomes is to use ordered pairs. For example, use the numbers 8 for pitchers and the letters A C for the catchers. (, A) (, A) (3, A) (4, A) (5, A) (6, A) (7, A) (8, A) (, B) (, B) (3, B) (4, B) (5, B) (6, B) (7, B) (8, B) (, C) (, C) (3, C) (4, C) (5, C) (6, C) (7, C) (8, C) There are 4 possible pitcher-catcher combinations. Probability is often expressed as a percent. A percent is a ratio of a number compared to 00. For example, 87% means : 00 or or A decimal can be converted to a percent by moving the decimal two places to the right. So, 0.4 can also be written as 40%. Another way to show the sample space is to use a tree diagram. You can use probability to predict the number of times an event will occur. Example 3 Solution SPORTS A softball player has had 4 hits in her first 60 times at bat. Predict her total number of hits in 330 at bats. First, use the outcomes that have already occurred to find the experimental probability of the player getting a hit each time at bat. P(hit) Then multiply that result by the total number of times at bat. 0.4(330) 3 Pitchers Catchers Outcomes A B C A B C A B C A B C A B C A B C A B C A B C (, A) (, B) (, C) (, A) (, B) (, C) (3, A) (3, B) (3, C) (4, A) (4, B) (4, C) (5, A) (5, B) (5, C) (6, A) (6, B) (6, C) (7, A) (7, B) (7, C) (8, A) (8, B) (8, C) Based on the player s first 60 times at bat, you can predict that she will get 3 hits in 330 at bats. As you increase the number of trials in a probability experiment, the experimental probability will probably get closer to the theoretical probability. For example, when tossing a fair coin, P(heads). The more often you toss the coin, the closer you will come to tossing an equal number of heads and tails. mathmatters3.com/extra_examples Lesson 9- Review Percents and Probability 385

7 The theoretical probability of an event, P(E), is the ratio of the number of favorable outcomes to the number of possible outcomes in the sample space. P(E) Example 4 Solution number of favorable outcomes number of possible outcomes Personal Tutor at mathmatters3.com CARD GAMES A card is picked at random from a standard deck of 5 cards. Find P(face card). There are 5 possible outcomes. There are favorable outcomes 4 jacks, 4 queens, and 4 kings. P(face card) Reading Math The odds in favor of an event are expressed as the ratio of the number of favorable outcomes to the number of unfavorable outcomes. For example, when you roll a die, the odds of getting a 4 are :5 because there is way the event can occur, and 5 ways it cannot. TRY THESE EXERCISES. SPORTS Of the first 500 fans to pass through the turnstiles at the stadium, 050 had reserved seats. What is the probability that the next person through will have a reserved seat? 0.7. WRITING MATH A person flips a penny, a nickel, and a dime. Each coin can land with heads up (H) or tails up (T). Make a tree diagram to show what different outcomes are possible. (H, H, H), (H, H, T), (H, T, H), (T, H, H), (H, T, T), (T, H, T), (T, T, H), (T, T, T) 3. You roll a die 60 times. Predict the number of times you will roll an even number greater than. 0 times 4. GAMES A spinner for a game is divided into ten equal sections numbered through 0. What is the probability of spinning 7 or higher? 5 PRACTICE EXERCISES For Extra Practice, see page 690. A die is rolled 00 times with the following results. Outcome Frequency What is the experimental probability of rolling each of the following results? a number less than What is the theoretical probability of rolling a number less than 4? CHAPTER INVESTIGATION Working with a partner, prepare to make your own baseball simulation game. To begin, gather batting statistics on at least 8 players. You may use statistics from the most recent baseball season or statistics from prior years. For each player, you will need to know the number At Bats (AB), Hits (H), Doubles (B), Triples (3B), Walks (BB) and Strike Outs (SO). This information is available in the newspaper, in sports magazines or on team websites. Answers will vary. 386 Chapter 9 Probability and Statistics

8 List all the elements of the sample space for each of the following experiments. 0. You flip a dime and a quarter. (H, T), (H, H), (T, H), (T, T). You spin each of these spinners once. A (A, ), (A, ), (A, 3), (B, ), (B, ), (B, 3), (C, ), (C, ), (C, 3) C B 3 Find the probability of each of the following:. CARD GAMES drawing a black jack from a standard deck of cards 6 3. rolling a die and getting a multiple of EDUCATION guessing correctly on one true-false question on a test EDUCATION guessing incorrectly on a multiple-choice question with four choices reaching into a drawer without looking and taking out a pair of black socks, when the drawer contains 3 pairs of black socks, pairs of white socks, pair of red socks, and pairs of blue socks SPORTS A basketball player has made 96 free-throws in his last 8 attempts. What is the probability he will be successful in his next attempt? How many successful free-throws do you predict this player will make in 500 attempts? 0.75; WRITING MATH You want to predict how many students in your school are right-handed. Describe how you would do it. Sample answer: Find the ratio of the number of righthanded students in your class to the number of students in your class. Then multiply by the total number of 9. WEATHER The weather forecaster predicts a 5% chance of rain in your area students in your tomorrow. Describe what this forecast means. Of past days when weather school. conditions were similar to those predicted for tomorrow, it rained 5% of the time. 0. TRANSPORTATION A bus breaks down while traveling between two cities that are 00 mi apart. What is the probability the breakdown is within 5 mi of either city? 0.5 EXTENDED PRACTICE EXERCISES Write whether each of the following probabilities can be determined experimentally or theoretically.. The probability that Player A will win when Player A plays Player B in tennis. experimentally. The probability that a person will win the state s lottery. 3. The probability that a family with 4 children has all boys. 4. The probability that it will snow on January 9. experimentally theoretically theoretically MIXED REVIEW EXERCISES Graph the image of triangle ABC with vertices at A(, ) B(4, ), and C(, 4), under each transformation from the original position. (Lesson 8-) See additional answers units up 6. reflected across the x-axis Graph the image of parallelogram PQRS with vertices at P(, ) Q(, 3), R(, 6), and S(, 4), under each transformation from the original position. (Lesson 8-) See additional answers units down 8. reflected across the x-axis mathmatters3.com/self_check_quiz Lesson 9- Review Percents and Probability 387

