Lines of Best Fit
Pediatricians
measure the size
and weight of babies
as a standard part of a
physical examination.
Parents often keep
measurements of
their babies
as well.
16.1 Where Do You Buy Your Books?
Drawing Lines of Best Fit ............................................ 831
16.2 Mia Is Growing Like a Weed!
Analyzing the Line of Best Fit .................................... 843
16.3 Stroop Test
Performing an Experiment ..........................................853
© 2011 Carnegie Learning
© 2011 Carnegie Learning
16.4 Human Chain: Shoulder Experiment
Using Technology to Determine a Linear Regression Equation .................................................. 863
16.5 Jumping
Correlation ................................................................. 873
829
Chapter 16 Overview
16.2
Drawing
Lines of Best
Fit
Analyzing
the Line of
Best Fit
16.3
Performing
an
Experiment
16.4
Using
Technology
to
Determine
a Linear
Regression
Equation
16.5
Correlation
8.SP.1
8.SP.2
8.SP.3
8.SP.1
8.SP.2
8.SP.3
8.SP.1
8.SP.2
8.SP.3
8.SP.1
8.SP.2
8.SP.3
8.SP.1
8.SP.2
8.SP.3
1
This lesson introduces the concept of line
of best fit and explains how although a
straight line will not pass through all of
the points on a scatter plot, a line can be
drawn to approximate the data as closely
as possible.
X
Technology
X
Talk the Talk
Highlights
Peer Analysis
Pacing
X
Questions ask students to identify the
slope and y-intercept of the line of best
fit in terms of a problem situation, and
then use their line of best fit to make
predictions.
This lesson explores the accuracy of a line
of best fit that models real-world data.
1
Questions ask students to use a line of best
fit equation to make predictions about a
related problem situation.
X
This lesson explores the line of best fit that
models the results of a class experiment.
1
1
1
829A • Chapter 16 Lines of Best Fit
Questions ask students to interpret
their line of best fit equation and make
predictions in regards to the experiment.
This lesson explores how to use a graphing
calculator to determine a linear regression
equation.
Questions ask students to compare the
linear regression equations from wrist and
shoulder experiments.
This lesson uses data collected by another
class experiment to determine whether a
data set is positively, negatively, or
not correlated.
Questions ask students to interpret the
correlation coefficient.
X
X
X
X
X
X
X
X
X
© 2011 Carnegie Learning
16.1
CCSS
Worked Examples
Lesson
Models
This chapter explores real-world bivariate data and the concept of line of best fit. Class experiments will be
conducted with the data recorded and plotted in a scatter plot. The line of best is then calculated and used
to make predictions.
Skills Practice Correlation for Chapter 16
Lesson
Problem
Set
Objectives
Vocabulary
16.1
16.2
16.3
© 2011 Carnegie Learning
16.4
Drawing Lines
of Best Fit
Analyzing
the Line of
Best Fit
Performing
an
Experiment
Using
Technology
to
Determine
a Linear
Regression
Equation
1–6
Create scatter plots representing tables
7 – 12
Draw lines of best fit on scatter plots
13 – 18
Draw lines of best fit then write the equation of the lines
19 – 24
Use given equations to answer questions
1–6
Estimate the equation of lines of best fit on scatter plots
7 – 12
Use given lines of best fit to make predictions 13 – 18
Compare given graphs to determine which lines is a better fit for data
19 – 22
Use given equations to answer questions
1–4
Create scatter plots from tables, estimate the equation of lines of best
fit, and use equations to answer questions
5–8
Write equations for lines of best fit
9 – 12
Use given equations to answer questions
13 – 16
Use given information about experiments to answer questions
1-6
Use given correlation coefficients to indicate how close data are to
being straight lines
7 - 12
Determine linear regression equations using a graphing calculator
13 - 16
Calculate linear regression equations using a graphing calculator and
given data from tables
17 - 20
Write corresponding linear regression equations using a graphing
calculator and given data from tables
Vocabulary
16.5
Correlation
1 - 10
Determine whether points have a positive correlation, a negative
correlation, or no correlation
11 - 14
Draw lines of best fit and explain what they indicate about the
relationships between two variables
15 - 20
Estimate equations for given lines of best fit
21 - 24
Use given equations to answer questions
Chapter 16 Lines of Best Fit • 829B
© 2011 Carnegie Learning
830 • Chapter 16 Lines of Best Fit
Where Do You
Buy Your Books?
Drawing Lines of Best Fit
Learning Goals
Key Terms
In this lesson, you will:





 Determine the definition of a line of best fit.
 Use a line of best fit to make predictions.
 Compare two lines of best fit.
Essential Ideas
• A scatter plot is a graph of data points.
• A line of best fit is a straight line that is as close to
as many points as possible, but does not have to go
through all the points on a scatter plot.
• Equations can be written for a line of best fit.
• A line of best fit can be used to make predictions
about data.
• A line of best fit and its equations are often referred
© 2011 Carnegie Learning
to as a model of the data.
line of best fit
model
trend line
interpolating
extrapolating
Common Core State Standards for
Mathematics
8.SP Statistics and Probability
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of
association between two quantities. Describe
patterns such as clustering, outliers, positive
or negative association, linear association, and
nonlinear association.
2. Know that straight lines are widely used to model
relationships between two quantitative variables.
For scatter plots that suggest a linear association,
informally fit a straight line, and informally assess
the model fit by judging the closeness of the data
points to the line.
3. Use the equation of a linear model to solve
problems in the context of bivariate measurement
data, interpreting the slope and intercept.
16.1 Drawing Lines of Best Fit • 831A
Overview
The terms line of best fit, trend line, interpolating, and extrapolating are defined in this lesson.
Students create two scatter plots to show percent of book sales from book stores and the internet for
time since 2004. They then will write an equation of the line of best fit for each scatter plot and draw it
© 2011 Carnegie Learning
in their plot. Using their line of best fit, students will predict the percent of book sales.
831B • Chapter 16 Lines of Best Fit
Warm Up
1. Solve for x.
50 5 3.5x 1 24.2
50 5 3.5x 1 24.2
50-24.2 5 3.5x 1 24.2 224.2
25.8 5 3.5x
x ¯ 7.37
2. Solve for x.
30 5 22.9x 1 50.3
30 5 22.9x 1 50.3
30 2 50.3 5 22.9x 1 50.3 2 50.3
220.3 5 22.9x
x57
3. Solve for y when x 5 6
y 5 3.5x 1 24.2
y 5 3.5x 1 24.2
y 5 3.5(6) 1 24.2
y 5 21 1 24.2
y 5 45.2
4. Solve for y when x 5 3
© 2011 Carnegie Learning
y 5 22.9 x 1 50.3
y 5 22.9x 1 50.3
y 5 22.9(3) 1 50.3
y 5 28.7 1 50.3
y 5 41.6
16.1 Drawing Lines of Best Fit • 831C
© 2011 Carnegie Learning
831D • Chapter 16 Lines of Best Fit
Where Do You
Buy Your Books?
Drawing Lines of Best Fit
Learning Goals
Key Terms
In this lesson, you will:
 Determine the definition of a line of best fit.
 line of best fit
 model
 Use a line of best fit to make predictions.
 Compare two lines of best fit.
 trend line
 interpolating
 extrapolating
E
-books are becoming more and more popular. First, the way people bought books changed a lot. Now, the actual books people read have changed too! E-books offer readers the opportunity to download books onto their reader devices. However, as trendy and convenient as e-books are, many technology experts feel that e-book reader devices may actually have a very short life. Because of new advancements in computers and cell phones, these same experts believe that e-books will be able to be read directly from portable laptop computers—computers not much bigger that a small stack of loose-leaf paper—and cell phones. What do you think are the advantages of combining technologies onto © 2011 Carnegie Learning
© 2011 Carnegie Learning
a single device? What are some disadvantages? Do you ever think the printed paper book will one day become completely obsolete?
16.1 Drawing Lines of Best Fit • 831
16.1 Drawing Lines of Best Fit • 831
Problem 1
Students analyze data for the
percent of book sales from a
book store from 2004 through
2010. They then create a
scatter plot to display data
with decreasing y-values as
time increases. A line of best
fit is drawn on the scatter plot
and two points are used to
determine the slope of the line
of best fit and the equation of
the line of best fit. Using the
equation for the line of best fit,
students will use the equation
to predict past and future sales.
Problem 1
Purchasing Books from Different Places
You can purchase books from many different places: a bookstore, a department store, the
Internet, a book club, and many other places. The source for purchasing books changes
as the available formats for books change.
Suppose the table shows the percent of book sales that came from bookstores for the
years 2004 through 2010.
Percent of Book Sales from Book Stores
Year
2004
2005
2006
2007
2008
2009
2010
Percent of
Total Sales
50.8
44.5
42.4
42.5
36.8
33.2
32.5
1. Identify the independent and dependent variables in this problem situation.
Time is the independent variable and percent of total sales is the dependent
variable.
Grouping
• Ask a student to read the
Because the x-coordinates represent time, you can define time as the number of years
introduction to Problem 1
aloud. Discuss the context
and complete Question 1
as a class.
since 2004. In this problem situation, you could represent 2004 as 0 on the x-axis.
2. How would you represent:
a. 2005?
• Have students complete
I would use 1 on the x-axis to represent 2005 because it is one year since 2004.
Questions 2 through 4 with
a partner. Then share the
responses as a class.
b. 2006?
I would use 3 since 2007 is 3 years since 2004.
your books from considered
a bookstore or a nonbookstore? Explain.
• Do these two types of stores
represent all the possible
sources where you can buy
books? Explain.
832 • Chapter 16 Lines of Best Fit
Share Phase, Question 2
• What year does the x-value of 0 represent on the graph of the scatter plot?
• What year does the x-value of 8 represent on the graph of the scatter plot?
• What year does the x-value of 24 represent on the graph of the scatter plot?
832 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
c. 2007?
Discuss Phase,
Question 1
• Is the place where you buy
© 2011 Carnegie Learning
I would use 2 because 2006 is 2 years since 2004.
Share Phase,
Questions 3 and 4
• What does the broken line
3. Create a scatter plot of the ordered pairs on the coordinate plane shown.
on the y-axis represent?
Why is it there?
circumstances under which
using a broken line on an
axis is appropriate?
Percent of Book Sales
from Bookstores
y
55
When you use 0
to indicate a particular
year, such as 2004, you should
indicate this on your graph with
the appropriate axis label. One
way to do this is to use a double
arrow: 0 ↔ 2004. You can
think of the double arrow as
meaning "is the same as."””
50
Percent of Total Sales
When you want to
show data points are
clustered together but
• What would the graph of
are away from the origin,
the scatter plot look like if
you can break the graph.
the broken line was not on
The squiggly line drawn
from the origin to the
the y-axis?
first interval shows a
• How would you describe the break in the graph.
45
40
35
30
• How does using a broken
0
0.0
line on the axis affect the
appearance of a graph?
x
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Time (years)
(0
2004)
4. Do all the points in your scatter plot lie on the same line? What does
this tell you about the percent of total sales as the time changes?
Grouping
The points do not lie on the same line. The percent of total sales is
Ask a student to read the
information following
Question 4 aloud. Discuss the
definitions and the following
worked examples as a class.
not changing at the same rate as the time changes.
Sometimes, it may seem that there is not a linear relationship between the
data points in a scatter plot. However, some of the data points may be
clustered where a straight line might pass. Although a straight line will not
pass through all of the points in your scatter plot, you can use a line to
approximate the data as closely as possible. This kind of line is called a line
© 2011 Carnegie Learning
© 2011 Carnegie Learning
of best fit. A line of best fit is a line that is as close to as many points as
possible, but doesn’t have to go through all the points. When you use a line
of best fit, the line and its equation are often referred to as a model of the
data, or a trend line.
When data is displayed with a scatter plot, constructing a line of best fit is
helpful to predict values that may not be displayed on the plot. You want to
begin by analyzing the data and ask yourself:
●
Does the data look like a line?
