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Unit 4 • Descriptive statisticsLesson 2: Working with Two Categorical and Quantitative Variables

NaMe:

Assessment

CCSS IP Math I Teacher Resource© Walch EducationU4-73

Pre-AssessmentCircle the letter of the best answer.

1. Ruby asks her classmates how many hours they sleep each night during the week, and separates the responses by gender in the two-way frequency table below. What is the joint frequency of males who sleep 8–10 hours?

GenderHours of sleep

4–6 6–8 8–10 10–12Male 5 14 8 2Female 11 10 5 1

a. 5

b. 8

c. 10

d. 14

2. Anna asks her friends which book they prefer in a trilogy. She separates the responses by age. What is the marginal frequency of Book 1?

AgePreferred book

Book 1 Book 2 Book 314 years old 8 5 1915 years old 10 12 716 years old 16 0 11

a. 17

b. 29

c. 34

d. 37

continued


NaMe:

Assessment

CCSS IP Math I Teacher ResourceU4-74

© Walch Education

3. Which equation could be used to approximate the data in the scatter plot below?

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2

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a. y = –1.4x + 3.5

b. y = 1.4x – 3.5

c. y = –1.4x – 3.5

d. y = 1.4x + 3.5

4. Sam created a residual plot to analyze a linear function fitted to data. The plot is below. What does the plot tell him about his line fitted to the data?

0 1 2 3 4 5 6 7 8 9 10

-20

-15

-10

-5

5

10

15

20

a. A linear function is not a good fit for the data.

b. A quadratic function is the best fit for the data.

c. An exponential function is the best fit for the data.

d. A linear function is a good fit for the data. continued


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Assessment


5. Which equation could be used to estimate the data in the scatter plot below?

0 1 2 3 4 5 6 7 8200

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600

800

1000

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1400

1600

1800

2000

2200

2400

a. y = 3x + 35

b. y x3 35= +

c. y x3 35( )= − +

d. y = 3x – 35

Lesson 2: Working with Two Categorical and Quantitative Variables

Unit 4 • Descriptive statistics

Instruction


© Walch Education

Common Core State Standards

S–ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.★

S–ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.★

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

b. Informally assess the fit of a function by plotting and analyzing residuals.

c. Fit a linear function for a scatter plot that suggests a linear association.

Essential Questions

1. When is a two-way frequency table a good way to present data?

2. Why is data represented in a scatter plot?

3. What does a residual plot display?

4. How can it be determined graphically that a line is a good estimate for a data set?

WORDS TO KNOW

conditional relative frequency the percentage of a joint frequency as compared to the total number of respondents, total number of people with a given characteristic, or the total number of times a specific response was given

function a relation of two variables where each input is assigned to one and only one output

joint frequency the number of times a specific response is given by people with a given characteristic; the cell values in a two-way frequency table

marginal frequency the total number of times a specific response is given, or the total number of people with a given characteristic


Instruction


residual the vertical distance between an observed data value and an estimated data value on a line of best fit

residual plot provides a visual representation of the residuals for a set of data; contains the points (x, residual for x)

scatter plot a graph of data in two variables on a coordinate plane, where each data pair is represented by a point

trend a pattern of behavior, usually observed over time or over multiple iterations

two-way frequency table a table that divides responses into categories, showing both a characteristic in the table rows and a characteristic in the table columns; values in cells are a count of the number of times each response was given by a respondent with a certain characteristic

Recommended Resources• Interactivate. “Finding Residuals.”

http://walch.com/rr/CAU4L2FindingResiduals

This site provides a discussion about residuals—what they are and how they are calculated—and gives an example.

• Interactivate. “Regression.”

http://walch.com/rr/CAU4L2Regression

This site allows users to plot points in a scatter plot and then have the computer generate a line of best fit. Users can also fit a line to the data using their own equation.

• MathIsFun.com. “Scatter Plots.”

http://walch.com/rr/CAU4L2ScatterPlots

This site explains how to create a scatter plot, draw a line of best fit, and analyze correlations. The site ends with a short, interactive quiz.

• VCE Further Maths. “Tutorial 15—Two-way Frequency Tables.”

http://walch.com/rr/CAU4L2TwoWayFrequencies

This site offers a ten-minute video tutorial of how to create two-way frequency tables.


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© Walch Education

Lesson 4.2.1: Summarizing Data Using Two-Way Frequency Tables

Warm-Up 4.2.1Elizabeth surveys 9th graders, 10th graders, and 11th graders in her school. She asks each student how many hours they spend doing homework each night. She records the responses in the table below.

GradeHours spent on homework

0–2 2–4 More than 49 38 12 2

10 21 25 911 14 18 20

1. How many 9th graders spend 0–2 hours on homework each night?


3. Which response was the most popular among 11th graders? 0–2 hours, 2–4 hours, or more than 4 hours?


Instruction


Lesson 4.2.1: Summarizing Data Using Two-Way Frequency TablesCommon Core State Standard

S–ID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.★

Warm-Up 4.2.1 DebriefElizabeth surveys 9th graders, 10th graders, and 11th graders in her school. She asks each student how many hours they spend doing homework each night. She records the responses in the table below.

GradeHours spent on homework

0–2 2–4 More than 49 38 12 2

10 21 25 911 14 18 20


Look for the row for grade 9, then the column for 0–2 hours. 38 9th graders spend 0–2 hours on homework each night.


Look for the row for grade 10, then the column for 2–4 hours. 25 10th graders spend 2–4 hours on homework each night.

3. Which response was the most popular among 11th graders? 0–2 hours, 2–4 hours, or more than 4 hours?

Look for the row for grade 11. Find the greatest response, and then identify the heading of the column with the greatest response. The most popular response was “more than 4” hours.

Connection to the Lesson

• In this lesson, students will read and interpret two-way frequency tables.

• This warm-up introduces students to data presented in a two-way frequency table.

• Students will translate from this informal introduction to more formal terminology.


Instruction


© Walch Education

Prerequisite Skills

This lesson requires the use of the following skills:

• creating tables with multiple rows and columns

• reading tables with multiple rows and columns

IntroductionInformation about people who are surveyed can be captured in two-way frequency tables. A two-way frequency table is a table of data that separates responses by a characteristic of the respondents.

Type of characteristicType of response

Response 1 Response 2Characteristic 1 a bCharacteristic 2 c d

Each cell in the table contains a count of the people with a given characteristic who gave each response. For example, in the table above a, b, c, and d would each be counts for the responses given by people with each characteristic. The sum of all the cells, a + b + c + d, is the total number of respondents. Two-way frequency tables help organize information and provide greater insight into features of a population being surveyed. A trend, or pattern in the data, can be examined using a two-way frequency table.

A joint frequency is the number of responses for a given characteristic. The entries in the cells of a two-way frequency table are joint frequencies. In the sample table, a, b, c, and d are each joint frequencies. A marginal frequency is the total number of times a response was given, or the total number of respondents with a given characteristic. This is the sum of either a row or a column in a two-way frequency table. In the sample table, a + b would be the marginal frequency of people with Characteristic 1.

A conditional relative frequency allows a comparison to be made for multiple responses in

a single row, single column, or table. Relative frequencies are expressed as a percentage, usually

written as a decimal. They are found by dividing the number of responses by either the total number

of people who gave that response, the total number of people with a given characteristic, or the total

number of respondents. In the sample table, a

a b+ is the relative frequency of Response 1 for people

with Characteristic 1.


Instruction


Key Concepts

• A two-way frequency table divides survey responses by characteristics of respondents.

• The number of times a response was given by people with a certain characteristic is called a joint frequency.

• A marginal frequency is the total number of times a response is given, or the total number of people with a certain characteristic.

• A conditional relative frequency expresses a number of responses as a percentage of the total number of respondents, the total number of people with a given characteristic, or the total number of times a specific response was given.

• Trends, or patterns of responses, can be identified by looking at the frequency of responses.

