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AK-1North Carolina Math 1

Teacher Resource/Student Workbook Answer Key© Walch Education

UNIT 2 • LINEAR FUNCTIONS

Answer KeyLesson 2.1: Proving Average Rate of Change (F–LE.1a•)

Warm-Up 2.11. BoatAwillbeworth$5,145after3years.2. BoatBwillbeworth$6,144after3years.3. BoatBwillbeworthmorethanBoatAafter3years.

Practice 2.1 A: Proving Average Rate of Change1. –0.92. –323. 46.744. –15.185. 1.8

6. 0.087. 0.088. 7.59. 13.65

10. 14.77

Practice 2.1 B: Proving Average Rate of Change1. –0.022. 203. 30.724. –5.475. 2.54

6. 0.137. 0.138. 149. 24.40

10. 26.98

Lesson 2.2: Recognizing Average Rate of Change (F–LE.1b•, F–LE.1c•)

Warm-Up 2.21. Therateofchangefrom1900to2000was2.03millionper

year.Therateofchangefrom2000to2010was3.5millionperyear.Therateofchangefor1900to2000waslessthantherateofchangefor2000to2010.

2. Highestrateofchange:Either1950to1960or1990to2000couldbedescribedasthe10-yearperiodwiththehighestrateofchange,becausetheyseemtohaveidenticalrates.Lowestrateofchange:1930to1940.Calculatetherateofchangeforeach10-yearperiod.

3. Answerswillvary.Sampleanswer:Thepopulationin2020couldreach348,000,000iftherateofchangecontinuesat3.5millionperyear.

Practice 2.2 A: Recognizing Average Rate of Change1. ≈0.001gallonsperfoot2. ≈0.001gallonsperfoot3. ≈0.8EurosperU.S.dollar4. ≈0.8EurosperU.S.dollar5. Yes,apredictionispossible.Sampleprediction:Therate

ofchangewouldbethesameastherateofchangeinquestions3and4becausethefunctionislinear.

6. ≈25visitorsperyear7. ≈29visitorsperyear8. ≈–271peopleperyear9. ≈–167peopleperyear

10. Therateofchangefortheinterval[2,9]issteeperthantheintervalfrom[14,20].Thepopulationisdecreasingatafasterratefortheinterval[2,9]thanfor[14,20].

Practice 2.2 B: Recognizing Average Rate of Change1. ≈0.06gallonsperdoor2. ≈0.06gallonsperdoor3. ≈0.97AustraliandollarsperU.S.dollar4. ≈0.97AustraliandollarsperU.S.dollar5. Yes,apredictionispossible.Sampleprediction:Therate

ofchangewouldbethesameastherateofchangeinquestions3and4becausethefunctionislinear.

6. ≈92campersperyear7. ≈150campersperyear8. ≈–160peopleperyear9. ≈–65peopleperyear

10. Therateofchangefortheinterval[1,6]issteeperthantheinterval[10,13].Thepopulationisdecreasingatafasterratefortheinterval[1,6]thanfor[10,13].

Lesson 2.4: Working with Parallel and Perpendicular Lines (G–GPE.5)

Warm-Up 2.41. Answersmayvary.Eachequationshouldhaveaslopeof

–5.Onepossibleanswerisy=–5x+2.y-interceptsmustbedifferentforlinestobeparallelandnotcoinciding.

2. Answersmayvary.Eachequationshouldhaveaslopeof–5.Onepossibleanswerisy=–5x+7.y-interceptsmustbedifferentforlinestobeparallelandnotcoinciding.

