Data Analysis. Notes. Introduction In this unit, students will learn how to analyze data using statistical

Transcription

1 Notes Data nalysis Introduction In this unit, students will learn how to analyze data using statistical analysis. This includes understanding sampling techniques, histograms, and box-and-whisker plots. Students will then learn how to use probability to predict outcomes. ssessment Options Unit 5 Test Pages of the Chapter Resource Masters may be used as a test or review for Unit 5. This assessment contains both multiple-choice and short answer items. Collecting and analyzing data allows you to make decisions and predictions about the future. In this unit, you will learn about statistics and probability. TestCheck and Worksheet Builder This CD-ROM can be used to create additional unit tests and review worksheets. Chapter 3 Statistics Chapter Probability 7 Unit 5 Data nalysis 7 Unit 5 Data nalysis

2 Teaching Suggestions Have students study the US TODY Snapshot. sk students what they can determine from the data in the graph. The population of the United States grew by 9 million in the years from 9 to. What might be needed to predict the population of the United States in the year 5? Sample answer: More information about the population growth rate and how much it has increased annually. dditional US TODY Snapshots appearing in Unit 5: Chapter 3 Cheaper wireless talk (p. 73) Chapter Women follow football on TV (p. 7) merica Counts! The U.S. government has been counting each person in the country since its first Census following independence was taken in 79. Befitting the first Census of the st century, the Census Bureau allowed Census questionnaires to be completed electronically for the first time. In this project, you will see how data analysis can be used to compare statistics about a state of your choice to other states in the United States. US TODY Snapshots U.S. population growth The U.S. population has more than doubled since 9. million U5-C-US 3 million Log on to Begin your WebQuest by reading the Task. Then continue working on your WebQuest as you study Unit 5. Lesson Page Source: U.S. Census Bureau By Marcy E. Mullins, US TODY Unit 5 Data nalysis 75 Internet Project WebQuest is an online project in which students do research on the Internet, gather data, and make presentations using word processing, graphing, page-making, or presentation software. In each chapter, students advance to the next step in their WebQuest. t the end of Chapter, the project culminates with a presentation of their findings. Teaching notes and sample answers are available in the WebQuest and Project Resources. Unit 5 Data nalysis 75

3 Statistics Chapter Overview and Pacing PCING (days) Regular Block Basic/ Basic/ verage dvanced verage dvanced Sampling and Bias (pp. 7 73) optional optional.5 Identify various sampling techniques. Recognize a biased sample. LESSON OBJECTIVES Introduction to Matrices (pp. 75 7) optional optional Organize data in matrices. Solve problems by adding or subtracting matrices or by multiplying by a scalar. Histograms (pp. 7 73) optional optional Interpret data displayed in histograms. Display data in histograms. Follow-Up: Use a graphing calculator to find an appropriate regression equation. Measures of Variation (pp ) optional optional Find the range of a set of data. Find the quartiles and interquartile range of a set of data. Box-and-Whisker Plots (pp ) optional 3 optional.5 Organize and use data in box-and-whisker plots. Organize and use data in parallel box-and-whisker plots. Follow-Up: Use tables to determine percentiles. Study Guide and Practice Test (pp ) optional optional.5 Standardized Test Practice (pp ) Chapter ssessment optional optional.5 TOTL 3 7 Pacing suggestions for the entire year can be found on pages T T. 7 Chapter 3 Statistics

4 Timesaving Tools Chapter Resource Manager ll-in-one Planner and Resource Center See pages T T3. Study Guide and Intervention CHPTER 3 RESOURCE MSTERS Practice (Skills and verage) Reading to Learn Mathematics Enrichment ssessment Prerequisite Skills Workbook pplications* Parent and Student Study Guide Workbook 5-Minute Check Transparencies Interactive Chalkboard lgepss: Tutorial Plus (lessons) Materials GCS graphing calculator , GCS, graphing calculator, SC 5 (Follow-Up: graphing calculator) 799 3, SC, (Preview: ruler) SM 3, 3 *Key to bbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 3 Statistics 7B

5 Mathematical Connections and Background Continuity of Instruction Prior Knowledge Students were introduced to analyzing data in tables and graphs, and to determining whether these data are misleading in Chapter. In Chapter, students learned to add, subtract, and multiply real numbers and how to represent data in line plots and stemand-leaf plots. They also learned to use the mean, median, and mode of data sets. In Chapter 5, students interpreted scatter plots and found lines of fit. This Chapter Students go beyond what they have already learned about statistics by identifying various sampling techniquesand interpreting data. They learn how to organize data in matrices, and manipulate the data by adding and subtracting matrices, or by scalar multiplication. Students then interpret data in histograms, and find the range, quartiles, and interquartile ranges of data sets. Finally students organize and use data in boxand-whisker plots. Future Connections Whether on television, in newspapers, or on the Internet, statistics are used to sway public opinion, to inform, or to persuade the public to buy a product. Being able to interpret these statistics is important to being able to make sound decisions. The decisions made based on statistics vary from the trivial, such as what shampoo to buy, to the most important, such as who to vote for in an upcoming election. Sampling and Bias To understand sampling techniques, students must first understand that a sample is a small portion of a larger group called a population. Samples are taken to represent a group because they are smaller and easier to survey. If an entire population is included in a sample, it is a census. Samples are used to find preferences or characteristics of a population. sample that is chosen without preference is a random sample. Random samples are chosen in different ways. simple random sample is exactly as it sounds, with members picked at random from a population without bias. If a population is first segregated into non-overlapping groups, from which random samples are taken, then it is a stratified random sample. n example would be if an algebra teacher randomly chose three people from each of his of her classes. systematic random sample is picked by following a certain pattern, such as picking every fifth person who walks by. Samples are biased if they favor one or more parts of a population. Biased samples include convenience samples, in which members of the sample are picked because they are convenient for the person taking the sample. nother example of a biased sample is a voluntary response sample. It is biased because the members of the sample only replied if they wanted to be included. Introduction to Matrices matrix is a rectangular array of numerical data arranged in regular rows and columns. The dimensions of a matrix are the number of rows and columns in the matrix. Each entry in a matrix is called an element. If two matrices have the same dimensions, then they can be added or subtracted by adding or subtracting the corresponding elements of the two matrices. If their dimensions are not the same, then they cannot be added or subtracted. Matrices can also be multiplied by a single real number called a scalar. In scalar multiplication, each member of a matrix is multiplied by the same scalar. 7C Chapter 3 Statistics

