4-8
Line of
of Best
Best Fit
Fit
4-8 Line
Warm Up
Lesson Presentation
Lesson Quiz
Holt
1 Algebra
HoltAlgebra
McDougal
Algebra11
McDougal
4-8
Line of Best Fit
Warm Up
Identify the slope and the y-intercept.
1. y = 2x + 1
m = 2, b = 1
m= 2
2. y = 2 x - 4
3 , b = -4
3
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
Holt McDougal Algebra 1
4-8
Line of Best Fit
Objectives
Determine a line of best fit for a set of linear
data. Determine and interpret the correlation
coefficient.
Holt McDougal Algebra 1
4-8
Line of Best Fit
Vocabulary
residual
least-squares line
line of best fit
linear regression
correlation coefficient
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
Helpful Hint
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.
Holt McDougal Algebra 1
4-8
Line of Best Fit
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
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
Y = - 1 x + 6 and y = -x + 8
2
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
Check It Out! Example 1 Continued
Find the residuals.
x+6:
y=– 1
2
Sum of squared residuals:
(–2)2 + (2)2 + (–2)2 + (2)2
4 + 4 + 4 + 4 = 16
Holt McDougal Algebra 1
4-8
Line of Best Fit
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
1
The line y = - 2 x + 6
is a better fit
Holt McDougal Algebra 1
4-8
Line of Best Fit
Lesson Quiz :
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