Common Errors/Misconceptions

• incorrectly locating frequencies in the table

• incorrectly calculating conditional relative frequencies by being inconsistent in the method used (dividing by the number of times a response was given, the number of people with a given characteristic, or the total number of respondents)


Instruction


© Walch Education

Guided Practice 4.2.1Example 1

Cameron surveys students in his school who play sports, and asks them which sport they prefer. He records the responses in the table below.

GenderPreferred sport

Baseball Soccer BasketballMale 49 52 16

Female 23 64 33

What is the joint frequency of male students who prefer soccer?

1. Look for the row of male students.

The characteristic “male” is in the first row of responses.

2. Look for the column with the response “soccer.”

The response “soccer” is in the second column of responses.

3. The frequency for the given characteristic and the given response is the joint frequency.

The cell in the first characteristic row and the second response column is 52.

The joint frequency of male students who prefer soccer is 52.


Instruction


Example 2

Abigail surveys students in different grades, and asks each student which pet they prefer. The responses are in the table below.

GradePreferred pet

Bird Cat Dog Fish9 3 49 53 22

10 7 36 64 10

What is the marginal frequency of each type of pet?

1. Sum the responses of people with each characteristic for the first pet type, “bird.”

3 people in grade 9 preferred birds, and 7 people in grade 10 preferred birds.

3 + 7 = 10 people who preferred birds

2. Sum the responses of people with each characteristic for the second pet type, “cat.”

49 people in grade 9 preferred cats, and 36 people in grade 10 preferred cats.

49 + 36 = 85 people who preferred cats

3. Sum the responses of people with each characteristic for the third pet type, “dog.”

53 people in grade 9 preferred dogs, and 64 people in grade 10 preferred dogs.

53 + 64 = 117 people who preferred dogs


Instruction


© Walch Education

4. Sum the responses of people with each characteristic for the fourth pet type, “fish.”

22 people in grade 9 preferred fish, and 10 people in grade 10 preferred fish.

22 + 10 = 32 people who preferred fish

5. Organize the marginal frequencies in a two-way frequency table.

Create a row and include the marginal frequencies of each response under the name of each response.

GradePreferred pet

Bird Cat Dog Fish9 3 49 53 22

10 7 36 64 10Total 10 85 117 32

Example 3

Ms. Scanlon surveys her students about the time they spend studying. She creates a table showing the amount of time students studied and the score each student earned on a recent test.

Time spent studying in hoursTest score

0–25 26–50 51–75 76–1000–2 2 8 12 22–4 0 10 8 244–6 1 0 2 96+ 0 0 1 4

Ms. Scanlon wants to understand the distribution of scores among all the students, and to get a sense of how students are performing and how much students are studying. Find the conditional relative frequencies as a percentage of the total number of students.


Instruction


1. Find the total number of students represented in the table by summing the joint frequencies.

2 + 8 + 12 + 2 + 0 + 10 + 8 + 24 + 1 + 0 + 2 + 9 + 0 + 0 + 1 + 4 = 83

2. Divide each joint frequency by the total number of students.

2

830.024

8

830.096

12

830.145

2

830.024

0

830

10

830.120

8

830.096

24

830.289

1

830.012

0

830

2

830.024

9

830.108

0

830

0

830

1

830.012

4

830.048

≈ ≈ ≈ ≈

≈ ≈ ≈ ≈

≈ ≈ ≈ ≈

≈ ≈ ≈ ≈

3. Represent the conditional joint frequencies in a new table.

Insert each conditional joint frequency in a table set up the same way as the two-way frequency table.

Time spent studying in hoursTest score

0–25 26–50 51–75 76–1000–2 0.024 0.096 0.145 0.0242–4 0 0.120 0.096 0.2894–6 0.012 0 0.024 0.1086+ 0 0 0.012 0.048


NaMe:


© Walch Education

Problem-Based Task 4.2.1: FunZone America SurveyFunZone America, an amusement park, collects information from park visitors. The park uses this information to determine how to attract certain guests to the park. For example, one summer FunZone America learned that 12–17-year-olds were most interested in roller coasters. When FunZone America wanted to try to get more 12–17-year-olds to visit the park, the park ran advertisements about roller coasters. The park surveyed recent visitors and recorded the information below. The three main attractions were roller coasters, shows, and the water park.

Visitor Age Favorite attraction Visitor Age Favorite attraction1 27 Roller coasters 26 31 Roller coasters2 30 Shows 27 12 Roller coasters3 18 Roller coasters 28 38 Water park4 35 Shows 29 29 Roller coasters5 31 Shows 30 28 Shows6 46 Roller coasters 31 16 Roller coasters7 25 Water park 32 47 Shows8 39 Shows 33 37 Shows9 8 Water park 34 9 Water park

10 14 Water park 35 48 Shows11 31 Shows 36 22 Water park12 25 Roller coasters 37 49 Roller coasters13 35 Shows 38 19 Roller coasters14 46 Roller coasters 39 53 Shows15 53 Roller coasters 40 15 Roller coasters16 27 Shows 41 16 Water park17 33 Water park 42 14 Shows18 34 Shows 43 39 Shows19 5 Shows 44 52 Shows20 41 Shows 45 20 Shows21 20 Roller coasters 46 33 Roller coasters22 24 Water park 47 21 Water park23 48 Shows 48 53 Water park24 34 Roller coasters 49 39 Roller coasters25 14 Shows 50 6 Shows

continued


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Create a two-way frequency table showing the joint frequencies of visitors with the following age ranges: 5–15, 16–25, 26–35, 36–45, 46–55. Include in the table the marginal frequency for the types of attractions and for the ages of the visitors. Are there any trends in the type of attractions preferred by each age group? Use conditional relative frequencies to describe your response.


NaMe:


© Walch Education

Problem-Based Task 4.2.1: FunZone America Survey

Coachinga. Sort the data first by attraction, then by age.

b. Count the number of attractions selected for each given age range. For example, to fill in the first joint frequency, count the number of people aged 5 through 15 who selected roller coasters. Fill in a two-way frequency table with this information.

c. Find the marginal frequencies. Sum each row, and sum each column.

d. Which type of conditional relative frequency would show the type of attraction preferred by each age group?

e. Calculate the conditional relative frequencies and put them in a table.

f. Look at the conditional relative frequencies for each age group. Is there an attraction that is preferred by each group?


Instruction


Problem-Based Task 4.2.1: FunZone America Survey

Coaching Sample Responsesa. Sort the data first by attraction, then by age.

Visitor Age Favorite attraction27 12 Roller coasters40 15 Roller coasters31 16 Roller coasters3 18 Roller coasters

38 19 Roller coasters21 20 Roller coasters12 25 Roller coasters1 27 Roller coasters

29 29 Roller coasters26 31 Roller coasters46 33 Roller coasters24 34 Roller coasters49 39 Roller coasters6 46 Roller coasters

14 46 Roller coasters37 49 Roller coasters15 53 Roller coasters

Visitor Age Favorite attraction9 8 Water park

34 9 Water park10 14 Water park41 16 Water park47 21 Water park36 22 Water park22 24 Water park7 25 Water park

17 33 Water park28 38 Water park48 53 Water park


Instruction


© Walch Education

Visitor Age Favorite attraction19 5 Shows50 6 Shows25 14 Shows42 14 Shows45 20 Shows16 27 Shows30 28 Shows2 30 Shows5 31 Shows

11 31 Shows18 34 Shows4 35 Shows

13 35 Shows33 37 Shows8 39 Shows

43 39 Shows20 41 Shows32 47 Shows23 48 Shows35 48 Shows44 52 Shows39 53 Shows

b. Count the number of attractions selected for each given age range. For example, to fill in the first joint frequency, count the number of people aged 5 through 15 who selected roller coasters. Fill in a two-way frequency table with this information.

Age rangeFavorite attraction

Roller coasters Shows Water park5–15 2 4 3

16–25 5 1 526–35 5 8 136–45 1 4 146–55 4 5 1


Instruction


c. Find the marginal frequencies. Sum each row, and sum each column.


TotalRoller coasters Shows Water park

5–15 2 4 3 916–25 5 1 5 1126–35 5 8 1 1436–45 1 4 1 646–55 4 5 1 10Total 17 22 11

d. Which type of conditional relative frequency would show the type of attraction preferred by each age group?