6 Histograms Imagine surveying people leaving a shopping mall and finding out that have spent $ or more, but less than $5. Six have spent $5 or more, but less than $, and have spent less than $5. These data can be represented in a two-column frequency table. The left column has three intervals, $ $5, $5 $, and $ $5. Then the next column shows the number of people who fit into each category. Each person is counted with a tally mark. The amount represented by the tallies for a category is called the frequency for each category. Now display the data from the frequency table as a bar graph. The horizontal axis would correspond to the left column of the table, showing three measurement classes; $ $5, $5 $, and $ $5. These must be organized in equal intervals. The vertical axis would correspond to the right column, and would display the frequency for each measurement class. This graph is known as a histogram. Histograms are used to compare data visually. You can quickly determine form looking at the bars which measurement class has the most, the least, etc. Box-and-Whisker Plots Use a box-and-whisker plot to graphically represent the measures of variation on a number line. The box portion of a box-and-whisker plot extends from the lower quartile to the upper quartile, with the median denoted within the box. The box represents the interquartile range. The whiskers extend from the lower quartile to the least value, and from the upper quartile to the greatest value. If either the greatest or least values are outliers, then the whiskers extend to the least or greatest values that are not outliers. The ends of the whiskers are the extreme values. Two sets of data can be compared by drawing two box-and-whisker plots above the same number line. This display is called parallel box-and-whisker plots. Measures of Variation Knowing how a set of data varies is often very helpful in interpreting the data. Mean, median, and mode, which were studied in Chapter, are measures of central tendency. Measures that describe the spread of the values in a set of data are called measures of variation. One such measure is the range, which is the difference between the greatest and least data values. Quartiles are another measure of variation. They are values that separate the data into four equal subsets. The lower quartile separates the lower half of the data into two equal parts. The upper quartile separates the upper half of the data into two equal parts. The interquartile range is the difference between the upper and lower quartiles. n outlier is a value in a set of data that is much less or much greater than the rest of the data. Chapter 3 Statistics 7D

7 and ssessment Type Student Edition Teacher Resources Technology/Internet SSESSMENT INTERVENTION Ongoing Prerequisite Skills, pp. 77, 7, 7, 73 Practice Quiz, p. 7 Practice Quiz, p. 73 Mixed Review Error nalysis Standardized Test Practice Open-Ended ssessment Chapter ssessment pp. 73, 7, 7, 73, 7 Find the Error, pp. 77, 733 pp. 73, 7, 73, 7, 7, 7, 73, 7, 79, Writing in Math, pp. 73, 7, 7, 73, 7 Open Ended, pp. 7, 77, 75, 733, 739 Standardized Test, p. 75 Study Guide, pp Practice Test, p Minute Check Transparencies Prerequisite Skills Workbook, pp., 9, 97 9 Quizzes, CRM pp. 5 Mid-Chapter Test, CRM p. 7 Study Guide and Intervention, CRM pp. 7 7, 77 7, , 799, 5 Cumulative Review, CRM p. Find the Error, TWE pp. 77, 733 Unlocking Misconceptions, TWE pp. 75, 73, 73 Tips for New Teachers, TWE p. 7 TWE pp Standardized Test Practice, CRM pp. 9 3 Modeling: TWE p. 7 Speaking: TWE pp. 73, 73 Writing: TWE pp. 7, 7 Open-Ended ssessment, CRM p. 3 Multiple-Choice Tests (Forms,, B), CRM pp. Free-Response Tests (Forms C, D, 3), CRM pp. 7 Vocabulary Test/Review, CRM p. lgepss: Tutorial Plus Standardized Test Practice CD-ROM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabulary_review Key to bbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters dditional Intervention Resources The Princeton Review s Cracking the ST & PST The Princeton Review s Cracking the CT LEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management System to keep student records (optional) Special banks are included for ST, CT, TIMSS, NEP, and End-of-Course tests. 7E Chapter 3 Statistics

8 Intervention Technology lgepss: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. lgebra Lesson lgepss Lesson Integration: Introduction to Statistics LEKS is an online mathematics learning system that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. or log on to Log on for student study help. For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. For chapter review, there is vocabulary review, test practice, and standardized test practice For more information on Intervention and ssessment, see pp. T T. Reading and Writing in Mathematics Glencoe lgebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Study Organizer, p. 77 Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 7, 77, 75, 733, 739) Reading Mathematics, p. 7 Writing in Math questions in every lesson, pp. 73, 7, 7, 73, 7 Reading Study Tip, pp. 73, 737 WebQuest, p. 7 Teacher Wraparound Edition Foldables Study Organizer, pp. 77, 75 Study Notebook suggestions, pp. 7, 7, 7, 75, 73, 7, 7 Modeling activities, p. 7 Speaking activities, pp. 73, 73 Writing activities, pp. 7, 7 Differentiated Instruction, (Verbal/Linguistic), p. 7 ELL Resources, pp. 7, 7, 7, 79, 7, 7, 735, 7, 75 dditional Resources Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 3 Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter 3 Resource Masters, pp. 75, 79, 797, 3, 9) Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources Hot Words/Hot Topics Sections.,.., 5. For more information on Reading and Writing in Mathematics, see pp. T T7. Chapter 3 Statistics 7F

9 Notes Statistics Have students read over the list of objectives and make a list of any words with which they are not familiar. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson. Lesson 3- Identify various sampling techniques. Lesson 3- Solve problems by adding or subtracting matrices or by multiplying by a scalar. Lesson 3-3 Interpret data displayed in histograms. Lesson 3- Find the range, quartiles, and interquartile range of a set of data. Lesson 3-5 Organize and use data in box-and-whisker plots. Key Vocabulary sample (p. 7) matrix (p. 75) histogram (p. 7) quartile (p. 73) box-and-whisker plot (p. 737) Each day statistics are reported in the newspapers, in magazines, on television, and on the radio. These data involve business, government, ecology, sports, and many other topics. basic knowledge of statistics allows you to interpret what you hear and read in the media. One important tool to help you understand the significance of a set of data is the box-and-whisker plot. You will draw and use a box-and-whisker plot for data involving NSCR racing in Lesson 3-5. NCTM Local Lesson Standards Objectives 3-, 5,,, 9 3-, 5,,, 9, 3-3, 5,,, 9, 3-3 5,,, 9 Follow-Up 3-, 5,,, 9, 3-5, 5,,, 9, 3-5, 5,, 9, Follow-Up Key to NCTM Standards: =Number & Operations, =lgebra, 3=Geometry, =Measurement, 5=Data nalysis & Probability, =Problem Solving, 7=Reasoning & Proof, =Communication, 9=Connections, =Representation 7 Chapter 3 Statistics Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 3 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 3 test. 7 Chapter 3 Statistics