9 9- Problem Solving Skills: Simulations Problem Sometimes a probability problem is too difficult to solve theoretically or experimentally. One way to solve such a problem is to model it with a simulation to estimate the probability. Simulations often use random numbers; these can be readily generated and recorded by a computer. You can also find random numbers by rolling dice, flipping coins, using numbered slips of paper, or spinning a spinner. MARKETING Each box of Batter-Up Pancake Mix contains one of 5 different classic baseball cards. Assuming that the company has evenly distributed the cards among the boxes, what is the probability that you will find all 5 cards if you buy 0 boxes of Batter-Up? Solve the Problem Work with a partner. Use 5 slips of paper numbered 5; each slip represents a box of cereal. Place the slips in a paper bag. Then draw one slip of paper from the bag, record its number, and place it back in the bag. Repeat the process until you have drawn and recorded 0 slips of paper. If you have drawn each of the 5 numbers at least once, consider the outcome of your experiment to be successful. If you have not drawn every number, the outcome is unsuccessful. Repeat the experiment 50 times, recording all results in a table. Indicate which trials are successful. Then write a ratio comparing successful outcomes to the total number of outcomes. This ratio will be an estimate of the probability of getting every card in the set when you buy 0 boxes of cereal. Problem Solving Strategies Guess and check Look for a pattern Solve a simpler problem Make a table, chart or list Use a picture, diagram or model Act it out Work backwards Eliminate possibilities Use an equation or formula TRY THESE EXERCISES Interactive Lab mathmatters3.com. COMPUTER SCIENCE A computer generates a list of random -digit numbers. Zero cannot be the first digit. What is the probability that a randomly chosen number from the list contains the digit? :5 (8 of the 90 possible numbers). MARKETING A candy company has placed 6 different prizes in its boxes. The prizes are uniformly distributed among the boxes of candy, only one per box. Describe a simulation you could do to estimate the probability of getting all 6 prizes in a -pack of candy. Answers will vary. 3. WRITING MATH Describe a simulation you could do to find out how many cards you would expect to have to draw from a standard deck to get two kings. Answers will vary. 4. TEST TAKING Suppose you are going to take a 0-question true-false test on the evolution of idiomatic phrases in Sri Lanka. You will need to guess each time, and you want to find out your chances of scoring 65% correct or better. Design a simulation to determine your chances. Hint: Use coin flipping. Answers will vary. 388 Chapter 9 Probability and Statistics

10 PRACTICE EXERCISES 5. A family wants to have 3 children. Do the following simulation to determine the probability that, if they do have 3 children, all 3 will be the same gender. a. Use 3 coins. Let heads girl and tails boy. Toss the coins and record the results. Repeat the coin tosses until you have recorded 50 sets of 3 tosses each. b. Count the successful outcomes those with either 3 heads or 3 tails. c. Write a ratio comparing successful outcomes with total outcomes. What is your experimental probability of having 3 children, all of whom are of the same gender? Answers will vary. 6. SPORTS One baseball player always arrives at the stadium between 5:30 P.M. and 6:30 P.M. for a night game. If batting practice always starts between 6:00 P.M. and 7:00 P.M., what is the probability on any given night that this player will arrive before batting practice begins? Design and do a simulation to find out. Answers will vary. 7. PROGRAMMING A pitcher throws strikes about 60% of the time. If he throws 80 pitches, how many might be strikes? The following computer programming statements can be used to simulate 80 pitches. S 0 The experiment begins with no successes. FOR I TO pitches 3 X RND() Generate a random decimal. 4 IF X.6 THEN S S If the decimal 0.6, increase S by. 5 NEXT I Simulate the next pitch. 6 PRINT S Total number of strikes. 7 END Use the program to simulate the problem. Then describe how you would adjust the program for a pitcher who throws strikes 40% of the time. Change line 4 to: IF X.4 THEN S S. 8. TALK ABOUT IT Petra is designing a simulation to determine the chance of guessing the correct answer on a multiple-choice test. Each item on the test has three choices. Petra plans to roll a 6-sided die to simulate random guesses. A roll of or will indicate choice A; a roll of 3 or 4, choice B; and a roll of 5 or 6, choice C. Will Petra s simulation work? Explain. Yes; the six-sided die can be used as long as each of the three choices is assigned an equal number of outcomes from the die roll. MIXED REVIEW EXERCISES Find the slope of each line using the given information. (Lesson 6-) 9. A(, ), B(5, 3) 4 0. C(7, ), D(3, ). E(, 8), F(3, 4) x y y x 8 4. x 4y 5. G(3, 5), H(3, 9) undefined 6. I(3, 5), J(3, 5) K(, ), L(8, ) 3 0 Solve each proportion. (Lesson 7-) 8. 5 x 5 9 x x x x x x x x x Five-step Plan Read Plan 3 Solve 4 Answer 5 Check Lesson 9- Problem Solving Skills: Simulations 389 3

11 Review and Practice Your Skills PRACTICE LESSON 9- A card is picked at random from a standard deck of 5 cards. Find each theoretical probability.. P(heart). P( jack) 3. P(red card ) 4. P(black ) P( or 3) 6. P(7 of hearts) 6 7. P(3 card 8) 8. P(king of clubs) You flip a coin four times. Find each theoretical probability P(no tails) 0. P(exactly one head). P( tails, heads) 8. P(4 tails) P( heads) 5 4. P(0 or head) 5 5. P(3 tails) 6. P(,, 3 or 4 heads) You roll a pair of dice and calculate the sum. Find each theoretical probability P(7) 8. P() 9. P(even) 0. P() P(). P(4 or 5) 7 3. P(6) P( 6) P(0,, or ) 6. P(9) 7. P( ) 8. P() A basketball player has made 48 free throws in her last 7 attempts. What is the probability she will be successful on her next attempt? How many free throws do you predict she will make in 600 attempts?, A car breaks down while traveling between two cities that are 300 mi apart. 3 What is the probability that the breakdown is within 8 mi of either city? or 0. 5 PRACTICE LESSON 9-3. A computer generates a list of random 3-digit numbers. Zero cannot be the first digit. What probability would you expect for a number in the list to 7 contain the digit 4? 5 3. Each box of Toasted Crunchies cereal contains a single prize. There are 4 different prizes uniformly distributed among all boxes of cereal at the production facility. Describe a simulation you could do to estimate the probability of getting all 4 prizes if you buy 0 boxes of Toasted Crunchies? Answers will vary. 33. Perform and document a simulation to determine the probability that a family having children will have boy and girl. Answers will vary. 34. Perform and document a simulation to find the chances of scoring 50% or higher on a 5 question multiple choice test. Each question has four choices, and you guess on each question. Answers will vary. 35. Agnes walks her dog each night outside the grounds of Tellco Corporation between 7:30 and 8:00 P.M. First shift workers leave the grounds between 7:30 and 8:30 P.M. each night. Describe a simulation using two spinners that you could use to calculate the probability of Agnes seeing first-shift workers leaving the grounds during her walk. Answers will vary. 36. Describe a simulation you could do to find out how many cards you would have to draw from a standard deck to get a pair of hearts. Answers will vary. 390 Chapter 9 Probability and Statistics

