2.6
Before
Now
Why?
Key Vocabulary
• scatter plot
• positive correlation
• negative correlation
• correlation
Draw Scatter Plots
and Best-Fitting Lines
You wrote equations of lines.
You will fit lines to data in scatter plots.
So you can model sports trends, as in Ex. 27.
A scatter plot is a graph of a set of data pairs (x, y). If y tends to increase as x
increases, then the data have a positive correlation. If y tends to decrease as
x increases, then the data have a negative correlation. If the points show no
obvious pattern, then the data have approximately no correlation.
y
y
coefficient
• best-fitting line
y
x
x
Positive
correlation
x
Approximately
no correlation
Negative
correlation
EXAMPLE 1
Describe correlation
TELEPHONES Describe the correlation shown by each scatter plot.
y
160
120
80
40
0
Cellular Phone Subscribers and
Corded Phone Sales,
1995–2003
Corded phone sales
(millions of dollars)
Cellular service regions
(thousands)
Cellular Phone Subscribers and
Cellular Service Regions,
1995–2003
0
40
80 120 160 x
Subscribers (millions)
y
550
450
350
250
0
40
80 120 160 x
Subscribers (millions)
Solution
The first scatter plot shows a positive correlation, because as the number of
cellular phone subscribers increased, the number of cellular service regions
tended to increase.
The second scatter plot shows a negative correlation, because as the number of
cellular phone subscribers increased, corded phone sales tended to decrease.
2.6 Draw Scatter Plots and Best-Fitting Lines
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CORRELATION COEFFICIENTS A correlation coefficient, denoted by r, is a number
from 21 to 1 that measures how well a line fits a set of data pairs (x, y). If r is near
1, the points lie close to a line with positive slope. If r is near 21, the points lie
close to a line with negative slope. If r is near 0, the points do not lie close to
any line.
r 5 21
Points lie near line
with a negative slope.
EXAMPLE 2
r50
r51
Points do not lie
near any line.
Points lie near line
with positive slope.
Estimate correlation coefficients
Tell whether the correlation coefficient for the data is closest to 21, 20.5,
0, 0.5, or 1.
a.
b.
y
150
c.
y
150
y
150
100
100
100
50
50
50
0
0
2
4
0
6x
0
2
4
0
6x
0
2
4
6x
Solution
a. The scatter plot shows a clear but fairly weak negative correlation. So, r
is between 0 and 21, but not too close to either one. The best estimate
given is r 5 20.5. (The actual value is r ø 20.46.)
b. The scatter plot shows approximately no correlation. So, the best
estimate given is r 5 0. (The actual value is r ø 20.02.)
c. The scatter plot shows a strong positive correlation. So, the best
estimate given is r 5 1. (The actual value is r ø 0.98.)
✓
GUIDED PRACTICE
for Examples 1 and 2
For each scatter plot, (a) tell whether the data have a positive correlation, a
negative correlation, or approximately no correlation, and (b) tell whether the
correlation coefficient is closest to 21, 20.5, 0, 0.5, or 1.
1.
2.
y
3.
y
y
100
100
100
50
50
50
0
0
2
4
6x
0
0
2
4
6x
0
0
2
4
6x
BEST-FITTING LINES If the correlation coefficient for a set of data is near 61, the
data can be reasonably modeled by a line. The best-fitting line is the line that lies
as close as possible to all the data points. You can approximate a best-fitting line
by graphing.
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Chapter 2 Linear Equations and Functions
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For Your Notebook
KEY CONCEPT
Approximating a Best-Fitting Line
Draw a scatter plot of the data.
STEP 1
STEP 2 Sketch the line that appears to follow most closely the trend given
by the data points. There should be about as many points above the
line as below it.
STEP 3 Choose two points on the line, and estimate the coordinates of each
point. These points do not have to be original data points.
STEP 4 Write an equation of the line that passes through the two points
from Step 3. This equation is a model for the data.
EXAMPLE 3
Approximate a best-fitting line
ALTERNATIVE-FUELED VEHICLES The table shows
the number y (in thousands) of alternative-fueled
vehicles in use in the United States x years after
1997. Approximate the best-fitting line for the data.
x
0
1
2
3
4
5
6
7
y
280
295
322
395
425
471
511
548
Solution
y
550
STEP 1 Draw a scatter plot of the data.
