Data Analysis Chapter 11 PDF

CHAPTER 11 Data Analysis Research involves asking questions, collecting information, and analysing the data to draw conclusions. Businesses and governments use research-based statistics to inform and persuade people. Park wardens use statistics to predict birth rates, death rates, and...

CHAPTER 11 Data Analysis Research involves asking questions, collecting information, and analysing the data to draw conclusions. Businesses and governments use research-based statistics to inform and persuade people. Park wardens use statistics to predict birth rates, death rates, and migration patterns of plant and animal populations. They also use statistics to predict the size of populations such as elk. In this chapter, you will learn about factors that influence the collection of data. You will learn about how to reduce the chances of invalid results from a survey. You will collect and display data, and draw conclusions based on the results. What You Will Learn to identify factors that can affect the collection of data to identify the difference between a population and a sample to decide whether to use the population or a sample to identify different types of samples to use data for making predictions to develop and carry out a research project to assess your research project Did You Know? The Cree name for elk is wapiti. 410 Chapter 11 NEL  Key Words population stratified sample sample systematic sample convenience voluntary response sample sample random sample biased sample Literacy Link A concept map can help you visually organize your understanding of math concepts. Create a concept map in your math journal or notebook. Make each oval large enough to write in. Leave enough space to draw additional ovals. As you work through the chapter, complete the concept map. Organize each term you learn by determining which of the three blue ovals it relates to. Then, attach and label an oval for the term and define it. Discuss your ideas with a classmate. You may wish to add to or correct what you have written. sample Data Analysis influencing population factors NEL Chapter 11 411  FOLDABLES TM Study Tool Materials Step 4 sheet of 11 × 17 paper ruler Use a quarter sheet of 8.5 × 11 paper to create a seven sheets of 8.5 × 11 paper pocket for storing additional terms. Staple the pocket scissors to the inside right flap of the Foldable as shown below. stapler Use index cards or cut strips of notebook paper to fit inside the pocket. Step 1 Chapte tI Step 5 r 11 Wha o Fold the long side of a sheet of Data ed t Analys Ne n Staple the booklets from Steps 2 and 3 into the O 11 × 17 paper in half. Pinch it is Work Foldable from Step 1 as shown. at the midpoint. Fold the outer edges of the paper to meet Key Words influencing factors at the midpoint. Label it population as shown. sample convenience sample Research Project Question random sample stratified sample Sampling and Data Collection Methods Step 2 Key Words population systematic sample Results Additional Terms voluntary sample response sample Fold the short side of a sheet of 8.5 × 11 Data Analysis convenience biased sample paper in half. On one side, use a ruler to sample Conclusion(s) draw a line 4 cm from the top. Then, draw random sample six more lines at 4-cm intervals. Cut stratified sample along the lines through one thickness Using the Foldable systematic sample of paper, forming eight tabs. Label the voluntary response sample tabs as shown. As you work through the chapter, define the Key Words biased sample beneath the tabs on the left panel. Record notes about the research project under the tabs on the centre panel. Use the pocket to store the definitions for Step 3 additional terms related to data analysis. Staple five sheets of 8.5 × 11 paper together along the Use the back of your Foldable to create a flow chart to top edge. Draw a line 15 cm from the bottom of the top track your progress on the research project. On the sheet. Cut across the entire width of the page at this front of the Foldable, keep track of what you need to mark. Make a line 12 cm up from the bottom of the work on. Check off each item as you deal with it. second sheet and cut across the width of the page at this mark. Similarly, cut off 9 cm from the third page, 6 cm from the fourth page, and 3 cm from the fifth page. Label as shown. Research Project Question Sampling and Data Collection Methods Results Data Analysis Conclusion(s) 412 Chapter 11 NEL Math Link Protecting and Managing Wildlife Elk Island National Park in Alberta is Canada’s largest fully enclosed park. It Literacy Link is home to the densest population of An ungulate is a ungulates in Canada. These ungulates mammal with hooves. include plains bison, wood bison, moose, One example is a deer, and elk. bison. Most ungulates use the tips of their Park wardens and naturalists play an hooves to hold their important role in studying, monitoring, body weight as they and managing the wildlife in the park. move. Val decided to study the population of the ungulates in Elk Island National Park. The Approximate Total Park Count Did You Know? table shows the data she collected from the park warden. Ungulate (2007 Fall Count) Population density is a measure of the Plains bison 425 1. a) Use at least two different methods to number of individuals display the data that Val collected. Wood bison 400 living in a defined b) Explain why you chose each method Elk 605 area. The population to display the data. density of ungulates Moose 300 in Elk Island National 2. What is the most common ungulate in the Deer 558 Park is second only to park? Explain. the density in the Serengeti Plains of 3. Elk Island National park has an area of 194 km2. What is the Africa. mean number of elk per square kilometre? Show your thinking. 4. What percent of the ungulate population do bison represent? Throughout this chapter, you will study an issue related to wildlife protection and management. As you do, you will develop your own research project. Your project will involve collecting, displaying, and analysing your data. You might study birth rates, death rates, or migration patterns of an animal population. Or, you might consider the effects of tourism, recreation, or environmental factors, such as pollution, on a population. You might study a wildlife issue in a park. If so, what park will you choose? Where is the park located? What issue interests you? Research at least two areas of interest. Web Link For more information about national parks, go to www.mathlinks9.ca and follow the links. NEL Math Link 413  11.1 Factors Affecting Data Collection Focus on... After this lesson, you will be able to... identify how bias, use of language, ethics, cost, time and timing, privacy, and cultural sensitivity may influence the collection of data write and analyse appropriate survey questions A Super Food employee provides samples of Bob Brat Sausages to customers between 5:00 p.m. and 5:30 p.m. for one day. Store employees asked customers who buy Bob Brat Sausages the following question. Did you buy Bob Brat Sausages because they are delicious, quick to prepare, or both delicious and quick to prepare? Literacy Link A survey is used to The table shows the results from the survey. collect opinions and/ or information. Bob Brat Sausages Delicious |||| | Quick to prepare |||| Literacy Link Both delicious and quick to prepare |||| ||| An advertising claim gives information A few weeks later, a store flyer made this advertising claim. about the performance of a Over 90% of shoppers love Bob Brat Sausages product or service. The because they taste great and save time! claim is designed to encourage you to buy. The claim may be true, What factors might have influenced the survey responses? How accurate false, or a little of both. do you think the advertising claim is? Explain. 