12 PRACTICE LESSON 9- LESSON 9- (, H), (4, H), (, T), (4, T), (, H), (5, H), (, T), (5, T), (3, H), (6, H), (3, T), (6, T) List all the elements of the sample space for each experiment. (Lesson 9-) 37. Tossing a quarter and a nickel. 38. Spinning each of these two spinners: (H, H), (T, H), (H, T), (T, T) 39. Rolling a die and tossing a dime. 40. A computer randomly generates a list of -digit numbers. Zero cannot be the first digit. What is the probability that the next number generated is a multiple of 3? (Lesson 9-) A C B D {(A, ), (B, ), (C, ), (D, ), (A, ), (B, ), (C, ), (D, ), (A, 3), (B, 3), (C, 3), (D, 3), (A, 4), (B, 4), (C, 4), (D, 4)} Workplace Knowhow Career Dietitian Dietetics has applications in many different career fields. Clinical dietitians plan meals and nutritional plans for groups such as schools and hospitals. Community dietitians inform the public on nutritional habits to prevent disease and promote healthy lifestyles. Consultant dietitians provide advice in the areas of sanitation and safety procedures. In the sports world, nutrition is important for maintaining muscle health and bone strength. As the dietitian for a baseball team, you need to determine whether the team members are getting enough calcium in their diets. To find out, you separate the players into three categories: infielders, outfielders, and pitchers. During a buffet, you chart the food selections of the players.. Find the average amount of calcium consumed by players in each group. A. Infielders B. Outfielders C. Pitchers B 300 mg RF 3 mg P 33 mg C 0 mg CF 97 mg P mg B, 705 mg; C, 57 B 86 mg LF 6 mg P3 84 mg mg; B, 437. mg; 3B 6 mg mg P4 58 mg 3B, mg; SS, SS 0 mg.75 mg 39.7 mg; RF, 04.8 mg. Your research shows that during lunch, calcium intake is mg; CF, of the amount mg; LF, consumed during the buffet. Breakfast amounts are 3 5 of the buffet amount mg; P, mg; P, How many milligrams of calcium is each player getting per day? 498. mg; P3, If 450 mg of calcium is recommended per player per day, which players are mg; P4, mg consuming too little calcium? B, SS, RF, P3 mathmatters3.com/mathworks Chapter 9 Review and Practice Your Skills 39

13 9-3 Compound Events Goals Find probabilities of compound events. Applications Sports, Games, Business Play this game with a partner. Use a pair of 6-sided dice.. Player A rolls first. If Player A rolls a 7, Player B wins the game. If not, Player B rolls.. If Player B rolls a 7 or an, Player A wins. If not, it is Player A s turn. Continue taking turns until there is a winner. 3. Play the game several times. Do you think one player has a better chance of winning? Answers will vary. BUILD UNDERSTANDING A compound event consists of two or more simple events. Compound events may involve finding the probability of one event and another event occurring. Or, the probability of one event or another event occurring. For example, when rolling a die, rolling a number that is even (event A) and greater than (event B) is written P(A and B). Rolling a number that is even or greater than is written P(A or B). If two events cannot occur at the same time, they are called mutually exclusive events. For example, it is impossible to draw from a standard deck of playing cards a card that is both a heart and a club. When two events A and B are mutually exclusive, the probability of the compound event A or B can be found using the formula P(A or B) P(A) P(B). Example SPORTS A standard deck of playing cards is used to simulate a baseball game. During the game, players draw a card at random. Spade number cards greater than 4 represent doubles. Home runs are represented by either a 3 or a queen. a. Find P(spade and a number card greater than 4). b. Find P(3 or queen). 39 Solution There are 5 possible outcomes. a. There are 6 outcomes in which spades are greater than 4: 5, 6, 7, 8, 9, and 0 of spades. 6 3 So, P(spade and number greater than 4) or The probability that the card will be a spade and a number greater than 4 is. 6 Chapter 9 Probability and Statistics

14 b. A card cannot be both a 3 and a queen at the same time, so the events are mutually exclusive. P(3 or queen) P(3) P(queen) or 5 3 The probability that the card will be a 3 or a queen is. 3 Events that can happen at the same time are not mutually exclusive. Example Solution GAMES You draw a card at random from a standard deck of playing cards. Find the probability that the card is a club or a jack. These are not mutually exclusive events, because a card can be both a club and a jack. There are 3 clubs, so P(club) There are 4 jacks, so P(jack). 5 However, club is a jack. P(club and jack). 5 P(club or jack) The probability that the card is a club or a jack is. 3 Example illustrates that when two events A and B are not mutually exclusive, the probability of A or B can be found using the formula P(A or B) P(A) P(B) P(A and B) Suppose you know the probability of event A. The set of outcomes in the sample space, but not in A, is called the complement of the event. Example 3 Solution P(not A) P(A) You select a marble from this jar without looking. You know 5 of the marbles are red and are blue. What 5 is the probability you will select neither a red nor a blue marble? Find the probability of selecting red or blue. P(red or blue) Personal Tutor at mathmatters3.com Check Understanding Classify each of the following pairs of events as mutually exclusive or not mutually exclusive.. drawing the 4 of clubs; drawing the 4 of spades. rolling two dice that show a sum of 8; rolling two dice and getting different numbers 3. tossing two coins and getting two tails; tossing two coins and getting two heads. mutually exclusive. not mutually exclusive 3. mutually exclusive mathmatters3.com/extra_examples Lesson 9-3 Compound Events 393

15 Find the probability of not selecting red or blue. P(neither red nor blue) P(red or blue) The probability that you will not select a red or blue marble is 3 5. TRY THESE EXERCISES. A die is rolled. Find the probability of rolling a or a. 3. Two coins are tossed. Find the probability that the coins show two heads or one tail and one head A card is drawn at random from a standard deck of playing cards. Find the probability that it is a 7 or a black card Two dice are rolled. Find the probability that the sum of the numbers is not greater than Each student in your class writes his or her full name on a piece of paper. The pieces are put in a box and one is chosen without looking. What is the probability that your name will not be chosen? number of pieces of paper number of pieces of paper PRACTICE EXERCISES For Extra Practice, see page 690. GAMES A player rolls two 6-sided dice. 6. List the sample space for the rolls. 7. Find the probability that the sum of the numbers rolled is odd and greater than Find the probability that the sum of the numbers rolled is either 8 or Find P(not even). 0. Find P(neither odd nor sum of 6) See additional answers. SPORTS Suppose you are on a team in the midst of a losing streak. The coach decides to shake up the line-up. He chooses the batting order by putting nine players names into a hat and pulling them out one by one. The player whose name is drawn first bats first, the second bats second, and so on.. What is the probability you will bat second or fourth? 9. What is the probability you will bat fifth, or in the first third of the batting order? What is the probability you will bat first, or in the first third of the batting order? 3 4. What is the probability you will bat in the last third of the batting order, or in an odd-numbered position? 3 5. BUSINESS Ms. Garrett plans to select a worker at random for a special training seminar. If there are 4 workers in sales, 6 in accounting and 5 in personnel, what is the probability that the worker will be from either sales or accounting? Chapter 9 Probability and Statistics

16 GAMES You spin the spinner shown. Find each probability. 6. spinning a or an odd number spinning an odd or an even number 8. spinning a multiple of or a multiple of 3 CARD GAMES To begin a game, the dealer draws a card at random from a standard deck of playing cards. 9. Find the probability that the card is a, a 5, or a face card. 0. Find the probability the card is a 7, an 8, or a red card. 5 6 A card is drawn at random from a standard deck of playing cards. For each event, estimate whether the probability is closer to,, or 0.. P(red or face card). P(, 3, or 4) 0 3. P(black and face card) 0 4. P(black, heart, or 7) PHOTOGRAPHY A team photo album contains photos of the players by themselves, the coaching staff by themselves, and the players and the coaches together. The players are in 5 of the photos and the coaches are in of the photos. In 6 photos, the players and coaches appear together. 5. How many photos are in the album? 6. If you open the album at random to one of the team photos, what is the probability that the photo shows only coaches? 7 EXTENDED PRACTICE EXERCISES 7. WRITING MATH Suppose you roll a pair of dice. Why are rolling a multiple of 6 and a multiple of 4 not mutually exclusive events? Sample answer: is a multiple of both numbers. 8. A pair of dice is rolled. What is the probability that the sum of the numbers is neither 5 nor a multiple of? SPORTS Batting average is found by dividing hits by at-bats. In 94, Ted Williams batted over.400 when he got 85 hits in 456 official at-bats for an average of.406. Suppose Ted Williams had more at-bat in 94. Based on his performance all season, what would you estimate the probability of his not getting a hit that time? CHAPTER INVESTIGATION For each player you selected, the number of hits (H) is equal to the sum of the singles (S), doubles (B), triples (3B), and home runs (HR). Since the number of singles is not usually reported as a separate statistic, calculate the number of singles (S) for each player using the formula: H (B 3B HR) S. You may want to use a computer spreadsheet. Answers will vary. MIXED REVIEW EXERCISES Evaluate each expression when a 5 and b 4. (Lesson -8) 3. a b a b ab a 3 b 35. a(a b ) ab ab a 3 b (a b ) 39. (a )(b )(a ) 0, a b(ab) (b ) 6 4. (b) mathmatters3.com/self_check_quiz Lesson 9-3 Compound Events 395

