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and one tail? 4. Two dice are rolled simultaneously. Each die may land with any one of the six numbers faceup. a. Use the counting principle to determine the number of outcomes in this sample space. b. Display the sample space by drawing a graph of the set of ordered pairs. 5. A sack contains four marbles: one red, one blue, one white, one green. One marble is drawn and placed on the table. Then a second marble is drawn and placed to the right of the first. a. How many possible marbles can be selected on the first draw? b. How many possible marbles can be selected on the second draw? c. How many possible ordered pairs of marbles can be drawn? d. Draw a tree diagram to show all of the possible outcomes for this experiment. Represent the marbles by R, B, W, and G. 6. A sack contains four marbles: one red, one blue, one white, one green. One marble is drawn, its color noted and then it is replaced in the sack. A second marble is drawn and its color noted to the right of the first. a. How many possible colors can be noted on the first draw? b. How many possible colors can be noted on the second draw? c. How many possible ordered pairs of colors can be noted? d. Draw a tree diagram to show all of the possible outcomes for this experiment. Represent the marbles by R, B, W, and G. 614 Probability 7. A fair coin and a six-sided die are tossed simultaneously. What is the probability of obtain- ing: a. a head on the coin? b. a 4 on the die? c. a head on the coin and a 4 on the die on a single throw? 8. A fair coin and a six-sided die are tossed simultaneously. What is the probability of obtain- ing on a single throw: a. a head and a 3? b. a head and an even number? c. a tail and a number less than 5? d. a tail and a number greater than 4? 9. Two fair coins are tossed. What is the probability that both land heads up? 10. Three fair coins are tossed. a. Find P(H, H, H). b. Find P(T, T, T). 11. A fair spinner contains congruent sectors, numbered 1 through 8. If the arrow is spun twice, find the probability |
that it lands: a. (7, 7) e. (2, 8) b. not (7, 7) f. not (2, 8) c. (not 7, not 7) g. (not 2, not 8) d. (7, not 7) or (not 7, 7) h. (2, not 8) or (not 2, 8) 12. A fair coin is tossed 50 times and lands heads up each time. What is the probability that it will land heads up on the next toss? Explain your answer. Applying Skills 13. Tell how many possible outfits consisting of one shirt and one pair of pants Terry can choose if he owns: a. 5 shirts and 2 pairs of pants b. 10 shirts and 4 pairs of pants c. 6 shirts and 6 pairs of pants 14. There are 10 doors into the school and eight staircases from the first floor to the second. How many possible ways are there for a student to go from outside the school to a classroom on the second floor? 15. A tennis club has 15 members: eight women and seven men. How many different teams may be formed consisting of one woman and one man on the team? 16. A dinner menu lists two soups, seven main courses, and three desserts. How many different meals consisting of one soup, one main course, and one dessert are possible? The Counting Principle, Sample Spaces, and Probability 615 17. The school cafeteria offers the menu shown. Main Course Dessert Beverage Pizza Yogurt Frankfurter Fruit salad Milk Juice Ham sandwich Jello Tuna sandwich Apple pie Veggie burger a. How many meals consisting of one main course, one dessert, and one beverage can be selected from this menu? b. Joe does not like ham and tuna. How many meals (again, one main course, one dessert, and one beverage) can Joe select, not having ham and not having tuna? c. If the pizza, frankfurters, yogurt, and fruit salad have been sold out, how many differ- ent meals can JoAnn select from the remaining menu? 18. A teacher gives a quiz consisting of three questions. Each question has as its answer either true (T) or false (F). a. Using T and F, draw a tree diagram to show all possible ways the questions can be answered. b. List this sample space as a set of ordered triples. 19. A test consists of multiple-choice questions. Each question has |
four choices. Tell how many possible ways there are to answer the questions on the test if the test consists of the following number of questions: a. 1 question b. 3 questions c. 5 questions d. n questions 20. Options on a bicycle include two types of handlebars, two types of seats, and a choice of 15 colors. The bike may also be ordered in ten-speed, in three-speed, or standard. How many possible versions of a bicycle can a customer choose from, if he selects a specific type of handlebars, type of seat, color, and speed? 21. A state issues license plates consisting of letters and numbers. There are 26 letters, and the letters may be repeated on a plate; there are 10 digits, and the digits may be repeated. Tell how many possible license plates the state may issue when a license consists of each of the following: a. 2 letters, followed by 3 numbers b. 2 numbers, followed by 3 letters c. 4 numbers, followed by 2 letters 22. In a school cafeteria, the menu rotates so that P(hamburger), P(apple pie), and 4 5 P(soup). The selection of menu items is random so that the appearance of hamburgers, apple pie, and soup are independent events. On any given day, what is the probability that the cafeteria offers hamburger, apple pie, and soup on the same menu? 2 3 1 4 616 Probability 23. A quiz consists of true-false questions only. Harry has not studied, and he guesses every answer. Find the probability that he will guess correctly to get a perfect score if the test conc. n questions sists of: a. 1 question b. 4 questions 24. The probability of the Tigers beating the Cougars is. The probability of the Tigers beating the Mustangs is. If the Tigers play one game with the Cougars and one game with the Mustangs, find the probability that the Tigers: a. win both games b. lose both games 1 4 2 3 25. On Main Street in Pittsford, there are two intersections that have traffic lights. The lights are not timed to accommodate traffic. They are independent of one another. At each of the intersections, P(red light).3 and P(green light).7 for cars traveling along Main Street. Find the probability that a car traveling on Main Street will be faced with each set of given conditions at the two traffic lights shown. a |
. Both lights are red. b. Both lights are green. c. The first light is red, and the second is green. d. The first light is green, and the second is red. e. At least one light is red, that is, not both lights are green. f. Both lights are the same color. 26. A manufacturer of radios knows that the probability of a defect in any of his products is If 10,000 radios are manufactured in January, how many are likely to be defective? 1 400. 27. Past records from the weather bureau indicate that the probability of rain in August on Cape Cod is. If Joan goes to Cape Cod for 2 weeks in August, how many days will it probably rain if the records hold true? 2 7 28. A nationwide fast-food chain has a promotion, distributing to customers 2,000,000 coupons for the prizes shown below. Each coupon awards the customer one of the following prizes. 1 Grand Prize: $25,000 cash 2 Second-Place Prizes: New car 100 Third-Place Prizes: New TV a. Find the probability of winning: Fourth-Place Prizes: Free meal Consolation Prizes: 25 cents off any purchase (1) the grand prize (2) a new car (3) a new TV b. If a customer has one coupon, what is the probability of winning one of the first three prizes (cash, a car, or a TV)? c. If the probability of winning a free meal is 1 400, how many coupons are marked as fourth-place prizes? d. How many coupons are marked “25 cents off any purchase”? e. If a customer has one coupon, what is the probability of not winning one of the first three prizes? 15-8 PROBABILITIES WITH TWO OR MORE ACTIVITIES Probabilities with Two or More Activities 617 Without Replacement Two cards are drawn at random from an ordinary pack of 52 cards. In this situation, a single card is drawn from a deck of 52 cards, and then a second card is drawn from the remaining 51 cards in the deck. What is the probability that both cards drawn are kings? On the first draw, there are four kings in the deck of 52 cards, so P(first king) 4 52 If a second card is drawn without replacing the first king selected, there are now only three kings in the 51 cards remaining. Therefore we are considering the probability of drawing a king, given that a king has already been drawn. |
P(second king) 3 51 By the counting principle for probabilities: P(both kings) P(first king) P(second king) 52 3 3 4 51 5 1 13 3 1 17 5 1 221 This result could also have been obtained using the counting principle. The number of elements in the sample space is 52 51. The number of elements in the event (two kings) is. Then, 4 3 3 P(both kings) 5 4 3 3 52 3 51 5 12 2,652 5 1 221 This is called a problem without replacement because the first king drawn was not placed back into the deck. These are dependent events because the probability of a king on the second draw depends on whether or not a king appeared on the first draw. In general, if A and B are two dependent events: P(A and B) P(A) P(B given that A has occurred) Earlier in the chapter we discussed P(A and B) where (A and B) is a single event that satisfies both conditions. Here (A and B) denotes two dependent events with A the outcome of one event and B the outcome of the other. The conditions of the problem will indicate which of these situations exists. When A and B are dependent events, P(A and B) can also be written as P(A, B). With Replacement A card is drawn at random from an ordinary deck, the card is placed back into the deck, and a second card is then drawn and replaced. In this situation, it is clear the deck contains 52 cards each time that a card is drawn and that the same card could be drawn twice. What is the probability that the card drawn each time is a king? 618 Probability On the first draw, there are four kings in the deck of 52 cards. P(first king) 4 52 If the first king drawn is now placed back into the deck, then, on the second draw, there are again four kings in the deck of 52 cards. P(second king) 4 52 By the counting principle for probabilities: P(both kings) P(first king) P(second king) 4 52 3 4 52 5 1 13 3 1 13 5 1 169 This is called a problem with replacement because the first card drawn was placed back into the deck. In this case, the events are independent because the probability of a king on the second draw does not depend on whether or not a king appeared on the first draw. Since the card drawn is replaced, the number |
of cards in the deck and the number of kings in the deck remain constant. In this case, P(B given that A has occurred) P(B). In general, if A and B are two independent events, P(A and B) P(A) P(B) Rolling two dice is similar to drawing two cards with replacement because the number of faces on each die remains constant, as did the number of cards in the deck. Typical problems with replacement include rolling dice, tossing coins (each coin always has two sides), and spinning arrows. KEEP IN MIND 1. If the problem does not specifically mention with replacement or without replacement, ask yourself: “Is this problem with or without replacement?” or “Are the events dependent or independent?” 2. For many compound events, the probability can be determined most easily by using the counting principle. 3. Every probability problem can always be solved by: • counting the number of elements in the sample space, n(S); • counting the number of outcomes in the event, n(E); • substituting these numbers in the probability formula, P(E) n(E) n(S) Probabilities with Two or More Activities 619 Conditional Probability The previous discussion involved the concept of conditional probability. For both dependent and independent events, in order to find the probability of A followed by B, it is necessary to calculate the probability that B occurs given that A has occurred. Notation for conditional probability is P(B given that A has occurred) P(B A). Then the following statement is true for both dependent and independent events: P(A and B) P(A) P(B A) If A and B independent events, P(B A) P(B). Therefore, for independent events: P(A and B) P(A) P(B A) P(A) P(B) The general formula P(A and B) P(A) P(B A) can be solved for P(B A): P(B A) P(A and B) P(A) For example, suppose a box contains one red marble, one blue marble, one green marble, and one yellow marble. Two marbles are drawn without replacement. Let R be the event {red marble} and Y be the event {yellow marble}. The probability of R is and the probability of Y is. If we want to find the probability of drawing a red marble followed by |
a yellow marble, R and Y are depen1 dent events. We need the probability of Y given that R has occurred, which is 3 since once the red marble has been drawn, only 3 marbles remain, one of which is yellow. 1 4 1 4 P(R followed by Y) P(R and Y) P(R) P(Y | R) 1 3 1 4 1 12 This result can be shown by displaying the sample space. (R, B) (B, R) (G, R) (Y, R) (R, G) (B, G) (G, B) (Y, B) (R, Y) (B, Y) (G, Y) (Y, G) There are 12 possible outcomes in the sample space and one of them is (R, Y). Therefore, P(R, Y).1 12 620 Probability EXAMPLE 1 Three fair dice are thrown. What is the probability that all three dice show a 5? Solution These are independent events. There are six possible faces that can come up on each die. 1 On the first die, there is one way to obtain a 5 so P(5). 6 1 On the second die, there is one way to obtain a 5 so P(5). 6 1 On the third die, there is one way to obtain a 5 so P(5). 6 By the counting principle: P(5 on each die) P(5 on first) P(5 on second) P(5 on third) 1 6 3 1 1 216 6 3 1 6 Answer EXAMPLE 2 If two cards are drawn from an ordinary deck without replacement, what is the probability that the cards form a pair (two cards of the same face value but different suits)? Solution These are dependent events. On the first draw, any card at all may be chosen, so: P(any card) 52 52 There are now 51 cards left in the deck. Of these 51, there are three that match the first card taken, to form a pair, so: P(second card forms a pair) 3 51 Then: P(pair) P(any card) P(second card forms a pair) EXAMPLE 3 52 52 3 3 51 1 3 1 17 Answer 1 17 A jar contains four white marbles and two blue marbles, all the same size. A marble is drawn at random and not replaced. A second marble is then drawn from the jar |
. Find the probability that: a. both marbles are white b. both marbles are blue c. both marbles are the same color Probabilities with Two or More Activities 621 Solution These are dependent events. a. On the first draw: P(white) 4 6 Since the white marble drawn is not replaced, five marbles, of which three are white, are left in the jar. On the second draw: Then: P(white given that the first was white) 3 5 P(both white) 6 3 3 4 5 5 12 30 or Answer 2 5 b. Start with a full jar of six marbles of which two are blue. On the first draw: P(blue) 2 6 Since the blue marble drawn is not replaced, five marbles, of which only one is blue, are left in the jar. On the second draw: P(blue given that the first was blue) 1 5 Then: P(both blue) 2 6 3 1 5 5 2 30 or 1 15 Answer c. If both marbles are the same color, both are white or both are blue. These are disjoint or mutually exclusive events: P(A or B) P(A) P(B) Therefore: P(both white or both blue) P(both white) P(both blue) 2 5 6 15 7 15 Answer 1 15 1 15 Note: By considering the complement of the set named in part c, we can easily determine the probability of drawing two marbles of different colors. P[not (both white or both blue)] 1 P(both white or both blue) 5 1 2 7 15 5 8 15 622 Probability EXAMPLE 4 A fair die is rolled. a. Find the probability that the die shows a 4 given that the die shows an even number. b. Find the probability that the die shows a 1 given that the die shows a num- ber less than 5. Solution a. Use the formula for conditional probability. EXAMPLE 5 P(4 given an even number) P(4 even) 5 P(4 and even) P(even) The event “4 and even” occurs whenever the outcome is 4. Therefore, P(4 and even) P(4). P(4 even) P(4 and even) P(even) 5 P(4) P(even) 5 1 6 3 6 5 1 3 Answer b. The event “1 and less than 5” occurs whenever |
the outcome is 1. Therefore, P(1 and less than 5) P(1). P(1 less than 5) P(1 and less than 5) P(less than 5) 5 P(1) P(less than 5) 5 1 6 5 6 5 1 5 Answer Fred has two quarters and one nickel in his pocket. The pocket has a hole in it, and a coin drops out. Fred picks up the coin and puts it back into his pocket. A few minutes later, a coin drops out of his pocket again. a. Draw a tree diagram or list the sample space for all possible pairs that are outcomes to describe the coins that fell. b. What is the probability that the same coin fell out of Fred’s pocket both times? a. What is the probability that the two coins that fell have a total value of 30 cents? b. What is the probability that a quarter fell out at least once? Solution a. Because there are two quarters, use subscripts. The three coins are {Q1, Q2, N}, where Q represents a quarter and N represents a nickel. This is a problem with replacement. The events are independent. {(Q1, Q1), (Q1, Q2), (Q1, N), (Q2, Q1), (Q2, Q2), (Q2, N), (N, Q1), (N, Q2), (N, N)} Q1 Q2 N Q1 Q2 N Q1 Q2 N Q1 Q2 N Probabilities with Two or More Activities 623 b. Of the nine outcomes, three name the same coin both times: (Q1, Q1), (Q2, Q2) and (N, N). Therefore: P(same coin) or Answer 3 9 1 3 c. Of the nine outcomes, the four that consist of a quarter and a nickel total 30 cents: (Q1, N), (Q2, N), (N, Q1), (N, Q2). P(total value of 30 cents) Answer 4 9 d. Of the nine outcomes, only (N, N) does not include a quarter. Eight contain at least one quarter, that is, one or more quarters. P(at least one quarter) Answer 8 9 EXAMPLE 6 Of Roosevelt High School’s 1,000 students, 300 are athletes, 200 are in the Honor Roll, and 120 play sports and are in |
the Honor Roll. What is the probability that a randomly chosen student who plays a sport is also in the Honor Roll? Solution Let A the event that the student is an athlete, and B the event that the student is in the Honor Roll. Then, A and B the event that a student is an athlete and is in the Honor Roll. Therefore, the conditional probability that the student is in the Honor Roll given that he or she is an athlete is: P(A and B) P(A) P(B A) 120 1,000 300 1,000 120 300 5.4 Answer EXERCISES Writing About Mathematics 1. The name of each person who attends a charity luncheon is placed in a box and names are drawn for first, second, and third prize. No one can win more than one prize. Is the probability of winning first prize greater than, equal to, or less than the probability of winning second prize? Explain your answer. 2. Six tiles numbered 1 through 6 are placed in a sack. Two tiles are drawn. Is the probability of drawing a pair of tiles whose numbers have a sum of 8 greater than, equal to, or less than the probability of obtaining a sum of 8 when rolling a pair of dice? Explain your answer. 624 Probability Developing Skills 3. A jar contains two red and five yellow marbles. If one marble is drawn at random, what is the probability that the marble drawn is: a. red? b. yellow? 4. A jar contains two red and five yellow marbles. A marble is drawn at random and then replaced. A second draw is made at random. Find the probability that: a. both marbles are red b. both marbles are yellow c. both marbles are the same color d. the marbles are different in color 5. A jar contains two red and five yellow marbles. A marble is drawn at random. Then without replacement, a second marble is drawn at random. Find the probability that: a. both marbles are red b. both marbles are yellow c. both marbles are the same color d. the marbles are different in color e. the second marble is red given that the first is yellow 6. In an experiment, an arrow is spun twice on a circular board containing four congruent sectors numbered 1 through 4. The arrow is equally likely to land on any one of the sectors. a. Indicate the sample space by drawing a tree diagram or writing a set of |
ordered pairs. b. Find the probability of spinning the digits 2 and 3 in that order. c. Find the probability that the same digit is spun both times. d. What is the probability that the first digit spun is larger than the second? 7. A jar contains nine orange disks and three blue disks. A girl chooses one at random and then, without replacing it, chooses another. Let O represent orange and B represent blue. a. Find the probability of each of the following outcomes: (1) (O, O) (2) (O, B) (3) (B, O) (4) (B, B) b. Now, find for the disks chosen, the probability that: (1) neither was orange (3) at least one was orange (5) at most one was orange (7) the second is orange given that the first was blue (2) only one was blue (4) they were the same color (6) they were the same disk 8. A card is drawn at random from a deck of 52 cards. Given that it is a red card, what is the probability that it is: a. a heart? b. a king? 9. A fair coin is tossed three times. a. What is the probability of getting all heads, given that the first toss is heads? b. What is the probability of getting all heads, given that the first two tosses are heads? Applying Skills 10. Sal has a bag of hard candies: three are lemon (L) and two are grape (G). He eats two of the candies while waiting for a bus, selecting them at random one after another. Probabilities with Two or More Activities 625 a. Using subscripts, draw a tree diagram or list the sample space of all possible outcomes showing which candies are eaten. b. Find the probability of each of the following outcomes: (1) both candies are lemon (3) the candies are the same flavor (2) neither candy is lemon (4) at least one candy is lemon c. What is the probability that the second candy that Sal ate was lemon given that the first was grape? 11. Carol has five children: three girls and two boys. One of her children was late for lunch. Later that day, one of her children was late for dinner. a. Indicate the sample space by a tree diagram or list of ordered pairs showing which children were late. b. If each child was equally likely to |
be late, find the probability of each outcome below. (1) Both children who were late were girls. (2) Both children who were late were boys. (3) The same child was late both times. (4) At least one of the children who was late was a boy. c. What is the probability that the child who was late for dinner was a boy given that the child who was late for lunch was a girl? 12. Several players start playing a game, each with a full deck of 52 cards. Each player draws two cards at random, one at a time without replacement. Find the probability that: a. Flo draws two jacks c. Jerry draws two red cards e. Ann does not draw a pair b. Frances draws two hearts d. Mary draws two picture cards f. Stephen draws two black kings g. Carrie draws a 10 on her second draw given that the first was a 5 h. Bill draws a heart and a club in that order i. Ann draws a king on her second draw given that the first was not a king 13. Saverio has four coins: a half dollar, a quarter, a dime, and a nickel. He chooses one of the coins at random and puts it in a bank. Later he chooses another coin and also puts that in the bank. a. Indicate the sample space of coins saved. b. If each coin is equally likely to be saved, find the probability that: (1) the coins saved will be worth a total of 35 cents (2) the coins saved will add to an even amount (3) the coins saved will include the half-dollar (4) the coins saved will be worth a total of less than 30 cents (5) the second coin saved will be worth more than 20 cents given that the first coin saved was worth more than 20 cents. (6) the second coin saved will be worth more than 20 cents given that the first coin saved was worth less than 20 cents. 626 Probability 14. Farmer Brown must wake up before sunrise to start his chores. Dressing in the dark, he reaches into a drawer and pulls out two loose socks. There are eight white socks and six red socks in the drawer. a. Find the probability that both socks are: (1) white (2) red (3) the same color (4) not the same color b. Find the minimum number of socks Farmer Brown must pull out of the drawer to guarantee that he will get a matching pair. 15 |