414 Chapter 11 NEL Explore Factors That Affect Survey Responses 1. From the store’s point of view, did Super Foods promote Bob Brat sausages well? Explain. 2. a) Did the survey ask a fair question? Explain your reasoning. b) Did the survey sample represent the whole population of shoppers? Explain. 3. Some people who read the flyer said the advertisement was misleading. Do you agree with them? Explain why or why not. Discuss your ideas with your group. Reflect and Check 4. Imagine you are hired as the store’s public relations consultant. a) What factors do you need to address to make the data collected about the sausages more accurate? b) Develop a survey question you would ask about Bob Brat Sausages. Who would you ask? Link the Ideas Several influencing factors affect how data are collected or how responses are obtained. These include: bias: Does the question show a preference for a specific product? use of language: Is the question presented in such a way that people understand what is being asked? ethics: Does the question refer to inappropriate behaviour? cost: Does the cost of the study outweigh the benefits? Literacy Link time and timing: Does the time the data were collected influence the Ethics involves results? Is the timing of the survey appropriate? judgments of right and wrong. For privacy: Do people have the right to refuse to answer? Are the responses example, cheating on kept confidential? a test is wrong. cultural sensitivity: Might the question offend people from different cultural groups? NEL 11.1 Factors Affecting Data Collection 415  Example 1: Identify Factors Affecting Data Collection Helen and Andre are reviewing the data collection methods used by a marketing company. For each situation, help them identify any influencing factors. Explain your reasoning. a) A sales representative stands in front of What is your favourite brand of a display of different kinds of toothpaste? For what reason(s)? toothpaste. He asks every person buying toothpaste the following question. b) Free samples of sunscreen are sent to every home in fall and winter. A mail reply card asks people if they would use the product again. c) A grocery store employee conducts a telephone survey of people living within 10 km of the store. To help determine what meat products to sell, she asks what type of red meat people prefer. d) A sales representative conducts a telephone survey. As she poses the question, a person receiving the call says, “I am not interested, thank you.” The sales representative responds, “Why not? Your input provides useful information.” She begins to repeat the survey question. e) Your school is under construction and is quite dusty and dirty. A survey is conducted about the environmental health of your school. The survey is done every four years. f) A sales representative sets up an online survey. The survey offers a free MP3 file of a song that was downloaded from the Internet to everyone who completes the survey. The company has not bought the rights to the song. Solution a) There is no bias. Standing in front of a toothpaste display allows the sales representative to survey people who are buying toothpaste. b) Fall and winter sun is less intense than spring and summer sun. People may be less likely to wear sunscreen during fall and winter. Therefore, these seasons are not likely the best choice for testing sunscreen. Sending sunscreen products during fall and winter is very costly for the company. In addition, asking people if they would use the product again assumes that they did use it. This may confuse people who have not tested the product. The language is unclear. c) The question is biased. It assumes that all people eat red meat, which is not true. Red meat is not acceptable to some people. For these people, this may be a culturally sensitive question. d) The sales representative does not respect the individual’s right to refuse to participate in the survey. In addition, any responses obtained under pressure may not truly represent the person’s opinion. e) This survey is poorly timed. When a school is under construction, some routines may be disturbed. People may express frustration with the temporary changes. This would reflect in a negative way on the survey. 416 Chapter 11 NEL  f) Offering a reward to participants in the survey is ethically wrong. In addition, it is not ethical to download music from the Internet without buying it. When people do this, musicians are not being paid for their work. Show You Know A marketing firm displayed two sports shirts at a grades 7 to 12 school with 800 students. The first ten grade 12 students who entered the school were asked the following question. Which one of these two sports shirts would you buy? A week later, the firm gave this flyer to the Over 78% of students at student council. Central High will buy Shirt B. What factors might have affected the data collected and the company’s conclusions? Explain. A: $29.95 B: $49.95 Example 2: Write Survey Questions Free of Influencing Factors A steel milling company conducts a survey. The proposed mill will produce 250 jobs and economic benefits for your community. Are a) Does the survey question you in favour of having a forward-thinking influence the results? Explain. steel mill in your community? YES NO b) Rewrite the question so that it is free of influencing factors. Solution a) Yes, the wording in the question may lead people to answer in a specific way. It implies that due to the economic benefits the company can offer, people should vote in favour of the mill. b) A better way to ask the question is, “Are you in favour of having a steel mill in your community?” Show You Know For each situation, explain whether the question has influencing factors. If it does, rewrite the question. a) Daniel is studying the population of polar bears in eastern Hudson Bay. He drafts the following question. Do you think the senseless slaughter of polar bears should be stopped? YES NO b) A recent study shows that 45% of Canadians are using the Internet to shop. Amy develops the following question for a class survey. What kinds of products and services do you buy online? NEL 11.1 Factors Affecting Data Collection 417  Key Ideas Survey questions should be worded so they are free from factors influencing the responses. The survey questions ask about 1. Do you buy food at the school cafeteria? YES NO purchases made at a school cafeteria. 2. If you responded NO, what reason best explains why you do not buy food at the cafeteria? Circle one. A The quality of the food is poor. B I do not like the food choices. C The cafeteria is in an inconvenient location. D The prices are too high. E Other (Please explain.) ______________________ Influencing factors include 1. In the past year, how many times did you buy food items at the bias, use of language, cafeteria? ____ ethics, cost, time and timing, privacy, and cultural sensitivity. Which box shows better 1. In the past week, did you buy any food items at the survey questions? Why? cafeteria? YES NO 2. If you responded YES, how many times did you buy each of the following food items? soup or salad ____ main course ____ drink ____ snack ____ Check Your Understanding Communicate the Ideas 1. Your friend is unclear what the term bias means. Develop an example to help explain the term. 2. Explain how influencing factors affect the collection of data. Give an example. 