10 Prerequisite Skills To be successful in this chapter, you ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 3. For Lesson 3- Use Logical Reasoning Find a counterexample for each statement. (For review, see Lesson -7.). If a b c, then a c. Sample answer: If a 5 and b, then c 3. However, If a flower is a rose, then it is red. Sample answer: It could be a yellow rose. 3. If Tara obeys the speed limit, then she will drive 5 miles per hour or less. 3. Sample answer: The speed. If a number is even, then it is divisible by. limit could be 55 mph, and Sample answer: is even, but not divisible by. Tara could be driving 5 mph. For Lesson 3- Find the Median Find the median for each set of data. (For review, see pages and 9.) 5., 7, 9, 5, 5, 59, 3 5.,,,, 9, 5,,,, 3,, 7, ,, 7, 7, 395, 355, 37, 35,, 375 For Lesson 3-5 Graph each set of numbers on a number line. Graph Numbers on a Number Line (For review, see Lesson -.). See margin.. {7, 9,, 3, } 9. {5, 7.5, 9,.5, 3}. {3.,., 5., 5.7,.}. {.3,., 3., 3.7,.5} This section provides a review of the basic concepts needed before beginning Chapter 3. Page references are included for additional student help. dditional review is provided in the Prerequisite Skills Workbook, pp., 9, and Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson. For Prerequisite Lesson Skill 3- Finding Sums and Differences, p Interpreting Graphs, p Finding the Median, p Graphing Numbers on a Number Line, p. 73 Stack sheets of paper with edges inch apart. 3 Stack Pages Crease and Staple Make this Foldable to help you organize information about statistics. Begin with three sheets of plain " by " paper. ll tabs should be the same size. Fold Up Bottom Edges Turn and Label nswers Staple along fold. Label the tabs with topics from the chapter. Statistics 3- Sampling and Bias 3- Matrices 3-3 Histograms 3- Measures of Variation 3-5 Box-and-Whisker Plots Reading and Writing notes and examples. s you read and study the chapter, use each page to write Chapter 3 Statistics 77 For more information about Foldables, see Teaching Mathematics with Foldables. TM Organization of Data and Statistics in Writing Students use their Foldables to take notes, define terms, record concepts, and write examples. On the back of the Foldable, have students record examples of statistics they see in everyday print newspapers, magazines, and advertisements. Note how writers use statistics to prove or disprove points of view, and discuss the ethical responsibilities writers have when using statistics. Chapter 3 Statistics 77

11 Lesson Notes Sampling and Bias Focus 5-Minute Check Transparency 3- Use as a quiz or review of Chapter. Mathematical Background notes are available for this lesson on p. 7C. is sampling important in manufacturing? sk students: Suppose the manufacturer produces CDs an hour, and takes a sample every hour. How many CDs would be sampled in an -hour day? What is an example of something you would sample at home? Sample answer: While cooking, you might sample gravy to check that it tastes good. Vocabulary sample population census random sample simple random sample stratified random sample systematic random sample biased sample convenience sample voluntary response sample Identify various sampling techniques. Recognize a biased sample. is sampling important in manufacturing? Manufacturing music CDs involves burning, or recording, copies from a master. However, not every burn is successful. It is costly and time-consuming to check every CD that is burned. Therefore, in order to monitor production, some CDs are picked at random and checked for defects. SMPLING TECHNIQUES When you wish to make an investigation, there are four ways that you can collect data. published data Use data that are already in a source like a newspaper or book. observational study Watch naturally occurring events and record the results. experiment Conduct an experiment and record the results. survey sk questions of a group of people and record the results. When performing an experiment or taking a survey, researchers often choose a sample. sample is some portion of a larger group, called the population, selected to represent that group. If all of the units within a population are included, it is called a census. Sample data are often used to estimate a characteristic within an entire population, such as voting preferences prior to elections. Population all of the light bulbs manufactured on a production line all of the water in a swimming pool all of the people in the United States Sample light bulbs selected from the production line a test tube of water from the pool 59 people from throughout the United States random sample of a population is selected so that it is representative of the entire population. The sample is chosen without any preference. There are several ways to pick a random sample. Random Samples Random Samples Type Definition Example Simple simple random sample is a sample The students in a class are each assigned a Random that is as likely to be chosen different number from to. Then three of the Sample as any other from the population. numbers are picked at random. Stratified Random Sample In a stratified random sample, the population is first divided into similar, nonoverlapping groups. simple random sample is then selected from each group. The students in a school are divided into freshman, sophomores, juniors, and seniors. Then two students are randomly selected from each group of students. Systematic In a systematic random sample, Every minutes, an item is pulled off the assembly line. Random the items are selected according or Sample to a specified time or item interval. Every twentieth item is pulled off the assembly line. 7 Chapter 3 Statistics Resource Manager Workbook and Reproducible Masters Chapter 3 Resource Masters Study Guide and Intervention, pp. 7 7 Skills Practice, p. 73 Practice, p. 7 Reading to Learn Mathematics, p. 75 Enrichment, p. 7 Parent and Student Study Guide Workbook, p. Transparencies 5-Minute Check Transparency 3- nswer Key Transparencies Technology Interactive Chalkboard

