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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)