How well can you use the strategies of focusing in and ignoring outliers to analyze a non-linear data set using a linear model?
Let's Review
In this lesson, you have learned strategies for analyzing non-linear data using a linear model. Look at this example before doing some practice problems on your own.
In the video at the beginning of this lesson, Shawn was trying to answer the following question.
How does the amount of sleep a person gets each night change as they get older?
Shawn gathered data comparing age with the amount of sleep people get on average.
Age (years)
2
5
8
13
15
18
22
26
30
50
65
Average Sleep Time (hours per night)
14
12
10
9
8.5
8.25
7
6.5
6
6.5
7.5
Then, he created a scatterplot of the data. The data in the scatterplot shows a non-linear association between the variables.
A non-linear scatter plot with the following attributes:
The x-axis is labeled Age (years).
The y-axis is labeled Average Sleep Time (hours per night).
The following ordered pairs are plotted: (2, 14), (5, 12), (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), (30, 6), (50, 6.5), and (65, 7.5).
How can Shawn use a linear model to analyze this non-linear data and answer his question?
Study the slideshow to see how to first create a linear model for the data by stating a domain or identifying outliers to ignore and then to use the linear model to answer Shawn's question.
To use a linear model to analyze non-linear data, you must find a linear association inside the non-linear association. One strategy is to focus on a specific part of the data that is linear. Another strategy is to ignore any points that are outliers.
Which strategy should Shawn use?
Focusing in allows you to create a linear model that represents a part of the data. Ignoring outliers allow you to create a linear model the represents all the data except for one or two points.
First, examine the scatterplot to see if there is any part of the data that has a linear association. If there is a linear section of the data that can be represented with a straight line, then focusing on a domain is the best strategy to use. If ignoring one or two data points allows the remaining data to be represented by a straight line, then ignoring outliers is the best strategy to use.
A non-linear scatter plot with the following attributes:
The x-axis is labeled Age (years).
The y-axis is labeled Average Sleep Time (hours per night).
The following ordered pairs are plotted: (2, 14), (5, 12), (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), (30, 6), (50, 6.5), and (65, 7.5).
The ordered pairs (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), and (30, 6) are highlighted.
The highlighted portion of the scatterplot shows the most linear portion of the data. There are too many points being excluded to be considered outliers in this case. Since the highlighted portion of the scatterplot represents data that can be modeled by a straight line, Shawn's data can be analyzed using a linear model by focusing on a domain.
The steps for focusing on a portion of the data that can be modeled with a straight line are shown in the table below. Click each step to see how to identify the domain of Shawn's scatterplot that can be analyzed using a linear model.
A non-linear scatter plot with the following attributes:
The x-axis is labeled Age (years).
The y-axis is labeled Average Sleep Time (hours per night).
The following ordered pairs are plotted: (2, 14), (5, 12), (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), (30, 6), (50, 6.5), and (65, 7.5).
The ordered pairs (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), and (30, 6) are highlighted.
The highlighted section of data has a linear association.
There are 7 data points included in the highlighted section. There are 4 points being excluded. Four is less than half the data, so it is acceptable to use the highlighted section for a linear model.
The domain of the highlighted section is based on \( x \)-values of the included data points. The highest and lowest \( x \)-values make the domain. These come from the data points that are farthest right and left in the included section.
The highest \( x \)-value of the included data is 30. The lowest \( x \)-value of the included data is 8.
The domain with a linear association is people of age 8 to 30.
Shawn can create a linear model for the data that represents the average sleep times for people between the ages of 8 and 30.
To make a linear model, Shawn will need to find the line of best fit for the linear data and calculate the equation in slope-intercept form. Then, he will be able to interpret the equation to answer his question.
The line of best fit for the data in the domain from age 8 to 30 is shown on the scatterplot below:
A non-linear scatter plot with the following attributes:
The x-axis is labeled Age (years).
The y-axis is labeled Average Sleep Time (hours per night).
The following ordered pairs are plotted: (2, 14), (5, 12), (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), (30, 6), (50, 6.5), and (65, 7.5).
A line of best fit is drawn through the ordered pairs (8, 10), (13, 9), (15, 8.5), (18, 8.25), (22, 7), (26, 6.5), and (30, 6).
The steps for calculating the equation in slope-intercept form of the line of best fit on a scatterplot are shown in the table below. Click each step to see how Shawn used it to create a linear model from the scatterplot.