We need to look at how the values in each row are distributed to understand which attraction is preferred by each age group. The conditional relative frequency that is the joint frequency divided by the number of people in each age group will show the percentage of each age group that preferred each attraction.

e. Calculate the conditional relative frequencies and put them in a table.

Divide each value by the total people in that age group. Make sure the sum of each row is 1.


TotalRoller coasters Shows Water park

5–152

90.22≈

4

90.44≈

3

90.33≈ 1.0

16–255

110.45≈

1

110.09≈

5

110.45≈ 1.0

26–355

140.36≈

8

140.57≈

1

140.07≈ 1.0

36–451

60.17≈

4

60.67≈

1

60.17≈ 1.0

46–554

100.4=

5

100.5=

1

100.1= 1.0


Instruction


© Walch Education

f. Look at the conditional relative frequencies for each age group. Is there an attraction that is preferred by each group?

The shows are preferred by ages 5–15, the roller coasters and water parks are equally preferred by ages 16–25, shows are preferred by ages 26–35, shows are preferred by ages 36–45, and shows are preferred by ages 46–55. There are some age groups with a strong preference for one attraction, such as the shows in the 36–45 age group. In others, there is only a slight preference, or there are two equally preferred attractions.

Recommended Closure Activity

Select one or more of the essential questions for a class discussion or as a journal entry prompt.


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Practice 4.2.1: Summarizing Data Using Two-Way Frequency TablesDylan asked his classmates about their favorite school subject, and wanted to see if there was any difference in the classes preferred by boys and girls. His data is recorded below. Use the data to answer the questions that follow.

Student Gender Favorite subject Student Gender Favorite subject1 Boy English 21 Boy Social studies2 Girl Math 22 Boy Science3 Girl Math 23 Girl Social studies4 Boy English 24 Girl Social studies5 Boy Science 25 Boy Social studies6 Girl Social studies 26 Boy English7 Boy Math 27 Boy Science8 Girl Math 28 Boy Science9 Girl Social studies 29 Girl English

10 Girl Math 30 Boy English11 Girl Math 31 Girl Science12 Boy Science 32 Girl Math13 Boy Social studies 33 Girl English14 Girl Social studies 34 Girl English15 Boy Math 35 Boy Science16 Girl Social studies 36 Girl Science17 Boy English 37 Boy Social studies18 Boy English 38 Boy English19 Boy Science 39 Girl Math20 Girl Science 40 Girl Math

1. Create a two-way frequency table showing the subjects preferred by students of each gender.

2. Find the marginal frequencies for each gender and for each subject. Include the marginal frequencies in the table.

3. What are the conditional frequencies relative to the total number of people surveyed? Include the values in a table.

4. What are the conditional frequencies relative to the total number of boys and the total number of girls?

5. Describe any trends in the subjects preferred by all students and the subjects preferred by boys versus girls. continued


NaMe:


© Walch Education

To better understand which type of cell phones people will purchase, a cell phone company collects information about its customers. Customers could select three of the following ages: under 25, 25–35, and over 35. Each customer indicated whether they used a basic phone or a smartphone. The information is recorded below. Use the data to answer the questions that follow.

Customer Age rangeType of cell phone used

Customer Age rangeType of cell phone used

1 25–35 smartphone 26 over 35 smartphone2 under 25 smartphone 27 under 25 smartphone3 under 25 smartphone 28 25–35 smartphone4 25–35 smartphone 29 25–35 smartphone5 25–35 smartphone 30 25–35 smartphone6 under 25 smartphone 31 over 35 basic phone7 over 35 smartphone 32 25–35 smartphone8 over 35 basic phone 33 under 25 smartphone9 25–35 smartphone 34 under 25 basic phone

10 25–35 basic phone 35 over 35 smartphone11 under 25 smartphone 36 under 25 smartphone12 over 35 basic phone 37 25–35 basic phone13 25–35 smartphone 38 25–35 basic phone14 over 35 smartphone 39 over 35 basic phone15 under 25 smartphone 40 under 25 smartphone16 under 25 smartphone 41 over 35 smartphone17 under 25 basic phone 42 under 25 basic phone18 under 25 smartphone 43 under 25 smartphone19 25–35 smartphone 44 25–35 basic phone20 25–35 smartphone 45 over 35 basic phone21 25–35 basic phone 46 25–35 smartphone22 25–35 smartphone 47 over 35 basic phone23 25–35 smartphone 48 25–35 smartphone24 under 25 smartphone 49 under 25 smartphone25 over 35 basic phone 50 over 35 smartphone

continued


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6. Create a two-way frequency table showing the phones used by customers of each age group.

7. Find the marginal frequencies for each age and for each phone type. Include the marginal frequencies in the table.

8. What are the conditional frequencies relative to the types of phones? Include the values in a table.

9. What are the conditional frequencies relative to all customers surveyed?

10. The cell phone company is thinking of creating a new phone. It wants to sell the cell phone type that is most popular to the age group that is most popular. Which type of cell phone should the company make, and to whom should the company sell it?


NaMe:


© Walch Education

Lesson 4.2.2: Solving Problems Given Functions Fitted to Data

Warm-Up 4.2.2A local dollar store sells goods for around $1, but the name is a little misleading these days with the rising costs of goods. You have kept track of the number of items you’ve bought and the prices you paid for that number of goods. You recorded your data in the table below. Use the table for problem 1.

Number of goods Cost in dollars ($)2 54 89 14

10 19

1. Plot each point in the table on a coordinate plane.

2. Yasin is a welder. For his job, he requires 1 hour to set up and then 3 hours for each project. The time it takes on his job to complete x projects in one day can be modeled by the function y = 3x + 1. Graph the function y = 3x + 1.


Instruction


Lesson 4.2.2: Solving Problems Given Functions Fitted to DataCommon Core State Standard


a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Warm-Up 4.2.2 DebriefA local dollar store sells goods for around $1, but the name is a little misleading these days with the rising costs of goods. You have kept track of the number of items you’ve bought and the prices you paid for that number of goods. You recorded your data in the table below. Use the table for problem 1.

Number of goods Cost in dollars ($)2 54 89 14

10 19

1. Plot each point in the table on a coordinate plane.

To plot each point, find the value of x along the x-axis (the horizontal axis), and then find the value of y along the y-axis (the vertical axis). Let the x-axis represent the number of goods, and the y-axis represent the cost in dollars.

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2

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6

8

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12

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20

Number of goods

Cos

t in

dolla

rs ($

)


Instruction


© Walch Education

2. Yasin is a welder. For his job, he requires 1 hour to set up and then 3 hours for each project. The time it takes on his job to complete x projects in one day can be modeled by the function y = 3x + 1. Graph the function y = 3x + 1.

A function of the form y = mx + b is a linear function, and the graph is a line. To graph a line, find two points on the line. Evaluate the equation at two values of x. Two easy values to use are 0 and 1.

y = 3(0) + 1 = 1

y = 3(1) + 1 = 4

Two points on the line are (0, 1) and (1, 4).

Graph the two points, and draw a line through the two points.

-4 -2 0 2 4

-4

-2

2

4


• In this lesson, students will examine the relationship between functions and data in a scatter plot.

• This warm-up will help students recall how to create a scatter plot given a data set.

• Students will also need to know how to graph a linear function given an algebraic equation.


Instruction


Prerequisite Skills


• plotting points on the coordinate plane, given data in a table

• plotting the graph of a linear function, given an equation

• plotting the graph of an exponential function, given an equation

• evaluating a function at a given input value

• solving a function for x given a y-value

• interpreting a function in a given context, using the graph or the equation

IntroductionData with two quantitative variables can be represented using a scatter plot. A scatter plot is a graph of data in two variables on a coordinate plane, where each data pair is represented by a point. Relationships between the two quantitative variables can be observed on the graph. A function is a relation of two variables where each input is assigned to one and only one output. Functions in two variables can be represented algebraically with an equation, or graphically on the coordinate plane. Graphing a function on the same coordinate plane as a scatter plot for a data set allows us to see if the function is a good estimation of the relationship between the two variables in the data set. The graph and the equation of the function can be used to estimate coordinate pairs that are not included in the data set.

