LESSON Lesson 26 OVERVIEW Understand Random …asms.psd202.org/documents/kpowell/1586215840.pdfOVERVIEW Lesson 26 Understand Randm ampes Lesson 26 Understand Random Samples Domain

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LESSON OVERVIEW

Lesson 26 Understand Random Samples


DomainStatistics and Probability

ClusterA. Use random sampling to draw

inferences about a population.

Standard7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Additional Standards7.RP.A.2a, 7.RP.A.3 (See page B3 for full text.)

Standards for Mathematical Practice (SMP)3 Construct viable arguments and

critique the reasoning of others.

4 Model with mathematics.

5 Use appropriate tools strategically.

CCSS Focus

Learning Progression

are only accurate if the sample is representative of the entire population. In addition, they see that the results of a survey of a representative sample is proportional to the results that would be found by surveying the entire population.

In Grade 8 students will construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.

In Grade 6 students were introduced to statistical questions and measures of center and variability. Students also developed a basic understanding of data. Students recognized patterns in data and determined the best way to display a set of data.

In this lesson students work with surveys. They learn that a sample population can be surveyed to make predictions, but those predictions

• random sample a sample in which every element in the population has an equal chance of being selected.

• population the entire group considered for a survey.

• biased sample a sample that does not represent the whole population.

Review the following key terms.

• distribute to give or spread out.

• generalize to apply information from one item or example to a larger group.

• point of view a way of thinking about something.

• sample items from a group.

• survey a set of questions used to gather information and make predictions.

Lesson Vocabulary

• Distinguish an accurate statistical question that will result in variable data from a question that will not.

• Determine if a relationship is proportional.

Prerequisite SkillsLesson Objectives

Content Objectives• Understand that a representative

sample can be used to make predictions about a large population.

• Describe different ways of finding a sample and determine which sample is the most representative of a given population.

• Create a representative sample and use it to make predictions about a population.

Language Objectives• Read and interpret information about

samples and differentiating between random and biased samples.

• Summarize information from text in writing or discussion.

• Participate in partner or group discussion to determine if samples are random or biased.

• Talk and write about random samples using terms such as random, bias, sample size, representation, fair chance, and different groups.

Lesson Pacing Guide

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Whole Class Instruction

Lesson 26


Small Group Differentiation

Teacher-Toolbox.com

Day 145–60 minutes

Toolbox: Interactive Tutorial*Random Samples

Practice and Problem SolvingAssign pages 281–282.

Introduction

• Think It Through Question 10 min• Think 15 min• Think 15 min• Reflect 5 min


Guided Instruction

Think About Identifying Random and Biased Samples• Let’s Explore the Idea 15 min• Let’s Talk About It 20 min• Try It Another Way 10 min



Guided Practice

Connect Ideas About Identifying Random and Biased Samples• Compare 5 min• Explain 5 min• Plan 5 min

Independent Practice

Apply Ideas About Identifying Random and Biased Samples• Put It Together 15 min• Intervention, On-Level, or Challenge

Activity 15 min


Toolbox: Lesson QuizLesson 26 Quiz

ReteachReady Prerequisite Lessons 45–90 min

Grade 6• Lesson 26 Understand Statistical Questions• Lesson 27 Measures of Center

and Variability

Teacher-led ActivitiesTools for Instruction 15–20 min

Grade 7• Random Sampling

Personalized Learning

i-Ready.com

Independenti-Ready Lessons* 15–20 min

Grade 7• Random Samples

* We continually update the Interactive Tutorials. Check the Teacher Toolbox for the most up-to-date offerings for this lesson.

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ELL English Language Development


Reading/Speaking Create an anchor chart for the unit by displaying the Academic Vocabulary. Read the words and have students repeat them chorally. Review the meaning of each word and ask students to illustrate the word or use it in a sentence. Work through Think It Through as a class. After each section, adapt the Co-constructed Word Bank routine to add terms to the anchor chart. Then ask students to Turn and Talk to choose and explain a term from the anchor chart. Provide these sentence frames:

• We choose the term .

• The term describes/means .

