Scatter Plots and Equations of Lines 5.4 - Weebly
[Pages:7]6-7
1. Plan
Objectives
1 To write an equation for a trend line and use it to make predictions
2 To write an equation for a line of best fit and use it to make predictions
Examples
1 Trend Line 2 Line of Best Fit
Math Background
Lines of best fit can be used to determine whether slopes for two sets of data are approximately equal, indicating a similarity in the relationships.
More Math Background: p. 306D
Lesson Planning and Resources
See p. 306E for a list of the resources that support this lesson.
PowerPoint
Bell Ringer Practice
Check Skills You'll Need For intervention, direct students to: Mean, Median, Mode, and Range Lesson 1-6: Example 4 Extra Skills and Word
Problem Practice, Ch. 1
6-7
Scatter Plots and
Equations of Lines
What You'll Learn
? To write an equation for a
trend line and use it to make predictions
? To write the equation for a
line of best fit and use it to make predictions
. . . And Why
To use a trend line to make a prediction, as in Example 1
Check Skills You'll Need
GO for Help Lesson 1-6
Use the data in each table to draw a scatter plot. 1?2. See back of book.
1. x
y
2. x
y
1
2
2 3
3
8
4
9
5 25
1 21
2 15
3 12
4
9
5
7
New Vocabulary ? line of best fit ? correlation coefficient
1 Writing an Equation for a Trend Line
In Chapter 1 you used scatter plots to determine how two sets of data are related. You can now write an equation for a trend line.
1 EXAMPLE Trend Line
Length and Wingspan of Hawks
WingType of Length span Hawk (in.) (in.)
Cooper's 21 36
Crane
21 41
Gray
18 38
Harris's
24 46
Roadside 16 31
Broadwinged
19 39
Shorttailed
17 35
Swanson's 19 46
SOURCE: Birds of North America
Birds Make a scatter plot of the data at the left. Draw a trend line and write its equation. Use the equation to predict the wingspan of a hawk that is 28 in. long.
Step 1 Make a scatter plot and draw a trend line. Estimate two points on the line.
Length and Wingspan of Hawks
44
Step 2 Write an equation of the trend line.
m
5
y2 x2
2 2
y1 x1
5
39 19
2 2
30 14
5
9 5
y - y1 = m(x - x1) Use point-slope form.
y - 30 = 95(x - 14)
Substitute
9 5
for
m
and (14, 30) for (x1, y1).
Wingspan (in.)
40
36
32
28 14 18 22 26 Length (in.)
Two points on the trend line are (14, 30) and (19, 39).
Step 3 Predict the wingspan of a hawk that is 28 in. long.
y - 30 = 95(28 - 14) Substitute 28 for x.
y - 30 = 95(14)
Simplify within the parentheses.
y - 30 = 25.2 y = 55.2
Multiply. Add 30 to each side.
The wingspan of a hawk 28 in. long is about 55 in.
350 Chapter 6 Linear Equations and Their Graphs
350
Special Needs L1 Some students may have difficulty plotting points accurately on a grid. Have them work with a partner to make sure each point is correctly positioned. Then have them discuss the position of trend lines.
learning style: visual
Below Level L2 To better understand a correlation coefficient, draw and discuss scatter plots that show positive and negative correlations with varying correlation coefficients.
learning style: visual
Quick Check 1 Graph the data below and draw a trend line. Find an equation for the trend line.
Estimate the number of Calories in a fast-food that has 14g of fat. See back of book.
Calories and Fat in Selected Fast-Food Meals
Fat (g)
6
7
10 19 20 27 36
Calories 276 260 220 388 430 550 633
12 Writing an Equation for a Line of Best Fit
The trend line that shows the relationship between two sets of data most accurately is called the line of best fit. A graphing calculator computes the equation of a line of best fit using a method called linear regression.
The graphing calculator also gives you the correlation coefficient r, which tells you how closely the equation models the data.
1
0
1
negative correlation
no correlation
positive correlation
When the data points cluster around a line, there is a strong correlation between the line and the data. So the nearer r is to 1 or -1, the more closely the data cluster around the line of best fit. In later chapters, you will learn how to find non-linear models which may better describe some data.