12 Ecology More than one twentieth of the area of Minnesota is covered by inland lakes. The largest lake is Red Lake, which covers 3 square miles. Source: World Book Encyclopedia Example Classify a Random Sample ECOLOGY Ten lakes are selected randomly from a list of all public-access lakes in Minnesota. Then liters of water are drawn from feet deep in each of the ten lakes. a. Identify the sample and suggest a population from which it was selected. The sample is ten -liter containers of lake water, one from each of lakes. The population is lake water from all of the public-access lakes in Minnesota. b. Classify the sample as simple, stratified, or systematic. This is a simple random sample. Each of the ten lakes was equally likely to have been chosen from the list. BISED SMPLE Random samples are unbiased. In a biased sample, one or more parts of a population are favored over others. Example Identify Sample as Biased or Unbiased Identify each sample as biased or unbiased. Explain your reasoning. a. MNUFCTURING Every th bolt is pulled from the production line and measured for length. The sample is chosen using a specified time interval. This is an unbiased sample because it is a systematic random sample. b. MUSIC Every tenth customer in line for a certain rock band s concert tickets is asked about his or her favorite rock band. The sample is a biased sample because customers in line for concert tickets are more likely to name the band giving the concert as a favorite band. Two popular forms of samples that are often biased include convenience samples and voluntary response samples. Type Definition Example Biased Samples convenience sample To check spoilage, a produce worker selects Convenience includes members of a apples from the top of the bin. The Sample population that are easily apples are unlikely to represent all of the accessed. apples in the bin. voluntary response radio call-in show records that 75% of its sample involves only callers voiced negative opinions about Voluntary those who want to a local football team. Those callers are Response participate in the unlikely to represent the entire local Sample sampling. population. Volunteer callers are more likely to have strong opinions and are typically more negative than the entire population. Example 3 Identify and Classify a Biased Sample BUSINESS The travel account records from of the departments in a corporation are to be reviewed. The accountant states that the first departments to voluntarily submit their records will be reviewed. a. Identify the sample and suggest a population from which it was selected. The sample is the travel account records from departments in the corporation. The population is the travel account records from all departments in the corporation. Teacher to Teacher Patricia Taepke Lesson 3- Sampling and Bias 79 South Hills H.S., West Covina, C "I copy pages 7 and 79 for my students to place in their lgebra Study Notebooks. These pages contain a great deal of vocabulary that they may use for future reference. It is arranged in a very concise manner." Teach SMPLING TECHNIQUES In-Class Example Teaching Tip Remind students that the sample is what is taken, and the population is the group from which the sample is taken. population does not have to be a group of people. RETIL Each day, a department store chain selects one male and one female shopper randomly from each of their 57 stores, and asks them survey questions about their shopping habits. a. Identify the sample and suggest a population from which it was selected. The sample is 57 male and 57 female shoppers each day. The population is shoppers in the chain s stores. b. Classify the sample as simple, stratified, or systematic. This is a stratified random sample. BISED SMPLE In-Class Example Power Point Power Point Identify each sample as biased or unbiased. Explain your reasoning. a. STUDENT COUNCIL The student council surveys the students in one classroom to decide the theme for the spring dance. The sample is biased because it includes only the students in one classroom. b. SCHOOL The Parent ssociation surveys the parents of every fifth student on the school roster to decide whether to hold a fundraiser. The sample is unbiased because the parents are picked using a systematic method. Lesson 3- Sampling and Bias 79

13 In-Class Examples Power Point 3 COMMUNITY The maintenance chairperson of a neighborhood association has been asked by the association to survey the residents of the neighborhood to find out when to hold a neighborhood clean up day. The chairperson decides to ask her immediate neighbors, and the neighbors in the houses directly across the street from her house. a. Identify the sample, and suggest a population from which it was selected. The sample is the chairperson s immediate neighbors and the neighbors across the street. The population is the residents of the neighborhood. b. Classify the sample as a convenience sample, or a voluntary response sample. This is a convenience sample because the chairperson asked only her closest neighbors. SCHOOL The high school Parent ssociation sent a letter to the parents of all graduating seniors asking them to return the enclosed ballot if they had a preference on where the graduation party was to be held. a. Identify the sample. The sample is a group of parents of the graduating seniors. b. Suggest a population from which the sample was selected. The population is all the parents of the graduating seniors. c. State whether the sample is unbiased (random) or biased. If unbiased, classify it as simple, stratified, or systematic. If biased, classify it as convenience or voluntary response. The sample is biased. It is a voluntary response sample. Concept Check Guided Practice GUIDED PRCTICE KEY Exercises Examples 7. a group of readers of a newspaper; all readers of the newspaper; biased; voluntary response 5. work from students; work from all students in the st period math class; biased; voluntary response 7 Chapter 3 Statistics b. Classify the sample as convenience or voluntary response. Since the departments voluntarily submit their records, this is a voluntary response sample. Example Identify the Sample NEWS REPORTING For an article in the school paper, Rafael needs to determine whether students in his school believe that an arts center should be added to the school. He polls 5 of his friends who sing in the choir. Twelve of them think the school needs an arts center, so Rafael reports that % of the students surveyed support the project. a. Identify the sample. The sample is a group of students from the choir. b. Suggest a population from which the sample was selected. The population for the survey is all of the students in the school. c. State whether the sample is unbiased (random) or biased. If unbiased, classify it as simple, stratified, or systematic. If biased, classify it as convenience or voluntary response. The sample was not randomly selected from the entire student body. So the reported support is not likey to be representative of the student body. The sample is biased. Since Rafael polled only his friends, it is a convenience sample.. Describe how the following three types of sampling techniques are similar and how they are different. See margin. simple random sample stratified random sample systematic random sample. Explain the difference between a convenience sample and a voluntary response sample. See margin. 3. OPEN ENDED Give an example of a biased sample. Sample answer: sk the members of the school s football team to name their favorite sport. Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response.. NEWSPPERS The local newspaper asks readers to write letters stating their preferred candidate for mayor. 5. SCHOOL teacher needs a sample of work from students in her first-period math class to display at the school open house. She selects the work of the first students who raise their hands.. BUSINESS hardware store wants to assess the strength of nails it sells. Store personnel select 5 boxes at random from among all of the boxes on the shelves. From each of the 5 boxes, they select one nail at random and subject it to a strength test. 5 nails; all nails on the store shelves; unbiased; stratified 7. SCHOOL class advisor hears complaints about an incorrect spelling of the school name on pencils sold at the school store. The advisor goes to the store and asks Namid to gather a sample of pencils and look for spelling errors. Namid grabs the closest box of pencils and counts out pencils from the top of the box. She checks the pencils, returns them to the box, and reports the results to the advisor. pencils; all pencils in the school store; biased; convenience Differentiated Instruction Visual/Spatial Place students in small groups. Give each group a number of different colored beads to serve as a population. Then, have the groups model the different types of random samples with the beads. For example, for stratified random samples, students must first divide the beads into groups by color and then take random beads from each group. Have students describe how they would take a systematic random sample. 7 Chapter 3 Statistics