Reading/Speaking Create an anchor chart for the unit by displaying the Academic Vocabulary. Call on students to explain or illustrate the meanings of the terms, providing support as needed. Work through Think It Through as a class. After each section, adapt the Co-constructed Word Bank routine to add terms to the anchor chart. Then ask students to Turn and Talk to choose terms from the anchor chart to use in a statement that summarizes some of the information from that section. Create a class summary of Think It Through from the statements.

Reading/Speaking Create an anchor chart for the unit by displaying the Academic Vocabulary. Call on students to explain or illustrate the meanings of the terms, providing support as needed. Work through each section of Think It Through as a class. After each section, adapt the Co-constructed Word Bank routine to add terms to the anchor chart. Then ask students to Turn and Talk to discuss relationships between the terms on the chart. Suggest they write the terms on sticky notes and sort them into categories about sampling that they choose.

ELP Levels 4–5ELP Levels 2–4ELP Levels 1–3

Prepare for Day 1: Use with Think It Through

Academic Vocabulary: A survey is a set of questions used to gather information and make predictions. Predictions are made by generalizing, or applying information from a survey to a larger group. A sample represents a larger group. To distribute means “to spread out.”

Speaking/Writing Work with small groups of students to discuss and complete the Let’s Talk About It problems. Model for students ways to use the language of the question in their answers. For example, ask:

• What is a random sample? A random sample is .

• Do you think this suggestion forms a random sample? I think this suggestion a random sample because .

Then read aloud the questions in Let’s Talk About It. Have students Turn and Talk to repeat the question, form an answer using the language of the question, and write their answers. Call on pairs to share their responses with the group.

Speaking/Writing Partner students to discuss and complete the Let’s Talk About It problems. Encourage students to incorporate terms from the previous lesson’s anchor chart into their discussions. As students discuss problem 7, it may be helpful to add the following terms to the bank: sample size and point of view.

After each partner has had the opportunity to explain their ideas, have them co-write answers in short sentences using terms from the question and the anchor chart. Call on partners to share their responses with the group.

Speaking/Writing Pair students to discuss the Let’s Talk About It problems. Encourage students to incorporate terms from the previous lesson’s anchor chart into their discussions. As students discuss problem 7, it may be helpful to add the following terms to the bank: sample size and point of view.

After each partner has had the opportunity to discuss ideas, have students compose their answers independently using complete sentences and terms from the anchor chart as appropriate. Call on individuals to share their responses with the group.


Prepare for Day 2: Use with Let’s Talk About It

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Lesson 26


Listening/Writing Display and read the characteristics of random samples:

• In a random sample, everyone has an equal chance of being chosen.

• A random sample is fair, not biased.

• Different opinions, or points of view, are represented by a random sample.

Then, work with students to discuss Connect problem 12. Use the Revoicing routine to validate student thinking and model fluent speech. Compare each idea to the characteristics. Ask:

• Will different groups be represented in the sample?

• Do all groups have an equal chance of being chosen?

Choose a plan and model drafting an answer.

Listening/Writing Organize students into small groups to discuss Connect problem 12. Review terms related to a random sample in the anchor chart, such as fair, biased, evenly distributed, equal chance, and points of view. Ask groups to use these terms to explain how they might choose a random sample that represents the population of their community. Have students consider the following questions:

• What groups make up our community?

• How can we make sure that the chosen sample is fair?

After listening to all ideas, have groups co-write a plan and share with the class.

Listening/Writing Organize students into small groups to create a list of characteristics of a random sample. Have students read Connect problem 12 and describe possible methods for conducting a random survey of the community as a group. Encourage students to critique the reasoning of the other group members by referring to the list they generated and asking questions such as the following:

• Will your sample include ?

• Is that method fair to ?

• Can you change your plan by ?

After talking with the group, have students revise their ideas and write answers independently.


Prepare for Day 3: Use with Connect, Problem 12

Introduction

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258 Lesson 26 Understand Random Samples

Students explore the idea that random samples are a way to get information about the whole population. Students explore random and biased samples.

• Work through Think It Through as a class.

• Introduce the Question at the top of the page.

• Talk about why the food service company would want the information.