Shawn chose to use the guide points (8, 10) and (50, 2) to find the slope of the line. He used the slope formula:
The slope is approximately \( - 0.19 \), and the \( y \)-intercept is approximately \( 11.52 \), so the equation of the linear model is \( y = - 0.19x + 11.52 \).
So, the linear model for the relationship between age and amount of sleep for people of ages 8 to 30 is:
\( y = - 0.19x + 11.52. \)
The linear model for the relationship between age and amount of sleep for people of ages 8 to 30 is:
\( y = - 0.19x + 11.52 \)
Use this linear model to answer the following question:
How does the amount of sleep a person gets each night change as they get older?
Shawn wants to know how sleep time changes as people get older. This is a question that asks how the two variables change, which means Shawn should analyze the slope. The slope is \( - \)0.19. The slope is negative, which means as age increases, the amount of sleep decreases. Each year a person ages, they will get 0.19 hours less sleep. To determine what 0.19 hours means, you could multiply by 60 to change hours into minutes.
\( 0.19 \times 60 = 11.4 \)
This means every year between age 8 and 30, a person's average amount of sleep each night will decrease by 11.4 minutes.
Think you got it?
Now it's your turn. Practice using a linear model to work with non-linear data by completing the activity below. Answer the question on each tab, then check your answer.
If you need graph paper, click below to download printable graph paper in Word or PDF format.
Identify the domain on the following scatterplot that can be analyzed using a linear model.
A non-linear scatter plot with the ordered pairs (0.25, 8.25), (1, 8.5), (2, 8), (3.5, 7.25), (4, 6.5), (4, 5.75), (4.5, 4.5), (4.5, 5.5), (5, 4), (5, 5), (5.5, 3.25), (7, 3), and (9, 2).
The domain that can be analyzed with a linear model includes \( x \)-values from 3.5 to 5.5.
If you need help arriving at this answer, click the Solution button.
Step 1: Look for a section of data on the scatterplot with a linear association.
The shaded portion of the data below is the most linear portion of the data.
A non-linear scatter plot with the ordered pairs (0.25, 8.25), (1, 8.5), (2, 8), (3.5, 7.25), (4, 6.5), (4, 5.75), (4.5, 4.5), (4.5, 5.5), (5, 4), (5, 5), (5.5, 3.25), (7, 3), and (9, 2). The ordered pairs (3.5, 7.25), (4, 6.5), (4, 5.75), (4.5, 4.5), (4.5, 5.5), (5, 4), (5, 5), and (5.5, 3.25) are highlighted.
Step 2: Check that less than half the data is being excluded.
There are 8 data points included in the highlighted section. There are 5 points being excluded. Five is less than half the data, so it is acceptable to use the highlighted section for a linear model.
Step 3: Identify and state the domain that has a linear association using the largest and smallest \( x \)-values of the linear data.
The domain of the highlighted section is based on \( x \)-values of the included data points. The highest and lowest \( x \)-values make the domain. These come from the data points that are farthest right and left in the included section.
The highest \( x \)-value of the included data is 5.5. The lowest \( x \)-value of the included data is 3.5.
The domain with a linear association is from 3.5 to 5.5.
On the following scatterplot, identify outliers that can be ignored to allow analysis with a linear model.
A non-linear scatter plot with the ordered pairs (2, 3.5), (3.5, 3), (4, 3.75), (4.5, 3.5), (5, 4), (5.75, 4.75), (6.25, 4.5), (6.5, 5), (7.25, 5.25), (7.75, 6), (8.25, 6.25), and (9.5, 5).
The outliers (2, 3.5) and (9.5, 5) can be ignored while creating a linear model for the data.
If you need help arriving at this answer, click the Solution button.
Step 1: Look for part of the association that could be linear.
The highlighted portion of the data shows a linear association.
A non-linear scatter plot with the ordered pairs (2, 3.5), (3.5, 3), (4, 3.75), (4.5, 3.5), (5, 4), (5.75, 4.75), (6.25, 4.5), (6.5, 5), (7.25, 5.25), (7.75, 6), (8.25, 6.25), and (9.5, 5). The ordered pairs (3.5, 3), (4, 3.75), (4.5, 3.5), (5, 4), (5.75, 4.75), (6.25, 4.5), (6.5, 5), (7.25, 5.25), (7.75, 6), and (8.25, 6.25) are highlighted.
