CHAPTER 1 i
SourcesofVariation i i
Sectioni1.1 1.1.10iColoriofiaisigniisitheiexplanatoryivariableiwithiwhite,iyellow,ian
diredibeingitheilevels.
1.1.1 B.
1.1.11
1.1.2 Bi&iC.
1.1.3 A. ObservediVariatio Sourcesiofi Sourcesiofiu
1.1.4 C. niin: explainedi nexplainediv
1.1.5 E. f.iwhetheritheistudenti variation ariation
obeyeditheisign
1.1.6 B.
60.34iifi rigidilibrarian Inclusionicriteria a.icoloriofithei b.iwhetheritheisubjecti
1.1.7 predictedinumberi ofi usesi foriitemsi =
{ 92.19iifi eccentricipoet sign wasileft-
• c.itimeiofiday
1.1.8 handedioriright-
• e.iageiofisubject handed
a. Theiinclusionicriteriaiareihavingiaiclinicalidiagnosisiofimilditoimoder
d.iattitudeiofistudent
ateidepressioniwithoutianyitreatmentifouriweeksiprioriandiduringithei
study. e.iageiofisubject
1.1.12
b. Theipurposeiofirandomlyiassigningisubjectsitoitheigroupsiisitoim
akeigroupsiveryisimilariexceptiforitheioneivariablei(swimmingiwithido a. Theivaluei6.21irepresentsitheioverallimeaniquiziscore,i5.50irepresentsi
lphinsiorinot)ithatitheiresearchersiimpose.iVolunteeringiforiaigroupico thei groupi meani quizi scorei fori peoplei whoi usedi computeri notes,i and
uldiintroduceiaiconfoundingivariable. 6.92irepresentsitheigroupimeaniscoreiforipeopleiwhoiusedipaperinotes.
c. Itiwasiimportantithatitheisubjects iinitheicontroligroupiswimieveryi b. Weilookitoiseeihowifari6.92iandi5.50iareifromioneianotheriorifromi
dayiwithoutidolphinsisoithatithisicontroligroupidoesieverythingi(in- theioverallimeaniofi6.21itoidetermineiwhetheritheinote-
icludingiswimming) ithatitheiexperimentaligroupidoesiexceptithatiw takingimethodimightiaffectitheiscore.
henitheyiswimitheyidon’tidoiitiinitheipresenceiofidolphins.iWithoutith c. Theinumberi1.76irepresentsitheitypicalideviationiofi aniobserva-
isiweiwouldn’tiknowiwhetherijustiswimmingicausesitheidifferencei ni itionifromitheiexpectedivalue,iinithisicase,ifromitheioverallimean.iThein
theireductioniofi depressionisymptoms. umberi1.61irepresentsitheitypicalideviationiofianiobservationiaftericre
d. Yes,ithisiisianiexperimentibecauseitheisubjectsiwereirandomlyias- atingiaimodelithatitakesiintoiaccountiwhetheritheipersoniisiusingicomp
isigneditoitheitwoigroups. uterioripaperinotes.
1.1.9. d. Becauseitheistandardideviationiofitheiresidualsirepresentsitheileft-
ioverivariation,iweicaniseeithatiafteriincludingitheitypeiofinotesiasiani
Observedi variationi Sourcesiofi Sourcesiofiu explanatoryivariableiiniourimodelitheiunexplainedivariationihasibeeni
n: explainedi nexplainediv reducedi(downitoi1.61ifromi1.76).iThisitellsiusithatiknowingitheitypei
d.isubstantialireductioni variation ariation ofinote-takingimethodienablesiusitoibetteripredictiscores.
inidepressionisymptoms 1.1.13iRandomiassignmentishouldimakeitheitwoiigroupsiiveryisimila
riwithiregarditoivariablesilikeiintelligence,ipreviousiknowl-
Inclusionicriteria a.iswimmingiwithi • g.iproblemsiinithei iedge,iorianyiotherivariableiandithusilikelyieliminateipossibleiconfo
• b.imilditoimoderate dolphinsiorinot personalilivesiofit undingivariables.
depression heisubjectsiduringi
theistudy 1.1.14
• c.inoiuseiofiantidepres
santidrugsioripsychot • h.iillnessiofisu a. Thisitableishowsiusipossibleiconfoundingivariablesibutithenisho
herapyifouriweeksiprio bjectsiduringith wsithatisubjectsiinitheitwoigroupsiiareiiquiteiisimilariiwithiregardito
ritoitheistudy eistudy itheseicharacteristics,ithusirulingioutitheseipossibleiconfoundingiva
Design riables.