Key Concepts

• Data with two quantitative variables can be represented graphically on a scatter plot.

• To create a scatter plot, plot each pair of data as a point on a coordinate plane.

• To compare a data set and a function, plot the function on the same coordinate plane as the scatter plot of a data set. The graph of the function should approximate the shape of the scatter plot.

• Evaluating or solving a function that has a similar shape as a data set can provide an estimate for data not included in the plotted data set.

• Solve a function algebraically by substituting a value for y and solving for x.


Instruction


© Walch Education

• Solve a function graphically by finding the point on the graph of the function with the known y-value, then finding the corresponding x-value of that point.

• Evaluate a function algebraically by replacing x with a known value and simplifying the expression to determine y.

• Evaluate a function graphically by finding the point on the graph of the function with the known x-value, then finding the corresponding y-value of that point.

• Graph a linear function by plotting two points and drawing a line through those two points.

• Graph an exponential function by plotting at least five points. Connect the points with a curve.


• confusing when to evaluate and when to solve a function

• using a linear function to estimate a relationship between two variables when an exponential function is a better fit

• using an exponential function to estimate a relationship between two variables when a linear function is a better fit

• confusing x and y when graphing data points or analyzing a graph


Instruction



Andrew wants to estimate his gas mileage, or miles traveled per gallon of gas used. He records the number of gallons of gas he purchased and the total miles he traveled with that gas.

Gallons Miles15 31317 34018 40119 42318 39217 37920 40819 43716 36620 416

Create a scatter plot showing the relationship between gallons of gas and miles driven. Which function is a better estimate for the function that relates gallons to miles: y = 10x or y = 22x? How is the equation of the function related to his gas mileage?


Instruction


© Walch Education

1. Plot each point on the coordinate plane.

Let the x-axis represent gallons and the y-axis represent miles.

0 2 4 6 8 10 12 14 16 18 20

40

80

120

160

200

240

280

320

360

400

440

Gallons

Mile

s


Instruction


2. Graph the function y = 10x on the coordinate plane.

It is a linear function, so only two points are needed to draw the line.

Evaluate the function at two values of x, such as 0 and 10, and draw a line through these points on the scatter plot.

y = 10x

y = 10(0) = 0 Substitute 0 for x.



0 2 4 6 8 10 12 14 16 18 20

40

80

120

160

200

240

280

320

360

400

440

y = 10x

Gallons

Mile

s


Instruction


© Walch Education

3. Graph the function y = 22x on the same coordinate plane.

This is also a linear function, so only two points are needed to draw the line.

Evaluate the function at two values of x, such as 0 and 10, and draw a line through these points on the scatter plot.




0 2 4 6 8 10 12 14 16 18 20

40

80

120

160

200

240

280

320

360

400

440

y = 22x

y = 10x

Gallons

Mile

s


Instruction


4. Look at the graph of the data and the functions.

Identify which function comes closer to the data values. This function is the better estimate for the data.

The graph of the function y = 22x goes through approximately the center of the points in the scatter plot. The function y = 10x is not steep enough to match the data values. The function y = 22x is a better estimate of the data.

5. Interpret the equation in the context of the problem, using the units of the x- and y-axes.

For a linear equation in the form y = mx + b, the slope (m) of the equation is the rate of change of the function, or the change in y over the change in x. The y-intercept (b) of the equation is the initial value.

In this example, y is miles and x is gallons. The slope is change inmiles

change in gallons.

For the equation y = 22x, the slope of 22 is equal to 22 miles

1 gallon.

The gas mileage of Andrew’s car is the miles driven per gallon of gas used. The gas mileage is equal to the slope of the line that fits the data.

Andrew’s car has a gas mileage of approximately 22 miles per gallon.


Instruction


© Walch Education

Example 2

The principal at Park High School records the total number of students each year. The table below shows the number of students for each of the last 8 years.

Year Number of students1 6302 6553 6904 7315 7526 8007 8448 930

Create a scatter plot showing the relationship between the year and the total number of students. Show that the function y = 600(1.05)x is a good estimate for the relationship between the year and the population. Approximately how many students will attend the high school in year 9?


Let the x-axis represent years and the y-axis represent the number of students.

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2. Graph y = 600(1.05)x on the coordinate plane.

Calculate the value of y for a few different values of x. Start with x = 0. Calculate the value of the function for at least four more x-values that are in the table of data.

x y0 600(1.05)0 = 6001 600(1.05)1 = 6303 600(1.05)3 = 694.5755 600(1.05)5 = 765.7697 600(1.05)7 = 855.260

Plot these points on the same coordinate plane. Connect the points with a curve.

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3. Compare the graph of the function to the scatter plot of the data.

The graph of the function appears to be very close to the points in the scatter plot. The function y = 600(1.05)x is a good estimate of the data.

4. Use the function to estimate the population in year 9.

Evaluate the function y = 600(1.05)x for year 9, when x = 9.

y = 600(1.05)9 = 930.797

The function y = 600(1.05)x is a good estimate of the population. There will be approximately 931 students in the school in year 9.

Example 3

The weights of oranges vary. Maria wants to come up with a way to estimate the number of oranges given a weight. She weighs oranges and records the weights in the table below.

Number of oranges Weight in pounds1 0.473 1.295 2.546 2.658 4.12

10 5.5712 7.1813 8.4814 7.07

Create a scatter plot showing the relationship between the number of oranges and the weight in pounds. Is the function y = 0.6x – 0.5 a good fit for the data? Maria has a bag of oranges that weighs 2 pounds. Approximately how many oranges are in the bag?


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Let the x-axis represent the number of oranges and the y-axis represent the weight in pounds.

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2. Graph the function y = 0.6x – 0.5 to determine if it is a good estimate for the data set.

Find two points on the line by evaluating the function at two values of x.

Two easy values to use are 0 and 1.

y = 0.6(0) – 0.5 = –0.5 Substitute 0 for x.

y = 0.6(1) – 0.5 = 0.1 Substitute 1 for x.

Two points on the line are (0, –0.5) and (1, 0.1).

Graph the two points on the same graph as the scatter plot, and then draw a line through the two points.

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3. Look at the relationship between the graph of the function and the graph of the data. Determine if the function closely resembles the graph of the data.

If a linear function is a good estimate for a data set, some of the data values will be above the line and some will be below the line. It appears that this equation is a good fit for the data.

4. Use the equation to estimate the number of oranges weighing 2 pounds.

In the equation y = 0.6x – 0.5, x is the number of oranges and y is the weight of the oranges. Solve the equation for y = 2 to estimate the number of oranges that weigh 2 pounds.

2 = 0.6x – 0.5 Set y equal to 2.

2.5 = 0.6x Add 0.5 to both sides of the equation.

4.2 ≈ x Divide both sides by 0.6.

Maria can use the equation y = 0.6x – 0.5 to estimate how many oranges have a given weight. 4 oranges weigh approximately 2 pounds.


NaMe:


© Walch Education

Problem-Based Task 4.2.2: Movie BuzzWord of mouth can be a great way to increase a movie’s popularity. A small local movie theater released a movie. On the first day, only 5 people saw the movie. They all loved it, and each told at least 5 more people to go see the movie. The second day of the movie’s release, many of the people who had been told to see the movie went to the theater. Each day, each person who viewed the movie told approximately 5 other people to go to the theater. The table below shows the number of people who viewed the movie in its first 4 days out.

Day Number of viewers1 52 273 1244 626

Create a scatter plot showing the number of viewers each day the movie played at the theater. Which type of function would best approximate the data? Two theater employees each try to determine a function to fit the data. One thinks that the equation y = 5x is a good fit for the data; the other thinks the equation y = 200x – 200 is a good fit for the data. Which function is a better fit? If this trend continues, approximately how many people will see the movie on the fifth day of the movie’s release?


