1. Linear Models and Scatter Plots Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 x 24 –2–2 – 4 y 2 4 6 A scatter plot represents data graphically using points plotted on a rectangular coordinate system. Example: (x, y) (1, – 4) (2, – 2) (3, – 1) (4, 0) (5, 2) (6, 4) Definition: Scatter plot

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Example: The average salary S (in millions of dollars) for professional baseball players from 1996 through 2002 is shown in the table. Let t = 6 represent the year 1996. Draw the scatter plot. Example: Scatter Plot YearSalary, S 19961.1 19971.3 19981.4 19991.6 20001.8 20012.1 20022.3 S 4 t 8 1 2 Year (6  1996) Salary (in millions of dollars)

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Graphing Utility: Scatter Plot YearSalary, S 19961.1 19971.3 19981.4 19991.6 20001.8 20012.1 20022.3 Graphing Utility: Draw the scatter plot that describes the data. Stat Plot Menu: Zoom Menu: Stat Edit Menu:

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 In a collection of ordered pairs (x, y), if y tends to increase as x increases, the collection has a positive correlation. Correlation x y x y x y If y tends to decrease as x increases, the collection has a negative correlation. positive correlation no correlation negative correlation

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Finding a linear model to represent the relationship described by a scatter pot is called fitting the line to data. Example: Fitting a Line to Data YearSalary, S 19961.1 19971.3 19981.4 19991.6 20001.8 20012.1 20022.3 4 t 8 S 1 2 Example: The table and scatter plot for the average salary S (in millions of dollars) for professional baseball players from 1996 through 2002 is shown. Let t = 6 represent the year 1996. Find the equation of the line. Example continues.

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Continued Example: Fitting a Line to Data YearSalary, S 19961.1 19971.3 19981.4 19991.6 20001.8 20012.1 20022.3 4 t 8 S 1 2 (6, 1.1) (12, 2.3) The equation of this line is S = 0.2t – 0.1. This line approximates the data. Example continued:

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Graphing Utility: Finding a Liner Model YearSalary, S 19961.1 19971.3 19981.4 19991.6 20001.8 20012.1 20022.3 Graphing Utility: Find a linear model that describes the data. Stat Menu: A linear model for this data is S = 0.2t – 0.14. This equation is very close to the equation found using two data points. Stat Edit:

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Correlation Coefficient The correlation coefficient (or r-value) of the data gives the measure of how well the model fits the data. The closer |r| is to 1, the better the points can be described by a line. r = 0.99 strong positive correlation r = – 0.93 negative correlation 0 13 2 0 52.6 52.3 0 13 r = 0.66 weak correlation 50 90 50 100