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Private: Learning Math: Data Analysis, Statistics, and Probability
Bivariate Data and Analysis Part C: Modeling Linear Relationships (35 minutes)
In This Part: How Square Can You Be?
In Parts A and B, you confirmed that there is a strong positive association between height and arm span. In Part C, we will investigate this association further. Note 2
The drawing below suggests that a person’s arm span should be the same as her or his height — in which case, a person could be considered a “square.” Is this correct?
Do most people have heights and arm spans that are approximately the same? That is, are most people “square?”
Problem C1
Why is this not the same as establishing an association between height and arm span?
Problem C2
Compare the measurements for the six heights and arm spans you collected, including your own. How many people are “squares” — i.e., their arm spans and heights are the same? For how many people are these measurements approximately the same?
To measure the differences between height and arm span, let’s look at the numerical differences between the two. In these problems, we will use “Height – Arm Span” as the measure of the difference between height and arm span.
Problem C3
Consider the difference:
Height – Arm Span
a. If you know only that this difference is positive, what does it tell you about a person? What does it not tell you?
b. If you know that this difference is negative, what does it tell you? What does it not tell you?
c. If you know that this difference is 0, what does it tell you?
In This Part: Analyzing the Differences
Here again is the data table for the 24 people we have been studying — but it now includes a column to show the difference between height and arm span for each person:
Problem C4
Let’s consider five of the people we have studied: Persons 1, 6, 9, 14, and 19. Use the table to determine the following:
a. Which of the five people have heights that are greater than their arm spans?
b. Which of the five people have heights that are less than their arm spans?
c. Which of the five has the greatest difference between height and arm span?
d. Which of the five has the smallest difference between height and arm span?
Problem C5
Use the table to determine the following:
a. How many of the 24 people have heights that are greater than their arm spans?
b. How many of the 24 people have heights that are less than their arm spans?
c. How many of the 24 people have heights that are equal to their arm spans?
d. Which six people are the closest to being square without being perfectly square?
e. Which five are the farthest from being square?
Problem C6
a. How many of the 24 people have heights and arm spans that differ by more than 6 cm?
b. How many people have heights and arm spans that differ by less than 3 cm?
In This Part: 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:
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 a relationship between two variables, such as height and arm span.
Problem C7
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
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
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
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?
Answer questions (c) and (d) by comparing those points to the line Height = Arm Span.
Here is the scatter plot of all 24 people and the graph of the line Height = Arm Span:
Problem C11
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 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.
Solutions
Problem C1
Even though we have established an association, we have not established a description of the nature of the relationship between height and arm span. This question seeks to investigate a specific relationship between arm span and height. Put another way, there are many positive associations (e.g., the association between years of job experience and salary), but the relationship between the variables is not that they are the same (i.e., “square”).
Problem C2
Answers will vary, but you should generally find the heights and arm spans to be approximately the same.
Problem C3
a. It tells you that this person’s height is greater than his or her arm span, and that this person is not “square.” It does not tell you the person’s exact height or arm span.
b. It tells you that this person’s height is less than his or her arm span, and that this person is not “square.” Again, it does not tell you this person’s exact height or arm span.
c. It tells you that this person’s height and arm span are equal, and that this person is “square.”
Problem C4
a. Two of the five people, Persons 1 and 6, have heights that are greater than their arm spans.
b. Two of the five people, Persons 14 and 19, have heights that are less than their arm spans.
c. Person 19 has the largest difference, 6 cm.
d. Person 9 has the smallest difference, 0 cm. Person 9 is “square.”
Problem C5
a. Nine people have heights that are greater than their arm spans.
b. Twelve people have heights that are less than their arm spans.
c. Three people have heights that are equal to their arm spans.
d. Persons 5, 6, 8, 14, 18, and 22 — the six people with the smallest non-zero difference (±1) in their heights and arm spans — come the closest to being a square without actually being a square.
e. Persons 24, 23, 11, 7, and 20 — the people with the greatest difference (positive or negative) between their heights and arm spans — are the most “non-square.”
Problem C6
a. Five people have heights and arm spans that differ by more than 6 cm.
b. Nine people have heights and arm spans that differ by less than 3 cm.
Problem C7
a. Person 1’s height is greater than his or her arm span, so the coordinates of that point will be above the line Height = Arm Span.
b. Person 9’s height is equal to his or her arm span, so the coordinates of that point will be on the line Height = Arm Span.
c. Person 19’s height is less than his or her arm span, so the coordinates of that point will be below the line Height = Arm Span.
d. Any point on the line Height = Arm Span represents a person who is “square.” Any points that are not on this line would indicate that a person’s height is either greater or less than that person’s arm span.
Problem C8
Since the difference between height and arm span is greater for Person 1 than it is for Person 6, the point for Person 1 should be farther from the line Height = Arm Span than the point for Person 6.
Problem C9
The vertical distance for Person 14 is 1 (|176 – 177| = |-1| = 1). The vertical distance for Person 19 is 6 (|182 – 188| = |-6| = 6). In each case the calculation is performed as |Height – Arm Span|.
Problem C10
a. The points for Persons 2 and 7 are above the line; therefore, their heights are greater than their arm spans.
b. The points for Persons 4 and 23 are below the line; therefore, their heights are less than their arm spans.
c. The point for Person 23 is the farthest from the line, vertically; therefore, Person 23 has the greatest difference between height and arm span.
d. The point for Person 2 is closest to the line, vertically; therefore, Person 2 has the smallest difference between height and arm span.
Problem C11
a. Nine points are above the line, so nine people have heights that are greater than their arm spans.
b. Twelve points are below the line, so 12 people have heights that are less than their arm spans.
c. Three points are on the line, so three people have heights that are equal to their arm spans.
d. The points that are farthest from the line represent people who have the greatest differences between heights and arm spans. (These are the points for Persons 11, 23, and 24.)
e. The six points that are closest to the line represent the smallest differences between heights and arm spans. (These are the points for Persons 5, 6, 8, 14, 18, and 22.)
