CorrectionKey=NL-B;CA-B CorrectionKey=NL-C;CA-C 22 …€¦ · · 2017-04-21from Various Sampling Methods You collect data about a population by surveying or studying some or all

Name Class Date

Resource Locker

Resource Locker

© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y • I

mag

e C

red

its:

©Ra

diu

s Im

ages

/Cor

bis

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

21 24 26 28 30 32 33 35 37 41

44 46 47 49 50 51 52 54 55 57

58 62 62 64 64 65 70 71 73 80

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.

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.

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.

Mean = 51 + 57 + 58 + 65 + 70 + 73

___ 6

= 374 _ 6

≈ 62

21, 24, 44, 46, 58, 62; mean = 21 + 24 + 44 + 46 + 58 + 62

___ 6

= 255 _ 6

≈ 43

30, 41, 50, 57, 64, 80; mean = 30 + 41 + 50 + 57 + 64 + 80

___ 6

= 322 _ 6

≈ 54

Module 22 1083 Lesson 1

22 . 1 Data-Gathering TechniquesEssential Question: Under what circumstances should a sample statistic be used as an

estimator of a population parameter?

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

A2_MNLESE385900_U9M22L1 1083 9/1/14 3:44 PM

Common Core Math StandardsThe student is expected to:

S-IC.1

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Mathematical Practices

MP.5 Using Tools

Language ObjectiveCreate a graphic organizer that shows the relationships among population, census, and parameter, as well as sample, sampling, and statistic.

HARDCOVER PAGES 793802

Turn to these pages to find this lesson in the hardcover student edition.

Data-Gathering Techniques

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 TASKView 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.

1083

HARDCOVER PAGES 793802

Turn to these pages to find this lesson in the hardcover student edition.

© H

ough

ton

Mif

flin

Har

cour

t Pub

lishi

ng C

omp

any

S-IC.1 Understand statistics as a process for making inferences about population parameters

based on a random sample from that population.

Name

Class Date

Resource

LockerResource

Locker

© H

ough

ton

Mif

flin

Har

cour

t Pub

lishi

ng C

omp

any

• Im

age

Cre

dit

s: ©

Rad

ius

Imag

es/C

orb

is

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

2124

2628

3032

3335

3741

4446

4749

5051

5254

5557

5862

6264

6465

7071

7380

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.

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.

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.

Mean = 51 + 57 + 58 + 65 + 70 + 73

__

_ 6

= 3

74 _ 6

≈ 62

21, 24, 44, 46, 58, 62; mean = 21 + 24 + 44 + 46 + 58 + 62

__

_ 6

= 2

55 _ 6

≈ 43

30, 41, 50, 57, 64, 80; mean = 30 + 41 + 50 + 57 + 64 + 80

__

_ 6

= 3

22 _ 6

≈ 54

Module 22

1083

Lesson 1

22 . 1 Data-Gathering Techniques

Essential Question: Under what circumstances should a sample statistic be used as an

estimator of a population parameter?

DO NOT EDIT--Changes must be made through "File info"

CorrectionKey=NL-B;CA-B

A2_MNLESE385900_U9M22L1 1083

9/1/14 10:38 AM

1083 Lesson 22 . 1

L E S S O N 22 . 1

DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-C;CA-C

© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Label the data in the table with the identifiers 1–10 for the first row, 11–20 for the second row, and 21–30 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.

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.

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?

Explain 1 Distinguishing Among Sampling MethodsThe 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 Description

Simple random sample Each individual in the population has an equal chance of being selected.

Self-selected sample Individuals volunteer to be part of the sample.

Convenience sample Individuals are selected based on how accessible they are.

Systematic sample Members of the sample are chosen according to a rule, such as every nth individual in the population.

Stratified sample The population is divided into groups, and individuals from each group are selected (typically through a random sample within each group).

Cluster sample 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).

Answers will vary. Sample answer based on the random identifiers shown on the calculator

screen: 71, 30, 51, 47, 62, 24; mean = 71 + 30 + 51 + 47 + 62 + 24

___ 6

= 285 _ 6

≈ 48

Population mean ≈ 49; simple random sample, systematic sample, convenience sample,

self-selected sample

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.


DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-A;CA-A


A2_MNLESE385900_U9M22L1 1084 4/4/14 7:06 PM

EXPLORE Finding the Mean of Samples Obtained from Various Sampling Methods

INTEGRATE TECHNOLOGYStudents have the option of completing the activity either in the book or online.

QUESTIONING STRATEGIESAre 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 STRATEGIESWhy 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 ERRORSStudents 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.

PROFESSIONAL DEVELOPMENT

Math BackgroundSampling 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.

Data-Gathering Techniques 1084


© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y • I

mag

e C

red

its:

©St

eph

en

Lam

/Get

ty Im

ages

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.