(7, 548)
STEP 2 Sketch the line that appears to
STEP 3 Choose two points that appear
to lie on the line. For the line
shown, you might choose
(1, 300), which is not an original
data point, and (7, 548), which is
an original data point.
STEP 4 Write an equation of the line.
First find the slope using the
points (1, 300) and (7, 548).
Number of vehicles
(thousands)
best fit the data. One possibility
is shown.
500
450
400
350
300
250
0
(1, 300)
0
2
4
6
8 x
Years since 1997
548 2 300 5 248 ø 41.3
m5}
}
721
6
Use point-slope form to write the equation. Choose (x1, y1) 5 (1, 300).
y 2 y1 5 m(x 2 x1)
y 2 300 5 41.3(x 2 1)
y ø 41.3x 1 259
Point-slope form
Substitute for m, x1, and y1.
Simplify.
c An approximation of the best-fitting line is y 5 41.3x 1 259.
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at classzone.com
2.6 Draw Scatter Plots and Best-Fitting Lines
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EXAMPLE 4
Use a line of fit to make a prediction
Use the equation of the line of fit from Example 3 to predict the number
of alternative-fueled vehicles in use in the United States in 2010.
Solution
Because 2010 is 13 years after 1997, substitute 13 for x in the equation from
Example 3.
y 5 41.3x 1 259 5 41.3(13) 1 259 ø 796
c You can predict that there will be about 796,000 alternative-fueled vehicles
in use in the United States in 2010.
LINEAR REGRESSION Many graphing calculators have a linear regression feature
that can be used to find the best-fitting line for a set of data.
EXAMPLE 5
Use a graphing calculator to find a best-fitting line
Use the linear regression feature on a graphing calculator to find an
equation of the best-fitting line for the data in Example 3.
Solution
FIND CORRELATION
If your calculator
does not display the
correlation coefficient
r when it displays the
regression equation,
you may need to select
DiagnosticOn from the
CATALOG menu.
STEP 1 Enter the data into two lists.
Press
and then select Edit.
Enter years since 1997 in L1 and
number of alternative-fueled
vehicles in L 2.
L1
L2
0
280
1
295
2
322
3
395
4
425
L1(2)=1
L3
STEP 3 Make a scatter plot of the
data pairs to see how well the
regression equation models the
data. Press
[STAT PLOT] to
set up your plot. Then select an
appropriate window for the graph.
STEP 2 Find an equation of the bestfitting (linear regression) line. Press
, choose the CALC menu, and
select LinReg(ax1b). The equation
can be rounded to y 5 40.9x 1 263.
LinReg
y=ax+b
a=40.86904762
b=262.83333333
r=.9929677507
STEP 4 Graph the regression equation
with the scatter plot by entering the
equation y 5 40.9x 1 263. The graph
(displayed in the window 0 ≤ x ≤ 8 and
200 ≤ y ≤ 600) shows that the line fits
the data well.
Plot1 Plot2 Plot3
On Off
Type
XList:L1
YList:L2
Mark:
+
c An equation of the best-fitting line is y 5 40.9x 1 263.
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✓
GUIDED PRACTICE
for Examples 3, 4, and 5
4. OIL PRODUCTION The table shows the U.S. daily oil production y (in
thousands of barrels) x years after 1994.
x
0
1
2
3
4
5
6
7
8
y
6660
6560
6470
6450
6250
5880
5820
5800
5750
a. Approximate the best-fitting line for the data.
b. Use your equation from part (a) to predict the daily oil production in 2009.
c. Use a graphing calculator to find and graph an equation of the best-fitting
line. Repeat the prediction from part (b) using this equation.
2.6
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 9, 11, and 25
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 16, 18, 21, and 28
5 MULTIPLE REPRESENTATIONS
Ex. 27
SKILL PRACTICE
1. VOCABULARY Copy and complete: A line that lies as close as possible to a set
of data points (x, y) is called the ? for the data points.
2. ★ WRITING Describe how to tell whether a set of data points shows a positive
correlation, a negative correlation, or approximately no correlation.
EXAMPLE 1
on p. 113
for Exs. 3–5
DESCRIBING CORRELATIONS Tell whether the data have a positive correlation,
a negative correlation, or approximately no correlation.