• Ask why surveying all the students from all 12 schools would be difficult.

• Suggest that it would be easy to survey 3 students from one of the schools. Have students describe problems with that plan.

SMP TIP Model with MathematicsWhen students study a sample of data instead of the entire population, they are learning to model with mathematics. As students work with samples, remind them that a sample is just a smaller version of the entire set of data, and it should be as accurate of a model as possible. (SMP 4)

• Discuss the terms random sample and population. Relate the way students use the words in everyday speech to their mathematical meanings.

• Have students tell why the first diagram shows a random sample, but the second diagram does not.

Mathematical Discourse 1 and 2

• As students share their responses to the Think questions, stress the idea that in a random sample, everyone has a fair chance to be chosen.

Mathematical Discourse 3

At A Glance

Step By Step

Mathematical Discourse

1 When people in charge make decisions based on a survey, why is it important for the sample to be random?

A random sample provides more diverse and less biased results. The information is likely to be representative of the general population.

2 In the example, what are some ways the food service company could survey students that would be unfair?

Students might say that the company could survey students from just one school or from just one group such as girls, athletes, or students on special diets.

3 How can the results of a random sample of students give you information about the entire population of students?

Listen for students to say that if a sample is random, then the fraction of students who choose each option in the sample will be approximately equal to the fraction of students who choose each option in the entire population.

Introduction

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Think It Through


A food service company supplies meals for 12 different schools. How might the company get information about students’ favorite lunches? Surveying every student in every school would take a lot of time and effort. It would be more efficient to survey a random sample that represents the whole group, or population. What makes a sample representative of the population?

Each rectangle below contains squares, circles, and triangles. The rectangle on the left shows the shapes scattered randomly. The rectangle on the right shows the shapes somewhat grouped. The circled group in each rectangle represents a sample.

In order for a sample to be considered random, every object or event has to have an equal chance of being selected. If a survey is given to the sample group, the ratio of each response to the sample size should be approximately equal to the ratio of each response to the size of the entire population. So, the percentage of people giving each response in the sample should be about equal to the percentage of people who would give each response in the entire population.

Look at the picture of the jar, which contains names of all students at Center School.

A sample that looks at the group from a sorted, uneven mix may not be representative of the population.

A sample that selects items from an evenly distributed group is representative of the population.

Think How do you find a random sample for a population?

Lesson 26Understand Random Samples

Underline the sentence that explains why selecting ten names from the top of the jar might be considered a biased sample.

How can you use samples to get information about a population?

258

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Lesson 26

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Suppose you pick ten slips of paper from the top of the jar. Maybe the slips at the top belong to the last class that put their names in the jar. Selecting from the top does not give the names at the bottom of the jar an equal chance. This would be a biased sample because it does not represent the whole population.

Suppose, instead, that you put your hand in and mix the slips all around each time you pick a name. With this method, all names get an equal chance of being selected. This is a random sample.

Here are some ways to select a representative random sample.

• Use a pattern, such as selecting every fourth person who enters the cafeteria.

• Use a method, such as drawing names out of a hat, where everyone has an equal chance of being selected.

• Divide the population into groups, such as by grade level, and randomly select people from each group.

Here are some ways of selecting a sample that might result in a biased sample.

• Let people volunteer to take a survey.

• Choose people who are easy to reach, such as the students who happen to be in the cafeteria when you are available to give surveys.

• Choose people as a group, such as students on the honor roll.

If the sample is random, then the ratio of each answer on a survey to the number of people in the sample group should be proportional to the ratio of each answer in the entire population. You can use data from a random sample to generalize about a population. Maybe about half of the students in the sample say that pizza is their favorite school lunch. You might predict that about half the population has the same preference. The data collected from the sample might be used for making menu choices and for determining food orders.

Reflect1 What do you think would be a good way to select a random sample of all the students at

the 12 schools in the school district mentioned on the previous page?

Think How can you use data collected from a random sample?

259

Possible answer: Survey every 10th student that enters the cafeteria at

each school.

Concept ExtensionExamine the result of bias in sampling.

• Have students line up by height. Make a list of their shoe sizes on the board. The sizes should be listed in the same order as the lineup.

