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Frequency Tables, Line Plots, and Histograms

Learn how to formulate a set of numbers with a specific median and mean, find the median and mode of data sets, and create frequency tables, line plots, and histograms.

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Frequency Tables, Line Plots, and Histograms

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  1. Sample answer 90, 92, 95, 110, 113 Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 Formulate a set of five different numbers whose median is 95 and whose mean is 100. 12-1

  2. Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 (For help, go to Lesson 3-3.) Find the median and mode of each data set. 1. 6, 9, 9, 5, 9 2. 73, 78, 77, 73, 79 3. 300, 100, 200, 150, 300 4. 3, 5, 7, 9, 3, 4, 6, 3, 7 Check Skills You’ll Need 12-1

  3. Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 Solutions 1. 5, 6, 9, 9, 9 2. 73, 73, 77, 78, 79 median = 9 median = 77 mode = 9 mode = 73 3. 100, 150, 200, 300, 300 4. 3, 3, 3, 4, 5, 6, 7, 7, 9 median = 200 median = 5 mode = 300 mode = 3 12-1

  4. List the numbers of hours in order. Use a tally mark for each result. Count the tally marks and record the frequency. Number Tally Frequency 1 6 |||| | 6 2 |||| | 3 7 |||| || 4 ||| 3 Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 Quick Check A survey asked 22 students how many hours of TV they watched daily. The results are shown. Display the data in a frequency table. 1 3 4 3 1 1 2 3 4 1 3 2 2 1 3 2 1 2 3 2 4 3 12-1

  5. “How many cases are you trying?” Number Frequency 0 3 1 5 2 4 3 5 4 4 Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 Twenty-one judges were asked how many cases they were trying on Monday. The frequency table below shows their responses. Display the data in a line plot. Then find the range. 12-1

  6. For a line plot, follow these steps 1 , 2 , and 3 . 3 Write a title that describes the data. Cases Tried by Judges x x x x x x x x x x x x x x x x x x x x x 2 Mark an x for each response. 0 1 2 3 4 1 Draw a number line with the choices below it. Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 (continued) The greatest value in the data set is 4 and the least value is 0. So the range is 4 – 0, or 4. Quick Check 12-1

  7. Pets Tally Frequency 0 IIII 5 1 IIII II 7 2 III 3 3 II 2 range: 12 Frequency Tables, Line Plots, and Histograms PRE-ALGEBRA LESSON 12-1 Solve. 1. Maria surveyed friends to find out how many pets each one has. The responses are below. Display the data in a frequency table. 0, 1, 1, 2, 3, 0, 1, 1, 3, 2, 1, 2, 1, 0, 1, 0, 0 2. Here are the numbers of books students read in the last month: 5, 4, 0, 12, 4, 5, 4, 3, 10, 5, 12, 3, 5, 7, 3, 10, 5, 0, 6. Display the data in a line plot. Then find the range. 12-1

  8. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 If the population of the United States is approximately 250 million and the median age is 33 yr, roughly how many people in the United States are younger than 33 years old? 125 million 12-2

  9. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 (For help, go to Lesson 3-3.) Find each median. 1. 12, 10, 11, 7, 9, 8, 10, 5 2. 4.5, 3.2, 6.3, 5.2, 5, 4.8, 6, 3.9 3. 55, 53, 67, 52, 50, 49, 51, 52, 52, 52 4. 101, 100, 100, 105, 102, 101 Check Skills You’ll Need 12-2

  10. 52 + 52 2 4.8 + 5 2 9 + 10 2 101 + 101 2 Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 Solutions 1. 5, 7, 8, 9, 10, 10, 11, 12 median = = 9.5 2. 3.2, 3.9, 4.5, 4.8, 5, 5.2, 6, 6.3 median = = 4.9 3. 49, 50, 51, 52, 52, 52, 52, 53, 55, 67 median = = 52 4. 100, 100, 101, 101, 102, 105 median = = 101 12-2

  11. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 The data below represent the wingspans in centimeters of captured birds. Make a box-and-whisker plot. 61 35 61 22 33 29 40 62 21 49 72 75 28 21 54 Step 1 Arrange the data in order from least to greatest. Find the median. 21 21 22 28 29 33 35 40 49 54 61 61 62 72 75 Step 2  Find the lower quartile and upper quartile, which are the medians of the lower and upper halves. 21 21 22 28 29 33 354049 54 61 61 62 72 75 lower quartile = 28 upper quartile = 61 12-2

