Linear Models and Scatter Plots 2015

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

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6 x 24 –2–2 – 4 y 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

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

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 Draw the scatter plot. Example: Scatter Plot YearSalary, S S 4 t Year (6  1996) Salary (in millions of dollars)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Graphing Utility: Scatter Plot YearSalary, S Graphing Utility: Draw the scatter plot that describes the data. Stat Plot Menu: Zoom Menu: Stat Edit Menu:

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 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 Find the equation of the line. Example continues.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Continued Example: Fitting a Line to Data YearSalary, S 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:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Graphing Utility: Finding a Liner Model YearSalary, S Graphing Utility: Find a linear model that describes the data. Stat Menu: A linear model for this data is S = 0.2t – 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:

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 r = 0.66 weak correlation

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Graphing Utility: Finding a Liner Model YearRadio Stations, R 19706, , , , , , ,058 A linear model for this data is R = t ,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 For the year 2010, how many radio stations does the model predict?

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Homework Pg all, 7-15 odds, odds