• Divide the class into four groups to find the mean, median, and range of four different data sets.

1. Group 1 will use all the data.

2. Group 2 will use the first 5 shoe sizes.

3. Group 3 will use the last 5 shoe sizes.

4. Group 4 will write all the shoe sizes on slips of paper, randomly choose 5 of the slips, and use those data to find the mean, median, and range.

• Have each group present their results. Compare the results. Decide which sample is the most representative of the

actual data. Discuss why it is the best sample.


3 Would someone ever use a biased sample on purpose? Why might someone do so?

Students might say that biased samples are used to exaggerate a product’s benefits or to show that a political candidate is more popular or successful than otherwise warranted.

4 Are biased samples always done on purpose? Explain your answer.

Students may say that sometimes people just don’t think a plan through completely and end up with a biased sample by accident.

• Read the discussion of samples.

• Talk about the difference between representative random samples and biased samples.

• Ask why mixing up the slips of paper would change the sample from biased to random.

• Read over the suggestions for selecting a representative random sample. Discuss why each would be fair or not fair to all the students.

• As you read about the samples that might be biased, have students describe the problem with each plan.

• Discuss the Think question. Have students provide examples of ways that data from a random sample can be used.

Concept Extension


• Have students read and reply to the Reflect question. As you talk over possible ways to choose a random sample, have students keep in mind that the selection process must give everyone a fair chance of being chosen.

Assign Practice and Problem Solving pages 281–282 after students have completed this section.

Step By Step

Mathematics PRACTICE AND PROBLEM SOLVING

Guided Instruction




Guided Instruction

Think About

Lesson 26

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Identifying Random and Biased Samples

Carla has a list of all 720 students in her middle school. She writes the name of each student on a slip of paper and puts each slip in a box. Then she pulls 30 names from the box to decide who she will survey about the upcoming school election.

2 How many students are in Carla’s sample?

How many students are in the population?

3 If 5 students in Carla’s sample say they plan to vote for James in the school election, how many students in the entire school do you expect will vote for James? Explain how you found your answer.

A graphing calculator or spreadsheet can be used to create a list of random numbers.

12 164 47 598 306 702

92 7 99 388 141 85

584 163 414 373 627 417

121 71 549 480 154 90

35 419 88 660 279 349

4 Describe one way you could use this list of numbers to choose the students for a sample.

5 Does using random numbers generated by a calculator to decide who to survey give everyone in the population an equal chance of being selected? Why or why not?

Let’s Explore the Idea Read the problem and answer the questions.

260

120 students; I solved the proportion 5 ··· 30 = x ···· 720 .

Possible answer: You could assign each student a number and then give the

survey to the students who match the random numbers.

Possible answer: This selection does give all students in the population a fair

chance. The graphing calculator creates the random number list. No people are

involved in selecting the sample, so there are no judgments being made.

30

720

Students consider how to conduct a random sample using a list of random numbers. Students consider different samples and determine which ones are representative, random samples and which ones exhibit bias.

Let’s Explore the Idea• Tell students that they will have time to work

individually on the problems on this page and then share their responses in groups. You may choose to work through the first problem together as a class.

• As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking.


• Ask why Carla does not survey all the students. Explain why her sample is random.

• Have students describe different groups that make up a student population and why sampling only those groups might not give accurate results.

• Help students see why using a list of random numbers might be easier than writing all 720 names on a slip of paper and drawing 30 of them.

SMP TIP Use ToolsGenerating a list of random numbers with a graphing calculator or spreadsheet is one way students can use appropriate tools strategically, making the task much easier. (SMP 5)

Student Misconception Alert Students should realize that choosing a sample randomly does not guarantee that it will be representative. It is unlikely that a sample of 30 will end up with all athletes or Grade 7 students, but it is possible.

At A Glance

Step By Step


1 Do you think Carla should survey 10 students chosen randomly from each grade or 30 students chosen randomly from the whole school? What are the advantages of each plan?

Students might say that 10 students from each grade would ensure that students from all grades are able to give their opinions. However, choosing 30 students from the entire school would be easier.

2 How does using a list of randomly created numbers to choose students keep the sample fair?

Students may note that a computer will not play favorites.


Lesson 26

261Lesson 26 Understand Random Samples

Let’s Talk About It• Organize students into pairs or groups. You

may choose to work through the first Let’s Talk About It problem together as a class.

• Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse questions to help support or extend students’ thinking.


• As students share their ideas about choosing representative samples, continue to emphasize that the idea of a random sample is to give everyone’s opinions a fair chance of being chosen.

Concept Extension

Try It Another Way• Direct student attention to Try It Another

Way. Have a volunteer from each group come to the board to explain their group’s responses to problems 8 and 9.


Step By Step


Concept ExtensionDetermine the fairness of a sample.

• Sketch a large rectangle on the board. In it, draw 10 circles, 8 triangles, 6 stars, and 2 hearts. Position the shapes in no particular order.

• Ask: If you randomly pick 2 shapes, how likely would you be to get a representative sample? What about 6 shapes? 10 shapes? Have students explain their reasons for each answer.

• Discuss why no sample guarantees that all shapes will be represented. Discuss why a larger sample size increases the likelihood of fairly representing a population.


3 Why does a larger sample size have a better chance of representing everyone’s point of view?

Students may explain that larger sample sizes contain more points of view. The larger the sample size, the better the chance that all points of view get heard.

4 If Carla surveys 100 students instead of 30 students, does she still have to make sure the sample is random? Explain your answer.

Students should emphasize that the sample should still be chosen randomly because surveying 100 athletes or 100 seventh-grade students will leave out other students.

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6 One of Carla’s friends suggests that she survey only eighth-graders because they are the oldest and probably know more about the election than younger students. Do you think this suggestion creates a random sample? Explain.

7 Another one of Carla’s friends suggests that she make the sample larger and survey 100 students. Which sample size is more likely to represent the population? Explain.

Try It Another Way Work with your group to decide if the methods for selecting a sample are fair or biased. Give reasons for your answers.

8 The events committee wants to survey students about a school dance. The committee is meeting in the gym, where the girls’ basketball team is practicing. They survey the players on the girls’ basketball team.

9 A store owner wants to survey customers about the products he sells. He programs the computer to select 100 customers from the mailing list and sends them each a survey.

Let’s Talk About It Solve the problems below as a group.

261

Possible answer: This method does not give everyone in the population an equal

chance, so it is not a good way to create a random sample.

Possible answer: The larger sample size of 100 students is more likely to

represent the population. Getting opinions from more people should give more

ideas about the opinions of the whole population. If you only survey 30 people,

there could be some important ideas that don’t even come up.

Possible answer: This sample is biased because it only includes girls who are

athletes. It does not give everyone in the population an equal chance of

being selected.

Possible answer: This is a random sample, since all customers have an equal

chance of being selected.

Guided Practice




Students demonstrate their understanding of representative random samples. Students read a situation that requires a survey and describe the characteristics of a representative sample. They also suggest a plan for a random sample as well as descriptions of biased samples.

• Discuss each Connect problem as a class using the discussion points outlined below.

Compare• Lead a class discussion about the merits of

each teacher’s selection method. Point out that there is more than one way to choose a representative random sample.

• Remind students that representative random samples must provide a fair chance for all types of students to be chosen.

Explain• Read the problem as a class. Discuss which

people, if any, would be left out using Julie’s plan.

Plan• Read the problem as a class. Discuss the

groups in your community whose opinions should be included in the survey. Discuss how to make sure all groups have a fair chance of being heard.

• Have students share their plans with the class. Have students note the good ideas as well as the problems with each plan. Have students modify their plans as needed.

SMP TIP Construct ArgumentsAs students explain their plan for a representative random survey, they must construct viable arguments and critique the reasoning of others. In addition to defending their plans, students should be able to listen to other students’ critiques and modify their plans as needed. (SMP 3)

At A Glance

Step By Step

See student facsimile page for possible student answers.

Scoring Rubrics

Part A

Points Expectations

2 The response should describe at least two different categories, e.g., age, gender, hours they attend the gym, or the type of activity preferred.