Practice 2.4 A: Working with Parallel and Perpendicular Lines

1. y=–2x+7

2. y x4

3

2

3=−

−

3. y x1

42= −

4. y = –x–15. 2 1.4 units≈

6. y x3

47=− +

7. y=–5x–34

8. y x1

510= +

9. y x4

32= −

10. TheshortestdistanceisthelinethatisperpendicularfromthetrainstationtoUnionStreet.Thisdistanceisapproximately647yards.

AK-2North Carolina Math 1 Teacher Resource/Student Workbook Answer Key

© Walch Education

Practice 2.4 B: Working with Parallel and Perpendicular Lines

1. y=3x–8

2. y x1

4=−

3. y x1

37= +

4. y=–2x+2

5.80

25or 1.79 units≈

6. y x1

2

7

2= −

7. y x1

3

7

3= +

8. y=–3x–169. y=–2x–3

10. Theshortestdistanceisthelinethatisperpendicular

fromthegrocerystoretoMapleStreet.Thisdistanceis81

102.85 units≈ or285yards.

Lesson 2.5: Interpreting Parameters (F–LE.5•)

Warm-Up 2.51. f(x)=2x+52. They-interceptis5.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

(0, 5)

(1, 7)

3. themembershipfee4. thecosttorenteachgame

Practice 2.5 A: Interpreting Parameters1. slope=7;y-intercept=52. growthfactor=2;verticalshift=33. slope=–2;y-intercept=104. startingquantity=2;growthfactor=35. startingquantity=3;growthfactor=2;verticalshift=56. f(x)=2x+10;slope=2;y-intercept=10

7. slope=1.25;y-intercept=58. Maxpicks18applesperminuteandstartedwith15apples

inhisbag.9. Youhad10antsinthecolonytostartwith,andthe

numberofantsdoublesevery36hours.10. f(x)=150(2x)+200;growthfactor=2;

startingamount=$150;verticalshift=$200

Practice 2.5 B: Interpreting Parameters1. slope=3;y-intercept=122. growthfactor=4;verticalshift=–83. slope=–6;y-intercept=134. growthfactor=2;startingamount=55. growthfactor=4;startingamount=2;verticalshift=96. f(x) =300(3x)+100;growthfactor=3;starting

amount=$300;verticalshift=$1007. slope=2.5;y-intercept=78. slope=3.75;y-intercept=129. Kendallpicks35strawberriesperminuteandstartedwith

20strawberriesinhisbasket.10. Therewere25antstostartwithandthepopulationgrows

atarateof3antsevery4days.

Lesson 2.6: Graphing the Set of All Solutions (A–REI.10)

Warm-Up 2.61. $52. $93. Theslopeofthegraphis2.ThegraphshowsthatMallory

canreceive$2onthey-axisforevery1dayonthex-axis.

Practice 2.6 A: Graphing the Set of All Solutions1.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10



2.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

3.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

4.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

5. {(0,1);(–2,4);(–3,8)}6. {(0,7);(8,6);(–8,8)}

7. Theprofitinyear7shouldbeabout$929.34.

0 1 2 3 4 5 6 7 8

100

200

300

400

500

600

700

800

900

1000

Years

Pro�

t in

dolla

rs

8. y=2x;for4batchesofcookies,Katyaneeds8eggs.

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Batches of cookies

Eggs

nee

ded

9.1

435.50y x= + ;at12months,thepricewillbe$38.50.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

32

33

34

35

36

37

38

39

40

Months

Pric

e pe

r sha

re


© Walch Education

10. Youwouldhavetopay$22.50for7.5gallonsofgas.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

5

10

15

20

25

30

35

40

45

50

55

Gallons of gas

Am

ount

pai

d

Practice 2.6 B: Graphing the Set of All Solutions1.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

2.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

3.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

4.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -10 1 2 3 4 5 6 7 8 9 10

-10-9-8-7-6-5-4-3-2-1

123456789

10

5. {(0,2);(2,5);(4,8)}6. {(–2,9);(–1,3);(0,1)}7. Thepainterwillhave35gallonsremainingafter6hours.