3.
y
5. y
60
30
6
40
20
4
20
10
2
4.
y
0
0
2
4
6
0
8x
0
2
4
6
0
8x
0
2
4
8x
6
6. REASONING Explain how you can determine the type of correlation for a set
of data pairs by examining the data in a table without drawing a scatter plot.
EXAMPLE 2
on p. 114
for Exs. 7–9
CORRELATION COEFFICIENTS Tell whether the correlation coefficient for the
data is closest to 21, 20.5, 0, 0.5, or 1.
7.
y
60
8.
y
60
9.
y
60
40
40
40
20
20
20
0
0
2
4
6
8 x
0
0
2
4
6
8 x
0
0
2
4
6
2.6 Draw Scatter Plots and Best-Fitting Lines
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8 x
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EXAMPLES
3 and 4
on pp. 115–116
for Exs. 10–15
BEST-FITTING LINES In Exercises 10–15, (a) draw a scatter plot of the data,
(b) approximate the best-fitting line, and (c) estimate y when x 5 20.
10.
12.
14.
11.
x
1
2
3
4
5
y
10
22
35
49
62
x
12
25
36
50
64
y
100
75
52
26
9
x
5.6
6.2
7
7.3
8.4
y
120
130
141
156
167
13.
15.
x
1
2
3
4
5
y
120
101
87
57
42
x
3
7
10
15
18
y
16
45
82
102
116
x
16
24
39
55
68
y
3.9
3.7
3.4
2.9
2.6
16. ★ MULTIPLE CHOICE Which equation best
y
models the data in the scatter plot?
20
A y 5 15
1 x 1 26
B y 5 2}
2
10
2 x 1 19
C y 5 2}
4 x 1 33
D y 5 2}
0
5
5
0
10
20
x
30
17. ERROR ANALYSIS The graph shows one
y
student’s approximation of the bestfitting line for the data in the scatter
plot. Describe and correct the error in the
student’s work.
40
20
2
4
6
x
8
18. ★ MULTIPLE CHOICE A set of data has correlation coefficient r. For which
value of r would the data points lie closest to a line?
A r 5 20.96
EXAMPLE 5
on p. 116
for Exs. 19–20
B r50
C r 5 0.38
D r 5 0.5
GRAPHING CALCULATOR In Exercises 19 and 20, use a graphing calculator to
find and graph an equation of the best-fitting line.
19.
20.
x
78
74
68
76
80
84
50
76
55
93
y
5.1
5.0
4.6
4.9
5.3
5.5
3.7
5.0
3.9
5.8
x
7000
7400
7800
8100
8500
8800
9200
9500
9800
y
56.0
54.5
51.9
50.0
47.3
45.6
43.1
41.6
39.9
21. ★ OPEN-ENDED MATH Give two real-life quantities that have (a) a positive
correlation, (b) a negative correlation, and (c) approximately no correlation.
22. REASONING A set of data pairs has correlation coefficient r 5 0.1. Is it logical
to use the best-fitting line to make predictions from the data? Explain.
23. CHALLENGE If x and y have a positive correlation and y and z have a negative
correlation, what can you say about the correlation between x and z? Explain.
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5 WORKED-OUT SOLUTIONS
Chapter 2 Linear
and Functions
on p. Equations
WS1
★
5 STANDARDIZED
TEST PRACTICE
5 MULTIPLE
REPRESENTATIONS
10/20/05 10:07:02 AM
PROBLEM SOLVING
GRAPHING CALCULATOR You may wish to use a graphing calculator to
complete the following Problem Solving exercises.
EXAMPLES
3, 4, and 5
on pp. 115–116
for Exs. 24–28
24. POPULATION The data pairs (x, y) give the population y (in millions) of Texas
x years after 1997. Approximate the best-fitting line for the data.
(0, 19.7), (1, 20.2), (2, 20.6), (3, 20.9), (4, 21.3), (5, 21.7), (6, 22.1), (7, 22.5)
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
25. TUITION The data pairs (x, y) give U.S. average annual public college tuition
y (in dollars) x years after 1997. Approximate the best-fitting line for the data.
(0, 2271), (1, 2360), (2, 2430), (3, 2506), (4, 2562), (5, 2727), (6, 2928)
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
26. PHYSICAL SCIENCE The diagram
shows the boiling point of water at
various elevations. Approximate the
best-fitting line for the data pairs
(x, y) where x represents the elevation
(in feet) and y represents the boiling
point (in degrees Fahrenheit). Then
use this line to estimate the boiling
point at an elevation of 14,000 feet.