●
Does the data seem to have a positive or negative correlation?
16.1 Drawing Lines of Best Fit • 833
16.1 Drawing Lines of Best Fit • 833
Let’s construct a line of best fit.
Step 1: Begin by plotting all the data.
Percent of Book Sales
from Bookstores
y
55
Percent of Total Sales
50
45
40
35
30
0
1
2
3 4
5 6
Time (years)
(0
2004)
7
8
x
Step 2: Draw a shape that encloses all of the data.
“Try to draw a
smooth and relatively
even shape.
Percent of Book Sales
from Bookstores
y
55
45
40
30
0
834 • Chapter 16 Lines of Best Fit
834 • Chapter 16 Lines of Best Fit
1
2
3 4
5 6
Time (years)
(0
2004)
7
8
x
© 2011 Carnegie Learning
35
© 2011 Carnegie Learning
Percent of Total Sales
50
Discuss Phase,
Steps 3 and 4
• Where did the ordered pair
(0, 49.2) come from? If you
were going to estimate the
coordinates of this point on
the scatter plot, what values
would you use?
Step 3: Draw a line that divides the enclosed area of the data in half.
Percent of Book Sales
from Bookstores
y
55
50 x
• Where did the ordered pair
Percent of Total Sales
(3, 40.5) come from? If you
were going to estimate the
coordinates of this point on
the scatter plot, what values
would you use?
45
x
40
35
30
• If you used your estimated
values of these two points,
what would be your
equation?
0
1
2
3 4
5 6
Time (years)
(0
2004)
7
8
x
Note that the line of best fit does not have to go through
any of the data values.
Step 4: Determine the equation of your line of
“The idea is that
you want to
identify a line
that has an equal
number of points
on either side.
best fit.
●
Begin by identifying two points on your trend line.
In this example, two points were chosen and
marked with an “x.” The estimated ordered pairs
are (0, 49.2) and (3, 40.5).
© 2011 Carnegie Learning
●
Calculate the slope of the line through your
two points.
●
49.2 2 40.5 5 ___
8.7
m 5 ___________
023
23
m 5 22.9
Write the equation of the line.
Let x represent the number of years since 2004, and y represent
the percent of all sales.
© 2011 Carnegie Learning
y 5 2.9x 1 49.2
16.1 Drawing Lines of Best Fit • 835
16.1 Drawing Lines of Best Fit • 835
Grouping
• Have students complete
It is possible to choose two different points and estimate those ordered pairs in a slightly
Questions 5 and 6 with a
partner. Then share the
responses as a class.
different way. Determining the line of best fit may lead to different equations depending
upon the estimated ordered pairs chosen to construct the line.
5. Identify the slope of the line of best fit and what it represents in this problem situation.
• Have students complete
The slope of the line of best fit is 22.9. The slope represents the change in percent
Questions 7 through 11 with
a partner. Then share the
responses as a class.
of book sales from bookstores per year.
6. Identify the y-intercept of the line of best fit and what it represents in this problem
situation.
The y-intercept is 49.2. The y-intercept represents the percent of book sales from
Share Phase,
Questions 5 and 6
• What would be the slope of
bookstores in 2004.
If you are predicting values that fall within the plotted values, you are interpolating. If you
are predicting values that fall outside the plotted values, you are extrapolating.
your line of best fit?
7. Use the line of best fit equation to predict the percent of book sales from bookstores:
• What would be the
a. in 2011.
y-intercept of your line of
best fit?
y 5 22.9(7) 1 49.2 5 28.9
About 29% of total sales from bookstores should occur in 2011.
b. in 2013.
Share Phase,
Question 7
• When the percent of sales
y 5 22.9(9) 1 49.2 5 23.1
About 23% of total sales from bookstores should occur in 2013.
of books from book stores
was 29%, what do you know
about the percent of sales of
books from other sources?
c. Explain how you determined your predictions.
First, I wrote the year as the number of years since 2004.
Then, I substituted this value for x-value into the equation
• When the percent of sales
8. Use the line of best fit equation to make predictions.
of books from book stores
was 23%, what do you know
about the percent of sales of
books from other sources?
a. In what year was the percent of book sales from
bookstores 60%?
60 5 22.9x 1 49.2
10.8 5 22.9x
23.7 ¯ x
2004 2 3.7 5 2000.3
Are printed book
sales trending
up or trending
down?
© 2011 Carnegie Learning
and solved for y.
836
•
Chapter 16
7719_CL_C3_CH16_pp829-886.indd 836
836 • Chapter 16 Lines of Best Fit
Lines of Best Fit
© 2011 Carnegie Learning
In 2000, bookstore sales were 60% of total sales.
11/04/15 5:32 PM
Share Phase,
Questions 8 through 11
• When the percent of sales
of books from book stores
was 60%, what do you know
about the percent of sales of
books from other sources?
b. In what year will the percent of book sales from bookstores be 20%?
20 5 22.9x 1 49.2
229.2 5 22.9x
10.1 ¯ x
2004 1 10.1 5 2014.1
Near the end of 2014, bookstore sales should be 20% of sales.
• When the percent of sales
of books from book stores
was 20%, what do you know
about the percent of sales of
books from other sources?
c. Explain how you determined your predictions.
First, I substituted the given value for y and solved for x. Then, I added the
value for x to 2004 to determine the year.
• What is the meaning of the
x-intercept in this situation?
9. Use the line of best fit equation to determine the percent of book sales from
bookstores in 2006.
• What year do you think the
y 5 22.9(2) 1 49.2 5 43.4
sales of books from book
stores will be 0%?
About 43% of the total book sales should come from bookstores.
• Why do you suppose that
the percent of sales of books
from books stores has
declined since 2004?
10. Compare your answer from Question 9 to the actual data from the table. What do
you notice?
The answer, about 43%, is fairly close to the actual data from the table,
which is 42.4%.
11. Do you think that the line of best fit model provides reasonable answers to the
questions posed in Questions 7 and 8? Explain your reasoning.
Yes. Because the line is reasonably close to all the data points, the model provides
© 2011 Carnegie Learning
© 2011 Carnegie Learning
reasonable answers.
16.1
7719_CL_C3_CH16_pp829-886.indd 837
Drawing Lines of Best Fit
•
837
13/04/15 11:31 AM
16.1 Drawing Lines of Best Fit • 837
Problem 2
Problem 2
Purchasing Books from the Internet
Suppose the table shows the percent of all book sales that came from the Internet for the
years 2004 through 2010.
Percent of Printed Book Sales from the Internet
Year
2004
2005
2006
2007
2008
2009
2010
Percent of
Total Sales
34.4
38.5
40.8
42.4
50.7
52.8
53.8
1. Identify the independent and dependent variables in this problem situation.
Time is the independent variable and percent of total sales is the
dependent variable.
Grouping
2. Create a scatter plot from the table data.
Have students complete
Questions 1 through 10 with
a partner. Then share the
responses as a class.
y
55
Percent of Printed Book Sales
from the Internet
Percent of Total Sales
50
Share Phase,
Questions 1 through 3
• How does the data in this
table look different than
the data in the table of
Problem 1?
45
40
35
30
• How does the scatter plot in
this problem look different
than the scatter plot in
Problem 1?
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Time (years)
(0
2004)
3. Analyze your scatter plot. Does the data look like a line?
If so, does the data seem to have a positive correlation
or negative correlation?
The data seems to form a line with a positive correlation.
838 • Chapter 16 Lines of Best Fit
838 • Chapter 16 Lines of Best Fit
x
Remember to
start by drawing a
shape around all the
data. Then, divide
that in half.
© 2011 Carnegie Learning
0
0.0
© 2011 Carnegie Learning
Problem 2 is very similar
mathematically to Problem 1.
Students analyze data for the
percent of book sales from
the internet. Students create
a scatter plot to display the
percent of sales data with
increasing y-values as time
increases. They will then make
predictions for time in the future
as well as for time in the past
using the equation of their line
of best fit.
Share Phase,
Questions 4 through 7
• What is the meaning of the
x-intercept in this situation?
4. Use a ruler to draw the line that best fits your data on the graph. Then, write the
equation of the line. Define your variables and include the units.
Answers will vary, but should be close to y 5 3.5x 1 34.3, where x is the number of
• When the percent of sales
years since 2004, and y is the percent of all sales.
of books from the Internet
was 59%, what do you know
about the percent of sales of
books from other sources?
• When the percent of sales
5. What does the slope of your line represent in this problem situation?
The slope represents the change in the percent of total sales per year.
6. What does the y-intercept represent in this problem situation?
of books from the Internet
was 66%, what do you know
about the percent of sales of
books from other sources?
The y-intercept represents the percent of total sales in 2004.
7. Use your equation to predict the percent of book sales from the Internet:
a. in 2011.
y 5 3.5(7) 1 34.3 5 58.8
About 59% of the total book sales should come from the Internet.
b. in 2013.
y 5 3.5(9) 1 34.3 5 65.8
About 66% of the total book sales should come from the Internet.
c. Explain how you calculated your answers.
First, I wrote the year as the number of years since 2004. Then, I substituted
© 2011 Carnegie Learning
© 2011 Carnegie Learning
this value for x into the equation and solved for y.
16.1 Drawing Lines of Best Fit • 839
16.1 Drawing Lines of Best Fit • 839
Share Phase,
Questions 8
through 10
• When the percent of sales
8. Use your equation to predict the year in which Internet sales will be a certain percent
of total book sales. Show all your work.
a. 60% of total book sales
of books from the Internet
was 60%, what do you know
about the percent of sales of
books from other sources?
60 5 3.5x 1 34.3
25.7 5 3.5x
7.3 < x
2004 1 7.3 5 2011.3
In 2011, Internet sales will be 60% of total sales.
• When the percent of sales
of books from the Internet
was 20%, what do you know
about the percent of sales of
books from other sources?
b. 20% of total book sales
20 5 3.5x 1 34.3
214.3 5 3.5x
24.1 < x
• When the percent of sales
2004 2 4.1 5 1999.9
Near the end of 1999, Internet sales were 20% of sales.
of books from the Internet
was 41%, what do you
know about the percent of
sales of books from all other
sources?
c. Explain how you calculated your answers.
First, I substituted the given value for y and solved for x. Then, I added the
value for x to 2004 to determine the year.
• What year do you think the
sales of books from the
Internet was 0%?
9. Compare the actual values to values using your equation.
a. How close is the value of the y-intercept to the actual value?
• What year do you think the
The y-intercept, 34.3%, is very close to the actual value, 34.4%.
sales of books from the
Internet will be 100%?
• Why do you suppose that
is this answer to the actual data?
y 5 3.5(2) 1 34.3 5 41.3
About 41% of the total sales should come from the Internet.
The answer, about 41%, is close to the actual data, 40.8%.
© 2011 Carnegie Learning
b. Use your equation to predict the percent of Internet book sales in 2006. How close
the percent of sales of
books from the Internet has
increased since 2004?
10. Do you think that your model provides reasonable answers to Questions 7 and 8?
Yes. Because the line is reasonably close to all the points, the model provides
reasonable answers.
840 • Chapter 16 Lines of Best Fit
840 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
Explain your reasoning.
Talk the Talk
Students compare the data
from Problems 1 and 2.
Talk the Talk
1. Do you think that the data from Problems 1 and 2 are related?
Grouping
Yes. As the bookstores’ percent of sales decrease, the Internet’s percent of
Have students complete
Questions 1 through 3 with
a partner. Then share the
responses as a class.
sales increase.
2. Which percent of book sales is changing faster: bookstore sales or Internet sales?
Explain your reasoning.
The Internet’s percent of sales is changing faster because that line
is steeper.
3. Which equation from Problem 1 and Problem 2 models its data better? Explain your
reasoning.
Answers will vary, but students should make a comparison between how well each
model fits its data. They should do this by checking how close each percent given
© 2011 Carnegie Learning
© 2011 Carnegie Learning
by their models from 2004 through 2010 is to the actual value given in the tables.
