521
d. Chi-Squarewithdf=48
e. 85.33
f. 0.0007
h. Decision:Rejectnull.
SolutiontoExercise11.13.28(p.505)
True
SolutiontoExercise11.13.29(p.505)
False
SolutiontoExercise11.13.30(p.505)
False
SolutiontoExercise11.13.31(p.505)
True
SolutiontoExercise11.13.32(p.505)
True
SolutiontoExercise11.13.33(p.505)
False
SolutiontoExercise11.13.34(p.506)
True
SolutiontoExercise11.13.35(p.506)
True
SolutiontoExercise11.13.36(p.506)
True
SolutiontoExercise11.13.37(p.506)
True
SolutiontoExercise11.13.38(p.506)
True
SolutiontoExercise11.13.39(p.506)
False
SolutiontoExercise11.13.40(p.506)
False
SolutiontoExercise11.13.41(p.506)
True
SolutionstoReview
SolutiontoExercise11.14.1(p.507)
(0.0424,0.0770)
SolutiontoExercise11.14.2(p.507)
2401
SolutiontoExercise11.14.4(p.507)
7.5
SolutiontoExercise11.14.5(p.507)
0.0122
SolutiontoExercise11.14.6(p.507)
N(7,0.63)
SolutiontoExercise11.14.7(p.507)
0.9911
SolutiontoExercise11.14.8(p.507)
B
SolutiontoExercise11.14.9(p.507)
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522
CHAPTER11. THECHI-SQUAREDISTRIBUTION
a. True
b. False
c. False
SolutiontoExercise11.14.11(p.508)
student-twithdf=15
SolutiontoExercise11.14.12(p.508)
(560.07,719.93)
SolutiontoExercise11.14.13(p.508)
quantitative-continuous
SolutiontoExercise11.14.14(p.508)
quantitative-discrete
SolutiontoExercise11.14.15(p.508)
b. P(4)
c. 0.0183
SolutiontoExercise11.14.16(p.508)
greaterthan
SolutiontoExercise11.14.17(p.508)
No;P(x=8)=0.0348
SolutiontoExercise11.14.18(p.508)
Youwilllose$5
SolutiontoExercise11.14.19(p.508)
Becca
SolutiontoExercise11.14.20(p.509)
14
SolutiontoExercise11.14.21(p.509)
•. Samplemean=3.2
•. Samplestandarddeviation=1.85
•. Median=3
•. Quartile1=2
•. Quartile3=5
•. IQR=3
SolutiontoExercise11.14.22(p.509)
d. z1.19
e. 0.1171
f. Donotrejectthenull
SolutiontoExercise11.14.23(p.510)
WeconcludethatthepatientdoeshavetheHIVviruswhen,infact,thepatientdoesnot.
SolutiontoExercise11.14.24(p.510)
c. z=2.21;p=0.0136
d. Rejectthenull
e. WeconcludethattheproportionofCalifornianprofessionalsthatwearjeanstoworkisgreaterthanthe
proportionofnon-Californianprofessionalswhen,infact,itisnotgreater.
f. WecannotconcludethattheproportionofCalifornianprofessionalsthatwearjeanstoworkisgreater
thantheproportionofnon-Californianprofessionalswhen,infact,itisgreater.
SolutiontoExercise11.14.25(p.510)
C
SolutiontoExercise11.14.26(p.510)
t
5
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Chapter12
LinearRegressionandCorrelation
12.1LinearRegressionandCorrelation
1
12.1.1StudentLearningOutcomes
Bytheendofthischapter,thestudentshouldbeableto:
 Discussbasicideasoflinearregressionandcorrelation.
 Createandinterpretalineofbestfit.
 Calculateandinterpretthecorrelationcoefficient.
 Calculateandinterpretoutliers.
12.1.2Introduction
Professionalsoftenwanttoknowhowtwoormorenumericvariablesarerelated. Forexample,istherea
relationshipbetweenthegradeonthesecondmathexamastudenttakesandthegradeonthefinalexam?
Ifthereisarelationship,whatisitandhowstrongistherelationship?
Inanotherexample,yourincomemaybedeterminedbyyoureducation,yourprofession,youryearsof
experience,andyourability.Theamountyoupayarepairpersonforlaborisoftendeterminedbyaninitial
amountplusanhourlyfee.Theseareallexamplesinwhichregressioncanbeused.
Thetypeofdatadescribedintheexamplesisbivariatedata-"bi"fortwovariables.Inreality,statisticians
usemultivariatedata,meaningmanyvariables.
Inthischapter,youwillbestudyingthesimplestformofregression,"linearregression"withoneindepen-
dentvariable(x).Thisinvolvesdatathatfitsalineintwodimensions.Youwillalsostudycorrelationwhich
measureshowstrongtherelationshipis.
12.2LinearEquations
2
Linearregressionfortwovariablesisbasedonalinearequationwithoneindependentvariable.Ithasthe
form:
y=a+bx
(12.1)
1
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2
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523
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524
CHAPTER12. LINEARREGRESSIONANDCORRELATION
whereaandbareconstantnumbers.
xistheindependentvariable,andyisthedependentvariable.Typically,youchooseavaluetosubstitute
fortheindependentvariableandthensolveforthedependentvariable.