For: Correlation Activity Use: Interactive Textbook, 6-7
For: Graphing calculator procedures
Web Code: ate-2122
2 EXAMPLE Line of Best Fit
Recreation Use a graphing calculator to find the equation of the line of best fit for the data at the right. What is the correlation coefficient to three decimal places?
Step 1 Use the EDIT feature of the screen on your graphing calculator.
Let 93 correspond to 1993 and 100 correspond to 2000. Enter the data for years and then the data for expenditures.
Step 2 Use the CALC feature in the screen. Find the equation for the
line of best fit.
LinReg y axb a 32.33333333 b 2671.666667 r2 .9929145373 r .9964509708
slope y-intercept
correlation coefficient
Recreation Expenditures
Dollars Year (billions) 1993 340
1994 369
1995 402
1996 430
1997 457
1998 489
1999 527
2000 574
SOURCE: Statistical Abstract of the United States. Go to for a data update. Web Code: atg-9041
The equation for the line of best fit is y = 32.33x - 2671.67 for values of a and b rounded to the nearest hundredth. The correlation coefficient is about 0.996.
Lesson 6-7 Scatter Plots and Equations of Lines 351
Advanced Learners L4 Make several scatter plots with different correlation coefficients. Students match each scatter plot with its corresponding correlation coefficient.
learning style: visual
English Language Learners ELL For Exercises 6-11, help students understand the term correlation by pointing out that it contains the word relation. The correlation coefficient tells you how closely related the equation and the data are.
learning style: verbal
2. Teach
Guided Instruction
PowerPoint
Additional Examples
1 Make a scatter plot to represent the data. Draw a trend line and write an equation for the trend line. Use the equation to predict the time needed to travel 32 miles on a bicycle. See back of book.
Speed on a Bicycle Trip
Miles
5 10 14 18 22
Time (min) 27 46 71 78 107
Graphing 2 EXAMPLE Calculator Hint
Point out that the equation of the line of best fit will be in the form y = ax + b or y = a + bx. The coefficient of x is always the slope and the constant is always the y-intercept.
PowerPoint
Additional Examples
2 Use a graphing calculator to
find the equation of the line of
best fit for the data below. What
is the correlation coefficient?
U.S. Crime Rate
See back
(per 100,000 inhabitants) of book.
Year No. of Crimes
1995
5275.9
1996
5086.6
1997
4930.0
1998
4619.3
1999
4266.8
[Source: Crime in the United States, 1999, FBI, Uniform Crime Reports]
Resources
? Daily Notetaking Guide 6-7 L3
? Daily Notetaking Guide 6-7--
Adapted Instruction
L1
Closure
Ask: What is the difference between a trend line and a line of best fit? A trend line is an approximation for a line of best fit.
351
3. Practice
Assignment Guide
1 A B 1-6, 12-13, 15, 19
2 A B 7-11, 14, 16-18
C Challenge
20
Test Prep Mixed Review
21-23 24-35
Homework Quick Check
To check students' understanding of key skills and concepts, go over Exercises 4, 10, 12, 15, 16.
Error Prevention!
Exercise 1?5 Some students may expect their trend lines to be exactly like those of their neighbors. Remind them that a trend line is not precise. Thus, their lines and equations may differ slightly.
Exercise 2 Ask students if these data have a strong correlation or a weak correlation.
GPS Guided Problem Solving
L3
Enrichment
L4
Reteaching
L2
Adapted Practice
L1
PNamreactice
Class
Date
L3
Practice 6-6
Scatter Plots and Equations of Lines
Decide whether the data in each scatter plot follow a linear pattern. If they do, find the equation of a trend line.
1.
y
21
20 15 10 5x
O2
2.
y
7
6
4 3
1O
x
4321 1 2 3 4 5 6
3.
y
140
120
100
80
60
40
20 O x
7654321 1 1
Use a graphing calculator to find the equation of the line of best fit for the following data. Find the value of the correlation coefficient r and determine if there is a strong correlation between the data.
4. x
y
5. x
y
6. x
y
1
7
1
6
1
5
2
5
2
15
4
8
3 -1
3 -5
8
3
4
3
4
1
13
10
5 -5
5 -2
19
13
Draw a scatter plot. Write the equation of the trend line.