14 indicates increased difficulty Practice and pply Homework Help For See Exercises Examples Extra Practice See page 9. Food Michigan leads the nation in cherry production by growing about 9 million pounds of cherries per year. Source: World Book Encyclopedia 9. a group of high-definition television sets; all high-definition television sets manufactured on one line during one shift; unbiased; systematic Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response.. SCHOOL Pieces of paper with the names of 3 sophomores are drawn from a hat containing identical pieces of paper with all sophomores names. 3 sophomores; all sophomores in the school; unbiased; simple 9. FOOD Twenty shoppers outside a fast-food restaurant are asked to name their preferred cola among two choices. shoppers; all shoppers; biased; convenience. RECYCLING n interviewer goes from house to house on weekdays between 9.M. and P.M. to determine how many people recycle. people who are home between 9.M. and P.M.; all people in the neighborhood; biased; convenience. POPULTION state is first divided into its counties and then people from each county are chosen at random. people from a state; all people in the state; unbiased; stratified. SCOOTERS scooter manufacturer is concerned about quality control. The manufacturer checks the first 5 scooters off the line in the morning and the last 5 off the line in the afternoon for defects. scooters; all scooters manufactured on a particular production line during one day; biased; convenience 3. SCHOOL To determine who will speak for her class at the school board meeting, Ms. Finchie used the numbers appearing next to her students names in her grade book. She writes each of the numbers on an identical piece of paper and shuffles the pieces of papers in a box. Without seeing the contents of the box, one student draws 3 pieces of paper from the box. The students whose numbers match the numbers chosen will speak for the class. 3 students; all of the students in Ms. Finchie s class; unbiased; simple. FRMING n -ounce jar was filled with corn from a storage silo by dipping the jar into the pile of corn. The corn in the jar was then analyzed for moisture content. an -oz jar of corn; all corn in the storage silo; biased; convenience 5. COURT The gender makeup of district court judges in the United States is to be estimated from a sample. ll judges are grouped geographically by federal reserve districts. Within each of the federal reserve districts, all judges names are assigned a distinct random number. In each district, the numbers are then listed in order. number between and inclusive is selected at random, and the judge with that number is selected. Then every th name after the first selected number is also included in the sample. a group of U.S. district court judges; all U.S. district court judges; unbiased; stratified. TELEVISION television station asks its viewers to share their opinions about a proposed golf course to be built just outside the city limits. Viewers can call one of two 9-numbers. One number represents a yes vote, and the other number represents a no vote. a group of people who watch a television station; all people who watch the television station; biased; voluntary response 7. GOVERNMENT To discuss leadership issues shared by all United States Senators, the President asks of his closest colleagues in the Senate to meet with him. U.S. Senators; all U.S. Senators; biased; convenience FOOD To sample the quality of the Bing cherries throughout the produce department, the produce manager picks up a handful of cherries from the edge of one case and checks to see if these cherries are spoiled. a handful of Bing cherries; all Bing cherries in the produce department; biased; convenience 9. MNUFCTURING During the manufacture of high-definition televisions, units are checked for defects. Within the first minutes of a work shift, a television is randomly chosen from the line of completed sets. For the rest of the shift, every 5th television on the line is checked for defects. Lesson 3- Sampling and Bias 7 3 Practice/pply Study Notebook Have students add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 3. include explanations on how to identify whether a sample is random, the type of random sample, and whether the sample is biased. include any other item(s) that they find helpful in mastering the skills in this lesson. bout the Exercises Odd/Even ssignments Exercises,, 5, 7, and are structured so that students practice the same concepts whether they are assigned odd or even problems. ssignment Guide Basic: 9 odd, 9 5 verage: 9 odd,, 3, 9 5 dvanced: even, 5 (optional: 5) nswers. ll three are unbiased samples. However, the methods for selecting each type of sample are different. In a simple random sample, a sample is as likely to be chosen as any other from the population. In a stratified random sample, the population is first divided into similar, nonoverlapping groups. Then a simple random sample is selected from each group. In a systematic random sample, the items are selected according to a specified time or item interval.. convenience sample is a biased sample that is determined based on the ease with which it is possible to gather the sample. voluntary sample is a biased sample composed of voluntary responses. Lesson 3- Sampling and Bias 7

