Explore Finding the Mean of Samples Obtained. Houghton Mifflin Harcourt Publishing Company Image Credits: Radius

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1 COMMON CORE Locker LESSON Data-Gathering Techniques Common Core Math Standards The student is expected to: COMMON CORE S-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Mathematical Practices COMMON CORE 22.1 MP.5 Using Tools Language Objective Create a graphic organizer that shows the relationships among population, census, and parameter, as well as sample, sampling, and statistic. ENGAGE Essential Question: Under what circumstances should a sample statistic be used as an estimator of a population parameter? The statistic needs to come from a representative sample, which is most likely to be obtained when the sampling method involves randomness. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how to obtain a representative sample of the student body in order to find out the types of foods most students would choose to eat. Then preview the Lesson Performance Task. Image Credits: Radius Images/Corbis Name Class Date 22.1 Data-Gathering Techniques Essential Question: Under what circumstances should a sample statistic be used as an estimator of a population parameter? Explore Finding the Mean of Samples Obtained from Various Sampling Methods You collect data about a population by surveying or studying some or all of the individuals in the population. When all the individuals in a population are surveyed or studied, the data-gathering technique is called a census. A parameter is a number that summarizes a characteristic of the population. When only some of the individuals in a population are surveyed or studied, the data-gathering technique is called sampling. A statistic is a number that summarizes a characteristic of a sample. Statistics can be used to estimate parameters. Consider the following table, which lists the salaries (in thousands of dollars) of all 30 employees at a small company. In this Explore, you will take samples from this population and compute the mean (the sum of the data divided by the sample size). Salaries at a Small Company Suppose the employees whose salaries are 51, 57, 58, 65, 70, and 73 volunteer to be in the sample. This is called a self-selected sample. Compute the sample s mean, rounding to the nearest whole number Mean = = _ Suppose the six salaries in the first two columns of the table are chosen. This is called a convenience sample because the data are easy to obtain. Record the salaries, and then compute the sample s mean, rounding to the nearest whole number , 24, 44, 46, 58, 62; mean = = _ Suppose every fifth salary in the list, reading from left to right in each row, is chosen. This is called a systematic sample. Record the salaries, and then compute the sample s mean, rounding to the nearest whole number , 41, 50, 57, 64, 80; mean = = _ Resource Locker Module Lesson 1 Name Class Date 22.1 Data-Gathering Techniques Essential Question: Under what circumstances should a sample statistic be used as an estimator of a population parameter? S-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Image Credits: Radius Images/Corbis Explore Finding the Mean of Samples Obtained from Various Sampling Methods You collect data about a population by surveying or studying some or all of the individuals in the population. When all the individuals in a population are surveyed or studied, the data-gathering technique is called a census. A parameter is a number that summarizes a characteristic of the population. When only some of the individuals in a population are surveyed or studied, the data-gathering technique is called sampling. A statistic is a number that summarizes a characteristic of a sample. Statistics can be used to estimate parameters. Consider the following table, which lists the salaries (in thousands of dollars) of all 30 employees at a small company. In this Explore, you will take samples from this population and compute the mean (the sum of the data divided by the sample size). Salaries at a Small Company Resource Suppose the employees whose salaries are 51, 57, 58, 65, 70, and 73 volunteer to be in the sample. This is called a self-selected sample. Compute the sample s mean, rounding to the nearest whole number Mean = = Suppose the six salaries in the first two columns of the table are chosen. This is called a convenience sample because the data are easy to obtain. Record the salaries, and then compute the sample s mean, rounding to the nearest whole number , 24, 44, 46, 58, 62; mean = = Suppose every fifth salary in the list, reading from left to right in each row, is chosen. This is called a systematic sample. Record the salaries, and then compute the sample s mean, rounding to the nearest whole number , 41, 50, 57, 64, 80; mean = = Module Lesson 1 HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition Lesson 22.1