. Tillie is approaching the toll booth on an expressway. She has three quarters and four dimes in her purse. She takes out two coins at random from her purse. Find the probability of each outcome. a. Both coins are quarters. b. Both coins are dimes. c. The coins are a dime and a quarter, in any order. d. The value of the coins is enough to pay the 35-cent toll. e. The value of the coins is not enough to pay the 35-cent toll given that one of the coins is a dime. f. The value of the coins was 35 cents given that one coin was a quarter. g. At least one of the coins picked was a quarter. 16. One hundred boys and one hundred girls were asked to name the current Secretary of State. Thirty boys and sixty girls knew the correct name. One of these boys and girls is selected at random. a. What is the probability that the person selected knew the correct name? b. What is the probability that the person selected is a girl, given that that person knew the correct name? c. What is the probability that the person selected knew the correct name, given that the person is a boy? 17. In a graduating class of 400 seniors, 200 were male and 200 were female. The students were asked if they had ever downloaded music from an online music store. 75 of the male students and 70 of the female students said that they had downloaded music. a. What is the probability that a randomly chosen senior has downloaded music? b. What is the probability that a randomly chosen senior has downloaded music given that the senior is male? 18. Gracie baked one dozen sugar cookies and two dozen brownies. She then topped two-thirds of the cookies and half the brownies with chocolate frosting. a. What is the probability that a randomly chosen treat is an unfrosted brownie? b. What is the probability that a baked good chosen at random is a cookie given that it is frosted? Permutations 627 19. In a game of Tic-Tac-Toe, the first player can put an X in any of the four corner squares, four edge squares, or the center square of the grid. The second player can then put an O in any of the eight remaining open squares. a. What is the probability that the second player will put an O in the center square given the first player has put an X in an edge square? b |
. If the first player puts an X in a corner square, what is the probability that the second player will put an O in a corner square? c. In a game of Tic-Tac-Toe, the first player puts an X in the center square and the second player puts an O in a corner square. What is the probability that the first player will put her next X in an edge square? 20. Of the 150 members of the high school marching band, 30 play the trumpet, 40 are in the jazz band, and 18 play the trumpet and are also in the jazz band. a. What is the probability that a randomly chosen member of the marching band plays the trumpet but is not in the jazz band? b. What is the probability that a randomly chosen member of the marching band is also in the jazz band but does not play the trumpet? 15-9 PERMUTATIONS A teacher has announced that Al, Betty, and Chris, three students in her class, will each give an oral report today. How many possible ways are there for the teacher to choose the order in which these students will give their reports? Al Betty Chris Betty Chris Al Chris Al Betty Chris Betty Chris Al Betty Al A tree diagram shows that there are six possible orders or arrangements. For example, Al, Betty, Chris is one possible arrangement; Al, Chris, Betty is another. Each of these arrangements is called a permutation. A permutation is an arrangement of objects or things in some specific order. (In discussing permutations, the words “objects” or “things” are used in a mathematical sense to include all elements in question, whether they are people, numbers, or inanimate objects.) The six possible permutations in this case may also be shown as a set of ordered triples. Here, we let A represent Al, B represent Betty, and C represent Chris: {(A, B, C), (A, C, B), (B, A, C), (B, C, A), (C, A, B), (C, B, A)} 628 Probability Let us see, from another point of view, why there are six possible orders. We know that any one of three students can be called on to give the first report. Once the first report is given, the teacher may call on any one of the two remaining students. After the second report is given, the teacher must call on the one remaining student. Using the counting principle, |
we see that there are 3 2 1 or 6 possible orders. Consider another situation. A chef is preparing a recipe with 10 ingredients. He puts all of one ingredient in a bowl, followed by all of another ingredient, and so on. How many possible orders are there for placing the 10 ingredients in a bowl? Using the counting principle, we have: 10,628,800 possible ways If there are more than 3 million possible ways of placing 10 ingredients in a bowl, can you imagine in how many ways 300 people who want to buy tickets for a football game can be arranged in a line? Using the counting principle, we find the number of possible orders to be 300 299 298 3 2 1. To symbolize such a product, we make use of the factorial symbol,!. We represent the product of these 300 numbers by the symbol 300!, read as “three hundred factorial” or “factorial 300.” Factorials In general, for any natural number n, we define n factorial or factorial n as follows: DEFINITION n! n(n 1)(n 2)(n 3) 3 2 1 Note that 1! is the natural number 1. Calculators can be used to evaluate factorials. On a graphing calculator, the factorial function is found by first pressing and then using the left arrow key to highlight the PRB menu. For example, to evaluate 5!, use the following sequence of keys: MATH ENTER: 5 MATH 4 ENTER DISPLAY: 5! 1 2 0 Of course, whether a calculator does or does not have a factorial function, we can always use repeated multiplication to evaluate a factorial: 5! 5 4 3 2 1 Permutations 629 When using a calculator, we must keep in mind that factorial numbers are usually very large. If the number of digits in a factorial exceeds the number of places in the display of the calculator, the calculator will shift from standard decimal notation to scientific notation. For example, evaluate 15! on a calculator: ENTER: 15 MATH 4 ENTER DISPLAY The number in the display can be written in scientific notation or in decimal notation. 1.307674368 E 12 1.307674368 1012 1,307,674,368,000 Representing Permutations We have said that permutations are arrangements of objects in different orders. For example, the number of different orders in which four people can board a bus is 4! or 4 3 2 1 or 24. |
There are 24 permutations, that is, 24 different orders or arrangements, of these four people, in which all four of them get on the bus. We may also represent this number of permutations by the symbol 4P4. The symbol 4P4 is read as: “the number of permutations of four objects taken four at a time.” Here, the letter P represents the word permutation. The small 4 written to the lower left of P tells us that four objects are avail- able to be used in an arrangement, (four people are waiting for a bus). The small 4 written to the lower right of P tells us how many of these objects are to be used in each arrangement, (four people getting on the bus). 4! 4 3 2 1 24. Thus, 4P4 Similarly, 5P5 In the next section, we will study examples where not all the objects are used in the arrangement. We will also examine a calculator key used with permutations. For now, we make the following observation: 5! 5 4 3 2 1 120. For any natural number n, the number of permutations of n objects taken n at a time can be represented as: nPn n! n(n 1)(n 2) 3 2 1 630 Probability EXAMPLE 1 Compute the value of each expression. a. 6! b. 2P2 c. 7! 3! Solution a. 6! 6 5 4 3 2 1 720 2! 2 1 2 b. 2P2 3! 3 3 2 3 1 c 840 Calculator Solution a. ENTER: 6 MATH 4 ENTER c. ENTER: 7 MATH 4 4 ENTER ENTER 3 MATH DISPLAY DISPLAY: 6! Answers a. 120 b. 2 c. 840 EXAMPLE 2 Paul wishes to call Virginia, but he has forgotten her unlisted telephone number. He knows that the exchange (the first three digits) is 555, and that the last four digits are 1, 4, 7, and 9, but he cannot remember their order. What is the maximum number of telephone calls that Paul may have to make in order to dial the correct number? Solution The telephone number is 555-. Since the last four digits will be an arrangement of 1, 4, 7, and 9, this is a permutation of four numbers, taken four at a time. 4P4 4! 4 3 2 1 24 possible orders Answer The maximum number of calls that Paul may have to make is 24 |
. Permutations That Use Some of the Elements At times, we deal with situations involving permutations in which we are given n objects, but we use fewer than n objects in each arrangement. For example, a teacher has announced that he will call students from the first row to explain homework problems at the board. The students in the first row are George, Helene, Jay, Karla, Lou, and Marta. If there are only two homework problems, and each problem is to be explained by a different student, in how many orders may the teacher select students to go to the board? We know that the first problem can be assigned to any of six students. Once this problem is explained, the second problem can be assigned to any of the five Permutations 631 remaining students. We use the counting principle to find the number of possible orders in which the selection can be made. 6 5 30 possible orders If there are three problems, then after the first two students have been selected, there are four students who could be selected to explain the third problem. Extend the counting principle to find the number of possible orders in which the selection can be made. 6 5 4 120 possible orders Note that the starting number is the number of persons in the group from which the selection is made. Each of the factors is one less than the preceding factor. The number of factors is the number of choices to be made. Using the language of permutations, we say that the number of permuta- tions of six objects taken three at a time is 120. The Symbols for Permutations In general, if we have a set of n different objects, and we make arrangements of r objects from this set, we represent the number of arrangements by the symbol nPr. The subscript, r, representing the number of factors being used, must be less than or equal to n, the total number of objects in the set. Thus: For numbers n and r, where r n, the permutation of n objects, taken r at a time, is found by the formula: nPr 5 n(n 2 1)(n 2 2)??? g r factors This formula can also be written as: nPr n(n 1)(n 2) (n r 1) Note that when there are r factors, the last factor is (n r 1). In the example given above, in which three students were selected from a group of six, n 6, r 3, and the last factor is n r 1 6 |
3 1 4. Permutations and the Calculator There are many ways to use a calculator to evaluate a permutation. In the three solutions presented here, we will evaluate the permutation 8P3, the order in which, from a set of 8 elements, 3 can be selected. 632 Probability METHOD 1. Use the Multiplication Key,. The number permutation is simply the product of factors. In 8P3, the first factor, 8, is multiplied by (8 1), or 7, and then by (8 2), or 6. Here, exactly three factors have been multiplied. ENTER: 8 7 6 ENTER DISPLAY: 8 * 7 * 6 3 3 6 This approach can be used for any permutation. In the general permutation nPr, where r n, the first factor n is multiplied by (n 1), and then by (n – 2), and so on, until exactly r factors have been multiplied. METHOD 2. Use the Factorial Function. When evaluating 8P3, it appears as if we start to evaluate 8! but then we stop after multiplying only three factors. There is a way to write the product 8 7 6 using factorials. As shown below, when we divide 8! by 5!, all but the first three factors will cancel, leaving or 336. Since the factorial in the denominator uses all but three factors, 8 3 5 can be used to find that factorial: 8 3 7 3 6 8P3! 5! 5 8! (8 2 3)! We now evaluate this fraction on a calculator. ENTER: 8 MATH 4 5 MATH 4 ENTER DISPLAY: 8! / 5! 3 3 6 This approach shows us that there is another formula that can be used for the general permutation nPr, where r n: nPr n! (n 2 r)! Permutations 633 METHOD 3. Use the Permutation Function. Calculators have a special function to evaluate permutations. On a graphing and then using the left arrow key calculator, nPr is found by first pressing MATH to highlight the PRB menu. The value of n is entered first, then the nPr symbol is entered followed by the value of r. ENTER: 8 MATH 2 3 ENTER DISPLAY: 8 n P r 3 3 3 6 EXAMPLE 3 Evaluate 6P2. Solution This is a permutation of six objects, taken two at a time. There are two possible formulas to |
use. 6P2 6 5 30 6P2 4 30 Calculator Solution ENTER: 6 MATH 2 2 ENTER DISPLAY: 6 n P r 2 3 0 Answer 6P2 = 30 EXAMPLE 4 How many three-letter “words” (arrangements of letters) can be formed from the letters L, O, G, I, C if each letter is used only once in a word? Solution Forming three-letter arrangements from a set of five letters is a permutation of five, taken three at a time. Thus: 5P3 5 4 3 60 5P3 Answer 60 words or 2! (5 2 3)! 5 5! 5 4 3 60 634 Probability EXAMPLE 5 A lottery ticket contains a four-digit number. How many possible four-digit numbers are there when: a. a digit may appear only once in the number? b. a digit may appear more than once in the number? Solution a. If a digit appears only once in a four-digit number, this is a permutation of 10 digits, taken four at a time. Thus: 10P4 10 9 8 7 5,040 b. If a digit may appear more than once, we can choose any of 10 digits for the first position, then any of 10 digits for the second position, and so forth. By the counting principle: 10 10 10 10 10,000 Answers a. 5,040 b. 10,000 EXERCISES Writing About Mathematics 1. Show that n! n(n 1)!. 2. Which of these two values, if either, is larger: 9P9 or 9P8? Explain your answer. Developing Skills In 3–17, compute the value of each expression. 3. 4! 8. 3P3 13. 20P2 4. 6! 9. 8P8 14. 11P4 5. 7! 10. 8! 5! 15. 7P6 6. 3! 2! 11. 6P3 16. 255P2 7. (3 2)! 12. 10P2 17. 999! (999 2 5)! In 18–21, in each case, how many three-letter arrangements can be formed if a letter is used only once? 18. LION 19. TIGER 20. MONKEY 21. LEOPARD 22. Write the following expressions in the order of their values, beginning with the smallest: 60P5, 45P6, 24P7 |
, 19P7. Permutations 635 Applying Skills 23. Using the letters E, M, I, T: a. How many arrangements of four letters can be found if each letter is used only once in the “word”? b. List these “words.” 24. In how many different ways can five students be arranged in a row? 25. In a game of cards, Gary held exactly one club, one diamond, one heart, and one spade. In how many different ways can Gary arrange these four cards in his hand? 26. There are nine players on a baseball team. The manager must establish a batting order for the players at each game. The pitcher will bat last. How many different batting orders are possible for the eight remaining players on the team? 27. There are 30 students in a class. Every day the teacher calls on different students to write homework problems on the board, with each problem done by only one student. In how many ways can the teacher call students to the board if the homework consists of: a. only 1 problem? b. 2 problems? c. 3 problems? 28. At the Olympics, three medals are given for each competition: gold, silver, and bronze. Tell how many possible winning orders there are for the gymnastic competition if the number of competitors is: a. 7 b. 9 c. 11 d. n 29. How many different ways are there to label the three vertices of a scalene triangle, using no letter more than once, when: a. we use the letters R, S, T? b. we use all the letters of the English alphabet? 30. A class of 31 students elects four people to office, namely, a president, vice president, secretary, and treasurer. In how many possible ways can four people be elected from this class? 31. How many possible ways are there to write two initials, using the letters of the English alphabet, if: a. an initial may appear only once in each pair? b. the same initial may be used twice? In 32–35: a. Write each answer in factorial form. b. Write each answer, after using a calculator, in scientific notation. 32. In how many different orders can 60 people line up to buy tickets at a theater? 33. We learn the alphabet in a certain order, starting with A, B, C, and ending with Z. How many possible orders are there for listing the letters of the |
English alphabet? 34. Twenty-five people are waiting for a bus. When the bus arrives, there is room for 18 people to board. In how many ways could 18 of the people who are waiting board the bus? 35. Forty people attend a party at which eight door prizes are to be awarded. In how many orders can the names of the winners be announced? 636 Probability 15-10 PERMUTATIONS WITH REPETITION How many different “words” or arrangements of four letters can be formed using each letter of the word PEAK? This is the number of permutations of four things, taken four at a time. Since 4! 4 3 2 1 24, there are 24 possible words or arrangements. 4P4 PEAK PAKE PKAE KPAE PAEK PEKA PKEA KPEA EAPK APKE AKPE KAEP AEPK EPKA EKPA KEAP EPAK AEKP AKEP KAPE APEK EAKP EKAP KEPA Now consider a related question. How many different words or arrange- ments of four letters can be formed using each letter of the word PEEK? This is an example of a permutation with repetition because the letter E is repeated in the word. We can try to list the different arrangements by simply replacing the A in each of the arrangements given for the word PEAK. Let the E from PEAK be E1 and the E that replaces A be E2. The arrangements can be written as follows: E1E2PK E2PKE1 E2KPE1 KE2E1P E2E1PK E1PKE2 E1KPE2 KE1E2P PE2E1K PE1KE2 PKE1E2 KPE1E2 E2PE1K E1E2KP E1KE2P KE1PE2 E1PE2K E2E1KP E2KE1P KE2PE1 PE1E2K PE2KE1 PKE2E1 KPE2E1 We have 24 different arrangements if we consider E1 to be different from E2. But they are not really different. Notice that if we consider the E’s to be the same, every word in the first column is the same as a word in the second column, every word in the third column is the same as a word in the fourth column, and every word in the |
fifth column is the same as a word in the sixth column. 24 Therefore, only the first, third, and fifth columns are different and there are 2 or 12 arrangements of four letters when two of them are the same. This is the number of arrangements of four letters divided by the number of arrangements of two letters. Now consider a third word. In how many ways can the letters of EEEK be arranged? We will use the 24 arrangements of the letters of PE1E2K and write E3 in place of P. E3E1E2K E3E2KE1 E3KE2E1 KE3E2E1 E1E2E3K E3E2E1K E2E3KE1 E3E1KE2 E3KE1E2 E2KE3E1 K E3E1E2 KE2E1E3 E2E1E3K E1E3KE2 E1K E3E2 KE1E2E3 E1E3E2K E2E1KE3 E2KE1E3 KE2E3E1 E2E3E1K E1E2KE3 E1KE2E3 KE1E3E2 Notice that each row is the same arrangement if we consider all the E’s to be the same letter. The 4! arrangements of four letters are in groups of 3! or 6, the number of different orders in which the 3 E’s can be arranged among themselves. Therefore the number of possible arrangements is: 3! 3 3 2 3 1 5 24 6 5 4 Permutations with Repetition 637 If we consider all E’s to be the same, the four arrangements are: EEEK EEKE EKEE KEEE In general, the number of permutations of n things, taken n at a time, with r of these things identical, is: n! r! EXAMPLE 1 How many six-digit numerals can be written using all of the following digits: 2, 2, 2, 2, 3, and 5? Solution This is the number of permutations of six things taken six at a time, with 4 of the digits 2, 2, 2, 2 identical. Therefore: 4! 4 3 3 3 2 3 1 4 MATH 4 5 6 3 5 5 30 4 ENTER Calculator Solution ENTER: 6 MATH DISPLAY: 6! / 4! 3 0 Answer 30 six |
-digit numerals EXAMPLE 2 Three children, Rita, Ann, and Marie, take turns doing the dishes each night of the week. At the beginning of each week they make a schedule. If Rita does the dishes three times, and Ann and Marie each do them twice, how many different schedules are possible? Solution We can think of this as an arrangement of the letters RRRAAMM, that is, an arrangement of seven letters (for the seven days of the week) with R appearing three times and A and M each appearing twice. Therefore we will divide 7! by 3!, the number of arrangements of Rita’s days; then by 2!, the number of arrangements of Ann’s days; and finally by 2! again, the number of arrangements of Marie’s days. Number of arrangements Answer 210 possible schedules 7! 3! 3 2! 3 2 210 638 Probability EXERCISES Writing About Mathematics 1. a. List the six different arrangements or permutations of the letters in the word TAR. b. Explain why exactly six arrangements are possible. 2. a. List the three different arrangements or permutations of the letters in the word TOT. b. Explain why exactly three arrangements are possible. Developing Skills In 3–6, how many five-letter permutations are there of the letters of each given word? 3. APPLE 4. ADDED 5. VIVID 6. TESTS In 7–14: a. How many different six-letter arrangements can be written using the letters in each given word? b. How many different arrangements begin with E? c. If an arrangement is chosen at random, what is the probability that it begins with E? 7. SIMPLE 8. FREEZE 9. SYSTEM 10. BETTER 11. SEEDED 12. DEEDED 13. TATTOO 14. ELEVEN In 15–22, find the number of distinct arrangements of the letters in each word. 15. STREETS 16. INSISTS 17. ESTEEMED 18. DESERVED 19. TENNESSEE 20. BOOKKEEPER 21. MISSISSIPPI 22. UNUSUALLY In 23–26, in each case find: a. How many different five-digit numerals can be written using all five digits listed? b. How many of the numerals formed from the given digits are greater than 12,000 and less than 13,000? c. If a num |
eral formed from the given digits is chosen at random, what is the probability that it is greater than 12,000 and less than 13,000? 23. 1, 2, 3, 4, 5 24. 1, 2, 2, 2, 2 25. 1, 1, 2, 2, 2 26. 2, 2, 2, 2, 3 In 27–31, when written without using exponents, a2x can be written as aax, axa, or xaa. How many different arrangements of letters are possible for each given expression when written without exponents? 27. b3y 28. a2b5 29. abx6 30. a2by7 31. a4b8 Applying Skills 32. A bookseller has 7 copies of a novel and 3 copies of a biography. In how many ways can these 10 books be arranged on a shelf? 33. In how many ways can 6 white flags and 3 blue flags be arranged one above another on a single rope on a flagstaff? Combinations 639 34. Florence has 6 blue beads, 8 white beads, and 4 green beads, all the same size. In how many ways can she string these beads on a chain to make a necklace? 35. Frances has 8 tulip bulbs, 10 daffodil bulbs, and 7 crocus bulbs. In how many ways can Frances plant these bulbs in a border along the edge of her garden? 36. Anna has 2 dozen Rollo bars and 1 dozen apples as treats for Halloween. In how many ways can Anna hand out 1 treat to each of 36 children who come to her door? 37. A dish of mixed nuts contains 7 almonds, 5 cashews, 3 filberts, and 4 peanuts. In how many different orders can Jerry eat the contents of the dish, one nut at a time? 38. Print your first and last names using capital letters. How many different arrangements of the letters in your full name are possible? 15-11 COMBINATIONS Comparing Permutations and Combinations A combination is a collection of things in which order is not important. Before we discuss combinations, let us start with a problem we know how to solve. Ann, Beth, Carlos, and Dava are the only members of a school club. In how many ways can they elect a president and a treasurer for the club? Any one of the 4 students can be elected as president. After this happens, any one of the 3 remaining students can be elected |
treasurer. Thus there are 4 3 or 12 possible outcomes. Using the initials to represent the students involved, we can write these 12 arrangements: (A, B) (B, A) (B, C) (C, B) (A, C) (C, A) (B, D) (D, B) (A, D) (D, A) (C, D) (D, C) These are the permutations. We could have found that there are 12 permutations by using the formula: Answer: 12 permutations 4P2 4 3 12 Now let us consider two problems of a different type involving the members of the same club. Ann, Beth, Carlos, and Dava are the only members of a school club. In how many ways can they choose two members to represent the club at a student council meeting? If we look carefully at the list of 12 possible selections given in the answer to problem 1, we can see that while (A, B) and (B, A) are two different choices for president and treasurer, sending Ann and Beth to the student council meet- 640 Probability ing is exactly the same as sending Beth and Ann. For this problem let us match up answers that consist of the same two persons. (A, B) ↔ (B, A) (A, C) ↔ (C, A) (A, D) ↔ (D, A) (B, C) ↔ (C, B) (B, D) ↔ (D, B) (C, D) ↔ (D, C) Although order is important in listing slates of officers in problem 1, there is no reason to consider the order of elements in this problem. In fact, if we think of two representatives to the student council as a set of two club members, we can list the sets of representatives as follows: {A, B} {A, C} {A, D} {B, C} {B, D} {C, D} From this list we can find the number of combinations of 4 things, taken 2 at a time, written in symbols as 4C2.. The answer to this problem is found by dividing the number of permutations of 4 things taken 2 at a time by 2!.Thus: Answer: 6 combinations 4C2 4 P 2 12 2 5 6 Ann, Beth, Carlos, and Dava are the only members of a school club. In how many ways |