15 Study Guide and Intervention, p. 7 (shown) and p. 7 NME DTE PERIOD 3- Study Guide and Intervention Sampling and Bias Sampling Techniques Suppose you want to survey students about their choice of radio stations. ll students make up the population you want to survey. sample is some portion of the larger group that you select to represent the entire group. census would include all students within the population. random sample of a population is selected so that it is representative of the entire population. Simple Random Sample Stratified Random Sample Systematic Random Sample a sample that is as likely to be chosen as another from a population population is first divided into similar, nonoverlapping groups. simple random sample is then chosen from each group. Items are selected according to a specified time or interval. Example SCHOOL Ten students Example DOOR PRIZES Each of are chosen randomly from each high the participants in a conference was school class to be on an advisory given a numbered name tag. committee with the principal. Twenty-five numbers were chosen at random to receive a door prize. a. Identify the sample and suggest a population from which it was chosen. a. Identify the sample and suggest a The sample is groups of students population from which it was chosen. each from the freshmen, sophomore, The sample was 5 participants of the junior, and senior classes. The population conference. The population was all of the is the entire student body of the school. participants of the conference. b. Classify the sample as simple, b. Classify the sample as simple, stratified, or systematic. stratified, or systematic. This is a stratified random sample because Since the numbers were chosen the population was first divided into randomly, this is a simple random nonoverlapping groups and then a random sample because each participant was sample was chosen from each group. equally likely to be chosen. Exercises Identify each sample, suggest a population from which it was selected, and classify the sample as simple, stratified, or systematic.. SCHOOL Each student in a class of. GRDENING gardener divided a lot 5 students was given a number at the into 5-square-foot sections. He then took beginning of the year. Periodically, the soil samples from each and tested the teacher chooses numbers at random to samples for mineral content. soil display their homework on the overhead samples from each section; entire projector. students; 5 students lot; stratified in the class; simple 3. SCHOOL One hundred students in the. SHOPPING Every tenth person leaving a lunch room are chosen for a survey. ll grocery store was asked if they would students in the school eat lunch at the participate in a community survey. every same time. students; all tenth person leaving a grocery students; simple store; all shoppers at the grocery store; systematic Skills Practice, p. 73 and NME DTE PERIOD 3- Practice (verage) Practice, Sampling and p. Bias 7 (shown) Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response.. GOVERNMENT t a town council meeting, the chair asks 5 citizens attending for their opinions on whether to approve rezoning for a residential area. 5 citizens of a town; all citizens of a town; biased; convenience. BOTNY To determine the extent of leaf blight in the maple trees at a nature preserve, a botanist divides the reserve into sections, randomly selects a -foot by -foot square in the section, and then examines all the maple trees in the section. the maple trees in a square area of each of sections at a nature preserve; all the maple trees at the nature preserve; unbiased; stratified 3. FINNCES To determine the popularity of online banking in the United States, a polling company sends a mail-in survey to 5 adults to see if they bank online, and if they do, how many times they bank online each month. 5 U.S. adults; all U.S. adults; biased; voluntary response. SHOES shoe manufacturer wants to check the quality of its shoes. Every twenty minutes, pairs of shoes are pulled off the assembly line for a thorough quality inspection. pairs of shoes every minutes on an assembly line; all the pairs of shoes coming down an assembly line; unbiased; systematic 5. BUSINESS To learn which benefits employees at a large company think are most important, the management has a computer select 5 employees at random. The employees are then interviewed by the Human Relations department. 5 employees of a company; all employees of a company; unbiased; simple. BUSINESS n insurance company checks every hundredth claim payment to ensure that claims have been processed correctly. every hundredth claim payment at an insurance company; all claim payments at an insurance company; unbiased; systematic 7. ENVIRONMENT Suppose you want to know if a manufacturing plant is discharging contaminants into a local river. Describe an unbiased way in which you could check the river water for contaminants. Sample answer: t a different time each day, take a -ounce sample of water from given locations just upstream and just downstream from where the plant discharges its wastes. Compare the samples for contaminants to see if any are entering the river from the discharge.. SCHOOL Suppose you want to know the issues most important to teachers at your school. Describe an unbiased way in which you could conduct your survey. Sample answer: Obtain a list of all teachers at the school. ssign each teacher a number, and then randomly select numbers. Interview each of the teachers assigned one of the selected numbers. Reading to Learn Mathematics, p NME DTE PERIOD Reading to Learn Mathematics Sampling and Bias ELL Pre-ctivity Why is sampling important in manufacturing? Read the introduction to Lesson 3- at the top of page 7 in your textbook. From what group are the CDs picked at random and then checked for defects? ll of the CDs that are burned. Reading the Lesson Suppose the principal at a school wants to use Saturdays as make-up days when school is closed for inclement weather. The principal selects and then polls a group of students to see if the student body supports the idea. Complete the sentences.. The student body is the population from which a sample of students is selected to be polled. If all the students are polled, it is called a census.. If all students are requested to enter school through the administration building and every twenty-fifth student is selected to be polled, then the sample is a systematic random sample. If only those students who are in the four classrooms closest to the principal s office are selected for the poll, then the sample is a convenience sample. If the principal announces a poll and then interviews the students who sign up to be interviewed, then the sample is a voluntary response sample. 3. Numbers can be assigned to all students and a computer can select 5 of the numbers at random. The students assigned those numbers would be polled. This would be a simple random sample. If students are first divided according to grade and then chosen at random from each group, then the sample is a stratified random sample.. ll random samples are unbiased since they are selected without preference for one unit of the population over another. biased sample favors one part or parts of the population over other parts. Helping You Remember 5. To remember what a stratified random sample is, look up the word stratified in a dictionary. What everyday meaning do you find that seems closest to the mathematical meaning presented in this lesson? Sample answer: to become formed into layers Lesson 3-. We know that the results are from a national survey conducted by Yankelovich Partners for Microsoft Corporation. 3. dditional information needed includes how the survey was conducted, how the survey respondents were selected, and the number of respondents. 7. Sample answer: Randomly pick 5 rows from each field of tomatoes and then pick a tomato every 5 ft along each row. 7 Chapter 3 Statistics 3- Enrichment, p. 7 Geometric Vanishing cts Puzzles of this type use a trick drawing. It appears that rearranging the pieces of each figure causes one or more squares to disappear. Make figures of your own on graph paper. Then explain the trick in each puzzle.. The rectangle has an area of 5 square. The square has an area of square units, units, but the square has an area of but the rectangle has an area of only only square units. 3 square units. C B B C B NME DTE PERIOD B C D Identify each sample, suggest a population from which it was selected, and state whether it is unbiased (random) or biased. If unbiased, classify the sample as simple, stratified, or systematic. If biased, classify as convenience or voluntary response.. BUSINESS To get reaction about a benefits package, a company uses a computer program to randomly pick one person from each of its departments. a group of employees; all employees of the company; unbiased; stratified. MOVIES magazine is trying to determine the most popular actor of the year. It asks its readers to mail the name of their favorite actor to the magazine s office. a group of readers of a magazine; all readers of the magazine; biased; voluntary response COLLEGE For Exercises and 3, use the following information. The graph at the right reveals that 5% of survey respondents did not have a formal financial plan for a child s college tuition.. Write a statement to describe what you do know about the sample. 3. What additional information would you like to have about the sample to determine whether the sample is biased?. SCHOOL Suppose you want to sample the opinion of the students in your school about a new dress code. Describe an unbiased way to conduct your survey. Sample answer: Get a copy of the school s list of students and call every th person on the list. 5. ELECTIONS Suppose you are running for mayor of your city and want to know if you are likely to be elected. Describe an unbiased way to poll the voters. Sample answer: Get a copy of the list of registered voters in the city and call every th person. Topics at Family. FMILY Study the graph at the right. Dinners Describe the information that is revealed in the graph. What information is there about the type or size of the sample? See margin. 7. FRMING Suppose you are a farmer and want to know if your tomato crop is ready to harvest. Describe an unbiased way to determine whether the crop is ready to harvest.. MNUFCTURING Suppose you want to know whether the infant car seats manufactured by your company meet the government standards for safety. Describe an unbiased way to determine whether the seats meet the standards. Sample answer: Every hour pull one infant seat from the end of the assembly line for testing. 9. CRITICL THINKING The following is a proposal for surveying a stratified random sample of the student body. Divide the student body according to those who are on the basketball team, those who are in the band, and those who are in the drama club. Then take a simple random sample from each of the three groups. Conduct the survey using this sample. Study the proposal. Describe its strengths and weaknesses. Is the sample a stratified random sample? Explain. See margin. Planning for Kids College Costs national survey asked U.S. parents: Do you have a formal financial plan or program that will provide for the future cost of your child s education? Not Sure 3% How the Day Was Family-Related News Plans For Tomorrow Current Events Yes % No 5% Source: Yankelovich Partners for Microsoft Corp. Source: National Pork Producers Council 9% 73% 5% % 7 Chapter 3 Statistics D C The triangle C actually has a height