17 9-4 Independent and Dependent Events Goals Find the probability of dependent and independent events. Applications Sports, Surveys, Scheduling Work with a partner. You will need a six-sided die and a coin.. Suppose one person rolls the die while the other tosses or flips the coin. What do you think the probability is of rolling a 3 and landing the coin heads up? Record your prediction.. Check your prediction by rolling the die and tossing the coin until you get both of these outcomes. 3. Share the results of your experiment with other groups. BUILD UNDERSTANDING When the outcome of one event does not affect the outcome of another event, the events are said to be independent. To find the probability of both events occurring, multiply the probabilities of each event. If A and B are independent events, then P(A and B) P(A) P(B) To emphasize that A and B do not characterize a single event, sometimes P(A and B) is written P(A, then B). Example Solution A bag contains 3 white softballs, yellow softballs, 3 green softballs, and 4 red softballs. You reach into the bag without looking and take out a ball. You replace it and then take out another ball at random. Find the probability that the first ball is red and the second ball is white. Because the first ball is replaced before the second is taken, the sample space of balls is the same for each event. The two events are independent. Multiply to find the probability that both will occur. P(red, then white) P(red) P(white) number of red balls total number of balls 3 4 The probability of picking red, then white, is. number of white balls total number of balls 396 Chapter 9 Probability and Statistics

18 When the outcome of one event is affected by the outcome of another, the events are said to be dependent. Example SPORTS Six teams the Panthers, Tigers, Lions, Bears, Cheetahs, and Elephants are in the lottery round for this year s draft picks. The name of each team is written on a card and placed in a box. To determine who gets the first lottery pick, one card will be drawn at random and not replaced. Then a second card will be drawn at random to determine the second pick. What is the probability that the Bears get the first draft choice and the Lions get the second draft choice? Check Understanding Are these events independent or dependent?. tossing a coin twice. picking two marbles from a bag without replacing the first one 3. choosing a captain and then choosing a cocaptain 4. rolling three dice Solution Because the first card is not replaced, the sample space for the second drawing has been changed. The second event is dependent on the first event. Probability of first event number of Bears cards P(Bears) total number of cards 6 Probability of second event P(Lions, after Bears) Multiply the probabilities. P(Lions, after Bears) 6 5 number of Lions cards new total number of cards The probability of drawing the Bears first and the Lions second is independent. dependent 3. dependent 4. independent Example 3 Personal Tutor at mathmatters3.com A bag contains 3 green marbles, red marbles, 4 yellow marbles, and black marble. Two are taken at random from the bag without replacement. Find P(green, then green). Solution number of green marbles 3 P(first green marble) total number of marbles 0 number of green marbles 3 P(second green marble) total number of marbles 0 9 Multiply the probabilities. 3 P(green, then green) The probability of picking green, then green, is. 5 mathmatters3.com/extra_examples Lesson 9-4 Independent and Dependent Events 397

19 TRY THESE EXERCISES A group of numbered cards contains three 3s, four 4s, and five 5s. Cards are picked at random, one at a time, and then replaced. Find each probability. 5. P(3, then 5). P(4, then odd number) 3. P(even, then not 4) SPORTS A baseball team has 0 pitchers, 3 catchers, 5 outfielders, and 7 infielders on its roster. Two players from this team will be chosen at random to represent the league in a tour of Japan. Find each probability. 4. P(pitcher, then catcher) 7 5. P(outfielder, then infielder) 0 0 SURVEYS A newspaper survey asked 00 men and 00 women whether they planned to vote for a proposed tax increase. Twenty men and 40 women said they are in support of the increase. A person from the survey is chosen at random. Find each probability What is the probability that the person chosen is in support of the tax increase? 0 7. What is the probability that the person is a woman in support of the increase? 5 8. What is the probability that the person is a man who is against the increase? 5 PRACTICE EXERCISES For Extra Practice, see page A box contains 3 red counters, 4 yellow counters, green counters, and blue counter. Counters are taken at random from the box, one at a time, and then replaced. Find each probability P(red, then yellow) 0. P(blue, then green). P(red, then not red) A drawer contains pairs of black socks, 3 pairs of brown socks, a pair of beige socks, and 6 pairs of white socks. One sock at a time is taken at random from the drawer and not replaced. Find each probability.. P(black, then black) 3. P(white, then black) 4. P(beige, then white) A billboard says EAT HERE NOW. Two letters fall off, one after the other. 5. What is the probability that both letters are vowels? 9 6. What is the probability that the first letter is an E, and the second letter is not 7 an E? You are given one ticket each to two soccer games at a stadium with 48,000 seats. What is the probability you will sit in Section D in the first game, and then Section A in the second game, if Section D has 4000 seats and Section A has 3000 seats? 9 SCHEDULING Liu and Michi plan to sign up for a drawing class next term. Drawing is offered during the first 4 periods of the day, and students are assigned randomly to classes. 8. What is the probability that Liu and Michi will have drawing together? 4 9. What is the probability that both students will have drawing during first period? 6 Chapter 9 Probability and Statistics Maracana Stadium, Brazil

20 Both spinners are spun. Find each probability. 0. P(red, red) 8. P(red, yellow) 6 7. P(green, not green) 4 8 A golf bag pocket contains 4 yellow golf balls, 3 white balls, green ball, and 4 red balls. You pull out one ball at a time, without replacing it. Find each probability. 3. P(red, then white, then yellow) 4. P(green, then red, then white) WRITING MATH Explain the difference between events that are mutually exclusive and those that are independent. Mutually exclusive events cannot occur at the same time. Independent events can occur at the same time, but neither event affects the other. 0 EXTENDED PRACTICE EXERCISES HISTORY Suppose that it is 944, and the Homestead Grays of the Negro National League are playing the Birmingham Black Barons of the Negro American League in a best two out of three series. Then suppose the Grays are the favored team, and the probability they will win any individual game is What is the probability that the Black Barons win a game? 7. What is the probability that the Grays win in two straight games? 8. What is the probability that the series goes for three games? 9. What is the probability that the Homestead Grays win the series? 30. DATA FILE Use the data on baseball on page 65. Suppose Davis had one more official at-bat in the 943 season and Wagner had one more official at-bat in the 948 season. What is the probability that both would have gone hitless? about CHAPTER INVESTIGATION When a baseball player comes to bat, the player can get a hit either a single, a double, a triple or a home run or the player can walk, strike out, or make an out some other way. For each player you have chosen, find the probability expressed as a percent that each event will occur. For example, to find the probability that a player will walk, divide the number of walks (BB) by the number of at bats (AB) and convert the decimal to a percent. Use a spreadsheet or calculator MIXED REVIEW EXERCISES Copy quadrilateral ABCD. Then draw its dilation image. (Lesson 8-3) See additional answers. 3. The center of dilation is the origin and the scale factor is The center of dilation is the origin and the scale factor is The center of dilation is point A and the scale factor is.5. y 4 D A C B 4 x mathmatters3.com/self_check_quiz Lesson 9-4 Independent and Dependent Events 399