can they choose a three-member committee to work on the club’s next project? Is order important to this answer? If 3 officers were to be elected, such as a president, a treasurer, and a secretary, then order would be important and the number of permutations would be needed. However, a committee is a set of people. In listing the elements of a set, order is not important. Compare the permutations and combinations of 4 persons taken 3 at a time: Permutations Combinations (A, B, C) (A, C, B) (B, A, C) (B, C, A) (C, A, B) (C, B, A) (A, B, D) (A, D, B) (B, A, D) (B, D, A) (D, A, B) (D, B, A) (A, C, D) (A, D, C) (C, A, D) (C, D, A) (D, A, C) (D, C, A) (B, C, D) (B, D, C) (C, B, D) (C, D, B) (D, B, C) (D, C, B) {A, B, C} {A, B, D} {A, C, D} {B, C, D} While there are 24 permutations, written as ordered triples, there are only 4 combinations, written as sets. For example, in the first row of permutations, there are 3! or or 6 ordered triples (slates of officers) including Ann, Beth, and Carlos. 3 3 2 3 1 However, there is only one set (committee) that includes these three persons. Therefore, the number of ways to select a three-person committee from a group of four persons is the number of combinations of 4 things taken 3 at a time. This number is found by dividing the number of permutations of 4 things taken 3 at a time by 3!, the number of arrangements of the three things. Combinations 641 Answer: 4 combinations 4C3 4 P 3 24 6 5 4 In general, for counting numbers n and r, where r n, the number of com- binations of n things taken r at a time is found by using the formula: P r nCr 5 n r! On a graphing calculator, |
the sequence of keys needed to find nCr is similar to that for nPr. The combination symbol is entry 3 in the PRB menu. ENTER: 4 MATH 3 3 ENTER DISPLAY: 4 n C r 3 4 Note: The notation also represents the number of combinations of n things taken r at a time. Thus: n r A B n r B A nCr or n r A B P n r r! Some Relationships Involving Combinations Given a group of 5 people, how many different 5-person committees can be formed? Common sense tells us that there is only 1 such committee, namely, the committee consisting of all 5 people. Using combinations, we see that 5C5 5 5 P 5 Also, P 3! 5 3 3 2 3 1 3 3C3 5 3 For any counting number n, nCn 3 3 2 3 1 5 1 and 1. 4C4 5 4 P 4 Given a group of 5 people, in how many different ways can we select a committee consisting of no people, or 0 people? Common sense tells us that there is 1. Let us agree to only 1 way to select no one. Thus, using combinations, 5C0 the following generalization: For any counting number n, nC0 1. 642 Probability In how many ways can we select a committee of 2 people from a group of 7 people? Since a committee is a combination, P 2! 5 7 3 6 2 Now, given a group of 7 people, in how many ways can 5 people not be appointed to the committee? Since each set of people not appointed is a combination: 2 3 1 5 21 7C2. 7 P 5 7C5 Notice that 7C2 = 7C5. In other words, starting with a group of 7 people, the number of sets of 2 people that can be selected is equal to the number of sets of 5 people that can be not selected. In the same way it can be shown that 7C3 In general, starting with n objects, the number of ways to choose r objects for a combination is equal to the number of ways to not choose (n – r) objects for the combination. 7C1, and that 7C7 2 3 1 3 1 5 21 7C4, that 7C6 7C0. For whole numbers n and r, where r n, nCr nCnr KEEP IN MIND PERMUTATIONS COMBINATIONS 1. Order is important. |
1. Order is not important. Think of ordered elements such as ordered pairs and ordered triples. 2. An arrangement or a slate of officers indicates a permutation. Think of sets. 2. A committee, or a selection of a group, indicates a combination. EXAMPLE 1 Evaluate: 10C3 Solution This is the number of combinations of 10 things, taken 3 at a time. Thus 10C3 10 P 3! 5 10 720 6 5 120 Calculator Solution ENTER: 10 MATH 3 3 ENTER DISPLAY Answer 120 Combinations 643 EXAMPLE 2 Evaluate: 25 23b a Solution (1) This is an alternative form for the number of combinations of 25 things, taken 23 at a time: 25C23. 25 23b a nCn r, 25C23 (2) Since nCr (3) Using 25C2, perform the shorter computation. Thus: 25C2. 25 23b a 5 25C23 5 25C2 5 25 3 24 2 3 1 5 300 Answer 300 EXAMPLE 3 There are 10 teachers in the science department. How many 4-person committees can be formed in the department if Mrs. Martens and Dr. Blumenthal, 2 of the teachers, must be on each committee? Solution Since Mrs. Martens and Dr. Blumenthal must be on each committee, the prob- lem becomes one of filling 2 positions on a committee from the remaining 8 teachers. 8C2 Answer 28 committees EXAMPLE 4 8 P 2 28 There are six points in a plane, no three of which are collinear. How many straight lines can be drawn using pairs of these three points? Solution Whether joining points A and B, or points B and A, only 1 line exists, namely,. Since order is not important here, this is a combination of 6 points, taken g AB 2 at a time. Answer 15 lines 6C2 6 P 2 15 644 Probability EXAMPLE 5 Lisa Dwyer is a teacher at a local high school. In her class, there are 10 boys and 20 girls. Find the number of ways in which Ms. Dwyer can select a team of 3 students from the class to work on a group project if the team consists of: a. any 3 students b. 1 boy and 2 girls c. 3 girls d. at least 2 girls Solution a. The class contains 10 boys and 20 girls, for a total of 30 students. Since order is not important on a team, |
this is a combination of 30 students, taken 3 at a time. 30 30C3 P 3! 5 30 3 29 3 28 3 b. This is a compound event. To find the number of ways to select 1 boy out of 10 boys for a team, use 10C1. To find the number of ways to select 2 girls out of 20 for the team, use 20C2. Then, by the counting principle, multiply the results. 3 3 2 3 1 5 4,060 1 3 20 3 19 10 2 3 1 c. This is another compound event, in which 0 boys out of 10 boys and 3 girls 10 190 1,900 10C1 20C2 out of 20 girls are selected. Recall that 10C0 = 1. Thus: 10C0 20C3 1 3 20 3 19 3 18 3 3 2 3 1 1,140 Note that this could also have been thought of as the simple event of selecting 3 girls out of 20 girls: 20 3 19 3 18 3 3 2 3 1 d. A team of at least 2 girls can consist of exactly 2 girls (see part b) or exactly 3 girls (see part c). Since these events are disjoint, add the solutions to parts b and c: 1,140 20C3 Answers a. 4,060 teams b. 1,900 teams c. 1,140 teams d. 3,040 teams 1,900 1,140 3,040 EXERCISES Writing About Mathematics 1. Explain the difference between a permutation and a combination. 2. A set of r letters is to be selected from the 26 letters of the English alphabet. For what value of r is the number of possible sets of numbers greatest? Combinations 645 Developing Skills In 3–14, evaluate each expression. 3. 15C2 7. 13C0 11. 7 3b a 4. 12C3 8. 14C14 12. 9 4b a 5. 10C4 9. 9C8 13. 17 17b a 6. 25C1 10. 200C198 14. 499 2 b a 15. Find the number of combinations of 6 things, taken 3 at a time. 16. How many different committees of 3 people can be chosen from a group of 9 people? 17. How many different subsets of exactly 7 elements can be formed from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}? 18. For each given |
number of non-collinear points in a plane, how many straight lines can be drawn? a. 3 b. 4 c. 5 d. 7 e. 8 f. n 19. Consider the following formulas: n! (n 2 r)! r! P n r r! (2) nCr (3) nCr (1) nCr a. Evaluate 8C3 using each of the three formulas. b. Evaluate 11C7 using each of the three formulas. c. Can all three formulas be used to find the combination of n things, taken r at a time? n(n 2 1)(n 2 2) c(n 2 r 1 1) r! Applying Skills 20. A coach selects players for a team. If, while making this first selection, the coach pays no attention to the positions that individuals will play, how many teams are possible? a. Of 14 candidates, Coach Richko needs 5 for a basketball team. b. Of 16 candidates, Coach Jones needs 11 for a football team. c. Of 13 candidates, Coach Greves needs 9 for a baseball team. 21. A disc jockey has 25 recordings at hand, but has time to play only 22 on the air. How many sets of 22 recordings can be selected? 22. There are 14 teachers in a mathematics department. a. How many 4-person committees can be formed in the department? b. How many 4-person committees can be formed if Mr. McDonough, 1 of the 14, must be on the committee? c. How many 4-person committees can be formed if Mr. Goldstein and Mrs. Friedel, 2 of the 14, must be on the committee? 646 Probability 23. There are 12 Republicans and 10 Democrats on a senate committee. From this group, a 3-person subcommittee is to be formed. Find the number of 3-person subcommittees that consist of: a. any members of the senate committee c. 1 Republican and 2 Democrats b. Democrats only d. at least 2 Democrats e. John Clark, who is a Democrat, and any 2 Republicans 24. Sue Bartling loves to read mystery books and car-repair manuals. On a visit to the library, Sue finds 9 new mystery books and 3 car-repair manuals. She borrows 4 of these books. Find the number of different sets of 4 books Sue can borrow if: a. all are mystery books c. only 1 is a mystery book b. exactly |
2 are mystery books d. all are car-repair manuals 25. Cards are drawn at random from a 52-card deck. Find the number of different 5-card poker hands possible consisting of: a. any 5 cards from the deck c. 4 queens and any other card e. 2 aces and 3 picture cards b. 3 aces and 2 kings d. 5 spades f. 5 jacks 26. How many committees consisting of 7 people or more can be formed from a group of 10 people? 27. There are 12 roses growing in Heather’s garden. How many different ways can Heather choose roses for a bouquet consisting of more than 8 roses? 15-12 PERMUTATIONS, COMBINATIONS, AND PROBABILITY In this section, a variety of probability questions are presented. In some cases, permutations should be used. In other cases, combinations should be used. When answering these questions, the following should be kept in mind: 1. If the question asks “How many?” or “In how many ways?” the answer will be a whole number. 2. If the question asks “What is the probability?” the answer will be a value between 0 and 1 inclusive. 3. If order is important, use permutations. 4. If order is not important, use combinations. In Examples 1–4, Ms. Fenstermacher must select 4 students to represent the class in a spelling bee. Her best students include 3 girls (Callie, Daretta, and Jessica) and 4 boys (Bandu, Carlos, Sanjit, and Uri). Permutations, Combinations, and Probability 647 EXAMPLE 1 Ms. Fenstermacher decides to select the 4 students from the 7 best by drawing names from a hat. How many different groups of 4 are possible? Solution In choosing a group of 4 students out of 7, order is not important. Therefore, use combinations. 7C4 Answer 35 groups EXAMPLE 2 7 P 4 35 What is the probability that the 4 students selected for the spelling bee will consist of 2 girls and 2 boys? Solution (1) The answer to Example 1 shows 35 possible groups, or n(S) 35. (2) Since order is not important in choosing groups, use combinations. The number of ways to choose 2 girls out of 3 is 3C2 = The number of ways to choose 2 boys out of 4 is 4C2 Event E is the |
compound event of choosing 2 girls and 2 boys, or 6. 3 6 18. n(E. 3C2 4C2 (3) Thus P(E) C 2 4 C 2 3 3 7C4 5 18 35. Answer P(2 girls, 2 boys) 18 35 EXAMPLE 3 The students chosen are Callie, Jessica, Carlos, and Sanjit. In how many orders can these 4 students be called upon in the spelling bee? Solution The number of ways in which 4 students can be called upon in a spelling bee means that someone is first, someone else is second, and so on. Since this is a problem about order, use permutations. 4P4 4! 4 3 2 1 24 Answer 24 orders 648 Probability EXAMPLE 4 For the group consisting of Callie, Jessica, Carlos, and Sanjit, what is the probability that the first 2 students called upon will be girls? Solution This question may be answered using various methods, two of which are shown below. METHOD 1. Counting Principle There are 2 girls out of 4 students who may be called first. Once a girl is called upon, only 1 girl remains in the 3 students not yet called upon. Apply the counting principle of probability. P(first 2 are girls) P(girl first) P(girl second) 2 4 3 1 3 5 2 12 5 1 6 METHOD 2. Permutations Let n(S) equal the number of ways to call 2 of the 4 students in order, and let n(E) equal the number of ways to call 2 of the 2 girls in order. P(E) n(E) n(S) 5 2 P 2 4P2 Answer P(first 2 are girls) 1 6 EXERCISES Writing About Mathematics 5 2 3 1 4 3 3 5 2 12 5 1 6 1. A committee of three persons is to be chosen from a group of eight persons. Olivia is one of the persons in that group. Olivia said that since 8C3 8C5, the probability that she will be chosen for that committee is equal to the probability that she will not be chosen. Do you agree with Olivia? Explain why or why not. 2. Four letters are to be selected at random from the alphabet. Jenna found the probability that the four letters followed S in alphabetical order by using permutations. Colin found the probability that the four letters followed S in alphabetical order by using combinations. Who was correct? Explain your |
answer. Applying Skills 3. The Art Club consists of 4 girls (Jennifer, Anna, Gloria, Teresa) and 2 boys (Mark and Dan). a. In how many ways can the club elect a president and a treasurer? b. Find the probability that the 2 officers elected are both girls. c. How many 2-person teams can be selected to work on a project? d. Find the probability that a 2-person team consists of: (1) 2 girls (2) 2 boys (3) 1 girl and 1 boy (4) Anna and Mark Permutations, Combinations, and Probability 649 4. A committee of 4 is to be chosen at random from 4 men and 3 women. a. How many different 4-member committees are possible? b. How many 4-member committees contain 3 men and 1 woman? c. What is the probability that a 4-member committee will contain exactly 1 woman? d. What is the probability that a man will be on the committee? e. What is the probability that the fourth person chosen is a woman given that 3 men have already been chosen? 5. A committee of 6 people is to be chosen from 9 available people. a. How many 6-person committees can be chosen? b. The committee, when chosen, has 4 students and 2 teachers. Find the probability that a 3-person subcommittee from this group includes: (1) students only (2) exactly 1 teacher (3) at least 2 students (4) a teacher given that 2 students have been chosen 6. A box of chocolate-covered candies contains 7 caramels and 3 creams all exactly the same in appearance. Jim selects 4 pieces of candy. a. Find the number of selections possible of 4 pieces of candy that include: (1) 4 caramels (2) 1 caramel and 3 creams (3) 2 caramels and 2 creams (4) any 4 pieces b. Find the probability that Jim’s selection included: (1) 4 caramels (2) 1 caramel and 3 creams (3) 2 caramels and 2 creams (4) no caramels (5) a second cream given that 2 caramels and a cream have been selected 7. Two cards are drawn at random from a 52-card deck without replacement. Find the proba- bility of drawing: a. the ace of clubs and jack of clubs in b. a red ace and a black jack |
in either order c. 2 jacks either order d. 2 clubs e. an ace and a jack in either order f. an ace and a jack in that order g. a heart given that the king of hearts h. a king given that a queen was was drawn drawn 8. Mrs. Carberry has 4 quarters and 3 nickels in her purse. If she takes 3 coins out of her purse without looking at them, find the probability that the 3 coins are worth: a. exactly 75 cents d. exactly 55 cents b. exactly 15 cents c. exactly 35 cents e. more than 10 cents f. less than 40 cents 650 Probability 9. A 3-digit numeral is formed by selecting digits at random from {2, 4, 6, 7} without repetition. Find the probability that the number formed: a. is less than 700 c. contains only even digits b. d. is greater than 600 is an even number 10. a. There are 10 runners on the track team. If 4 runners are needed for a relay race, how many different relay teams are possible? b. Once the relay team is chosen, in how many different orders can the 4 runners run the race? c. If Nicolette is on the relay team, what is the probability that she will lead off the race? 11. Lou Grant is an editor at a newspaper employing 10 reporters and 3 photographers. a. If Lou selects 2 reporters and 1 photographer to cover a story, from how many possible 3-person teams can he choose? b. If Lou hands out 1 assignment per reporter, in how many ways can he assign the first 3 stories to his 10 reporters? c. If Lou plans to give the first story to Rossi, a reporter, in how many ways can he now assign the first 3 stories? d. If 3 out of 10 reporters are chosen at random to cover a story, what is the probability that Rossi is on this team? 12. Chris, Willie, Tim, Matt, Juan, Bob, and Steven audition for roles in the school play. a. If 2 male roles in the play are those of the hero and the clown, in how many ways can the director select 2 of the 7 boys for these roles? b. Chris and Willie got the 2 leading male roles. (1) In how many ways can the director select a group of 3 of the remaining 5 boys to work in a crowd scene? (2) How many of these groups of 3 will include Tim? (3 |
) Find the probability that Tim is in the crowd scene. 13. There are 8 candidates for 3 seats in the student government. The candidates include 3 boys (Alberto, Peter, Thomas) and 5 girls (Elizabeth, Maria, Joanna, Rosa, Danielle). If all candidates have an equal chance of winning, find the probability that the winners include: a. 3 boys c. 1 boy and 2 girls b. 3 girls d. at least 2 girls e. Maria, Peter, and anyone else f. Danielle and any other two candidates g. Alberto, Elizabeth, and Rosa h. Rosa, Peter, and Thomas Chapter Summary 651 14. A gumball machine contains 6 lemon-, 4 lime-, 3 cherry-, and 2 orange-flavored gumballs. Five coins are put into the machine, and 5 gumballs are obtained. a. How many different sets of 5 gumballs are possible? b. How many of these will contain 2 lemon and 3 lime gumballs? c. Find the probability that the 5 gumballs dispensed by the machine include: (1) 2 lemon and 3 lime (2) 3 cherry and 2 orange (3) 2 lemon, 2 lime, and 1 orange (4) lemon only (5) lime only (6) no lemon 15. The letters in the word HOLIDAY are rearranged at random. a. How many 7-letter words can be formed? b. Find the probability that the first 2 letters are vowels. c. Find the probability of no vowels in the first 3 letters. 16. At a bus stop, 5 people enter a bus that has only 3 empty seats. a. In how many different ways can 3 of the 5 people occupy these empty seats? b. If Mrs. Costa is 1 of the 5 people, what is the probability that she will not get a seat? c. If Ann and Bill are 2 of the 5 people, what is the probability that they both will get seats? CHAPTER SUMMARY Probability is a branch of mathematics in which the chance of an event happening is assigned a numerical value that predicts how likely that event is to occur. Empirical probability may be defined as the most accurate scientific estimate, based on a large number of trials, of the cumulative relative frequency of an event happening. An outcome is a result of some activity or experiment. A sample space is a set of all possible outcomes for the activity. An event is a subset of the sample space. The theoretical probability of an event is |
the number of ways that the event can occur, divided by the total number of possibilities in the sample space. If P(E) represents the probability of event E, n(E) represents the number of outcomes in event E and n(S) represents the number of outcomes in the sample space S. The formula, which applies to fair and unbiased objects and situations is: P(E) n(E) n(S) The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. The probability of any event E must be equal to or greater than 0, and less than or equal to 1: 0 P(E) 1 652 Probability For the shared outcomes of events A and B: n(A d B) n(S) If two events A and C are mutually exclusive: P(A and B) P(A or C) P(A) P(C) If two events A and B are not mutually exclusive: P(A or B) P(A) P(B) P(A and B) For any event A: P(A) P(not A) 1 The sum of the probabilities of all possible singleton outcomes for any sam- ple space must always equal 1. The Counting Principle: If one activity can occur in any of m ways and, following this, a second activity can occur in any of n ways, then both activities can occur in the order given in m n ways. The Counting Principle for Probability: E and F are independent events. The probability of event E is m (0 m 1) and the probability of event F is n (0 n 1), the probability of the event in which E and F occur jointly is the product m n. When the result of one activity in no way influences the result of a second activity, the results of these activities are called independent events. Two events are called dependent events when the result of one activity influences the result of a second activity. If A and B are two events, conditional probability is the probability of B given that A has occurred. The notation for conditional probability is P(B A). P(B A) P(A and B) P(A) The general result P(A and B) P(A) P(B A) is true for both dependent and independent events since for independent events, P(B A) P(B). A permutation is an arrangement of objects in some specific order. |
The num- ber of permutations of n objects taken n at a time, nPn, is equal to n factorial: nPn = n! n (n 1) (n 2) 3 2 1 The number of permutations of n objects taken r at a time, where r n, is equal to: nPr 5 n(n 2 1)(n 2 2) h r factors or nPr n! (n 2 r)! A permutation of n objects taken n at a time in which r are identical is equal.n! r! to A combination is a set of objects in which order is not important, as in a committee. The formula for the number of combinations of n objects taken r at a time (r n) is Review Exercises 653 nCr P n r r! From this formula, it can be shown that nCn nCnr. Permutations and combinations are used to evaluate n(S) and n(E) in prob- 1, and nCr 1, nC0 ability problems. VOCABULARY 15-1 Probability • Empirical study • Relative frequency • Cumulative rela- tive frequency • Converge • Empirical probability • Trial • Experiment • Fair and unbiased objects • Biased objects • Die • Standard deck of cards 15-2 Outcome • Sample space • Event • Favorable event • Unfavorable event • Singleton event • Theoretical probability • Calculated probability • Uniform probability • Equally likely outcomes • Random selection 15-3 Impossibility • Certainty • Subscript 15-5 Mutually exclusive events 15-7 Compound event • Tree diagram • List of ordered pairs • Graph of ordered pairs • Counting Principle • Independent events 15-8 Without replacement • Dependent events • With replacement • Conditional probability 15-9 Permutation • Factorial symbol (!) • n factorial 15-11 Combination REVIEW EXERCISES 1. In a dish of jellybeans, some are black. Aaron takes a jellybean from the dish at random without looking at what color he is taking. Jake chooses the same color jellybean that Aaron takes. Is the probability that Aaron takes a black jellybean the same as the probability that Jake takes a black jellybean? Explain your answer. 2. The probability that Aaron takes a black jellybean from the dish is 8 25 we conclude that there are 25 jellybeans in the dish and 8 of them are black? Explain why |
or why not.. Can 654 Probability 3. The numerical value of nPr is the product of r factors. Express the smallest of those factors in terms of n and r. 4. If P(A).4, P(B).3, and P(A B).2, find P(A B). 5. If P(A).5, P(B).2, and P(A B).5, find P(A B). In 6–11, evaluate each expression: 6. 8! 7. 5P5 8. 12P2 9. 5C5 10. 12C3 11. 40C38 12. How many 7-letter words can be formed from the letters in UNUSUAL if each letter is used only once in a word? 13. A SYZYGY is a nearly straight-line configuration of three celestial bodies such as the Earth, Moon, and Sun during an eclipse. a. How many different 6-letter arrangements can be made using the let- ters in the word SYZYGY? b. Find the probability that the first letter in an arrangement of SYZYGY is (1) G (2) Y (3) a vowel (4) not a vowel 14. From a list of 10 books, Gwen selects 4 to read over the summer. In how many ways can Gwen make a selection of 4 books? 15. Mrs. Moskowitz, the librarian, checks out the 4 books Gwen has chosen. In how many different orders can the librarian stamp these 4 books? 16. A coach lists all possible teams of 5 that could be chosen from 8 candi- dates. How many different teams can he list? 17. The probability that Greg will get a hit the next time at bat is 35%. What is the probability that Greg will not get a hit? 18. a. How many 3-digit numerals can be formed using the digits 2, 3, 4, 5, 6, 7, and 8 if repetition is not allowed? b. What is the probability that such a 3-digit numeral is greater than 400? 19. a. If a 4-member committee is formed from 3 girls and 6 boys in a club, how many committees can be formed? b. If the members of the committee are chosen at random, find the proba- bility that the committee consists of: (1) 2 girls and 2 boys ( |
