1. Linear Models and Scatter Plots 2015

2. Objectives Interpret correlation Use a graphing calculator to find linear models and make predictions about data. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

3. HWQ 8/11/15 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

4. Extra practice: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

5. Interesting result: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

6. 6 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

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 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

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 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)

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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:

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 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.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 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:

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 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. What does the model predict Baseball players salaries to be in 2015? Stat Edit:

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 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

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Graphing Utility: Finding a Liner Model YearRadio Stations, R 19706,760 19757,744 19808,566 198510,359 199010,788 199511,834 200013,058 A linear model for this data is R = 209.257t +6733.857. 15,104 You Try: The number R of U.S. radio stations for selected years is shown in the table. Use a graphing utility to find a linear model that describes the data. Let t=0 correspond to 1970. For the year 2010, how many radio stations does the model predict?

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Homework Pg. 77 3-6 all, 7-15 odds, 27-37 odds