Be prepared to share your solutions and methods.
16.1 Drawing Lines of Best Fit • 841
16.1 Drawing Lines of Best Fit • 841
Follow Up
Assignment
Use the Assignment for Lesson 16.1 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 16.1 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 16.
Check for Students’ Understanding
Estimate the equation of the line of best fit shown in each scatter plot.
1.
y
20
18
16
14
12
10
8
6
4
2
0
2
4
6
8
10 12 14 16 18 20 x
y 2y
y 2 y1 5 m (x 2 x1) m = _______
​ x2 2 x1 ​ 
5 _______
​  2 2 1 
 ​ 
52
0.5 2 0
2
1
y 2 1 5 2(x 2 0)
y 5 2x 1 1
842 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
Answers will vary but should be close to the following equation:
2.
y
20
18
16
14
12
10
8
6
4
2
0
2
4
6
8
10 12 14 16 18 20 x
Answers will vary but should be close to the following equation:
y 2y
y 2 y1 5 m (x 2 x1) m 5 _______
​ x2 2 x1 ​ 
5 ________
​ 15 2 11 
 ​5 1
14 2 10
2
1
y 2 11 5 1(x 2 10)
© 2011 Carnegie Learning
y5x11
16.1 Drawing Lines of Best Fit • 842A
© 2011 Carnegie Learning
842B • Chapter 16 Lines of Best Fit
Mia’s Growing
Like a Weed!
Analyzing the Line of Best Fit
Learning Goals
In this lesson, you will:




Create a scatter plot.
Draw a line of best fit.
Write an equation of a line of best fit.
Use a line of best fit to make predictions.
Essential Ideas
• A scatter plot is a graph of data points.
• A line of best fit is a straight line that is as close to
as many points as possible, but does not have to go
through all the points on a scatter plot.
• Equations can be written for a line of best fit.
• A line of best fit can be used to make predictions
about data.
• A line of best fit and its equations are often referred
© 2011 Carnegie Learning
to as a model of the data.
Common Core State Standards for
Mathematics
8.SP Statistics and Probability
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of
association between two quantities. Describe
patterns such as clustering, outliers, positive
or negative association, linear association, and
nonlinear association.
2. Know that straight lines are widely used to model
relationships between two quantitative variables.
For scatter plots that suggest a linear association,
informally fit a straight line, and informally assess
the model fit by judging the closeness of the data
points to the line.
3. Use the equation of a linear model to solve
problems in the context of bivariate measurement
data, interpreting the slope and intercept.
16.2 Analyzing the Line of Best Fit • 843A
Overview
Students create a scatter plot for age and height, and a scatter plot for age and weight. They will draw
the line of best fit and determine the equation of the line of best fit for each scatter plot. Students
then make predictions for height and for weight based on age using the equation of each line of
© 2011 Carnegie Learning
best fit.
843B • Chapter 16 Lines of Best Fit
Warm Up
Complete each statement.
1. If a line has a positive slope, then as the x- values increase, the y-values will _________________.
increase
2. If a line has a negative slope, then as the x-values increase, the y-value will _________________.
decrease
3. If a line has a slope of 7, then as the x-values increase by one unit, the y-values will
_______________ by ____________.
increase by 7 units
4. If a line has a slope of 7, then as the x-values increase by 5 units, the y-values will
_______________ by ____________.
© 2011 Carnegie Learning
increase by 35 units
16.2 Analyzing the Line of Best Fit • 843C
© 2011 Carnegie Learning
843D • Chapter 16 Lines of Best Fit
Mia Is Growing
Like a Weed!
Analyzing the Line of Best Fit
Learning Goals
In this lesson, you will:
 Create a scatter plot.
 Draw a line of best fit.
 Write an equation of a line of best fit.
 Use a line of best fit to make predictions.
T
he birth of an endangered species is reason to celebrate! And on July 9, 2005, the birth of a new bundle of joy took Washington, D.C. by storm. The birth of Tai Shan, the first giant panda to be born at the National Zoo, stole front page headlines from other events in the world. And when he made his first appearance in December 2005, 13,000 tickets were distributed to the public—all to get a glimpse of Tai Shan! But while he was gaining star attention by the people of D.C., zoologists were carefully monitoring his diet, behavior, weight, length, and exercise. Why do you think zoologists were so interested in Tai Shan’s daily activities and growth? What are other newborn creatures that are monitored © 2011 Carnegie Learning
© 2011 Carnegie Learning
constantly during their first years of life? Do you think there is a special type of doctor that only takes care of babies?
16.2 Analyzing the Line of Best Fit • 843
16.2 Analyzing the Line of Best Fit • 843
Problem 1
Problem 1
How Quickly is She Growing?
1. Mia was born a healthy, happy baby girl to the Sanchez family. At each doctor’s visit,
Mia’s height and weight were recorded. Her records from birth until she was 18
months old are shown in the table.
Weight
(lbs)
0.0
6.1
1.0
8.1
1.8
10.0
Grouping
2.3
10.3
Ask a student to read the
information at the beginning of
Question 1 aloud. Discuss the
context and complete
Question 1, part (a) as a class.
4.0
13.7
6.0
17.0
8.0
21.0
10.0
22.0
12.0
23.0
15.0
23.0
18.0
25.1
a. Consider the relationship between Mia’s age and her weight. What happens to
Mia’s weight as she gets older?
Mia’s weight increases as she gets older.
844 • Chapter 16 Lines of Best Fit
844 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
Age
(months)
© 2011 Carnegie Learning
Students analyze a table of
data of a child’s growth rate for
weight. They will consider the
relationships between age and
weight, and determine the unit
rate to compare her weight at
different ages. Weight and age
are then graphed using a
scatter plot. Finally, students
will approximate the line of best
fit and use this information to
predict trends in weight.
Grouping
Have students complete
Questions 1, part (b) through 3
with a partner. Then share the
responses as a class.
b. Write a unit rate that compares Mia’s weight change to her change in age from age
4 months to age 6 months. Explain how you calculated your answer.
3.3 pounds ____________
1.65 pounds
17 pounds 2 13.7 pounds ___________
_________________________
5
5
6 months 2 4 months
2 months
1 month
Mia’s weight changed by 1.65 pounds per month
between age 4 months and age 6 months.
Share Phase,
Question 1, parts (b)
through (d)
• What would you expect to
I divided the difference in weight by
the change in age.
happen to a baby’s weight as
she got older?
• Would she grow heavier at a
Remember, a unit
rate is a comparison of
two measurements in
which the denominator
has a value of
one unit.
c. Write a unit rate that compares Mia’s weight change
to her change in age from 6 months to 8 months. Explain
how you calculated your answer.
constant rate, a faster rate, or
a slower rate during her first
month of life or her fifteenth
month of life? Explain.
4 pounds _________
2 pounds
21 pounds 2 17 pounds _________
_______________________
5
5
8 months 2 6 months
2 months
1 month
Mia’s weight changed by 2 pounds per month between
6 months and 8 months.
I divided the difference in weight by the change in age.
• How can you check your
conjectures using the data
given for Mia?
d. Is Mia gaining weight faster from 4 months to 6 months, or
from 6 months to 8 months? Explain your reasoning.
Mia is gaining weight faster from 6 months to 8 months
because the rate of her change in weight during this time is
greater than the rate of her change in weight from 4 months
© 2011 Carnegie Learning
© 2011 Carnegie Learning
to 6 months.
16.2 Analyzing the Line of Best Fit • 845
16.2 Analyzing the Line of Best Fit • 845
Share Phase,
Questions 2 and 3
• How many points on the
2. Create a scatter plot that shows Mia’s age as the independent variable and her weight
as the dependent variable.
scatter plot are above your
line of best fit?
y
Mia’s Weight over Time
30
• How many points on the
28
26
scatter plot are below your
line of best fit?
24
22
Weight (lbs)
20
• How many points on the
scatter plot are on your line
of best fit?
18
16
14
12
10
8
6
4
Grouping
2
0
Have students complete
Questions 4 through 9 with
a partner. Then share the
responses as a class.
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
x
Age (months)
3. Do all the points in your scatter plot lie on the same line? What does this tell you
about Mia’s weight change as time changes? Explain your reasoning.
No. All the points do not lie on the same line. This means that Mia’s weight is not
changing at the same rate as time changes.
Share Phase,
Questions 4 and 5
• What is the slope of your line
4. Use a ruler to draw the line that best fits the data on the graph.
The graph of the line should be close to y 5 1.08x 1 8.61.
of best fit?
The equation should be close to
y 5 1.08x 1 8.61, where x is the age in
• What ordered pairs did you
months, and y is the weight in pounds.
use to determine the slope of
the line of best fit?
• How does your equation for
the line of best fit compare to
your classmate’s equations?
846 • Chapter 16 Lines of Best Fit
846 • Chapter 16 Lines of Best Fit
Remember, after
you draw the line, pick
two points from your line
to write the equation.
© 2011 Carnegie Learning
define your variables and include the units.
with respect to the problem
situation?
© 2011 Carnegie Learning
5. Write the equation of your line. Be sure to
• What does the slope mean
6. According to your line, approximately how many pounds did Mia gain each month
from the time she was born until she was 18 months old? How did you determine
your answer?
Mia gained about 1.08 pounds each month. The slope of my line of best fit is about
1.08 pounds per month.
7. If Mia continues to grow at this rate, how much will she weigh when she is:
a. 2 years old?
y 5 1.08(24) 1 8.61 5 34.53
Mia will weigh about 34.53 pounds.
b. 5 years old?
y 5 1.08(60) 1 8.61 5 73.41
Mia will weigh about 73.4 pounds.
c. 18 years old?
y 5 1.08(216) 1 8.61 5 241.89
Mia will weigh about 241.9 pounds.
8. Do all your answers to Question 7 make sense? Explain your reasoning.
No. The weights seem unreasonable. Some 2-year-olds might reach 34.5 pounds,
but the other two weights seem unreasonable. Most adults will never reach a
© 2011 Carnegie Learning
© 2011 Carnegie Learning
weight of 241 pounds.
9. What can you conclude about the accuracy of your model?
The model seems to be approximately accurate for the first 2 years, but not
beyond that.
16.2 Analyzing the Line of Best Fit • 847
16.2 Analyzing the Line of Best Fit • 847
Problem 2
A table of values that describes
Mia’s age and height is given.
Students use the table of data
to create a scatter plot for the
child’s height over time. They
will draw a line of best fit and
analyze the graph. They then
write the equation of the line of
best fit and use the equation to
predict future heights.
Problem 2
How Does Mia’s Height Change?
Analyze the table shown with the data of Mia’s age and her height.
Age
(months)
Height
(in.)
0.0
17.9
1.0
20.5
1.8
21.0
2.3
21.8
4.0
25.0
6.0
25.8
8.0
27.0
10.0
27.0
12.0
29.3
15.0
30.5
18.0
32.5
Grouping
Have students complete
Questions 1 through 8 with
a partner. Then share the
responses as a class.
Share Phase,
Question 1
• What is the same in Problem
2 as in Problem 1?
• What is different in Problem 2
than in Problem 1?
1. Consider the relationship between Mia’s age and her height. What happens to Mia’s
• What would you expect to
• Would her height increase
at a constant rate, a faster
rate, or a slower rate during
her first month of life or
her fifteenth month of life?
Explain.
• What do you think is the
range of typical heights for
most adults? Describe the
range of heights using units
of inches.
848 • Chapter 16 Lines of Best Fit
848 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
Mia’s height increases as she gets older.
© 2011 Carnegie Learning
height as she gets older?
happen to a baby’s height as
she got older?
Share Phase,
Questions 2 through 5
• How many points on the
scatter plot are above your
line of best fit?
2. Create a scatter plot that shows Mia’s age as the independent variable and her height
as the dependent variable. First, label the axes to represent the independent and
dependent variables. Next, choose the appropriate intervals for your scatter plot.
y
• How many points on the
Mia’s Height over Time
45
42
scatter plot are below your
line of best fit?
39
36
33
• How many points on the
Height (in.)
30
scatter plot are on your line
of best fit?
• What is the slope of your line
of best fit?
27
24
21
18
15
12
9
6
• What does the slope mean
3
0
with respect to the problem
situation?
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
x
Age (months)
3. Can these data be exactly represented by a linear equation? Explain your reasoning.
• What ordered pairs did you
No. All the points do not lie in a straight line.
use to determine the slope of
the line of best fit?
• How does your equation for
4. Use a ruler to draw the line that best fits your data on your graph. Then, write the
the line of best fit compare to
your classmate’s equations?
equation of your line. Be sure to define your variables and include the units.
The equation should be close to y 5 0.73x 1 20.08, where x is the age in months,
and y is the height in inches.
5. According to your line, approximately how many inches did Mia grow each month
© 2011 Carnegie Learning
© 2011 Carnegie Learning
from the time she was born until she was 18 months old? How did you determine
your answer?
Mia grew about 0.73 inch each month. The answer is given by the slope of
the line in Question 5.
16.2 Analyzing the Line of Best Fit • 849
16.2 Analyzing the Line of Best Fit • 849
6. If Mia continues to grow at this rate, how tall will she be when she is:
a. 2 years old?
y 5 0.73(24) 1 20.08 5 37.6
Mia will be about 37.6 inches tall.
b. 5 years old?
y 5 0.73(60) 1 20.08 5 63.88
Mia will be about 63.9 inches tall.
c. 18 years old?
y 5 0.73(216) 1 20.08 5 177.76
Mia will be about 177.8 inches tall.
7. Do all of your answers to Question 6 make sense? Explain your reasoning.
No. The heights at age 5 years and age 18 years seem to be unreasonable because
most 5-year-olds are not over 5 feet tall. Also, no adult humans have ever been
almost 15 feet tall.
8. What can you conclude about the accuracy of your model?
850 • Chapter 16 Lines of Best Fit
850 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning
The model seems to be accurate for the first 2 years, but not beyond that.
Problem 3
Students are given additional
data values for the data table.
They plot these values and
compare them to their predicted
values. Students then update
their graphs and lines of best fit
with the additional data. They
will write an equation of best fit
and use their equation to make
predictions about the height
and weight of the child when
she is 18 years old.
Problem 3
More Data about Mia
1.
The table shows Mia’s growth from age 2 to age 5 __
2
Grouping
Have students complete
Questions 1 through 6 with
a partner. Then share the
responses as a class.
Age
(years)
Age
(months)
Weight
(lbs)
Height
(in.)
2.0
24
27.3
34.5
2.5
30
30.0
35.8
3.0
36
32.0
36.6
3.5
42
33.0
38.0
4.5
54
39.0
42.0
5.5
66
44.0
45.0
1. Complete the table shown by converting each age from years to months.
Share Phase,
Questions 1 and 2
• What unit was given for Mia’s
2. The scatter plot shown relates Mia’s age to her weight. Include the new data from the
table on the scatter plot. Then, draw the line of best fit and determine the equation of
the line.
y
age in Problem 1?
56
ages in Problem 3 into
months?
• How many months are in a
year? Is that an exact value?
© 2011 Carnegie Learning
• What units were given for
Mia’s weight in Problem 1?
What units were given for
Mia’s height in Problem 3?
• Why is a line of best fit called
52
48
44
40
Weight (lbs)
• How can you convert the
© 2011 Carnegie Learning
• What unit was given for Mia’s
age in Problem 3?
Mia’s Weight over Time
60
36
32
28
24
20
16
12
8
4
0
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
x
Age (months)
The equation should be close to y 5 0.52x 1 12.47.
16.2 Analyzing the Line of Best Fit • 851
that? What is fitting best?
16.2 Analyzing the Line of Best Fit • 851
Share Phase,
Questions 3 through 6
• How many points on each
3. The scatter plot shown relates Mia’s age to her height. Include the new data from the
table on the scatter plot. Then, draw the line of best fit and determine the equation of
scatter plot are above your
line of best fit?
the line.
y
• How many points on each
60
Mia’s Height over Time
56
scatter plot are below your
line of best fit?
52
48
44
• How many points on each
Height (in.)
40
scatter plot are on your line
of best fit?
• What is the slope of each line
of best fit?
36
32
28
24
20
16
12
8
• What does the slope mean
4
0
with respect to the problem
situation?
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
x
Age (months)
The equation should be close to y 5 0.38x 1 22.66.
• What ordered pairs did you
use to determine the slope of
each line of best fit?
4. Use your new lines of best fit to determine Mia’s height and weight when she is
18 years old.
y 5 0.52(216) 1 12.47 5 124.79
• How do your equations
y 5 0.38(216) 1 22.66 5 104.74
for each line of best fit
compare to your classmate’s
equations?
According to the lines of best fit, Mia will weigh about 124.8 pounds and be about
104.7 inches tall.
• Is it reasonable for the line of
5. How do these predictions compare to the predictions in Problems 1 and 2? Are these
best fit to be accurate on a
limited domain but not for the
set of all real numbers?
predictions reasonable? Explain your reasoning.
• How can you calculate the
line of best fit?
6. Do you think that extending the lines of best fit for Mia’s weight and height over time
made sense to make predictions about her weight and height beyond 6 years?
• How can you use a line of
No. The rate of Mia’s height and weight will change over time, but not at a
best fit to predict future
values?
constant rate.
Be prepared to share your solutions and methods.
852 • Chapter 16 Lines of Best Fit
852 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
but the height is still unreasonable.
© 2011 Carnegie Learning
The weight seems to be more reasonable than the prediction in Problems 1 and 2,
Follow Up
Assignment
Use the Assignment for Lesson 16.2 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 16.2 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 16.
Check for Students’ Understanding
1. Use the given data to create a scatter plot.
x
y
3
7
4
6
5
6
8
2
9
3
y
9
8
7
6
5
© 2011 Carnegie Learning
4
3
2
1
0
1
2
3
4
5
6
7
8
9
x
2. Draw a line of best fit on the scatter plot.
3. Determine the equation for the line of best fit.
Answers will vary.
The equation should be close to the equation y 5 20.79x 1 9.39.
16.2 Analyzing the Line of Best Fit • 852A
© 2011 Carnegie Learning
852B • Chapter 16 Lines of Best Fit
Stroop Test
Performing an Experiment
Learning Goals
In this lesson, you will:
 Perform an experiment.
 Write and use the equations of lines of best fit.
 Compare results of an experiment.
Essential Ideas
• A Stroop Test studies a person’s perception of words
and colors by using lists of color words that are
written in colors.
• A scatter plot is a graph of data points.
• A line of best fit is a straight line that is as close to
as many points as possible, but does not have to go
through all the points on a scatter plot.
• Equations can be written for a line of best fit.
• A line of best fit can be used to make predictions
© 2011 Carnegie Learning
about data.
Common Core State Standards for
Mathematics
1. Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of
association between two quantities. Describe
patterns such as clustering, outliers, positive
or negative association, linear association, and
nonlinear association.
2. Know that straight lines are widely used to model
relationships between two quantitative variables.
For scatter plots that suggest a linear association,
informally fit a straight line, and informally assess
the model fit by judging the closeness of the data
points to the line.
3. Use the equation of a linear model to solve
problems in the context of bivariate measurement
data, interpreting the slope and intercept.
8.SP Statistics and Probability
Investigate patterns of association in bivariate data.
Materials
Matching Lists of different lengths
Non-matching Lists of different lengths
Stopwatches (one for each time keeper)
16.3 Performing an Experiment • 853A
Overview
The Stroop Test studies a person’s perception of words and colors by using lists of color words (red,
green, black, and blue) that are written in one of the four colors. Students conduct the Stroop Test
experiment to gather data. They will calculate the mean time for various matching and non-matching
lists of words and create scatter plots of the list length versus the amount of time. Students then draw
the line of best fit for each scatter plot and make predictions for the amount of time based on the list
length using the equations of the lines of best fit.
Materials
Lists of words using different color inks will need to be prepared before this lesson is used. Several
lists will need to be made to conduct this test for each group of students performing the experiment.
Each list should be different lengths. Half of the lists should have the name of the color matching the
color of ink in which it is printed and the other half of the lists should have the name of the color not
matching the color of the ink in which it is printed.
© 2011 Carnegie Learning
Also, more than one stop watch or watches with a second hand will be needed for each group.
853B • Chapter 16 Lines of Best Fit
Warm Up
1. Use the given data to create a scatter plot.
x
y
5
0
4
2
3
3
2
5
9
3
y
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
x
2. Draw a line of best fit on the scatter plot.
3. Determine the equation for the line of best fit.
© 2011 Carnegie Learning
Answers will vary.
The equation should be close to the equation y 5 20.20x 1 3.51.
16.3 Performing an Experiment • 853C
© 2011 Carnegie Learning
853D • Chapter 16 Lines of Best Fit
Stroop Test
Performing an Experiment
Learning Goals
In this lesson, you will:
 Perform an experiment.
 Write and use the equations of lines of best fit.
 Compare results of an experiment.
T
he purpose of cognitive psychology is to understand and explain how the human brain works. One way to study how the brain works is to have human subjects perform experiments. The Stroop Test is one such experiment. The Stroop Test studies a person’s perception of words and colors by using lists of color words (red, green, black, and blue) that are written in one of the four colors. A person who participates in the Stroop Test receives one of two lists, a matching list or a non-matching list, with a varying number of words. In a matching list, the ink color matches the color of the word. In a non-matching list, the ink color does not match the color of the word.
© 2011 Carnegie Learning
© 2011 Carnegie Learning
The person participating in the experiment is given either kind of list. The person says aloud the ink color in which each word is written. The time it takes for the person to say the ink color for the list and the number of words in the list are recorded. The experiment is repeatedly performed with different people until enough data are collected to make a conclusion about the experiment. Why do you think a Stroop Test is important to study? Why do you think scientists and psychologists would be interested in the results?
16.3 Performing an Experiment • 853
16.3 Performing an Experiment • 853
Problem 1
Students gather data and
analyze the data for the Stroop
Test; the data is usually quite
linear. They create scatter plots
of the data, and determine the
line of best fit for each situation.
Students will then use their
equations to make predictions
and consider the accuracy of
their line of best fit.
Problem 1
Running a Stroop Test
In this lesson, you will perform the Stroop Test and calculate a line of best fit to make
predictions.
1. Before you perform this experiment, what results would you expect to see for either
the matching lists or non-matching lists? How do you think the results for the
matching lists will compare to the non-matching lists?
Please note: the answers supplied for the experiment are meant to be a sample.
As either list gets longer, the time increases.
It will probably take longer to say the ink color for words in the non-matching lists
than it will take to say the ink color for words in the matching lists.
Materials
Matching Lists
Non-Matching Lists
2. Identify the independent variable and the dependent variable in this problem situation.
The list length (number of words) is the independent variable.
Stopwatches
The amount of time to say the list aloud is the dependent variable.
Grouping
Have students complete
Questions 1 and 2 with a
partner. Then share the
responses as a class.
© 2011 Carnegie Learning
Share Phase,
Questions 1 and 2
• What will this experiment
measure?
• Do you think that the data
that you gather will be
somewhat linear?
will be a different amount of
time for the non-matching
lists than for the matching
lists?
• Why will you need to use
854 • Chapter 16 Lines of Best Fit
more than one timer?
• Why do you suppose the
colors red, green, black, and
blue were chosen to conduct
this test?
• What are other colors that
could have been used?
854 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
• Why do you think that there
Grouping
• Ask a student to read
Question 3 aloud. Discuss
the steps to completing the
experiment as a class.
3. Perform a Stroop Test and record your data in the tables shown. Make sure to
conduct three trials of the Stroop Test for each matching and non-matching list. Then,
vary the test by lengthening or shortening the number of words in the matching and
• Have students complete the
non-matching lists. Be sure to record the matching or non-matching list’s data in the
experiment and Questions 3
and 4 as directed.
correct table. You will complete the last column of the table in Question 4.
The data values listed are meant to be sample data. Your students’ responses
will vary.
Matching Lists
© 2011 Carnegie Learning
© 2011 Carnegie Learning
A trial is the
number of times
you conduct an
experiment.
List length
(words)
Time 1
(seconds)
Time 2
(seconds)
Time 3
(seconds)
Mean Time
(seconds)
10
7.20
7.22
7.13
7.18
15
9.59
9.22
9.43
9.41
6
3.35
3.22
3.43
3.33
12
5.74
5.77
5.79
5.77
20
14.34
13.92
13.56
13.94
26
12.12
12.20
12.01
12.11
13
5.63
5.75
5.60
5.66
8
3.37
3.06
3.33
3.25
11
4.00
4.40
4.20
4.20
5
3.20
3.28
3.36
3.28
Why should you
record three trials for
each time you vary the length
of the Stroop Test?
16.3 Performing an Experiment • 855
16.3 Performing an Experiment • 855
Share Phase,
Question 4
How did you determine the
mean time in seconds for
each list?
Non-Matching Lists
List length
(words)
Time 1
(seconds)
Time 2
(seconds)
Time 3
(seconds)
Mean Time
(seconds)
8
6.85
6.78
6.49
6.71
13
9.76
9.87
11.09
10.24
12
9.91
9.41
9.20
9.51
6
6.70
6.68
6.53
6.64
10
8.12
8.50
8.27
8.30
26
21.31
21.85
21.90
21.69
20
16.72
16.70
14.60
16.01
7
8.49
7.63
7.12
7.75
15
10.01
9.99
10.03
10.01
11
4.90
4.94
4.98
4.94
4. In the fifth column of each table, record the mean time in seconds for each list. Round
856 • Chapter 16 Lines of Best Fit
856 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning
your answers to the nearest hundredth.
Grouping
Have students complete
Questions 5 through 10 with
a partner. Then share the
responses as a class.
5. Write the ordered pairs from the matching lists table that show the average time as
the dependent variable, and the average list length (number of words) as the
independent variable.
The ordered pairs for the matching list are (10, 7.18), (15, 9.41), (6, 3.33), (12, 5.77),
(20, 13.94), (26, 12.11), (13, 5.66), (8, 3.25), (11, 4.2), and (5, 3.28).
Share Phase,
Questions 5 through 8
• What two points did you use
to determine the slope of the
lines of best fit?
6. Create a scatter plot of the ordered pairs on the grid shown. First, label the axes to
represent the independent and dependent variables. Next, choose the appropriate
intervals for your scatter plot. Finally, name your scatter plot.
y
Matching Lists
22.5
21.0
• What is the y-intercept for
19.5
18.0
the line of best fit?
16.5
Time (seconds)
• What is the equation of the
line of best fit?
• What are the units for the
y-intercept in your line of
best fit?
15.0
13.5
12.0
10.5
9.0
7.5
6.0
4.5
3.0
• What does the x-value
1.5
0.0
represent in your line of
best fit?
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
x
List Length (words)
7. Use a ruler to draw the line of best fit. Then, write the equation of your line.
• What are the units for the
The graph of the line should be close to y 5 0.54x 1 0.06.
x-value?
© 2011 Carnegie Learning
represent in your line of
best fit?
© 2011 Carnegie Learning
• What does the y-value
8. State the y-intercept of your line. What does the y-intercept represent in
this situation?
The y-intercept is 0.06. It represents the amount of time it takes to read 0 words.
16.3 Performing an Experiment • 857
16.3 Performing an Experiment • 857
Share Phase,
Question 9
9. State the slope of your line. What does the slope represent in this situation?
What are the units for the slope
in your line of best fit?
The slope is 0.54. It represents the amount of time it takes to say the ink color for
a new word.
Share Phase,
Question 10
• How can you use your line of
10. Use your equation to answer each question.
a. About how many seconds should it take a person to say the ink color of a
matching list of 25 words?
best fit to predict an amount
of time if given a matching
list length?
y 5 0.54(25) 1 0.06 5 13.56
It should take a person about 13.56 seconds to say a matching list of 25 words.
• How can you use your line of
best fit to predict a matching
list length if given an amount
of time?
b. About how many seconds should it take a person to say the ink color of a
matching list of 10 words?
y 5 0.54(10) 1 0.06 5 5.46
It should take a person about 5.46 seconds to say a matching list of 10 words.
c. About how many words should a person be able to say the ink color from a
matching list in 2 minutes?
120 5 0.54x 1 0.06
119.94 5 0.54x
222.11 ¯ x
matching list in 2 minutes.
d. About how many words should a person be able to say the ink color from a
matching list in 5 minutes?
300 5 0.54x 1 0.06
© 2011 Carnegie Learning
A person should be able to say the ink color for approximately 222 words from a
299.94 5 0.54x
A person should be able to say the ink color for approximately 555 words from a
matching list in 5 minutes.
858 • Chapter 16 Lines of Best Fit
858 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
555.44 ¯ x
Grouping
Have students complete
Questions 11 through 16 with
a partner. Then share the
responses as a class.
11. Write the ordered pairs from the non-matching lists table that show the average time
as the dependent variable, and the list length as the independent variable.
The ordered pairs for the non-matching list are (8, 6.71), (13, 10.24), (12, 9.51),
(6, 6.64), (10, 8.3), (26, 21.69), (20, 16.01), (7, 7.75), (15, 10.01), and (11, 4.94).
Share Phase,
Questions 11
through 15
• What is a line of best fit?
• Why were the times for the
12. Create a scatter plot of the ordered pairs for the non-matching lists on the grid. First,
label the axes to represent the independent and dependent variables. Next, choose
the appropriate intervals for your scatter plot. Finally, name your scatter plot.
Non-Matching Lists
y
30
28
26
non-matching lists longer
than for the matching lists?
24
Time (seconds)
22
• What may have made the
results less accurate than
they could have been?
20
18
16
14
12
10
8
• How can you calculate a line
6
of best fit?
4
2
0
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30
x
List Length (words)
13. Use a ruler to draw the line of best fit. Then, write the equation of your line.
© 2011 Carnegie Learning
The graph of the line should be close to y 5 0.76x 1 0.42.
14. State the y-intercept of your line. What does the y-intercept represent in
this situation?
The y-intercept is 0.42. It represents the amount of time it takes to say the
ink color for 0 words.
15. State the slope of your line. What does the slope represent in this situation?
© 2011 Carnegie Learning
Sample Answer: The slope is 0.76. It represents the amount of time it takes to say
the ink color for a new word from the non-matching list.
16.3 Performing an Experiment • 859
16.3 Performing an Experiment • 859
Share Phase,
Question 16
16. Use your equation to answer each question.
How can you use a line of best
fit to predict future values?
a. About how many seconds should it take a person to say the color of the ink for a
non-matching list of 25 words?
y 5 0.76(25) 1 0.42 5 19.42
It should take a person about 19.42 seconds to say a non-matching list of
Note
25 words.
The difference between the
slopes in the lines of best fit is
the average amount of time for
the students’ brains to override
their impulse to read a word.
The unit for the difference is
seconds per word.
b. About how many seconds should it take a person to say the color of the ink for a
non-matching list of 10 words?
y 5 0.76(10) 1 0.42 5 8.02
It should take a person about 8.02 seconds to say a non-matching list of
10 words.
c. About how many words should a person be able to say the ink color from a
non-matching list in 2 minutes?
120 5 0.76x 1 0.42
119.58 5 0.76x
157.34 ¯ x
A person should be able to say the ink color for a non-matching list for about
d. About how many words should a person be able to say the ink color from a
non-matching list in 5 minutes?
300 5 0.76x 1 0.42
299.58 5 0.76x
394.18 ¯ x
A person should be able to say the ink color of a non-matching list for about
© 2011 Carnegie Learning
157 words from a non-matching list in 2 minutes.
860 • Chapter 16 Lines of Best Fit
860 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
394 words from a non-matching list in 5 minutes.
Talk the Talk
Students answer questions
related to the data gathered
from performing the experiment.
Talk the Talk
1. Compare your results for the matching lists to the results for the non-matching lists.
Do your results seem reasonable? Explain your reasoning.
Grouping
According to the models, a person takes more time to say the ink color of a
Have students complete
Questions 1 through 3 with
a partner. Then share the
responses as a class.
non-matching list than a matching list when the lists are the same length.
This seems reasonable because it should take longer to say the ink color that
does not match the written word.
2. Are the results what you expected? Explain your reasoning.
No. I thought there would be a greater difference in time between matching and
non-matching lists of the same length.
3. What conclusions do you think a cognitive psychologist might draw from your
experiment results?
The psychologist might conclude that the ink color and the written color word are
pieces of information that can conflict in a person’s mind. The psychologist might
© 2011 Carnegie Learning
also conclude that if a person can say the ink color of either list in approximately
the same amount of time, the person’s brain can easily block certain information.
© 2011 Carnegie Learning
Be prepared to share your solutions and methods.
16.3 Performing an Experiment • 861
16.3 Performing an Experiment • 861
Follow Up
Assignment
Use the Assignment for Lesson 16.3 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 16.3 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 16.
Check for Students’ Understanding
Two experiments are conducted to compare how long it takes inkjet printers to print in black-andwhite and how long it takes them to print in color. The number of pages printed using black-and-white
can be expressed by the line of best fit pb 5 33.8t 1 5.3, and the number of pages printed using color
can be expressed by the line of best fit pc 5 21t 1 2.7, where p is the total number of pages printed,
and t is the time in minutes. If you only have 15 minutes to use an inkjet printer, how many more
black-and-white pages could you print than color pages?
Pages printed in black-and-white:
p 5 33.8(15) 1 5.3 5 507 1 5.3 5 512.3
Pages printed in color:
p 5 21(15) 1 2.7 5 315 1 2.7 5 317.7
© 2011 Carnegie Learning
You can print 512.3 2 317.7 5 194.6 (approximately 195) more black-and-white pages in
15 minutes than color pages.
862 • Chapter 16 Lines of Best Fit
Human Chain:
Shoulder Experiment
Using Technology to Determine
a Linear Regression Equation
Learning Goals
Key Terms
In this lesson, you will:
 linear regression
 linear regression equation
 Perform an experiment.
 Use technology to determine a linear
regression equation.
 Use a linear regression equation to
predict results.
Essential Ideas
• Technology can be used to calculate the line of
best fit.
8.SP Statistics and Probability
• The linear regression equation is the equation used
by a calculator or spreadsheet program to find the
line of best fit.
• The least squares method is the method used by a
calculator or spreadsheet program to find the line of
best fit.
• The correlation coefficient indicates how close the
data are to forming a straight line.
• A linear regression equation can be used to predict
© 2011 Carnegie Learning
the results of an experiment.
Materials
Stopwatch
Graphing Calculators
Common Core State Standards for
Mathematics
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of
association between two quantities. Describe
patterns such as clustering, outliers, positive
or negative association, linear association, and
nonlinear association.
2. Know that straight lines are widely used to model
relationships between two quantitative variables.
For scatter plots that suggest a linear association,
informally fit a straight line, and informally assess
the model fit by judging the closeness of the data
points to the line.
3. Use the equation of a linear model to solve
problems in the context of bivariate measurement
data, interpreting the slope and intercept.
16.4 Using Technology to Determine a Linear Regression Equation • 863A
Overview
The terms linear regression and linear regression equation are defined in this lesson. Students conduct
an experiment to gather data. They will then create a scatter plot of the amount of time compared
to the number of people in a chain. A calculator is used to calculate the linear regression equation.
Students then predict values for time and for chain length for given values using the linear
© 2011 Carnegie Learning
regression equation.
863B • Chapter 16 Lines of Best Fit
Warm Up
Plot the data in the table on the coordinate plane shown. Draw the line of best fit. Write the equation
for the line of best fit.
1.
x
y
4
1
5
3
2
4
1
3
8
2
y
4
3
2
1
0
1
2
3
4
5
6
7
8
9
x
Answers will vary but should be close to the equation y 5 20.20x 1 3.4.
© 2011 Carnegie Learning
2.
x
y
6
30
5
40
4
80
3
80
2
90
y
120
110
100
90
80
70
60
50
40
30
20
10
0
1
2
3
4
5
6
x
Answers will vary but should be close to the equation y 5 216x 1 128.
16.4 Using Technology to Determine a Linear Regression Equation • 863C
© 2011 Carnegie Learning
863D • Chapter 16 Lines of Best Fit
Human Chain:
Shoulder Experiment
Using Technology to Determine
a Linear Regression Equation
Learning Goals
Key Terms
In this lesson, you will:
 linear regression
 Perform an experiment.
 Use technology to determine a linear  linear regression equation
regression equation.
 Use a linear regression equation to predict results.
A
s you learned previously, distance can affect the nerve impulses that affect your central nervous system. But does distance affect how people and businesses operate? In the past, distance was a crucial factor on how businesses operated. For example, there was no such term as “telecommuting,” which is another way of describing when a person who works from a remote location not inside a company’s office. Distance was also a factor for professional sports teams not to expand to the West Coast until the mid 1950s. So, for most of the time, distance was a factor for many decisions, but just how long will distance remain being a © 2011 Carnegie Learning
© 2011 Carnegie Learning
factor in business decisions with the technological advances?
16.4 Using Technology to Determine a Linear Regression Equation • 863
16.4 Using Technology to Determine a Linear Regression Equation • 863
Problem 1
Students gather another set of
data to analyze and model with
a linear regression equation.
They will create a scatter
plot and calculate the linear
regression equation using
technology. Students then
analyze their linear regression
equations and use their linear
regression equation to predict
values for times given various
lengths of chains.
Problem 1
A Chain Connected at the Shoulder
Previously, your class performed an experiment to measure the speed of your nerve
impulses. Now, your class is going to perform an experiment to see if distance can affect
how quick your nerve impulses travel.
Like the previous experiment you and your classmates will form a circular chain. This time,
each student will gently hold the shoulder of the student to his or her right. One student
begins the chain, and another student ends the chain. Also, one student must be the
timekeeper. This experiment is very similar to the wrist experiment. The group members
must keep their eyes closed. To begin, the timekeeper says, “Go,” and the first student
carefully but quickly squeezes the next student’s shoulder. This next student squeezes a
shoulder, and so on. Once the last student’s shoulder is squeezed, he or she says, “Stop,”
and lets go of the next student’s shoulder. The amount of time it takes to complete the
chain is recorded by the timekeeper.
Materials
1. How do you think this experiment’s results will be different from the results in the
wrist experiment?
Stopwatch
Please note the answers supplied for the experiment
Graphing Calculator
are meant to be a sample—your students’ answers
will vary from the sample answers provided within
this lesson.
Grouping
It should take less time because the distance
from a person’s wrist on one hand to his or her
Ask a student to read the
information before Question 1
aloud. Discuss the experiment
and complete Questions 1 and
2 as a class.
other hand is greater than the distance from a
equation in this human chain
shoulder experiment different
from the linear regression
equation that you calculated
in the human chain wrist
experiment?
equation in this human chain
shoulder experiment similar
to the linear regression
equation that you calculated
in the human chain wrist
experiment?
864 • Chapter 16 Lines of Best Fit
864 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning
person’s shoulder to the hand on the other arm.
Share Phase,
Question 1
• How is the linear regression
• How is the linear regression
Remember, to
accurately
perform the experiment,
you should use a
stopwatch.
2. Perform the experiment 10 times, using different chain lengths each time. Perform
3 trials for each chain that involves a different number of students. Record the data in
the table. Then, determine the mean time of each row and record the results in the
Grouping
© 2011 Carnegie Learning
Have students complete
Questions 3 through 5 with
a partner. Then share the
responses as a class.
© 2011 Carnegie Learning
table. Round your averages to the nearest hundredth, if necessary.
Chain Length
(students)
Time 1
(seconds)
Time 2
(seconds)
Time 3
(seconds)
Mean Time
(seconds)
15
4.79
4.10
4.11
4.33
10
2.60
2.41
2.23
2.41
5
1.41
1.28
1.18
1.29
30
6.45
6.84
7.01
6.77
45
10.95
11.23
11.07
11.08
8
2.16
2.40
1.96
2.17
20
4.72
4.54
4.63
4.63
27
6.03
6.24
6.26
6.18
18
4.47
4.29
4.32
4.36
10
2.23
2.15
2.16
2.18
3. Write the ordered pairs from the table with the chain length as the independent
variable, and the mean time as the dependent variable.
The ordered pairs are (15, 4.33), (10, 2.41), (5, 1.29), (30, 6.77), (45, 11.08), (8, 2.17),
(20, 4.63), (27, 6.18), (20, 4.36), and (10, 2.18).
16.4 Using Technology to Determine a Linear Regression Equation • 865
16.4 Using Technology to Determine a Linear Regression Equation • 865
Share Phase,
Questions 4 and 5
• What is the meaning and
4. Create a scatter plot of the ordered pairs on the grid shown. Do not forget to label
each axis and name your scatter plot.
what are the units of the
y-value in your linear
regression equation from
the wrist experiment? What
about from the shoulder
experiment?
y
30
Shoulder Experiment
28
26
24
Time (seconds)
22
• What is the meaning and
what are the units of the
x- value in your linear
regression equation from
the wrist experiment? What
about from the shoulder
experiment?
20
18
16
14
12
10
8
6
4
2
0
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
x
Chain Length (students)
5. Did you need to break your graph? Why or why not.
No. I did not need to break my graph because the data values spread out through
the graph and many data points occurred near the origin.
Grouping
So far, you have drawn the line of best fit for a data set. You have also determined the
equation of the line of best fit. Now, you will use a graphing calculator to determine a line of
Ask a student to read the
information following Question
5 aloud. Discuss the definitions
and complete the steps to using
a graphic calculator as a class.
best fit. The graphing calculator uses a method and an equation for a line of best fit called
the linear regression equation. Throughout this chapter, you have been performing linear
regression, which is to model the relationship of two variables in a data set by drawing a
866 • Chapter 16 Lines of Best Fit
866 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning
line of best fit. The linear regression equation is the equation for the line of best fit.
To determine the linear regression equation for the ordered pairs of a data set you must
first enter the data into your calculator.
Step 1: Press STAT . The EDIT , and 1: Edit selection should be highlighted
on your screen.
Step 2: Key in your data points from your experiment. Let the chain length data
values be entered into L1, and let the mean time data be entered in L2 for each
ordered pair.
L1
L2
45
8
20
27
18
10
---
11.08
2.17
4.63
6.18
4.36
2.18
L3
2
L2(11)=
When you have completed entering your data points in your calculator, you then need the
graphing calculator to plot the points in a scatter plot.
Step 3: Press 2nd and then Y5 . The Stat Plots and 1: Plot 1 should be
highlighted. Press ENTER and select the first type of graph.
Step 4: Next, press WINDOW . You must select the least and greatest distances along
© 2011 Carnegie Learning
© 2011 Carnegie Learning
the x- and y-axes. Select Xmin 5 0, Xmax 5 60 , and Xscl 5 5. Then, perform
the same steps for the y-axis. Finally, press GRAPH .
16.4 Using Technology to Determine a Linear Regression Equation • 867
16.4 Using Technology to Determine a Linear Regression Equation • 867
Next, you will use your graphing calculator to determine the
linear regression equation.
Step 1: Press STAT and use your arrow key to scroll to
CALC . Use your arrow key and scroll to 4:
LineReg(ax 1 b) and then press ENTER .
Step 2: Use lists L1 and L2 for your lists. Do this by
pressing 2nd and L1 and press , for the
To ensure
your calculator will display
the value of r, “``Diagnostics,
must be turned on. Press 2nd
and 0 to display Catalog. Scroll
to the Diagnostics On and then
press ENTER twice.
The calculator should display
the word Done.
x-coordinate data points. Then, put in the
y-coordinate data points by pressing 2nd and L2 .
LinReg
y = ax+b
a =.2362746199
b = 0.978491456
r2 =.9868599065
r =.9934082275
To determine the linear regression equation, simply substitute the values for the
terms a and b in the equation.
On your screen, you should also see a value for the variable r. The variable, r, is
used to represent the correlation coefficient. The correlation coefficient shows
how close the data points are to forming a straight line.
If the data set has a positive correlation, then r has a value between 0 and 1. The
closer the data is to forming a straight line, the closer the value of r is to 1.
If the data has a negative correlation, then r has a value between 0 and 21. The
closer the data is to forming a straight line, the closer the value of r is to 21.
Grouping
6. Use a graphing calculator to determine the linear regression equation, and correlation
coefficient for the data. If necessary, round the values a and b to the nearest
Have students complete
Questions 6 through 10 with
a partner. Then share the
responses as a class.
hundredth. Then, graph the line on your coordinate plane in Question 4.
The linear regression equation is y 5 0.24x 1 0.06.
868 • Chapter 16 Lines of Best Fit
868 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
As a shortcut, you can say: 21 # r # 1
© 2011 Carnegie Learning
If there is no linear relationship in the data, the value of r is 0.
Share Phase,
Questions 7 through 9
• What is the meaning
and what are the units of
the slope in your linear
regression equation from
the wrist experiment? What
about from the shoulder
experiment?
7. How close are the data to forming a straight line? Explain how you determined
your answer.
The data are very close to forming a straight line because the r-value is
approximately 0.992.
• What is the meaning and
8. What is the slope of the linear regression equation? What does the slope mean in the
problem situation?
what are the units of the
y-intercept in your linear
regression equation from
the wrist experiment? What
about from the shoulder
experiment?
The slope is 0.24.
The slope indicates the number of seconds added to the time for each student
added to the chain.
9. Use your linear regression equation and your calculator to determine the number of
seconds it should take to perform the experiment if:
• What is the correlation
a. 100 people are in the chain.
coefficient for your linear
regression equation for
the wrists experiment?
What about the shoulder
experiment? Why would they
be different?
y 5 0.24(100) 1 0.06 5 24.06
It should take about 24 seconds.
b. 50 people are in the chain.
Do you
remember how
to convert different
units of measure to
the same measure?
You will need to do
this to complete
this task.
• How can you use the linear
© 2011 Carnegie Learning
regression equation to
predict the time for a given
chain length?
y 5 0.24(50) 1 0.06 5 12.06
It should take about 12 seconds.
c. 10,000 people are in the chain.
y 5 0.24(10,000) 1 0.06 5 2400.06
It should take about 2400 seconds.
d. 6.9 billion people (the world’s population) are in the chain.
y 5 0.24(6,900,000,000) 1 0.06 5 1,656,000,000.1
It should take about 1,656,000,000 seconds.
e. Write your answer to part (d) in years.
© 2011 Carnegie Learning
1 day _________
1 yr
1 hr 3 ______
1 min 3 _______
3
¯ 52.51 years
1,656,000,000 sec 3 _______
60 sec 60 min
24 hr
365 days
It should take about 53 years.
16.4 Using Technology to Determine a Linear Regression Equation • 869
16.4 Using Technology to Determine a Linear Regression Equation • 869
Share Phase,
Question 10
• How can you use the linear
10. Use your linear regression equation to determine the length of the chain if it takes:
a. 1 hour to complete the chain.
regression equation to
predict the chain length for a
given time?
60 min 3 _______
60 sec 5 3600 sec
1 hr 3 _______
1 hr
1 min
3600 5 0.24x1 0.06
3599.94 5 0.24x
• How accurate will the
14,999.75 ¯ x
results be if you use the
linear regression equation to
predict values?
There were about 15,000 people in the chain.
• How do you measure the
accuracy of the line?
b. 1 day to complete the chain.