2) 1 girl and 3 boys (3) 4 boys c. What is the probability that the fourth member chosen is a girl if the first three are 2 boys and a girl? d. What is the probability that a boy is on the committee? Review Exercises 655 20. From a 52-card deck, 2 cards are drawn at random without replacement. Find the probability of selecting: a. a ten and a king, in that order c. a ten and a king, in either order b. 2 tens d. 2 spades e. a king as the second card if the king of hearts is the first card selected. 21. If a card is drawn at random from a 52-card deck, find the probability that the card is: a. an eight or a queen b. an eight or a club c. red or an eight e. red and a club d. not a club f. not an eight or not a club 22. From an urn, 3 marbles are drawn at random with no replacement. Find the probability that the 3 marbles are the same color if the urn contains, in each case, the given marbles: a. 3 red and 2 white c. 3 red and 4 blue e. 10 blue b. 2 red and 2 blue d. 4 white and 5 blue f. 2 red, 1 white, 3 blue 23. From a class of girls and boys, the probability that one student chosen at random to answer a question will be a girl is. If four boys leave the class, the probability that a student chosen at random to answer a question will be a girl is. How many boys and girls are there in the class before the four boys leave? 2 5 1 3 24. A committee of 5 is to be chosen from 4 men and 3 women. a. How many different 5-person committees are possible? b. Find the probability that the committee includes: (1) 2 men and 3 women (3) at least 2 women (2) 3 men and 2 women (4) all women c. In how many ways can this 5-person committee select a chairperson and a secretary? d. If Hilary and Helene are on the committee, what is the probability that one is selected as chairperson and the other as secretary? 25. Assume that P(male) P(female). In a family of three children, what is the probability that all three children are of the same gender? 26. a. Let n |
(S) the number of five-letter words that can be formed using the letters in the word RADIO. Find n(S). b. Let n(E) the number of five-letter words that can be formed using the letters in the word RADIO, in which the first letter is a vowel Find n(E). 656 Probability c. If the letters in RADIO are rearranged at random, find the probability that the first letter is a vowel by using the answers to parts a and b, d. Find the answer to part c by a more direct method. In 27–32, if the letters in each given word are rearranged at random, use two different methods to find the probability that the first letter in the word is A. 27. RADAR 28. CANVAS 29. AZALEA 30. AA 31. DEFINE 32. CANOE 33. Twenty-four women and eighteen men are standing in a ticket line. What is the probability that the first five persons in line are women? (The answer need not be simplified.) 34. A four-digit code consists of numbers selected from the set of even digits: 0, 2, 4, 6, 8. No digit is used more than once in any code and the code can begin with 0. What is the probability that the code is a number less than 4,000? 35. In a class there are 4 more boys than girls. A student from the class is chosen at random. The probability that the student is a boy is. How many girls and how many boys are there in the class? 3 5 36. The number of seniors in the chess club is 2 less than twice the number of juniors, and the number of sophomores is 7 more than the number of juniors. If one person is selected at random to represent the chess club at a 2 tournament, the probability that a senior is chosen is. Find the number of 5 students from each class who are members of the club. 37. In a dish, Annie has 16 plain chocolates and 34 candy-coated chocolates, of which 4 are blue, 12 are purple, 15 are green, and 3 are gray. a. What is the probability that Annie will randomly choose a plain choco- late followed by a blue chocolate? b. Annie eats 2 green chocolates and then passes the dish to Bob. What is the probability that he will randomly choose a purple chocolate? 38. Find the probability that the |
next patient of the doctor described in the chapter opener on page 575 will need either a flu shot or a pneumonia shot. Exploration Rachael had a box of disks, all the same size and shape. She removed 20 disks from the box, marked them, and then returned them to the box. After mixing the marked disks with the others in the box, she removed a handful of disks and recorded the total number of disks and the number of marked disks. Then she Cumulative Review 657 returned the disks to the box, mixed them, and removed another handful. She repeated this last step eight times. The chart below shows her results. Disks Marked Disks 12` 3 10 4 15 5 11 4 8 3 10 2 7 2 12 5 9 3 13 5 a. Use the data to estimate the number of disks in the box. b. Repeat Rachael’s experiment using a box containing an unknown number of disks. Compare your estimate with the actual number of disks in the box. c. Explain how this procedure could be used to estimate the number of fish in a pond. CUMULATIVE REVIEW CHAPTERS 1–15 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. Which of the following is a rational number? (1) 169 " (2) 12 " (3) 9 1 4 " (4) p 2. The area of a trapezoid is 21 square inches and the measure of its height is 6.0 inches. The sum of the lengths of the bases is (1) 3.5 inches (3) 7.0 inches (2) 5.3 inches 3. A point on the line whose equation is y 3x 1 is (2) (1, –2) (1) (4, 1) (3) (2, 1) (4) 14 inches (4) (1, 2) 4. Which of the following is the graph of an exponential function? (1) (2) y y x x (3) y (4) y x x 658 Probability 5. The height of a cone is 12.0 centimeters and the radius of its base is 2.30 centimeters. What is the volume of the cone to three significant digits? (1) 52.2 square centimeters (2) 52.3 square centimeters (3) 66.5 square centimeters (4) 66.6 square centimeters 6. If 12 |
3(x 1) 7x, the value of x is (1) 1.5 (2) 4.5 (3) 0.9 (4) 3.75 7. The product of 4x–2 and 5x3 is (1) 20x–6 (2) 20x (3) 9x–6 (4) 9x 8. There are 12 members of the basketball team. At each game, the coach selects a group of five team members to start the game. For how many games could the coach make different selections? (1) 12! (2) 5! (3) 12! 5! (4) 12! 7!? 5! 9. Which of the following is an example of direct variation? (1) the area of any square compared to the length of its side (2) the perimeter of any square compared to the length of its side (3) the time it takes to drive 500 miles compared to the speed (4) the temperature compared to the time of day 10. Which of the following intervals represents the solution set of the inequal- ity (1) (3, 7) 23 # 2x11, 7? (2) (2, 3) (3) [–3, 7) (4) [–2, 3) Part II Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 11. The vertices of parallelogram ABCD are A(0, 0), B(7, 0), (12, 8), and D(5, 8). Find, to the nearest degree the measure of DAB. 12. Marty bought 5 pounds of apples for 98 cents a pound. A week later, she bought 7 pounds of apples for 74 cents a pound. What was the average price per pound that Marty paid for apples? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 13. Each student in an international study group speaks English, Japanese, or Spanish. Of these students, 100 speak English, 50 speak Japanese, 100 speak Spanish, |
45 speak English and Spanish, 10 speak English and Cumulative Review 659 Japanese, 13 speak Spanish and Japanese, and 5 speak all three languages. How many students do not speak English? 14. Abe, Brian, and Carmela share the responsibility of caring for the family pets. During a seven-day week, Abe and Brian each take three days and Carmela the other one. In how many different orders can the days of a week be assigned? Part IV Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. 15. Mrs. Martinez is buying a sweater that is on sale for 25% off of the origi- nal price. She has a coupon that gives her an additional 20% off of the sale price. Her final purchase price is what percent of the original price? 16. The area of a rectangular parcel of land is 720 square meters. The length of the land is 4 meters less than twice the width. a. Write an equation that can be used to find the dimensions of the land. b. Solve the equation written in part a to find the dimensions of the land. STATISTICS Every four years, each major political party in the United States holds a convention to select the party’s nominee for President of the United States. Before these conventions are held, each candidate assembles a staff whose job is to plan a successful campaign.This plan relies heavily on statistics: on the collection and organization of data, on the results of opinion polls, and on information about the factors that influence the way people vote. At the same time, newspaper reporters and television commentators assemble other data to keep the public informed on the progress of the candidates. Election campaigns are just one example of the use of statistics to organize data in a way that enables us to use available information to evaluate the current situation and to plan for the future. CHAPTER 16 CHAPTER TABLE OF CONTENTS 16-1 Collecting Data 16-2 Organizing Data 16-3 The Histogram 16-4 The Mean, the Median, and the Mode 16-5 Measures of Central Tendency and Grouped Data 16-6 Quartiles, Percentiles, and Cumulative Frequency 16-7 Bivariate Statistics Chapter Summary Vocabulary Review Exercises Cumulative Review 660 16-1 COLLECTING DATA Collecting |
Data 661 In our daily lives, we often deal with problems that involve many related items of numerical information called data. For example, in the daily newspaper we can find data dealing with sports, with business, with politics, or with the weather. Statistics is the study of numerical data. There are three typical steps in a statistical study: STEP 1. The collection of data. STEP 2. The organization of these data into tables, charts, and graphs. STEP 3. The drawing of conclusions from an analysis of these data. When these three steps, which describe and summarize the formation and use of a set of data, are included in a statistical study, the study is often called descriptive statistics. You will study these steps in this first course. In some cases, a fourth step, in which the analyzed data are used to predict trends and future events, is added. Data can be either qualitative or quantitative. For example, a restaurant may ask customers to rate the meal that was served as excellent, very good, good, fair, or poor. This is a qualitative evaluation. Or the restaurant may wish to make a record of each customer tip at different times of the day. This is a quantitative evaluation, which lends itself more readily to further statistical analysis. Data can be collected in a number of ways, including the following: 1. A written questionnaire or list of questions that a person can answer by checking one of several categories or supplying written responses. Categories to be checked may be either qualitative or quantitative. Written responses are usually qualitative. 2. An interview, either in person or by telephone, in which answers are given verbally and responses are recorded by the person asking the questions. Verbal answers are usually qualitative. 3. A log or a diary, such as a hospital chart or an hourly recording of the out- door temperature, in which a person records information on a regular basis. This type of information is usually quantitative. Note: Not all numerical data are quantitative data. For instance, a researcher wishes to investigate the eye color of the population of a certain island. The researcher assigns “blue” to 0, “black” to 1, “brown” to 2, and so on. The resulting data, although numerical, are qualitative since it represents eye color and the assignment was arbitrary. 662 Statistics Sampling A statistical study may be useful in situations such as the following: 1. A doctor wants to know how effective a new medicine will be in curing a disease. 2. A quality-control |
team wants to know the expected life span of flashlight batteries made by its company. 3. A company advertising on television wants to know the most frequently watched TV shows so that its ads will be seen by the greatest number of people. When a statistical study is conducted, it is not always possible to obtain information about every person, object, or situation to which the study applies. Unlike a census, in which every person is counted, some statistical studies use only a sample, or portion, of the items being investigated. To find effective medicines, pharmaceutical companies usually conduct tests in which a sample, or small group, of the patients having the disease under study receive the medicine. If the manufacturer of flashlight batteries tested the life span of every battery made, the warehouse would soon be filled with dead batteries. The manufacturer tests only a sample of the batteries to determine their average life span. An advertiser cannot contact every person owning a TV set to determine which shows are being watched. Instead, the advertiser studies TV ratings released by a firm that conducts polls based on a small sample of TV viewers. For any statistical study, whether based on a census or a sample, to be useful, data must be collected carefully and correctly. Poorly designed sampling techniques result in bias, that is, the tendency to favor a selection of certain members of the population which, in turn, produces unreliable conclusions. Techniques of Sampling We must be careful when choosing samples: 1. The sample must be fair or unbiased, to reflect the entire population being studied. To know what an apple pie tastes like, it is not necessary to eat the entire pie. Eating a sample, such as a piece of the apple pie, would be a fair way of knowing how the pie tastes. However, eating only the crust or only the apples would be an unfair sample that would not tell us what the entire pie tastes like. 2. The sample must contain a reasonable number of the items being tested or counted. If a medicine is generally effective, it must work for many people. The sample tested cannot include only one or two patients. Similarly, the manufacturer of flashlight batteries cannot make claims based on testing five or 10 batteries. A better sample might include 100 batteries. Collecting Data 663 3. Patterns of sampling or random selection should be employed in a study. The manufacturer of flashlight batteries might test every 1,000th battery to come off the assembly line. Or, the batteries to be tested might be selected at random. These techniques will help to make the sample, or the small |
group, representative of the entire population of items being studied. From the study of the small group, reasonable conclusions can be drawn about the entire group. EXAMPLE 1 To determine which television programs are the most popular in a large city, a poll is conducted by selecting people at random at a street corner and interviewing them. Outside of which location would the interviewer be most likely to find an unbiased sample? (1) a ball park (2) a concert hall (3) a supermarket Solution People outside a ball park may be going to a game or purchasing tickets for a game in the future; this sample may be biased in favor of sports programs. Similarly, those outside a concert hall may favor musical or cultural programs. The best (that is, the fairest) sample or cross section of people for the three choices given would probably be found outside a supermarket. Answer (3) Experimental Design So far we have focused on data collection. In an experiment, a researcher imposes a treatment on one or more groups. The treatment group receives the treatment, while the control group does not. For instance, consider an experiment of a new medicine for weight loss. Only the treatment group is given the medicine, and conditions are kept as similar as possible for both groups. In particular, both groups are given the same diet and exercise. Also, both groups are of large enough size and are chosen such that they are comprised of representative samples of the general population. However, it is often not enough to have just a control group and a treatment group. The researcher must keep in mind that people often tend to respond to any treatment. This is called the placebo effect. In such cases, subjects would report that the treatment worked even when it is ineffective. To account for the placebo effect, researchers use a group that is given a placebo or a dummy treatment. Of course, subjects in the experimental and placebo groups should not know which group they are in (otherwise, psychology will again confound the results). The practice of not letting people know whether or not they have been given the real treatment is called blinding, and experiments using blinding are said to be single-blind experiments. When the variable of interest is hard to measure or 664 Statistics define, double-blind experiments are needed. For example, consider an experiment measuring the effectiveness of a drug for attention deficit disorder. The problem is that “attention deficiency” is difficult to define, and so a researcher with a bias towards a particular conclusion may interpret the results of the placebo and treatment groups differently |
. To avoid such problems, the researchers working directly with the test subjects are not told which group a subject belongs to. Interpreting Graphs of Data Oftentimes embellishments to graphs distort the perception of the data, and so you must exercise care when interpreting graphs of data. 1. Two- and three-dimensional figures. As the graph on the right shows, graphs using two- or three-dimensional figures can distort small changes in the data. The lengths show the decrease in crime, but since our eyes tend to focus on the areas, the total decrease appears greater than it really is. The reason is because linear changes are increased in higher dimensions. For instance, if a length doubles in value, say from x to 2x, the area of a square with sides of length x will increase by x2 → (2x)2 4x2, CRIME RATE IN THE U.S. 1990 = 14,475,613 POLICE POLICE 1995 = 13,862,727 POLICE 2000 = 11,876,669 a four-fold increase. Similarly, the volume of a cube with edges of length x will increase by x3 → (2x)3 8x3, an eight-fold increase! 2. Horizontal and vertical scales. The scales used on the vertical and horizontal axes can exaggerate, diminish, and/or distort the nature of the change in the data. For instance, in the graph on the left of the following page, the total change in weight is less than a pound, which is negligible for an adult human. However, the scale used apparently amplifies this amount. While on the right, the unequal horizontal scale makes the population growth appear linear. AVERAGE WEIGHT OF SUBJECTS OVER 6-MONTH PERIOD 200.0 199.9 199.8 199.7 199.6 199.5 199.4 199.3 199.2 199.1 199. Collecting Data 665 POPULATION OF ANYTOWN, U.S ( 500 400 300 200 100 Month Year EXERCISES Writing About Mathematics 1. A census attempts to count every person. Explain why a census may be unreliable. 2. A sample of a new soap powder was left at each home in a small town. The occupants were asked to try the powder and return a questionnaire evaluating the product. To encourage the return of the questionnaire, the company promised to send a coupon for a free box of the soap powder to each person |
who responded. Do you think that the questionnaires that were returned represent a fair sample of all of the persons who tried the soap? Explain why or why not. Developing Skills In 3–10, determine if each variable is quantitative or qualitative. 3. Political affiliation 4. Opinions of students on a new music album 5. SAT scores 6. Nationality 7. Cholesterol level 8. Class membership (freshman, sophomore, etc.) 9. Height 10. Number of times the word “alligator” is used in an essay. In 11–18, in each case a sample of students is to be selected and the height of each student is to be measured to determine the average height of a student in high school. For each sample: a. Tell whether the sample is biased or unbiased. b. If the sample is biased, explain how this might affect the outcome of the survey. 11. The basketball team 13. All 14-year-old students 12. The senior class 14. All girls 666 Statistics 15. Every tenth person selected from an alphabetical list of all students 16. Every fifth person selected from an alphabetical list of all boys 17. The first three students who report to the nurse on Monday 18. The first three students who enter each homeroom on Tuesday In 19–24, in each case the Student Organization wishes to interview a sample of students to determine the general interests of the student body. Two questions will be asked: “Do you want more pep rallies for sports events? Do you want more dances?” For each location, tell whether the Student Organization would find an unbiased sample at that place. If the sample is biased, explain how this might influence the result of the survey. 19. The gym, after a game 20. The library 21. The lunchroom 22. The cheerleaders’ meeting 23. The next meeting of the Junior Prom committee 24. A homeroom section chosen at random 25. A statistical study is useful when reliable data are collected. At times, however, people may exaggerate or lie when answering a question. Of the six questions that follow, find the three questions that will most probably produce the largest number of unreliable answers. (1) What is your height? (3) What is your age? (5) What is your income? (2) What is your weight? (4) In which state do you live? (6) How many people are in your family? 26. List the three steps |
necessary to conduct a statistical study. 27. Explain why the graph below is misleading. SUMMER OLYMPIC GAMES CHAMPIONS 100-METER RACE 1988 Carl Lewis (USA) 9.92 sec 1992 Linford Christie (GBR) 9.96 sec 1996 Donovan Bailey (CAN) 9.84 sec 2000 Maurice Green (USA) 9.87 sec 2004 Justin Gatlin (USA) 9.85 sec Organizing Data 667 28. Investigators at the University of Kalamazoo were interested in determining whether or not women can determine a man’s preference for children based on the way that he looks. Researchers asked a group of 20 male volunteers whether or not they liked children. The researchers then showed photographs of the faces of the men to a group of 10 female volunteers and asked them to pick out which men they thought liked children. The women correctly identified over 90% of the men who said they liked children. The researchers concluded that women could identify a man’s preference for children based on the way that he looks. Identify potential problems with this experiment. Hands-On Activity Collect quantitative data for a statistical study. 1. Decide the topic of the study. What data will you collect? 2. Decide how the data will be collected. What will be the source(s) of that data? a. Questionnaires b. Personal interviews c. Telephone interviews d. Published materials from sources such as almanacs or newspapers. 3. Collect the data. How many values are necessary to obtain reliable information? Keep the data that you collect to use as you learn more about statistical studies. 16-2 ORGANIZING DATA Data are often collected in an unorganized and random manner. For example, a teacher recorded the number of days each of 25 students in her class was absent last month. These absences were as follows: 0, 3, 1, 0, 4, 2, 1, 3, 5, 0, 2, 0, 0, 0, 4, 0, 1, 1, 2, 1, 0, 7, 3, 1, 0 How many students were absent fewer than 2 days? What was the number of days for which the most students were absent? How many students were absent more than 5 days? To answer questions such as these, we find it helpful to organize the data. One method of organizing data is to write it as an ordered list. In order from least to greatest, the absences become: 0, 0, 0, 0 |
, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 7 We can immediately observe certain facts from this ordered list: more students were absent 0 days than any other number of days, the same number of students were absent 5 and 7 days. However, for more a quantitative analysis, it is useful to make a table. 668 Statistics Preparing a Table In the left column of the accompanying table, we list the data values (in this case the number of absences) in order. We start with the largest number, 7, at the top and go down to the smallest number, 0. For each occurrence of a data value, we place a tally mark, |, in the row for that number. For example, the first data value in the teacher’s list is 0, so we place a tally in the 0 row; the second value is 3, so we place a tally in the 3 row. We follow this procedure until a tally for each data value is recorded in the proper row. To simplify counting, we write every fifth tally as a diagonal mark passing through the first four tallies:. Once the data have been organized, we can count the number of tally marks in each row and add a column for the frequency, that is, the number of times that a value occurs in the set of data. When there are no tally marks in a row, as for the row showing 6 absences, the frequency is 0.The sum of all of the frequencies is called the total frequency. In this case, the total frequency is 25. (It is always wise to check the total frequency to be sure that no data value was overlooked or duplicated in tallying.) From the table, called a frequency distribution table, it is now easy to see that 15 students were absent fewer than 2 days, that more students were absent 0 days (9) than any other number of days, and that 1 student was absent more than 5 days. Grouped Data Absences Tally 7 6 5 4 3 2 1 0 Absences Tally Frequency Total frequency 25 A teacher marked a set of 32 test papers. The grades or scores earned by the students were as follows: 90, 85, 74, 86, 65, 62, 100, 95, 77, 82, 50, 83, 77, 93, 73, 72, 98, 66, 45, 100, |
50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72, 75 Organizing Data 669 Because of the large number of different scores, it is convenient to organize these data into groups or intervals, which must be equal in size. Here we will use six intervals: 41–50, 51–60, 61–70, 71–80, 81–90, 91–100. Each interval has a length of 10, found by subtracting the starting point of an interval from the starting point of the next higher interval. Interval Tally Frequency For each test score, we now place a tally mark in the row for the interval that includes that score. For example, the first two scores in the list above are 90 and 85, so we place two tally marks in the interval 81–90. The next score is 74, so we place a tally mark in the interval 71–80. When all of the scores have been tallied, we write the frequency for each interval. This table, containing a set of intervals and the corresponding frequency for each interval, is an example of grouped data. 91–100 81–90 71–80 61–70 51–60 41–50 6 8 11 4 0 3 When unorganized data are grouped into intervals, we must follow certain rules in setting up the intervals: 1. The intervals must cover the complete range of values. The range is the difference between the highest and lowest values. 2. The intervals must be equal in size. 3. The number of intervals should be between 5 and 15. The use of too many or too few intervals does not make for effective grouping of data. We usually use a large number of intervals, for example, 15, only when we have a very large set of data, such as hundreds of test scores. 4. Every data value to be tallied must fall into one and only one interval. Thus, the intervals should not overlap. When an interval ends with a counting number, the following interval begins with the next counting number. 5. The intervals must be listed in order, either highest to lowest or lowest to highest. 670 Statistics These rules tell us that there are many ways to set up tables, all of them correct, for the same set of data. For example, here is another correct way to group the 32 unorganized test scores given at the beginning of this section. Note that the length of the interval here is 8. Interval Tally Frequency 93–100 85–92 77–84 |
69–76 61–68 53–60 45–52 6 4 9 7 3 0 3 Constructing a Stem-and-Leaf Diagram Another method of displaying data is called a stem-and-leaf diagram. The stemand-leaf diagram groups the data without losing the individual data values. A group of 30 students were asked to record the length of time, in minutes, spent on math homework yesterday. They reported the following data: 38, 15, 22, 20, 25, 44, 5, 40, 38, 22, 20, 35, 20, 0, 36, 27, 37, 26, 33, 25, 17, 45, 22, 30, 18, 48, 12, 10, 24, 27 To construct a stem-and-leaf diagram for the lengths of time given, we begin by choosing part of the data values to be the stem. Since every score is a one- or two-digit number, we will choose the tens digit as a convenient stem. For the one-digit numbers, 0 and 5, the stem is 0; for the other data values, the stem is 1, 2, 3, or 4. Then the units digit will be the leaf. We construct the diagram as follows: STEP 1. List the stems, starting with 4, under one another Stem Leaf to the left of a vertical line beneath a crossbar. STEP 2. Enter each score by writing its leaf (the units digit) to the right of the vertical line, following the appropriate stem (its tens value). For example, enter 38 by writing 8 to the right of the vertical line, after stem 3. 4 3 2 1 0 Stem Leaf 8 4 3 2 1 0 STEP 3. Add the other scores to the diagram until all are entered. Organizing Data 671 Stem Leaf STEP 4. Arrange the leaves in order after Stem Leaf each stem. STEP 5. Add a key to demonstrate the meaning of each value in the diagram Key: 3 0 30 EXAMPLE 1 The following data consist of the weights, in kilograms, of a group of 30 students: 70, 43, 48, 72, 53, 81, 76, 54, 58, 64, 51, 53, 75, 62, 84, 67, 72, 80, 88, 65, 60, 43, 53, 42, 57, 61, 55, 75, 82, 71 a. Organize the data in a table. Use five intervals starting with 40–49. b |
. Based on the grouped data, which interval contains the greatest number of students? c. How many students weigh less than 70 kilograms? Solution a. Interval Tally Frequency (number) 80–89 70–79 60–69 50–59 40–49 5 7 6 8 4 b. The interval 50–59 contains the greatest number of students, 8. Answer c. The three lowest intervals, namely 40–49, 50–59, and 60–69, show weights less than 70 kilograms. Add the frequencies in these three intervals: 4 8 6 18 Answer 672 Statistics EXAMPLE 2 Draw a stem-and-leaf diagram for the data in Example 1. Solution Let the tens digit be the stem and the units digit the leaf. (1) Enter the data values in the (2) Arrange the leaves in numerical given order: Stem Leaf 3) Add a key indicating unit of measure: order after each stem: Stem Leaf Key: 5 1 51 kg EXERCISES Writing About Mathematics 1. Of the examples given above, which gives more information about the data: the table or the stem-and-leaf diagram? Explain your answer. 2. A set of data ranges from 2 to 654. What stem can be used for this set of data when drawing a stem-and-leaf diagram? What leaves would be used with this stem? Explain your choices. Developing Skills 3. a. Copy and complete the table to group the data, which represent the heights, in centime- ters, of 36 students: 162, 173, 178, 181, 155, 162, 168, 147, 180, 171, 168, 183, 157, 158, 180, 164, 160, 171, 183, 174, 166, 175, 169, 180, 149, 170, 150, 158, 162, 175, 171, 163, 158, 163, 164, 177 b. Use the grouped data to answer the following questions: (1) How many students are less than 160 centimeters in height? (2) How many students are 160 centimeters or more in height? Interval Tally Frequency 180–189 170–179 160–169 150–159 140–149 (3) Which interval contains the greatest number of students? (4) Which interval contains the least number of students? Organizing Data 673 c. Display the data in a stem-and-leaf diagram. Use the first two digits of the numbers as the stems. d. What is the range of |
the data? e. How many students are taller than 175 centimeters? 4. a. Copy and complete the table to group the data, which gives the lifespan, in hours, of 50 flashlight batteries: 73, 81, 92, 80, 108, 76, 84, 102, 58, 72, 82, 100, 70, 72, 95, 105, 75, 84, 101, 62, 63, 104, 97, 85, 106, 72, 57, 85, 82, 90, 54, 75, 80, 52, 87, 91, 85, 103, 78, 79, 91, 70, 88, 73, 67, 101, 96, 84, 53, 86 b. Use the grouped data to answer the following questions: (1) How many flashlight batteries lasted for 80 or more hours? Interval Tally Frequency 50–59 60–69 70–79 80–89 90–99 (2) How many flashlight batteries lasted fewer than 80 100–109 hours? (3) Which interval contains the greatest number of batteries? (4) Which interval contains the least number of batteries? c. Display the data in a stem-and-leaf diagram. Use the digits from 5 through 10 as the stems. d. What is the range of the data? e. What is the probability that a battery selected at random lasted more than 100 hours? 5. The following data consist of the hours spent each week watching television, as reported by a group of 38 teenagers: 13, 20, 17, 36, 25, 21, 9, 32, 20, 17, 12, 19, 5, 8, 11, 28, 25, 18, 19, 22, 4, 6, 0, 10, 16, 3, 27, 31, 15, 18, 20, 17, 3, 6, 19, 25, 4, 7 a. Construct a table to group these data, using intervals of 0–4, 5–9, 10–14, 15–19, 20–24, 25–29, 30–34, and 35–39. b. Construct a table to group these data, using intervals of 0–7, 8–15, 16–23, 24–31, and 32–39. c. Display the data in a stem-and-leaf diagram. d. What is the range of the data? e. What is the probability that a teenager, selected at random from this group, spends less than 4 hours watching television each week? 6. |
The following data show test scores for 30 students: 90, 83, 87, 71, 62, 46, 67, 72, 75, 100, 93, 81, 74, 75, 82, 83, 83, 84, 92, 58, 95, 98, 81, 88, 72, 59, 95, 50, 73, 93 674 Statistics a. Construct a table, using intervals of length 10 starting with 91–100. b. Construct a table, using intervals of length 12 starting with 89–100. c. For the grouped data in part a, which interval contains the greatest number of stu- dents? d. For the grouped data in part b, which interval contains the greatest number of stu- dents? e. Do the answers for parts c and d indicate the same general region of test scores, such as “scores in the eighties”? Explain your answer. 7. For the ungrouped data from Exercise 5, tell why each of the following sets of intervals is not correct for grouping the data. a. Interval b. Interval c. Interval d. Interval 25–38 13–24 0–12 30–39 20–29 10–19 5–9 0–4 32–40 24–32 16–24 8–16 0–8 33–40 25–32 17–24 9–16 1–8 Hands-On Activity Organize the data that you collected in the Hands-On Activity for Section 16-1. 1. Use a stem-and-leaf diagram. a. Decide what will be used as stems. b. Decide what will be used as leaves. c. Construct the diagram. d. Check that the number of leaves in the diagram equals the number of values in the data collected. 2. Use a frequency table. a. How many intervals will be used? b. What will be the length of each interval? c. What will be the starting and ending points of each interval? Check that the intervals do not overlap, are equal in size, and that every value falls into only one interval. d. Tally the data. e. List the frequency for each interval. f. Check that the total frequency equals the number of values in the data collected. 3. Decide which method of organization is better for your data. Explain your choice. Keep your organized data to work with as you learn more about statistics. The Histogram 675 16-3 THE HISTOG |
RAM In Section 16-2 we organized data by grouping them into intervals of equal length. After the data have been organized, a graph can be used to visualize the intervals and their frequencies. The table below shows the distribution of test scores for 32 students in a class. The data have been organized into six intervals of length 10. Test Scores (Intervals) Frequency (Number of Scores) 91–100 81–90 71–80 61–70 51–60 41–50 6 8 11 4 0 3 We can use a histogram to display the data graphically. A histogram is a vertical bar graph in which each interval is represented by the width of the bar and the frequency of the interval is represented by the height of the bar. The bars are placed next to each other to show that, as one interval ends, the next interval begins 12 11 10 TEST SCORES OF 32 STUDENTS 41–50 51–60 61–70 71–80 81–90 91–100 Test scores (intervals) In the above histogram, the intervals are listed on the horizontal axis in the order of increasing scores, and the frequency scale is shown on the vertical axis. The first bar shows that 3 students had test scores in the interval 41–50. Since no student scored in the interval 51–60, there is no bar for this interval. Then, 4 students scored between 61 and 70; 11 between 71 and 80; 8 between 81 and 90; and 6 between 91 and 100. 676 Statistics Except for an interval having a frequency of 0, the interval 51–60 in this example, there are no gaps between the bars drawn in a histogram. Since the histogram displays the frequency, or number of data values, in each interval, we sometimes call this graph a frequency histogram. A graphing calculator can display a frequency histogram from the data on a frequency distribution table. (1) Clear L1 and L2 with the ClrList function by pressing STAT 4 2nd L1, 2nd L2 ENTER. (2) Press STAT 1 to edit the lists. L1 will L1 L2 contain the minimum value of each interval. Move the cursor to the first entry position in L1. Type the value and then. Type the next value and press ENTER ENTER then press until all the minimum values of the intervals have been entered.. Repeat this process 91 81 71 61 51 41 ------ ------ 6 8 11 4 0 3 L2(7) = 2 L3 ------ (3 |
) Repeat the process to enter the frequencies that correspond to each inter- val in L2. (4) Clear any functions in the Y= menu. (5) Turn on Plot1 from the STAT PLOT menu, and configure it to graph a histogram. Make sure to also set Xlist to L1 and Freq to L2. ENTER: STAT PLOT ENTER 2nd 1 L1 ENTER 2nd 2nd L2 (6) In the WINDOW menu, accessed by, enter Xmin as 31, WINDOW pressing the length of one interval less than the smallest interval value and Xmax as 110, the length of one interval more than the largest interval value. Enter Xscl as 10, the length of the interval. The Ymin is 0 and Ymax is 12 to be greater than the largest frequency = Ty The Histogram 677 (7) Press GRAPH to draw the graph. We can P 1 : L 1, L 2 view the frequency (n) associated with each interval by pressing TRACE. Use the left and right arrow keys to move between intervals EXAMPLE 1 The table on the right represents the number of miles per gallon of gasoline obtained by 40 drivers of compact cars. Construct a frequency histogram based on the data. Solution (1) Draw and label a vertical scale to show fre- quencies. The scale starts at 0 and increases to include the highest frequency in any one interval (here, it is 11). (2) Draw and label intervals of equal length on a horizontal scale. Label the horizontal scale, telling what the numbers represent. Interval Frequency 16–19 20–23 24–27 28–31 32–35 36–39 40–43 5 11 8 5 7 3 1 (3) Draw the bars vertically, leaving no gaps between the intervals 12 11 10 16–19 20–23 24–27 28–31 32–35 36–39 40–43 Mileage (miles per gallon) for compact cars 678 Statistics Calculator Solution (1) Press 1 STAT to edit the lists and enter the minimum value of each interval into L1: 16, 20, 24, 28, 32, 36, 40. Use the arrow key to move into L2, and enter the corresponding frequencies: 5, 11, 8, 5, 7, 3, 1. (2) Go to the STAT PLOT menu and choose Plot1 by pressing 2nd STAT PLOT 1. Move the cursor with the arrow keys, then press |
ENTER to select On and the histogram. Type 2nd L1 into Xlist and 2nd L2 into Freq. (3) Set the Window. Each interval has length 4, so set Xmin to 12 (4 less than the smallest interval value), Xmax to 44 (4 more than the largest interval value), and Xscl to 4. Make Ymin 0 and Ymax 12 to be greater than the largest frequency4) Draw the graph by pressing GRAPH. Press TRACE and use the right and left arrow keys to show the frequencies, the heights of the vertical bars EXAMPLE 2 Use the histogram constructed in Example 1 to answer the following questions: a. In what interval is the greatest frequency found? b. What is the number (or frequency) of cars reporting mileages between 28 and 31 miles per gallon? c. For what interval are the fewest cars reported? d. How many of the cars reported mileage greater than 31 miles per gallon? e. What percent of the cars reported mileage from 24 to 27 miles per gallon? Solution a. 20–23 b. 5 c. 40–43 d. Add the frequencies for the three highest intervals. The interval 32–35 has a frequency of 7; 36–39 a frequency of 3; 40–43 a frequency of 1: 7 3 1 11. e. The interval 24–27 has a frequency of 8. The total frequency for this survey is 40. 8 40 5 1 5 20%. Answers a. 20–23 b. 5 c. 40–43 d. 11 e. 20% The Histogram 679 EXERCISES Writing About Mathematics 1. Compare a stem-and-leaf diagram with a frequency histogram. In what ways are they alike and in what ways are they different? 2. If the data in Example 1 had been grouped into intervals with a lowest interval of 16–20, what would be the endpoints for the other intervals? Would you be able to determine the frequency for each new interval? Explain why or why not. Developing Skills In 3–5, in each case, construct a frequency histogram for the grouped data. Use graph paper or a graphing calculator. 3. Interval Frequency 4. Interval Frequency 5. Interval Frequency 91–100 81–90 71–80 61–70 51–60 5 9 7 2 4 30–34 25–29 20–24 15–19 10–14 5–9 5 10 10 12 0 2 1–3 |
4–6 7–9 10–12 13–15 16–18 24 30 28 41 19 8 6. For the table of grouped data given in Exercise 5, answer the following questions: a. What is the total frequency in the table? b. What interval contains the greatest frequency? c. The number of data values reported for the interval 4–6 is what percent of the total number of data values? d. How many data values from 10 through 18 were reported? Applying Skills 7. Towering Ted McGurn is the star of the school’s basketball team. The number of points scored by Ted in his last 20 games are as follows: 36, 32, 28, 30, 33, 36, 24, 33, 29, 30, 30, 25, 34, 36, 34, 31, 36, 29, 30, 34 a. Copy and complete the table to find the frequency for each interval. b. Construct a frequency histogram based on the data found in part a. c. Which interval contains the greatest frequency? d. In how many games did Ted score 32 or more points? e. In what percent of these 20 games did Ted score fewer than 26 points? Interval Tally Frequency 35–37 32–34 29–31 26–28 23–25 680 Statistics 8. Thirty students on the track team were timed in the 200-meter dash. Each student’s time was recorded to the nearest tenth of a second. Their times are as follows: 29.3, 31.2, 28.5, 37.6, 30.9, 26.0, 32.4, 31.8, 36.6, 35.0, 38.0, 37.0, 22.8, 35.2, 35.8, 37.7, 38.1, 34.0, 34.1, 28.8, 29.6, 26.9, 36.9, 39.6, 29.9, 30.0, 36.0, 36.1, 38.2, 37.8 a. Copy and complete the table to find the frequency in each interval. b. Construct a frequency histogram for the given data. c. Determine the number of students who ran the 200- meter dash in under 29 seconds. d. If a student on the track team is chosen at random, what is the probability that he or she ran the 200meter dash in fewer than 29 seconds? Interval Tally Frequency 37 |
.0–40.9 33.0–36.9 29.0–32.9 25.0–28.9 21.0–24.9 Hands-On-Activity Construct a histogram to display the data that you collected and organized in the Hands-On Activities for Sections 16-1 and 16-2. 1. Draw the histogram on graph paper. 2. Follow the steps in this section to display the histogram on a graphing calculator. 16-4 THE MEAN, THE MEDIAN, AND THE MODE In a statistical study, after we have collected the data, organized them, and presented them graphically, we then analyze the data and summarize our findings. To do this, we often look for a representative, or typical, score. Averages in Arithmetic In your previous study of arithmetic, you learned how to find the average of two or more numbers. For example, to find the average of 17, 25, and 30: STEP 1. Add these three numbers: 17 25 30 72. STEP 2. Divide this sum by 3 since there are three numbers: 72 3 24. The average of the three numbers is 24. Averages in Statistics The word average has many different meanings. For example, there is an average of test scores, a batting average, the average television viewer, an average intelligence, and the average size of a family. These averages are not found by The Mean, the Median, and the Mode 681 the same rule or procedure. Because of this confusion, in statistics we speak of measures of central tendency. These measures are numbers that usually fall somewhere in the center of a set of organized data. We will discuss three measures of central tendency: the mean, the median, and the mode. The Mean In statistics, the arithmetic average previously studied is called the mean of a set of numbers. It is also called the arithmetic mean or the numerical average. The mean is found in the same way as the arithmetic average is found. Procedure To find the mean of a set of n numbers, add the numbers and divide the sum by n.The symbol used for the mean is x–. For example, if Ralph’s grades on five tests in science during this marking period are 93, 80, 86, 72, and 94, he can find the mean of his test grades as follows: STEP 1. Add the five data values: 93 80 86 72 94 425. STEP 2. Divide this sum by 5, the number of tests: 425 5 |
85. The mean (arithmetic average) is 85. Let us consider another example. In a car wash, there are seven employees whose ages are 17, 19, 20, 17, 46, 17, and 18. What is the mean of the ages of these employees? Here, we add the seven ages to get a sum of 154. Then, 154 7 22. While the mean age of 22 is the correct answer, this measure does not truly represent the data. Only one person is older than 22, while six people are under 22. For this reason, we will look at another measure of central tendency that will eliminate the extreme case (the employee aged 46) that is distorting the data. The Median The median is the middle value for a set of data arranged in numerical order. For example, the median of the ages 17, 19, 20, 17, 46, 17, and 18 for the car-wash employees can be found in the following manner: STEP 1. Arrange the ages in numerical order: STEP 2. Find the middle number: 17, 17, 17, 18, 19, 20, 46 17, 17, 17, 18, 19, 20, 46 ↑ The median is 18 because there are three ages less than 18 and three ages greater than 18. The median, 18, is a better indication of the typical age of the 682 Statistics employees than the mean, 22, because there are so many younger people working at the car wash. Now, let us suppose that one of the car-wash employees has a birthday, and her age changes from 17 to 18. What is now the median age? STEP 1. Arrange the ages in numerical order: STEP 2. Find the middle number: 17, 17, 18, 18, 19, 20, 46 17, 17, 18, 18, 19, 20, 46 ↑ The median, or middle value, is again 18. We can no longer say that there are three ages less than 18 because one of the three youngest employees is now 18. We can say, however, that: 1. the median is 18 because there are three ages less than or equal to 18 and three ages greater than or equal to 18; or 2. the median is 18 because, when the data values are arranged in numerical order, there are three values below this median, or middle number, and three values above it. Recently, the car wash hired a new employee whose age is 21. The data now include eight ages, an even number, |
so there is no middle value. What is now the median age? STEP 1. Arrange the ages in numerical order: STEP 2. There is no single middle number. 17, 17, 18, 18, 19, 20, 21, 46 17, 17, 18, 18, 19, 20, 21, 46 ↑ ↑ 2 5 181 2 18 1 19 Find the two middle numbers: STEP 3. Find the mean (arithmetic average) of the two middle numbers: 181 The median is now 2 and four ages greater than. There are four ages less than this center value of 181 2. 181 2 Procedure To find the median of a set of n numbers: 1. Arrange the numbers in numerical order. 2. If n is odd, find the middle number.This number is the median. 3. If n is even, find the mean (arithmetic average) of the two middle numbers. This average is the median. The Mode The mode is the data value that appears most often in a given set of data. It is usually best to arrange the data in numerical order before finding the mode. The Mean, the Median, and the Mode 683 Let us consider some examples of finding the mode: 1. The ages of employees in a car wash are 17, 17, 17, 18, 19, 20, 46. The mode, which is the number appearing most often, is 17. 2. The number of hours each of six students spent reading a book are 6, 6, 8, 11, 14, 21. The mode, or number appearing most frequently, is 6. In this case, however, the mode is not a useful measure of central tendency. A better indication is given by the mean or the median. 3. The number of photographs printed from each of Renee’s last six rolls of film are 8, 8, 9, 11, 11, and 12. Since 8 appears twice and 11 appears twice, we say that there are two modes: 8 and 11. We do not take the average of these two numbers since the mode tells us where most of the scores appear. We simply report both numbers. When two modes appear within a set of data, we say that the data are bimodal. 4. The number of people living in each house on Meryl’s street are 2, 2, 3, 3, 4, 5, 5, 6, 8. These data have three modes: 2, 3, and 5. 5. Ralph |