Example12.1
Thefollowingexamplesarelinearequations.
y=3+2x
(12.2)
y0.01+1.2x
(12.3)
Thegraphofalinearequationoftheformy=a+bxisastraightline.Anylinethatisnotverticalcanbe
describedbythisequation.
Example12.2
Figure12.1:Graphoftheequationy1+2x.
Linearequationsofthisformoccurinapplicationsoflifesciences,socialsciences,psychology,business,
economics,physicalsciences,mathematics,andotherareas.
Example12.3
Aaron’sWordProcessingService(AWPS)doeswordprocessing. Itsrateis$32perhourplusa
$31.50one-timecharge. Thetotalcosttoacustomerdependsonthenumberofhoursittakesto
dothewordprocessingjob.
Problem
Findtheequationthatexpressesthetotalcostintermsofthenumberofhoursrequiredtofinish
thewordprocessingjob.
Solution
Letx=thenumberofhoursittakestogetthejobdone.
Lety=thetotalcosttothecustomer.
The$31.50isafixedcost. Ifittakesxhourstocompletethejob,then(32)(x)isthecostofthe
wordprocessingonly.Thetotalcostis:
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525
y=31.50+32x
12.3SlopeandY-InterceptofaLinearEquation
3
Forthelinearequationy=a+bx,b=slopeanda=y-intercept.
Fromalgebrarecallthattheslopeisanumberthatdescribesthesteepnessofalineandthey-interceptis
theycoordinateofthepoint(0,a)wherethelinecrossesthey-axis.
(a)
(b)
(c)
Figure12.2:Threepossiblegraphsofy=a+bx.(a)If0,thelineslopesupwardtotheright.(b)If
0,thelineishorizontal.(c)If0,thelineslopesdownwardtotheright.
Example12.4
Svetlanatutorstomakeextramoneyfor college. . Foreachtutoringsession,shechargesaone
timefeeof$25plus$15perhouroftutoring.Alinearequationthatexpressesthetotalamountof
moneySvetlanaearnsforeachsessionshetutorsisy=25+15x.
Problem
What are theindependent anddependentvariables? ? What t isthe y-interceptandwhatisthe
slope?Interpretthemusingcompletesentences.
Solution
Theindependentvariable(x)isthenumberofhoursSvetlanatutorseachsession.Thedependent
variable(y)istheamount,indollars,Svetlanaearnsforeachsession.
They-interceptis25(a=25). Atthestartofthetutoringsession,Svetlanachargesaone-timefee
of$25(thisiswhenx=0). Theslopeis15(b=15). Foreachsession,Svetlanaearns$15foreach
hourshetutors.
3
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526
CHAPTER12. LINEARREGRESSIONANDCORRELATION
12.4ScatterPlots
4
Beforewetakeupthediscussionoflinearregressionandcorrelation,weneedtoexamineawaytodisplay
the relationbetweentwovariablesxand y. The e most commonandeasiest wayisascatter plot. The
followingexampleillustratesascatterplot.
Example12.5
Fromanarticle inthe WallStreet Journal: : InEurope e and Asia, m-commerce ispopular. . M-
commerceusershavespecialmobilephonesthatworklikeelectronicwalletsaswellasprovide
phoneandInternetservices.UserscandoeverythingfrompayingforparkingtobuyingaTVset
orsodafromamachinetobankingtocheckingsportsscoresontheInternet. Fortheyears2000
through2004,wastherearelationshipbetweentheyearandthenumberofm-commerceusers?
Constructascatterplot.Letx=theyearandlety=thenumberofm-commerceusers,inmillions.
x(year)
y(#ofusers)
2000
0.5
2002
20.0
2003
33.0
2004
47.0
(a)
(b)
Figure12.3: (a)Tableshowingthenumberofm-commerceusers(inmillions)byyear. (b)Scatterplot
showingthenumberofm-commerceusers(inmillions)byyear.
Ascatterplotshowsthedirectionandstrengthofarelationshipbetweenthevariables. Acleardirection
happenswhenthereiseither:
 Highvaluesofonevariableoccurringwithhighvaluesoftheothervariableorlowvaluesofone
variableoccurringwithlowvaluesoftheothervariable.
 Highvaluesofonevariableoccurringwithlowvaluesoftheothervariable.
Youcandeterminethestrengthoftherelationshipbylookingatthescatterplotandseeinghowclosethe
pointsaretoaline,apowerfunction,anexponentialfunction,ortosomeothertypeoffunction.
Whenyoulookatascatterplot,youwanttonoticetheoverallpatternandanydeviationsfromthepattern.
Thefollowingscatterplotexamplesillustratetheseconcepts.