2 Label the data in the table with the identifiers 1 10 for the first row, for the second row, and for the third row. Then use a graphing calculator s random integer generator to generate six identifiers between 1 and 30, as shown. (If any identifiers are repeated, simply generate replacements for them until you have six unique identifiers.) This is called a simple random sample. Record the corresponding salaries, and then compute the sample s mean, rounding to the nearest whole number. Answers will vary. Sample answer based on the random identifiers shown on the calculator screen: 71, 30, 51, 47, 62, 24; mean = = _ Reflect 1. Compute the mean of the population. Then list the four samples from best to worst in terms of how well each sample mean estimates the population mean. Population mean 49; simple random sample, systematic sample, convenience sample, self-selected sample 2. With the way the table is organized, both the convenience sample and the systematic sample have means that are not too far from the population mean. Why? The data are organized in ascending order. So, choosing every fifth salary for the systematic sample gets some low, some medium, and some high salaries. The same is true when choosing the six salaries in the first two columns of the table for the convenience sample. Explain 1 Distinguishing Among Sampling Methods The goal of sampling is to obtain a representative sample, because the statistic obtained from the sample is a good estimator of the corresponding population parameter. Some sampling methods can result in biased samples that may not be representative of the population and can produce statistics that lead to inaccurate conclusions about the corresponding population parameters. Sampling Method Simple random sample Self-selected sample Convenience sample Systematic sample Stratified sample Cluster sample Description Each individual in the population has an equal chance of being selected. Individuals volunteer to be part of the sample. Individuals are selected based on how accessible they are. Members of the sample are chosen according to a rule, such as every nth individual in the population. The population is divided into groups, and individuals from each group are selected (typically through a random sample within each group). The population is divided into groups, some of the groups are randomly selected, and either all the individuals in the selected groups are selected or just some of the individuals from the selected groups are selected (typically through a random sample within each selected group). Module Lesson 1 PROFESSIONAL DEVELOPMENT Math Background Sampling involves making observations of or gathering data from a population, either the entire population (a census) or a part of the population (a sample). The goal is to gain knowledge about a population, often to make predictions, or to persuade. Predictions based on statistics have a certain amount of error associated with them. Error can come about from the sampling method used, whether intentional (for example, when using statistics to persuade) or unintentional. A frequent source of error in statistical predictions is bias, which makes it more likely that certain subsets of the population are over-represented in a sample. Sampling methods can affect the statistics gathered and, therefore, the predictions that are based on them. EXPLORE Finding the Mean of Samples Obtained from Various Sampling Methods INTEGRATE TECHNOLOGY Students have the option of completing the activity either in the book or online. QUESTIONING STRATEGIES Are you trying to find a statistic or a parameter? trying to find a statistic based upon the sample to find the parameter for the population EXPLAIN 1 Distinguishing Among Sampling Methods CONNECT VOCABULARY Relate the word stratified to the English prefixes strati- and strato-, meaning layer. Students may be aware that the stratosphere is a layer of the atmosphere. QUESTIONING STRATEGIES Why might the sample be representative of the population? Why might it not be representative? It might be representative because it is a random sample. It might not be representative because the sample is small. AVOID COMMON ERRORS Students may prefer to use convenience samples because they are easier. Point out that the results from some convenience samples may be biased, but that other convenience samples may produce good results. For example, if their math class consists of all juniors and they wish to know the percent of students going to the junior prom, using their class as a sample might produce good results. Data-Gathering Techniques 1084

3 The Explore showed that simple random samples are likely to be representative of a population (as are other sampling methods that involve randomness) and are therefore preferred over sampling methods that don t involve randomness. Example 1 Identify the population, classify the sampling methods, and decide whether the sampling methods could result in a biased sample. Explain your reasoning. The officials of the National Football League (NFL) want to know how the players feel about some proposed changes to the NFL rules. They decide to ask a sample of 100 players. a. The officials choose the first 100 players who volunteer their opinions. b. The officials randomly choose 3 or 4 players from each of the 32 teams in the NFL. c. The officials have a computer randomly generate a list of 100 players