3. Shunta and Susan are discussing how to choose the top five lunch specials for the cafeteria menu. Each develops a different survey question. They decide to survey all the students. Shunta: What are Susan: Which are your top five favourite lunch your top five specials from this menu? Circle your choices. favourite lunch LUNCH SPECIALS specials? tomato soup and salad grilled cheese hamburger chicken salad sandwich vegetable lasagna turkey hot dog vegetarian chili pepperoni pizza Which survey question do you prefer? Explain why. 418 Chapter 11 NEL Practise For help with #4 and #5, refer to Example 1 on pages For help with #6 to #8, refer to Example 2 on page 417. 416-417. 6. In each case, describe the effect of any 4. In each case, identify and describe any influencing factors on the collection of data. factors that may affect the collection of data. Then, write an improved survey question. a) Survey members of the soccer team about a) A sales representative asks the following new uniforms for the volleyball team. question. b) At a truck rally, ask drivers what type Which do you prefer? of vehicle they prefer to drive. A Cola c) Provide samples of a new granola bar at B Diet Cola all conferences and conventions in your community. Ask people who attend the following question. b) An opposition party member asks the following question. What is your favourite among the new granola bars you tried in the past month? Is the current prime minister not the best prime minister in Canadian history? YES NO d) Ask customers in a sports store the following question. c) A small business develops the following Invincible Bikes are the most sturdy and question. expertly designed bikes on the market. What brand will you buy? Do you know about the Hands-On Repair Company and the maintenance your appliances and tools need? YES NO 5. For each situation, identify whether there is bias. If so, identify the bias. Then, rewrite the statement to correct it. d) Jennifer asks students the following question. a) Ask all horse owners if they are willing to pay higher horse-boarding fees. What is your parents’ total income? b) Ask owners of horses boarded in a stable in the city centre if this location is annoying. c) Ask horseback riders if they would support building a public park on the site of their stable. NEL 11.1 Factors Affecting Data Collection 419  7. Rewrite each survey question without any Apply influencing factors. 9. For each situation, write two different a) Sam asks riders of all-terrain vehicles the survey questions that may have resulted following question. in each conclusion. Do you support closing some riding trails a) Most juice lovers prefer apple and orange. to save some endangered animals? b) Yellow is the most popular shirt colour. YES NO c) Four out of five doctors surveyed strongly support a healthy, natural-food diet. b) A marketer surveys all the people entering a movie theatre. 10. Write a survey question for each situation. Identify who you would ask to participate Who is your favourite male movie star? in the survey. A Brad Pitt B Keanu Reeves a) You want to find out which sport teens C Matt Damon like best. D Other _____________________ b) You want to find out if price or brand is more important when buying a cell phone. c) You want to find out which media source c) A student asks people at an airport the people trust most to give them accurate following question. information. Do you think flying is still the cheapest way to travel a long distance? 11. Rewrite each survey question so that it YES NO collects more helpful data. a) If you are a juice drinker, would you consider switching to Crystal Juice? 8. Rewrite each survey question so there are no YES NO influencing factors. a) b) Do you like to watch hockey, the only great Which of the following cough medicines sport? have you tried? YES NO b) Most people choose chocolate, but what is your favourite flavour of ice cream? c) c) Would you run if you came across a moose? A recent survey shows that 42% of teens YES NO use the Internet to watch TV. What TV shows do you watch most often online? d) Are you satisfied with your Internet access? Circle one response. A Excellent B Good C Poor 420 Chapter 11 NEL 12. For each case, identify the influencing 14. a) You have been hired to develop Arctic factors that may affect the collection of data. adventure tours. Develop three survey Then, write a survey question that is free of questions to help determine the activities influencing factors and is clearly written. that appeal to tourists. a) Teens are asked about which clothing b) Exchange your questions with a items they have bought at an expensive classmate. Critique each other’s questions store in the past year. for clarity and the presence of any b) The members of a golf club are asked if influencing factors. they are in favour of a proposed highway. c) Based on the feedback, revise your survey It will reduce traffic jams by going questions so they are clearly written and through the golf course. free of influencing factors. Extend 13. a) Write two survey questions that relate to a topic of your choice, such as sports, fashion, movies, or games. Develop a question that contains bias. Develop a different question that is free of influencing factors. b) Use the first question to survey 20 friends or classmates. Use the second question to 15. When interpreting the results of a political survey 20 different friends or classmates. poll, why is it important to know the source of the poll? c) Compare the results. Identify the bias in the first question. Explain how its wording may have affected the results. For your research project, select a wildlife protection I am interested or management topic. Climate Change in the effects of climate change a) Use a graphic organizer such as the concept melting on Arctic map shown here to help you organize sea ice walrus mammals, in ideas for your topic. Record subtopics and particular, the connecting details and facts. threatened beluga whale. Arctic mammals b) Choose a subtopic to study. c) Consider what data you need to obtain polar bear narwhal related to your topic. Then, write three possible research questions. beluga whale d) Exchange your questions with a classmate shifts in habitat loss food supply to check that they are free of influencing factors. Revise your questions if necessary. rising water temperature Web Link Research Questions: How does melting sea ice threaten Arctic mammals? For information about national and Which Arctic mammals are most threatened by melting sea ice? provincial parks in Canada, go to What are the population trends for beluga whales in Canada? www.mathlinks9.ca and follow the links. NEL 11.1 Factors Affecting Data Collection 421  11.2 Collecting Data Focus On … After this lesson you will be able to... identify the difference between Who will w a population and a at the Ju in best female vo sample no Award calist s? identify different types of samples justify using a population or a sample for given situations determine whether results from a sample can be applied to a population Do Internet forums accurately reflect the opinions of all their readers? Do Did You Know? these people share the same opinions as those who do not read the forums? An Internet forum allows people to How can you reduce the chances of making inaccurate predictions from participate in online a survey? discussions. There are forums for many topics, including Explore Using Survey Data to Make Predictions sports, fashion, and politics. Work as a class. Develop a question to determine the opinions of students at your school about a topic of your choice. For a topic, you might choose favourite foods, sports, actors, or musicians. 1. Write and edit your survey question. 2. Survey everyone in your class. 3. a) Organize the results. b) Based on the results of the class survey, predict the whole school’s response to your question. 4. Does your prediction accurately reflect the opinions of all students in your school? Explain. 422 Chapter 11 NEL Reflect and Check 5. Is your class a population or a sample ? Explain. population all of the individuals in 6. How else might you choose people for your survey to reflect the the group being studied for example, the opinions of all students in your school? population in a federal election is all eligible 7. How can you reduce the chances of making inaccurate predictions voters using a survey? sample any group of individuals Link the Ideas selected from the population There are several different types of samples. for example, a sample convenience sample of the population in a a sample created by choosing individuals from the population who are federal election might easy to access be 100 individuals chosen from each random sample province or territory a sample created by choosing a specific number of individuals randomly from the whole population. Random means that each individual has an equal chance of being chosen. As a result, a random sample is likely to represent the whole population. Data from a random sample can be used to make predictions about the population. Stratified samples and systematic samples are types of random samples. stratified sample systematic sample a sample created by dividing the a sample created by choosing whole population into distinct individuals at fixed intervals groups, and then choosing the same from an ordered list of the fraction of members from each group whole population voluntary response sample a sample created by inviting the whole population to participate Example 1: Identify the Population Identify the population for each situation. Then, state whether you would survey the population or a sample of the population. Explain your reasoning. a) A bicycle store owner wants to know which brand of mountain bike her customers prefer. b) The school board wants to know how many hours of homework students do each day. c) A candle manufacturer wants to know how many of its candles are made with flaws. NEL 11.2 Collecting Data 423  Solution a) The population is the store’s customers. It depends on the size of the store. A small store might survey all of its customers. A large, busy store would likely survey a sample of customers. For them it would be time-consuming and costly to survey all of the customers. b) The population is students in schools within the school board. The school board would likely survey a stratified sample of its students. They would want to include the same fraction of students at various grade levels because the amount of homework done varies from grade to grade. c) The population is all candles made by the company. A small company might check each candle made. A large company might check every tenth candle. It would be costly and time-consuming to check every item. Show You Know For each scenario, identify the population. Then, indicate whether you would survey the population or a sample. Explain your reasoning. a) The Royal Garden restaurant needs to know which main dish its customers favour. b) Stephan wants to find out if teachers in Canada prefer to wear glasses or contact lenses. c) A junior hockey team wants to find out why some people who bought season tickets last year are not buying them this year. Example 2: Identify a Sample For each situation, describe how the sample could be selected. Identify the type of sample. a) A teacher wishes to get feedback from her class about the school dance. She plans to survey 5 students out of a class of 30. b) A telephone company wants to determine whether a fitness centre would be well used by its 3000 employees. The company plans to survey 300 employees. c) A chain store is trying to decide whether Location Population to open a store in Camrose, Alberta. The Camrose 16 000 company decides to survey people in Camrose Bashaw 825 and three nearby towns. The population of Tofield 1 876 each location is shown in the table. d) A marketing research company mails surveys Daysland 876 to all of the adult residents in a town. The survey asks about brands of consumer products. The residents are asked to mail their responses in a prepaid envelope. e) A restaurant owner wants to know the favourite pizza topping of his customers. He plans to survey every customer who orders a pizza in his restaurant between 5:00 p.m. and 10:00 p.m. one evening. 424 Chapter 11 NEL  Solution You can use different types of samples in a survey. These are some possible solutions. a) The teacher could put all the students’ names in a box and draw five names. This is a random sample. b) The company plans to survey 300 out of their 3000 employees. To ensure that the sample fairly represents the population, the company might interview every tenth person on the payroll list. This is a systematic sample. This type of sample is time and cost efficient. c) Since the city has more people who use the company’s products than TOFIELD the nearby towns, the company could survey 25% of the population in each location. This is a stratified sample. Since 25% of each group is surveyed, the same proportion of each town is represented in the sample. In this case, the company would survey 4000 people from CAMROSE Camrose, 206 people from Bashaw, 469 people from Tofield, and 219 people from Daysland. DAYSLAND d) The marketing research company is inviting all residents to participate. This is a voluntary response sample. This sample may not represent the population because only those who are interested BASHAW will respond. e) This is a convenience sample. It is not random since only customers For each situation, is who order pizza are surveyed. However, the sample does target there a different type customers who will provide useful input. These customers are easily of sample that might accessible. The sample provides the owner with information right be used? If so, explain what type of sample away and costs no extra money. and how you would organize it. Show You Know a) For each scenario, what type of sample would you use? How would you select the sample? A marketing firm plans to conduct a telephone survey in a city of 800 000 people. The survey asks whether there is interest in a new art gallery. A student wants to know the most popular cell phone provider that grade 9 students use. b) In each case, identify the type of sample. A coach puts the names of all of the basketball players into a hat and draws one name for a free basketball. A questionnaire is sent to every ninth person on an alphabetical list of a store’s credit card customers. The student council invites all students to provide ideas for