60 min 3 _______
60 sec 5 86,400 sec
24 hr 3 _______
1 day 3 ______
1 day
1 hr
1 min
86,400 5 0.24x 1 0.06
86,399.94 5 0.24x
359,999.75 ¯ x
There were about 360,000 people in the chain.
c. 1 year to complete the chain.
365 days
60 min 3 _______
60 sec 5 31,536,000 sec
24 hr 3 _______
1 yr 3 _________ 3 ______
1 yr
1 day
1 hr
1 min
31,536,000 5 0.24x 1 0.06
870 • Chapter 16 Lines of Best Fit
870 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
131,399,999.8 ¯ x
There were about 131,400,000 people in the chain.
© 2011 Carnegie Learning
31,535,999.94 5 0.24x
Talk the Talk
Students enter the data from
the wrist experiment in a
previous lesson into a graphing
calculator to determine the
linear regression equation.
They will then graph the linear
regression equation from
the wrist experiment and the
shoulder experiment on the
same coordinate plane and
compare the slopes.
Talk the Talk
1. How does this experiment differ from the wrist experiment you previously conducted?
In this experiment, I might think that the nerve impulses do not travel as far, so
they are quicker than the wrist experiment.
2. Enter the data from the wrist experiment into your calculator.
a. Use your calculator to determine the liner regression equation for the
wrist experiment.
b. Graph the equation on the grid shown.
c. Graph the equation for the shoulder experiment on the grid shown.
Grouping
y
30
Have students complete
Questions 1 through 3 with
a partner. Then share the
responses as a class.
Wrist and Shoulder Experiment
Title your
graph and label
your axes.
28
26
24
Time (seconds)
22
20
18
16
Wrist Experiment
14
12
10
8
6
4
Shoulder Experiment
2
0
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
x
Chain Length (students)
3. What do the slopes in each experiment (wrist and shoulder) represent?
© 2011 Carnegie Learning
© 2011 Carnegie Learning
What do you think is the cause of the difference between the slopes?
The slope indicates the number of seconds added to the time for each
person added to the chain. The difference is caused by the distance
nerve impulses have to travel in each experiment. The distance from a
person’s wrist on one hand to his or her other hand is greater than the
distance from a person’s shoulder to the hand on the other arm. So,
the number of seconds per person is less in the second situation.
Be prepared to share your solutions and methods.
16.4 Using Technology to Determine a Linear Regression Equation • 871
16.4 Using Technology to Determine a Linear Regression Equation • 871
Follow Up
Assignment
Use the Assignment for Lesson 16.4 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 16.4 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 16.
Check for Students’ Understanding
Use a graphing calculator and the data from each table to calculate the linear regression equation.
Round any answers to the nearest hundredth where necessary. Sketch the screen shot that contains
the scatter plot and the regression equation.
1.
x
y
4
1
5
3
2
4
1
3
8
2
y
4
3
1
0
1
2
3
4
5
6
7
8
9
x
Answers will vary but should be close to the equation y 5 20.20x 1 3.4.
872 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
2
2.
x
y
6
30
5
40
4
80
3
80
2
90
y
120
110
100
90
80
70
60
50
40
30
20
10
0
1
2
3
4
5
6
x
Answers will vary but should be close to the equation y 5 216x 1 128.
3. Compare the equations you wrote in Questions 1 and 2 above to the equations you wrote in the
warm up activity in this lesson. Are they close?
© 2011 Carnegie Learning
The regression equations in this activity are more accurate.
16.4 Using Technology to Determine a Linear Regression Equation • 872A
© 2011 Carnegie Learning
872B • Chapter 16 Lines of Best Fit
Jumping
Correlation
Learning Goals
In this lesson, you will:




Perform an experiment.
Draw a line of best fit.
Write and use an equation of a line of best fit.
Determine whether data are positively correlated, negatively correlated, or not correlated.
Essential Ideas
• A line of best fit is a straight line that is as close to
as many points as possible, but does not have to go
through all the points on a scatter plot.
• Equations can be written for a line of best fit.
• A line of best fit can be used to make predictions
about data.
• There is a correlation between the x- and y- values
when it is appropriate to use a line of best fit to
approximate a collection of points.
• The points are positively correlated when the line of
© 2011 Carnegie Learning
best fit has a positive slope.
• The points are negatively correlated when the line of
best fit has a negative slope.
Materials
Measuring Tape
Graphing Calculator
Common Core State Standards for
Mathematics
8.SP Statistics and Probability
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of
association between two quantities. Describe
patterns such as clustering, outliers, positive
or negative association, linear association, and
nonlinear association.
2. Know that straight lines are widely used to model
relationships between two quantitative variables.
For scatter plots that suggest a linear association,
informally fit a straight line, and informally assess
the model fit by judging the closeness of the data
points to the line.
3. Use the equation of a linear model to solve
problems in the context of bivariate measurement
data, interpreting the slope and intercept.
16.5 Correlation • 873A
Overview
An experiment is used to explore the relationship between the height of a student and the height
which a student can jump. Students conduct an experiment to gather data, record the data in a table,
use the table to write ordered pairs, and use the ordered pairs to graph the scatter plot. A graphing
calculator is used to determine the line of best fit and students draw the line of best fit on the scatter
plot. In the second problem, the term correlation is introduced and lines with positive correlations,
negative correlations, and no correlation are described. Using the r-value, or correlation coefficient,
© 2011 Carnegie Learning
students identify the most accurate correlation coefficient for various scatter plots given.
873B • Chapter 16 Lines of Best Fit
Warm Up
Sketch a scatter plot that matches the given description.
1. Data that may have a linear regression with a positive slope.
y
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
x
2. Data that may have a linear regression with a negative slope.
y
7
6
5
4
3
2
1
1
2
3
4
5
6
7
x
© 2011 Carnegie Learning
0
16.5 Correlation • 873C
© 2011 Carnegie Learning
873D • Chapter 16 Lines of Best Fit
Jumping
Correlation
Learning Goals
In this lesson, you will:
 Perform an experiment.
 Draw a line of best fit.  Write and use an equation of a line of best fit.
 Determine whether data are positively correlated, negatively correlated, or not correlated.
J
umping is probably something you do during sports like basketball or during track-and-field events like the long jump. However, imagine if jumping was your usual way of travel! Well for certain creatures, jumping is as natural for them as walking is to humans. Crickets and grasshoppers jump to get from different blades of grass or other vegetation to another location—and jumping is important to get away from predators. Of course frogs and jumping spiders jump to attack prey for their next meals! And still, one of the largest leapers in the animal kingdom is the kangaroo. Native to Australia, kangaroos can leap for miles! Even though these leapers are both insects, arachnids, reptiles, and mammals, what special bodily © 2011 Carnegie Learning
© 2011 Carnegie Learning
feature do they each share?
16.5 Correlation • 873
16.5 Correlation • 873
Problem 1
Problem 1
How High Can You Jump?
There is a debate whether a person’s height affects how high a person can jump.
In this lesson, you will conduct an experiment, collect, and analyze the data to
determine the answer.
1. Create a table that you can use for your experiment. The table should include columns
for the person’s name, the person’s height in inches, and the height jumped in inches.
Please note the answers supplied for the experiment students will perform are
meant to be a sample.
Measuring Tape
Graphing Calculator
Grouping
Ask a student to read the
introduction to Problem 1 aloud.
Discuss the directions and
complete Questions 1 through 3
as a class
Discuss Phase,
Introduction
• What will this experiment
Person Height
(inches)
Jump Height
(inches)
Student A
67
13
Student B
65
9.5
Student C
71
7.5
Student D
67
10.5
Student E
62
12
Student F
68
9
Student G
60
5
Student H
75
20
Student I
73
19.5
Student J
72
18
measure?
• How can you measure the
heights that the students
jump?
• Which suggestion will give
the most accurate way
to measure the heights
jumped?
874 • Chapter 16 Lines of Best Fit
• Do you think the data that you gather will be somewhat linear?
• What units will you use to measure the heights of the students? What units will
you use to measure the heights jumped?
• How accurate will you be able to measure the heights jumped?
• Do you feel that the height that a person can jump would depend on the height
of the person? If so, how do you think that they are related?
874 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
Materials
Name
How will you
measure the
height each person
can jump?
© 2011 Carnegie Learning
In this experiment, students
collect data for the heights
jumped by ten students in
the class. They enter the data
collected in a table, identify the
independent and dependent
variables, and write the ordered
pairs from the data in the table.
Line breaks are used when
scaling the x-axis and students
create a scatter plot using the
ordered pairs. Using a graphing
calculator, they will determine
and draw the line of best fit
and interpret the regression
equation with respect to the
problem situation.
2. Perform the experiment with your classmates and record the results in the table.
Describe how you measured the height each person jumped.
Answers will vary.
3. Identify the independent and dependent variables in this problem situation.
The independent variable is the person’s height. The dependent variable is the
person’s jump height.
4. Write the ordered pairs from your table.
Grouping
The ordered pairs are (67, 13), (65, 9.5), (71, 7.5), (67, 10.5), (62, 12), (68, 9), (60, 5),
Have students complete
Questions 4 through 6 with
a partner. Then share the
responses as a class.
© 2011 Carnegie Learning
© 2011 Carnegie Learning
(75, 20), (73, 19.5), and (72, 18).
16.5 Correlation • 875
16.5 Correlation • 875
Share Phase,
Worked Example
• When is it appropriate to
create a break in the scale of
a graph?
When data points are clustered together but are far from the origin, it can be hard
to draw a line of best fit. This is especially true if the portion of the graph from the
• Would it be appropriate to
origin is included in the graph. To help you draw a line of best fit, you can break
break both the horizontal
scale and the vertical scale
of the same graph?
the graph and show the portion where the data appear. In fact, there have been
many occurrences where the graph has been broken in previous lessons.
Consider the two graphs shown that display the same data.
• Can you think of an
example where it would be
appropriate to break both
the horizontal scale and the
vertical scale of the same
graph?
• Does using a break in the
scale of the graph appear to
spread the data out? Explain.
y
30
27
24
When you
break a graph, draw a
squiggly line from the
origin to the first
interval on the x-axis.
It actually looks like
you did BREAK the
graph!
21
18
15
12
9
6
3
0
0
15 30 45 60 75 90 105 120 135 150
x
y
30
27
24
21
18
15
This symbol
indicates a break
in the graph.
12
9
6
0
90 96 102 108 114 120 126 132 138 144
x
The data values all occur greater than 90 and less than 150. By breaking the graph,
you can make it easier to draw a line of best fit. Therefore you can show a break in
the scale for the x-axis so that the portion of the graph containing the data appears.
876 • Chapter 16 Lines of Best Fit
876 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
0
© 2011 Carnegie Learning
3
Share Phase,
Questions 5 and 6
Is it easier to draw a line of best
fit with or without the break in
the scale of the graph? Explain.
5. What is/are the advantage(s) of using a graph with a break in it?
The advantages are that I can see all the data, easily draw a line of best fit, and
determine the slope of the line.
6. What is/are the disadvantage(s) of using a graph with a break in it?
One disadvantage is that I cannot see the y-intercept.
Grouping
7. Create a scatter plot of the data you collected. For this
Have students complete
Questions 7 through 11 with
a partner. Then share the
responses as a class.
scatter plot, make sure you break the graph to make it
easier to draw a line of best fit for your data.
y
30
If you are
using your
graphing calculator, you
can adjust your Window
view to see the data
better.
A Person’s Height and Jumping Height
28
26
24
Jump Height (in.)
22
20
18
16
14
12
10
8
6
4
2
0
0 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
x
Person Height (in.)
8. Use your graphing calculator to determine the line of
best fit.
© 2011 Carnegie Learning
© 2011 Carnegie Learning
See graph. The graph of the line should be close to
y5 0.8x2 42.
9. Describe the slope of your line.
The slope is positive.
How can you
write the equation
of the line when
you can't see the
y -intercept?
