Curve Fitting with Linear Models

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Curve Fitting with Linear Models
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Write the equation of the line passing
through each pair of passing points in
slope-intercept form. 1. (5, 1), (0, 3) 2.
(8, 5), (8, 7) Use the equation y 0.2x 4.
Find x for each given value of y. 3. y 7
4. y 3.5
x 15
x 2.5
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Objectives
Fit scatter plot data using linear models with
and without technology. Use linear models to
make predictions.
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Vocabulary
regression correlation line of best
fit correlation coefficient
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Researchers, such as anthropologists, are often
interested in how two measurements are related.
The statistical study of the relationship between
variables is called regression.
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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.
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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.
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Example 1 Meteorology Application
Albany and Sydney are about the same distance
from the equator. Make a scatter plot with
Albanys temperature as the independent variable.
Name the type of correlation. Then sketch a line
of best fit and find its equation.
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Example 1 Continued
Step 1 Plot the data points.
Step 2 Identify the correlation.
Notice that the data set is negatively
correlatedas the temperature rises in Albany, it
falls in Sydney.











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Example 1 Continued
Step 3 Sketch a line of best fit.
Draw a line that splits the data evenly above and
below.











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Example 1 Continued
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.
y y1 m(x x1)
Point-slope form.
Substitute.
y 64 0.46(x 35)
y 0.46x 80.1
Simplify.
An equation that models the data is y 0.46x
80.1.
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Check It Out! Example 1
Make a scatter plot for this set of data.
Identify the correlation, sketch a line of best
fit, and find its equation.
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Check It Out! Example 1 Continued
Step 1 Plot the data points.
Step 2 Identify the correlation.
Notice that the data set is positively
correlatedas time increases, more points are
scored










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Check It Out! Example 1 Continued
Step 3 Sketch a line of best fit.
Draw a line that splits the data evenly above and
below.










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Check It Out! Example 1 Continued
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)
Point-slope form.
y 10 0.75(x 20)
Substitute.
y 0.75x 5
Simplify.
A possible answer is p 0.75x 5.
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The correlation coefficient r is a measure of how
well the data set is fit by a model.
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You can use a graphing calculator to perform a
linear regression and find the correlation
coefficient r.
To display the correlation coefficient r, you may
have to turn on the diagnostic mode. To do this,
press and choose the
DiagnosticOn mode.
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Example 2 Anthropology Application
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.
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Example 2 Continued
a. Make a scatter plot of the data with femur
length as the independent variable.








The scatter plot is shown at right.
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Example 2 Continued
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
4LinReg. The equation of the line of best fit is
h 2.91l 54.04.
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Example 2 Continued
The slope is about 2.91, so for each 1 cm
increase in femur length, the predicted increase
in a human beings height is 2.91 cm.
The correlation coefficient is r 0.986 which
indicates a strong positive correlation.
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Example 2 Continued
c. A mans femur is 41 cm long. Predict the
mans height.
The equation of the line of best fit is h
2.91l 54.04. Use the equation to predict the
mans height. For a 41-cm-long femur,
Substitute 41 for l.
h 2.91(41) 54.04
h 173.35
The height of a man with a 41-cm-long femur would
be about 173 cm.
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Check It Out! Example 2
The gas mileage for randomly selected cars based
upon engine horsepower is given in the table.
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Check It Out! Example 2 Continued
a. Make a scatter plot of the data with
horsepower as the independent variable.








The scatter plot is shown on the right.


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Check It Out! Example 2 Continued
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
4LinReg. The equation of the line of best fit is
y 0.15x 47.5.
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Check It Out! Example 2 Continued
The slope is about 0.15, so for each 1 unit
increase in horsepower, gas mileage drops 0.15
mi/gal.
The correlation coefficient is r 0.916, which
indicates a strong negative correlation.
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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.
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Example 3 Meteorology Application
Find the following for this data on average
temperature and rainfall for eight months in
Boston, MA.
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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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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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Example 3 Continued
c. Predict the temperature when the rainfall
is 86 mm. How accurate do you think your
prediction is?
Rainfall is the dependent variable.
86 0.35x 106.4
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.
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Check It Out! Example 3
Find the following information for this data set
on the number of grams of fat and the number of
calories in sandwiches served at Daves Deli.
Use the equation of the line of best fit to
predict the number of grams of fat in a sandwich
with 420 Calories. How close is your answer to
the value given in the table?
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Check It Out! Example 3
a. Make a scatter plot of the data with fat
as the independent variable.
The scatter plot is shown on the right.
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Check It Out! Example 3
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 0.682. The
equation of the line of best fit is y 11.1x
309.8.
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Check It Out! Example 3
c. Predict the amount of fat in a sandwich
with 420 Calories. How accurate do you think
your prediction is?
420 11.1x 309.8
Calories is the dependent variable.
110.2 11.1x
9.9 x
The line predicts 10 grams of fat. This is not
close to the 15 g in the table.
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Lesson Quiz Part I
Use the table for Problems 13.
1. Make a scatter plot with mass as the
independent variable.
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Lesson Quiz Part II
2. Find the correlation coefficient and the
equation of the line of best fit on your scatter
plot. Draw the line of best fit on your
scatter plot.
r 0.67 y 0.07x 5.24
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Lesson Quiz Part III
3. Predict the weight of a 40 tire. How
accurate do you think your prediction is?
646 g the scatter plot and value of r show that
price is not a good predictor of weight.