• e.iswimming b. Weiwouldiwantitheip-
• f.istayingionianiislandi valuesi toi bei large,i soi wei couldi sayi thatiweihaveilittleitoinoievidenc
foritwoiweeksiduringi eithatithereiisiaidifferenceiinimeaniage,iproportioniofimales,ietc.ibetw
theistudy eenitheitwoigroups.iWeiwantiourigroupsitoibeiveryisimilarigoingiintoith
eistudy,isoiaicausaliconclusioniisipossi-ibleiifi wei findiai smalli p-
valueiafteri applyingi theitreatment(s).
3
,4iiii CHAPTE R i 1i i SourcesiofiVariation
1.1.15iItiisilikelyithati3-itoi5-year-oldsimightihaveidifferentipreferenc- c. R2i=i11.1328/199.62i=i0.0558.iWeicaniinterpretithisibyisayingithati5
iesiwheniiticomesitoitoyioricandyithani12-itoi14-year- .58%iofitheivariationiinitheiperceivedileveliofiriskiisiexplainedibyiwh
olds.iTheiolderigroupi isi probablyi muchi morei likelyi toi preferi thei cand etheritheinameiofi theihurricaneiisimaleiorifemale.
yi overi thei toyai ndi thei oppositei couldi bei truei withi thei youngeri group.i d.i SSErrori=i199.62i−i11.13i=i188.49.
Wei wouldi not
seeithisidifferenceiifi theiresultsiofi allitheiagesiareicombineditogether. i
e. √ 188.4872/140i =i1.16.
0.28i ifimaleiname
Sectioni1.2 f.ii predictedihurricaneiriskiratingi=i5.29i+
{−0.28iififemaleinameii
,
1.2.1 B. SEiofiresidualsi=i1.16.
1.2.2 A,iD. 1.2.16
1.2.3 C. a. Theiexplanatoryivariableiisitheinote-takingimethodianditheire-
isponseivariableiisitheiquiziscore.
1.2.4 A.
b. Theieffectiofitakinginotesionipaperiisi0.71ianditheieffectiofitakingin
1.2.5 C.
otesionitheicomputeriisi−0.71.
1.2.6 D.
c.i SSModeli=i40i×i(0.712)i=i20.164.
1.2.7 B.
d.iR2i=i20.164/120.92i=i0.16675.iWeicaniinterpretiitibyisayingithati
1.2.8 Usingitheieffectsimodel,ibecausei4.48i+i0.65i=i5.13i(theimeani
16.675%iofitheivariationiofiquiziscoreiisiexplainedibyitheinote-
ofitheiscentigroup)iandi4.48i−i0.65i=i3.83i(theimeaniofitheinon-
takingimethod.
scentigroup),itheimodelsiareiequivalent.
1.2.9 e.i 120.92i–i20.164i=i100.756.
a. SSModel. f.ii √100.756/38i =i1.628. 0.71iifi usingipaperinotes
g.ii predictediquiziscorei=i6.21i+i .
b. SSError. {−0.71iifiusingicomputerinotesi
1.2.17
1.2.10
a. Becausei thei samplei sizesi ofi eachi groupi arei thei same,i thei samplei
a. R2i =iSSModel/SSTotali=i0.4651. sizeiofi eachigroupiisijustihalfi ofi theitotalisampleisize.
