Focus on Math Concepts Lesson 26 Part 1: Introduction Understand Random Samples · 2019. 11....

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Understand Random SamplesLesson 26 Part 1: Introduction

Focus on Math Concepts

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

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

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

In order for a sample to be considered random, every object or event has to have an equal chance of being selected. Look at the picture of the jar, which contains names of all students at Center School.

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

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

A sample that looks at the group from a random, even mix is representative of the population.

CCLS7.SP.1

©Curriculum Associates, LLC Copying is not permitted.249L26: Understand Random Samples

Lesson 26Part 1: Introduction

Suppose you pick ten slips of paper from the top of the jar. Do all names have an equal chance of being selected? 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. You do this each time you pick the ten names. With this method, all names get an equal chance of being selected. This would be 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.

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

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.

Reflect

1 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?

Part 2: Guided Instruction Lesson 26

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L26: Understand Random Samples250

Explore It

Read the problem and answer the questions.

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 What are different attributes of students or different groups of students that should be represented in the sample?

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

12 164 47 598 306 70292 7 99 388 141 85

584 163 414 373 627 417121 71 549 480 154 9035 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 a calculator to decide who to survey give everyone in the population an equal chance of being selected? Why or why not?

Part 2: Guided Instruction Lesson 26


Talk About It

Solve the problems below as a group.

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.

Part 3: Guided Practice Lesson 26



Connect It

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.

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.

Mr. Burr sends the four students sitting closest to the door.

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 find a random sample of 100 people from your community to complete a survey about recycling.

Part 4: Common Core Performance Task Lesson 26


Put It Together

13 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.

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

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

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

Lesson 26

L26: Understand Random Samples 253©Curriculum Associates, LLC Copying is not permitted.

Understand Random Samples(Student Book pages 248–253)

Lesson 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.

Prerequisite skiLLs • Students are able to distinguish an accurate

statistical question that will result in variable data from a question that will not.

voCabuLaryrandom 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

the Learning ProgressionIn elementary school, students develop a basic understanding of data and how to display it. In Grade 7, students develop a deeper understanding of variability and data distributions, using numerical measures of center and spread. Students recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

In this lesson, students work with surveys. They learn that you can survey a sample population to make predictions, but those predictions are only accurate if the sample is representative of 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.


CCLs Focus

7.SP.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.

stanDarDs For MatheMatiCaL PraCtiCe: SMP 3–5 (see page A9 for full text)

Toolbox Teacher-Toolbox.com

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Prerequisite Skills

7.SP.1

Ready Lessons

Tools for Instruction

Interactive Tutorials ✓

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Lesson 26Part 1: introduction

at a gLanCeStudents explore the idea that random samples are a way to get information about the whole population.

steP by steP• 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 a problem.

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

• 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.

• 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.

sMP tip: When students study a sample of data instead of the entire population, they are learning to model with mathematics (SMP 4). 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.

• 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.

• 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.

Mathematical Discourse

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Understand Random SamplesLesson 26 Part 1: introduction


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

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

think How do you fi nd a random sample for a population?

In order for a sample to be considered random, every object or event has to have an equal chance of being selected. Look at the picture of the jar, which contains names of all students at Center School.

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

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

A sample that looks at the group from a random, even mix is representative of the population.

CCLs7.sP.1

L26: Understand Random Samples 255©Curriculum Associates, LLC Copying is not permitted.


at a gLanCeStudents explore random and biased samples.

steP by steP• 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.

• 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.

examine 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.

Group 1 will use all the data.

Group 2 will use the first 5 shoes sizes.

Group 3 will use the last 5 shoe sizes.

Group 4 will write all the shoes 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.

Concept extension

• 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.

• 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.




Suppose you pick ten slips of paper from the top of the jar. Do all names have an equal chance of being selected? 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. You do this each time you pick the ten names. With this method, all names get an equal chance of being selected. This would be 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.

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

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.

reflect

1 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?

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

each school.

256©Curriculum Associates, LLC Copying is not permitted.

L26: Understand Random Samples

Lesson 26Part 2: guided instruction

at a gLanCeStudents consider how to conduct a random sample using a list of random numbers.

steP by steP• Tell students that they will have time to work

individually on the Explore It 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.

• Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson.

sMP tip: Generating a list of random numbers with a graphing calculator or spreadsheet is one way students can use appropriate tools strategically (SMP 5). Note that once students master the use of the technology, it makes the task much easier.

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.

• 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.

• 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.


Part 2: guided instruction Lesson 26



explore it

read the problem and answer the questions.

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 What are diff erent attributes of students or diff erent groups of students that should be represented in the sample?