16 Standardized Test Practice Maintain Your Skills Mixed Review Getting Ready for the Next Lesson 3. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See pp B. Why is sampling important in manufacturing? Include the following in your answer: an unbiased way to pick which CDs to check, and a biased way to pick which CDs to check. 3. To predict the candidate who will win the seat in city council, which method would give the newspaper the most accurate result? B sk every 5th person that passes a reporter in the mall. B Use a list of registered voters and call every th person. C Publish a survey and ask readers to reply. D sk reporters at the newspaper. 3. cookie manufacturer plans to make a new type of cookie and wants to know if people will buy these cookies. For accurate results, which method should they use? D sk visitors to their factory to evaluate the cookie. B Place a sample of the new cookie with their other cookies, and ask people to answer a questionnaire about the cookie. C Take samples to a school, and ask students to raise their hands if they like the cookie. D Divide the United States into regions. Then pick 3 cities in each region at random, and conduct a taste test in each of the cities. Solve each equation. (Lesson -9) y y r r r 35. m m 3 m 3 5 Simplify. (Lesson -) 35 t 3. 5 a x 37. a 5 t 3. x a 5t a 7 a t x t GEOMETRY What is the perimeter of BC? (Lesson -) cm cm 35 cm 5 cm Solve each equation by using the Quadratic Formula. pproximate any irrational roots to the nearest tenth. (Lesson -). x x,. b 5 9b. d 9d 3.3,. Find each product. (Lesson -7) 3, 3. (y 5)(y 7). (c 3)(c 7) 5. (x )(x ) y y 35 c c x x 3 BSIC SKILL Find each sum or difference Lesson 3- Sampling and Bias 73 ssess Open-Ended ssessment Speaking Pass out newspapers or news magazines and have students scan the articles for the results of opinion polls. When students find such results, have them identify the sample and population for the poll. Then have them describe how the people conducting the poll could make sure the sample was not biased. Getting Ready for Lesson 3- BSIC SKILL Students will learn how to organize data in matrices, and how to add and subtract matrices in Lesson 3-. It is important that students understand basic addition and subtraction in order to add and subtract matrices. Use Exercises 5 to determine your students familiarity with basic addition and subtraction. nswer 9. It is a good idea to divide the school population into groups and to take a simple random sample from each group. The problem that prevents this from being a legitimate stratified random sample is the way the three groups are formed. The three groups probably do not represent all students. The students who do not participate in any of these three activities will not be represented in the survey. Other students may be involved in two or three of these activities. These students will be more likely to be chosen for the survey. nswer. The graph shows four phrases with a percent associated with each phrase. We can assume that the percents indicate the percent of respondents who said the indicated topic was discussed during family dinners. Based on the sum of the percents, respondents must have been able to choose or state more than one topic. We do not know how many respondents there were, whether the respondents selected topics from a list of choices or stated their own topics, whether there were any restrictions that may have existed about the topics, and the time period of the family dinners considered in this survey (a night, a week, a month, or more). Lesson 3- Sampling and Bias 73

17 Reading Mathematics Getting Started Before using this page, ask students if they have ever asked for permission from their parents to do something, and tried to influence the way their parents answered. Have volunteers describe some of the methods they use to influence their parents decisions. Teach Biased Questions Discuss with students why the two questions about sales tax on Internet purchases might have elicited different responses. Explain that the reason for saying yes to question two is that the question points out that the tax would have been paid at a store purchase. People are more likely to agree to spending money that they would have otherwise spent elsewhere, so the question is biased. ssess Study Notebook sk students to summarize what they have learned about asking biased questions in their study notebooks. ELL English Language Learners may benefit from writing key concepts from this activity in their Study Notebooks in their native language and then in English. nswers a. This question will bias people toward answering yes because it gives them a reason to think that recycling will help alleviate a shortage in resources. Survey Questions Even though taking a random sample eliminates bias or favoritism in the choice of a sample, questions may be worded to influence people s thoughts in a desired direction. Two different surveys on Internet sales tax had different results. Question Question Should there be sales tax on Do you think people should or should purchases made on the Internet? not be required to pay the same sales tax for purchases made over the Internet that they would if they had bought the item in person at a local store? Yes 3% No 5% Notice the difference in Questions and. Question includes more information. Pointing out that customers pay sales tax for items bought at a local store may give the people answering the survey a reason to answer yes. sking the question in that way probably led people to answer the way they did. Because they are random samples, the results of both of these surveys are accurate. However, the results could be used in a misleading way by someone with an interest in the issue. For example, an Internet retailer would prefer to state the results of Question. Be sure to think about survey questions carefully to make sure that you interpret the results correctly. Reading to Learn For Exercises, tell whether each question is likely to bias the results. Explain your reasoning. 3. See margin.. On a survey on environmental issues: a. Due to diminishing resources, should a law be made to require recycling? b. Should the government require citizens to participate in recycling efforts?. On a survey on education: a. Should schools fund extracurricular sports programs? b. The budget of the River Valley School District is short of funds. Should taxes be raised in order for the district to fund extracurricular sports programs? 3. Suppose you want to determine whether to serve hamburgers or pizza at the class party. a. Write a survey question that would likely produce biased results. b. Write a survey question that would likely produce unbiased results. 7 Investigating Slope-Intercept Form 7 Chapter 3 Statistics Don't know or refused to answer % b. This question will bias people toward answering no because most citizens are against the government making laws that require certain behaviors. a. This question is not biased. It does not lead to a yes or no answer. b. This question will bias people toward answering no because most people do not want taxes to be raised. Yes 5% No % Don't know or refused to answer 7% 3a. Sample answer: Since we had hamburgers at the last party, would you prefer pizza for the next party? 3b. Sample answer: Would you prefer hamburgers or pizza for the class party? 7 Chapter 3 Statistics