4
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527
(a)PositiveLinearPattern(Strong)
(b)LinearPatternw/OneDeviation
Figure12.4
(a)NegativeLinearPattern(Strong)
(b)NegativeLinearPattern(Weak)
Figure12.5
(a)ExponentialGrowthPattern
(b)NoPattern
Figure12.6
Inthischapter,weareinterestedinscatterplotsthatshowalinearpattern.Linearpatternsarequitecom-
mon. Thelinearrelationshipisstrongifthepointsareclosetoastraightline. . Ifwethinkthatthepoints
showalinearrelationship,wewouldliketodrawalineonthescatterplot. Thislinecanbecalculated
throughaprocesscalledlinearregression. However,weonlycalculatearegressionlineifoneofthevari-
ableshelpstoexplainorpredicttheothervariable. Ifxistheindependentvariableandythedependent
variable,thenwecanusearegressionlinetopredictyforagivenvalueofx.
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528
CHAPTER12. LINEARREGRESSIONANDCORRELATION
12.5TheRegressionEquation
5
Datararelyfitastraightlineexactly.Usually,youmustbesatisfiedwithroughpredictions.Typically,you
haveasetofdatawhosescatterplotappearsto"fit"astraightline.ThisiscalledaLineofBestFitorLeast
SquaresLine.
12.5.1OptionalCollaborativeClassroomActivity
Ifyouknowaperson’spinky(smallest)fingerlength,doyouthinkyoucouldpredictthatperson’sheight?
Collectdatafromyourclass(pinkyfingerlength,ininches). Theindependentvariable,x,ispinkyfinger
lengthandthedependentvariable,y,isheight.
Foreachsetofdata,plotthepointsongraphpaper. Makeyourgraphbigenoughandusearuler. Then
"byeye"drawalinethatappearsto"fit"thedata.Foryourline,picktwoconvenientpointsandusethem
tofindtheslopeoftheline.Findthey-interceptofthelinebyextendingyourlinessotheycrossthey-axis.
Usingtheslopesandthey-intercepts,writeyourequationof"bestfit". Doyouthinkeveryonewillhave
thesameequation?Whyorwhynot?
Usingyourequation,whatisthepredictedheightforapinkylengthof2.5inches?
Example12.6
Arandomsampleof11statisticsstudentsproducedthefollowingdatawherexisthethirdexam
score,outof80,andyisthefinalexamscore,outof200.Canyoupredictthefinalexamscoreofa
randomstudentifyouknowthethirdexamscore?
5
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529
x(thirdexamscore)
y(finalexamscore)
65
175
67
133
71
185
71
163
66
126
75
198
67
153
70
163
71
159
69
151
69
159
(a)
(b)
Figure12.7: (a)Tableshowingthescoresonthefinalexambasedonscoresfromthethirdexam.(b)Scatter
plotshowingthescoresonthefinalexambasedonscoresfromthethirdexam.
Thethirdexamscore,x,istheindependentvariableandthefinalexamscore,y,isthedependentvariable.
Wewillplotaregressionlinethatbest"fits"thedata. Ifeachofyouweretofitaline"byeye",youwould
drawdifferentlines.Wecanusewhatiscalledaleast-squaresregressionlinetoobtainthebestfitline.
Considerthefollowingdiagram. Eachpointofdataisofthetheform(x,y)andeachpointofthelineof
bestfitusingleast-squareslinearregressionhastheform
x,
^
y
!
.
The
^
y
isread"yhat"andistheestimatedvalueofy.Itisthevalueofyobtainedusingtheregressionline.
Itisnotgenerallyequaltoyfromdata.
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530
CHAPTER12. LINEARREGRESSIONANDCORRELATION
Figure12.8
Thetermy
0
^
y
0
e
0
iscalledthe"error"orresidual. Itisnotanerrorinthesenseofamistake. . The
absolutevalueofaresidualmeasurestheverticaldistancebetweentheactualvalueofyandtheestimated
valueofy.Inotherwords,itmeasurestheverticaldistancebetweentheactualdatapointandthepredicted
pointontheline.
Ifthe observeddatapointliesabove the line, theresidualispositive, andthe lineunderestimatesthe
actualdatavaluefory.Iftheobserveddatapointliesbelowtheline,theresidualisnegative,andtheline
overestimatesthatactualdatavaluefory.
Inthediagramabove,y
0
^
y
0
=e
0
istheresidualforthepointshown. Herethepointliesabovetheline
andtheresidualispositive.
e=theGreekletterepsilon
Foreachdatapoint,youcancalculatetheresidualsorerrors,y
i
^
y
i
=e
i
fori=1,2,3,...,11.
Eachjejisaverticaldistance.
Fortheexampleaboutthethirdexamscoresandthefinalexamscoresforthe11statisticsstudents,there
are11datapoints.Therefore,thereare11evalues.Ifyousquareeacheandadd,youget
(e
1
)
2
+(e
2
)
2
+...+(e
11
)
2
=
11
S
i=1
e2
ThisiscalledtheSumofSquaredErrors(SSE).
Usingcalculus,youcandeterminethevaluesofaandbthatmaketheSSEaminimum. Whenyoumake
theSSEaminimum,youhavedeterminedthepointsthatareonthelineofbestfit.Itturnsoutthattheline
ofbestfithastheequation:
^
y=
a+bx
(12.4)
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