21 Review and Practice Your Skills PRACTICE LESSON 9-3 A card is picked at random from a standard deck of 5 cards. Find each probability. 3. P(heart and face card). P(jack or queen) 3. P(red or black card) P(black two) 5. P( or 3) 6. P(7 of hearts) P( card 5) 8. P(king of clubs) 9. P(club and (ten or king)) 3 5 You flip a coin four times. Find each theoretical probability P(exactly one head). P( tails, heads) 3. P(3 or 4 tails) P(more than heads) 4. P(0 or head) 5. P(3 tails) You roll a pair of dice. Find each theoretical probability. 6. P(sum 7) 7. P(sum ) 8. P(both even) 6 9. P( is rolled) 8 0. P(4 or 5 is rolled) 5 4. P(sum ) 36. P(sum is odd) P(sum 6) 4. P(sum 0,, or ) 5 5. P(sum is even and 7) 8 6. P(sum is odd and ) 7. P(values are equal) A spinner has 0 equal sectors numbered 0. Are spinning a multiple of 4 and multiple of 9 mutually exclusive events? Explain. No; 36 is a multiple of both. 9. You spin a spinner with 8 equal sectors, numbered 8. What is the probability of spinning a number that is neither odd nor greater than 6? 3 8 PRACTICE LESSON A drawer contains 7 red shirts, 5 blue shirts, and 4 white shirts. One shirt at a time is taken at random from the drawer and not replaced. Find each probability P(red, then blue) 3. P(red, then not white) 3. P(white, then blue) P(both white) 34. P(not blue, then not red) P(both not blue) A box contains 4 red cards, 5 black cards, 0 green cards, and blue cards. Cards are taken at random from the box, one at a time, and then replaced. Find each probability P(red, then red) 37. P(red, then green, then blue) P(not red, then green) P(black, then black, then not green) P(not green in each of three draws) P(black, then not blue) P(red, then red, then red, then red) P(blue, then blue, then black) 9 4, You are given one ticket each to two hockey games in an arena with 8,000 seats. What is the probability that you will sit in Section B in the first game, and then Section C in the second game, if Section B has 4500 seats and Section C has 3000 seats? 4 Chapter 9 Probability and Statistics

22 PRACTICE LESSON 9- LESSON 9-4 List all the elements of the sample space for each of the following experiments. (Lesson 9-) 45. tossing a quarter and a nickel {HH, HT, TH, TT} 46. choosing a month of the year {,, 3, 4, 5, 6, 7, 8, 9, 0,, } 47. choosing the day of the week 48. choosing a letter of the alphabet {Sun., Mon., Tues., Wed., Thurs., Fri., Sat.} {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z} 49. Conduct and document a simulation using a six-question multiple choice test. If each question has three choices for answers, and you guess on each question, what are your chances of getting 3 or more questions correct? (Lesson 9-) Answers will vary. Three coins are tossed. Find each probability. (Lesson 9-3) 50. P(at least one heads) 7 5. P(no tails) 5. P(0 or tails) P(two tails) P(all three coins the same) 55. P( or heads) A box contains 00 cards, numbered from 00. Cards are taken at random from the box, one at a time, and not replaced. Find each probability. (Lesson 9-4) 56. P(even number, then odd number) P(multiple of 3, then 50) P(45, then 45) P(99, then 00) P(number 40, then number 80) 6. P(prime number, then prime number) Mid-Chapter Quiz. How many outcomes are there for an outfit chosen from three pairs of pants and five shirts? (Lesson 9-) 5. A basketball player has made 60 out of 5 attempts. How many shots is he likely to make in 500 attempts? (Lesson 9-) A family has five children. What is P(three boys and two girls)? (Lesson 9-) 6 A card is picked at random from a standard deck of 5 cards. (Lesson 9-3) 4. Find P(heart and less than 5) 5. Find P(heart or less than 5) Find P(7 or king) 7. Find P(neither 5 nor diamond) Find P(neither 3 nor red) 6 3 A bag contains 7 green marbles, 3 red marbles, and 5 blue marbles. A marble is picked and replaced. Then another marble is picked. (Lesson 9-4) 7 9. Find P(green, then red) 0. Find P(two red marbles) From the same bag, a marble is picked and not replaced. Then another marble is picked. (Lesson 9-4). Find P(green, then red). Find P(two red marbles) Chapter 9 Review and Practice Your Skills 40

23 9-5 Permutations and Combinations Goals Find the number of permutations and combinations of a set. Applications Travel, Sports, Office Work Work with a partner to answer the questions. You have just applied for your first set of license plates. Each license plate has 3 different letters followed by 3 different digits. The letters and digits are assigned randomly by the Department of Motor Vehicles.. How many arrangements of 3 different letters are possible? 5,600. How many arrangements of 3 different digits are possible? Remember, to include 0 as a digit Suppose you had hoped that your plate would read ACE 3. What is the probability that you will receive this plate? How do you know? 5, ,3,000 BUILD UNDERSTANDING Thus far, you have used either a tree diagram or a set of ordered pairs to find the number of outcomes in a sample space. But sometimes the sample space is too large to use either of these methods. Another way to find the total number of outcomes is to use the fundamental counting principle. This principle can be used to calculate the number of ways two or more events can happen in succession. It states that, if an event A can occur in m ways and an event B can occur in n ways, then events A and B can happen in m n ways. Example Solution TRAVEL Suppose the Carthage College Women s Basketball team will travel on a road trip to Grand Rapids, Peoria, and Battle Creek. They can go from Kenosha to Grand Rapids by car, train, or bus, then from Grand Rapids to Peoria by bus, train, or plane, and from there to Battle Creek by car, bus, train, or plane. Finally, from Battle Creek, they can either take the bus or the train back to Kenosha. How many different routes are possible on this road trip? Use the fundamental counting principle. The 3 possible routes for the first leg of the trip are car, train, or bus. Then they have 3 possible routes for the second leg of the trip, 4 for the third leg of the trip, and for the return trip to Kenosha Seventy-two different routes are possible. 40 Chapter 9 Probability and Statistics