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 .

a. This is a sample because . This method

[is/isn’t] likely to result in a biased sample because

b. This is a sample because . This method

[is/isn’t] likely to result in a biased sample because .

c. This is a sample because

. This method [is/isn’t] likely to result in a biased sample because

.

the students at the school

systematic the rule of every 25th student is used

convenience

simple random

of being chosen

the students are easily accessible

each student has an equally likely chance

the students are chosen randomly

it does not include students who bring their lunch.

a wide range of students will be surveyed.



A2_MNLESE385900_U9M22L1 1085 4/4/14 7:06 PM

COLLABORATIVE LEARNING

Whole Class ActivityAsk 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.

1085 Lesson 22 . 1


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

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.

Explain 2 Making Predictions from a Random SampleIn 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.

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 120 1

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.




A2_MNLESE385900_U9M22L1 1086 4/4/14 7:06 PM

DIFFERENTIATE INSTRUCTION

Kinesthetic ExperienceMaking 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.

EXPLAIN 2 Making Predictions from a Random Sample

QUESTIONING STRATEGIESHow 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

INTEGRATE TECHNOLOGYA 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.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

A 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 __ 25 = 0.4 or 40%

Proportion of runners among those who get regular cardio exercise: 5 __ 10 = 0.5 or 50%

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: 5000 · 0.4 · 0.5 = 1000

B 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: minutes

Mean frequency of exercise for those who get regular cardio exercise: 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: minutes ∙ /week ∙ 1 hour _ 60 minutes ≈ hours/week

Reflect

4. Discussion How much confidence do you have in the predictions made from the results of the survey? Explain your reasoning.

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 10 5 2 14 27 3 18 5

Number of lift rides 12 15 6 18 10 6 15 9

Number of days 4 16 7 12 19 14 25 13

Number of lift rides 11 13 14 10 8 6 15 18

48

3.7

3.748 3

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.

Mean number of days skied: 12.125; mean number of lift rides taken per day: 11.625; predicted number of lift rides taken by a pass holder per season:

12.125 days · 11.625 lift rides __ day

≈ 141 lift rides


DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A

A2_MNLESE385900_U9M22L1 1087 4/4/14 7:05 PM

LANGUAGE SUPPORT

Communicate MathHave 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.

Population Sample↓ ↓

Census Sampling

↓ ↓Parameter Statistic

ELABORATE INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.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 ERRORSStudents 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.

SUMMARIZE THE LESSONWhat 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.

1087 Lesson 22 . 1


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

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.

7. Why are there different statistics for numerical and categorical data?

8. Essential Question Check-In Explain the difference between a parameter and a statistic.

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.

b. Survey all students in three randomly selected homerooms.

c. Survey 20 randomly selected freshmen, 20 randomly selected sophomores, 20 randomly selected juniors, and 20 randomly selected seniors.

d. Survey those who ask the council president for a questionnaire.

e. Survey a random selection of those who happen to be in the cafeteria at noon.

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.

• Online Homework• Hints and Help• Extra Practice

Evaluate: Homework and Practice

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.

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).

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.

Systematic sample

Convenience sample

Self-selected sample

Stratified sample

Cluster sample

The population consists of all residents of the neighborhood.


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-C;CA-C

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-C;CA-C

A2_MNLESE385900_U9M22L1 1088 6/8/15 11:47 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–2 1 Recall of Information MP.3 Logic

3 2 Skills/Concepts MP.3 Logic

4 1 Recall of Information MP.3 Logic

5–7 2 Skills/Concepts MP.3 Logic

8–9 1 Recall of Information MP.3 Logic

10–11 2 Skills/Concepts MP.4 Modeling

12–13 2 Skills/Concepts MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore Finding the Mean of Samples Obtained from Various Sampling Methods

Exercise 1

Example 1Distinguishing Among Sampling Methods

Exercises 2–9

Example 2Making Predictions from a Random Sample

Exercises 10–11

INTEGRATE MATHEMATICAL PROCESSESFocus on ReasoningMP.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.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

b. Which sampling method is most likely to result in a representative sample of the population? Explain.

c. Describe another sampling method that is likely to result in a representative sample of the population.

d. Describe the categorical and numerical data that the officers of the neighborhood association want to gather.

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.

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.

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.

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.

Specialty Number of Doctors

Dermatology 40

Geriatrics 120

Oncology 140

Pediatrics 100

Surgery 100

Possible answer: Randomly select names from the association’s roster.

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.

Interest in beautifying the neighborhood is categorical. The amounts residents are willing to contribute is numerical.

This sampling method is likely not to be biased since the sample is random and all of the freshmen are available to be chosen.

This sampling method could be biased since friends of the club members may be willing to pay more than others.

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.

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.



A2_MNLESE385900_U9M22L1 1089 4/4/14 7:05 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

14 3 Strategic Thinking MP.3 Logic

AVOID COMMON ERRORSStudents may believe they are calculating parameters when they make predictions, but actual parameters cannot be found through sampling.

COLLABORATIVE LEARNINGHave 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.

1089 Lesson 22 . 1


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

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.

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.

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.

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.

c. Predict the number of cars that have a 3.6 L V6 engine and an automatic transmission. Show your calculations.

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

The population consists of all computers at the assembly plant. This is a simple random sample.

