561
c. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y=
a+bx
d. Findthecorrelationcoefficient.Isitsignificant?
e. Ifthelaundrydetergentweresoldina40ouncesize,findtheestimatedcost.
f. Ifthelaundrydetergentweresoldina90ouncesize,findtheestimatedcost.
g. Usethetwopointsin(e)and(f)toplottheleastsquareslineonyourgraphfrom(a).
h. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
i. Arethereanyoutliersintheabovedata?
j. Istheleastsquaresline validforpredictingwhata300ouncesizeofthelaundrydetergent
wouldcost?Whyorwhynot?
k. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.10
a. Completetheabovetableforthecostperounceofthedifferentsizes.
b. Using“Size”asthe independentvariableand “Costper ounce”asthedependent variable,
makeascatterplotofthedata.
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Isitsignificant?
f. Ifthelaundrydetergentweresoldina40ouncesize,findtheestimatedcostperounce.
g. Ifthelaundrydetergentweresoldina90ouncesize,findtheestimatedcostperounce.
h. Usethetwopointsin(f)and(g)toplottheleastsquareslineonyourgraphfrom(b).
i. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
j. Arethereanyoutliersintheabovedata?
k. Istheleastsquareslinevalidforpredictingwhata300ouncesizeofthelaundrydetergent
wouldcostperounce?Whyorwhynot?
l. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.11
(Solutiononp.580.)
AccordingtoflyerbyaPrudentialInsuranceCompanyrepresentative,thecostsofapproximate
probatefeesandtaxesforselectednettaxableestatesareasfollows:
NetTaxableEstate($)
ApproximateProbateFeesandTaxes($)
600,000
30,000
750,000
92,500
1,000,000
203,000
1,500,000
438,000
2,000,000
688,000
2,500,000
1,037,000
3,000,000
1,350,000
Table12.12
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Makeascatterplotofthedata.
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562
CHAPTER12. LINEARREGRESSIONANDCORRELATION
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Isitsignificant?
f. Findtheestimatedtotalcostforanettaxableestateof$1,000,000.Findthecostfor$2,500,000.
g. Usethetwopointsin(f)toplottheleastsquareslineonyourgraphfrom(b).
h. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
i. Arethereanyoutliersintheabovedata?
j. Basedontheabove,whatwouldbetheprobatefeesandtaxesforanestatethatdoesnothave
anyassets?
k. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.12
ThefollowingareadvertisedsalepricesofcolortelevisionsatAnderson’s.
Size(inches)
SalePrice($)
9
147
20
197
27
297
31
447
35
1177
40
2177
60
2497
Table12.13
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Makeascatterplotofthedata.
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y=
a+bx
e. Findthecorrelationcoefficient.Isitsignificant?
f. Findtheestimatedsalepricefora32inchtelevision.Findthecostfora50inchtelevision.
g. Usethetwopointsin(f)toplottheleastsquareslineonyourgraphfrom(b).
h. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
i. Arethereanyoutliersintheabovedata?
j. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.13
(Solutiononp.580.)
BelowaretheaverageheightsforAmericanboys.(Source:Physician’sHandbook,1990)
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563
Age(years)
Height(cm)
birth
50.8
2
83.8
3
91.4
5
106.6
7
119.3
10
137.1
14
157.5
Table12.14
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Makeascatterplotofthedata.
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Isitsignificant?
f. Findtheestimatedaverageheightforaoneyear–old.Findtheestimatedaverageheightforan
elevenyear–old.
g. Usethetwopointsin(f)toplottheleastsquareslineonyourgraphfrom(b).
h. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
i. Arethereanyoutliersintheabovedata?
j. Usetheleastsquareslinetoestimatetheaverageheightforasixty–twoyear–oldman.Doyou
thinkthatyouranswerisreasonable?Whyorwhynot?
k. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.14
ThefollowingchartgivesthegoldmedaltimesforeveryotherSummerOlympicsforthewomen’s
100meterfreestyle(swimming).
Year
Time(seconds)
1912
82.2
1924
72.4
1932
66.8
1952
66.8
1960
61.2
1968
60.0
1976
55.65
1984
55.92
1992
54.64
2000
53.8
2008
53.1
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564
CHAPTER12. LINEARREGRESSIONANDCORRELATION
Table12.15
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Makeascatterplotofthedata.