from a database of all NFL players. The population consists of the players in the NFL. a. This is a self-selected sample because the players volunteer their opinions. This could result in a biased sample because the players who feel strongly about the rules would be the first ones to volunteer and get their opinions counted. b. This is a stratified sample because the players are separated by team and randomly chosen from each team. This is not likely to be biased since the players are chosen randomly and are taken from each team. Image Credits: Stephen Lam/Getty Images c. This is a simple random sample because each player has an equally likely chance of being chosen. This is not likely to be biased since the players are chosen randomly. Administrators at your school want to know if students think that more vegetarian items should be added to the lunch menu. a. The administrators survey every 25th student who enters the cafeteria during the lunch period. b. The administrators survey the first 50 students who get in the lunch line to buy lunch. c. The administrators use a randomly generated list of 50 students from a master list of all students. The population consists of the students at the school. a. This is a systematic sample because the rule of every 25th student is used. This method [is/isn t] likely to result in a biased sample because a wide range of students will be surveyed. b. This is a convenience sample because the students are easily accessible. This method [is/isn t] likely to result in a biased sample because it does not include students who bring their. lunch. c. This is a simple random sample because each student has an equally likely chance of being chosen. This method [is/isn t] likely to result in a biased sample because the students are chosen randomly. Module Lesson 1 COLLABORATIVE LEARNING Whole Class Activity Ask a few students in the class whether they are more interested in music or in art. Use their responses to make a prediction about the whole class. Then ask half the students in the class the same question, and make a second prediction. Finally, direct the question to the entire class and tally the results. Discuss how accurate each prediction is, and whether it is surprising or not that one prediction is more accurate than another Lesson 22.1

4 Your Turn Identify the population, classify the sampling methods, and decide whether the sampling methods could result in a biased sample. Explain your reasoning. 3. A local newspaper conducts a survey to find out if adult residents of the city think the use of hand-held cell phones while driving in the city should be banned. a. The newspaper sends a text message to a random selection of 1000 subscribers whose cell phones are listed in the paper s subscription database. b. Using the 10 neighborhoods into which the city is divided, the newspaper randomly contacts 100 adults living in each of the neighborhoods. The population consists of adult residents of the city. a. Simple random sample of subscribers with cell phones; everyone has an equally likely chance of being chosen. Likely to be biased; people who do not use cell phones (or did not give their cell phone numbers to the newspaper) are left out of the survey. b. Stratified sample; individuals are grouped and a random sample from each group is selected. Not likely to result in a biased sample; a wide range of residents is surveyed. Explain 2 Making Predictions from a Random Sample In statistics, you work with data. Data can be numerical, such as heights or salaries, or categorical, such as eye color or political affiliation. While a statistic like the mean is appropriate for numerical data, an appropriate statistic for categorical data is a proportion, which is the relative frequency of a category. EXPLAIN 2 Making Predictions from a Random Sample QUESTIONING STRATEGIES How could the sampling method be improved to get a more representative sample? Increase the number of people surveyed, and verify that the people are truthful in their responses. What makes a prediction likely to be reliable? randomness and a sample representative of the population, plus correct calculations Example 2 A community health center surveyed a small random sample of adults in the community about their exercise habits. The survey asked whether the person engages in regular cardio exercise (running, walking, swimming, or other) and, if so, what the duration and frequency of exercise are. Of the 25 people surveyed, 10 said that they do engage in regular cardio exercise. The table lists the data for those 10 people. Calculate statistics from the sample, and use the statistics to make predictions about the exercise habits of the approximately 5000 adults living in the community. Type of exercise Duration (minutes spent exercising) Frequency (times per week) Running 30 4 Walking 20 5 Running 40 3 Running 60 6 Swimming 40 4 Other 90 2 Running 30 3 Walking 20 5 Running 30 4 Other Module Lesson 1 DIFFERENTIATE INSTRUCTION Kinesthetic Experience Making flash cards will help students learn the language of statistics. They can add to their cards as the unit progresses. In addition to using the cards