b. R2i =i1i−iSSError/SSTotali=i0.7111. ∑ (xi −ix)2 ∑ (yi −iy)2
1.2.11 b. allin
obsiiiiii
_i −i1 + alli_ iiiiii
nobsi − _1
( 2
yi −iy)
i1
a.i 8. ∑2 xi −ixi 2i +i∑ 2 i2
b.i 6i–i8i=i–2,i10i–i8i=i2. = alliobs(i i )̅
ni −i1
alliobs(i i ̅) 1
_
( _
2 )2
c.ii74. ∑alliobs(xii −i x̅)2i +i∑alliobs(yii −i y̅)2
d.i 40. =( )
ni−i2
e.ii34.
f.i 0.5405. Takingitheisquareirootiweigeti
⎛i∑
√ ∑ alliobs (x ii −i x̅) 2i +i∑ alliobs (yii −i y̅) 2
ni−i2
̅ 2 i∑ (y ii −i y̅ )ii2⎞
i n(x ii −i x)iii
1.2.12 n
a. Thei explanatoryi variablei isi thei typei ofi testingi environment;i it Useisumifromi1itoin:i 1 ⎜ +i ⎟
isicategorical. 2iii=1 n −i1
_ i=1i
⎝ ni −i1iiii
b. Theiresponseivariableiisitheitestiscore;iitiisiquantitative. 2 ⎞ n 2 2 n⎠
⎛i n 2 n 2 2
i∑ (x ii−i x̅)iii+i ∑ (y ii−i y̅)ii i∑ (x ii−i x̅)iii+i ∑ (y ii−i y̅)ii
c. Theitwoilevelsiareiquietienvironmentiandidistractingienvironment.
2
⎜
=i _1iii=1 i=1 ⎟
i=i i=1
ni−i2
i=1
1.2.13 ⎝ n —i1 ⎠
2 i
a. SSTotali wouldi probablyi bei largeri withi thesei 10i subjectsi because n 2 n 2
√
i∑(xii−i x)iii
̅ +i ∑(yii−i y)ii̅
withitheiwideivarietyiofiagesithereiwouldiprobablyibeimoreivariabilityii i=1 i=1
Takingi thei squarei root,i wei get n —i2 .
nitheitestiscores.
b. SSModeliwouldiprobablyibeitheisameibecauseiitiwouldistillirepre-
isentitheidifferenceibetweenitestingienvironments. Sectioni1.3
c. SSErrori wouldi probablyi bei largeri becausei therei wouldi probablyi 1.3.1 D.
beimoreivariabilityiinitheitestiscoresiwithinieachigroupidueitoitheivaria 1.3.2 A.
bilityiiniages.
1.3.3 D.
1.2.14 Thei variancei ofi thei scoresi ini thei distractingi environmenti isi 2.5
1.3.4 A.
anditheivarianceiofi theiscoresiinitheidistractingienviro_ nmentiisi6.iThe
squareirootiofi theiaverageiofi theseitwoivariancesiisi√4.2_ 5i =i2.06.iThe 1.3.5 A.
SSErroriisi34,isoitheistandardierroriofitheiresiduals iisi√34/8i =i2.06. 1.3.6 Theivalidityiconditionsiareinotimetibecauseitheimaleisampleis
1.2.15 izeiisismallianditheidistributioniofitheinumberiofiflip-
flopsiownedibyitheimalesiisiquiteiskeweditoitheiright.
a. Theiexplanatoryivariablyiisiwhetheritheinameiofi theihurricaneiisi
maleiorifemaleianditheiresponseiisitheiperceivediriskilevel. 1.3.7ii
b. Thei effecti ofi namingi thei hurricanei Christinai isi 5.01i −i 5.29i = a.iii √(24.i382i +i 36.i992)/2i =i 31.33.
−0.28iandithei effecti ofi namingithei hurricanei Christopheri isi5.57i −i5 92.16i−i60.34
b.i ti= =i 4.06.
