Lesson 28 Introduction M.8.25 Scatter lots · Independent ractice Practice 246 ©Curriculum Associates, LLC Copying is not permitted. Lesson 28 Lesson 28 Scatter Plots Identifying
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Introduction
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Use What You Know
Lesson 28 Scatter Plots
Lesson 28Scatter Plots
In previous lessons, you learned how to represent relationships between two variables on the coordinate plane. Take a look at this problem.
Mr. Finley wanted to see if there was a relationship between students’ scores on the final exam in algebra and the amount of time they spent studying for it. Students studied for the test by doing practice problems on a website that recorded the number of minutes they spent studying. He recorded the number of minutes of studying and test scores in a table. Does the table show a relationship?
Use the math you already know to solve the problem.
a. What is the highest score on the exam and the corresponding
studying time?
b. What is the lowest score on the exam and the corresponding
studying time?
c. What is the approximate mean test score and mean studying time,
to the nearest tenth?
d. Are there any ordered pairs that are surprising to you? Explain.
e. Do you think that spending more time studying helped these students do better on the test? Why or why not?
Minutes Studied
Test Score
60 85
65 85
70 90
90 100
88 94
99 98
86 38
99 99
96 95
45 70
30 60
56 72
30 35
40 52
25 30
M.8.25
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Find Out More
Lesson 28 Scatter Plots
A scatter plot is a graph of ordered pairs in the coordinate plane that represents data. Each ordered pair reports information about two variables. Mr. Finley’s data compares minutes a student spent studying (the x-coordinate) with that student’s test score (the y-coordinate). The study of two related variables is called bivariate data analysis.
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10 20 30 40 50
Minutes Studied
Test
Sco
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60 70 80 90 100
Scatter plots show characteristics of data sets that are not easily seen in tables. For example, notice the clustering above. Points outside the main cluster of data stand out. The ordered pair (86, 38) is outside the main cluster. This point might be considered an outlier.
Scatter plots also help reveal the type of association there may be between two variables. The test scores and time spent studying show a positive association because as time spent studying increased, test scores tended to increase as well. A positive association is represented by an upward trend from left to right in the scatter plot. A negative association shows a downward trend from left to right; one variable decreases as the other increases.
In this scatter plot, the trend of the data could be modeled by a straight line through the cluster of points. So, you could describe the relationship shown in the graph as being roughly linear. This is not true for all scatter plots. Some may show nonlinear relationships.
Reflect1 Is it surprising that there is a positive association between the variables in Mr. Finley’s
data set rather than a negative association? Why or why not?
Modeled and Guided Instruction
Learn About
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Lesson 28
Lesson 28 Scatter Plots
Identifying Positive and Negative Associations
Read the problem below. Then explore relationships in bivariate data sets.
Mr. Finley wondered if there was a relationship between the number of texts students sent the day before the final exam and their test scores. He asked students to volunteer the data from their cell phones, and then he compared it against their test scores. How can he identify trends?
Model It You can plot the data in a table and scatter plot to show clusters and associations.
Add a horizontal line to represent the mean score and a vertical line to represent the mean number of texts. Look for trends by finding the quadrants that contain large clusters of points.
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40 45 50 55 60
Number of Texts
Test
Sco
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65 70 75 80 85 90
mean number of texts
mean score on test
Model It You can also cut the scatter plot into vertical slices and compute the mean test score for each slice.
The triangles point to the mean test score for each slice to its right. This representation gives more information about the trend of the data.
Number of Texts Score on Test
50 85
52 85
57 90
38 100
50 94
40 98
75 38
39 99
45 95
67 70
72 60
62 72
85 35
78 52
87 30
Number of Texts
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40
32−39 40−47 48−55 56−63 64−71 72−79 80−87
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Sco
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100
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Connect It Now you will analyze the data set from the previous page.
2 What is the greatest number of text messages sent? What is the least number sent?
3 Are there any points that lie outside the general trend of the data? Explain. You may want to compare the data on the previous page with the scatter plot in the introduction.
4 Does the scatter plot in the first Model It show a positive association or a negative
association? Explain.
5 How does showing a horizontal line for one mean and a vertical line for the other mean
help support your answer to Problem 4?
6 Describe the trend of the means in the sliced scatter plot, which is in the second Model It. How does the trend of the means help support your answer to Problem 4?