NaMe:


Problem-Based Task 4.2.2: Movie Buzz

Coachinga. Create a scatter plot of the given data.

b. Look at the shape of the scatter plot. Is the data linear or exponential?

c. Graph the functions y = 5x and y = 200x – 200 on the scatter plot with the data.

d. Which function is a better fit for this data?

e. Use your equation to estimate the number of people who will see the movie on the fifth day.


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Problem-Based Task 4.2.2: Movie Buzz

Coaching Sample Responsesa. Create a scatter plot of the given data.

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b. Look at the shape of the scatter plot. Is the data linear or exponential?

If the data were linear, the slope between each pair of points would be the same. As the x-values increase by 1, the y-values increase by different amounts. The change in y between the first two points is much less than the change in y between the second and third points. The change in y between the third and fourth points is greater than the change in y between the second and third points. The shape of the scatter plot is curved.

c. Graph the functions y = 5x and y = 200x – 200 on the scatter plot with the data.

To create a graph of the exponential function y = 5x, evaluate the function at a few values in the domain of the data set. The original data set contains the values 1, 2, 3, and 4 in the domain; evaluate the function at these values.

x y = 5x

1 51 = 52 52 = 253 53 = 1254 54 = 625


Instruction


Plot each of these values on the scatter plot, and connect the lines with a curve.

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The graph of a linear equation is a line. To plot a linear equation, find two points on the line. Two values to evaluate are x = 1 and x = 3.

y = 200(1) – 200 = 0 Substitute 1 for x.


Plot the points (1, 0) and (3, 400) on the same graph as the scatter plot and draw a straight line through the points.


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d. Which function is a better fit for this data?

Look at the shape of each graph. The first graph of y = 5x comes very close to the data set, and matches the shape of the data. This function is a better fit for the data.

e. Use your equation to estimate the number of people who will watch the movie on the fifth day.

The x-values are the days, so replace x in the equation y = 5x with 5. Evaluate the expression to find y.

y = 5(5) = 3125

On the fifth day, there will be approximately 3,125 viewers if this pattern continues.




NaMe:


Practice 4.2.2: Solving Problems Given Functions Fitted to DataRacecar tracks vary in length. A racecar driver records the time it takes him to circle various tracks once at top speed. The distance of the track and his time to circle each track once are listed in the table below. Use the data to answer the questions that follow.

Time in minutes Track length in miles0.42 1.50.15 0.530.42 1.50.43 1.40.82 2.50.31 10.56 20.26 0.90.15 0.50.75 2.7

1. Create a scatter plot of the data set.

2. Would a linear or exponential function be a better estimate for the data? Explain.

3. Which equation is a better fit for the data: y = 2.3x or y = 3.3x? Use a graph to support your answer.

4. Approximately how long would it take the driver to circle a track that is 1.8 miles long?

5. It takes the driver 0.6 minutes to circle a track. Approximately how long is the track?

continued


NaMe:


© Walch Education

The value of a car decreases over time. Bethan buys a car for $20,000. Each year, she determines how much her car is worth. She records the value of her car each year in the table below. Use the data to answer the questions that follow.

Year Value in dollars ($)1 20,0002 16,0003 14,5004 13,2005 12,0006 11,0007 10,000

6. Create a scatter plot showing the value of her car over time.

7. Would a linear or exponential function be a better estimate for the data? Explain.

8. Is y = 20,000(1.10)x or y = 20,000(0.90)x a good estimate for the data? Use your graph to explain why or why not.

9. Bethan wants to sell her car when it’s worth approximately $9,000. After how many years should Bethan sell it? Use your graph to explain your answer.

10. How much will her car be worth in 12 years?


NaMe:


Lesson 4.2.3: Analyzing Residuals

Warm-Up 4.2.3Felicia is learning to ride a unicycle. She started at her house, which is at the origin. She went 2 blocks east and wasn’t able to successfully ride the unicycle. Then she started going north toward her friend’s house. After many failed attempts and falling off for 4 blocks, she had success for 2 blocks.

1. Plot the points (2, 4) and (2, 6) on a coordinate plane.

2. Find the distance between the two points. How far did Felicia successfully ride her unicycle?

Felicia is going to sell unicycle pins to raise money for medical research. She spent $2 of her own money to buy the pins and will sell each pin for $4. Her revenue can be modeled by the function y = 4x – 2.

3. Plot the function y = 4x – 2 over the domain of all real numbers.


Instruction


© Walch Education

Lesson 4.2.3: Analyzing ResidualsCommon Core State Standard


b. Informally assess the fit of a function by plotting and analyzing residuals.

Warm-Up 4.2.3 DebriefFelicia is learning to ride a unicycle. She started at her house, which is at the origin. She went 2 blocks east and wasn’t able to successfully ride the unicycle. Then she started going north toward her friend’s house. After many failed attempts and falling off for 4 blocks, she had success for 2 blocks.

1. Plot the points (2, 4) and (2, 6) on a coordinate plane.

The first number in each ordered pair is the x-value, and the second number is the y-value.

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2. Find the distance between the two points. How far did Felicia successfully ride her unicycle?

Look at the location of the points on the coordinate plane. The two points have the same x-value, so the only distance between the two points is a vertical distance. The distance is the absolute value of the difference between the two y-values: |6 – 4| = 2. Felicia rode 2 blocks.


Instruction


Felicia is going to sell unicycle pins to raise money for medical research. She spent $2 of her own money to buy the pins and will sell each pin for $4. Her revenue can be modeled by the function y = 4x – 2.

3. Plot the function y = 4x – 2 over the domain of all real numbers.

An equation of the form y = mx + b is a linear function. The graph is a line, so only two points are needed to create the graph. Evaluate the function at two values of x to find two points on the line. For example, evaluate the function at x = 0 and x = 1.

y = 4(0) – 2 = –2 Substitute 0 for x.


Two points on the line are (0, –2) and (1, 2).

Plot these points, and then draw a line through them.

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• In this lesson, students will analyze the fit of linear functions to a set of data.

• Students will need to be familiar with calculating the vertical distance between two points on the coordinate plane.

• Students will also need to be able to plot points and graphs of linear functions.


Instruction


© Walch Education

Prerequisite Skills


• graphing points on the coordinate plane

• calculating the vertical distance between two points on the coordinate plane

• graphing linear functions on the coordinate plane

IntroductionThe fit of a linear function to a set of data can be assessed by analyzing residuals. A residual is the vertical distance between an observed data value and an estimated data value on a line of best fit. Representing residuals on a residual plot provides a visual representation of the residuals for a set of data. A residual plot contains the points: (x, residual for x). A random residual plot, with both positive and negative residual values, indicates that the line is a good fit for the data. If the residual plot follows a pattern, such as a U-shape, the line is likely not a good fit for the data.

Key Concepts

• A residual is the distance between an observed data point and an estimated data value on a line of best fit. For the observed data point (x, y) and the estimated data value on a line of best fit (x, y

0), the residual is y – y

0.

• A residual plot is a plot of each x-value and its corresponding residual. For the observed data point (x, y) and the estimated data value on a line of best fit (x, y

0), the point on a residual plot

is (x, y – y0).

• A residual plot with a random pattern indicates that the line of best fit is a good approximation for the data.

• A residual plot with a U-shape indicates that the line of best fit is not a good approximation for the data.


• incorrectly finding the residual

• incorrectly plotting points on the residual plot


Instruction



Pablo’s science class is growing plants. He recorded the height of his plant each day for 10 days. The plant’s height, in centimeters, over that time is listed in the table below.

Day Height in centimeters1 32 5.13 7.24 8.85 10.56 12.57 148 15.99 17.3

10 18.9

Pablo determines that the function y = 1.73x + 1.87 is a good fit for the data. How close is his estimate to the actual data? Approximately how much does the plant grow each day?


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1. Create a scatter plot of the data.

Let the x-axis represent days and the y-axis represent height in centimeters.

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2. Draw the line of best fit through two of the data points.

A good line of best fit will have some points below the line and some above the line. Use the graph to initially determine if the function is a good fit for the data.

