Line of Best Fit 4-8 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 Holt McDougal Algebra 1

Warm Up Identify the slope and the y-intercept. 1. y = 2x + 1 m = 2, b = 1 3 2 m= 3 2 , b = -4 Identify the correlation (positive, negative, or none) that you would expect to see between each pair of data sets. 3. a person’s height and shoe size pos 4. the age of a car and its value neg

Objectives Determine a line of best fit for a set of linear data. Determine and interpret the correlation coefficient.

Vocabulary residual least-squares line line of best fit linear regression correlation coefficient

A residual is the signed vertical distance between a data point and a line of fit. The least-squares line for a data set is the line of fit for which the sum of the squares of the residuals is as small as possible. A line of best fit is the line that comes closest to all of the points in the data set, using a given process. Linear regression is a process of finding the least-squares line.

The correlation coefficient is a number r, where -1 ≤ r ≤ 1, that describes how closely the points in a scatter plot cluster around a line of best fit.

In By using squares of residuals, positive and negative residuals do not “cancel out” and residuals with squares greater than 1 have a magnified effect on the sum. Helpful Hint

Example1: Calculating Residuals Two lines of fit for this data are y = 2x + 2 and y = x + 4. For each line, find the sum of the squares of the residuals. Which line is a better fit? X 1 2 3 4 Y 7 5 6 9

Example1: Continued Find the residuals y = x + 4: Sum of squared residuals: (2)2 + (–1)2 + (–1)2 + (1)2 4 + 1 + 1 + 1 = 7

Example1: Continued Find the residuals y = 2x + 2: Sum of squared residuals: (3)2 + (–1)2 + (–2)2 + (-1)2 9 + 1 + 4 + 1 = 15 The line y = x + 4 is a better fit.

Check It Out! Example 1 Two lines of fit for this data are For each line, find the sum of the squares of the residuals. Which line is a better fit? Y = - 2 1 x + 6 and y = -x + 8

Check It Out! Example 1 Continued Find the residuals. 2 1 y = – x + 6 : Sum of squared residuals: (–2)2 + (2)2 + (–2)2 + (2)2 4 + 4 + 4 + 4 = 16

Check It Out! Example 1 Continued Find the residuals. y = –x + 8: Sum of squared residuals: (–3)2 + (2)2 + (–1)2 + (4)2 9 + 4 + 1 + 16 = 30 y = - 2 1 The line x + 6 is a better fit

Lesson Quiz : The table shows time spent on homework and number of incorrect quiz answers for several students. Two lines of fit are y = -x + 11 and y = -0.5x + 8. Find the sum of the squares of the residuals for each line. Which line is a better fit? 18; 20; y = -x + 11