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Isthedecreaseintimessignificant?
f. Findtheestimatedgoldmedaltimefor1932.Findtheestimatedtimefor1984.
g. Whyaretheanswersfrom(f)differentfromthechartvalues?
h. Usethetwopointsin(f)toplottheleastsquareslineonyourgraphfrom(b).
i. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
j. UsetheleastsquareslinetoestimatethegoldmedaltimeforthenextSummerOlympics. Do
youthinkthatyouranswerisreasonable?Whyorwhynot?
Thenextthreequestionsusethefollowingstateinformation.
State
#lettersinname
Year entered the
Union
Rankforentering
theUnion
Area
(square
miles)
Alabama
7
1819
22
52,423
Colorado
1876
38
104,100
Hawaii
1959
50
10,932
Iowa
1846
29
56,276
Maryland
1788
7
12,407
Missouri
1821
24
69,709
NewJersey
1787
3
8,722
Ohio
1803
17
44,828
SouthCarolina
13
1788
8
32,008
Utah
1896
45
84,904
Wisconsin
1848
30
65,499
Table12.16
Exercise12.13.15
(Solutiononp.581.)
Weareinterestedinwhetherornotthenumberoflettersinastatenamedependsupontheyear
thestateenteredtheUnion.
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Makeascatterplotofthedata.
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Whatdoesitimplyaboutthesignificanceoftherelationship?
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565
f. Findtheestimatednumberofletters(tothenearestinteger)astatewouldhaveifitentered
theUnionin1900. Findtheestimatednumberoflettersastatewouldhaveifitenteredthe
Unionin1940.
g. Usethetwopointsin(f)toplottheleastsquareslineonyourgraphfrom(b).
h. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
i. UsetheleastsquareslinetoestimatethenumberoflettersanewstatethatenterstheUnionthis
yearwouldhave.Cantheleastsquareslinebeusedtopredictit?Whyorwhynot?
Exercise12.13.16
Weareinterestedinwhetherthereisarelationshipbetweentherankingofastateandtheareaof
thestate.
a. Letrankbetheindependentvariableandareabethedependentvariable.
b. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata.
c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Whatdoesitimplyaboutthesignificanceoftherelationship?
f. FindtheestimatedareasforAlabamaandforColorado.Aretheyclosetotheactualareas?
g. Usethetwopointsin(f)toplottheleastsquareslineonyourgraphfrom(b).
h. Doesitappearthatalineisthebestwaytofitthedata?Whyorwhynot?
i. Arethereanyoutliers?
j. UsetheleastsquareslinetoestimatetheareaofanewstatethatenterstheUnion.Cantheleast
squareslinebeusedtopredictit?Whyorwhynot?
k. Delete“Hawaii”andsubstitute“Alaska”for it. . Alaskaisthe e fortiethstate withanareaof
656,424squaremiles.
l. Calculatethenewleastsquaresline.
m. FindtheestimatedareaforAlabama.Isitclosertotheactualareawiththisnewleastsquares
lineorwiththepreviousonethatincludedHawaii?Whydoyouthinkthat’sthecase?
n. Doyouthinkthat,ingeneral,newerstatesarelargerthantheoriginalstates?
Exercise12.13.17
(Solutiononp.581.)
Weareinterestedinwhetherthereisarelationshipbetweentherankofastateandtheyearit
enteredtheUnion.
a. Letyearbetheindependentvariableandrankbethedependentvariable.
b. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata.
c. Whymusttherelationshipbepositivebetweenthevariables?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Whatdoesitimplyaboutthesignificanceoftherelationship?
f. Let’ssayafifty-firststateenteredtheunion. Basedupontheleastsquaresline,whenshould
thathaveoccurred?
g. Usingtheleastsquaresline,howmanystatesdowecurrentlyhave?
h. Whyisn’ttheleastsquareslineagoodestimatorforthisyear?
Exercise12.13.18
BelowarethepercentsoftheU.S.laborforce(excludingself-employedandunemployed)that
aremembersofaunion.Weareinterestedinwhetherthedecreaseissignificant.(Source:Bureau
ofLaborStatistics,U.S.Dept.ofLabor)
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566
CHAPTER12. LINEARREGRESSIONANDCORRELATION
Year
Percent
1945
35.5
1950
31.5
1960
31.4
1970
27.3
1980
21.9
1993
15.8
2011
11.8
Table12.17
a. Letyearbetheindependentvariableandpercentbethedependentvariable.
b. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata.
c. Whywilltherelationshipbetweenthevariablesbenegative?
d. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
e. Findthecorrelationcoefficient.Whatdoesitimplyaboutthesignificanceoftherelationship?
f. Basedonyouranswerto(e),doyouthinkthattherelationshipcanbesaidtobedecreasing?
g. Ifthetrendcontinues,whenwilltherenolongerbeanyunionmembers? Doyouthinkthat
willhappen?