Step 2: Make sure only one or two points are ignored as possible outliers.
There are only two points that are possible outliers. This is an acceptable number of outliers to ignore while creating a linear model.
Step 3: Identify the outlier points that will be ignored by their coordinate pairs.
A non-linear scatter plot with the ordered pairs (2, 3.5), (3.5, 3), (4, 3.75), (4.5, 3.5), (5, 4), (5.75, 4.75), (6.25, 4.5), (6.5, 5), (7.25, 5.25), (7.75, 6), (8.25, 6.25), and (9.5, 5). The ordered pairs (3.5, 3), (4, 3.75), (4.5, 3.5), (5, 4), (5.75, 4.75), (6.25, 4.5), (6.5, 5), (7.25, 5.25), (7.75, 6), and (8.25, 6.25) are highlighted and the ordered pairs (2, 3.5) and (9.5, 5) are circled.
The two outlier points are (2, 3.5) and (9.5, 5).
The data in the table shows the percent of a slice of bread that is covered in mold compared to how long the bread had been sitting out.
Time the Bread Sat Out (days)
5
9
15
21
19
17
21
24
28
Percent of Bread Covered in Mold
10
18
20
37
29
21
43
50
85
Create a scatterplot to visualize the data. If the association is non-linear, identify either the domain that can be used or the outliers that can be ignored. Then, find the equation of a linear model for the data.
The data has a linear association for the domain from 15 to 24.
The equation of the linear model for the domain from 15 to 24 is:
\( y = 3.57x - 36.41 \)
If you need help arriving at this answer, click the Solution button.
The data is shown on the scatterplot below:
A non-linear scatter plot with the following attributes:
The x-axis labeled Time the Bread Sat Out (days).
The y-axis labeled Percent of Bread Covered in Mold.
The ordered pairs (5, 10), (9, 18), (15, 20), (21, 37), (19, 29), (17, 21), (21, 43), (24, 50), and (28, 85) are plotted.
The association is non-linear, but there is a linear portion of the data, which appears as if it could be analyzed by focusing on a domain.
Step 1: Look for a section of data on the scatterplot with a linear association.
A non-linear scatter plot with the following attributes:
The x-axis labeled Time the Bread Sat Out (days).
The y-axis labeled Percent of Bread Covered in Mold.
The ordered pairs (5, 10), (9, 18), (15, 20), (21, 37), (19, 29), (17, 21), (21, 43), (24, 50), and (28, 85) are plotted.
The ordered pairs (15, 20), (21, 37), (19, 29), (17, 21), (21, 43), and (24, 50) are highlighted.
The shaded portion of the data has a linear association.
Step 2: Check that less than half the data is being excluded.
There are 6 data points included in the highlighted section. There are 3 points being excluded. Three is less than half the data, so it is acceptable to use the highlighted section for a linear model.
Step 3: Identify and state the domain that has a linear association using the largest and smallest \( x \)-values of the linear data.
The domain of the highlighted section is based on \( x \)-values of the included data points. The highest and lowest \( x \)-values make the domain. These come from the data points that are farthest right and left in the included section.
The highest \( x \)-value of the included data is 24. The lowest \( x \)-value of the included data is 15.
The domain with a linear association is from 15 to 24.
The linear model for the domain from 15 to 24 is shown on the scatterplot below:
A non-linear scatter plot with the following attributes:
The x-axis labeled Time the Bread Sat Out (days).
The y-axis labeled Percent of Bread Covered in Mold.
The ordered pairs (5, 10), (9, 18), (15, 20), (21, 37), (19, 29), (17, 21), (21, 43), (24, 50), and (28, 85) are plotted.
A line of best fit is drawn through the ordered pairs (15, 20), (21, 37), (19, 29), (17, 21), (21, 43), and (24, 50.
Your line may be slightly different from the one shown but should be centered and balanced on the data. Check your line using the distance method.
Use the steps below to calculate the equation in slope-intercept form of the line of best fit shown on the scatterplot.
Step 1: Choose two points on the line, and calculate the slope using the slope formula.
The guide points (13, 10) and (20, 35) are used here.
The \( y \)-intercept is not visible on the graph, so it is calculated using the guide point (13, 10) and the slope in the slope-intercept form equation.
Step 3: Write the final equation in slope-intercept form.
The equation of the linear model for the domain from 15 to 24 is:
\( y = 3.57x - 36.41 \)
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