The population consists of people who go to the movie theater. This is a systematic sample.

Proportion of cars with an automatic

transmission: 4 __ 10 = 0.4 or 40%

Predicted number of cars with an automatic

transmission: 120 · 0.4 = 48

Proportion of cars with the 3.6 L V6 engine: 6 __ 10 = 0.6 or 60%

Predicted number of cars with the 3.6 L V6 engine: 120 · 0.6 = 72

Proportion of cars with a 3.6 L V6 engine and an automatic

transmission: 3 __ 10 = 0.3 or 30%

Predicted number of cars with a 3.6 L V6 engine and an automatic

transmission: 120 · 0.3 = 36




A2_MNLESE385900_U9M22L1 1090 4/4/14 7:05 PM

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.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y • I

mag

e C

red

its:

©Be

rnd

Tsch

aker

t Ph

oto

gra

phy

/Ala

my

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.

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.

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.

Proportion of adults interested in attending plays: 12 __ 50 = 0.24 or 24%

Proportion of adults who prefer dramas among those who are interested

in attending plays: 3 __ 12 = 0.25 or 25%

Predicted number of adults who are interested in attending plays that

are dramas: 15,000 · 0.24 · 0.25 = 900



A2_MNLESE385900_U9M22L1 1091 4/4/14 7:05 PM

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.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.

1091 Lesson 22 . 1


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

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.

11. Match each description of a sample on the left with a sampling technique on the right.

A. A television reporter asks people walking by on the street to answer a question about an upcoming election. Simple random sample

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.

Self-selected sample

C. A television reporter contacts voters randomly chosen from a complete list of registered voters to ask about an upcoming election.

Convenience sample

D. A television reporter contacts every 100th voter from a complete list of registered voters to ask about an upcoming election.

Systematic sample

E. A television reporter contacts registered voters randomly chosen from every voting precinct to ask about an upcoming election.

Stratified sample

F. A television reporter asks viewers to call in their response to a question about an upcoming election. 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?

b. Was the reporter’s data-gathering technique a census or a sample? Explain.

c. Are the data representative of the entire student body? Why or why not?

Mean number of people attending plays: 29 __ 12 ≈ 2.4; mean number of plays attended: 25 __ 12 ≈ 2.1

Predicted number of households with an adult having an interest in attending plays: 7000 ·

0.24 = 1680

Predicted number of tickets sold: 1680 households · 2.4 people _______

household · 2.1 tickets _______ person = 8467.2 tickets

Predicted revenue from ticket sales: 8467.2 tickets · $40 ____

ticket = $338,688

C

F

A

D

E

B

The reporter gathered numerical data.

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.

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.




A2_MNLESE385900_U9M22L1 1092 4/4/14 7:05 PM

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.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.



© H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

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.

b. A researcher records the gender of each subject.

c. A researcher records the amount of time each subject spends using an electronic device during a day.

d. A researcher records whether each subject is a young adult, a middle-aged adult, or a senior.

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 68.9.

109 52 62 72 110 61 51 50 100 50

54 104 54 111 74 73 77 68 65 66

108 53 27 75 52 117 76 60 64 67

73 36 103 71 67 60 59 26 80 61

38 63 35 112 75 68 51 72 79 62

58 105 55 53 118 57 101 66 116 31

29 57 74 33 102 69 28 71 30 58

39 55 34 120 64 114 70 113 78 63

107 37 56 25 76 70 69 77 30 115

56 40 106 32 119 65 80 78 79 59

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.

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.

The data are continuous because time can be measured in hours, minutes, or even seconds. Another example of continuous data: subject’s height.

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).

Answers will vary. The mean for any given sample is likely to fall between 57.5 and 80.3.



A2_MNLESE385900_U9M22L1 1093 4/4/14 7:05 PM

PEERTOPEER DISCUSSIONGive 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.

JOURNALHave 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.

1093 Lesson 22 . 1


© H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

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

3 times 4

2 times 2

1 time 4

0 times 12

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 ___ 25 = 0.72, so the predicted number of students in the school who are satisfied is 600 · 0.72 = 432. From the second table, the mean number of lunches brought to school by a student in a typical week is

2 · 5 + 1 · 4 + 4 · 3 + 2 · 2 + 4 · 1 + 12 · 0 ________________________________ 25 = 1.36, so the predicted number

of lunches brought to school by all students is 600 · 1.36 = 816.



A2_MNLESE385900_U9M22L1 1094 4/4/14 7:05 PM

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 of miscalculation allowed in computing data; margin of

error = ± 1 _

√ ― n , where n represents the number of data values in the set; margin

of error = ± 1 _

√ ―― 2400 ≈ ±0.02, or about 2%.

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 STRATEGIESUnder 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 PRACTICESFocus on ReasoningMP.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.

Scoring Rubric2 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.



Documents

CorrectionKey=NL-B;CA-B CorrectionKey=NL-C;CA-C 22 …€¦ · · 2017-04-21from Various Sampling Methods You collect data about a population by surveying or studying some or all