Thenexttwoquestionsrefertothefollowinginformation:Thedatabelowreflectsthe1991-92Reunion
ClassGiving.(Source:SUNYAlbanyalumnimagazine)
ClassYear
AverageGift
TotalGiving
1922
41.67
125
1927
60.75
1,215
1932
83.82
3,772
1937
87.84
5,710
1947
88.27
6,003
1952
76.14
5,254
1957
52.29
4,393
1962
57.80
4,451
1972
42.68
18,093
1976
49.39
22,473
1981
46.87
20,997
1986
37.03
12,590
Table12.18
Exercise12.13.19
(Solutiononp.581.)
Wewillusethecolumns“classyear”and“totalgiving”forallquestions,unlessotherwisestated.
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567
a. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata.
b. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
c. Findthecorrelationcoefficient.Whatdoesitimplyaboutthesignificanceoftherelationship?
d. Fortheclassof1930,predictthetotalclassgift.
e. Fortheclassof1964,predictthetotalclassgift.
f. Fortheclassof1850,predictthetotalclassgift.Whydoesn’tthisvaluemakeanysense?
Exercise12.13.20
Wewillusethecolumns“classyear”and“averagegift”forallquestions,unlessotherwisestated.
a. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata.
b. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
c. Findthecorrelationcoefficient.Whatdoesitimplyaboutthesignificanceoftherelationship?
d. Fortheclassof1930,predicttheaverageclassgift.
e. Fortheclassof1964,predicttheaverageclassgift.
f. Fortheclassof2010,predicttheaverageclassgift.Whydoesn’tthisvaluemakeanysense?
Exercise12.13.21
(Solutiononp.581.)
Weareinterestedinexploringtherelationshipbetweentheweightofavehicleanditsfueleffi-
ciency(gasolinemileage). Thedatainthetableshowtheweights,inpounds,andfuelefficiency,
measuredinmilespergallon,forasampleof12vehicles.
Weight
FuelEfficiency
2715
24
2570
28
2610
29
2750
38
3000
25
3410
22
3640
20
3700
26
3880
21
3900
18
4060
18
4710
15
Table12.19
a. Graphascatterplotofthedata.
b. Findthecorrelationcoefficientanddetermineifitissignificant.
c. Findtheequationofthebestfitline.
d. Writethesentencethatinterpretsthemeaningoftheslopeofthelineinthecontextofthedata.
e. Whatpercentofthevariationinfuelefficiencyisexplainedbythevariationintheweightofthe
vehicles,usingtheregressionline?(Stateyouranswerinacompletesentenceinthecontext
ofthedata.)
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568
CHAPTER12. LINEARREGRESSIONANDCORRELATION
f. Accuratelygraphthebestfitlineonyourscatterplot.
g. Forthevehiclethatweights3000pounds,findtheresidual(y-yhat). Doesthevaluepredicted
bythelineunderestimateoroverestimatetheobserveddatavalue?
h. Identifyanyoutliers,usingeitherthegraphicalornumericalproceduredemonstratedinthe
textbook.
i. Theoutlierisahybridcarthatrunsongasolineandelectrictechnology,butallothervehicles
inthesamplehaveenginesthatusegasolineonly. Explainwhyitwouldbeappropriateto
removetheoutlierfromthedatainthissituation.Removetheoutlierfromthesampledata.
Findthenewcorrelationcoefficient,coefficientofdetermination,andbestfitline.
j. Comparethecorrelationcoefficientsandcoefficientsofdeterminationbeforeandafterremoving
theoutlier,andexplainincompletesentenceswhatthesenumbersindicateabouthowthe
modelhaschanged.
Exercise12.13.22
(Solutiononp.581.)
ThefourdatasetsbelowwerecreatedbystatisticianFrancisAnscomb. Theyshowwhyitisim-
portanttoexaminethescatterplotsforyourdata,inadditiontofindingthecorrelationcoefficient,
inordertoevaluatetheappropriatenessoffittingalinearmodel.