8. x y 1 17 2 20 3 22 4 26 5 28 6 31
9.
U.S. Union
Year Membership
(millions)
1988
17.00
1989
16.96
1990
16.74
1991
16.57
1992
16.39
10. x y 1 18 2 20 3 24 4 30 5 28 6 33
1993
16.60
1994
16.75
1995
16.36
1996
16.27
1997
16.11
1998
16.21
Source: World Almanac 2000, p. 154.
7. x
y
12 28
15 50
18 14
21 28
24 36
11.
U.S.
Year Unemployment
Rate (%)
1988
5.5
1989
5.3
1990
5.6
1991
6.8
1992
7.5
1993
6.9
1994
6.1
1995
5.6
1996
5.4
1997
4.9
1998
4.5
Source: World Almanac 2000, p. 145.
? Pearson Education, Inc. All rights reserved.
Quick Check 2 Find the equation of the line of best fit. Let 95 correspond to 1995 and 103
correspond to 2003. What is the correlation coefficient to three decimal places?
2000 2001 2002 2003
y 0.52x ? 43.99; 0.990
EXERCISES
For more exercises, see Extra Skill and Word Problem Practice.
Practice and Problem Solving
A Practice by Example
1?6. Trend lines may vary. Samples Find an equation of a reasonable trend line for each scatter plot. are given.
Example 1
1.
(page 350)
GO
for Help
65 60
Cable TV
2.
Animal Longevity
and Gestation
600
Subscribers (millions)
55
400
50
200
Gestation (days)
0 '90 '92 '94 '96 '98 y ? 52.5 Y2e(xar? 91)
0 10 20 30 40
Longevity (years) y ? 100 15.71(x ? 5)
Response Speed (s)
Points per Game
3. 22 20
NBA Players
4. 80 60 40
Memory Test
18
20
16
0 70 72 74 76 Games Played
y ? 16.4 0.64(x ? 69.9)
0 12345 Study Time (min)
y ? 85 ?15.25(x ? 1)
Revenue (billions of dollars)
5. 10.0 9.0 8.0 7.0
5. Graph the data in the table below for the attendance and revenue at theme parks. Find an equation for a trend line of the data. See left.
Attendance and Revenue at U.S. Theme Parks
Year
1995 1996 1997 1998 1999 2000 2001 2002 2003
Attendance (millions)
280 290 300 300 309 317 319 324 322
270 290 310 330 Attendance (millions)
Revenue (billions of dollars)
7.4 7.9 8.4 8.7 9.1 9.6 9.6 9.9 10.3
y ? 7.5 0.06(x ? 280)
SOURCE: International Association of Amusement Parks and Attractions. Go to for a data update. Web Code: atg-9041
352 Chapter 6 Linear Equations and Their Graphs
352
Average Annual Precipitation (in.)
6.
y
58
54
50
46
6. Graph the data for the average July temperature and the annual precipitation of the cities in the table below. Find an equation for the line of best fit of the data. Estimate the average rainfall for a city with average July temperature of 758 F.
Precipitation and Temperature in Selected Eastern Cities
City New York
Average July Temperature (?F)
76.4
Average Annual Precipitation (in.)
42.82
42
Baltimore
76.8
41.84
38
x
O 72 76 80 84
Average July
Temperature (F)
Atlanta
78.6
Jacksonville
81.3
Washington, D.C.
78.9
Boston
73.5
48.61 52.76 39.00 43.81
y 1.6x ? 80; 40 in.
Miami
SOURCE: Time Almanac
82.5
57.55
Example 2 (page 351)
Graphing Calculator Use a graphing calculator to find the equation of the line of best fit for the data. Find the value of the correlation coefficient r to three decimal places.
7.
Average Temperatures in Northern Latitudes
Latitude (? N) 0 10 20 30 40 50 60 70 80
Temp. (?F) 79.2 80.1 77.5 68.7 57.4 42.4 30.0 12.7 1.0 y ?1.06x ? 92.31.; ?0.970
8.