’s test scores in science are 72, 80, 86, 93, and 94. Here, every number appears the same number of times, once. Since no number appears more often than the others, we define such data as having no mode. Procedure To find the mode for a set of data, find the number or numbers that occur most often. 1. If one number appears most often in the data, that number is the mode. 2. If two or more numbers appear more often than all other data values, and these numbers appear with the same frequency, then each of these numbers is a mode. 3. If each number in a set of data occurs with the same frequency, there is no mode. KEEP IN MIND Three measures of central tendency are: 1. The mean, or mean average, found by adding n data values and then divid- ing the sum by n. 2. The median, or middle score, found when the data are arranged in numeri- cal order. 3. The mode, or the value that appears most often. A graphing calculator can be used to arrange the data in numerical order and to find the mean and the median. The calculator solution in the following example lists the keystrokes needed to do this. 684 Statistics EXAMPLE 1 The weights, in pounds, of five players on the basketball team are 195, 168, 174, 182, and 181. Find the average weight of a player on this team. Solution The word average, by itself, indicates the mean. Therefore: (1) Add the five weights: 195 168 174 182 181 900. (2) Divide the sum by 5, the number of players: 900 5 180. Calculator Solution Enter the data into list L1. Then use 1-Var Stats from the STAT CALC menu to display information about this set of data. ENTER: STAT ENTER ENTER DISPLAY< n = 5 The first value given is, the mean. x2 Answer 180 pounds The second value given is Σx 900. The symbol Σ represents a sum and Σx 900 can be read as “The sum of the values of x is 900.” The list shows other values related to this set of data. The arrow at the bottom of the display indicates that more entries follow what appears on the screen. These can be displayed by pressing the down arrow. One of these is the median (Med 181). The display also shows that there are 5 data values (n = 5). Others we |
will use in later sections in this chapter and in more advanced courses. EXAMPLE 2 Renaldo has marks of 75, 82, and 90 on three mathematics tests. What mark must he obtain on the next test to have an average of exactly 85 for the four math tests? The Mean, the Median, and the Mode 685 Solution The word average, by itself, indicates the mean. Let x Renaldo’s mark on the fourth test. The sum of the four test marks divided by 4 is 85. 75 1 82 1 90 1 x 4 5 85 247 1 x 4 5 85 247 1 x 5 340 x 5 93 Check 75 1 82 1 90 1 93 4 5? 85 340 4 5? 85 85 5? 85 ✔ Answer Renaldo must obtain a mark of 93 on his fourth math test. EXAMPLE 3 Find the median for each distribution. a. 4, 2, 5, 5, 1 b. 9, 8, 8, 7, 4, 3, 3, 2, 0, 0 Solution a. Arrange the data in numerical order: The median is the middle value: 1, 2, 4, 5, 5 1, 2, 4, 5, 5 ↑ Answer median 4 b. Since there is an even number of values, there are two middle values. Find the mean (average) of these two middle values: 9, 8, 8, 7, 4, 3, 3, 2, 0 31 2 Answer median or 3.5 31 2 EXAMPLE 4 Find the mode for each distribution. a. 2, 9, 3, 7, 3 b. 3, 4, 5, 4, 3, 7, 2 c. 1, 2, 3, 4, 5, 6, 7 Solution a. Arrange the data in numerical order: 2, 3, 3, 7, 9. The mode, or most frequent value, is 3. b. Arrange the data in numerical order: 2, 3, 3, 4, 4, 5, 7. Both 3 and 4 appear twice. There are two modes. c. Every value occurs the same number of times in the data set 1, 2, 3, 4, 5, 6, 7. There is no mode. Answers a. The mode is 3. b. The modes are 3 and 4. c. There is no mode. 686 Statistics Linear Transformations of Data Multiplying each data value by the same constant or adding the same constant to each data value is an |
example of a linear transformation of a set of data. Let us start by examining additive transformations. For instance, consider the data 2, 2, 3, 4, 5. If 10 is added to each data value, the data set becomes: 12, 12, 13, 14, 15 Notice that every measure of central tendency has been shifted to the right by 10 units old mean old median 3 old mode 2 5 3.2 12 1 12 1 13 1 14 1 15 5 new mean new median 13 new mode 12 5 13.2 In fact, this result is valid for any additive transformation of a data set. In general: If x–, d, and o are the mean, median, and mode of a set of data and the constant c is added to each data value, then x– c, d c, and o c are the mean, median, and mode of the transformed data. It can be also shown that a similar result holds for multiplicative transfor- mations, that is: If x–, d, and o are the mean, median, and mode of a set of data and each data value is multiplied by the nonzero constant c, then cx–, cd, and co are the mean, median, and mode of the transformed data. EXAMPLE 5 In Ms. Huan’s Algebra class, the average score on the most recent quiz was 65. Being in a generous mood, Ms. Huan decided to curve the quiz by adding 10 points to each quiz score. What will be the new average score for the class? Answer 65 10 75 points EXERCISES Writing About Mathematics 1. On her first two math tests, Rene received grades of 67 and 79. Her mean (average) grade for these two tests was 73. On her third test she received a grade of 91. Rene found the mean of 73 and 91 and said that her mean for the three tests was 82. Do you agree with Rene? Explain why or why not. The Mean, the Median, and the Mode 687 2. Carlos said that when there are n numbers in a set of data and n is an odd number, the th number when the data are arranged in order. Do you agree with n 1 1 median is the 2 Carlos? Explain why or why not. Developing Skills 3. For each set of data, find the mean. a. 7, 3, 5, 11, 9 23 53 4 4 71 2 c.,,, 51 |
2 41 2 4. Find the median for each set of data., b. 22, 38, 18, 14, 22, 30 d. 1.00, 0.01, 1.10, 0.12, 1.00, 1.03 a. 1, 2, 5, 3, 4 c. 3, 8, 12, 7, 1, 0, 4 e. 3.2, 8.7, 1.4 b. 2, 9, 2, 9, 7 d. 80, 83, 97, 79, 25 f. 2.00, 0.20, 2.20, 0.02, 2.02 g. 21, 24, 23, 22, 20, 24, 23, 21, 22, 23 h. 5, 7, 9, 3, 8, 7, 5, 6 5. What is the median for the digits 1, 2, 3,..., 9? 6. What is the median for the counting numbers from 1 through 100? 7. Find the mode for each distribution. a. 2, 2, 3, 4, 8 c. 2, 2, 8, 8, 8 e. 2, 2, 3, 8, 8, 9, 9 g. 1, 2, 3, 2, 1, 2, 3, 2, 1 b. 2, 2, 3, 8, 8 d. 2, 3, 4, 7, 8 f. 1, 2, 1, 2, 1, 2, 1 h. 3, 19, 21, 75, 0, 6 i. 3, 2, 7, 6, 2, 7, 3, 1, 4, 2, 7, 5 j. 19, 21, 18, 23, 19, 22, 18, 19, 20 8. A set of data consists of six numbers: 7, 8, 8, 9, 9, and x. Find the mode for these six numbers when: a. x 9 b. x 8 c. x 7 d. x 6 9. A set of data consists of the values 2, 4, 5, x, 5, 4. Find a possible value of x such that: a. there is no mode because all scores appear an equal number of times b. there is only one mode c. there are two modes 10. For the set of data 5, 5, 6, 7, 7, which statement is true? (1) mean mode (2) median mode (3) |
mean median (4) mean median 11. For the set of data 8, 8, 9, 10, 15, which statement is true? (1) mean median (2) mean mode (3) median mode (4) mean median 688 Statistics 12. When the data consists of 3, 4, 5, 4, 3, 4, 5, which statement is true? (1) mean median (2) mean mode (3) median mode (4) mean median 13. For which set of data is there no mode? (1) 2, 1, 3, 1, 2 (2) 1, 2, 3, 3, 3 (3) 1, 2, 4, 3, 5 (4) 2, 2, 3, 3, 3 14. For which set of data is there more than one mode? (1) 8, 7, 7, 8, 7 (2) 8, 7, 4, 5, 6 (3) 8, 7, 5, 7, 6, 5 (4) 1, 2, 2, 3, 3, 3 15. For which set of data does the median equal the mode? (1) 3, 3, 4, 5, 6 (2) 3, 3, 4, 5 (3) 3, 3, 4 (4) 3, 4 16. For which set of data will the mean, median, and mode all be equal? (1) 1, 2, 5, 5, 7 (2) 1, 2, 5, 5, 8, 9 (3) 1, 1, 1, 2, 5 (4) 1, 1, 2 17. The median of the following data is 11: 2, 5, 9, 11, 40, 3, 4, 5, 10, 45, 32, 40, 67, 7, 11, 9, 20, 34, 5, 1, 8, 15, 16, 19, 39 a. If 4 is subtracted from each data value, what is the median of the transformed data set? b. If the largest data value is doubled and the smallest data value is halved, what is the median of the new data set? 18. The mean of the following data is 37.625: 3, 0, 1, 7, 8, 11, 31, 15, 99, 98, 92, 81, 85, 87, 55, 54, 34, 27, 26, 21, 14, 17, 19, 18 If each |
data value is multiplied by 2 and increased by 5, what is the mean of the transformed data set? 19. Three consecutive integers can be represented by x, x 1, and x 2. The average of these consecutive integers is 32. What are the three integers? 20. Three consecutive even integers can be represented by x, x 2, and x 4. The average of these consecutive even integers is 20. Find the integers. 21. The mean of three numbers is 31. The second is 1 more than twice the first. The third is 4 less than 3 times the first. Find the numbers. Applying Skills 22. Sid received grades of 92, 84, and 70 on three tests. Find his test average. 23. Sarah’s grades were 80 on each of two of her tests and 90 on each of three other tests. Find her test average. 24. Louise received a grade of x on each of two of her tests and of y on each of three other tests. Represent her average for all the tests in terms of x and y. The Mean, the Median, and the Mode 689 25. Andy has grades of 84, 65, and 76 on three social studies tests. What grade must he obtain on the next test to have an average of exactly 80 for the four tests? 26. Rosemary has grades of 90, 90, 92, and 78 on four English tests. What grade must she obtain on the next test so that her average for the five tests will be 90? 27. The first three test scores are shown below for each of four students. A fourth test will be given and averages taken for all four tests. Each student hopes to maintain an average of 85. Find the score needed by each student on the fourth test to have an 85 average, or explain why such an average is not possible. a. Pat: 78, 80, 100 c. Helen: 90, 92, 95 b. Bernice: 79, 80, 81 d. Al: 65, 80, 80 28. The average weight of Sue, Pam, and Nancy is 55 kilograms. a. What is the total weight of the three girls? b. Agnes weighs 60 kilograms. What is the average weight of the four girls: Sue, Pam, Nancy, and Agnes? 29. For the first 6 days of a week, the average rainfall in Chicago was 1.2 inches. On the last day of the week, 1.9 inches of rain fell. What was the average rainfall for |
the week? 30. If the heights, in centimeters, of a group of students are 180, 180, 173, 170, and 167, what is the mean height of these students? 31. What is the median age of a family whose members are 42, 38, 14, 13, 10, and 8 years old? 32. What is the median age of a class in which 14 students are 14 years old and 16 students are 15 years old? 33. In a charity collection, ten people gave amounts of $1, $2, $1, $1, $3, $1, $2, $1, $1, and $1.50. What was the median donation? 34. The test scores for an examination were 62, 67, 67, 70, 90, 93, and 98. What is the median test score? 35. The weekly salaries of six employees in a small firm are $440, $445, $445, $450, $450, and $620. a. For these six salaries, find: (1) the mean (2) the median (3) the mode b. If negotiations for new salaries are in session and you represent management, which measure of central tendency will you use as the average salary? Explain your answer. c. If negotiations are in session and you represent the labor union, which measure of cen- tral tendency will you use as an average salary? Explain your answer. 36. In a certain school district, bus service is provided for students living at least miles from 11 2 school. The distances, rounded to the nearest half mile, from school to home for ten students are 0,, and 10 miles., 1, 1, 1, 1,,, 1 2 1 2 11 2 31 2 a. For these data, find: (1) the mean (2) the median (3) the mode b. How many of the ten students are entitled to bus service? c. Explain why the mean is not a good measure of central tendency to describe the aver- age distance between home and school for these students. 690 Statistics 37. Last month, a carpenter used 12 boxes of nails each of which contained nails of only one size. The sizes marked on the boxes were: 4 in., a. For these data, find: (1) the mean (2) the median (3) the mode 4 in., 3 4 in., 3 4 in., 3 4 in., 3 4 in., 3 4 |
in., 3 3 4 in., 3 1 in., 1 in., 2 in., 2 in. b. Describe the average-size nail used by the carpenter, using at least one of these mea- sures of central tendency. Explain your answer. Hands-On Activity Find the mean, the median, and the mode for the data that you collected in the Hands-On Activity for Section 16-1. It may be necessary to go back to your original data to do this. 16-5 MEASURES OF CENTRAL TENDENCY AND GROUPED DATA Intervals of Length 1 In a statistical study, when the range is small, we can use intervals of length 1 to group the data. For example, each member of a class of 25 students reported the number of books he or she read during the first half of the school year. The data are as follows: 5, 3, 5, 3, 1, 8, 2, 4, 2, 6, 3, 8, 8, 5, 3, 4, 5, 8, 5, 3, 3, 5, 6, 2, 3 These data, for which the values range from 1 to 8, can be organized into a table such as the one shown at the right, with each value representing an interval. Since 25 students were included in this study, the total frequency, N, is 25. We can use this table, with intervals of length 1, to find the mode, median, and mean for these data. Interval Frequency 25 Mode of a Set of Grouped Data Since the greatest frequency, 7, appears for interval 3, the mode for the data is 3. In general: For a set of grouped data, the mode is the value of the interval that contains the greatest frequency. Measures of Central Tendency and Grouped Data 691 Median of a Set of Grouped Data We have learned that the median for a set of data in numerical order is the middle value. For these 25 numbers, there are 12 numbers greater than or equal to the median, and 12 numbers less than or equal to the median. Therefore, when the numbers are written in numerical order, the median is the 13th number from either end. 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 8, 8, 8, 8 ↑– The median is 4. When the data are |
grouped in the table shown earlier, a simple counting procedure can be used to find the median, the 13th number. When we add the frequencies of the first four intervals, starting at the top, we find that these intervals include data for: 4 0 2 6 12 students Therefore, the next lower interval (with frequency greater than 0) must include the median, the value for the 13th student. This is the interval for the data value 4. When we add the frequencies of the first three intervals, starting at the bot- tom, we find that these intervals include data for: 1 3 7 11 students The next higher interval contains two scores, one for the 12th student and one that is the median, or the value for the 13th student. Again this is the interval for the data value 4. In general: For a set of grouped data, the median is the value of the interval that con- tains the middle data value. Mean of a Set of Grouped Data By adding the four 8’s in the ungrouped data, we see that four students, reading eight books each, have read 8 8 8 8 or 32 books. We can arrive at this same number by using the grouped intervals in the table: we multiply the four 8’s by the frequency 4. Thus,. Applying this multiplication shortcut to each row of the table, we obtain the third column of the following table: (4)(8) 5 32 692 Statistics Interval Frequency (Interval) (Frequency = 25 8 4 32 7 0 0 6 2 12 5 6 30 4 2 8 3 7 21 2 3 6 1 1 1 Total 110 The total (110) represents the sum of all 25 pieces of data. We can check this by adding the 25 scores in the unorganized data. Finally, to find the mean, we divide the total number, 110, by the number of items, 25. Thus, the mean for the data is: 110 25 4.4. Procedure To find the mean for N values in a table of grouped data when the length of each interval is 1: 1. For each interval, multiply the interval value by its corresponding frequency. 2. Find the sum of these products. 3. Divide this sum by the total frequency, N. Calculator Solution for Grouped Data The calculator can be used to find the mean and median for the grouped data shown above. Enter the number of books read by each student into L1 and the frequency for each number of books into L |
2. Then use the 1-Var Stats from the STAT CALC menu to display information about the data. ENTER: STAT ENTER 2nd L1, 2nd L2 ENTER DISPLAY< n = 2 5 Measures of Central Tendency and Grouped Data 693 The display shows that the mean, x–, is 4.4, the sum of the number of books read is 110, and the number of students, the total frequency, N, is 25. Use the down arrow to display the median, Med 4. Intervals Other Than Length 1 There are specific mathematical procedures to find the mean, median, and mode for grouped data with intervals other than length 1, but we will not study them at this time. Instead, we will simply identify the intervals that contain some of these measures of central tendency. For example, a small industrial plant surveyed 50 workers to find the number of miles each person commuted to work. The commuting distances were reported, to the nearest mile, as follows: 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 9, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 14, 15, 17, 17, 18, 22, 23, 25, 28, 30, 32, 32, 33, 34, 34, 36, 37, 37, 52 These data are organized into a table with intervals of length 10, as follows: Interval (commuting distance) Frequency (number of workers) 50–59 40–49 30–39 20–29 10–19 0–9 1 0 9 4 15 21 N 50 Modal Interval In the table, interval 0–9 contains the greatest frequency, 21. We say that interval 0–9 is the group mode, or modal interval, because this group of numbers has the greatest frequency. The modal interval is not the same as the mode. The modal interval is a group of numbers; the mode is usually a single number. For this example, the original data (before being placed into the table) show that the number appearing most often is 10. Hence, the mode is 10. The modal interval, which is 0–9, tells us that, of the six intervals in the table, the most frequently occurring commuting distance is 0 to 9 miles. Both the mode and the modal interval depend on the concept |
of greatest frequency. For the mode, we look for a single number that has the greatest frequency. For the modal interval, we look for the interval that has the greatest frequency. 694 Statistics Interval Containing the Median To find the interval containing the median, we follow the procedure described earlier in this section. For 50 numbers, the median, or middle number, will be at a point where 25 numbers are at or above the median and 25 are at or below it. Count the frequencies in the table from the uppermost interval and move downward. We add 1 0 9 4 14. Since there are 15 numbers in the next lower interval, and, we see that the 25th number will be reached somewhere in that interval, 10–19. 14 1 15 5 29 Count from the bottom interval and move up. We have 21 numbers in the first interval. Since there are 15 numbers in the next higher interval, and 21 15 36, we see that 25th number will be reached somewhere in that interval, 10–19. This is the same result that we obtained when we moved downward. The interval containing the median for this grouping is 10–19. In this course, we will not deal with problems in which the median is not found in any interval. Interval Containing the Mean When data are grouped using intervals of length other than 1, there is no simple procedure to identify the interval containing the mean. However, the mean can be approximated by assuming that the data are equally distributed throughout each interval. The mean is then found by using the midpoint of each interval as the value of each entry in the interval. This problem is studied in higher-level courses. EXAMPLE 1 In the table, the data indicate the heights, in inches, of 17 basketball players. For these data find: a. the mode b. the median c. the mean Height (inches) Frequency (number) Solution a. The greatest frequency, 5, occurs for the height of 75 inches. The mode, or height appearing most often, is 75. 77 76 75 2 0 5 74 3 73 b. For 17 players, the median is the 9th number, so there are 8 heights greater than or equal to the median and 8 heights less than or equal to the median. Counting the frequencies going down, we have 2 0 5 7. Since the frequency of the next interval is 3, the 8th, 9th, and 10th heights are in this interval, 74. Counting the frequencies going |
up, we have 1 2 4 7. Again, the frequency of the next interval is 3, and the 8th, 9th, and 10th heights are in this interval. The 9th height, the median, is 74. 71 72 2 1 4 Measures of Central Tendency and Grouped Data 695 c. (1) Multiply each height by its corresponding frequency: 75 5 375 76 0 0 71 1 71 72 2 144 77 2 154 73 4 292 (2) Find the total of these products: 154 0 375 222 292 144 71 1,258 74 3 222 (3) Divide this total, 1,258, by the total frequency, 17 to obtain the mean: 1258 17 74 Calculator Solution Clear any previous data that may be stored in L1 and L2. Enter the heights of the players into L1 and the frequencies into L2. Then use 1-Var Stats from the STAT CALC menu to display information about the data. The screen will show the mean, x–. Press the down arrow key to display the median. ENTER: STAT ENTER 2nd L1, 2nd L2 ENTER DISPLAY Answers a. mode 75 b. median 74 c. mean 74 EXERCISES Writing About Mathematics 1. The median for a set of 50 data values is the average of the 25th and 26th data values when the data is in numerical order. What must be true if the median is equal to one of the data values? Explain your answer. 2. What must be true about a set of data if the median is not one of the data values? Explain your answer. 696 Statistics Developing Skills In 3–5, the data are grouped in each table in intervals of length 1. Find: a. the total frequency b. the mean c. the median d. the mode 3. Interval Frequency 4. Interval Frequency 5. Interval Frequency 10 15 16 17 18 19 20 3 2 4 1 5 6 25 24 23 22 21 20 19 4 0 3 2 4 5 2 In 6–8, the data are grouped in each table in intervals other than length 1. Find: a. the total frequency b. the interval that contains the median c. the modal interval 6. Interval Frequency 7. Interval Frequency 8. Interval Frequency 55–64 45–54 35–44 25–34 15–24 3 8 7 6 2 4–9 10–15 16–21 22–27 28–33 34–39 12 13 |
9 12 15 10 126–150 101–125 76–100 51–75 26–50 1–25 4 6 6 3 7 2 Applying Skills 9. On a test consisting of 20 questions, 15 students received the following scores: 17, 14, 16, 18, 17, 19, 15, 15, 16, 13, 17, 12, 18, 16, 17 a. Make a frequency table for these students listing scores from 12 to 20. b. Find the median score. c. Find the mode. d. Find the mean. Measures of Central Tendency and Grouped Data 697 10. A questionnaire was distributed to 100 people. The table shows the time taken, in minutes, to complete the questionnaire. a. For this set of data, find: (1) the mean (2) the Interval Frequency median (3) the mode b. How are the three measures found in part a related for these data? 6 5 4 3 2 12 20 36 20 12 11. A storeowner kept a tally of the sizes of suits purchased in the store, as shown in the table. a. For this set of data, find: (1) the total frequency (3) the median (2) the mean (4) the mode b. Which measure of central tendency should the store- owner use to describe the average suit sold? Size of Suit Number Sold (interval) (frequency) 48 46 44 42 40 38 36 34 1 1 3 5 3 8 2 2 12. Test scores for a class of 20 students are as follows: 93, 84, 97, 98, 100, 78, 86, 100, 85, 92, 72, 55, 91, 90, 75, 94, 83, 60, 81, 95 a. Organize the data in a table using 51–60 as the smallest interval. b. Find the modal interval. c. Find the interval that contains the median. 13. The following data consist of the weights, in pounds, of 35 adults: 176, 154, 161, 125, 138, 142, 108, 115, 187, 158, 168, 162 135, 120, 134, 190, 195, 117, 142, 133, 138, 151, 150, 168 172, 115, 148, 112, 123, 137, 186, 171, 166, 166, 179 a. Organize the data in a table, using 100–119 as the smallest interval. b. Construct a frequency histogram based on the grouped data. c |
