Sources of Variation v v
Sectionv1.1 1.1.10v Colorvofvavsignvisvthevexplanatoryvvariablevwithvwhite,vyellow
,vandvredvbeingvthevlevels.
1.1.1 B.
1.1.11
1.1.2 Bv&vC.
1.1.3 A. ObservedvVariatio Sourcesvof Sourcesvofvu
1.1.4 C. nvin: vexplaine nexplainedvv
1.1.5 E. f.vwhethervthevstudent dvvariatio ariation
vobeyedvthevsign n
1.1.6 B.
60.34vifvrigidvlibrarian Inclusionv criteria a.vcolorvofvthevsi b.vwhethervthevsubject
1.1.7 predictedvnumbervofv usesvforvitemsv=
{92.19vifv eccentricv poet gn vwasvleft-
• c.vtimevofvday
1.1.8 handedvorvright-
• e.vagevofv subject handed
a. Thevinclusionvcriteriavarevhavingvavclinicalvdiagnosisvofvmildvtov
d.vattitudevofvstudent
moderatevdepressionvwithoutvanyvtreatmentvfourvweeksvpriorvandvdu
ringvthevstudy. e.vagevofv subject
1.1.12
b. Thevpurposevofvrandomlyvassigningvsubjectsvtovthevgroupsvisvtov
makevgroupsvveryvsimilarvexceptvforvthevonevvariablev(swimmingvwit a. Thevvaluev6.21vrepresentsvthevoverallvmeanvquizvscore,v5.50vrepresent
hvdolphinsvorvnot)vthatvthevresearchersvimpose.vVolunteeringvforvavgro svthevgroupvmeanvquizvscorevforvpeoplevwhovusedvcomputervnotes,va
upvcouldvintroducevavconfoundingvvariable. nd
6.92vrepresentsvthevgroupvmeanvscorevforvpeoplevwhovusedvpapervnotes.
c. Itvwasvimportantvthatvthevsubjectsvinvthevcontrolvgroupvswimvever
yvdayvwithoutvdolphinsvsovthatvthisvcontrolvgroupvdoesveverythingv(i b. Wevlookvtovseevhowvfarv6.92vandv5.50varevfromvonevanothervorvfro
n- mvthevoverallvmeanvofv6.21vtovdeterminevwhethervthevnote-
vcluding vswimming)vthat vthe vexperimental vgroup vdoes vexceptvthatvw
takingvmethodvmightvaffectvthevscore.
henvtheyvswimvtheyvdon’tvdovitvinvthevpresencevofvdolphins.vWithoutv c. Thevnumberv1.76vrepresentsvthevtypicalvdeviationvofvanvobserva-
thisvwevwouldn’tvknowvwhethervjustvswimmingvcausesvthevdifferenc vtion vfromvthe vexpected vvalue, vinvthisvcase, vfromvthe voverallvmean. vT
evinvthevreductionvofvdepressionvsymptoms. hevnumberv1.61vrepresentsvthevtypicalvdeviationvofvanvobservationvaf
d. Yes,vthisvisvanvexperimentvbecausevthevsubjectsvwerevrandomlyvas tervcreatingvavmodelvthatvtakesvintovaccountvwhethervthevpersonvisvus
-vsignedvtovthevtwovgroups. ingvcomputervorvpapervnotes.
d. Becausevthevstandardvdeviationvofvthevresidualsvrepresentsvthevleft
1.1.9.
-
Observedvvariationvi Sourcesvof Sourcesvofvu vover vvariation,vwe vcanvsee vthatvaftervincluding vthe vtype vofvnotesvas v
n: vexplaine nexplainedv anvexplanatoryvvariablevinvourvmodelvthevunexplainedvvariationvhasvb
d.vsubstantialvreductionv dvvariatio variation eenvreducedv(downvtov1.61vfromv1.76).vThisvtellsvusvthatvknowingvthe
vtype vof vnote-takingvmethod venablesvus vtovbettervpredictvscores.
invdepressionvsymptoms n
1.1.13vRandomv assignmentv shouldv makev thev twov groupsv veryvsi
Inclusionv criteria a.vswimmingvwithvd • g.vproblemsvinvthe milarvwithvregardvtovvariablesvlikevintelligence,vpreviousvknowl-
• b.vmildvtovmoderate olphinsvorvnot vpersonalvlivesvofvt
vedge, vor vany vothervvariablevand vthusvlikelyveliminatevpossible vconfo
depression hevsubjectsvduringv undingvvariables.
• c.vnovusevofvantidepre thevstudy
1.1.14
ssantvdrugsvorvpsycho • h.villnessvofvsu
therapyvfourvweeksvpr bjects vduringvt a. Thisvtablevshowsvusvpossiblevconfounding vvariablesvbutvthenvsho
iorvtovthevstudy hevstudy wsv thatv subjectsv inv thev twov groupsv arev quitev similarv withvregar
Design dvtovthesevcharacteristics,vthusvrulingvoutvthesevpossiblevconfounding
• e.vswimming vvariables.
• f.vstayingvonvanvislan b. Wev wouldv wantv thev p-
dvforvtwovweeksvduri valuesv tov bev large,v sov wev couldv sayv thatvwevhavevlittlevtovnovevide
ngvthevstudy ncevthatvtherevisvavdifferencevinvmeanvage,vproportionvofvmales,vetc.vb
etweenvthevtwovgroups. vWevwantvourvgroupsvtovbevveryvsimilarvgoingv
intovthevstudy,vsovavcausalvconclusionvisvpossi-
vble vifvwevfind va vsmallvp-value vaftervapplying vthevtreatment(s).