0 1 2 3 4 5 6 7 85

10

15

20

25

30

35

40

45

50

55

60

65

70

Hours

Gal

lons

of p

aint



8. Therewereabout1.6gramsofbacteriaafter60hours.

0 10 20 30 40 50 60 70 80 90 100 110 1202

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

Hours

Gra

ms

9. Enricowouldhavetoride49milesonthefifthday.

0 1 2 3 4

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

Days

Mile

s

10. Itwilltakejustmorethan24monthsforMr.Samuelson’ssavingstoexceedhiscosts.

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42

500

1000

1500

2000

2500

3000

3500

4000

Months

Savi

ngs

in d

olla

rsLesson 2.7: Sequences As Functions (F–IF.3)

Warm-Up 2.71. Saturday,May72. Friday,May133. Thursday,May19

Practice 2.7 A: Sequences As Functions1. 282. 233.

0 1 2 3 4 5 6 7

-5-4-3-2-1

123456789

1011121314151617181920an

n


© Walch Education

4.

0 1 2 3 4 5 6 7-2-1

123456789

1011121314151617181920212223242526272829303132

an

n

5. 756. 1047. a

5=162

0 1 2 3 4 5 6 7

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500an

n

8. a5=45,a

6=53

0 1 2 3 4 5 6 72468

1012141618202224262830323436384042444648505254an

n

9. 2:01p.m.10. Thefifthwaterstationwillbeatthe19-milemark.

Practice 2.7 B: Sequences As Functions1. 322. 22

0 1 2 3 4 5 6 7123456789

10111213141516171819202122232425an

n



3. 53

0 1 2 3 4 5 6 72468

101214161820222426283032343638404244464850525456586062646668an

n

4. 395. 326. 2977. a

5=21,a

6=25

0 1 2 3 4 5 6 71

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26an

n

8. a5=32,a

6=64

0 1 2 3 4 5 6 7 8 948

12162024283236404448525660646872768084889296

100104108112116120124128an

n

9. 5:53a.m.10. Thelargestcontainerholds32gallons;thesmallest

containerholds2gallons.

Lesson 2.8: Building Functions from Context (F–BF.1a•)

Warm-Up 2.81. c=4p;$42. d=55t;330miles

3. t=4s;120students4. p=12h;36pages

Practice 2.8 A: Building Functions from Context1. f(x) =19+12x2. f(x)=–5x+303. f(x)=132–22x4. f(x)=12•(5)x –1

5. f(x)=32•(0.5)x

6. f(x)=2+x7. f(x)=3x –1

8. f(x)=4x –1

9. f(x) = 150+25(x–1)10. f(x)=10,000(1.10)x

Practice 2.8 B: Building Functions from Context1. f(x)=15x2. f(x)=–18x+2603. f(x)=7+9x4. f(x)=6•4x –1

5. f(x)=60•(0.9)x

6. f(x)=x7. f(x)=4x–38. f(x)=2xorf(x)=2•2x –1

9. f(x) =10+0.25x10. f(x)=15,000•(0.80)x


© Walch Education

Lesson 2.12: Solving Problems Given Functions Fitted to Data (S–ID.6a•)

Warm-Up 2.121.

0 2 4 6 8 10 12 14 16 18

2

4

6

8

10

12

14

16

18

Cost

in d

olla

rs ($

)

Number of goods

2.

Practice 2.12 A: Solving Problems Given Functions Fitted to Data

1.

Trac

k le

ngth

in m

iles

Time in minutes

2. Thescatterplotcouldbeestimatedusingastraightline.Alinearfunctionisabetterfitforthedata.

3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

y = 3.3x

y = 2.3x

Trac

k le

ngth

in m

iles

Time in minutes

y=3.3xisabetterfitforthedatabecauseitmorecloselyfollowsthetrendinthepoints.

4. 0.545minutes5. 1.98miles



6.

0 1 2 3 4 5 6 7 8

2000

4000

6000

8000

1 104

1.2 104

1.4 104

1.6 104

1.8 104

2 104Va

lue

of c

ar, i

n do

llars

($)

Year

7. Anexponentialfunction;thegraphisacurve,andtheverticaldistancebetweenthepointsisdecreasingasthehorizontaldistanceremainsconstant.