  12. Step 3 Draw a number line. Mark the least and greatest values, the median, and the quartiles. Draw a box from the first to the third quartiles. Mark the median with a vertical segment. Draw whiskers from the box to the least and greatest values. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 (continued) Quick Check 12-2

  13. Draw a number line for both sets of data. Use the range of data points to choose a scale. Draw the second box-and-whisker plot below the first one. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 Use box-and-whisker plots to compare test scores from two math classes. Class A: 92, 84, 76, 68, 90, 67, 82, 71, 79, 85, 79 Class B: 78, 93, 81, 98, 69, 95, 74, 87, 81, 75, 83 Quick Check 12-2

  14. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 Describe the data in the box-and-whisker plot. The lowest score is 55 and the highest is 85. One fourth of the scores are at or below 66 and one fourth of the scores are at or above 80. Half of the scores are at or between 66 and 80 and thus within 10 points of the median, 76. Quick Check 12-2

  15. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 The plots below compare the percents of students who were eligible to those who participated in extracurricular activities in one school from 1992 to 2002. What conclusions can you draw? About 95% of the students were eligible to participate in extracurricular activities. Around 60% of the students did participate. A little less than two thirds of the eligible students participated in extracurricular activities. Quick Check 12-2

  16. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 Solve. 1. Use the data to make a box-and-whisker plot. Student heights (in.) are: 60, 66, 59, 67, 68, 63, 62, 61, 69, 64, 61. a. What is the median height? b. Between what heights do 50% of the students fall? 63 in. between 61 in. and 67 in. 12-2

  17. Box-and-Whisker Plots PRE-ALGEBRA LESSON 12-2 2. The box-and-whisker plots below compare prices for the same items at Mary’s Discount Store and Ed’s Clothing. What conclusions can you draw? Prices at the discount store are more tightly grouped around the median price of $25. Half the items cost from $18 to $45. For less expensive items, there is not much difference in the prices at the two stores. For more expensive items, the discount store offers lower prices. 12-2

  18. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 Use estimation to create a circle graph. Twenty percent of the students surveyed always prefer a salad. Thirty percent always prefer a hamburger. The rest prefer a hamburger sometimes and a salad sometimes. Check students’ graphs. The graphs should show 20% (72°) salad, 30% (108°) hamburger, and 50% (180°) salad or hamburger. 12-3

  19. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 (For help, go to Lesson 8-3.) Find the slope of AB in each graph. 1.2. Check Skills You’ll Need 12-3

  20. 1 2 1 2 Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 Solutions 1.2. 12-3

  21. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 Which title would be more appropriate for the graph below: “Texas Overwhelms California” or “Areas of California and Texas”? Explain. 12-3

  22. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 (continued) Because of the break in the vertical axis, the bar for Texas appears to be more than six times the height of the bar for California. Actually, the area of Texas is about 267,000 mi2, which is not even two times the area of California, which is about 159,000 mi2. The title “Texas Overwhelms California” could be misleading. “Areas of Texas and California” better describes the information in the graph. Quick Check 12-3

  23. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 Study the graphs below. Which graph gives the impression of a sharper increase in rainfall from March to April? Explain. 12-3

  24. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 (continued) In the second graph, the months are closer together and the rainfall amounts are farther apart than in the first graph. Thus the line appears to climb more rapidly from March to April in the second graph. Quick Check 12-3

  25. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 What makes the graph misleading? Explain. The “cake” on the right has much more than two times the area of the cake on the left. Quick Check 12-3

  26. Using Graphs to Persuade PRE-ALGEBRA LESSON 12-3 Solve. 1. Name two ways you can use graphs to give different impressions of the same data. 2. What makes this graph misleading? Explain. Answers may vary. Sample: Use breaks in scales; vary the size of scale intervals. Both dimensions of the “photo” on the right were changed. This gives the impression that the difference in costs is much greater than it really is. 12-3

  27. 1 18 1 12 0 ; ; Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 If you roll two standard number cubes, what is the probability that the sum will be 1? 11? Less than 4? 12-4

  28. Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 (For help, go to Lesson 6-4.) A bag has 5 blue (B) chips, 4 red (R) chips, and 3 tan (T) chips. Find each probability for choosing a chip at random from the bag. 1.P(R) 2.P(not R) 3.P(B) 4.P(R or B) 5.P(T) 6. P(B or T) Check Skills You’ll Need 12-4

