Learning Objectives for Section 1.3 Linear Regression The student will be able to calculate slope as a rate of change. The.
Download ReportTranscript Learning Objectives for Section 1.3 Linear Regression The student will be able to calculate slope as a rate of change. The.
Slide 1
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 2
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 3
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 4
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 5
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 6
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 7
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 8
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 9
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 10
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 11
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 12
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 13
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 14
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 15
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 16
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 17
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 18
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 2
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 3
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 4
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 5
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 6
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 7
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 8
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 9
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 10
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 11
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 12
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 13
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 14
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 15
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 16
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 17
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
18
Slide 18
Learning Objectives for Section 1.3
Linear Regression
The student will be able to calculate slope as a rate of change.
The student will be able to calculate linear regression using a
calculator.
1
Mathematical Modeling
MATHEMATICAL MODELING is the process of using
mathematics to solve real-world problems. This process can
be broken down into three steps:
1. Construct the mathematical model, a problem whose solution will
provide information about the real-world problem.
2. Solve the mathematical model.
3. Interpret the solution to the mathematical model in terms of the
original real-world problem.
In this section we will discuss one of the simplest mathematical
models, a linear equation.
2
Slope as a Rate of Change
If x and y are related by the equation y = mx + b, where m and b
are constants with m not equal to zero, then x and y are linearly
related. If (x1, y1) and (x2, y2) are two distinct points on this line,
then the slope of the line is
y2 y1 y
m
x2 x1 x
This ratio is called the RATE OF CHANGE of y with respect
to x.
3
Slope as a Rate of Change
Since the slope of a line is unique, the rate of change of two
linearly related variables is constant.
Some examples of familiar rates of change are miles per hour
and revolutions per minute.
4
Example 1: Rate of Change
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
1. State the rate of change of the function, and describe what
this value signifies within the context of this scenario.
5
Example 1: Rate of Change
(cont.)
The following linear equation expresses the number of municipal
golf courses in the U.S. t years after 1975.
G = 30.8t + 1550
2. State the vertical intercept of this function, and describe what
this value signifies within the context of this scenario.
6
Linear Regression
In real world applications we often encounter numerical data
in the form of a table. The powerful mathematical tool,
regression analysis, can be used to analyze numerical data.
In general, regression analysis is a process for finding a
function that best fits a set of data points.
In the next example, we use a linear model obtained by using
linear regression on a graphing calculator.
7
Regression Notes
Regression: a process used to relate two quantitative variables.
Independent variable: the x variable (or explanatory variable)
Dependent variable: the y variable
To interpret the scatterplot, identify the following:
Form
Direction
Strength
8
Form
Form: the function that best describes the relationship between
the two variables.
Some possible forms would be linear, quadratic, cubic,
exponential, or logarithmic.
9
Direction
Direction: a positive or negative direction can be found when
looking at linear regression lines only.
The direction is found by looking at the sign of the slope.
10
Strength
Strength: how closely the points in the data are gathered around
the form.
11
Making Predictions
Predictions should only be made for values of x within the span
of the x-values in the data set. Predictions made outside the
data set are called extrapolations, which can be dangerous and
ridiculous, thus extrapolating is not recommended.
To make a prediction within the span of the x-values, hit then
.
Next, arrow up or down until the regression equation
appears in the upper-left hand corner then type in the x-value
and hit .
12
Example of Linear Regression
Prices for emerald-shaped diamonds taken from an on-line
trader are given in the following table. Find the linear model
that best fits this data.
Weight (carats)
0.5
0.6
0.7
0.8
0.9
Price
$1,677
$2,353
$2,718
$3,218
$3,982
13
Scatter Plots
Enter these values into the lists in a graphing calculator as
shown below .
14
Scatter Plots
We can plot the data points in the previous example on a
Cartesian coordinate plane, either by hand or using a graphing
calculator. If we use the calculator, we obtain the following
plot:
Price of diamond
(thousands)
Weight (tenths of a carat)
15
Example of Linear Regression
(continued)
Based on the scatterplot, the data appears to be linearly
correlated; thus, we can choose linear regression from the
statistics menu, we obtain the second screen, which gives the
equation of best fit.
The linear equation of best fit
is y = 5475x - 1042.9.
16
Scatter Plots
We can plot the graph of our line of best fit on top of the
scatter plot:
y = 5475x - 1042.9
Price of emerald
(thousands)
Weight (tenths of a carat)
17
Making a Prediction
If it is known that the pricing model holds for diamonds up to 1.5
carats, predict the price of an emerald-shaped diamond that
weighs 1.3 carats.
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