. In what interval is the median for these grouped data? d. What is the modal interval? 698 Statistics 16-6 QUARTILES, PERCENTILES, AND CUMULATIVE FREQUENCY Quartiles When the values in a set of data are listed in numerical order, the median separates the values into two equal parts. The numbers that separate the set into four equal parts are called quartiles. To find the quartile values, we first divide the set of data into two equal parts and then divide each of these parts into two equal parts. The heights, in inches, of 20 students are shown in the following list. The median, which is the average of the 10th and 11th data values, is shown here enclosed in a box. Lower half |______________________________| 53, 60, 61, 63, 64, 65, 65, 65, 65, 66, Upper half |_______________________________| 66, 67, 67, 68, 69, 70, 70, 71, 71, 73 ↑ 66 Median Ten heights are listed in the lower half, 53–66. The middle value for these 10 heights is the average of the 5th and 6th values from the lower end, or 64.5. This value separates the lower half into two equal parts. Ten heights are also listed in the upper half, 66–73. The middle value for these 10 heights is the average of the 5th and 6th values from the upper end, or 69.5. This value separates the upper half into two equal parts. The 20 data values are now separated into four equal parts, or quarters. |_______________| 53, 60, 61, 63, 64, |_______________| 65, 65, 65, 65, 66, ↑ 64.5 |_______________| 66, 67, 67, 68, 69, |_______________| 70, 70, 71, 71, 73 ↑ 69.5 ↑ 66 First quartile Median Second quartile Third quartile The numbers that separate the data into four equal parts are the quartiles. For this set of data: 1. Since one quarter of the heights are less than or equal to 64.5 inches, 64.5 is the lower quartile, or first quartile. 2. Since two quarters of the heights are less than or equal to 66 inches, 66 is the second quartile. The second quartile is always the same as the median. 3. Since three quarters of the heights are less than or equal to 69. |
5 inches, 69.5 is the upper quartile, or third quartile. Note: The quartiles are sometimes denoted Q1, Q2, and Q3. Quartiles, Percentiles, and Cumulative Frequency 699 Procedure To find the quartile values for a set of data: 1. Arrange the data in ascending order from left to right. 2. Find the median for the set of data.The median is the second quartile value. 3. Find the middle value for the lower half of the data.This number is the first, or lower, quartile value. 4. Find the middle value for the upper half of the data.This number is the third, or upper, quartile value. Note that when finding the first quartile, use all of the data values less than or equal to the median, but do not include the median in the calculation. Similarly, when finding the third quartile, use all of the data values greater than or equal to the median, but do not include the median in the calculation. Constructing a Box-and-Whisker Plot A box-and-whisker plot is a diagram that uses the quartile values, together with the maximum and minimum values, to display information about a set of data. To draw a box-and-whisker plot, we use the following steps. STEP 1. Draw a scale with numbers from the minimum to the maximum value of a set of data. For example, for the set of heights of the 20 students, the scale should include the numbers from 53 to 73. STEP 2. Above the scale, place dots to represent the five numbers that are the statistical summary for this set of data: the minimum value, the first quartile, the median, the third quartile, and the maximum value. For the heights of the 20 students, these numbers are 53, 64.5, 66, 69.5 and 73. 50 55 60 65 70 75 STEP 3. Draw a box between the dots that represent the lower and upper quartiles, and a vertical line in the box through the point that represents the median. 50 55 60 65 70 75 700 Statistics STEP 4. Add the whiskers by drawing a line segment joining the dots that represent the minimum data value and the lower quartile, and a second line segment joining the dots that represent the maximum data value and the upper quartile. 50 55 60 65 70 75 The box indicates the ranges of the middle half of the set of data |
. The long whisker at the left shows us that the data are more scattered at the lower than at the higher end. A graphing calculator can display a boxand-whisker plot. Enter the data in L1, then go to the STAT PLOT menu to select the type of graph to draw. ENTER: 2nd STAT PLOT 1 ENTER ENTER 2nd L1 ALPHA = Ty Now display the box-and-whisker plot by P 1 : L 1 entering ZOOM 9. We can press TRACE and the right and left arrow keys to display the minimum value, first quartile, median, third quartile, and maximum value. The five statistical summary can also be displayed in 1-Var Stats. Scroll down to the last five values. ENTER: STAT ENTER ENTER EXAMPLE 1 Find the five statistical summary for the following set of data: 8, 5, 12, 9, 6, 2, 14, 7, 10, 17, 11, 8, 14, 5 Solution (1) Arrange the data in numerical order: Quartiles, Percentiles, and Cumulative Frequency 701 2, 5, 5, 6, 7, 8, 8, 9,10, 11, 12, 14, 14, 17 We can see that 2 is the minimum value and 17 is the maximum value. (2) Find the median. Since there are 14 data values in the set, the median is the average of the 7th and 8th values. Therefore, 8.5 is the second quartile. Median 8 1 9 2 8.5 (3) Find the first quartile. There are seven values less than 8.5. The middle value is the 4th value from the lower end of the set of data, 6. Therefore, 6 is the first, or lower, quartile. (4) Find the third quartile. There are seven values greater than 8.5. The middle value is the 4th value from the upper end of the set of data, 12. Therefore, 12 is the third, or upper, quartile. Answer The minimum is 2, first quartile is 6, the second quartile is 8.5, the third quartile is 12, and the maximum is 17. Note: The quartiles 6, 8.5, and 12 separate the data values into four equal parts even though the original number of data values, 14, is not divisible by 4: |_____| 2, 5, 5, 6 |
, |_____| |________| 7, 8, 8, 9, 10, 11, ↑ 8.5 |_________| 14, 14, 17 12, The first and third quartile values, 6 and 12, are data values. If we think of each of these as a half data value in the groups that they separate, each group contains 3 data values, which is 25% of the total. 1 2 Percentiles A percentile is a number that tells us what percent of the total number of data values lies at or below a given measure. Let us consider again the set of data values representing the heights of 20 students. What is the percentile rank of 65? To find out, we separate the data into the values that are less than or equal to 65 and those that are greater than or equal to 65, so that the four 65’s in the set are divided equally between the two groups: 53, 60, 61, 63, 64, 65, 65, 65, 65, 66, 66, 67, 67, 68, 69, 70, 70, 71, 71, 73 Half of 4, or 2, of the 65’s are in the lower group and half are in the upper group. 702 Statistics Since there are seven data values in the lower group, we find what percent 7 is of 20, the total number of values: 7 20 5 0.35 5 35% Therefore, 65 is at the 35th percentile. To find the percentile rank of 69, we separate the data into the values that are less than or equal to 69 and those that are greater or equal to 69: 53, 60, 61, 63, 64, 65, 65, 65, 65, 66, 66, 67, 67, 68, 69, 70, 70, 71, 71, 73 Because 69 occurs only once, we will include it as half of a data value in the 141 lower group and half of a data value in the upper group. Therefore, there are 2 or 14.5 data values in the lower group. 14.5 20 5 0.725 5 72.5% Because percentiles are usually not written using fractions, we say that 69 is at the 73rd percentile. EXAMPLE 2 Find the percentile rank of 87 in the following set of 30 marks: 56, 65, 65, 67, 72, 73, 75, 77, 77, 78, 78, 78, 80, 80, 80, 82, 83, 85, 85, 85, 86 |
, 87, 87, 87, 88, 90, 92, 93, 95, 98 Solution (1) Find the sum of the number of marks less than 87 and half of the number of 87’s: Number of marks less than 87 21 Half of the number of 87’s (0.5 3) 1.5 22.5 (2) Divide the sum by the total number of marks: (3) Change the decimal value to a percent: 0.75 75%. 22.5 30 5 0.75 Answer: A mark of 87 is at the 75th percentile. Note: 87 is also the upper quartile mark. Cumulative Frequency In a school, a final examination was given to all 240 students taking biology. The test grades of these students were then grouped into a table. At the same time, a histogram of the results was constructed, as shown below. Interval (test scores) Frequency (number) 91–100 81–90 71–80 61–70 51–60 45 60 75 40 20 y c n e u q e r F 80 60 40 20 0 Quartiles, Percentiles, and Cumulative Frequency 703 HISTOGRAM 51–60 61–70 71–80 81–90 91–100 Test scores From the table and the histogram, we can see that 20 students scored in the interval 51–60, 40 students scored in the interval 61–70, and so forth. We can use these data to construct a new type of histogram that will answer the question, “How many students scored below a certain grade?” By answering the following questions, we will gather some information before constructing the new histogram: 1. How many students scored 60 or less on the test? From the lowest interval, 51–60, we know that 20 students scored 60 or less. 2. How many students scored 70 or less on the test? By adding the frequencies for the two lowest intervals, 51–60 and 61–70, we see that 20 40, or 60, students scored 70 or less. 3. How many students scored 80 or less on the test? By adding the frequencies for the three lowest intervals, 51–60, 61–70, and 71–80, we see that 20 40 75, or 135, students scored 80 or less. 4. How many students scored 90 or less on the test? Here, we add the frequencies in the four lowest intervals. Thus, 20 40 75 60, or 195, |
students scored 90 or less. 5. How many students scored 100 or less on the test? By adding the five lowest frequencies, 20 40 75 60 45, we see that 240 students scored 100 or less. This result makes sense because 240 students took the test and all of them scored 100 or less. Constructing a Cumulative Frequency Histogram The answers to the five questions we have just asked were found by adding, or accumulating, the frequencies for the intervals in the grouped data to find the cumulative frequency. The accumulation of data starts with the lowest interval of data values, in this case, the lowest test scores. The histogram that displays these accumulated figures is called a cumulative frequency histogram. 704 Statistics Interval (test scores) Frequency (number) Cumulative Frequency CUMULATIVE FREQUENCY HISTOGRAM 240 91–100 81–90 71–80 61–70 51–60 45 60 75 40 20 240 195 135 60 20 CUMULATIVE FREQUENCY HISTOGRAM 240 100% 210 180 150 120 90 60 30 75% 50% 25% 51–60 51–70 51–80 51–90 51–100 Test scores 210 180 150 120 90 60 30 0 51–60 51–70 51–80 51–90 51–100 Test scores To find the cumulative frequency for each interval, we add the frequency for that interval to the frequencies for the intervals with lower values. To draw a cumulative frequency histogram, we use the cumulative frequencies to determine the heights of the bars. For our example of the 240 biology students and their scores, the frequency scale for the cumulative frequency histogram goes from 0 to 240 (the total frequency for all of the data). We can replace the scale of the cumulative frequency histogram shown above with a different one that expresses the cumulative frequency in percents. Since 240 students represent 100% of the students taking the biology test, we write 100% to correspond to a cumulative frequency of 240. Similarly, since 0 students represent 0% of the students taking the biology test, we write 0% to correspond to a cumulative frequency of 0. If we divide the percent scale into four equal parts, we can label the three added divisions as 25%, 50%, and 75%. Quartiles, Percentiles, and Cumulative Frequency 705 Thus the graph relates each cumulative frequency to a percent of the total number of biology students. For example, 120 students (half of the total number) corresponds to 50%. Let us use the percent scale to answer the question, |
“What percent of the students scored 70 or below on the test?” The height of each bar represents both the number of students and the percent of the students who had scores at or below the largest number in the interval represented by that bar. Since 25%, or a quarter, of the scores were 70 or below, we say that 70 is an approximate value for the lower quartile, or the 25th percentile. CUMULATIVE FREQUENCY HISTOGRAM 240 100 210 180 150 120 90 60 30 0 75% 56% 50% 25% 8% 0% 51–60 51–70 51–80 51–90 51–100 Test scores From the histogram, we can see that about 56% of the students had scores at or below 80. Thus, the second quartile, the median, is in the 51–80 interval. For these data, the upper quartile is in the 51–90 interval. From the histogram, we can also conveniently read the approximate percentiles for the scores that are the end values of the intervals. For example, to find the percentile for a score of 60, the right-end score of the first interval, we draw a horizontal line segment from the height of the first interval to the percent scale, as shown by the dashed line in the histogram above. The fact that the horizontal line crosses the percent scale at about one-third the distance between 0% and 25% tells us that approximately 8% of the students scored 60 or below 60. Thus, the 8th percentile is a good estimate for a score of 60. 706 Statistics EXAMPLE 3 A reporter for the local newspaper is preparing an article on the ice cream stores in the area. She listed the following prices for a two-scoop cone at 15 stores. $2.48, $2.57, $2.30, $2.79, $2.25, $3.00, $2.82, $2.75, $2.55, $2.98, $2.53, $2.40, $2.80, $2.50, $2.65 a. List the data in a stem-and-leaf diagram. b. Find the median. c. Find the first and third quartiles. d. Construct a box-and-whisker plot. e. Draw a cumulative frequency histogram. f. Find the percentile rank of a price of $2.75. Solution a |
. The first two digits in each price will be the stem. The lowest price is $2.25 and the highest price is $3.00. b. Since there are 15 prices, the median is the 8th from the top or from the bottom. The median is $2.57. c. The middle value of the set of numbers below the median is the first quartile. That price is $2.48. The middle value of the set of numbers above the median is the third quartile. That price is $2.80. Stem Leaf 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2. Key: 2.9 8 $2.98 d. Use a scale from $2.25 to $3.00. Place dots at $2.48, $2.57, and $2.80 for the first quartile, the median, and the third quartile. Draw the box around the quartiles with a vertical line through the median. Add the whiskers. 2.25 2.50 2.75 3.00 Interval Frequency Cumulative Frequency 3.00–3.09 2.90–2.99 2.80–2.89 2.70–2.79 2.60–2.69 2.50–2.59 2.40–2.49 2.30–2.39 2.20–2.29 1 1 2 2 1 4 2 1 1 15 14 13 11 9 8 4 2 1 Quartiles, Percentiles, and Cumulative Frequency 707 CUMULATIVE FREQUENCY HISTOGRAM 15 12.20–2.29 2.20–2.39 2.20–2.49 2.20–2.59 2.20–2.69 2.20–2.79 2.20–2.89 2.20–2.99 2.20–3.09 e. Make a cumulative frequency table and draw the histogram. Use 2.20–2.29 Price of a Two-Scoop Cone as the smallest interval. f. There are 9 data values below $2.75. Add 1 2 Percentile rank: 0.63 192 15 for the data value $2.75. – 63% Answers a. Diagram b. median $2.57 c. first quartile $2.48; third quartile $2.80 d |
. Diagram e. Diagram f. 63rd percentile Note: A cumulative frequency histogram can be drawn on a calculator just like a regular histogram. In list L2, where we previously entered the frequencies for each individual interval, we now enter each cumulative frequency. EXERCISES Writing About Mathematics 1. a. Is it possible to determine the percentile rank of a given score if the set of scores is arranged in a stem-and-leaf diagram? Explain why or why not. b. Is it possible to determine the percentile rank of a given score if the set of scores is shown on a cumulative frequency histogram? Explain why or why not. 2. A set of data consisting of 23 consecutive numbers is written in numerical order from left to right. a. The number that is the first quartile is in which position from the left? b. The number that is the third quartile is in which position from the left? 708 Statistics Developing Skills In 3–6, for each set of data: a. Find the five numbers of the statistical summary b. Draw a box-andwhisker plot. 3. 12, 17, 20, 21, 25, 27, 29, 30, 32, 33, 33, 37, 40, 42, 44 4. 67, 70, 72, 77, 78, 78, 80, 84, 86, 88, 90, 92 5. 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 5, 7, 9, 9 6. 3.6, 4.0, 4.2, 4.3, 4.5, 4.8, 4.9, 5.0 In 7–9, data are grouped into tables. For each set of data: a. Construct a cumulative frequency histogram. b. Find the interval in which the lower quartile lies. c. Find the interval in which the median lies. d. Find the interval in which the upper quartile lies. 7. Interval Frequency 8. Interval Frequency 9. Interval Frequency 41–50 31–40 21–30 11–20 1–10 8 5 2 5 4 25–29 20–24 15–19 10–14 5–9 3 1 3 9 9 1–4 5–8 9–12 13–16 17–20 4 3 7 2 2 10. For the data given in the table: a. Construct a cumulative frequency his |
- togram. b. In what interval is the median? c. The value 10 occurs twice in the data. What is the percentile rank of 10? 11. For the data given in the table: a. Construct a cumulative frequency his- togram. b. In what interval is the median? c. In what interval is the upper quartile? d. What percent of scores are 17 or less? e. In what interval is the 25th percentile? Interval Frequency 21–25 16–20 11–15 6–10 1–5 5 4 6 3 2 Interval Frequency 33–37 28–32 23–27 18–22 13–17 8–12 3–7 4 3 7 12 8 5 1 Applying Skills 12. A group of 400 students were asked to state the number of minutes that each spends watching television in 1 day. The cumulative frequency histogram shown below summarizes the responses as percents. a. What percent of the students questioned watch television for 90 minutes or less each day? b. How many of the students watch television for 90 minutes or less each day? c. In what interval is the upper quartile? d. In what interval is the lower quar- tile? e. If one of these students is picked at random, what is the probability that he or she watches 30 minutes or less of television each day? Quartiles, Percentiles, and Cumulative Frequency 709 CUMULATIVE FREQUENCY HISTOGRAM 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 1–30 1–60 1–90 1–120 1–150 1–180 Number of minutes 13. A journalism student was doing a study of the readability of the daily newspaper. She chose several paragraphs at random and listed the number of letters in each of 88 words. She prepared the following chart. a. Copy the chart, adding a column that lists the cumulative frequency b. Find the median. c. Find the first and third quartiles. d. Construct a box-and whisker plot. e. Draw a cumulative frequency his- togram. f. Find the percentile rank of a word with 7 letters. Number of letters Frequency 1 2 3 4 5 6 7 8 9 10 4 14 20 20 3 18 5 2 1 1 14. Cecilia’s average for 4 years is 86. Her average is the upper quartile for her class of 250 students. At most, how many students in her class |
have averages that are less than Cecilia’s? 710 Statistics 15. In the table at the right, data are given for the heights, in inches, of 22 football players. a. Copy and complete the table. b. Draw a cumulative frequency histogram. c. Find the height that is the lower quartile. d. Find the height that is the upper quartile. Height (inches) Frequency Cumulative Frequency 77 76 75 74 73 72 71 2 2 7 5 3 2 1 16. The lower quartile for a set of data was 40. These data consisted of the heights, in inches, of 680 children. At most, how many of these children measured more than 40 inches? In 17 and 18, select, in each case, the numeral preceding the correct answer. 17. On a standardized test, Sally scored at the 80th percentile. This means that (1) Sally answered 80 questions correctly. (2) Sally answered 80% of the questions correctly. (3) Of the students who took the test, about 80% had the same score as Sally. (4) Of the students who took the test, at least 80% had scores that were less than or equal to Sally’s score. 18. For a set of data consisting of test scores, the 50th percentile is 87. Which of the following could be false? (1) 50% of the scores are 87. (2) 50% of the scores are 87 or less. (3) Half of the scores are at least 87. (4) The median is 87. 16-7 BIVARIATE STATISTICS We have been studying univariate statistics or statistics that involve a single set of numbers. Statistics are often used to study the relationship between two different sets of values. For example, a dietician may want to study the relationship between the number of calories from fat in a person’s diet and the level of cholesterol in that person’s blood, or a merchant may want to study the relationship between the amount spent on advertising and gross sales. Although these examples involving two-valued statistics or bivariate statistics require complex statistical methods, we can investigate some of the properties of similar but simpler problems by looking at graphs and by using a graphing calculator. A graph that shows the pairs of values in the data as points in the plane is called a scatter plot. Bivariate Statistics 711 Correlation We will consider five cases of two-valued statistics to investigate the relationship or correlation between |
the variables based on their scatter plots. CASE 1 approximate a straight line that has a positive slope. The data has positive linear correlation. The points in the scatter plot A driver recorded the number of gallons of gasoline used and the number of miles driven each time she filled the tank. In this example, there is both correlation and causation since the increase in the number of miles driven causes the number of gallons of gasoline needed to increase. Gallons 7.2 5.8 7.0 5.5 5.6 7.1 6.0 4.4 5.0 6.2 4.7 5.7 Miles 240 188 226 193 187 235 202 145 167 212 154 188 This scatter plot can be duplicated on your graphing calculator. Enter the number of miles as L1 and the number of gallons of gasoline as corresponding entries in L2. The miles will be graphed as x-values and the gallons of gasoline as y-values. First, turn on Plot 1: 250 225 s e l i M 200 175 150 3 4 5 6 7 8 9 Gallons of Gasoline ENTER: 2nd STAT PLOT 1 ENTER ENTER 2nd L1 2nd L2 DISPLAY = Ty : +. 712 Statistics Now use ZoomStat from the ZOOM menu to construct a window that will include all values of x and y. ENTER: ZOOM 9 DISPLAY: The data has moderate positive correlation. The points in a scatter plot CASE 2 do not lie in a straight line but there is a general tendency for the values of y to increase as the values of x increase. Last month, each student in an English class was required to choose a book to read. The teacher recorded, for each student in the class, the number of days spent reading the book and the number of pages in the book. Days 8 14 12 26 9 17 28 13 15 30 18 20 Pages 225 300 298 356 200 412 205 215 310 357 209 250 Days 29 22 17 14 11 14 22 19 16 7 18 30 Pages 314 288 256 225 232 256 300 305 276 172 318 480 While the books with more pages may have required more time, some students read more rapidly and some spent more time each day reading. The graph shows that, in general, as the number of days needed to read a book increased, the number of pages that were read also increased. 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Days 500 475 450 425 400 375 350 325 300 275 250 225 200 175 s e g a |
P Bivariate Statistics 713 CASE 3 The data has no correlation. Before giving a test, a teacher asked each student how many minutes each had spent the night before preparing for the test. After correcting the test, she prepared the table below which compares the number of minutes of study to the number of correct answers. Minutes of Study Correct Answers 20 15 40 5 10 25 30 12 15 10 3 19 16 6 12 3 5 5 20 35 40 8 16 14 The graph shows that there is no correlation between the time spent studying just before the test and the number of correct answers on the test 20 18 16 14 12 10 8 6 4 2 0 5 10 15 20 25 30 35 40 45 Minutes of Study The data has moderate negative correlation. The points in a scatter plot CASE 4 do not lie in a straight line but there is a general tendency for the values of y to decrease as the values of x increase. A group of children go to an after-school program at a local youth club. The director of the program keeps a record, shown below, of the time, in minutes, each student spends playing video games and doing homework. Games Homework 20 50 30 60 90 10 60 40 30 40 50 35 70 15 40 30 80 30 60 10 In this instance, the unit of measure, minutes, is found in the problem rather than in the table. To create meaningful graphs, always include a unit of measure on the horizontal and vertical axes. 714 Statistics 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 Minutes Playing Games The graph shows that, in general, as the number of minutes spent playing video games increases, the number of minutes spent doing homework decreases. CASE 5 approximate a straight line that has a negative slope. The data has negative linear correlation. The points in the scatter plot A long-distance truck driver travels 500 miles each day. As he passes through different areas on his trip, his average speed and the length of time he drives each day vary. The chart below shows a record of average speed and time for a 10-day period. Speed Time 50 10 64 68 60 54 66 70 62 64 58 7.9 7.5 8.5 9.0 7.0 7.1 8.0 8.2 9.0 11 10.5 10 9.5 9 8.5 8 7.5 7 6. 50 55 60 65 70 75 Speed in Miles per Hour In this case, the increase in the average speed causes the time required to drive a |
fixed distance to decrease. This example indicates both negative correlation and causation. It is important to note that correlation is not the same as causation. Correlation is an indication of the strength of the linear relationship or association between the variables, but it does not mean that changes in one variable are the cause of changes in the other. For example, suppose a study found there was a strong positive correlation between the number of pages in the daily newspaper and the number of voters who turn out for an election. One would not be correct in concluding that a greater number of pages causes a greater turnout. Rather, it is likely that the urgency of the issues is reflected in the increase of both the size of the newspaper and the size of the turnout. Bivariate Statistics 715 Another example where there is no causation occurs in time series or data that is collected at regular intervals over time. For instance, the population of the U.S. recorded every ten years is an example of a time series. In this case, we cannot say that time causes a change in the population. All we can do is note a general trend, if any. Line of Best Fit When it makes sense to consider one variable as the independent variable and the other as the dependent variable, and the data has a linear correlation (even if it is only moderate correlation), the data can be represented by a line of best fit. For example, we can write an equation for the data in Case 1. Enter the data into L1 and L2 if it is not already there. Find the mean values for x, the number of miles driven, and for y, the number of gallons of gasoline used. Then use 2-Var Stats from the STAT CALC menu: ENTER: STAT 2 ENTER DISPLAY< n = 1 2 The calculator gives x– 5.85 and, by pressing the down arrow key, y– 194.75. We will use these mean values, (5.85, 194.75), as one of the points on our line. We will choose one other data point, for example (7.1, 235), as a second point and write the equation of the line using the slope-intercept form y mx b. First find the slope: m y2 2 y1 x2 2 x1 5 194.75 2 235 5.85 2 7.1 5 240.25 21.25 5 32.2 Now use one of the points to find the y-intercept: 194.75 32.2(5.85 |
) b 194.75 188.37 b 6.38 b Round the values to three significant digits. A possible equation for a line of best fit is y 32.2x 6.38. The calculator can also be used to find a line called the regression line to fit a bivariate set of data. Use the LinReg(ax+b) function in the STAT CALC menu. 716 Statistics ENTER: STAT 4 ENTER DISPLAY If we round the values to three significant digits, the equation of the regression line is y 32.7x 3.59. In this case, the difference between these two equations is negligible. However, this is not always the case. The regression line is a special line of best fit that minimizes the square of the vertical distances to each data point. We can compare these two equations with the actual data. Graph the scatter plot of the data using ZoomStat. Then write the two equations in the Y= menu. ENTER: Y 32.2 X,T,,n 6.38 ENTER 32.7 X,T,,n 3.59 GRAPH DISPLAY: Notice that the lines are very close and do approximate the data. Note 1: The equation of the line of best fit is very sensitive to rounding. Try to round the coefficients of the line of best fit to at least three significant digits or to whatever the test question asks. Note 2: A line of best fit is appropriate only for data that exhibit a linear pattern. In more advanced courses, you will learn how to deal with nonlinear patterns. These equations can be used to predict values. For example, if the driver has driven 250 miles before filling the tank, how many gallons of gasoline should be needed? We will use the equation from the calculator. y 32.7x 3.59 250 32.7x 3.59 246.41 32.7x 7.535474006 x Bivariate Statistics 717 It is reasonable to say that the driver can expect to need about 7.5 gallons of gasoline. What we just did is called extrapolation, that is, using the line of best fit to make a prediction outside of the range of data values. Using the line of best fit to make a prediction within the given range of data values is called interpolation. In general, interpolation is usually safe, while care should be taken when extrapolating. The observed correlation pattern may not be valid outside of the given range of data values. For example, consider the scatter plot of the population |
of a town shown below. The population grew at a constant rate during the years in which the data was gathered. However, we do not expect the population to continue to grow forever, and thus, it may not be possible to extrapolate far into the future. 280 270 260 250 240 230 220 210 200 190 180 170 160 ) 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Year Keep in Mind In general, when a given relationship involves two sets of data: 1. In some cases a straight line, a line of best fit, can be drawn to approxi- mate the relationship between the data sets. 2. If a line of best fit has a positive slope, the data has positive linear correla- tion. 3. If the line of best fit has a negative slope, the data has negative linear cor- relation. 4. A line of best fit can be drawn through (x–, y–), the point whose coordinates are the means of the given data. Any data point that appears to lie on or near the line of best fit can be used as a second point to write the equation. 718 Statistics 5. A calculator can be used to find the regression line as the line of best fit. 6. When the graphed data points are so scattered that it is not possible to draw a straight line that approximates the given relationship, the data has no correlation. To study bivariate data without using a graphing calculator: 1. Make a table that lists the data. 2. Plot the data as points on a graph. 3. Find the mean of each set of data and locate the point (x–, y–) on the graph. 4. Draw a line that best approximates the data. 5. Choose the point (x–, y–) and one other point or any two points that are on or close to the line that you drew. Use these points to write an equation of the line. 6. Use the equation of the line to predict related outcomes. To study bivariate data using a graphing calculator: 1. Enter the data into L1 and L2 or any two lists in the memory of the calcu- lator. 2. Use STAT PLOT to turn on a plot and to choose the type of plot needed. Enter the names of the lists in which the data is stored and choose the mark to be used for each data point. 3. Use ZoomStat to choose a viewing window that shows all of the data points. 4. Find the |
regression line using LinReg(ax+b) from the STAT CALC menu. 5. Enter the equation of the regression line in the Y= menu and use to show the relationship between the data and the regression line. GRAPH 6. Use the equation of the line to predict related outcomes. In this course, we have found a line of best fit by finding a line that seems to represent the data or by using a calculator. In more advanced courses in statistics, you will learn detailed methods for finding the line of best fit. The table below shows the number of calories and the number of grams of carbohydrates in a half-cup serving of ten different canned or frozen vegetables. Carbohydrates 9 23 Calories 45 100 4 20 5 19 25 110 8 35 12 50 7 30 13 70 17 80 EXAMPLE 1 Bivariate Statistics 719 a. Draw a scatter plot on graph paper. Let the horizontal axis represent grams of carbohydrates and the vertical axis represent the number of calories. b. Find the mean number of grams of carbohydrates in a serving of vegetables and the mean number of calories in a serving of vegetables. c. On the graph, draw a line that approximates the data in the table, and deter- mine its equation. d. Enter the data in L1 and L2 on your calculator and find the linear regression equation, LinReg(ax+b). e. Use each equation to find the expected number of calories in a serving of vegetables with 20 grams of carbohydrates. Compare the answers. Solution a. s e i r o l a C 120 110 100 90 80 70 60 50 40 30 20 0 5 10 15 20 25 Grams of Carbohydrates b. Enter the number of grams of carbohydrates in L1 and the number of calo- ries in L2. Find x– and y–, using 2-Var Stats. ENTER: STAT 2 ENTER The means of the x- and y-coordinates are x– 11.7 and y– 56.5. Locate the point (11.7, 56.5) on the graph. 720 Statistics c. 120 110 100 90 80 70 60 50 40 30 20 x, y) 10 20 5 Grams of Carbohydrates 15 25 The line we have drawn seems to go through the point (4, 20). We will use this point and the point with the mean values, (11.7, 56.5), to write an equation of a line of best fit. m 5 56.5 2 20 11. |
7 2 4 5 36.5 7.7 < 4.74 y 5 mx 1 b 20 5 36.5 7.7 (4) 1 b An equation of a best fit line is y 4.74x 1.05. 1.05 < b d. The data are in L1 and L2. ENTER: STAT 4 ENTER DISPLAY The equation of the regression line is y = 4.89x 0.660. e. Let x 20. Use the equation from c. y 4.74x 1.05 y 4.74(20) 1.05 y 95.85 Use the equation from d. y = 4.89x 0.660 y 4.89(20) 0.660 y 97.14 The two equations give very similar results. It would be reasonable to say that we could expect the number of calories to be about 96 or close to 100. Bivariate Statistics 721 EXERCISES Writing About Mathematics 1. a. Give an example of a set of bivariate data that has negative correlation. b. Do you think that the change in the independent variable in your example causes the change in the dependent variable? 2. Explain the purpose of finding a line of best fit. Applying Skills 3. When Gina bought a new car, she decided to keep a record of how much gas she uses. Each time she puts gas in the car, she records the number of gallons of gas purchased and the number of miles driven since the last fill-up. Her record for the first 2 months is as follows: Gallons of gas 10 12 9 6 11 10 8 12 10 7 Miles driven 324 375 290 190 345 336 250 375 330 225 a. Draw a scatter plot of the data. Let the horizontal axis represent the number of gallons of gas and the vertical axis represent the number of miles driven. b. Does the data have positive, negative, or no correlation? c. Is this a causal relationship? d. Find the mean number of gallons of gasoline per fill-up. e. Find the mean number of miles driven between fill-ups. f. Locate the point that represents the mean number of gallons of gasoline and the mean number of miles driven. Use (0, 0) as a second point. Draw a line through these two points to approximate the data in the table. g. Use the line drawn in part d to approximate the number of miles Gina could drive on 3 gallons of gasoline. 4. Gemma made a record of the cost |
and length of each of the 14 long-distance telephone calls that she made in the past month. Her record is given below. Minutes 3.7 1.0 19.6 0.8 4.3 34.8 2.9 Cost $0.35 $0.11 $2.12 $0.09 $0.47 $3.78 $0.24 Minutes 2.5 7.1 10.9 5.8 1.5 1.4 8.0 Cost $0.27 $0.79 $1.21 $0.65 $0.20 $0.17 $0.89 a. Draw a scatter plot of the data on graph paper. Let the horizontal axis represent the number of minutes, and the vertical axis represent the cost of the call. b. Does the data have positive, negative, or no correlation? c. Is this a causal relationship? 722 Statistics d. Find the mean number of minutes per call. e. Find the mean cost of the calls. f. On the graph, draw a line of best fit for the data in the table and write its equation. g. Use a calculator to find the equation of the regression line. h. Approximate the cost of a call that lasted 14 minutes using the equation written in d. i. Approximate the cost of a call that lasted 14 minutes using the equation written in e. 5. A local store did a study comparing the cost of a head of lettuce with the number of heads sold in one day. Each week, for five weeks, the price was changed and the average number of heads of lettuce sold per day was recorded. The data is shown in the chart below. Cost per Head of Lettuce $1.50 $1.25 $0.90 $1.75 $0.50 No. of Heads Sold 48 52 70 42 88 a. Draw a scatter plot of the data. Let the horizontal axis represent the cost of a head of lettuce and the vertical axis represent the number of heads sold. b. Does the data have positive, negative, or no correlation? c. Is this a causal relationship? d. Find the mean cost per head. e. Find the mean number of heads sold. f. On the graph, draw a line that approximates the data in the table. g. What appears to be the result of raising the price of a head of lettuce? 6. The chart below shows the recorded heights in inches and weights in pounds for the |
last 24 persons who enrolled in a health club. Height Weight Height Weight Height Weight Height Weight 69 67 63 73 71 79 160 160 135 185 215 225 75 76 70 73 68 74 180 155 175 170 190 190 66 66 68 68 72 77 145 130 160 140 170 195 71 66 67 78 72 69 165 155 140 210 160 145 a. Draw a scatter plot on graph paper to display the data. b. Does the data have positive, negative, or no linear correlation? c. Is this a causal relationship? d. Draw and find the equation of a line of best fit. Use (77, 195) as a second point. e. Use a calculator to find the linear regression line. Bivariate Statistics 723 f. According to the equation written in d., if the next person who enrolls in the health club is 62 inches tall, what would be the expected weight of that person? g. According to the equation written in e., if the next person who enrolls in the health club weighs 200 pounds, what would be the expected height of that person? 7. The chart below shows the number of millions of cellular telephones in use in the United States by year from 1994 to 2003. Year Phones ’94 24.1 ’95 33.8 ’96 44.0 ’97 55.3 ’98 69.2 ’99 86.0 ’00 ’01 ’02 ’03 109.5 128.3 140.8 158.7 Let L1 be the number of years after 1990: 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. a. Draw a scatter plot on graph paper to display the data. b. Does the data have positive, negative, or no linear correlation? c. Is this a causal relationship? d. Draw and find the equation of a line of best fit. e. On the graph, draw a line that approximates the data in the table, and determine its equation. Use (6, 44.0) as a second point. f. If the line of best fit is approximately correct for years beyond 2003, estimate how many cellular phones will be in use in 2007. 8. The chart below shows, for the last 20 Supreme Court Justices to have left the court before 2000, the age at which the judge was nominated and the number of years as a Supreme Court judge. Age Years Age Years 47 15 50 33 64 16 56 16 62 24 62 16 62 |
17 59 7 59 24 50 18 55 4 56 7 54 3 57 13 45 31 49 6 43 23 49 12 56 5 62 1 a. Draw a scatter plot on graph paper to display the data. b. Does the data have positive, negative, or no linear correlation? c. Is there a causal relationship? d. Draw and find an equation of a line of best fit. e. Use a calculator to find the linear regression line. f. Do you think that the data, the line of best fit, the regression line, or none of these could be used to approximate the number of years as a Supreme Court justice for the next person to retire from that office? 724 Statistics 9. A cook was trying different recipes for potato salad and comparing the amount of dressing with the number of potatoes given in the recipe. The following data was recorded. Number of Potatoes Cups of Dressing 7 11 2 4 7 8 2 3 4 8 11 4 6 1 7 13 4 5 11 8 4 3 4 a. Draw a scatter plot on graph paper to display the data. b. Does the data have positive, negative, or no linear correlation? c. Draw and find the equation of a line of best fit. Use d. Use a calculator to find the linear regression line. as a second point. 4, 7 8 A B e. According to the equation written in c., if the cook needs to use 10 potatoes to have enough salad, approximately how many cups of dressing are needed? CHAPTER SUMMARY Statistics is the study of numerical data. In a statistical study, data are col- lected, organized into tables and graphs, and analyzed to draw conclusions. Data can either be quantitative or qualitative. Quantitative data represents counts or measurements. Qualitative data represents categories or qualities. In an experiment, a researcher imposes a treatment on one or more groups. The treatment group receives the treatment, while the control group does not. Tables and stem-and-leaf diagrams are used to organize data. A table should have between five and fifteen intervals that include all data values, are of equal size, and do not overlap. A histogram is a bar graph in which the height of a bar represents the fre- quency of the data values represented by that bar. A cumulative frequency histogram is a bar graph in which the height of the bar represents the total frequency of the data values that are less than or equal to the upper endpoint of that bar. The mean, median, and mode are three measures of central |
tendency. The mean is the sum of the data values divided by the total frequency. The median is the middle value when the data values are placed in numerical order. The mode is the data value that has the largest frequency. Quartile values separate the data into four equal parts. A box-and-whisker plot displays a set of data values using the minimum, the first quartile, the median, the third quartile, and the maximum as significant measures. The percentile rank tells what percent of the data values lie at or below a given measure. In two-valued statistics or bivariate statistics, a relation between two different sets of data is studied. The data can be graphed on a scatter plot. The data may have positive, negative, or no correlation. Data that has positive or negative linear correlation can be represented by a line of best fit. The line of best fit can be used to predict values not in the included data set. Interpolation is predicting within the given data range. Extrapolation is predicting outside of the given data range. Review Exercises 725 VOCABULARY 16-1 Data • Statistics • Descriptive statistics • Qualitative data • Quantitative data • Census • Sample • Bias • Experiment • Treatment group • Control group • Placebo effect • Placebo • Blinding • Singleblind experiment • Double-blind experiment 16-2 Tally • Frequency • Total frequency • Frequency distribution table • Group • Interval • Grouped data • Range • Stem-and-leaf diagram • 16-3 Histogram • Frequency histogram 16-4 Average • Measures of central tendency • Mean • Arithmetic mean • Numerical average • Median • Mode • Bimodal • Linear transformation of a data set 16-5 Group mode • Modal interval 16-6 Quartile • Lower quartile • First quartile • Second quartile • Upper quartile • Third quartile • Box-and-whisker plot • Five statistical summary • Percentile • Cumulative frequency • Cumulative frequency histogram 16-7 Univariate statistics • Two-valued statistics • Bivariate statistics • Scatter plot • Correlation • Causation • Time series • Line of best fit • Regression line • Extrapolation • Interpolation REVIEW EXERCISES 1. Courtney said that the mean of a set of consecutive integers is the same as the median and that the mean can be found by adding the smallest and the largest numbers and dividing the sum by 2 |
. Do you agree with Courtney? Explain why or why not. 2. A set of data contains N numbers arranged in numerical order. a. When is the median one of the numbers in the set of data? b. When is the median not one of the numbers in the set of data? 3. For each of the following sets of data, find: a. the mean b. the median c. the mode (if one exists) (1) 3, 4, 3, 4, 3, 5 (3) 9, 3, 2, 8, 3, 3 (2) 1, 3, 5, 7, 1, 2, 4 (4) 9, 3, 2, 3, 8, 2, 7 4. Express, in terms of y, the mean of 3y 2 and 7y 18. 726 Statistics 5. For the following data: 78, 91, 60, 65, 81, 72, 78, 80, 65, 63, 59, 78, 78, 54, 87, 75, 77 a. Use a stem-and-leaf diagram to organize the data. b. Draw a histogram, using 50–59 as the lowest interval. c. Draw a cumulative frequency histogram. d. Draw a box-and-whisker plot. 6. The weights, in kilograms, of five adults are 53, 72, 68, 70, and 72. a. Find: (1) the mean (2) the median (3) the mode b. If each of the adults lost 5 kilograms, find, for the new set of weights: (1) the mean (2) the median (3) the mode 7. Steve’s test scores are 82, 94, and 91. What grade must Steve earn on a fourth test so that the mean of his four scores will be exactly 90? 8. From Monday to Saturday of a week in May, the recorded high temperature readings were 72°, 75°, 79°, 83°, 83°, and 88°. For these data, find: a. the mean b. the median c. the mode 9. Paul worked the following numbers of hours each week over a 20-week period: 15, 3, 7, 6, 2, 14, 9, 25, 8, 12, 8, 8, 15, 0, 8, 12, 28, 10, 14, 10 a. Organize the data in a frequency table, using 0–5 as the lowest |
interval. b. Draw a frequency histogram of the data. c. In what interval does the median lie? d. Which interval contains the lower quartile? Score Frequency Cumulative Frequency 10. The table shows the scores of 25 test papers. a. Is the data univariate or bivariate? b. Find the mean score. c. Find the median score. d. Find the mode. e. Copy and complete the table. 100 f. Draw a cumulative frequency histogram. g. Find the percentile rank of 90. 60 70 80 90 1 9 8 2 5 h. What is the probability that a paper chosen at random has a score of 80? Review Exercises 727 11. The electoral votes cast for the winning presidential candidate in elections from 1900 to 2004 are as follows: 292, 336, 321, 435, 277, 404, 382, 444, 472, 523, 449, 432, 303, 442, 457, 303, 486, 301, 520, 297, 489, 525, 426, 370, 379, 271, 286 a. Organize the data in a stem-and-leaf diagram. (Use the first digit as the stems, and the last two digits as the leaves.) b. Find the median number of electoral votes cast for the winning candi- date. c. Find the first-quartile and third-quartile values. d. Draw a box-and-whisker plot to display the data. 12. The ages of 21 high school students are shown in the table at the right. a. What is the median age? b. What is the percentile rank of age 15? c. When the ages of these 21 students are combined with the ages of 20 additional students, the median age remains unchanged. What is the smallest possible number of students under 16 in the second group? 13. For each variable, determine if it is qualitative or Age Frequency 18 17 16 15 14 13 1 4 2 7 2 5 quantitative. a. Major in college b. GPA in college c. Wind speed of a hurricane d. Temperature of a rodent e. Yearly profit of a corporation f. Number of students late to class g. Zip code h. Employment status 14. Researchers looked into a possible relationship between alcoholism and pneumonia. They conducted a study of 100 current alcoholics, 50 former alcoholics, and 1,000 non-alcoholics who were hospitalized for a mild form of pneumonia. The researchers found that 30% of alcohol |
ics and 30% of former alcoholics, versus only 15% of the non-alcoholics developed a more dangerous form of pneumonia. The researchers concluded that alcoholism raises the risk for developing pneumonia. Discuss possible problems with this study. 728 Statistics 15. Aurora buys oranges every week. The accompanying table lists the weights and the costs of her last 10 purchases of oranges. Weight (lb) 2.2 1.2 3.6 4.5 1.0 2.5 1.8 5.0 3.5 1.7 Cost ($) 1.22 0.60 1.04 1.58 0.50 0.89 0.95 1.88 1.46 0.70 a. Is the data univariate or bivariate? b. Draw a scatter plot of the data on graph paper. Let the horizontal axis represent the weights of the oranges and the vertical axis the costs. c. Is there a correlation between the weight and the cost of the oranges? If so, is it positive or negative? d. If the price is determined by the number of oranges purchased, do the variables have a causal relationship? Explain your answer. e. On the graph, draw a line of best fit that approximates the data in the table and write its equation. f. Use the equation written in d to approximate the cost of 4 pounds of oranges. 16. Explain why the graph on the right is misleading. (Hint: In accounting, numbers enclosed by parentheses denote negative numbers.) NET INCOME OF XYZ COMPANY (in thousands) $12,000 $8,050 $7,100 $5,123 $(4,000) 2002 2003 2004 2005 2006 Exploration a. Marny took the SAT in 2004 and scored a 1370. She was in the 94th percentile. Jordan took the SAT in 2000 and scored 1370. He was in the 95th percentile. Explain how this is possible. b. Taylor’s class rank stayed the same even though he had a cumulative grade point average of 3.4 one semester and 3.8 the next semester. Explain how this is possible. CUMULATIVE REVIEW Part I Cumulative Review 729 CHAPTERS 1–16 Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. When the domain is the set of integers, the solution set of the inequality 0, 0.1x 2 0.4 # 0.2 (1) { } is ( |
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