  29. 2 3 2 3 1 3 1 4 3 4 favorable outcomes all possible outcomes favorable outcomes all possible outcomes favorable outcomes all possible outcomes favorable outcomes all possible outcomes favorable outcomes all possible outcomes favorable outcomes all possible outcomes drawing a red chip 12 = = = drawing a chip that is not red 12 = = = drawing a blue chip 12 = = drawing a red or blue chip 12 drawing a tan chip 12 = = = drawing a blue or tan chip 12 = = = = = = 4 12 3 12 5 12 8 12 9 12 8 12 Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 Solutions 1. 2. 3. 4. 5. 6. 12-4

  30. mayonnaise Each branch of the “tree” represents one choice—for example, wheat-ham-mayonnaise. ham mustard wheat mayonnaise turkey mustard mayonnaise ham mustard white mayonnaise turkey mustard Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 The school cafeteria sells sandwiches for which you can choose one item from each of the following categories: two breads (wheat or white), two meats (ham or turkey), and two condiments (mayonnaise or mustard). Draw a tree diagram to find the number of sandwich choices. There are 8 possible sandwich choices. Quick Check 12-4

  31. first digit, possible choices second digit, possible choices numbers, possible choices 5 • 5 = 25 Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 How many two-digit numbers can be formed for which the first digit is odd and the second digit is even? There are 25 possible two-digit numbers in which the first digit is odd and the second digit is even. Quick Check 12-4

  32. right right wrong right wrong wrong number of favorable outcomes number of possible outcomes Use the probability formula. P(event) = 1 4 = 1 4 The probability of guessing correctly on two true/false questions is . Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 Quick Check Use a tree diagram to show the sample space for guessing right or wrong on two true-false questions. Then find the probability of guessing correctly on both questions. The tree diagram shows there are four possible outcomes, one of which is guessing correctly on both questions. 12-4

  33. 1st digit outcomes 10 2nd digit outcomes 10 3rd digit outcomes 10 4th digit outcomes 10 5th digit outcomes 10 total outcomes = 100,000 • • • • number of favorable outcomes number of possible outcomes P(winning number) = = 1 20,000 The probability is , or . 5 100,000 5 100,000 Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 In some state lotteries, the winning number is made up of five digits chosen at random. Suppose a player buys 5 tickets with different numbers. What is the probability that the player has a winning number? First find the number of possible outcomes. For each digit, there are 10 possible outcomes, 0 through 9. Then find the probability when there are five favorable outcomes. Quick Check 12-4

  34. 1 6 Counting Outcomes and Theoretical Probability PRE-ALGEBRA LESSON 12-4 Use the following information for Questions 1 and 2. In a game, a number cube is tossed to determine the number of spaces to move, and a coin is tossed to determine forward or backward movement. 1. How many possible outcomes are there? 2. What is the theoretical probability you will move four spaces? 3. How many different three-digit whole numbers are possible using the digits 1, 2, 3, 4, and 5? 12 125 12-4

  35. 3 4 12 15 2 21 1 15 1 2 5 2 5 6 3 5 2 or Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Multiply. Express the answers in lowest terms. 7 10 b. c.  a.   3 12-5

  36. 4 7 5 9 3 6 2 4 4 8 1 5 3 5 8 9 1 4 9 10 2 10 4 10 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 (For help, go to Lesson 5-4.) Multiply. 1. • 2. • 3. • 4. • 5. • 6. • Check Skills You’ll Need 12-5

  37. 1 8 2 7 4 5 5 18 8 100 72 90 2 16 12 42 20 72 3 25 2 25 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Solutions 1. 2. 3. 4. 5. 6. = = = = = 12-5

  38. 1 6 P(5) = There is one 5 among 6 numbers on a number cube. 3 6 P(less than 4) = There are three numbers less than 4 on a number cube. P(5, then less than 4) = P(5) • P(less than 4) 1 6 3 6 = • 3 36 1 12 1 12 = , or The probability of rolling 5 and then a number less than 4 is . Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Quick Check You roll a number cube once. Then you roll it again. What is the probability that you get 5 on the first roll and a number less than 4 on the second roll? 12-5