Try It Use what you just learned about associations between variables to solve this problem. Show your work on a separate sheet of either lined paper or graph paper.
7 Describe a situation where a negative association would exist between two variables. Explain why you would expect a negative association. Then make up a data set with 15 ordered pairs and graph them in a scatter plot.
Guided Practice
Practice
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Lesson 28
Lesson 28 Scatter Plots
Identifying Positive and Negative Associations
Read the situation described below. Then solve problems 8–10.
Example
Mr. Finley’s class tried a new website to study for the next test. Mr. Finley again compared time spent studying against test scores using the data displays below.
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Test
Sco
re 80
60
40
20
0
40 50 60
Minutes Studied
70 80 90 100
Test
Sco
re
Minutes Studied
100
50
0
36−47
48−59
60−71
72−83
84−95
96−108
Does there seem to be an association between time spent studying on the new website and test scores?
Solution
8 Do you think Mr. Finley should encourage students to study algebra on the website mentioned in the introduction or on this new website?
Solution
Pair/ShareName another real-world pair of variables that might have no association with each other.
Possible answer: There is no clear upward or downward trend in the data displays. Points appear to be randomly scattered throughout the scatter plot. About the same number of points fall within each quadrant. The mean for each slice does not increase or decrease with a consistent trend.
Scatter plots help us decide what kind of association might exist between two variables.
You can cut a scatter plot into four parts by making a horizontal line to show the mean y-value and a vertical line for the mean x-value.
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9 Look at the quadrants in the graph on the previous page. What percentage of the points fall in each of the four quadrants? What do these percentages tell you about the association between the two variables?
Show your work.
Solution
10 Which pair of variables would you expect to have a negative association?
A The amount of money you make in a week and the number of hours you work during the week.
B A person’s height and the number of pets he or she owns.
C The speed at which a person drives to work and the time it takes her to get to work.
D The number of friends you have and the number of birthday parties you attend in a year.
Brian chose B as the correct answer. Why is Choice B incorrect?
Pair/ShareWhat would the graph on the previous page look like if two of the quadrants contained 0% of the total number of points?
Pair/ShareHow would you help Brian understand his error?
Find the number of points in each quadrant and the total number of points on the graph. Then divide the number in each quadrant by the total number of points on the graph.
You can often predict if a pair of real-world variables will have positive or negative association by thinking about the context.
Independent Practice
Practice
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Lesson 28
Lesson 28 Scatter Plots
Identifying Positive and Negative Associations
Solve the problems.
1 Students at Central Middle School decided to gather survey data from their classmates on a number of different questions. Which pair of variables from the survey would be most likely to have no association with one another?
A number of hours spent studying per week and grade point average
B height in inches and age in months
C number of books owned and number of visits to the doctor’s office in one year
D height and weight
2 The scatter plot below represents 2011 data comparing the average teacher salary and average student/teacher ratios for a number of school districts in the state of Vermont (source: education.vt.gov).
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Aver
age
Stud
ent/
Teac
her R
atio
7.5
7.8
9.5
10.5
11.5
12.5
13.5
48 53 58
Average Teacher Salary(thousands of dollars)
63
Choose True or False for each statement.
a. An outlier exists at the point representing a salary of approximately $62,000. True False
b. There is a negative linear association between average salaries and average student/teacher ratios. True False
c. The smallest school district has an average student/ teacher ratio of approximately 8.6. True False
d. There is a positive linear association between average salaries and average student/teacher ratios. True False
e. The school district with an average salary of $48,000 and a student/teacher ratio of 13 is an outlier compared to districts paying less than $48,000. True False
Self Check
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Go back and see what you can check off on the Self Check on page 239.
3 Each scatter plot below can be described by one of the following types of association.
Positive linear Negative linear Nonlinear None
Write the correct association below each scatter plot.
4 Invent your own set of bivariate data. Use the blank table and the scatter plot to both create and show your data. The data set should meet the following conditions:
• All x-values must be between 0 and 20; all y-values must be between 0 and 15.
• The association between the variables should be both negative and linear.
• One point should be an outlier; label that point “outlier.”
Explain how your data set meets the above conditions by referring to specific features of the scatter plot and data set.
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210 4 6 8 10 12 14 16 183 5 7 9 11 13 15 17 19 20
x
y
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70 80 90 100 110 120 60
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70 80 90 100 110 120 60
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70 80 90 100 110 120 60
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