1 The response only considers one characteristic.

0 There is no response, or the characteristic mentioned is irrelevant to the problem.

Guided Practice

Connect

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Lesson 26



Talk through these problems as a class, then write your answers below.

10 Compare The teachers in a school are asked to send four students from their homerooms to represent the class.

Method 1

Ms. Rose puts the names of all the girls in a box

and chooses two without looking. Then she does the same for the boys’ names.

Method 2

Mr. Burr sends the four students sitting closest to

the door.

Method 3

Mrs. Rosati puts the names of all the students

in a box, mixes the names, and pulls out four names

without looking.

Compare the selection methods. Do you think each one creates a random or biased sample? Which is more likely to be representative? Explain.

11 Explain The producers of a television singing contest are conducting a survey on their website to see who viewers think should win the competition. Julie says that this method will create a random sample of the people who watch the show. Do you agree?

12 Plan Describe a way to fi nd a random sample of 100 people from your community to complete a survey about recycling.

262

Possible answer: Ms. Rose’s and Mrs. Rosati’s methods create a random sample.

Ms. Rose’s method might be more representative because it ensures that two

girls and two boys will be selected. Mr. Burr’s method is biased since all students

do not have an equal chance of being selected.

Possible answer: I disagree. There are likely many people who watch the show

who don’t go to the website or access to the Internet. All viewers do not have an

equal chance of expressing their opinions this way.

Possible answer: Look at a map of the town and select several streets from

different parts of the town. Create a list of random numbers and use the

numbers to select street addresses for residents that will get the survey.



263

Lesson 26


Put It Together• Direct students to complete the Put It

Together task independently.

• Point out that there is more than one way to create a random sample, and there are many ways to create biased samples.

• As students work on their own, walk around to assess their progress and understanding, answer their questions, and give additional support.

• If time permits, have students share their plans for a representative survey, their ideas for biased samples, and their answers to Part D.


Step By Step


See student facsimile page for possible student answers.

Parts B and C

Points Expectations

2 For both questions, at least two categories of gym members are identified and described.

1 For Part B, a method for randomly surveying the entire gym membership is described, or a method for surveying different categories is described, but the method is slightly biased. For Part C, only one biased sample is described.

0 No method is given, or the methods that are given are incorrect.

Part D

Points Expectations

2 The student accurately calculated the number of gym members who would be interested in yoga, and the student explains how they found the answer.

1 The incorrect answer is given, but the method and explanation are correct, or the correct answer is given with an incorrect or missing explanation.

0 No response is given or the answer and explanation given are incorrect.

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Apply

Lesson 26



13 Put It Together Use what you have learned to complete the task.

The manager at Fitness Forever wants to add some new types of fitness classes and possibly remove others. He wants to offer a variety of classes that will appeal to all gym members.

Part A What would you consider to be a representative sample of this population?

Part B Describe how you would create a random sample of the gym population to participate in a survey about fi tness classes.

Part C Describe at least two diff erent samples for this population that could be considered biased and explain why they might be biased.

Part D Katelyn did a survey of 200 randomly-selected gym members, and found that 72 of them are interested in yoga classes. The gym has 1,000 members. About how many gym members would be interested in yoga? Explain how you found your answer.

263

Possible answer: A representative sample would be a group that every gym

member has an equal chance of being in.

Possible answer: I would want the sample to have an equal number of men and

women from different age groups. I would sort the membership list into males

and females. Then I would sort males into two or three age categories and

females into two or three age categories. Then, I would select a random sample

from each of the sorted groups.

Possible answer: A sample that includes men and women ages 18 to 25 would be

a biased sample. It does not give anyone over the age of 25 a chance of being

selected. A sample that includes women ages 18 to 50 would be biased. It does

not give men an equal chance of being selected.

Possible answer: About 360 gym members would be interested in yoga; I know that

72 out of 200 is 36%, and 36% of 1,000 is 360.

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LESSON Lesson 26 OVERVIEW Understand Random …asms.psd202.org/documents/kpowell/1586215840.pdfOVERVIEW Lesson 26 Understand Randm ampes Lesson 26 Understand Random Samples Domain