27.
MULTIPLE REPRESENTATIONS The table shows the numbers of
countries that participated in the Winter Olympics from 1980 to 2002.
Year
Countries
1980
1984
1988
1992
1994
1998
2002
37
49
57
64
67
72
77
a. Making a List Use the table to make a list of data pairs (x, y) where x
represents years since 1980 and y represents the number of countries.
b. Drawing a Graph Draw a scatter plot of the data pairs from part (a).
c. Writing an Equation Write an equation that approximates the best-fitting
line, and use it to predict the number of participating countries in 2014.
28. ★ EXTENDED RESPONSE The table shows manufacturers’ shipments
(in millions) of cassettes and CDs in the United States from 1988 to 2002.
Year
1988
1990
1992
1994
1996
1998
2000
2002
Cassettes
450.1
442.2
336.4
345.4
225.3
158.5
76.0
31.1
CDs
149.7
286.5
407.5
662.1
778.9
847.0
942.5
803.3
a. Draw a scatter plot of the data pairs (year, shipments of cassettes).
Describe the correlation shown by the scatter plot.
b. Draw a scatter plot of the data pairs (year, shipments of CDs).
Describe the correlation shown by the scatter plot.
c. Describe the correlation between cassette shipments and CD shipments.
What real-world factors might account for this?
2.6 Draw Scatter Plots and Best-Fitting Lines
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29. CHALLENGE Data from some countries in North America show a positive
correlation between the average life expectancy in a country and the
number of personal computers per capita in that country.
a. Make a conjecture about the reason for the positive correlation between
life expectancy and number of personal computers per capita.
b. Is it reasonable to conclude from the data that giving residents of a
country more personal computers will lengthen their lives? Explain.
MIXED REVIEW
Solve the equation for y. Then find the value of y for the given value of x. (p. 26)
30. 2x 2 y 5 10; x 5 8
31. 6y 1 x 5 25; x 5 1
32. x 2 4y 5 3; x 5 23
33. 23x 1 4y 1 5 5 0; x 5 22
34. 20.5y 1 0.25x 5 2; x 5 4
35. xy 2 4x 5 9; x 5 6
Evaluate the function for the given value of x. (p. 72)
PREVIEW
Prepare for
Lesson 2.7
in Exs. 36–41.
36. f(x) 5 2x 1 7; f(9)
37. f(x) 5 24x 2 11; f(25)
38. f(x) 5 14 2 x; f (22)
39. f(x) 5 x 2 10; f(10)
40. f(x) 5 26 2 x; f (4)
41. f(x) 5 2x 1 8 2 1; f (23)
42. y 5 x 1 8
43. y 5 2x 2 14
44. y 5 5x 1 9
45. 2x 1 y 5 1
46. 3x 2 2y 5 24
47. x 1 3y 5 15
Graph the equation. (p. 89)
QUIZ for Lessons 2.4–2.6
Write an equation of the line that satisfies the given conditions. (p. 98)
1. m 5 25, b 5 3
2. m 5 2, b 5 12
3. m 5 4, passes through (23, 6)
4. m 5 27, passes through (1, 24)
5. passes through (0, 7) and (23, 22)
6. passes through (29, 9) and (29, 0)
Write and graph a direct variation equation that has the given ordered pair as a
solution. (p. 107)
7. (1, 2)
8. (22, 8)
9. (5, 216)
10. (12, 4)
The variables x and y vary directly. Write an equation that relates x and y. Then
find y when x 5 8. (p. 107)
11. x 5 4, y 5 12
12. x 5 23, y 5 28
13. x 5 40, y 5 25
14. x 5 12, y 5 2
15. CONCERT TICKETS The table shows the average price of a concert ticket to
one of the top 50 musical touring acts for the years 1999–2004. Write an
equation that approximates the best-fitting line for the data pairs (x, y). Use
the equation to predict the average price of a ticket in 2010. (p. 113)
Years since 1999, x
Ticket price (dollars), y
120
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0
1
2
3
4
5
38.56
44.80
46.69
50.81
51.81
58.71
PRACTICE
Lesson 2.6, p. 1011
Chapter 2EXTRA
Linear Equations
andfor
Functions
ONLINE QUIZ at classzone.com
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