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3. Find the residuals for each data point.

The residual for each data point is the difference between the observed value and the estimated value using a line of best fit. Evaluate the equation of the line at each value of x.

x y = 1.73x + 1.871 y = 1.73(1) + 1.87 = 3.62 y = 1.73(2) + 1.87 = 5.333 y = 1.73(3) + 1.87 = 7.064 y = 1.73(4) + 1.87 = 8.795 y = 1.73(5) + 1.87 = 10.526 y = 1.73(6) + 1.87 = 12.257 y = 1.73(7) + 1.87 = 13.988 y = 1.73(8) + 1.87 = 15.719 y = 1.73(9) + 1.87 = 17.44

10 y = 1.73(10) + 1.87 = 19.17

Next, find the difference between each observed value and each calculated value for each value of x.

x Residual1 3 – 3.6 = –0.62 5.1 – 5.33 = –0.233 7.2 – 7.06 = 0.144 8.8 – 8.79 = 0.015 10.5 – 10.52 = –0.026 12.5 – 12.25 = 0.257 14 – 13.98 = 0.028 15.9 – 15.71 = 0.199 17.3 – 17.44 = –0.14

10 18.9 – 19.17 = –0.27


Instruction


4. Plot the residuals on a residual plot.

Plot the points (x, residual for x).

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5. Describe the fit of the line based on the shape of the residual plot.

The plot of the residuals appears to be random, with some negative and some positive values. This indicates that the line is a good line of fit.

6. Use the equation to estimate the centimeters grown each day.

The change in the height per day is the centimeters grown each day. In the equation of the line, the slope is the change in height per day. The plant is growing approximately 1.73 centimeters each day.


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Example 2

Lindsay created the table below showing the population of fruit flies over the last 10 weeks.

Week Number of flies1 502 783 984 1225 1536 1917 2388 2989 373

10 466

She estimates that the population of fruit flies can be represented by the equation y = 46x – 40. Using residuals, determine if her representation is a good estimate.

1. Find the estimated population at each x-value.

Evaluate the equation at each value of x.

x y = 46x – 401 46(1) – 40 = 62 46(2) – 40 = 523 46(3) – 40 = 984 46(4) – 40 = 1445 46(5) – 40 = 1906 46(6) – 40 = 2367 46(7) – 40 = 2828 46(8) – 40 = 3289 46(9) – 40 = 374

10 46(10) – 40 = 420


Instruction


2. Find the residuals by finding each difference between the observed population and estimated population.

x Residual1 50 – 6 = 442 78 – 52 = 263 98 – 98 = 04 122 – 144 = –225 153 – 190 = –376 191 – 236 = –457 238 – 282 = –448 298 – 328 = –309 373 – 374 = –1

10 466 – 420 = 46

3. Create a residual plot.


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4. Analyze the residual plot to determine if the equation is a good estimate for the population.

The residual plot has a U-shape. This indicates that a non-linear estimation would be a better fit for this data set.

The shape of the residual plot indicates that the equation y = 46x – 40 is not a good estimate for this data set.

Example 3

Anthony is traveling across the country by car. He keeps track of the hours he has driven and total miles he has traveled in the table below.

Hours Miles1 383 1704 2348 390

11 49512 52815 69917 76720 857

Anthony uses the equation y = 42.64x + 42.12 to estimate his total miles driven after any number of hours. Use a residual plot to determine how well the line fits the data. Approximately how many miles had Anthony driven after 13 hours?

Identify any constants.

The number that does not change in the expression is 10; therefore, 10 is a constant.


Instruction



Let the x-axis represent hours and the y-axis represent miles.

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2. Plot the line of the equation Anthony used to estimate the total miles driven.

To graph a linear equation, find two points on the line. Then draw a straight line through the two points. Two easy values of x to use are 0 and 1.

y = 42.64(0) + 42.12 = 42.12 Substitute 0 for x.


Two points on the line are (0, 42.12) and (1, 84.76).

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3. Find the residuals.

Evaluate the line of best fit for each value of x.

x y = 42.64x + 42.121 42.64(1) + 42.12 = 84.763 42.64(3) + 42.12 = 170.044 42.64(4) + 42.12 = 212.688 42.64(8) + 42.12 = 383.24

11 42.64(11) + 42.12 = 511.1612 42.64(12) + 42.12 = 553.815 42.64(15) + 42.12 = 681.7217 42.64(17) + 42.12 = 76720 42.64(20) + 42.12 = 894.92

4. Find the difference between each observed distance and estimated distance.

x Residual1 38 – 84.76 = –46.763 170 – 170.04 = –0.044 234 – 212.68 = 21.328 390 – 383.24 = 6.76

11 495 – 511.16 = –16.1612 528 – 553.8 = –25.815 699 – 681.72 = 17.2817 767 – 767 = 020 857 – 894.92 = –37.92


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5. Create a residual plot.


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The residual plot has a random shape, indicating that the linear function is a good estimate for the data.

7. Use the equation to estimate the total miles driven when the time equals 13 hours.

In the line of best fit, x = hours driven and y = total miles driven. Evaluate the function at x = 13 to estimate the total miles driven after 13 hours.


After 13 hours, Anthony had driven approximately 596 miles.


NaMe:


Problem-Based Task 4.2.3: Estimating SalariesMarcy surveys 10 people who work at a software company. She asks each person how many years they have worked, and what their estimated salary was last year. Her results are in the table below.

Years of experience Salary in dollars ($)1 52,8102 61,6153 80,8314 77,2015 136,5666 100,7078 135,460

10 208,88911 228,83113 209,726

Marcy believes that the salaries can be estimated using the equation y = 14,000x + 34,000. Is her line a good fit for the data? Marcy estimates that her salary should be $130,000. Approximately how many years of work experience does Marcy have?


NaMe:


© Walch Education

Problem-Based Task 4.2.3: Estimating Salaries

Coachinga. Create a scatter plot of the data.

b. Plot the line of best fit on the scatter plot.

c. Does it appear that the line is a good fit for the data?

d. Calculate the estimated salary for each x-value.

e. Find the residual for each x-value, or the difference between the observed and estimated salaries for each value of x.

f. Create a residual plot.

g. Is the line a good fit for the data?

h. If Marcy estimates her salary should be $130,000, how can the years she has worked be approximated using the equation?


Instruction


Problem-Based Task 4.2.3: Estimating Salaries

Coaching Sample Responsesa. Create a scatter plot of the data.

Let the x-axis represent years of experience and the y-axis represent salary.

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b. Plot the line of best fit on the scatter plot.

Find the value of two points on the line, and then draw a line through those two points. Two easy values of x to use are 0 and 1.

y = 14,000(0) + 34,000 = 34,000 Substitute 0 for x.

y = 14,000(1) + 34,000 = 48,000 Substitute 1 for x.

The two points are (0, 34,000) and (1, 48,000).


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c. Does it appear that the line is a good fit for the data?

A good line of best fit goes through the middle of the point set, where some of the observed values are above the line and some are below the line. It appears that the line is a good fit for the data.

d. Calculate the estimated salary for each x-value.

Evaluate the equation y = 14,000x + 34,000 for each observed x-value.

x y = 14,000x + 34,0001 14,000(1) + 34,000 = 48,0002 14,000(2) + 34,000 = 62,0003 14,000(3) + 34,000 = 76,0004 14,000(4) + 34,000 = 90,0005 14,000(5) + 34,000 = 104,0006 14,000(6) + 34,000 = 118,0008 14,000(8) + 34,000 = 146,000

10 14,000(10) + 34,000 = 174,00011 14,000(11) + 34,000 = 188,00013 14,000(13) + 34,000 = 216,000


Instruction


e. Find the residual for each x-value, or the difference between the observed and estimated salaries for each value of x.

x Residual1 52,810 – 48,000 = 48102 61,615 – 62,000 = –3853 80,813 – 76,000 = 48314 77,201 – 90,000 = –12,7995 136,566 – 104,000 = 32,5666 100,707 – 118,000 = –17,2938 135,460 – 146,000 = –10,540

10 208,889 – 174,000 = 34,88911 228,831 – 188,000 = 40,83113 209,726 – 216,000 = –6274

f. Create a residual plot.