for study, students can play games with the flash cards; for example, have one student give clues to a second, who must guess what is on the card. INTEGRATE TECHNOLOGY A graphing calculator or spreadsheet can be used to calculate statistics and make projections for the parameters. CONNECT VOCABULARY Ask students whether bias in data reporting and bias in society are connected. Note that bias in society is generally intentional, while bias in data may be unintentional. CONNECT VOCABULARY Ask students to answer the following questions using this vocabulary: representative, unbiased, population, statistic, parameter. What are samples used for and why? A sample is used to make a prediction about a population when a census is impractical. What should be true of a sample in order to make predictions and estimates? A sample should be both unbiased and representative of every group in the population. Data-Gathering Techniques 1086

5 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Have students consider a situation in which two different random samples are used, resulting in very different estimates for the population parameters. Students should consider using both samples together to comprise a larger sample. AVOID COMMON ERRORS Students may confuse a sample with a census, and assume that they have calculated a statistic when they have actually found the parameter itself for an entire population. Remind students that if every member of a population is sampled, the sample and population are the same. A B Calculate the proportion of adults who get regular cardio exercise and the proportion of runners among those who get regular cardio exercise. Use the proportions to predict the number of runners among all adults living in the community. Proportion of adults who get regular cardio exercise: 10 = 0.4 or 40% 25 Proportion of runners among those who get regular cardio exercise: 5 = 0.5 or 50% 10 To predict the number of runners in the community, multiply the number of adults in the community by the proportion of adults who get regular cardio exercise and then by the proportion of runners among those who get regular cardio exercise. Predicted number of runners in the community: = 1000 Calculate the mean duration of exercise for those who get regular cardio exercise and the mean frequency of exercise for those who get regular cardio exercise. Use the means to predict, for those who get regular cardio exercise, the number of hours spent exercising each week. Show your calculations and include units. Mean duration of exercise for those who get regular cardio exercise: 48 minutes Mean frequency of exercise for those who get regular cardio exercise: 3.7 times per week To predict the number of hours spent exercising, multiply the mean duration of exercise (in minutes) for those who get regular cardio exercise by the mean frequency of exercise (in times per week) for those who get regular cardio exercise. This product will be in minutes per week. To convert to hours per week, also multiply by the conversion factor _ 1 hour 60 minutes. Predicted time spent exercising: 48 minutes 3.7 /week _ 1 hour 60 minutes 3 hours/week Reflect 4. Discussion How much confidence do you have in the predictions made from the results of the survey? Explain your reasoning. Possible answer: The sample was random, which makes the results representative of the population, but the sample size was small, which may not make the results accurate. SUMMARIZE THE LESSON What are the different methods for gathering data about a population? A census, in which information is gathered from every individual in the population, can be used. If a census is impractical, a sampling method can be used. Sampling techniques include random, self-selected, convenience, systematic, stratified, and cluster. Your Turn 5. A ski resort uses the information gained from scanning season ski passes in lift lines to determine how many days out of each season the pass holders ski and how many lift rides they take each day. The table lists the data for a random sample of 16 pass holders. Calculate the mean number of days skied and the mean number of lift rides taken per day. Use the means to predict the number of lift rides taken per season by a pass holder. Number of days Number of lift rides Number of days Number of lift rides Mean number of days skied: ; mean number of lift rides taken per day: ; predicted number of lift rides taken by a pass holder per season: lift rides days 141 lift rides day Module Lesson 1 LANGUAGE SUPPORT Communicate Math Have students work in pairs to complete a graphic organizer of relationships like the one below using this list of words: census, sample, sampling, population, statistic, parameter. Students should be able to verbalize the relationships Lesson 22.1 Population Census Parameter Sample Sampling Statistic