Retail Department Store Sales (billions of dollars)
Year
1980 1985 1990 1994 1995 1996 1997 1998
Sales
86 126 166 217 231 245 261 279
SOURCE: Statistical Abstract of the United States. Go to for an update. Web Code: atg-9041
y 10.60x ? 772.66; 0.991
9. Olympic 5000-Meter Men's Gold Medal Speed Skating Times
Year
1980 1984 1988 1992 1994 1998 2002
Time (seconds)
422
432
404
420
395
382
402
SOURCE: International Skating Union
y ?1.63x ? 556.76; ?0.725
10.
Average Male Lung Power
Respiration (breaths/min)
50
30
25
20
18
16
14
Real-World Connection
Heart Rate (beats/min)
200 150 140 130 120 110 100
The 500-meter men's speed skating race has been an Olympic event since 1924.
SOURCE: Encyclopeadia Britannica y 2.64x ? 70.51; 0.990
11.
Wind Chill Temperature for 15 mi/h Wind
Air Temp. (?F)
35 30 25 20 15 10 5 0
Wind-Chill Temp. (?F) 16 9 2 5 11 18 25 31
y 1.35x ? 31.42; 1.000
Lesson 6-7 Scatter Plots and Equations of Lines 353
Connection to Geography Exercise 7 Longitudes and
latitudes make up a coordinate system used in designating the location of places on the surface of Earth. Latitude gives location north or south of the equator. It is expressed by angle measurements ranging from 0? at the equator to 90? at the poles.
Exercise 11 The r-value rounds to 1,000 since its value is 0.999808967.
353
Connection to Geometry
Exercise 12 The circumference of a circle divided by its diameter equals p.
pages 352?356 Exercises 12a. y
28
14
O 2 6 10 x
13a.
y
144
Female (millions)
140
136
132
128
x
O 122 126 130 134 138
Male (millions)
b. Answers may vary. Sample: y 0.939x ? 13.8
d. Answers may vary. Sample: No, the year is too far in the future.
14a. Check students' work.
b. 1
1 20a.
y
Stopping Distance (ft)
132 66
O 16 32 x Speed (mi/h)
y 4.82x ? 29.65
B Apply Your Skills
12c. Answers may vary. Sample: The slope is the approximate ratio of the circumference to the diameter.
Population Growth
1790
1860
Today
More than 2 persons per square mile
15. Answers may vary. Sample: Pos. slope; as temp. increases, more students are absent.
12. Geometry Students measured the diameters and circumferences of the tops of GPS a variety of cylinders. Below is the data that they collected.
Cylinder Tops
Diameter (cm)
3 3 5 6 8 8 9.5 10 10 12
Circumference (cm) 9.3 9.5 16 18.8 25 25.6 29.5 31.5 30.9 39.5
a. Graph the data. See margin. b. Find the equation of a trend line.
c. What does the slope of the equation mean? See left.
y 3.25x ? 1
d. Find the diameter of a cylinder with a circumference of 45 cm. about 14 cm
13a?b. See margin.
13. Population Use the data at the right.
Estimated Population of
a. Graph the data for the male and female
the United States (thousands)
populations of the United States. b. Find the equation of a trend line. c. Use your equation to predict the number
Year Male Female 1991 122,956 129,197
of females if the number of males were to increase to 150,000,000. 154,650,000 d. Critical Thinking Would it be reasonable
1992 124,424 130,606 1993 125,788 131,995
to predict the population in 2025 from these data? Explain. See margin.
14. a. Open-Ended Make a table of data for a linear function. Use a graphing calculator
1994 1995 1996
127,049 128,294 129,504
133,278 134,510 135,724
to find the equation of the line of best fit.
1997 130,783 137,001
b. What is the correlation coefficient for your linear data? a?b. See margin.
1998 132,030 138,218
15. Writing What kind of trend line do you think 1999 133,277 139,414
data for the following comparison would be
2000 138,054 143,368
likely to show? Explain. See left. temperature and the number of students
SOURCE: U.S. Census Bureau. Go to
for a data update. Web Code: atg-9041
absent from school
16. Graphing Calculator A school collected data on math and science grades of nine randomly selected students.
Student 1 2 3 4 5 6 7 8 9
Math
76 89 84 79 94 71 79 91 84
Science 82 94 89 89 94 84 68 89 84
16a. y 0.61x ? 35.31
b. Answers may vary. Sample: No; small set of data with weak correlation
a. Use a graphing calculator to find the equation of the line of best fit for the data. a?b. See left.
b. Critical Thinking Should the equation for the line of best fit be used to predict grades? Explain.