8. Thegraphofy=19,000(1.10)xisnotagoodestimateforthedatabecauseitdoesn’tapproachanyofthedatapointsonthescatterplot.However,thegraphofy=19,000(0.90)xisagoodestimatebecauseitgoesthroughorcloselyapproachesallofthedatapoints.

0 1 2 3 4 5 6 7 82000

4000

6000

8000

1 104

1.2 104

1.4 104

1.6 104

1.8 104

2 104

2.2 104

2.4 104

2.6 104

2.8 104

3 104

y = 19, 000 • (1.10)x

y = 19, 000 • (0.90)x

Valu

e of

car

, in

dolla

rs ($

)

Year

9. 7.1years,becauselookingatthetrendinthegraph,thecurvefallsthrough$9,000atabout7.1years.

0 1 2 3 4 5 6 7 82000

4000

6000

8000

1 104

1.2 104

1.4 104

1.6 104

1.8 104

2 104

2.2 104

2.4 104

2.6 104

2.8 104

3 104

Valu

e of

car

, in

dolla

rs ($

)

Year

10. approximately$5,366

Practice 2.12 B: Solving Problems Given Functions Fitted to Data

1.

0 1 2 3 4 5 6 720

40

60

80

100

120

140

160

180

200

220

240

New

vis

itors

Day

2. Anexponentialfunction;thegraphisacurve,andtheverticaldistancebetweenthepointsisincreasingasthehorizontaldistanceremainsconstant.


© Walch Education

3.

0 1 2 3 4 5 6 720

40

60

80

100

120

140

160

180

200

220

240

y = 4x y = 3x

New

vis

itors

Day

Thefunctiony=3xisabetterfitbecauseitmorecloselyfollowsthedatatrend.

4. onday65. approximately2,187visitors6.

0 1 2 3 4 5

10

20

30

40

50

60

70

80

90

100

Sand

wic

hes

rem

aini

ng

Hours open

7. Thescatterplotcouldbeestimatedusingastraightline.Alinearfunctionisabetterfitforthedata.

8.

0 1 2 3 4 5 6

10

20

30

40

50

60

70

80

90

100

y = -3.8x + 92

y = -5.8x + 99

Sand

wic

hes

rem

aini

ng

Hours open

y=–5.8x+99,becausethecurvefollowsthetrendinthedata.

9. approximately98sandwiches10. approximately17hours

Lesson 2.13: Analyzing Residuals (S–ID.6b•)

Warm-Up 2.131.

0 1 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

9

2. Thedistanceistheabsolutevalueofthedifferencebetweenthetwoy-values:|6–4|=2.



3.

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

1

2

3

4

Practice 2.13 A: Analyzing Residuals1.

0 5 10 15 20 25 30 35 40 45 50 55

10

20

30

40

50

60

70

80

90

100

110

Dee

r

Acres

2.

0 5 10 15 20 25 30 35 40 45 50 55 60

10

20

30

40

50

60

70

80

90

100

110

120

y = 2x + 22

Dee

r

Acres

3. Yes;itfollowstheshapeofthedata.4.

0 5 10 15 20 25 30 35 40 45 50 55

-40

-30

-20

-10

10

20

30

40

50

60

Theplotappearsrandom,soalinearfunctionislikelyagoodfitforthedata.


© Walch Education

5.

0 1 2 3 4 5 6 7 8 950

100

150

200

250

300

350

400

450

500

550

600

650

700

750Ac

coun

t bal

ance

in d

olla

rs ($

)

Years

6.

0 1 2 3 4 5 6 7 8 950

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Acco

unt b

alan

ce in

dol

lars

($)

Years

7. Yes;thelineappearstofollowtheshapeofthedata.

8.

0 1 2 3 4 5 6 7 8 9

-8

-6

-4

-2

2

4

6

8

Thoughitappearedthatthelinefitthedata,theU-shapeof theresidualplotindicatesthatalinearfunctionisnotagoodfitforthedata.

9.