18 Introduction to Matrices Lesson Notes Vocabulary matrix dimensions row column element scalar multiplication Organize data in matrices. Solve problems by adding or subtracting matrices or by multiplying by a scalar. ORGNIZE DT IN MTRICES If you have ever used a spreadsheet program on the computer, you have worked with matrices. matrix is a rectangular arrangement of numbers in rows and columns. matrix is usually described by its dimensions, or the number of rows and columns, with the number of rows stated first. Each entry in a matrix is called an element. Example are matrices used to organize data? To determine the best type of aircraft to use for certain flights, the management of an airline company considers the following aircraft operating statistics. ircraft B77- DC-- MD- 3- Number of Seats irborne Speed (mph) Source: ir Transport ssociation of merica Possible Flight Distance (miles) Name Dimensions of Matrices Fuel per Hour (gallons) Operating Cost per Hour (dollars) The table has rows and columns of information. When we concentrate only on the numerical information, we see an array with rows and 5 columns This array of numbers is called a matrix State the dimensions of each matrix. Then identify the position of the circled element in each matrix. a. [ 5 ] b. 3 This matrix has row and 3 columns. Therefore, it is a This matrix has 3 rows and -by-3 matrix. columns. Therefore, it is a The circled element is in 3-by- matrix. the first row and the second The circled element is in the column. third row and the first column Focus 5-Minute Check Transparency 3- Use as a quiz or review of Lesson 3-. Mathematical Background notes are available for this lesson on p. 7C. Building on Prior Knowledge In Chapter, students learned how to add and subtract rational numbers using a number line. In this lesson, students will use this skill to add and subtract rational numbers in matrices. are matrices used to organize data? sk students: Which lines of numbers are the rows of the matrix? The rows go from left to right. Which lines of numbers are the columns? The columns go from top to bottom. In what instances might putting numerical data from a table into a matrix be beneficial? Sample answer: Putting numerical data into a matrix might make it easier to perform calculations on the data. Lesson 3- Introduction to Matrices 75 Resource Manager Workbook and Reproducible Masters Chapter 3 Resource Masters Study Guide and Intervention, pp Skills Practice, p. 79 Practice, p. 79 Reading to Learn Mathematics, p. 79 Enrichment, p. 79 ssessment, p. 5 Graphing Calculator and Spreadsheet Masters, p. 7 Parent and Student Study Guide Workbook, p. Transparencies 5-Minute Check Transparency 3- nswer Key Transparencies Technology Interactive Chalkboard Lesson x-x Lesson Title 75

19 Teach ORGNIZE DT IN MTRICES In-Class Example a. State the dimensions of each matrix. Then identify the position of the circled element in each matrix by ; second row, first column b by ; first row, fourth column MTRIX OPERTIONS In-Class Example Power Point Power Point Find each sum. If the sum does not exist, write impossible. 3 7 a b. 3 7 impossible College Football Each year the National Football Foundation awards the Macrthur Bowl to the number one college football team. The bowl is made of about ounces of silver and represents a stadium with rows of seats. Source: ESPN Information Please Sports lmanac 7 Chapter 3 Statistics Two matrices are equal only if they have the same dimensions and each element of one matrix is equal to the corresponding element in the other matrix MTRIX OPERTIONS If two matrices have the same dimensions, you can add or subtract them. To do this, add or subtract corresponding elements of the two matrices. Example dd Matrices If the sum does not exist, write impossible. a. B If, B, and C, find each sum. B Substitution b. B C () Definition of matrix addition (3) Simplify. 7 () B C Substitution 5 3 Since B is a -by-3 matrix and C is a -by- matrix, the matrices do not have the same dimensions. Therefore, it is impossible to add these matrices. ddition and subtraction of matrices can be used to solve real-world problems. Example 3 Subtract Matrices COLLEGE FOOTBLL The Division I- college football teams with the five best records during the 99s are listed below. Overall Record Bowl Record Wins Losses Ties Wins Losses Ties Florida State 9 3 Florida State Nebraska Nebraska 5 5 Marshall 5 Marshall Florida Florida 5 Tennessee 99 Tennessee Use subtraction of matrices to determine the regular season records of these teams during the decade Interactive Chalkboard PowerPoint Presentations This CD-ROM is a customizable Microsoft PowerPoint presentation that includes: Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition dditional, Your Turn exercises for each example The 5-Minute Check Transparencies Hot links to Glencoe Online Study Tools 7 Chapter 3 Statistics

20 Concept Check. -by- matrix has rows and columns, and a -by- matrix has rows and columns. So, the regular season records of the teams can be described as follows. Regular Season Record Wins Losses Ties Florida State Nebraska Marshall Florida Tennessee You can multiply any matrix by a constant called a scalar. This is called scalar multiplication. When scalar multiplication is performed, each element is multiplied by the scalar and a new matrix is formed. Example If T 3 3 a b c ma mb mc m d e f Scalar Multiplication of a Matrix Who is correct? Explain your reasoning. Estrella; Hiroshi did not multiply each element of the matrix by 5. nswer. Sample answer: md me Perform Scalar Multiplication, find 3T. 3T 3 Substitution 3() 3() 3() 3() 3(3) 3() Simplify. 3 9 Definition of scalar multiplication. Describe the difference between a -by- matrix and a -by- matrix OPEN ENDED Write two matrices whose sum is 9. See margin FIND THE ERROR Hiroshi and Estrella are finding 5 5. Hiroshi = 5 7 mf Estrella = 5 Lesson 3- Introduction to Matrices 77 In-Class Examples 3 Teaching Tip Explain that in this example, the overall record includes the regular season record, plus the bowl (postseason) record. COLLEGE FOOTBLL The Division - current football coaches with the five best overall records as of are listed below. Overall Record Coach Won Lost Tied Joe Paterno Bobby Bowden 35 7 Lou Holtz 7 Jackie Sherrill 7 93 Ken Hatfield 7 Bowl Record Coach Won Lost Tied Joe Paterno 9 Bobby Bowden 7 Lou Holtz Jackie Sherrill Ken Hatfield Source: NC Use subtraction of matrices to determine the regular season records of these coaches. Regular Season Record Power Point Coach Won Lost Tied Joe Paterno 3 Bobby Bowden 9 3 Lou Holtz 3 5 Jackie Sherrill 7 Ken Hatfield 3 9 If R 5 3, find 5R. 5 5 FIND THE ERROR Have students identify the difference between the two results first, which will help them pinpoint the error. Lesson 3- Introduction to Matrices 77