6 Elaborate 6. Name a sampling method that is more likely to produce a representative sample and a sampling method that is more likely to produce a biased sample. Answers may vary. Possible answer: A simple random sample is more likely to produce a representative sample. A self-selected sample is more likely to produce a biased sample. EVALUATE 7. Why are there different statistics for numerical and categorical data? Numerical data are numbers that can be ordered (to find the median), summed (to find the mean), and so on. Categorical data are categories, and you cannot perform operations on categories. Instead, you can count the number of individuals in a category (to find the frequency) or find the fraction of all individuals in a category (to obtain the relative frequency). 8. Essential Question Check-In Explain the difference between a parameter and a statistic. A parameter is a number that summarizes a population and is obtained from a census. A statistic is a number that summarizes a sample taken from a population. Evaluate: Homework and Practice 1. A student council wants to know whether students would like the council to sponsor a mid-winter dance or a mid-winter carnival this year. Classify each sampling method. a. Survey every tenth student on the school s roster. Systematic sample b. Survey all students in three randomly selected homerooms. Cluster sample c. Survey 20 randomly selected freshmen, 20 randomly selected sophomores, 20 randomly selected juniors, and 20 randomly selected seniors. Stratified sample d. Survey those who ask the council president for a questionnaire. Self-selected sample e. Survey a random selection of those who happen to be in the cafeteria at noon. Convenience sample Online Homework Hints and Help Extra Practice 2. The officers of a neighborhood association want to know whether residents are interested in beautifying the neighborhood and, if so, how much money they are willing to contribute toward the costs involved. The officers are considering two methods for gathering data: Method A: Call and survey every tenth resident on the association s roster. Method B: Randomly select and survey 10 residents from among those who come to the neighborhood block party. a. Identify the population. The population consists of all residents of the neighborhood. ASSIGNMENT GUIDE Concepts and Skills Explore Finding the Mean of Samples Obtained from Various Sampling Methods Example 1 Distinguishing Among Sampling Methods Example 2 Making Predictions from a Random Sample Practice Exercise 1 Exercises 2 9 Exercises INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning MP.2 Students should understand the difference between categorical data and numerical data, and that statistics, such as frequency and mode, can be found for categorical data. Module Lesson 1 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices Recall of Information MP.3 Logic 3 2 Skills/Concepts MP.3 Logic 4 1 Recall of Information MP.3 Logic Skills/Concepts MP.3 Logic Recall of Information MP.3 Logic Skills/Concepts MP.4 Modeling Skills/Concepts MP.3 Logic Data-Gathering Techniques 1088

7 AVOID COMMON ERRORS Students may believe they are calculating parameters when they make predictions, but actual parameters cannot be found through sampling. COLLABORATIVE LEARNING Have students design a survey to answer a question about the faculty of the school. Students should come up with one question and two sampling methods that could be used one that is likely to be representative of the population and one that is not. Tell them to make up results and base a prediction on each method. b. Which sampling method is most likely to result in a representative sample of the population? Explain. Method A is most likely to result in a representative sample. If method B is used, those who do not attend the block party would have no chance of being represented. c. Describe another sampling method that is likely to result in a representative sample of the population. Possible answer: Randomly select names from the association s roster. d. Describe the categorical and numerical data that the officers of the neighborhood association want to gather. Interest in beautifying the neighborhood is categorical. The amounts residents are willing to contribute is numerical. Decide whether the sampling method could result in a biased sample. Explain your reasoning. 3. On the first day of school, all of the incoming freshmen attend an orientation program. Afterward, the principal wants to learn the opinions of the freshmen regarding the orientation. She decides to ask 25 freshmen as they leave the auditorium to complete a questionnaire. This sampling method is likely not to be biased since the sample is random and all of the freshmen are available to be chosen. 4. The members of the school drama club want to know how much students are willing to pay for a ticket to one of their productions. They decide that each member of the drama club should ask 5 of his or her friends what they are each willing to pay. This sampling method could be biased since friends of the club members may be willing to pay more than others. 