0 2 4 6 8 10 12 14 16 18 20 22 24

20

40

60

80

100

120

140

160

180

200

220

240Ti

me

in m

inut

es

Distance in miles



10.

0 2 4 6 8 10 12 14 16 18 20 22 24

-25

-20

-15

-10

-5

5

10

15

Theresidualplotisrandom,soalinearfunctionisagoodfit.

Practice 2.13 B: Analyzing Residuals1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

60

120

180

240

300

360

420

480

540

Hei

ght i

n m

eter

s

Time in seconds

2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

60

120

180

240

300

360

420

480

540

Hei

ght i

n m

eter

s

Time in seconds 3. Itcomesclosetothedata,butthelinedoesnotfollowthe

shapeofthedata.4.

0 1 2 3 4 5 6 7 8 9 10 11 12

-30

-20

-10

10

20

30

40

50

60

70

80

TheresidualplothasaU-shape,indicatingalinearfunctionisnotthebestfitforthedata.


© Walch Education

5.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 152

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34H

eigh

t in

inch

es

Age in months

6.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 152

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

Hei

ght i

n in

ches

Age in months

7. Yes;itfollowstheshapeofthedataset.

8.

0 2 4 6 8 10 12 14

-4

-3

-2

-1

1

2

3

4

Whenlookingattheresidualplot,theshapeappearstobeaU.Alinearfunctionisnotagoodfitforthedata.

9.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 705

10

15

20

25

30

35

40

45

50

55

60

65

70A

rm s

pan

in in

ches

Height in inches



10.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

-4

-3

-2

-1

1

2

3

4

Theresidualplotisrandom,soalinearfunctionisagoodfit.

Lesson 2.14: Interpreting Slope and y-intercept (S–ID.7•)

Warm-Up 2.141. Slope=16;y-intercept=(0,0)2. y=16x3. –14. –2,orthepoint(0,–2)

Practice 2.14 A: Interpreting Slope and y-intercept1.

0 1 2 3 4 5 6 7 8 9100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

New

hom

es

Year

2. Answersmayvary;equationsshouldbeclosetoy=150x–120.

3. Thenumberofhomesbuilteachyearisincreasingby150.Inthiscase,whenx=0,y=–120.Thisvaluedoesnotmakesenseincontext,sincethenumberofnewhomesbuiltinayearcannotbenegative.

4. Answersmayvary;equationsshouldbeclosetoy=17.25x+7.75.

5. IttakesMadelineapproximately17minutestocompleteeachhomeworkassignment.Inthiscase,whenMadelinehasnoassignments,hertimetocompletetheassignmentswouldbe0.They-interceptdoesnotmakesenseinthiscontext.

6.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 282

4

6

8

10

12

14

16

18

20

22

24

26

28

Free

thro

ws

mad

e

Free throws attempted

7. Answersmayvary;equationsshouldbeclosetoy=0.77x–0.7.

8. Willmakesapproximately77%ofhisfreethrows.Inthiscontext,ifWilldoesnotattemptanyfreethrows,hewouldnotmakeanyfreethrows;whenx=0,yshouldequal0.They-interceptisnotrelevantinthiscontext.


10. Ittakestheconstructioncompanyapproximately7.6weekstoconstructabuildingforeachfloorofthebuilding.Inthiscontext,ifthebuildingdoesn’thaveanyfloors,thetimetoconstructthebuildingwouldbe0;whenx=0,yshouldequal0.They-interceptisnotrelevantinthiscontext.

Practice 2.14 B: Interpreting Slope and y-intercept1.

02 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

1

2

3

4

5

6

7

8

9

Num

ber o

f win

ners

Number of players



© Walch Education

3. Aplayerwinsthegameapproximately19%ofthetime.Inthiscase,whentherearenoplayers,therewouldbenowinners;whenx=0,thenyshouldequal0.They-interceptisnotrelevantinthiscontext.

