Holt McDougal Algebra 1

4-8 Line of Best Fit 4-8 Line of Best Fit

Holt Algebra 1

Lesson Quiz Lesson Presentation

Warm Up

Holt McDougal Algebra 1

Holt McDougal Algebra 1

4-8 Line of Best Fit

Determine a line of best fit for a set of linear

data. Determine and interpret the correlation

coefficient.

Objectives

Holt McDougal Algebra 1

4-8 Line of Best Fit

residual

least-squares line

line of best fit

linear regression

correlation coefficient

Vocabulary

Holt McDougal Algebra 1

4-8 Line of Best Fit

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.

Holt McDougal Algebra 1

4-8 Line of Best Fit

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.

Holt McDougal Algebra 1

4-8 Line of Best Fit Example1: Calculating Residuals

X 1 2 3 4 Y 7 5 6 9

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?

Holt McDougal Algebra 1

4-8 Line of Best Fit 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

Holt McDougal Algebra 1

4-8 Line of Best Fit 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.

Holt McDougal Algebra 1

4-8 Line of Best 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

Holt McDougal Algebra 1

4-8 Line of Best Fit Check It Out! Example 1 Continued

Find the residuals.

Sum of squared residuals:

(–2)2 + (2)2 + (–2)2 + (2)2

4 + 4 + 4 + 4 = 16

21y = – x + 6 :

Holt McDougal Algebra 1

4-8 Line of Best Fit Check It Out! Example 1 Continued

Find the residuals.

Sum of squared residuals:

(–3)2 + (2)2 + (–1)2 + (4)2

9 + 4 + 1 + 16 = 30

y = –x + 8:

y = - 2 1

The line x + 6 is a better fit

Holt McDougal Algebra 1

4-8 Line of Best Fit

r-values close to 1 or –1 indicate a very strong correlation. The closer r is to 0, the weaker the correlation.

Helpful Hint

Holt McDougal Algebra 1

4-8 Line of Best Fit Lesson Quiz : Part-1

The table shows time spent on homework and number of incorrect quiz answers for several students.

1.  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

Holt McDougal Algebra 1

4-8 Line of Best Fit Lesson Quiz : Part-2

2. Find an equation for a line of best fit. Interpret the meaning of the slope and y-intercept. Use your equation to predict the number of incorrect answers for 5 hours of study.

y ≈ -0.8x + 10.3; slope: for every hour of study, the number of incorrect answers decreases by 0.8; y-int.: a student who studies for 0 h will get 10.3 incorrect answers; 6.3

Holt McDougal Algebra 1

4-8 Line of Best Fit Lesson Quiz : Part-3

3. How well does the line of best fit represent the data? Explain.

fairly well (r ≈ -0.79)