5. A medical conference has 500 participating doctors. The table lists the doctors specialties. A researcher wants to survey a sample of 25 of the doctors to get their opinions on proposed new rules for health care providers. Explain why it may be better for the researcher to use a stratified sample rather than a simple random sample. Doctors from different specialties may have different opinions about the new rules. Using a simple random sample means that dermatologists, as a group, have less of a chance of having their opinions surveyed than any other group, and oncologists have a greater chance than any other group. Using a stratified sample guarantees that the opinions all of five specialties are surveyed. Specialty 6. A researcher wants to conduct a face-to-face survey of 100 farmers in a large agricultural state to get their opinions about the risks and rewards of farming. The researcher has limited time and budget. Explain why it may be better for the researcher to use a cluster sample based on counties in the state rather than a simple random sample. By sampling a handful of counties (a cluster sample) and surveying farmers only within those counties, the researcher limits the time and travel costs for the survey. Number of Doctors Dermatology 40 Geriatrics 120 Oncology 140 Pediatrics 100 Surgery 100 Module Lesson 1 Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 14 3 Strategic Thinking MP.3 Logic 1089 Lesson 22.1

8 Identify the population and the sampling method. 7. A quality control inspector at a computer assembly plant needs to estimate the number of defective computers in a group of 250 computers. He tests 25 randomly chosen computers. The population consists of all computers at the assembly plant. This is a simple random sample. 8. The manager of a movie theater wants to know how the movie viewers feel about the new stadium seating at the theater. She asks every 30th person who exits the theater each Saturday night for a month. The population consists of people who go to the movie theater. This is a systematic sample. CONNECT VOCABULARY Help students distinguish among the many terms used for the types of data covered in this lesson. Make connections between the idea of categorical data and the descriptions of categories, such as gender, age, ethnicity, and so on. 9. Eric is interested in purchasing a used sports car. He selects the make and model of the car at a website that locates all used cars of that make and model for sale within a certain distance of his home. The website delivers a list of 120 cars that meet his criteria. Eric randomly selects 10 of those cars and records what type of engine and transmission they have, as shown in the table. a. If Eric can only drive cars with automatic transmissions, predict the number of such cars on the website s list. Show your calculations. Proportion of cars with an automatic transmission: 4 = 0.4 or 40% 10 Predicted number of cars with an automatic transmission: = 48 Engine (L = liter) Transmission 3.6 L V6 Manual 3.6 L V6 Automatic 6.2 L V8 Manual 3.6 L V6 Manual 6.2 L Supercharged V8 Manual 3.6 L V6 Automatic 6.2 L V8 Manual 3.6 L V6 Automatic 3.6 L V6 Manual 6.2 L V8 Automatic b. Eric knows that the 3.6 L V6 engine takes regular gasoline, but the 6.2 L V8 engine requires premium gasoline. To minimize his fuel costs, he wants a car with the 3.6 L V6 engine. Predict the number of such cars on the website s list. Show your calculations. Proportion of cars with the 3.6 L V6 engine: 6 = 0.6 or 60% 10 Predicted number of cars with the 3.6 L V6 engine: = 72 c. Predict the number of cars that have a 3.6 L V6 engine and an automatic transmission. Show your calculations. Proportion of cars with a 3.6 L V6 engine and an automatic 3 transmission: = 0.3 or 30% 10 Predicted number of cars with a 3.6 L V6 engine and an automatic transmission: = 36 Module Lesson 1 Data-Gathering Techniques 1090

9 INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Explain to students that, although self-selected surveys are not likely to be representative of a general population, they can be useful in determining the opinions of people who already have an interest in an issue. Self-selected surveys also have the benefit of eliminating bias that might stem from direct contact with a surveyor. These two facts can make self-selected surveys especially useful for politicians. Image Credits: Bernd Tschakert Photography/Alamy 10. A community theater association plans to produce three plays for the upcoming season. The association surveys a random sample of the approximately 7000 households in the community to see if an adult member of the household is interested in attending plays and, if so, what type of plays the person prefers (comedy, drama, or musical), how many members of the household (including the person surveyed) might attend plays, and how many of the three plays those household members might attend. Of the 50 adults surveyed, 12 indicated an interest in attending plays. The table lists the data for those 12 people. Preferred Type of Play Number of People Attending Number of Plays Attending Comedy 2 1 Musical 3 2 Musical 1 2 Drama 2 3 Comedy 3 2 Comedy 2 3 Musical 4 1 Drama 2 3 Comedy 2 2 Musical 2 3 Comedy 5 1 Drama 1 2 a. Describe the categorical and numerical data gathered in the survey. Interest in attending plays and preferred type of play are categorical data. Number of household members who might attend plays and number of plays they might attend are numerical data. b. Calculate the proportion of adults who indicated an interest in attending plays and calculate the proportion of adults who prefer dramas among those who are interested in attending plays. If approximately 15,000 adults live in the community, predict the number of adults who are interested in attending plays that are dramas. Show your calculations. Proportion of adults interested in attending plays: 12 = 0.24 or 24% 50 Proportion of adults who prefer dramas among those who are interested in attending plays: 3 = 0.25 or 25% 12 Predicted number of adults who are interested in attending plays that are dramas: 15, = 900 Module Lesson Lesson 22.1