4.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 14010

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

Min

utes

to d

eliv

er

Number of packages


6. Ittakesthecompanyapproximately0.87minutestodelivereachpackage.Inthiscase,iftherearenopackagestodeliver,itwouldtakenotimetodeliverthepackage;whenx=0,yshouldequal0.They-interceptisnotrelevantinthiscontext.


8. Eachcowproducesapproximately18litersofmilkeachday.Inthiscase,iftherearenocows,thereisnomilkproduction;whenx=0,yshouldequal0.They-interceptisnotrelevantinthiscontext.


10. Aperson’sheightincreasesbyapproximately1.8inchesforeachshoesize.Inthiscase,ifashoesizeis0,thereisnoheight;whenx=0,yshouldequal0.They-interceptisnotrelevantinthiscontext.

Lesson 2.15: Calculating and Interpreting the Correlation Coefficient (S–ID.8•)

Warm-Up 2.151.

0 1 2 3 4 5 6 7 8 9 10 11

5

10

15

20

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35

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New

use

rs, i

n th

ousa

nds

Number of weeks

2. Thegraphofalinearfunctionisaline.Theshapeofthedatainthegraphislinearbecausealinecouldbedrawnonthegraphwithanapproximatelyequalnumberofpointsaboveandbelowtheline.Thedataappearstohavealinearrelationship.

Practice 2.15 A: Calculating and Interpreting the Correlation Coefficient

1. strongpositivelinearcorrelation2. weakpositivelinearcorrelation3. strongnegativelinearcorrelation4. nocorrelation,orveryweaknegativelinearcorrelation5.

500 52 54 56 58 60 62 64 66 68 70 72 74 76 7820

40

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120

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Plan

ts s

old

Average temperature (°F)

6. Thereappearstobeapositivelinearcorrelationbetweentemperatureandplantsales.

7. r=0.848,whichindicatesthereisastrongpositivelinearcorrelationbetweentemperatureandplantsales.



8.

0 100 200 300 400 500 600 700 800 900

20

40

60

80

100

120

140Av

erag

e po

ol p

arty

att

enda

nce

Number of children

9. Poolpartyattendanceappearstodecreaseasthenumberofchildrenincreases;theremaybeanegativelinearcorrelationbetweenthedata.

10. r=–0.779;thereisanegativelinearcorrelationbetweenthedata.

Practice 2.15 B: Calculating and Interpreting the Correlation Coefficient

1. weaknegativelinearcorrelation2. strongpositivelinearcorrelation3. nocorrelation4. strongnegativelinearcorrelation5.

0 50 100 150 200 250 300 350 400 450 5002000

4000

6000

8000

1 1041.2 1041.4 1041.6 1041.8 1042 104

2.2 1042.4 1042.6 1042.8 104

Poun

ds o

f lug

gage

Number of passengers

6. Thereappearstobeapositivelinearcorrelationbetweennumberofpassengersandpoundsofluggage.

7. r=0.92;thereisaverystrongpositivelinearcorrelationbetweenthedata.

8.

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5500

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9500

Uni

que

web

site

vis

itors

Magazines sold

9. Theredoesnotappeartobearelationshipbetweenthedata.10. r=–0.168;thereisnorelationshipbetweenthedata,ora

veryweaknegativelinearcorrelation.

Lesson 2.16: Distinguishing Between Correlation and Causation (S–ID.9•)

Warm-Up 2.161. Theshapeofalineargraphisastraightline.Thedata

seemstofollowastraightline,anditappearsthatthereisalinearcorrelationbetweenxandy.

2.

0 1 2 3 4 5 6 7 8 9 10 114

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Mis

take

s m

ade

durin

g a

perf

orm

ance

Minutes spent practicing per day

3. Usingacalculator,r=–0.97.Thecorrelationcoefficientisverycloseto–1,sothereisastrongnegativelinearcorrelationbetweenxandy.


© Walch Education

Practice 2.16 A: Distinguishing Between Correlation and Causation

1.