24 An arrangement of items in a particular order is called a permutation. For the four letters M, A, T, H, there are 4 different 4-letter permutations. MATH MAHT MHAT MHTA MTHA MTAH AMTH AMHT AHTM AHMT ATHM ATMH TAMH TAHM TMAH TMHA THAM THMA HAMT HATM HMAT HMTA HTAM HTMA You can use the fundamental counting principle to find the number of permutations of any group of items. For each arrangement of M, A, T, H, there are 4 choices for the first letter, 3 choices for the second, choices for the third, and choice for the fourth can be written in factorial notation as 4! In general, the number of permutations of n different items is written n! and read as n factorial. Example Solution In how many different ways can you arrange your math, science, social studies, language arts, and literature anthology books in a row on a shelf? There are five books. Find the number of permutations of five items. number of permutations of five items 5! or 0 There are 0 different ways to line up five books on a shelf. Sometimes you need only part of a set of items, such as selecting two of nine players. The number of permutations of n different items, taken r items at a time, with no repetitions, is written n P r. Use the formula below to find the number of permutations when only part of a set is used. n! np r (n r)! Example 3 SPORTS Eight teams enter a tournament. How many different arrangements of first-, second-, and third-place winners are possible? Check Understanding What is wrong with the notation 4 P 5? A group of 4 items cannot be taken 5 at a time. Solution n! Use the formula: n P r (n r)! 8! 8P 3 (8 3)! There are 336 ways for teams to finish first, second, and third. Cancel common factors to simplify the computation. mathmatters3.com/extra_examples Lesson 9-5 Permutations and Combinations 403

25 For each situation described in the preceding Examples, the order of the items in consideration is important. A set of items in no particular order is called a combination. The number of combinations of n different items, taken r items at a time, where 0 r n, is written n C r. You can use the formula below to find the number of combinations of a set of items. n! nc r (n r)!r! Example 4 Solution How many different four-person ensembles can be chosen from a pool of 0 musicians? There are 0 people from which to pick, 4 at a time. So, n 0 and r 4. Use the formula: n! nc r (n r)!r! 0! 0C 4 (0 4)!4! Personal Tutor at mathmatters3.com ( )(4 3 ) Cancel common factors to simplify the 4 computation. 0 There can be 0 different four-person ensembles. Math: Who, Where, When? Although several 6thand 7th-century mathematicians, notably Pascal and Fermat, investigated probability, Jacques Bernoulli is considered by some to be the founder of probability theory. His book, Ars Conjectandi, published in 73, is the first book devoted entirely to the subject of probability. This book contains a theory of combinations, essentially the same as we understand it today, as well as the first appearance, with today s meaning, of the word permutation. TRY THESE EXERCISES. SPORTS A manager is choosing her infield from among 4 third-base players, 3 shortstops, second-base players, and 5 first-base players. How many different ways can an infield be chosen? 0 Tell whether each question involves a permutation or a combination. Then solve.. In how many different ways can you arrange the letters a, c, e, g, i, k, and? permutation; How many different selections of three tapes can be made by a consumer choosing from among a collection of six tapes? combination; 0 4. In how many different ways can a starting lineup of 5 players be selected from a group of basketball players? combination; In how many different ways can 4 winners be chosen from 5 contestants? permutation; 3,760 PRACTICE EXERCISES For Extra Practice, see page OFFICE WORK A secretary has to create ten new customer files. In how many different orders can he do this? 0! or 3,68, HIRING Six applicants apply for two jobs. How many different outcomes are possible? Chapter 9 Probability and Statistics

26 8. Ralph is a tour guide. In how many ways can he choose 3 museums to visit from the 8 museums in a city? In how many ways can a disk jockey select 5 of the 0 top hits? 5, ERROR ALERT On a test, students must choose 3 out of 5 essay questions to answer. Dale calculates that there are 60 ways to do this. What has Dale done wrong? 0. Dale has found the number of. SPORTS Bill, Phil, and Jill are among players competing for 3 permutations instead spots on a table-tennis team. Every player has an equal chance of of combinations. In making the team. Find the probability that all three will make the this situation, order team. doesn t matter. There 0 are only 0. What is n, if n P 3 0? 6 combinations. 3. Suppose that license plates contain three different letters. What is the probability that Meg s plates will spell her name? 5, SPORTS How many ways can a batting order be made for 9 players if you know that one player has already been designated to bat first and another to bat fourth? 7! or WRITING MATH Find the number of permutations of the letters in the word shutout. Explain how you did it. 60; Sample answer: Divide 7! by!!!!! 6. Two students out of 8 will be chosen to speak at a school assembly. How many different outcomes are possible? 8 DATA FILE For Exercises 7 8, use the information about the All-American Girls Professional Baseball League on page Suppose you could interview all eight women who were batting champions of the All-American Girls Professional Baseball League to discover what playing in this league was like. How many orders for these interviews would be possible? 8! or 40,30 8. If you were to interview only four of the eight women, what is the probability that you would first interview a player from either Fort Wayne or Rockford? 4 EXTENDED PRACTICE EXERCISES 9. Compare the values of 8 C 5 and 8 C 3. What do you notice? 0. Find the values of 7 C 4 and 7 C 3. What do you notice?. What can you say about the sum of the number of items taken at one time for each combination shown in Exercises 9 and 0? They equal the total number of items.. CRITICAL THINKING Use what you have discovered to quickly find 67 C ,905 MIXED REVIEW EXERCISES Add. (Lesson 8-5) Find the measure of each angle. (Lesson 3-) They are the same. Both have values of 35. Technology Note Some calculators have a factorial key, marked. To find 7!, enter 7, then. On graphing calculators, you can choose the factorial function from a displayed mathematical menu. This menu may also include operations for finding permutations and combinations. 5. BZC 5 6. CZD AZC A Z D 4 mathmatters3.com/self_check_quiz Lesson 9-5 Permutations and Combinations B (4x 9) (3x 6) C

27 9-6 Scatter Plots and Box-and-Whisker Plots Goals Interpret and make scatter plots and box-andwhisker plots. Applications Manufacturing, Sales, Sports Work in groups of 3 4 students. Find out your classmates favorite music performer. Make a list of ten popular music groups or artists. Using a rating scale of 0, survey 5 students. Make a graph to display your findings. Compare findings and graphs with those of classmates. Answers will vary. BUILD UNDERSTANDING Data can be presented in many ways. Graphs are useful because they can help identify characteristics of data. Recall that a histogram shows frequencies of intervals of data. A stem-and-leaf plot shows all data ordered as in a frequency table, but also visually, as in a bar graph. It shows how data are clustered. A scatter plot is another type of visual display used to explore the relationship between two sets of data, represented by unconnected points on a grid. Example Solution MANUFACTURING The scatter plot shows the relationship between years of experience and hourly pay at one factory. a. Why are the scales different? b. What does each represent? c. Find the hourly pay of an employee with 8 years of experience. d. Describe the relationship between experience and pay? a. There are two different sets of data hourly pay and years of experience. b. Each shows the hourly pay given the years of experience. c. $0.50 d. Hourly pay usually increases with years of work experience. Hourly Pay in Dollars Factory Wages Years of Experience A pattern may emerge that shows a relationship between the two sets of data. If data clusters around a line of best fit, or trend line, from the bottom left upward to the top right of the graph, this shows a positive correlation between the sets of data. If the line slopes downward from left to right, it indicates a negative correlation between the data. 406 Chapter 9 Probability and Statistics