Let the x-axis represent years worked and the y-axis represent the residual values. Make sure the y-axis includes all values in the range of residuals. The least residual is –17,293, and the greatest residual is 40,831, so the y-axis must include both these values.

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g. Is the line a good fit for the data?

The residual plot is random, which indicates that the line is a good fit for the data.

h. If Marcy estimates her salary should be $130,000, how can the years she has worked be approximated using the equation?

y is the salary, and x is the years worked. Substitute the salary estimate for y, and solve for x using the line of best fit.

y = 14,000x + 34,000 Equation

130,000 = 14,000x + 34,000 Substitute values for x and y.

96,000 = 14,000x Subtract 34,000 from both sides.

6.86 ≈ x Divide both sides by 14,000.

Round up to the nearest whole year.

Marcy has approximately 7 years of work experience.




NaMe:


Practice 4.2.3: Analyzing ResidualsTo understand the density of a deer population, Lewis counts the deer in different areas of a forest. He records the deer in each portion of the forest below. Use the data for problems 1–4.

Acres of forest Deer population5 108 0

10 014 4220 10022 6630 9045 18050 10058 116

1. Create a scatter plot showing the deer population in each acreage.

2. Lewis states that the population can be estimated using the equation y = 2x + 22. Draw the line of the equation on the scatter plot.

3. Does it appear that this line is a good fit for the data? Explain.

4. Use a residual plot to determine if a linear function is a good fit for the data.

continued


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© Walch Education

Skylar has a savings account. She records the balance in the account each year. Use the data for problems 5–8.

Years Account balance in dollars ($)2 5513 5784 6085 6386 6707 7048 7399 776

5. Create a scatter plot of the account balances.

6. Skylar estimates that the account balance can be represented by the equation y = 32x + 483. Draw the line of the equation on the scatter plot.

7. Does it appear that this line is a good fit for the data? Explain.

8. Use a residual plot to determine if a linear function is a good fit for the data.

Esmeralda is training for a marathon. She records the distance and time of her recent runs in the table below. Use the data for problems 9 and 10.

Distance in miles Time in minutes10 12011 115.5

12.5 121.2515 168

16.8 169.6819 163.421 224.722 228.824 230.4

9. Create a scatter plot of the running times.

10. Esmeralda determines that her time can be approximated using the equation y = 9x + 19. Use a residual plot to determine if a linear function is a good fit for the data.


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Lesson 4.2.4: Fitting Linear Functions to Data

Warm-Up 4.2.4The data table below shows temperatures in degrees Fahrenheit taken at 7:00 a.m. and noon on 8 different days throughout the year in a small town in Siberia. Use the table to complete problems 1 and 2. Then use your knowledge of equations to answer the remaining questions.

7:00 a.m. Noon0 –31 –12 15 77 11

10 1716 2920 37

1. Plot the points on a scatter plot.

2. Describe the shape of the points.

3. If x = the number of students in class and y = the number of index cards a teacher needs to purchase if every students needs 8, and she wants a couple of extra cards, what is the slope of the line with the equation y = 8x + 2 that models this scenario?

4. What is the graph of the equation y = –x + 1?


Instruction


© Walch Education

Lesson 4.2.4: Fitting Linear Functions to DataCommon Core State Standard


c. Fit a linear function for a scatter plot that suggests a linear association.

Warm-Up 4.2.4 DebriefThe data table below shows temperatures in degrees Fahrenheit taken at 7:00 a.m. and noon on 8 different days throughout the year in a small town in Siberia. Use the table to complete problems 1 and 2. Then use your knowledge of equations to answer the remaining questions.

7:00 a.m. Noon0 –31 –12 15 77 11

10 1716 2920 37

1. Plot the points on a scatter plot.

Plot each point in the form (x, y).

02 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 402

4

6

8

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40


Instruction


2. Describe the shape of the points.

The points appear to be on a straight line. The points are linear.

3. If x = the number of students in class and y = the number of index cards a teacher needs to purchase if every students needs 8, and she wants a couple of extra cards, what is the slope of the line with the equation y = 8x + 2 that models this scenario?

For a line in the form y = mx + b, m is the slope and b is the y-intercept. The slope of the line is 8.

4. What is the graph of the equation y = –x + 1?

The equation is in the form y = mx + b, so the graph will be a line. To graph a line, find two points on the line. Evaluate the function at two values of x. Easy values of x to use are 0 and 1.

y = –(0) + 1 = 1 Substitute 0 for x.

y = –(1) + 1 = 0 Substitute 1 for x.


Graph the two points and draw a line through them.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5

-4

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-1

1

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4

5


Instruction


© Walch Education


• In this lesson, students will recognize that a set of points in a scatter plot can be approximated with a linear function.

• Students will need to understand that the general shape of a linear function is a line.

• Students will make connections between linear graphs and equations, and will need to understand how the slope of a line relates to the equation of the line.

• Students will also need to know how the equation of a line relates to the graph of a line, and will be finding equations of lines based on two points.


Instruction


Prerequisite Skills


• plotting points on a coordinate plane

• creating the graph of a line given the equation of a line

• finding the equation of a line using two points on the line

• knowing how the slope and y-intercept of a line are related to the equation of a line in point-slope form

• understanding that the general shape of the graph of a linear function is a line

IntroductionThe relationship between two variables can be estimated using a function. The equation can be used to estimate values that are not in the observed data set. To determine which type of equation should be used for a data set, first create a scatter plot of the data. Data that has a linear shape, or can be approximated by a straight line, can be fitted to a linear equation. Points in the data set can be used to find a linear equation that is a good approximation for the data. Only two points are needed to draw a line.

Drawing a line through two data points on the same coordinate plane as the scatter plot helps display how well the line matches the data set. If the line is a good fit for the data, some data points will be above the line and some data points will be below the line. After creating a graphical representation of a line that fits the data, find the equation of this line using the two known points on the line. Use the two known points on the line to calculate the slope and y-intercept of the line.

Key Concepts

• A scatter plot that can be estimated with a linear function will look approximately like a line.

• A line through two points in the scatter plot can be used to find a linear function that fits the data.

• If a line is a good fit for a data set, some of the data points will be above the line and some will be below the line.

• The general equation of a line in point-slope form is y = mx + b, where m is the slope and b is the y-intercept.

• To find the equation of a line with two known points, calculate the slope and y-intercept of a line through the two points.


Instruction


© Walch Education

• Slope is the change in y divided by the change in x; a line through the points (x1, y

1) and (x

2, y

2)

has a slope of y y

x x

−−

2 1

2 1

.

• To find the y-intercept, or b in the equation y = mx + b, replace m with the calculated slope, and replace x and y with values of x and y from a point on the line. Then solve the equation for b.

• For example, for a line with a slope of 2 containing the point (1, –3), m = 2, y = –3, and x = 1; –3 = (2)(1) + b, and –5 = b.


• thinking that a line is a good estimate for data that is not linear

• drawing a line that is not a good fit for the data, and calculating the equation of this line

• miscalculating the slope of a line using two points on the line

• incorrectly calculating the y-intercept when finding the equation of a line given a graph


Instruction



A weather team records the weather each hour after sunrise one morning in May. The hours after sunrise and the temperature in degrees Fahrenheit are in the table below.

Hours after sunrise Temperature in ºF0 521 532 563 574 605 636 647 67

Can the temperature 0–7 hours after sunrise be represented by a linear function? If yes, find the equation of the function.


Let the x-axis represent the hours after sunrise and the y-axis represent the temperature in degrees Fahrenheit.

0 1 2 3 4 5 6 7 85

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Hours after sunrise

Tem

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ture

(˚F)


Instruction


© Walch Education

2. Determine if the data can be represented by a linear function.

The graph of a linear equation is a line. If the data looks like it could fit a line, then a linear equation could be used to represent the data.

The temperatures appear to increase in a line, and a linear equation could be used to represent the data set.