10 c. For an adult with an interest in attending plays, calculate the mean number of household members who might attend plays and the mean number of plays that those household members might attend. Round each mean to the nearest tenth. If the theater association plans to sells tickets to the plays for $40 each, predict the amount of revenue from ticket sales. Show your calculations and include units. Mean number of people attending plays: ; mean number of plays attended: Predicted number of households with an adult having an interest in attending plays: = 1680 Predicted number of tickets sold: 1680 households 2.4 people household 2.1 person tickets = tickets Predicted revenue from ticket sales: tickets $40 ticket = $338, Match each description of a sample on the left with a sampling technique on the right. INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students research what a Gallup poll is, and write a short report describing the poll and the purpose of the organization that is responsible for Gallup polls. A. A television reporter asks people walking by on the street to answer a question about an upcoming election. B. A television reporter randomly selects voting precincts and then contacts voters randomly chosen from a list of registered voters residing in those precincts to ask about an upcoming election. C. A television reporter contacts voters randomly chosen from a complete list of registered voters to ask about an upcoming election. D. A television reporter contacts every 100th voter from a complete list of registered voters to ask about an upcoming election. E. A television reporter contacts registered voters randomly chosen from every voting precinct to ask about an upcoming election. F. A television reporter asks viewers to call in their response to a question about an upcoming election. C F A D E B Simple random sample Self-selected sample Convenience sample Systematic sample Stratified sample Cluster sample H.O.T. Focus on Higher Order Thinking 12. Critical Thinking A reporter for a high school newspaper asked all members of the school s track team how many miles they run each week. a. What type of data did the reporter gather? The reporter gathered numerical data. b. Was the reporter s data-gathering technique a census or a sample? Explain. It was a census of the track team unless the reporter s intended population is the entire student body, in which case it was a sample. c. Are the data representative of the entire student body? Why or why not? No, the data are not representative of the entire student body; members of the track team most likely run more than members of the entire student body. Module Lesson 1 Data-Gathering Techniques 1092

11 PEER-TO-PEER DISCUSSION Give each student six index cards. Tell them to draw a diagram from the Types of Samples chart on one side of each card, and then write the corresponding name of the sample on the other side. Have students pair up and take turns quizzing each other with the cards, with one student showing the diagram and the other student giving the name of the sample type. Students can take the cards home and use them as flash cards to study. JOURNAL Have students write a journal entry discussing the different types of sampling random, self-selected, convenience, systematic, stratified, and cluster. The entry should include a description and an example of each type, a reason why you might use each type, and a statement about how representative of the entire population each type is likely to be. 13. Communicate Mathematical Ideas Categorical data can be nominal or ordinal. Nominal data refer to categories that do not have any natural ordering, while ordinal data refer to categories that do have an order. Similarly, numerical data can be discrete or continuous. Discrete data are typically counts or scores (which cannot be made more precise), while continuous data are typically measurements (which can be made more precise). For each description of a set of data, identify whether the data are nominal, ordinal, discrete, or continuous. Explain your reasoning. Also give another example of the same type of data. a. A researcher records how many people live in a subject s household. The data are discrete because how many people live in household is a count. Another example of discrete data: number of televisions in a household. b. A researcher records the gender of each subject. The data are nominal because the categories of male and female do not have an order. Another example of nominal data: subject s eye color. c. A researcher records the amount of time each subject spends using an electronic device during a day. The data are continuous because time can be measured in hours, minutes, or even seconds. Another example of continuous data: subject s height. d. A researcher records whether each subject is a young adult, a middle-aged adult, or a senior. The data are ordinal because the categories have an order from youngest to oldest. Another example of ordinal data: the highest educational degree that a subject has attained (high school diploma, associate s degree, bachelor s degree, and so on). 