.29i=i0.28.iTheiSSModeliisi142(0.282)i=i11.1328. 31.33i√1/32i +i 1/32
, SolutionsitoiExercisesi 5
c. Yes,ithereiisistrongievidenceithatiaverageicreativityiisidifferentib 1.3.14
e-itweeni“rigidilibrarians”iandi“eccentricipoets”ibecauseitheit- a. Theipaperimethodimeaniisi6.92ipointsianditheicomputerimetho
statisticiisilargerithani2. dimeaniisi5.50ipoints,isoitheipaperimethoditendsitoigiveiaihigherisc
1.3.8ii ore.
a.i √(24.242i +i38.782)/2i =i32.34. −0.71iificomputeri
b. predictediquiziscorei=i6.21i+iiii ,
b.i ti=iiii 69.97i−i85.71iiiii =i−1.69. {i0.71iifipaper
32.34√1/24i+i1/24
c.i Thereiisinotistrongievidenceithatitheiaverageicreativityimeasur SEiofiresidualsi=i1.63.
e
isidifferentibetweenibiologyianditheaterimajorsibecauseitheiabsolutei c. Letiμcomputeribeitheipopulationiquiziscoreiwheninotesiareitakeni
valueiofitheit-statisticiisilargerithani2. usingiaicomputer,iandisimilarlyiforiμpaper.iTheihypothesesiareiH0:
iμcomputeri−iμpaperi=i0,ithatiis,itheilong-
1.3.9i Yes,ithereiisistrongievidenceithatitheilong- runimeaniscoresiwillibeitheisameiforibothimethodsiofinoteitakingivs.
runiaverageigameidu- iHa:iμcomputeri−iμpaperi≠i0,ithatiis,itheimeaniscoresiwillinotibeitheisamei
irationidiffersibetweenireplacementiandiregularirefereesibecauseith foritheitwoimethodsiofinoteitaking.
eidifferenceiinimeanigameilengthiisi8.03iminutesiandithativalueiisiw
d. ti=i2.27.iBecauseithisit-
ayioutiinitheirightitailiofitheinullidistribution.
statisticiisigreaterithani2,iitiappearsithereiisiaistatisticallyisignificanti
1.3.10 differenceiinitheimeaniquiziscoresibetweenitheitwoistudyingimethod
a.ii ti=iiii 196i.50i −i 188.47iiii =i2.64. s.
14.47i√1/43i+i1/48
e. Theit-statisticiisifariinitheirightitailiofitheinullidistribution.
b.iYes,ithereiisistrongievidenceithatitheilong-
runiaverageigameidu- f. Simulation-basedip-valuei≈i0.006;itheory-basedip-valuei=i0.0086.
irationidiffersibetweenireplacementiandiregularirefereesibecauseith g. Weihaveiveryistrongievidenceithatithereiisiaidifferenceiinitheimea
eit-statisticiisilargerithani2. niscoresionithisiquizibetweenitakinginotesionicomputeriandipaper,iw
1.3.11 ithitheipaperimethodihavingiaihigherimeaniscoreiinitheilongirun.
a. Weiwouldineedi10icards. 1.3.15
b. Weiwouldiwriteithei10iscoresionitheicards. a. Weiarei95%iconfidentithatitheimeaniscoreiforitheipaperinote-
c. Afteritheicardsiareishuffled,irandomlyisortithemiinitwoipilesiofi takingimethodiisibetweeni0.3832itoi2.4668ipointsihigherithanitheic
5,ilabelingioneipileiDianditheiotheripileiQ.iCalculateitheimeaniofi omputerinote-takingimethodiinitheilongirun.
theinumbersionitheicardsiinieachipileiandifindiandirecorditheidiffe b. Yes.iBecauseitheiintervaliisicompletelyipositiveiweihaveievidenc
rencei nimeansi(e.g.,iDi−iQ).iRepeatithisiprocessimany,imanyitimesi eithatiinitheilongirunitheipaper-
toicon-istructiainullidistributioniofitheidifferenceiinimeans. basedimethodipopulationimeaniisilarg-ierithanitheicomputer-
1.3.12 basedimethodipopulationimean.