  39. P(a seed grows) = 50%, or 0.50 Write the percent as a decimal. Substitute. = 0.50 • 0.50 Multiply. = 0.25 Write 0.25 as a percent. = 25% Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Bluebonnets grow wild in the southwestern United States. Under the best conditions in the wild, each bluebonnet seed has a 20% probability of growing. Suppose you plant bluebonnet seeds in your garden and use a fertilizer that increases to 50% the probability that a seed will grow. If you select two seeds at random, what is the probability that both will grow in your garden? P(two seeds grow) = P(a seed grows) • P(a seed grows) Quick Check The probability that two seeds will grow is 25%. 12-5

  40. 2 5 P(boy) = Two of five students are boys. If a boy’s name is drawn, one of the four remaining students is a boy. 1 4 P(boy after boy) = 2 5 1 4 Substitute. = • 2 20 1 10 1 10 Simplify. = , or The probability that both representatives will be boys is . Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Three girls and two boys volunteer to represent their class at a school assembly. The teacher selects one name and then another from a bag containing the five students’ names. What is the probability that both representatives will be boys? P(boy, then boy) = P(boy) • P(boy after boy) Quick Check 12-5

  41. 1 12 Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Solve. 1. You roll a number cube once. Then you roll it again. What is the probability that you get 6 on the first roll and a number greater than 3 on the second roll? 2. Suppose there are three white marbles and three black marbles in a bag and you want to remove two marbles. What is the probability that you will select a white marble and then a black marble? Express your answer as a percent. 30% 12-5

  42. 7 22 5 33 ; Independent and Dependent Events PRE-ALGEBRA LESSON 12-5 Solve. 3. Each of five girls and seven boys wants to be one of the two announcers for a variety show. To be fair, a teacher puts the names of the twelve students in a hat and draws two. What is the probability that the teacher will draw the names of two boys? Of two girls? 12-5

  43. Permutations and Combinations PRE-ALGEBRA LESSON 12-6 Bianca’s family needs to choose exterior paint for their new house. The wall colors are white, green, and beige. The trim colors are white, green, blue, and cocoa. How many combinations of wall color and trim are possible? 12 12-6

  44. Permutations and Combinations PRE-ALGEBRA LESSON 12-6 (For help, go to Lesson 12-4.) Use the Counting Principle to find the number of outcomes. 1. Roll 2 number cubes. 2. Choose three different letters. 3. Select a month and a day of the week. 4. Toss a coin 4 times. Check Skills You’ll Need 12-6

  45. Permutations and Combinations PRE-ALGEBRA LESSON 12-6 Solutions 1.first numbersecond number 6 outcomes 6 outcomes 6  6 = 36 2.first lettersecond letterthird letter (different from 1st)(different from 1st & 2nd) 26 choices 25 choices 24 choices 26  25  24 = 15,600 12-6

  46. Permutations and Combinations PRE-ALGEBRA LESSON 12-6 Solutions (continued) 3.monthday 12 choices 7 choices 12  7 = 84 4.first tosssecond tossthird tossfourth toss 2 outcomes 2 outcomes 2 outcomes 2 outcomes 2  2  2  2 = 16 12-6

  47. 1st letter 5 choices 5 2nd letter 4 choices 4 3rd letter 3 choices 3 4th letter 2 choices 2 5th letter 1 choice 1 = 120 • • • • Permutations and Combinations PRE-ALGEBRA LESSON 12-6 Find the number of permutations possible for the letters H, O, M, E, and S. There are 120 permutations of the letters H, O, M, E, and S. Quick Check 12-6

  48. 7 studentsChoose 3. 7P3 Permutations and Combinations PRE-ALGEBRA LESSON 12-6 In how many ways can you line up 3 students chosen from 7 students for a photograph? = 7 • 6 • 5 = 210 Simplify You can line up 3 students from 7 in 210 ways. Quick Check 12-6

  49. State Area (mi2) Alabama Colorado Maine Oregon Texas 50,750 103,729 30,865 96,003 261,914 Permutations and Combinations PRE-ALGEBRA LESSON 12-6 In how many ways can you choose two states from the table when you write reports about the areas of states? Make an organized list of all the combinations. 12-6

  50. Permutations and Combinations PRE-ALGEBRA LESSON 12-6 (continued) AL, COAL, MEAL, ORAL, TXUse abbreviations of each CO, MECO, ORCO, TXstate’s name. First, list all ME, ORME, TXpairs containing Alabama. OR, TXContinue until every pair of states is listed. There are ten ways to choose two states from a list of five. Quick Check 12-6

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