20,000

30,000

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60,000

70,000

80,000

90,000

100,000

110,000

120,000

130,000

140,000

150,000

160,000

170,000

0 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000

Prod

ucts

sol

d

Advertising spending ($)

2. Thegraphappearstoberandom,withsomeincreaseinnumbersofproductssoldasadvertisingdollarsincrease.

3. r=0.311;thisindicatesthatthereisaweakpositivelinearcorrelationbetweenadvertisingdollarsspentandthenumberofproductssold.

4. Thecorrelationbetweenthetwovaluesisweak,anditisdifficulttodetermineifthereisanycausalrelationshipbetweenadvertisingdollarsspentandthenumberofproductssold.

5.

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 801

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Coun

trie

s vi

site

d

Age of client

6. Asageincreases,numberofcountriesalsoincreases;thegraphfollowsalinearshape.

7. r=0.912;thisindicatesthatthereisastrongpositivelinearcorrelationbetweenageandnumberofcountriesvisited.

8. Eventhoughthereisastrongcorrelationbetweenageandnumberofcountriesvisited,thisdoesnotmeanthatageisresponsibleforthenumberofcountriesvisited.Anolderpersonhashadmoreopportunitytotravel,buttheageoftheclientdoesnotimpactthenumberofcountriestheclienthaschosentovisit.Itisnotlikelythatthereisacausalrelationshipbetweenageandnumberofcountriesvisited.

9. Astemperatureincreases,timedecreases.Thegraphgenerallyhasalinearshapewithanegativeslope.

10. Afewfactorsinfluencethetimeittakeswatertoboil:volumeofwater,heatappliedtowater,distributionofwaterwithinitscontainer,andairpressure.Thestudentsattempttokeepthreeofthesefactorsconsistentthroughouttheexperiment,byusingidenticalcontainers,performingtheexperimentinthesameclassroom,andusingthesamevolumeofwater.Thedecreaseintimeisthereforelikelyduetotheincreaseintemperature,anditislikelythatthereisacausalrelationshipbetweentemperatureappliedandtimetoboil.

Practice 2.16 B: Distinguishing Between Correlation and Causation

1.

0 20 40 60 80 100 120 140 160 18020

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60

80

100

120

140

160

180

200M

illio

ns o

f dol

lars

ear

ned

in U

.S. t

heat

ers

Millions of dollars invested

2. Thegraphappearstoberandom,withsomeincreaseindollarsearnedasdollarsinvestedincreases.

3. r=0.342;thisindicatesthatthereisaweakpositivelinearcorrelationbetweendollarsinvestedanddollarsearnedbyamovie.

4. Thecorrelationbetweenthetwovaluesisweak,anditisdifficulttodetermineifthereisanycausalrelationshipbetweendollarsinvestedanddollarsearnedinU.S.theaters.



5.

0 1 2 3 4 5 6 7 8481216202428323640444852566064687276

Pairs

of s

hoes

sol

d

Salespeople

6. Thepairsofshoessoldincreasesasthenumberofsalespeopleincreases,andthegraphappearstofollowalinearshape.

7. r=0.892;thisindicatesthatthereisastrongpositivelinearcorrelationbetweennumberofsalespeopleandpairsofshoessold.

8. Sincecustomersaredependentonasalespersontohelpthemgetshoesinthecorrectsize,andthereisastrongcorrelationbetweennumberofsalespeopleandpairsofshoessold,itislikelythatthereisacausalrelationshipbetweenthetwovalues.Otherfactorsalsoinfluencethesales,buthavingmoresalespeopledirectlyimpactsthenumberofpeoplewhoareabletogetshoesinthecorrectsize,leadingtogreatershoesales.Notethateveniftherearealargenumberofsalespeople,salescannotincreaseunlesstherearecustomers.

9. Astheageofcellphoneusersincreases,thenumberofappsdecreases.Thegraphfollowsapproximatelyalinearshapewithanegativeslope.

10. Eventhoughthereappearstobeacorrelationbetweenageandnumberofapps,itisunlikelythatagecausesthereducednumberofcellphoneapps.