17. Graphing Calculator Use a graphing calculator to find the equation of the line of best fit for the data below. Predict sales of greeting cards in the year 2010.
Greeting Card Sales
Year
1989 1990 1991 1992 1993 1994 1995 1996 1997 1998
Sales (billions)
$4.2 $4.6 $5.0 $5.3 $5.6 $5.9 $6.3 $6.8 $7.3 $7.5
SOURCE: Greeting Card Association
y 0.37x ? 28.66; $12.04 billion
354 Chapter 6 Linear Equations and Their Graphs
354
GO nline
Homework Help
Visit: Web Code: ate-0607
C Challenge
20c. The speed is much faster than those speeds used to find the equation of a trend line.
18. a. Data Collection Find two sets of data that you could display in a scatter plot, such as the number of boys and girls in each class in your school, or the population and the number of airports in some states. Then graph the data.
b. Find the equation of a trend line. c. Use the equation to predict another value that could be on your scatter plot. d. What is the correlation coefficient? a?d. Check students' work.
19. Another way you can find a line of best fit is the median-median method. The graph below shows how this method works. The points in red indicate the original data.
y 8
y 8
8 yLine between
Median
first and last
6
6 points
6 median points
4
4
2
2
x 0 2468 0 246
Divide the data into three groups of equal size.
Find and plot the median points, (x-median, y-median).
4 Line of
2
best fit
x
x
8 0 2468
Find the line parallel to the line
between the first and last
median
points
and
1 3
of
the
way
to the middle median point.
a. Estimate two coordinates on the purple line in the graph at the right above.
Find the equation of the line of best fit. (2, 3) and (6, 6); y 0.75x ? 1.5
b. Graphing Calculator You can use a graphing calculator to find the line
of best fit with the median-median method. Below are the coordinates
of the points graphed in red. Use the EDIT feature of the
screen
on your graphing calculator. Use the Med-Med feature to find a line of
best fit. y 0.75x ? 1.21
(1, 2), (1, 3), (2, 3), (2, 4), (3, 2), (3, 3), (4, 3), (5, 1), (6, 6), (7, 9), (8, 8), (8, 7)
20. a. Make a scatter plot of the data below. Then find the equation of the line of best fit. See margin. Car Stopping Distances
Speed (mi/h)
10 15 20 25 30 35 40 45
Stopping Distances (ft) 27 44 63 85 109 136 164 196
b. Use your equation to predict the stopping distance at 90 mi/h. 404 ft c. Critical Thinking The actual stopping distance at 90 mi/h is close to 584 ft.
Why do you think this is not close to your prediction? See left.
Test Prep
Multiple Choice
21. A horizontal line passes through (5, -2). Which other point does it also
pass through? B
A. (5, 2)
B. (-5, -2)
C. (-5, 2)
D. (5, 0)
22. Which of the following equations has a graph that contains the ordered
pairs (-3, 4) and (1, -4)? H
F. x + 2y = 8 G. 2x - y = 4
H. 2x + y = -2 J. x - 2y = -6
lesson quiz, , Web Code: ata-0607
Lesson 6-7 Scatter Plots and Equations of Lines 355
4. Assess & Reteach
PowerPoint
Lesson Quiz
Number of Households in the U.S.
Year Households (millions)
1975
71.1
1980
80.8
1985
86.8
1990
93.3
1995
99.0
[Source: U.S. Census Bureau, Current Population Reports. From Statistical Abstract of the United States, 2000]
1. Graph the data in a scatter plot. Draw a trend line.
y 100
95 90 85 80 75 70
Household (millions)
O '75 '80 '85 '90 '95 '00
Year
2. Write an equation for the trend line. y ? 86.8 1.3(x ? 85)
3. Predict the number of households in the U.S. in 2005. (Use 105 for x.) about 112.8 million households
4. Use a calculator to find the line of best fit for the data. y 1.366x ? 29.91
5. What is the correlation coefficient? 0.9943767027
Alternative Assessment
Have students search for data that would be suitable for a scatter plot. Each student should graph the data, find a trend line, and then use a graphing calculator to find the line of best fit and the correlation coefficient.
355
Test Prep
Resources For additional practice with a variety of test item formats: ? Standardized Test Prep, p. 369 ? Test-Taking Strategies, p. 364 ? Test-Taking Strategies with
Transparencies
Alternative Method Exercise 22 Many times there is
more than one way to solve a problem. For Exercise 22, you can substitute both points into each equation or use the two points to find an equation.
Use this Checkpoint Quiz to check students' understanding of the skills and concepts of Lessons 6-5 through 6-7.
Resources Grab & Go ? Checkpoint Quiz 2
Elderly (millions)
pages 352?356 Exercises
23. [4] a.
y
24
12
x
O
70 90
Year (19?)
b ? c. Answers may vary. Samples:
b. Let 1960 60. Two points on line are (62, 18) and (90, 30). y 0.429x ? 8.6
c. y 0.429(105) ? 8.6; 36.4 million people
356
Extended Response
23. The table right shows the number of elderly in the United States from 1960 through 2000. a. Graph the data and draw a trend line. b. Write an equation for the trend line you drew. c. Predict the elderly population in the United States in 2005. Show your work. a?c. See margin.
Mixed Review
U.S. Elderly Population
Year 1960
Elderly (millions)
16.560
1970
19.980
1980
25.550
1990
31.235
2000
34.709
SOURCE: Statistical Abstract of the United States. Go to for a data update.
Web Code: atg-9041
Lesson 6-6
GO
for Help
Lesson 4-4
30. x S 134 31. x R 5 32. x S ?1
Checkpoint Quiz 2
Write the equation for the line that is parallel to the given line and that passes
through the given point. y ? 3 5(x ? 2)
y ? 4 ?23 (x ? 1)
24. y = 5x + 1; (2,-3) 25. y = -x - 9; (0, 5)
26. 2x + 3y = 9; (-1, 4)
27. y = -12x; (3, -4)
y ? 5 ?x 28. y = -2x + 3; (-2, -1)
Solvye?ea4chin?eq12u(axli?ty.33) 0?32.
y ? 1 ?2(x ? 2) See left
29.
y y
= ?
223x+237(x; (?-11, )2)
30. 1 + 5x + 1 . x + 9 31. 7x + 3 , 2x + 28
32. 4x + 4 . 2 + 2x
33. 4x + 3 # 2x - 7 x K ?5
34. -x + 5 , 3x - 1
x
S
3 2
35. 2x . 7x - 3 - 4x xR3
Checkpoint Quiz 2
Lessons 6-5 through 6-7
1. y ? 4 ?14 (x ? 3) 2. y ? 3 18x
3. y ?5
1?3. See left. Write an equation for the line through the given point that has the given slope.
1. (3, 4); m = 214
2. (0, -3); m = 18
3. (-7, -5); m = 0
4. Write an equation for the line through the points (2, -6) and (-1, -4). y ? 6 ?23 (x ? 2)
Write an equation of the line that is parallel to the given line and that passes
through the given point.
5. x + y = 3; (5, 4) y ? 4 ?(x ? 5) 6. 3x + 2y = 1; (-2, 6) y ? 6 ?32 (x ? 2)
Write an equation of the line that is perpendicular to the given line and that passes
through the given point.
7.
y
=
- 4x
+
2; (0, 2)
y
?
2
1 4
x
9. Find the equation for a trend line
for the data at the right. Answers may vary. Sample: y 5.33x ? 1.34
8.
y y
= ?
2 3 2
x + 6; (-6, ?32 (x ?
2) 6)
x1 2 3 4 5 6 7
y 7 12 19 20 28 33 40
10. y ?6.07x ? 62.71
10. Graphing Calculator Use a graphing calculator to find the equation for the line of best fit for the data at the right. See left.
x1 2 3 4 5 6 7 y 54 52 45 40 33 27 18
356 Chapter 6 Linear Equations and Their Graphs
[3] appropriate methods but one computational error
[2] incorrect points used correctly OR points used incorrectly;
function written appropriately, given previous results. [1] correct function, without work shown
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