Key Concepts
After this lesson student will be able to:
- Understand association.
- Understand correlation.
- Write the equation of a trend line.
- Interpret trend lines.
Scatter Plots
Model and discuss
Nicholas plotted data points to represent the relationship between screen size and the cost of television sets. Everything about televisions is the same except for the screen size.

- Describe any patterns you see.
- What does this set of points talk about the relationship of screen size and the cost of television?
- Where do you think the point for a 46-inch television would be on the graph? How about a 60-inch TV? Explain?
Solution:
- As the screen size of the television set increases, the cost also increases.
- The relationship of the screen size and the cost of the television is approximately linear.
- The cost of a 46-inch TV should be somewhere in between the cost of the 43- and 48-inch TV.
So, we can approximate its cost at about $450.
The cost of a 60-inch TV should be greater than that of a 50-inch TV.
So, we approximate the cost of a 60-inch TV to be approximately more than $700.
Understand Association
Example 1.
- What is the relationship between the hours after sunrise, x, and the temperature, y, shown in the scatter plot?

Solution:
As the number of hours after sunrise increases, so does the temperature.
When y-values tend to increase as x-values increase, the two data sets have a positive association.
- What is the relationship between the hours after sunset, x, and the temperature, y, shown in the scatter plot?

Solution:
As the number of hours after sunset increases, the temperature decreases.
When y-values tend to decrease as x-values increase, the two data sets have a negative association.
- What is the relationship between the hours after sunset, x, and the amount of rain, y, shown in the scatter plot?

Solution:
There is no relationship between the amount of rainfall and the number of hours after sunset.
When there is no general relationship between x-values and y-values, the two data sets have no association.
Try it
- Describe the type of association each scatter plot shows.

Solution:
If the y-values increase as the x-values increase, then there is a positive association.
If the y-values decrease as the x-values increase, then there is a negative association.
If there is no relationship between the x- and y-values, then there is no association.
- As the x-values increase, the y-values also increase, so there is a positive association.
- There is no relationship between the x- and y-values, so there is no association.
Understand correlation
Example 2
- How can the relationship between the hours after sunrise, x, and the temperature, y, be modeled?

Solution:
The data points on the scatter plot approximates a line. Sketch a trend line that best fits the data to determine whether a linear function can model the relationship.
The scatter plot suggests a linear relationship.
There is a positive correlation between hours after sunrise and the temperature.
When data with a negative association are modeled with a line, there is a negative correlation.
If the data do not have an association, they cannot be modeled with a linear function.
Try it
- How can the relationship between the hours after sunset, x, and the temperature, y, be modeled?
If the relationship is modeled with a linear function, describe the correlation between the two data sets.

Solution:
The data points on the scatter plot approximate a line, so, it suggests a linear relationship. As the number of hours after sunset increases, the temperature decreases so there is a negative correlation between hours after sunset and the temperature.
Linear, negative correlation
Write the equation of a trend line
Example 3
What trend line models the data in the scatter plot?
Solution:
A trend line models the data in a scatter plot by showing the general direction of the data. A trend line fits the data as closely as possible.
Step 1:
Sketch a trend line for the data.
A trend line approximates a balance of
points above and below that line.

Step 2:
Write the equation of this trend line.
Select two points on the trend line to find the slope.

Use the slope and one of the points to write the equation in slope intercept form.
y – 50 = 32.1(x-2)
y = 32.1x – 14.2
This trend line that models the data is y = 32.1x – 14.2
This trend line is one of many possible trend lines.
Try it
3.
- What trend line, in slope-intercept form, models the data from example 2 try it?
- Explain why there could be no points on a trend line, yet the line models the data?
Solution:
- Depending on the trend line drawn, a possible two points to be used are (0, 71) and (4, 66).
The slope of the trend line is:
m= 66−71 / 4−0
m = −5 / 4
m=-1.25
Use the slope and one of the points to write the equation in slope-intercept form:
y — y₁ = m (x − x₁)
y – 71 = -1.25(x – 0)
y = -1.25x + 71
- Because a trend line approximates a balance of points above and below the line.
Interpret trend lines
Example 4.
The table shows the amount of time required to download a 100-megabyte file for various internet speeds. Assuming the trend continues, how long would it take to download the 100-megabyte file if the internet speed is 75 kilobytes per second?