28 Example Solution Personal Tutor at mathmatters3.com SALES Use the scatter plot at the right for these questions. a. What can you say about the correlation between the age of a car and its resale value? b. Predict the resale value of an 8-year-old car with an original cost of $5,000. a. The trend line slopes downward from upper left to lower right, so there is a negative correlation between a car s age and its resale value. b. Extend the pattern. A reasonable assumption would be for the resale value to be about 30% of the original cost for an 8-year-old car. So, a car that cost $5,000 originally might sell for about $4500 after 8 years. Another way to display data is with a box-and-whisker plot, also known as box plot. This plot shows how data are dispersed around a median, but does not show each specific item in the data. By examining a box-and-whisker plot, you can tell if data are clustered closely together or spread far apart. A box-and-whisker plot shows both the median and the extremes of a set of data. It also shows the median of the lower half of the data, called the lower quartile, and the median of the upper half of the data, called the upper quartile. Both quartiles include the median if the data contain an odd number of items. Car Value as % of Sticker Price Car Resale Values Age of Car 9 0 Example 3 Solution SPORTS Joe DiMaggio played center field for the New York Yankees for 3 years. During each year of his career, he hit the following number of home runs: 9, 46, 3, 30, 3, 30,, 5, 0, 39, 4, 3, and. Make a box-and-whisker plot for this data. Write the data in numerical order. Find the least and greatest values, the median, the lower quartile, and the upper quartile. least value median greatest value lower quartile upper quartile Use points to mark the values below a number line. Draw a box that starts and stops at the lower and upper quartiles, and a vertical line at the point for the median. Then draw whiskers, or line segments, from each end of the box to the least and greatest values. Finally, give your graph a title. DiMaggio s Home Runs Check Understanding In a box-and-whisker plot, what percent of a set of data is represented by the box? By the whisker to the right of the box? 50%; 5% mathmatters3.com/extra_examples Lesson 9-6 Scatter Plots and Box-and-Whisker Plots 407

29 Box-and-whisker plots can be used to compare sets of data. Example 4 Use the box-and-whisker plots below to answer questions about the math test scores of two different classes. Ms. Cotter s Class Mr. Pascal s Class Test Scores From Two Classes Solution a. Which class had the higher median score? b. What was the lower quartile in Mr. Pascal s class? c. Which class had its scores grouped more closely around its median? d. For which class were the lowest scores clustered more closely? e. Which class, as a whole, scored better on the test? a. Ms. Cotter s b. about 7. c. Mr. Pascal s; the range of the middle 50% of the scores is about.8. The range for the middle 50% in Ms. Cotter s class is about 3.5. d. Ms. Cotter s; the range for the lowest quarter is about. In Mr. Pascal s class, it is about.. e. Ms. Cotter s Reading Math The abbreviations Q, Q, and Q 3 are sometimes used for the lower quartile, the median, and the upper quartile. The interquartile range is the difference between the values of the upper and lower quartiles. TRY THESE EXERCISES FITNESS Use the scatter plot at the right for Exercises 3.. What is the weight of the student who is 67 in. tall? about 30 lbs. Does the scatter plot show a positive or negative correlation? positive 3. Give an estimate of the height of a student who weighs 45 lb in. 4. SPORTS Make a box-and-whisker plot for the following data. Height in Inches Heights and Weights Weight in Pounds Check students plots; TOP PRICES OF TICKETS TO look for median; 37.5; SPORTING EVENTS (IN DOLLARS) upper quartile: 45; lower 45, 55, 40, 60, 5, 5, 35, 30, 0, 40 quartile: 5; extremes: 0 and Use the box-and-whisker plot you made in Exercise 4. Are the data clustered more closely above or below the median? above Technology Note Use a graphing calculator to make a box-andwhisker plot.. Enter the data as a list using the STAT feature.. Select the STATPLOT menu and choose Plot. Under Type, select the box-andwhisker plot diagram. 3. Adjust the window dimensions if necessary and press GRAPH. 4. Use the TRACE feature to find the median and upper and lower quartiles. 408 Chapter 9 Probability and Statistics

30 PRACTICE EXERCISES For Extra Practice, see page 69. SPORTS This table shows how many points a basketball player scored during his career. Use this information for Exercises Make a scatter plot. 7. What is the range of this player s scoring average? 8. Does your scatter plot show a positive correlation, a negative correlation, or no correlation? No correction SPORTS These box-and-whisker plots show batting averages for 3 baseball teams. Artichokes Check students drawings. Player Batting Averages 0 points Age Scoring Average Onions Meatballs Which team has the highest median batting average? 0. Which team has the smallest range of batting averages? Artichokes Onions. WRITING MATH Why is the right whisker for the Meatballs longer than the left whisker? There is a greater range of batting averages in the upper 5% of its players than in the lower 5%.. CHAPTER INVESTIGATION Create a 0-by-0 table for each player. Number both the columns and rows from to 0. The table represents all the possible outcomes for a player at bat. Using the percents you calculated, fill in the cells of the table with appropriate abbreviations. For example, in 998 Gary Sheffield hit a home run in 5% of his at bats. To create a table for Sheffield, you would write HR in any 5 cells. Answers will vary. EXTENDED PRACTICE EXERCISES Choose the graph you think works best to display the data described. Answers will vary. Sample answers are given. 3. MANUFACTURING To show the relationship between the percent of polyester in an article of clothing and the price of the article of clothing scatterplot 4. To show that the test scores in your class clustered around the middle-most score box-and-whisker-plot 5. WRITING MATH Is it possible for the mean of a set of data to fall outside the box part of a box-and-whisker plot? Explain. yes, when there are data values far from the median MIXED REVIEW EXERCISES Multiply. (Lesson 8-5) mathmatters3.com/self_check_quiz Lesson 9-6 Scatter Plots and Box-and-Whisker Plots 409

31 Review and Practice Your Skills PRACTICE LESSON 9-5 Evaluate.. 5 P P 7 40, P P P P P 7 7,97, P 6 60, C C C 0. 6 C C C C 5 6. C In how many ways can the positions of president, vice president, and secretary be chosen from a club containing 0 members? In how many ways can a committee of three people be chosen from a club containing 0 members? In how many ways can a volleyball coach choose 6 starters from a team of 4 players? In how many ways can a disc jockey play 3 of the top ten hits?. In how many ways can first-place and runner-up winners be chosen from 5 entrants in a contest? 0. In how many ways can the numbers,, 3, 4, and 5 be arranged in a 5-digit password? 0 3. What is n, if n P 7? 4. Find the values of 0 C 4 and 0 C 6. What do you notice? 0, 0; They are equal. PRACTICE LESSON 9-6 This table shows the appraised value of a house over time. Age (years) Value (thousands) Make a scatter plot of the data. 6. What is the range of appraised values? See additional answers. $54, Does your scatter plot show a positive correlation, a negative correlation, or no correlation? positive 8. Draw a box-and-whisker plot that has the following attributes: (Lesson 9-6) a. range of 87 d. upper quartile of 6 b. median value of 35 e. range of middle 50% of data of 38 c. low value of 07 See additional answers. 9. Draw a box-and-whisker plot that has the same value for its maximum value and its upper quartile. Define a collection of data points that would yield this type of plot. (Lesson 9-6) Answers will vary. 40 Chapter 9 Probability and Statistics