activities. NEL 11.2 Collecting Data 425  Key Ideas A population is the whole group of individuals being studied. It is not always practical or cost effective to survey everyone in a large population. You might survey a sample of the population. A sample is any part of the population. A random sample ensures that all people have an equal chance of being selected for a study. You can use data from a random sample to make predictions about the population. Systematic samples and stratified samples are types of random samples. Voluntary response samples and convenience samples are types of non-random samples. ten people from the city, random sample five people from the town 50 employees picked randomly stratified sample from a list of all employess convenience sample everyone who calls into a radio talk show voluntary response sample every third house after a random starting point systematic sample everyone who orders a sandwich combo between 5 p.m. and 7 p.m. Check Your Understanding Communicate the Ideas 1. Your friend sends you this e-mail. Write your response. FROM: Nick TO: Jonah I am confused about populations and samples. Would you please explain the difference? It would help if you gave an example of each. Nick 2. A group can be the population or a sample, depending on the survey question. Do you agree? Give examples to support your opinion. 3. Kim and Scott want to know how many people took public transit to come to the sold-out concert. The seating capacity at the venue is 18 000 people. Kim: Scott: I plan to ask the I plan to ask first 20 people the first who arrive. 200 people. a) Whose sample would provide a more accurate representation of the population? Explain why. b) Recommend a different type of sample that would give more accurate results. Explain your reasoning. 426 Chapter 11 NEL Practise For help with #4 and #5, refer to Example 1 on For help with #6 and #7, refer to Example 2 on pages 423-424. pages 424-425. 4. Identify the population for each survey 6. For each situation in #5, what type of question. Indicate whether the population sample would you use? How would you or a sample should be used for the survey. select the sample? Justify your choice. a) 7. For each context, identify and describe the Which band rocks like no other? sample you would select for a survey. a) A radio talk-show host wants listeners’ b) views on a proposed by-law about Who will be next year’s grade 10 watering lawns and gardens. representative on student council? b) A province wants to select 25 schools to participate in a new physical education c) program. Which styles of soccer shirts should be c) A marketing firm wants to know teens’ sold at Sports R Us? favourite magazines. d) A reporter wants to ask people d) downtown about their plans to What brand of shampoo do you prefer? participate in the Big Valley Jamboree. Did You Know? The Big Valley Jamboree, which is held in Camrose, 5. In each case, identify the population. State Alberta, is Canada’s biggest country music show. It was whether you would survey the population or a voted the Canadian Country Music Association’s Country sample of the population. Justify your choice. Music Event of the Year in 2006. a) A newspaper wants to know the online activities of Internet users at home. Apply b) The principal wants to know if people 8. For each context, would you recommend associated with the school are in favour surveying the population or a sample? of school uniforms. Justify your choice. c) An electronics store needs to find out a) You want to determine the air quality in whether its customers are satisfied with hospitals in Edmonton, Alberta. the repairs and services department. b) You want to know post-secondary plans d) The municipal government wants to of grade 9 students. determine if bus transit is needed for c) You want to test the quality of people with special needs. parachutes. d) You want to test the quality of bike tires. 9. Kristi wants to create a menu for a family picnic that will appeal to adults, teens, and children. Her family includes 20 adults, 8 teens, and 12 children. If she has time to talk to only 10 people, how should she choose her sample? Explain. NEL 11.2 Collecting Data 427 10. Jason, a member of the Graduation 13. Erin plans to survey her friends to determine Committee, plans to ask each student who the average number of children per enters the cafeteria the following questions. household in Canada. Is this a random sample? Explain your reasoning. What is your favourite paint colour for the cafeteria walls? ________ 14. Ben asked 50 people at random in a mall Should the cafeteria be used for the following question. graduation? YES NO Are you allergic to any of these animals? A cats a) Identify the population. B dogs C birds b) Identify the sample. D gerbils c) Will the results of his survey accurately E hedgehogs represent the population? Explain. d) Is Jason correct in using the same sample for both questions? Explain your thinking. His results are displayed. Animal Allergy Frequency Total 11. The student council plans to survey students Cats |||| |||| |||| |||| |||| | 26 about how best to spend the budget for activities. Enzo prefers to spend the money Dogs |||| |||| || 12 on baseball equipment. He decides to Birds |||| ||| 8 randomly survey students at a baseball game. Gerbils |||| 4 a) Is there a bias in Enzo’s sample? If so, Hedgehogs 0 what is the bias? a) Ben made the following predictions b) Describe a sample that would reflect the based on his data. Do you agree with overall opinion of students. Explain your each prediction? Explain. thinking. Almost 25% of the population is allergic to dogs. Hedgehogs do not cause allergies. b) Improve the survey question. Explain your reasoning. Extend 15. Search various media for information about a recent survey. Use sources such as magazine and newspaper articles, or radio, television, and Internet reports. 12. Anita and Cindi are asked to find out what type of mural to paint in the entrance of a) Identify and comment on the population their office tower. There are 1400 and the sample. employees. Cindi proposes using a random b) Are the predictions valid for the sample of 20 employees. Anita suggests population? Explain your reasoning. using a stratified sample to get input from c) Was the survey well conducted? What every department. Whose sampling method improvements, if any, would you is better? Explain your reasoning. recommend? 428 Chapter 11 NEL  THE DAILY NEWS 16. Five firefighters conducted a survey to assess how well the fire department is performing. They asked local residents the following Survey Reveals Majority Very question. Satisfied With Fire Department Which of the following choices best describes your opinion of how well the fire department is doing? a) How might the wording of the question A Excellent affect the collection of data? Rewrite B Very Good the question to produce more accurate C Good responses. D Poor b) What else may have influenced the collection of data? Describe how to A week later, the local paper has the conduct a survey that would reflect headline as shown. opinions more accurately. For your research project, choose one of the research questions you wrote for the Math Link on page 421. a) What is your question? b) Write a hypothesis that clearly states what you want to prove or disprove. Literacy Link c) Identify and describe the population for your question. A hypothesis is a d) To answer your question, you will need to find data from studies and statement put surveys that have already been done. You will do the research in the Math forward to guide Link on page 439. What sampling methods do you think would be used to an investigation. collect data related to your question? Explain your reasoning. An example is shown. Research Question: What are the The beluga whale populations in Canada population trends for beluga whales can be organized into seven groups. in Canada? We predict that all beluga whale 7 populations in Canada are 6 threatened. 