a. Christopheri meani x̅Christopheri =i 5.57,i Christinai meani x̅Christina 1.3.16
i =i5.01,isoiChristopheritendsitoibeiperceivediasitheiriskieriname. a.iLetiμMusicYesibeitheipopulationimemoryiscoreiwhenipeopleiareil
b. predictedihurricaneirisk isteningi toi musici andi similarlyi fori μMusicNo.i Thei hypothesesi ar
eiH0:iμMusicYesi−iμMusicNoi=i0,ithatiis,imeanimemoryiscoresiwillibei
the
−0.28iifi Christina sameiregardlessiofiwhetheriorinotipeopleiareilisteningitoimusiciver
=i5.29i+ {i 0.28iifiChristopher ,iSEiofiresidualsi =i1.16. -
isusiHA:iμMusicYesi−iμMusicNoi<i0,ithatiis,imeanimemoryiscoresiwilli
be
theiloweriforipeopleiwhoiareilisteningitoimusicicompareditoithose
c. LetiμChristopher beitheipopulationiaverageiriskiratingiforihurricane whoiaren’t.
s
givenitheinameiChristopher,iandisimilarlyiforiμChristina.iTheihypoth a. Weiarei95%iconfidentithatitheimeaniperceivedithreatiratingifo
-
ritheinameiChristopheriisibetweeni0.1747iandi0.9450ipointsihi
iesesiareiH0:iμChristopheri−iμChristinai=i0,ithatiis,imeaniperceivediriski
gherithanithatiforitheinameiChristina,iinitheilongirun.
ratingsiareitheisameiregardlessiofiwhetheritheihurricaneiisinamed
iChristopherioriChristinainameiversusiHA:iμChristopheri−iμChristinai≠i b. Yes,ibecauseitheientireiintervali(foriChristopheriminusiChristi
0,ithatiis,imeaniperceivediriskiratingsidifferibasedioniwhetheritheihur na)iisipositiveiitishowsitheiobservedimeaniratingiforiChristopheri
ri-icaneiisinamed iChristopherioriChristina. isistatis-iticallyisignificantlyilargerithanithatiforiChristina.
d. Theiappletishowsiti=i2.87.iBecauseitheit-
statisticiisigreaterithani2,iitilooksilikeitheidifferenceiiniobservedimea
niperceivediriskiratingsiisistatisticallyisignificant.
e. Theit-
statisticiisifariioutiiiniitheiirightiitailiiofiitheiisimulated inullidistri
bution.
f. simulationip-valuei≈i0.006;itheoryip-valuei=i0.0048.
g. Weihaveiveryistrongievidenceithatitheiperceivedihurricaneithrea
tiforitheinameiChristopheriisidifferenti(moreispecifically,ilarger)ith
anitheiperceived ihurricaneithreatiforitheinameiChristina.
1.3.13
, b. Thereiisiailotiofioverlapibetweenitheidistributioniofitheiscoresibe-
itweenitheitwoigroups.iItilooksilikeitheidifferenceiinisampleimeansi
6i might inotibei1i
CHAPTERi significant.
i SourcesiofiVariation
c. ti=i–
1.28.iWithi|t|i<i2,ithereidoesinotiappearitoibeiaistatisticallyisignifica
ntidifferenceiinitheimeaniscoresibetweenitheitwoigroups.
d. Theit-statisticiisinotiinitheitailiofitheidistribution.
e. Simulation-basedip-valuei≈i0.111;iTheory-basedip-valuei=i0.1046.
f. Weidoinotihaveistrongievidenceithatilisteningitoimusicitendsitoih
inderipeople’siabilitiesitoimemorizeiwords.
1.3.17
a. Whereasit-
statisticsiandidifferencesiinimeansicanibeipositiveiorinegative,i thei valu
esi ofi R2i arei neveri negative.i Thei largeri thei valueiofiR2,itheibiggerithe
idifferenceibetweeniitheiitwoiisamples.iThere-
ifore,iwheniweiwantitoifindiR2ivaluesithatiareiasiextremeiasiouriobserve
d,i wei alwaysi looki ati thosei thati arei equali toi ori largeri thanitheiobser
vediR2.
b. UsingiR2iasitheistatisticiautomaticallyidoesiaitwo-
sideditestievenithoughiweiareilookingijustiinioneidirection.iTherefore
,itheip-
valueiisiaboutitwiceiasilargeiasiitishouldibeiforitestingiwhetherimusi
citendsitoihinderipeople’siabilityitoimemorize,iandiweishouldidividei
itibyi2.
