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Learning Math Home
Data Session 7, Part C: Modeling Linear Relationships
 
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Session 7, Part C:
Modeling Linear Relationships

In This Part: How Square Can You Be? | Analyzing the Differences | Using a Scatter Plot

A scatter plot is also useful in investigating the nature of the relationship between height and arm span. Here is the scatter plot of the 24 heights and arm spans:

Consider these people from the data table:

Person #

Arm Span

Height

Height - Arm Span

1

156

162

6

6

161

162

1

9

170

170

0

14

177

176

-1

19

188

182

-6

The scatter plot below shows the five points for these people together with a graph of the line Height = Arm Span. We draw such lines to explore potential models for describing relationship between two variables, such as height and arm span.

Problem C7

Solution  

a. 

Why is the point for Person 1 above the line Height = Arm Span?

b. 

Why is the point for Person 9 on the line Height = Arm Span?

c. 

Why is the point for Person 19 below the line Height = Arm Span?

d. 

Why is it helpful to draw the line where Height = Arm Span? How does this line help us analyze differences?


 

Problem C8

Solution  

The points for Person 1 and Person 6 are both above the line. Why is the point for Person 1 farther away from the line?

Person 1 has an arm span of 156 cm and a height of 162 cm, which is the coordinate point (156, 162) in the scatter plot. A hypothetical person represented by the coordinate point (156, 156) would be on the line Height = Arm Span, as shown below:

The vertical distance from a point to the line is the absolute value of the difference in y-coordinates from the first point and the point on the line directly above (or below) that point. In this case, Person 1's point (156, 162) is six above the line's point (156, 156). Therefore, the vertical distance from Person 1's point to the line is 6.

Put another way, the vertical distance from (156,162) to the line Height = Arm Span is the magnitude (or absolute value) of the difference between the height and the arm span.

The vertical distance is: |Height - Arm Span| = |162 - 156| = |6| = 6.

In a similar way, the vertical distance from the point for Person 6 (which is [161,162]) to the line Height = Arm Span is: |Height - Arm Span| = |162 - 161| = |1| = 1.


 

Problem C9

Solution  

The points for Persons 14 and 19 are both below the line Height = Arm Span. Determine the vertical distance from each of their points to the line.

The following scatter plot shows four points corresponding to four new people and the graph of the line Height = Arm Span:


 

Problem C10

Solution  

Consider the four points corresponding to Persons 2, 4, 7, and 23. Use the scatter plot to determine the following:

a. 

Which of the four people have heights greater than their arm spans?

b. 

Which of the four people have heights that are less than their arm spans?

c. 

Which of the four has the greatest difference between height and arm span?

d. 

Which of the four has the smallest difference between height and arm span?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Answer questions (c) and (d) by comparing those points to the line Height = Arm Span.   Close Tip

 
 

Here is the scatter plot of all 24 people and the graph of the line Height = Arm Span:


 

Problem C11

Solution  

Use the scatter plot to help you answer these questions.

a. 

How many of the 24 people have heights greater than their arm spans?

b. 

How many of the 24 people have heights less than their arm spans?

c. 

How many of the 24 people have heights equal to their arm spans?

d. 

Which three points represent the greatest differences between height and arm span?

e. 

Other than the points that fall on the line Height = Arm Span, which six points represent the smallest differences between height and arm span?



video thumbnail
 

Video Segment
In this video segment, Professor Kader draws the line Y = X on the class's scatter plot and asks participants to consider points in relation to this line.

If you're using a VCR, you can find this segment on the session video approximately 9 minutes and 0 seconds after the Annenberg Media logo.

 

Next > Part D: Fitting Lines to Data

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