5 2 We hypothesize that there are few 4 3 differences in the population trends for the whale populations in Canada. 1 1. St. Lawrence 5. Cumberland 2. Ungava Bay Sound 3. Eastern Hudson Bay 6. High Arctic 4. Western Hudson Bay 7. Beaufort Sea Digital rights not available. We expect that studies will use samples of beluga whale populations since there are large numbers of whales spread out over large areas. NEL 11.2 Collecting Data 429  11.3 Probability in Society Focus On... After this lesson, you will be able to... identify and explain assumptions linked to probabilities explain decisions based on probabilities When planning an outdoor event, knowing the probability of a rain shower or a thunderstorm can be helpful. Forecasters often use probability to express their degree of certainty in the chance of a weather event occurring. For example, a 30% probability of rain today means that there are 3 chances in 10 that you will get wet today. To produce a probability forecast, the forecaster studies the current weather situation, including wind and moisture patterns and determines how these patterns will change over time. What assumptions might a weather forecaster make when making a probability forecast? Explore Making Decisions Based on Probability You are a town planner and need to know if people want walking trails or bike paths along the nearby river. There are 15 000 people in the town. You decide to survey 1500 people. Literacy Link 1. Identify the population. An assumption is something taken for 2. Describe how you might select the sample and how you could conduct granted, as if it were the survey. true. 3. a) As a result of the survey, suppose 60% of people prefer bike paths. What prediction would you make about the preference of the town’s population? b) What assumptions did you make in your prediction? Reflect and Check 4. a) Exchange your sample with the one of a classmate. Does your classmate’s sample represent the population? Explain your reasoning. b) Based on the sample, what decision will you make about walking trails or bike paths? 430 Chapter 11 NEL Link the Ideas Example 1: Use a Sample to Make a Prediction About the Population Ruth wants to determine the most common eye colour of students. All grade 12 students in five of seven high schools in a city recorded their eye Web Link colour. A total of 2300 students were surveyed. The results are shown in For information about the table. gene types that determine eye colour, go Eye Colour Total to www.mathlinks9.ca and follow the links. Brown 1656 Blue 483 Digital rights not available. Green 115 Other 46 a) From the results, predict how many of the 7200 students at the local college will have brown eyes. Show your thinking. b) Is your prediction reasonable? Explain why or why not. Solution a) Of the 2300 students surveyed, 1656 students have brown eyes. Strategies 1656 × 100 Percent with brown eyes = _____ Estimate and Check 2300 = 72 Calculate 72% of 7200. Over half of 0.72 × 7200 = 5184 the surveyed students have brown Based on the results, approximately 5200 students at the local college eyes. This means that will have brown eyes. at least 3600 students at the college have b) No, not necessarily. An assumption was made that the high school brown eyes. sample represents the college population. The sample represents the local grade 12 population by surveying all students in five out of seven schools. The sample may not represent the college student population. A college often has many students who come from other parts of the province or territory, and other provinces and countries. In this case, the sample could be a biased sample. As a biased sample result, the prediction may not be valid for the college population. does not represent the population You can be more confident that the prediction is reasonable by revising can make survey results the prediction statement to include a limitation. “If the student population inaccurate in the high schools and the college are similar, based on the results, there are approximately 5200 college students who have brown eyes.” NEL 11.3 Probability in Society 431  Literacy Link Show You Know To generalize means Use the information in Example 1 to help answer the following. to make a broad statement from a) Predict how many of the local college students have blue eyes or known facts. green eyes. Show your work. b) Can you generalize the results from the sample to the local college population? Explain. c) What limitation might you include to make your prediction more accurate? Did You Know? Advertising and marketing agencies Example 2: Avoid Making a False Prediction often generalize Mr. Krutz gave an assignment worth 30 marks. After marking the first results from a sample five papers, he was concerned that the students did not understand the that does not represent the assignment. He predicted most students would not do well. population. They do this in order to promote a product or service. Consumers need to consider advertising claims carefully. a) Based on Mr. Krutz’s sample, predict the “average” mark for the whole class on the assignment as a percent. b) The scores for all 30 students in the class are: 20 15 18 19 18 16 17 23 24 30 22 24 21 20 24 25 19 24 15 28 27 28 22 24 19 13 28 22 24 20 Why does Mr. Krutz’s sample lead him to make a false prediction? 432 Chapter 11 NEL  Solution a) To predict the average mark, Mr. Krutz could use the measures of Strategies central tendency. Organize, Analyse, Mean: and Solve 20 + 15 + 18 + 19 + 18 Mean = _______________________ 5 = 18 The mean is 18. Median: The median is 18. Mode: The mode is 18. The mean, median, and mode for the sample scores are 18. Based on the sample statistics, the “average” mark on the assignment is predicted to be 18 × 100 or 60%. ___ 30 b) The mean, median, and mode for the class How did Mr. Krutz population are 22, 22, and 24, respectively. determine these values? Mr. Krutz assumed that the sample consisting of the first five papers was representative of the entire class. This is false. The mean score in the sample is 60%. The mean score in the population is approximately 73%. The most frequent score is 80%. Mr. Krutz may have considered too few members of the class. The sample does not represent the population. Show You Know Use the data in Example 2. What if Mr. Krutz had marked the last five papers first? Do they give a more reasonable prediction of the class average? Explain. NEL 11.3 Probability in Society 433  Example 3: Make a Decision Based on Probability A youth association surveys its 400 members about their preferred activity. There are 100 members in each of four groups. The activities were chosen from a youth activities resource. The table displays the survey results. Group Swimming Rock Climbing Watching Movies Bowling Total Red 14 9 40 37 100 Blue 11 19 59 11 100 Green 27 12 57 4 100 Yellow 13 24 44 19 100 a) What is the probability that a member of any group will choose swimming? Based on this, predict how many of the 400 members will choose swimming. b) What assumptions did you make? c) Based on the survey results, predict the probability that a member will choose swimming. d) Compare your answers for parts a) and c). Explain any differences. Solution a) The theoretical probability of choosing any one of the four activities is equally likely. P(swimming) = __ 1 or 25% 4 The probability that a member of any group will choose swimming is __ 1. 4 Since there are 400 members, 100 will probably choose swimming. b) The assumptions are as follows: Every activity has an equal chance of being selected. Members have an equal level of interest in each activity. c) The survey results reflect the experimental probability. 14 + 11 + 27 + 13 P(swimming) = __________________ 400 65 = ____ 65 ____ = 16.25% 400 400 The experimental probability that a member will choose swimming is 16.25%. d) 25% > 16.25%. The theoretical probability is greater than the experimental probability. Watching movies has the greatest probability. The group of students who answered this survey appear to prefer watching movies to swimming, rock climbing, or bowling. The experimental probability gives a truer reflection of the youths’ interests. 434 Chapter 11 NEL  Show You Know Refer to the information in Example 3. a) Based on the survey results, what is the probability that a member will choose watching movies? bowling? b) If you were the youth coordinator planning the activities, how would you determine the favourite activity? Explain your reasoning. Key Ideas A biased sample can make survey results inaccurate. A biased sample may When a sample represents the population, you can generalize the result when a large results to the population. percent of individuals in a sample refuse to You can use experimental probability and theoretical probability participate in a survey. to help make decisions based on probability. Check Your Understanding Communicate the Ideas 1. Kelly is confused about the difference between a sample that represents the population and one that does not. Use an example to help explain the difference to him. 2. Use the cartoon to explain how a sample might result in a false prediction. 3. How might you use experimental probability and theoretical probability to help make a decision about what flavours of ice cream to offer at a sport tournament? Practise For help with #4 and #5, refer to Example 1 on page 431. 5. A toothpick factory samples every 100th 4. A light bulb factory samples light bulbs as toothpick for damage. The sample shows a they come off the assembly line. A random 0.17% probability of damage. How many sample shows that 1 bulb out of every 20 is toothpicks would you predict to be damaged defective. In a run of 1380 bulbs, the quality in the daily production of 2.4 million manager predicts that 69 bulbs will be toothpicks? Include any assumptions you defective. What assumptions did the quality made in your prediction. manager make in his prediction? NEL 11.3 Probability in Society 435 For help with #6 and #7, refer to Example 2 on 9. A movie rental company has five types of pages 432-433. movies. They are drama, comedy, horror, 6. A cafeteria supervisor asked three students action, and science fiction/fantasy movies. who are vegetarians about their preference a) What is the theoretical probability that for a lunch menu. All three chose garden a person will choose a comedy? salad, tomato soup, and garlic bread. The b) What assumptions did you make? supervisor plans to serve their menu choice c) The table displays the movie preferences the next day, thinking that it will sell well. from a random survey of 50 customers. a) Did the sample lead the supervisor to Predict the probability that a customer make a false prediction? Explain. will choose a comedy movie. b) If the prediction is false, explain how Movie Type Responses you might make a more accurate one. Drama 15 7. A manufacturer makes the following claim Comedy 10 about the mass of its health bars. Horror 12 Action 11 Science fiction/Fantasy 2 Each bar has a mass d) Compare your answers for parts a) and c). of at least 50 g. Explain any differences. Erika and Brett weighed ten health bars to e) About how many rentals out of a total check the claim. Three bars had a mass less of 2000 movies will be drama movies? than 50 g and one bar had a mass of exactly 50 g. The students predicted that 30% of the Apply health bars made by the company would not meet the claim. 10. Miya received the following scores from ten a) Did the sample lead the students to make judges in a skating competition. Skating a false prediction? Explain. performances are given a score out of 10. b) If the prediction is false, explain how you Judge 1 2 3 4 5 6 7 8 9 10 might make a more accurate one. Score 8.5 6 6.5 6.5 6.5 7 6 6.5 4.5 7 a) Calculate Miya’s mean For help with #8 and #9, refer to Example 3 on page 434. score based on all ten 8. Greenville, a town with 4000 people, is judges. having an election for mayor. A reporter b) Use the first three polled 40 people and found 53% chose judges’ scores as a Candidate A, 23% chose Candidate B, and sample. Calculate the the rest chose Candidate C. mean. a) How many people polled chose Candidate C? c) Use the last three b) What is the theoretical probability that judges’ scores as a a voter will choose Candidate A? What sample. Calculate the assumptions did you make? mean. c) Compare the experimental and theoretical d) Compare the mean from probabilities of Candidate A winning. each sample to the mean for d) The reporter predicts that Candidate A all judges. Are the samples will win the election. Do you agree with a good predictor for Miya’s his prediction? Explain your reasoning. overall score? Explain. 436 Chapter 11 NEL 11. Jack wants to know the weekly earnings of a) In the article, what are the assumptions grade 9 students who work part-time in the associated with each probability? Is each summer. He surveyed five grade 9 students. probability accurate? Explain. Here are the results: $75, $120, $45, $250, b) In your opinion, is there a bias against and $85. young drivers? Explain your reasoning. a) Is this a biased sample? Explain your c) “Decisions about car insurance costs are reasoning. based on a combination of experimental b) Jack concluded that grade 9 students earn probability, theoretical probability, and an average of $115 per week. Do you biased judgment.” Do you agree or agree with his conclusion? Explain. disagree with this statement? Explain your reasoning. 12. Colin read an article Trial Boys Girls that claims that more 1 40 60 girls are born than boys. Web Link 2 43 57 Colin predicted that a For information about road safety, risk factors, and factors couple has a 50% 3 50 50 affecting car insurance rates, go to www.mathlinks9.ca and 4 46 54 follow the links. chance of having a boy. He tested the prediction 5 48 52 by tossing a coin 14. Cathy and John are waiting for the bus. 6 58 42 100 times for each of John predicts that one of the next five 7 50 50 vehicles to pass the bus stop will be a 10 trials. Here are the results. 8 49 51 minivan. Cathy predicts two of the next five Do these experimental 9 50 50 vehicles will be minivans. John made his results confirm Colin’s 10 53 47 prediction based on five types of vehicles on prediction or the article’s the road: cars, sport utility vehicles (SUVs), claim? Show your thinking. buses, minivans, and trucks. a) How do you think John and Cathy made 13. The graph shows the number of collision their predictions? claims per 100 vehicles insured for male b) John and Cathy decided to test their drivers in Alberta, in 2004. predictions by conducting a survey. They d Number of Collision Claims Per observed vehicles passing the bus stop 12 100 Vehicles Insured for 1 h at the same time each day for five Claim Frequency 10 days. The table shows their results. 8 6 Type Day 1 Day 2 Day 3 Day 4 Day 5 4 Car 30 28 25 27 25 2 SUV 20 25 18 22 18 0 19–20 21–22 25–35 36–45 46–55 a Male Drivers by Age Group Bus 4 4 3 4 4 Many insurance companies charge drivers Minivan 8 10 12 9 7 under the age of 25 higher insurance Truck 10 7 11 8 10 premiums based on the probability of From the results, whose prediction was accidents. Find information about car more accurate? Explain. insurance costs based on the probability c) In one day, 800 vehicles passed the bus of collision. Paste the article into your stop. Based on the survey results, predict notebook. the number of trucks. NEL 11.3 Probability in Society 437 15. The Jackson family is celebrating the coming c) Of the 160 students interviewed, birth of triplets. They currently have two 20 students played a musical instrument boys. Mrs. Jackson is hoping for three girls. and were on a sports team. Predict how a) What is the theoretical probability that she many of the 2100 youth group members will have three girls? Show your thinking. do not play a musical instrument or play on a sports team. b) The Jacksons used a random number generator to simulate the situation. d) What assumptions did you make in They decide that 1 indicates a girl and part c)? 0 indicates a boy. The table shows the results for ten trials. What is the experimental probability of three girls? Trial Experimental Results 1 0 0 0 2 0 1 1 3 0 1 0 4 1 0 1 5 1 0 1 6 1 1 0 7 1 0 0 8 1 1 0 17. Search magazines, newspapers, or the Internet. 9 0 0 0 Look for an article that uses probability to make a prediction for a population. 10 0 1 0 a) Identify the assumptions associated with c) Compare the experimental probability the probability. and the theoretical probability. b) Explain the limitations of each d) The boys predict that their mother will assumption. have three more boys. Do you agree with their prediction? Justify your answer. c) In your opinion, is the prediction accurate? Justify your answer. e) What assumptions did you make? 18. Search magazines, newspapers, or the Extend Internet for an issue of personal interest. 16. A random sample of 160 students out a) Take a stand on the issue. Write an of 2100 participants in a summer youth argument that includes a probability program responded to a survey. statement to support your stand. Use methods of your choice to display your Yes No data. Do you play a musical instrument? 40 120 b) Take the opposite stand. Using the same Do you play on a sports team? 25 135 data, or new data, write an argument that includes a probability statement to a) Does the sample represent the support your new stand. population? Explain. c) Present your arguments to a classmate. b) Based on the data, what is the probability Have your classmate point out the that a participant, chosen at random, will strengths and weaknesses of both play a musical instrument? arguments. 438 Chapter 11 NEL  For your research project, collect data from studies and surveys that have been done. Use sources such as scientific publications and the Internet. Depending on the data that you find, you may need to revise your research question. a) Describe the data you will look for. Where will you look? b) Record notes for at least three studies related to your question. Include the following information for each study: Describe the sampling method used. Did it involve the population or a sample? For a study involving a sample, discuss whether the results can be generalized to the population. Describe the method used to collect the data. Summarize the results. Describe any assumptions that were made. Explain the limitations of each assumption. Discuss the accuracy of any predictions made about the population. Provide complete source information. A summary of a whale population study is shown. We researched a study done by the Committee on the Status of Endangered Wildlife in Canada. The table shows the estimated whale populations. Population Estimated Compared Beluga Whale Population to Population (2004) Trend Pre-Whaling St. Lawrence 950 low endangered stable or increasing Ungava Bay too small very low, may no longer to estimate or may no exist longer exist Eastern 2000 low decreasing rapidly Hudson Bay Western 23 000 large unknown Hudson Bay Cumberland 1500 low endangered Sound stable or increasing High Arctic 20 000 unknown unknown Beaufort Sea 39 000 large not threatened increasing Smith, T. G. “COSEWIC Assessment and Update Status Report on the Beluga Whale Delphinapterus leucas in Canada.” 2004. Committee on the Status of Endangered Wildlife in Canada. Ottawa. 20 Oct 2008. NEL 11.3 Probability in Society 439  11.4 Developing and Implementing a Project Plan Focus on... Developing and carrying out a research project requires careful planning. After this lesson, you During the Math Links on pages 421, 429, and 439, you worked on Step 1, will be able to... as shown in the flow chart. Use your work to continue to develop your develop a research project plan. In this section, you will also develop a rubric, carry out your project plan plan, and then assess your project using your rubric. complete a research Use the flow chart to help organize your research project and carry out project according to your plan. a plan, draw conclusions, and Step 1: Develop the project plan. communicate  Write the research question. findings  Write the hypothesis. self-assess a research  Describe the population. project by applying a  Describe how you will collect data. rubric  Record notes for at least three studies related to the research question. Materials blank Research Project Step 2: Create a rubric to assess your project. Rubric Step 3: Continue to develop the project plan.  Describe how you will display the data.  Describe how you will analyse the data.  Describe how you will present your findings. Step 4: Complete the project according to your plan.  Display the data.  Analyse the data.  Draw a conclusion or make a prediction.  Evaluate the research results. Step 5: Present your findings. Step 6: Self-assess your project. Step 1 Complete the research for your project. 440 Chapter 11 NEL Step 2 Develop a rubric so that you know what is expected. Use the following example and a copy of the blank Research Project Rubric to help develop your own. List the criteria in column 1. You may find it useful to order the criteria according to the sequence of the project. For each criterion, record an indicator for each of four levels of performance. See how the first row in the example below is completed to give you ideas for your own rubric. – Level 1 reflects work that shows little evidence of expected results. – Level 2 reflects work that meets the minimum expected standard. – Level 3 reflects work that meets the expected standard. – Level 4 reflects work that is beyond the expected standard. Criteria Level 1 Level 2 Level 3 Level 4 Planning not clear and not fairly clear but not mostly clear and very clear