14. Analyze Relationships The grid represents the entire population of 100 trees in an apple orchard. The values in the grid show the number of kilograms of apples produced by each tree during one year. Given the data, obtain three random samples of size 20 from the population and find the mean of each sample. Discuss how the means of those samples compare with the population mean, which is Answers will vary. The mean for any given sample is likely to fall between 57.5 and Module Lesson Lesson 22.1

12 Lesson Performance Task Think about your school s cafeteria and the food it serves. Suppose you are given the opportunity to conduct a survey about the cafeteria. a. Identify the population to be surveyed. b. Write one or more survey questions. For each question, state whether it will generate numerical data or categorical data. c. Assuming that you aren t able to conduct a census of the population, describe how you could obtain a representative sample of the population. d. Suppose you asked a random sample of 25 students in your school whether they were satisfied with cafeteria lunches and how often in a typical week they brought their own lunches. The tables give the results of the survey. If the school has 600 students, use the results to predict the number of students who are satisfied with cafeteria lunches and the number of lunches brought to school in a typical week. Satisfied with cafeteria lunches? Bring own lunches how often in a week? Response Number Response Number Yes 18 5 times 2 No 7 4 times 1 a. Answers may vary. Possible answer: All students who attend the school. (Alternatively, the population might be only students who sometimes or always eat the cafeteria s lunches.) b. Answers may vary. Possible answers: The question Are you satisfied with cafeteria lunches? would generate categorical data, while the question In a typical week, how often do you bring your own lunch? would generate numerical data. c. Answers may vary. Possible answer: If a complete roster of students is available, assign numerical IDs to the students and use a random number generator to obtain a list of IDs. Otherwise, use a systematic sample of students in the cafeteria during each of the school s lunch periods. d. From the first table, the proportion of students who are satisfied with cafeteria lunches is 18 = 0.72, so the 25 predicted number of students in the school who are satisfied is = 432. From the second table, the mean number of lunches brought to school by a student in a typical week is = 1.36, so the predicted number 25 of lunches brought to school by all students is = times 4 2 times 2 1 time 4 0 times 12 CONNECT VOCABULARY Discuss the fact that, in an ice cream parlor, paint store, or fabric store, a customer can ask for a sample before deciding to make a purchase. Ask students how this type of sample is similar to polling a sample of a population. Students should make the connection that, in both situations, a small portion of the whole is evaluated in order to make a decision about the whole. QUESTIONING STRATEGIES Under what circumstances would you take a census rather than a sample of the school population? Possible answer: If the food supplier needed to prepare specific lunches for those students who have food allergies, the chef would need to know how many such lunches to prepare. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss why you would not want to poll the first 30 students entering the cafeteria at a specific time of day. Look for student responses such as the following: The first 30 students might be closest to the cafeteria, or the hungriest, or have different needs. They would be more likely to be the same age or to arrive in groups of the same gender or from the same location. Using data from such a poll would create a bias in favor of that group s preferences. Module Lesson 1 EXTENSION ACTIVITY Have students research the meaning of the expression margin of error and a formula for finding it when sampling n people. Then have them find the percent margin of error for a poll of 2400 people. Have them consider standard error margins for political surveys, for example. Note that margin of error will be discussed further in Lesson 24.3, in the study of data distributions. A margin of error is the amount 1_ of miscalculation allowed in computing data; margin of error = ±, where n represents the number of data values in the set; margin n of error = ± 1_ ±0.02, or about 2% Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Data-Gathering Techniques 1094