3. Draw a line to estimate the data set.

Two points in the data set can be used to draw a line that estimates that data. When the line is drawn, some of the data values should be above the line, and some should be below the line.

A line through (2, 56) and (6, 64) looks like a good fit for the data.

0 1 2 3 4 5 6 7 85

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Hours after sunrise

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Instruction


4. Find the equation of the line.

The general equation of a line in point-slope form is y = mx + b, where m is the slope, and b is the y-intercept.

Find the slope, m, of the line through the two chosen points. The slope

is y

x

change in

change in. For any two points (x

1, y

1) and (x

2, y

2), the slope is

y y

x x

−−

2 1

2 1

.

For the two points (2, 56) and (6, 64), the slope is −−

=64 56

6 22 .

Next, find the y-intercept, b. Use the general equation of a line to solve for b. Substitute x and y from a known point on the line, and replace m with the calculated slope.

y = mx + b

For the point (2, 56): 56 = 2(2) + b; b = 52

Replace m and b with the calculated values in the general equation of a line.

y = 2x + 52

The temperature between 0 and 7 hours after sunrise can be approximated with the equation y = 2x + 52.


Instruction


© Walch Education

Example 2

To learn more about the performance of an engine, engineers conduct tests and record the time it takes the car to reach certain speeds. A car starts from a stop and accelerates to 75 miles per hour. The table below shows the time, in seconds, after the car starts to accelerate and the speed it reaches at each time.

Time in seconds Speed in miles per hour0 01 2.32 6.63 13.54 22.45 32.26 44.27 57.88 74.6

Can the speed between 0 and 8 seconds be represented by a linear function? If yes, find the equation of the function.


Instruction



Let the x-axis represent the time, in seconds, and the y-axis represent the speed.

0 1 2 3 4 5 6 7 8 9 10

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80

Time

Spee

d


The x-values of each point are increasing by 1 unit. The y-values of each point are increasing by greater amounts as x gets larger. The first two points are close together, but the last two points show a large change in the speed. A curved graph has been created.

This data should not be approximated using a line, and therefore should not be represented by a linear equation.

No, the speed should not be represented by a linear equation.


Instruction


© Walch Education

Example 3

Automated tractors can mow lawns without being driven by a person. A company runs trials using fields of different sizes, and records the amount of time it takes the tractor to mow each field. The field sizes are measured in acres.

Acres Time in hours5 157 10

10 2217 32.318 46.820 3422 39.625 7530 7040 112

Can the time to mow acres of a field be represented by a linear function? If yes, find the equation of the function.


Instruction



Let the x-axis represent the acres and the y-axis represent the time in hours.

0 5 10 15 20 25 30 35 40

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Acres

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The graph of a linear equation is a line. If the data looks like it could fit a line, then a linear equation could be used to represent the data.

The temperatures appear to increase in a line, and a linear equation could be used to represent the data set.


Instruction


© Walch Education


Two points in the data set can be used to draw a line that estimates the data. When the line is drawn, some of the data values should be above the line, and some should be below the line.

A line through (7, 10) and (40, 112) looks like a good fit for the data.

0 5 10 15 20 25 30 35 40

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Instruction


4. Find the equation of the line.

The general equation of a line in point-slope form is y = mx + b, where m is the slope, and b is the y-intercept.

Find the slope, m, of the line through the two chosen points. The

slope is y

x

change in


1, y

1) and (x

2, y

2), the slope is

y y

x x

−−

2 1

2 1

.


=112 10

40 73.1 .


y = mx + b

For the point (7, 10): 10 = 3.1(7) + b; b = –12


y = 3.1x – 12

The amount of time to mow the acres of a field can be represented using the equation y = 3.1x – 12.


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© Walch Education

Problem-Based Task 4.2.4: Lion Cub BirthsA zoologist studies different prides, or groups of lions, living throughout Africa. He records the number of adult females in each pride, and the number of newborn cubs. His results are in the table that follows.

Adult females Cubs6 5

13 77 6

17 914 73 1

10 67 44 3

15 88 53 0

13 812 711 714 96 4

The zoologist would like to use this information estimate the number of cubs born each year. He would like an equation that relates the number of adult females to the number of newborn cubs. Can this relationship be estimated using a linear function? If yes, find the equation of the function.


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Problem-Based Task 4.2.4: Lion Cub Births

Coachinga. Create a scatter plot of the data set.

b. What is the shape of the graph of a linear function?

c. Can this data set be estimated using the graph of a linear function?

d. Draw a line to estimate the data set.

e. Find the equation of the line.


Instruction


© Walch Education

Problem-Based Task 4.2.4: Lion Cub Births

Coaching Sample Responsesa. Create a scatter plot of the data set.

Let the x-axis represent the number of adult female lions, and the y-axis represent the number of cubs.

0 2 4 6 8 10 12 14 16

2

4

6

8

10

12

14

16

Adult females

Cub

s

b. What is the shape of the graph of a linear function?

The data seems to show that as the x-values increase, the y-values also increase. The graph looks like a line.

c. Can this data set be estimated using the graph of a linear function?

The graph of a linear function is a line. Since the data appears to follow a line, the data could be approximated using a linear function.

d. Draw a line to estimate the data set.

Two points in the data set can be used to draw a line that estimates that data. When the line is drawn, some of the data values should be above the line, and some should be below the line. A line through (8, 5) and (13, 8) looks like a good fit for the data.


Instruction


e. Find the equation of the line.

The general equation of a line in point-slope form is y = mx + b, where m is the slope, and b is the y-intercept. Find the slope, m, of the line through the two chosen points.

The slope is y

x

change in


1, y

1) and (x

2, y

2), the slope is

y y

x x

−−

2 1

2 1

.


=8 5

13 80.6 .


y = mx + b

For the point (8, 5): 5 = 0.6(8) + b; b = 0.2


y = 0.6x + 0.2

The equation of the line is y = 0.6x + 0.2.




NaMe:


© Walch Education

Practice 4.2.4: Fitting Linear Functions to DataEach of Mrs. Jackson’s students records the number of hours he or she studied for a recent quiz. Mrs. Jackson then compared this time to the score earned by each student. Her data is in the table below. Use the data for problems 1–4.

Time studied, in hours Score earned, out of 1004.5 902.5 693 705 851 43

4.5 853.5 735 985 1002 46

1.5 561.5 484 69


2. Describe the shape of the data.


4. Find the equation of the line that estimates the relationship between the hours studied and the score earned.

continued


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A clothing store manager conducts research on how many articles of clothing each customer purchases. The manager is trying to understand if there is a relationship between the number of items tried on in the dressing room and the number of items purchased. The data for 10 customers is in the table below. Use the data for problems 5 and 6.

Number of items tried on Number of items purchased12 814 01 42 5

13 212 1114 71 30 40 1

5. Create a scatter plot of the data set. Describe the shape of the data.

6. Can the data be represented using a linear equation? If yes, find the equation. If no, explain why not.

continued


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© Walch Education

To learn more about the relationship between years of schooling and yearly income, a company surveys 20 people with jobs. Each person identifies the number of years he or she attended school and his or her current yearly income. Use the data for problems 7 and 8.

Years of schooling Income in dollars ($)11 16,00019 91,50010 20,00010 19,50015 49,0009 10,000

10 13,00011 18,50018 81,00015 49,50017 78,00017 69,50012 26,0008 5,000

14 51,00016 69,00018 88,00020 91,50016 72,50019 77,500

7. Create a scatter plot of the data set, and draw a line to fit the data.

8. Find the equation to estimate the relationship between years of schooling and income.

continued


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The table below shows the cost of lunch at a high school each year for the last 10 years. Use the data for problems 9 and 10.

Year Cost of lunch, in dollars ($)1 1.102 1.203 1.314 1.435 1.546 1.647 1.758 1.899 2.01

10 2.12

9. Create a scatter plot of the data set, and draw a line to fit the data.

10. Find the equation to estimate the relationship between the year and the cost of lunch.

name: unit 4 • descriptive statistics lesson 2: working ...unit+4+lesson+2.pdfunit 4 •...

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