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

12 164 47 598 306 70292 7 99 388 141 85

584 163 414 373 627 417121 71 549 480 154 9035 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 a calculator to decide who to survey give everyone in the population an equal chance of being selected? Why or why not?

30

720

Possible answer: the sample should represent students with different opinions

and interests. it should also include students from different groups, such as

boys, girls, grades, athletes, and musicians.

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.

257©Curriculum Associates, LLC Copying is not permitted.L26: Understand Random Samples

Lesson 26Part 2: guided instruction

at a gLanCeStudents consider different samples and determine which ones are representative, random samples and which ones contain bias.

steP by steP• Organize students into pairs or groups. You may

choose to work through the first 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 making samples representative, continue to emphasize that the idea of a random sample is to give everyone’s opinions a fair chance of being chosen.

• 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.

Determine 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.

visual Model• 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.

• 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.


Part 2: guided instruction Lesson 26


talk about it

solve the problems below as a group.

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.

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.



Lesson 26Part 3: guided Practice

at a gLanCeStudents demonstrate their understanding of representative random samples.

steP by steP• Discuss each Connect It 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: As students explain their plan for a representative random survey, they must construct viable arguments and critique the reasoning of others (SMP 3). In addition to defending their plans, students should be able to listen to other students’ critiques and modify their plans as needed.

Part 3: guided Practice Lesson 26



Connect it

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.

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.

Mr. Burr sends the four students sitting closest to the door.

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.

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.

259©Curriculum Associates, LLC Copying is not permitted.L26: Understand Random Samples

Lesson 26Part 4: Common Core Performance task

at a gLanCeStudents 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.

steP by steP• 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 as well as biased plans.

sCoring rubriCsSee student facsimile page for possible student answers.

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.

Part 4: Common Core Performance task Lesson 26


Put it together

13 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.

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

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

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

Possible answer: a representative sample would include both males and

females and people from different age groups. it should also include

members that take fitness classes and those that use the equipment.

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 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.

B Points expectations

2At least two categories of gym members are identified, and a way to sample each category randomly is described.

1

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.

0 No method is given, or a method that is biased and unrepresentative is described.

C Points expectations

2 At least two unrepresentative or biased samples are described.

1 One unrepresentative or biased sample is described.

0 No response is given or the method(s) described are representative.


Differentiated instruction


Lesson 26

Challenge activity

intervention activity on-Level activityexplore bias in samples.

• Announce that the school board wants to conduct a survey to find out how much homework students should have. The board wants a wide variety of people to respond.

• Ask students to think about different possibilities for the sample survey population. Have students describe samples that favor the opinions of various groups such as parents, teachers, good students, or struggling students. Have them identify the particular bias in the selection of each sample.

• Finally, as a class, develop a sampling method that would provide a representative, random sample that includes all groups.

relate the concept of fairness to representative random sampling.

• Announce that the class is planning a field trip. Say that you will choose a committee of 7 students to decide where to go. You want everyone to have a fair chance of being on the committee.

• Tell students you might write only boys’ names on slips of paper and choose 7. Have students discuss whether this is a fair method.

• Suggest several other biased plans, such as choosing the tallest students or students who love animals. Each time, have students discuss whether the selection is fair.

• Introduce the word biased into the discussion. Have students develop a selection procedure that is not biased. As students describe the plan, encourage them to use words such as random, representative, and unbiased.

study the effect of sample size.

Materials: two number cubes numbered 1–6, pencil, two sheets of paper, scissors, paper bag

• Display to students the table shown below. Put students into groups of four. Tell each group to copy the table onto one of the sheets of paper.

• Now have the group cut the other piece of paper into 35 equal-sized slips. Have each group use the number cubes to generate 35 random numbers between 2 and 12. As each number is rolled, have one student record it neatly on a blank slip until all 35 slips bear a number. Have each group find the mean and median of the entire data set and record it in their table.

• Have each group put all of the slips into the bag. Have students in each group take turns collecting four random samples, the conditions of which are specified below. After each student collects four samples, have him or her find the mean and the median of each of those samples.

• The four sample sizes are as follows: sample one, 3 numbers; sample two, 8 numbers; sample three, 13 numbers; sample four, 18 numbers.

• After students have completed their calculations, have them compare the mean and median of the samples with the mean and median of the entire data set. Lead a discussion as to whether and how sample size appears to affect the accuracy of the mean and median of the data.

sample size Entire Data Set 3 3 3 3 8 8 8 8 13 13 13 13 18 18 18 18

Mean

Median

Focus on Math Concepts Lesson 26 Part 1: Introduction Understand Random Samples · 2019. 11....

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