Set1
Set2
Set3
Set4
x
y
x
y
x
y
x
y
10
8.04
10
9.14
10
7.46
8
6.58
8
6.95
8
8.14
8
6.77
8
5.76
13
7.58
13
8.74
13
12.74
8
7.71
9
8.81
9
8.77
9
7.11
8
8.84
11
8.33
11
9.26
11
7.81
8
8.47
14
9.96
14
8.10
14
8.84
8
7.04
6
7.24
6
6.13
6
6.08
8
5.25
4
4.26
4
3.10
4
5.39
19
12.50
12
10.84
12
9.13
12
8.15
8
5.56
7
4.82
7
7.26
7
6.42
8
7.91
5
5.68
5
4.74
5
5.73
8
6.89
Table12.20
a.Foreachdataset,findtheleastsquaresregressionlineandthecorrelationcoefficient.Whatdid
youdiscoveraboutthelinesandvaluesofr?
Foreachdataset,createascatterplotandgraphtheleastsquaresregressionline.Usethegraphs
toanswerthefollowingquestions:
b. Forwhichdatasetdoesitappearthatacurvewouldbeamoreappropriatemodelthanaline?
c. Whichdatasethasaninfluentialpoint (pointclosetooronthelinethatgreatlyinfluencesthe
bestfitline)?
d. Whichdatasethasanoutlier(obviouslyvisibleonthescatterplotwithbestfitlinegraphed)?
e. Whichdatasetappearstobethemostappropriatetomodelusingtheleastsquaresregression
line?
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569
12.13.1Trythesemultiplechoicequestions
Exercise12.13.23
(Solutiononp.582.)
Acorrelationcoefficientof-0.95meansthereisa____________betweenthetwovariables.
A. Strongpositivecorrelation
B. Weaknegativecorrelation
C. Strongnegativecorrelation
D. NoCorrelation
Exercise12.13.24
(Solutiononp.582.)
AccordingtothedatareportedbytheNewYorkStateDepartmentofHealthregardingWestNile
Virus (http://www.health.state.ny.us/nysdoh/westnile/update/update.htm)for r the years2000-
2008,theleastsquareslineequationforthenumberofreporteddeadbirds(x)versusthenumber
ofhumanWestNileviruscases(y)is
^
y
10.2638+0.0491x.Ifthenumberofdeadbirdsreported
inayearis732,howmanyhumancasesofWestNileviruscanbeexpected?r=0.5490
A. Nopredictioncanbemade.
B. 19.6
C. 15
D. 38.1
Thenextthreequestionsrefertothefollowingdata:(showingthenumberofhurricanesbycategoryto
directlystrikethemainlandU.S.eachdecade)obtainedfromwww.nhc.noaa.gov/gifs/table6.gif
15
Amajor
hurricaneisonewithastrengthratingof3,4or5.
Decade
TotalNumberofHurricanes
NumberofMajorHurricanes
1941-1950
24
10
1951-1960
17
8
1961-1970
14
6
1971-1980
12
4
1981-1990
15
5
1991-2000
14
5
2001–2004
9
3
Table12.21
Exercise12.13.25
(Solutiononp.582.)
Usingonlycompleteddecades(1941–2000),calculatetheleastsquareslineforthenumberof
majorhurricanesexpectedbaseduponthetotalnumberofhurricanes.
A.
^
y
1.67x+0.5
B.
^
y
=0.5x 1.67
C.
^
y
=0.94x 1.67
D.
^
y
2x+1
15
http://www.nhc.noaa.gov/gifs/table6.gif
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570
CHAPTER12. LINEARREGRESSIONANDCORRELATION
Exercise12.13.26
(Solutiononp.582.)
Thecorrelationcoefficientis0.942.Isthisconsideredsignificant?Whyorwhynot?
A. No,because0.942isgreaterthanthecriticalvalueof0.707
B. Yes,because0.942isgreaterthanthecriticalvalueof0.707
C. No,because0942isgreaterthanthecriticalvalueof0.811
D. Yes,because0.942isgreaterthanthecriticalvalueof0.811
Exercise12.13.27
(Solutiononp.582.)
Thedatafor2001-2004show9hurricaneshavehitthemainlandUnitedStates.Thelineofbestfit
predicts2.83majorhurricanestohitmainlandU.S.Cantheleastsquareslinebeusedtomakethis
prediction?
A. No,because9liesoutsidetheindependentvariablevalues
B. Yes,because,infact,therehavebeen3majorhurricanesthisdecade
C. No,because2.83liesoutsidethedependentvariablevalues
D. Yes,becausehowelsecouldwepredictwhatisgoingtohappenthisdecade.
**Exercises21and22contributedbyRobertaBloom
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