32 PRACTICE LESSON 9- LESSON 9-6 A spinner with 8 equal sectors labeled A through H is spun. Find each probability. (Lesson 9-) 30. P(spinning E) 3. P(spinning vowel) 3. P(spinning H) P(spinning a letter before F) P(spinning B or G) 35. P(spinning M) Describe a simulation you could do to find out how many cards you would expect to have to draw from a standard deck to get three clubs. (Lesson 9-) Answers will vary. For each situation, tell whether order does or does not matter. (Lesson 9-5) 37. You are selecting three-number combinations for school lockers. 38. You are selecting five books to check out from the library. 39. You are choosing the 9 starters on a baseball team. does does not does 40. You are choosing a 5-member committee from your leadership board. does not 8 0 Workplace Knowhow Career Physical Therapist Physical therapists work with people who have been injured. They improve mobility, relieve pain and prevent or limit permanent physical disabilities. To relieve pain and treat injuries, physical therapists use massages, electrical stimulation, hot and cold packs and traction. The sports world depends on physical therapists to help athletes who are injured during practices, exercise sessions or games. Some injuries require surgery, but many can be treated by rest followed by proper exercise. Physical therapists often travel with teams.. As a physical therapist, you have treated 36 injuries during the past year. Of the total number, 08 injuries were caused by an improper warm-up. What percent of the injuries resulted from an improper warm-up? about 46%. A team of 45 players suffered 5 ankle injuries during the season. If the squad is increased to 5 players, how many ankle injuries would you expect to see during a season? 7 or 8 ankle injuries 3. A physical therapist employs 4 kinds of massage, 3 kinds of baths, type of electrical stimulation, and 8 exercise programs. If each injury is treated with all four types of therapies, how many combinations of therapies does this therapist offer? You are examining the player files for 30 players who have been injured during the season. For this sport, out of 3 injuries are to the knee and out of of these cases require surgery. What is the probability that the first file you select will belong to a player requiring knee surgery? 6 mathmatters3.com/mathworks Chapter 9 Review and Practice Your Skills 4

33 9-7 Standard Deviation Goals Find the variance, standard deviation and z-scores for a set of data. Use standard deviation to interpret data. Applications Sports, Test-Taking, Education Work in groups of 4 5 students. Answers will vary.. Collect a set of data about students in class, such as heights, arm lengths, head circumference, lengths of thumbs and so on.. Study the data. Look for new ways to describe the data. Instead of focusing on central tendencies, study how the data are spread out, or dispersed. 3. Consider these questions: How much do individual values in your data differ from the greatest value? The least value? The mean, median or mode of the values? 4. Share your results with your classmates. BUILD UNDERSTANDING Statistics that show how data is spread out are called measures of dispersion. For example, you know that the range of a set of data is the difference between the largest and smallest item. Variance is another measure of dispersion. The variance of a set of numbers is the mean of the squared differences between each number in the set and the mean of all numbers in the set. For the set of numbers x, x,... x n, with a mean of m, use this formula. v Example Solution (x m) (x m)... (x n m) n SPORTS During a basketball tournament, the five starters for the Bulldogs made the following number of 3-pointers: Bowen, 3; White, 4; Fillmore, 5; Graham 6; and Bonilla, 7. Find the variance for the set of numbers.. Divide the sum of scores by 5 to find the mean, m. (m 5). Find the difference between each number and the mean. Then find the square of each difference. 3. Find the mean of all the squares in Step The variance is. The standard deviation, s, of a set of numbers is the square root of the variance. number x m (x m) () () Chapter 9 Probability and Statistics

34 Example Solution Find the standard deviation for the set of numbers in Example. Find the square root of the variance. Example 3 Solution.4 The standard deviation is.4 Personal Tutor at mathmatters3.com Test A Test B Molly took two tests. On which did she score better, compared with others in her class? Molly s score Mean score Standard deviation 8 0. Compare both of her scores with the mean. She scored 0 points higher than the mean on both tests.. Use the standard deviation. In Test A, Molly s score was 0, or.5 standard deviations above the mean score. 8 In Test B, it was 0, or standard deviations above the mean score. 0 Relative to her classmates, Molly scored better on Test A. The number of standard deviations between a score and the mean score is indicated by a z-score. Molly s z-score was.5 on Test A. A score below the mean would have a negative z-score. TRY THESE EXERCISES Compute the variance and standard deviation for each set of data.. 4, 4, 4, 4, 4. 7, 3, 5, 9, 3..7, 4.7, 6.7, 8.7, 0.7 0; 0 8; 8; PRACTICE EXERCISES For Extra Practice, see page 69. Compute the variance and standard deviation for each set of data. 4. 6, 6, 6, 6 0; ,.5, 3.5, 4.5, 5.5 ; 6. 4., 9., 4., 9., , 4, 9.4, 6.5, ; 5 50; 5 8. Raymond took two tests. On the first test, his score was 45, while the mean score was 55 and the standard deviation was 5. On the second test, his score was 55, while the mean score was 65 and the standard deviation was 0. On which test did Raymond score better, relative to the scores of his classmates? second test 9. On a science test taken by 8 students, the mean score was 8.5. The standard deviation for the scores was 5.3. What was the sum of all the scores? WRITING MATH What can you say about the relationship between the standard deviation of a set of scores and how spread out the scores are? Generally, the higher the standard deviation, the more spread out the scores are. mathmatters3.com/extra_examples Lesson 9-7 Standard Deviation 43

35 A visual display that shows the relative frequency of data is called a frequency distribution. A histogram is often used for this purpose.. Find the mean, median, mode, variance, and standard deviation to the nearest whole number for the data presented below. Frequency Scores mean: 38; median: 40; mode: 30; variance: 56; standard deviation: 6 TEST TAKING Find out how well your classmates would score on a test on which they had to guess the answer to every question. Work in a small group of 3 4 students. Make up a 0-question multiple-choice test using the topic of obscure and unimportant sports data. Use an almanac, a sports encyclopedia, a book of records, or any other source to find facts unfamiliar to anyone in your class. Ask everybody to take the test. Then grade the test as a class. Record and analyze the results. For Exercises 9, answers will vary.. Find the range, mean, median and mode for the scores. 3. Find the standard deviation. 4. Make a visual display of the scores. Have the class retake the test. Then analyze the results. 5. Find the range, mean, median and mode for the new scores. 6. Find the standard deviation. 7. Make a visual display of the scores. Compare both sets of results. 8. On which test did the class perform better? Explain. 9. On which test did you perform better relative to your classmates? What was your z-score on that test? 0. CHAPTER INVESTIGATION Play the baseball simulation game. Draw a baseball diamond and use coins for markers. To play, two people choose nine baseball players each. Put the players tables in batting order. 44 Use 0-sided polyhedral dice or a deck of standard playing cards with the kings, queens and jacks removed. Either roll two dice or draw two cards from the deck. The first die or card indicates the row on the player s table. The second die or card indicates the column. Find the cell at the intersection of the row and column to see what happens in the game. If the baseball player gets a hit, place a marker in the appropriate place on the baseball diamond. Keep score as you would in a real baseball game. Play nine innings. Did your team do as well as you expected? Check students work. Chapter 9 Probability and Statistics