16.5 Correlation • 877
16.5 Correlation • 877
Share Phase,
Questions 10 and 11
• What function of a graphing
10. What does your graph and line of best fit indicate about the relationship between a
person’s height and how high a person can jump?
calculator has to be turned
on to view the r-values
(correlation coefficient)?
The graph and line of fit seem to indicate that the taller a person is, the higher the
person can jump.
• What is the correlation
coefficient in your regression
equation? Is your regression
equation considered a
good fit?
11. What factors could cause an incorrect jump height measurement or an inconsistency
in the measurements? Describe situations in which these errors or inconsistencies
could occur.
Some inconsistencies could occur due to inconsistent measurement of jump
height, the types of shoes each person wears, the physical ability of each person,
and the type of clothing each person wears. For example, if one person was
wearing sandals and another person was wearing sneakers, the person wearing
sandals would probably not be able to jump as high as the person
wearing sneakers.
Problem 2
Correlation
In many situations, you will have to determine if it is appropriate to use a line of best fit to
approximate a collection of points. When it is appropriate, you say that there is a correlation
between the x- and y-values. Otherwise, you can say that there is no correlation. When the
line of best fit has a positive slope, the points are positively correlated. When the line of best
fit has a negative slope, the points are negatively correlated.
1. Describe the correlation of the points in your graph. Explain your reasoning.
The points are positively correlated because the line of best fit has a
positive slope.
878 • Chapter 16 Lines of Best Fit
Discuss Phase, Question 1
• What is the difference between a positive association and a positive
correlation?
Grouping
Ask a student to read the
introduction to Problem 2 aloud.
Discuss this information and
complete Question 1 as a class.
• What is the difference between a negative association and a negative correlation?
• What is the difference between no association and no correlation?
• How would you describe the slope of the line of best fit in the previous problem?
• How would you describe the correlation of a line of best fit with a positive slope?
878 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
In this problem, students are
informally introduced to the
concept of correlation and
develop an intuitive sense
of correlation. The concepts
of positive correlation,
negative correlation, and no
correlation are described.
Students will classify the type
of correlation that exists for
the data represented in given
scatter plots by choosing the
most appropriate correlation
coefficient.
© 2011 Carnegie Learning
Problem 2
Grouping
Have students complete
Question 2 with a partner. Then
share the responses as a class.
2. Determine whether the points in each scatter plot have a positive correlation, a
negative correlation, or no correlation. Then determine which of the values of r you
think is most accurate. Explain why you chose your answer.
a.
y
8
Share Phase,
Question 2
• What does it mean for two
6
5
• What does it mean for two
quantities to be correlated?
b.
3
Since the data are negatively correlated,
2
the value of r must be negative. Also,
1
because the data are close to forming a
1
2
3
4
5
6
7
8
x
y
8
the most accurate.
●
r 5 0.7
●
r 5 20.7
●
r 5 0.07
●
r 5 20.07
7
6
5
4
The correct answer is r 5 0.7.
3
Since the data are positively correlated,
2
the value of r must be positive.
1
Also, because the data are fairly close
to forming a straight line, a value of 0.7
0
1
2
3
4
5
6
7
8
x
would be the most accurate.
Positive correlation
• You create a scatter plot of
© 2011 Carnegie Learning
r 5 20.09
Negative correlation
the x- values increase, on
the line of best fit, what type
of slope does the line have?
What type of correlation do
the data have?
another situation that
you think has a negative
correlation?
●
straight line, a value of 20.9 would be
• If the y- values decrease as
• What is an example of
r 5 0.09
The correct answer is r 5 20.9.
0
the x- values increase on the
line of best fit, what type of
slope does the line have?
What type of correlation do
the data have?
another situation that
you think has a positive
correlation?
r 5 20.9
●
4
• If the y- values increase as
• What is an example of
r 5 0.9
●
7
quantities to be related?
c.
© 2011 Carnegie Learning
data that shows the amount
of time since you made an
ice cream cone and the
amount of ice cream that
remains in the cone. What
type of correlation do the
data have?
●
y
8
●
r51
●
r 5 0.5
●
r 5 0.01
7
6
The correct answer is r 5 0.01.
5
Since there is not a linear relationship in
4
the data, the value of r is close to 0,
3
and 0.01 is close to 0.
2
1
0
1
2
3
4
5
6
7
8
x
No correlation
16.5 Correlation • 879
• What an example of a
situation that you think has
no correlation?
16.5 Correlation • 879
Talk the Talk
Students answer questions
about correlation and a line of
best fit.
Talk the Talk
1. Some lines of best fit model their data better than other lines of best fit. If a line of
Grouping
best fit models the data very well, what would you expect to see in a graph of the
data and the line?
Have students complete
Questions 1 and 2 with a
partner. Then share the
responses as a class.
The line is very close to all of the points.
2. Describe the graph of a collection of points that has no correlation.
Be prepared to share your solutions and methods.
880 • Chapter 16 Lines of Best Fit
880 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning
The data do not fall in a straight line.
Follow Up
Assignment
Use the Assignment for Lesson 16.5 in the Student Assignments book. See the Teacher’s Resources
and Assessments book for answers.
Skills Practice
Refer to the Skills Practice worksheet for Lesson 16.5 in the Student Assignments book for additional
resources. See the Teacher’s Resources and Assessments book for answers.
Assessment
See the Assessments provided in the Teacher’s Resources and Assessments book for Chapter 16.
Check for Students’ Understanding
Determine whether the points in each scatter plot have a positive correlation, a negative correlation, or
no correlation. Explain your answer.
1.
y
20
20
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
© 2011 Carnegie Learning
2.
y
2
4
6
8
10 12 14 16 18 20 x
0
2
4
6
8
10 12 14 16 18 20 x
The points in this scatter plot have a
The points in this scatter plot have a
negative correlation because the line of
positive correlation because the line of
best fit has a negative slope.
best fit has a positive slope.
16.5 Correlation • 880A
© 2011 Carnegie Learning
880B • Chapter 16 Lines of Best Fit
Chapter 16
Summary
Key Terms
     extrapolating (16.1)
 linear regression (16.4)
 linear regression equation (16.4)
line of best fit (16.1)
model (16.1)
trend line (16.1)
interpolating (16.1)
Creating Scatter Plots and Drawing a Line of Best Fit
A scatter plot is a graph of data points. A line of best fit approximates the data in a scatter
plot as closely as possible. A line of best fit does not have to pass through all or any of the
data points. To determine how to construct a line of best fit, begin by plotting all the data.
Next, draw a shape that encloses all of the data. Then, draw a line that divides the
enclosed area of the data in half.
Example
Candice is an environmental engineer who is measuring the temperature of the ocean at
different depths. Her results are listed in the table. A scatter plot is shown from the
© 2011 Carnegie Learning
© 2011 Carnegie Learning
information in the table. A line of best fit is also shown on the graph.
Depth
(in meters)
0
100
200
300
400
500
600
700
800
900
Temperature
(°F)
81
76
73
70
66
61
56
52
48
43
Make sure you
stay organized
and not scattered!
It is easier for your
brain to understand
neat, organized
information.
Chapter 16 Summary • 881
Ocean Water Temperature
y
80
76
72
Temperature (ºF)
68
64
60
56
52
48
44
40
0
0
100 200 300 400 500 600 700 800 900 x
Water Depth (in meters)
Writing a Line of Best Fit Equation and Making Predictions
A line of best fit equation is usually written using the slope-intercept form of a line. The
slope-intercept form of a line is y 5 mx 1 b, where m represents the slope of the line and
b represents the y-intercept of the line. The equation for the line of best fit can be used to
make predictions about the related problem.
Example
An equation can be written that represents the line of best fit from the previous example.
That equation can be used to predict the water temperature at depths of 1000 feet and
1100 feet.
The line goes through the points (200, 73) and (500, 61). The slope of the line is
61 2 73 5 2____
12 5 20.04.
__________
500 2 200
300
73 5 20.04(200) 1 b
73 5 28 1 b
81 5 b
Therefore, the equation of the line of best fit is y 5 20.04x 1 81, where y represents the
water temperature in degrees Fahrenheit, and x represents the ocean depth in meters.
The water temperature at a depth of 1000 feet is predicted to be
y 5 20.04(1000) 1 81 5 240 1 81 5 41°F.
The water temperature at a depth of 1100 feet is predicted to be
y 5 20.04(1100) 1 81 5 244 1 81 5 37°F.
882 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
y 5 mx 1 b
© 2011 Carnegie Learning
Use the slope and one of the data points to determine the y-intercept.
Performing an Experiment and Comparing the Results
Lines of best fit can be used to model the results of experiments. The Stroop Test studies
a person’s perception of words and colors by using lists of color words (red, green, black,
and blue) that are written in one of the four colors. In a matching list, the color of the ink
matches the color of the word. In a non-matching list, the color of the ink does not match
the color of the word.
Example
Jin conducts an experiment using matching lists and non-matching lists. The matching list
consists of a list of repeated digits. The number of digits on each line matches the
numerical value of the repeated digit. For instance, one line in the list reads 5 5 5 5 5.
The non-matching list also consists of a list of repeated digits. The number of digits on
each line does not match the numerical value of the repeated digit. For instance, one line
in the non-matching list reads 7 7 7 7. In the experiment, Jin gives either the matching
list or the non-matching list to a person and asks them to say aloud the number of
digits in each row. Jin has determined that the line of best fit for the matching list data is
y 5 0.3x 1 0.2, where y represents the time in seconds it takes to read a list consisting
of x lines of digits. The line of best fit for the non-matching list is y 5 0.8x 1 0.2.
The slope in each line of best fit represents the predicted amount of time it takes to read
each line in the given list. The y-intercept represents the amount of time it takes to read
0 lines.
The predicted amount of time it takes a person to read a matching list containing
20 lines is
y 5 0.3(20) 1 0.2
y 5 6 1 0.2
y 5 6.20 seconds
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© 2011 Carnegie Learning
The predicted amount of time it takes a person to read a non-matching list containing
20 lines is
y 5 0.8(20) 1 0.2
y 5 16 1 0.2
y 5 16.20 seconds
The predicted times make sense based on the experiment. It should take longer to read a
non-matching list of digits than to read a matching list of digits because the person must
count the digits in each line of the non-matching list.
Chapter 16 Summary • 883
Using Technology to Determine a Linear Regression Equation
Calculators and spreadsheet programs use methods to determine the line of best fit.
The resulting equation for the line of best fit is called the linear regression equation. The
calculator can also determine the correlation coefficient, r, which indicates how close the
data are to forming a straight line. The value of r is between 0 and 1 if the linear regression
equation has a positive slope. The value of r is between 0 and 21 if the linear regression
equation has a negative slope. The closer r is to 1 (if the data are positively correlated) or
21 (if the data are negatively correlated), the closer the data are to being in a straight line.
Example
Sabrina uses her calculator to determine the linear regression equation for a set of data.
The display screen of her calculator is shown. In this case, the value of a represents the
slope, and the value of b represents the y-intercept.
/LQ5HJ
\ D[E
D E U U The linear regression equation is y 5 21.33x 1 60.29. The data are negatively correlated,
because the slope of the linear regression equation is negative and the value of r is also
negative. The data are very close to being in a straight line because the value for r is very
884 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning
close to 21.
Determining the Correlation of Data
When the line of best fit representing a set of data points on a scatter plot has a positive
slope, the points are positively correlated. When the line of best fit has a negative slope,
the points are negatively correlated. When there appears to be no relationship between
the x-values and the y-values and there is not a suitable line of best fit to represent the
data, the points have no correlation.
Example
The scatter plots shown give an example of points that are positively correlated, points
that are negatively correlated, and points that have no correlation.
y
y
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
1
2
3
4
5
6
7
8
x
9
1
Negative Correlation
2
3
4
5
6
7
8
9
x
No Correlation
y
9
8
7
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© 2011 Carnegie Learning
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
x
Positive Correlation
Chapter 16 Summary • 885
886 • Chapter 16 Lines of Best Fit
© 2011 Carnegie Learning
© 2011 Carnegie Learning