Solution:
Step 1
Make a scatter plot of the data and sketch a trend line.

Step 2
Find the equation of the trend line.
Select the two points on the trend line to find the slope:
(40, 6) and (55, 4)
= 6−4 / 40−55 ≈ -0.13
Use the slope and one of the points to write the equation for the trend line.
y – 6 = -0.13(x – 40)
y = -0.13x + 11.2
The equation of the trend line is y = -0.13x + 11.2.
Step 3
Use the equation of the linear model to find the y-value that corresponds to x = 75.
y = -0.13x + 11.2
y = -0.13(75) + 11.2
y = 1.45
The download time would be 1.45 sec.
Try it
- What is the x-intercept of the trend line? Is that possible in a real–world situation? Explain.
Solution:
From example 4, we have the equation: y = – 0.13x + 11.2
To find the x-intercept, we set y = 0 and solve for x:
0 = – 0.13x + 11.2
0.13x = 11.2
x ≈ 86
This represents an internet speed of 86 KB/s, which is a possible internet speed but since the corresponding time is 0 minutes, then this is not possible in a real-world situation.
- The table shows the maximum recommended viewing distances y, in feet, for an HDTV with screen size x, in inches. What trend line models the data shown in the table? What does the slope of the trend line represent?

Solution:
Make a scatter plot of the data and sketch a trend line.

Find the equation of the trend line.
Select two points on the trend line to find the slope.
Here, let’s use (60, 12.5) and (40, 12.5):
m= 12.5−8.3 / 60−40
m=0.21
Use the slope and one of the points to write the equation in slope-intercept form:
y – y₁ = m (x − x₁)
y – 8.3 = 0.21(x – 40)
y – 8.3 = 0.21x – 8.4
y = 0.21x – 0.1
The slope means that the maximum viewing distance increases by about 0.21 ft as the screen size of an HDTV increases by 1 inch.
Check your knowledge
- Describe the type of association between x and y for each set of data. Explain.

- Describe the type of association between x and y for the set of data. Explain.

- How could be finding the y-intercept of a trend line for a data set help you determine the usefulness of the trend line as a model?
- Describe the type of association each scatter plot shows.

- Make a scatter plot of the data. Describe the type of association that the scatter plot shows.

- Make a scatter plot of the data. Draw a trend line and write its equation.

Check your knowledge – Answers
- If you observe, the graph y-values increase as x -value increases. So, the graph has a positive association.
- As the x-values increase, the y-values decrease, so, there is a negative association.
- The y-intercept gives us the initial value of the situation; so by using it, we can determine if the trend line is a good fit for the data set.
- If you observe the graph, the y-values increase and decrease as x-value increases. So, it has no association.
- Scatter plot of the given data is from the data, y-values decrease as x-value increase, then it has a negative association.

- Find the equation of the trend line. Select the two points on the trend line to find the slope. Here, I decided to use (4, 5) and (10, 10):
m= 10−5 / 10−4
m= 5 / 6
Use the slope and one of the points to write the equation in slope-intercept form:
y-y₁ = m (x – x₁)
y-5= 5 / 6(x-4)
y – 5 = 5x / 6 – 5X4 / 6
y – 5 = 5x / 6 – 10 / 3
y =5x / 6 + 5 / 3

Exercise
- Describe the type of association between x and y for each set of data. Explain




- Tell whether the data show a positive, a negative, or no relationship. Television size and price

- Make a scatter plot of the data. Tell whether the data shows a positive, a negative or, no relationship.

Concept Summary
Scatter Plots and Trend Lines
Positive Association


Negative Association



Concept Map

Related topics
Composite Figures – Area and Volume
A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]
Read More >>Special Right Triangles: Types, Formulas, with Solved Examples.
Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]
Read More >>Ways to Simplify Algebraic Expressions
Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]
Read More >>How to Solve Right Triangles?
In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]
Read More >>Other topics

Comments: