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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