2-7 Curve Fitting with Linear Models
Objectives
Fit scatter plot data using linear models
with and without technology.
Use linear models to make predictions.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
A scatter plot is helpful in understanding the form,
direction, and strength of the relationship between two
variables. Correlation is the strength and direction of
the linear relationship between the two variables.
If there is a strong linear relationship between two variables, a line of
best fit, or a line that best fits the data, can be used to make predictions.
Helpful Hint
Try to have about the same number of points
above and below the line of best fit.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Albany and Sydney are
about the same distance
from the equator. Make
a scatter plot with
Albany’s temperature as
the independent
variable. Name the type
of correlation. Then
sketch a line of best fit
and find its equation.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Step 1 Plot the data points.
Step 2 Identify the correlation.
Notice that the data set is
negatively correlated–as the
temperature rises in Albany, it
falls in Sydney.
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Holt Algebra 2
2-7 Curve Fitting with Linear Models
Step 3 Sketch a line of best fit.
Draw a line that splits
the data evenly above
and below.
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Holt Algebra 2
2-7 Curve Fitting with Linear Models
Step 4 Identify two points on the line.
For this data, you might select (35, 64) and
(85, 41).
Step 5 Find the slope of the line that models the
data.
Use the point-slope form.
Point-slope form.
y – y1= m(x – x1)
y – 64 = –0.46(x – 35)
y = –0.46x + 80.1
Substitute.
Simplify.
An equation that models the data is y = –0.46x + 80.1.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Make a scatter plot for this set of data.
Identify the correlation, sketch a line of best
fit, and find its equation.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Step 1 Plot the data points.
Step 2 Identify the correlation.
Notice that the data set is
positively correlated–as time
increases, more points are
scored
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Step 3 Sketch a line of best fit.
Draw a line that splits
the data evenly above
and below.
Holt Algebra 2
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2-7 Curve Fitting with Linear Models
Step 4 Identify two points on the line.
For this data, you might select (20, 10) and (40, 25).
Step 5 Find the slope of the line that models the data.
Use the point-slope form.
y – y1= m(x – x1)
y – 10 = 0.75(x – 20)
y = 0.75x – 5
Point-slope form.
Substitute.
Simplify.
A possible answer is p = 0.75x + 5.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Anthropologists can
use the femur, or
thighbone, to estimate
the height of a human
being. The table shows
the results of a
randomly selected
sample.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
a. Make a scatter
plot of the data
with femur
length as the
independent
variable.
The scatter plot is
shown at right.
Holt Algebra 2
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2-7 Curve Fitting with Linear Models
b. Find the line of best fit. Interpret the slope of the
line of best fit in the context of the problem.
Enter the data into lists L1
and L2 on a graphing
calculator. Use the linear
regression feature by
pressing STAT, choosing
CALC, and selecting
4:LinReg. The equation of
the line of best fit is
h ≈ 2.91l + 54.04.
The slope is about 2.91, so for each 1 cm increase in
femur length, the predicted increase in a human being’s
height is 2.91 cm.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
c. A man’s femur is 41 cm long. Predict the
man’s height.
The equation of the line of best fit is
h ≈ 2.91l + 54.04. Use the equation to predict the
man’s height.
For a 41-cm-long femur,
h ≈ 2.91(41) + 54.04 Substitute 41 for l.
h ≈ 173.35
The height of a man with a 41-cm-long femur
would be about 173 cm.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
The gas mileage for randomly selected cars based
upon engine horsepower is given in the table.
a. Make a scatter
plot of the data
with horsepower
as the independent
variable.
Holt Algebra 2
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2-7 Curve Fitting with Linear Models
b. Find the correlation coefficient r and the line of
best fit. Interpret the slope of the line of best
fit in the context of the problem.
Enter the data into lists L1
and L2 on a graphing
calculator. Use the linear
regression feature by
pressing STAT, choosing
CALC, and selecting
4:LinReg. The equation of
the line of best fit is
y ≈ –0.15x + 47.5.
The slope is about –0.15, so for each 1 unit increase in
horsepower, gas mileage drops ≈ 0.15 mi/gal.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Check It Out! Example 2 Continued
c. Predict the gas mileage for a 210-horsepower
engine.
The equation of the line of best fit is
y ≈ –0.15x + 47.5. Use the equation to predict
the gas mileage. For a 210-horsepower engine,
y ≈ –0.15(210) + 47.50.
Substitute 210 for x.
y ≈ 16
The mileage for a 210-horsepower engine would be
about 16.0 mi/gal.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Example 3: Meteorology Application
Find the following for
this data on average
temperature and
rainfall for eight
months in Boston, MA.
Holt Algebra 2
2-7 Curve Fitting with Linear Models
Example 3 Continued
a. Make a scatter plot of the data with
temperature as the independent variable.
The scatter plot is
shown on the right.
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Holt Algebra 2
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2-7 Curve Fitting with Linear Models
Example 3 Continued
b. Find the correlation coefficient and the
equation of the line of best fit. Draw the line of
best fit on your scatter plot.
The correlation
coefficient is
r = –0.703.
The equation of the
line of best fit is
y ≈ –0.35x + 106.4.
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Holt Algebra 2
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2-7 Curve Fitting with Linear Models
Example 3 Continued
c. Predict the temperature when the rainfall
is 86 mm. How accurate do you think
your prediction is?
86 ≈ –0.35x + 106.4 Rainfall is the dependent variable.
–20.4 ≈ –0.35x
58.3 ≈ x
The line predicts 58.3F, but the scatter plot and the
value of r show that temperature by itself is not an
accurate predictor of rainfall.
Holt Algebra 2