, 3
v 1
,4v CHAPTER v 1v v SourcesvofvVariation
1.1.15vItvisvlikelyvthatv3-vtov5-year- c. R2v=v11.1328/199.62 v=v0.0558. vWevcanvinterpretvthisvbyvsayingvth
oldsvmightvhavevdifferentvpreferenc- atv5.58%vofvthevvariationvinvthevperceivedvlevelvofvriskvisvexplainedv
ves vwhen vitvcomes vtovtoy vor vcandy vthan v12- vtov14-year- byvwhethervthevnamevofvthevhurricanevisvmalevorvfemale.
olds.vThevoldervgroupvisvprobablyvmuchvmorevlikelyvtovprefervthevca d. SSErrorv =v199.62v−v11.13v=v188.49.
ndyvovervthevtoyvandvthevoppositevcouldvbevtruevwithvthevyoungervgr
oup.vWevwouldvnot
seevthisvdifferencevifvthevresultsvofvallvthevagesvarevcombinedvtogether.
v
e.v √ 188.4872/140v=v1.16 0.28v ifvmalevname
.
Sectionv1.2 f.v v predictedv hurricanev riskv ratingv =v 5.29v +v
{−0.28v ifv femalev namev
,
1.2.1 B. SEvofvresidualsv=v1.16.
1.2.2 A,vD. 1.2.16
1.2.3 C. a. Thevexplanatoryvvariablevisvthevnote-takingvmethodvandvthevre-
v sponse vvariablevis vthevquizvscore.
1.2.4 A.
b. Theveffectvofvtakingvnotesvonvpapervisv0.71vandvtheveffectvofvtaki
1.2.5 C.
ngvnotesvonvthevcomputervisv−0.71.
1.2.6 D.
c. SSModelv =v 40v ×v (0.712)v =v 20.164.
1.2.7 B.
d. R2v =v20.164/120.92v=v0.16675.vWevcanvinterpretvitvbyvsayingvtha
1.2.8 Usingvtheveffectsvmodel,vbecausev4.48v+v0.65v=v5.13v(thevmean tv16.675%vofvthevvariationvofvquizvscorevisvexplainedvbyvthevnote-
vof vthevscent vgroup) vand v4.48 v− v0.65v= v3.83 v(thevmean vof vthe vnon-
takingvmethod.
scentvgroup),vthevmodelsvarevequivalent.
e. 120.92v–v20.164v=v100.756.
1.2.9 f. √100.756/38 v =v 1.628.
a. SSModel. 0.71vifvusingvpapervnotes
g. predictedvquizvscorev=v6.21v+v .
b. SSError. {−0.71vifvusingvcomputervnotesv
1.2.17
1.2.10
a. Becausevthevsamplevsizesvofveachvgroupvarevthevsame,vthevsample
a. R2v =v SSModel/SSTotalv =v 0.4651. vsizevofveachvgroup visvjustvhalfvof vthevtotalvsamplevsize.
b. R2v =v 1v −v SSError/SSTotalv =v 0.7111. ∑ (xv −vx)2 ∑ (yv −vy)2
allvobsv v i ̅ v v ̅ _1
b. + allvobsv
vi
v
1.2.11 nv−v1 nv−v1
( _
2 _
2 )2
a. 8. ∑allvobs(xiv −vx̅) +v∑allvobs(yiv −v y̅)2v v _1
2v
b. v68v–
v=v8 v= v–2, v10v–
v2. n
=
( _v−v1
)2
2
c. 74.
∑allvobs(xiv −vx)̅ +v∑allvobs(yiv −v y̅)
2v 2
d. 40. =v( )
nv−v2
e. 34.
Takingv thev squarev rootv wev getv ∑allvobs(xiv −vx̅) +v∑allvobs(yiv −v y̅
2v
f. 0.5405.
√ )2
nv−v2
⎛v n n
2⎞
⎜ ⎟
1.2.12 2
a. Thev explanatoryv variablev isv thev typev ofv testingv environment;v it Usevsumvfromv1vtovn:v v∑v(vxivv −vx)̅ v + v∑v(yv viv−v y
1
_v i=1 i=1
vn vn
⎝ ̅)2v v −v1
2
2v −v1v v ⎠
isvcategorical.
b. Thevresponsevvariablevisvthevtestvscore;vitvisvquantitative. ⎛v n n 2⎞ n
2
n
2
2
c. Thevtwovlevelsvarevquietvenvironmentvandvdistractingvenvironment. v∑(x iv−v x̅)v +v ∑(y iv−v y )̅v ∑(xiv −v x)̅ +v ∑(yiv −v y̅)
1.2.13
1v
⎝
⎜
=v_2v i=1 vnv
i=1
−v1
⎟⎠
v=
vi=1 i=1
nv−v2
2
√
n n
a. SSTotalv wouldv probablyv bev largerv withv thesev 10v subjectsv because ∑(xiv −v x)̅ 2 + ∑(yiv−v y)̅ 2
withvthevwidevvarietyvofvagesvtherevwouldvprobablyvbevmorevvariabilit vmalevor vfemalevandvthe vresponse visvthevperceived vriskvlevel.
nv−v
yvinvthevtestvscores. b. Thev effectv ofv namingv thev hurricanev Christina v isv 5.01v −v 5.29v =
2
b. SSModelvwouldvprobablyvbevthevsamevbecausevitvwouldvstillvrepre −0.28v andv thev effectv ofv namingv thev hurricanev Christopherv isv 5.57v −
-vsentvthevdifferencevbetweenvtestingvenvironments. 5.29v=v0.28.vThevSSModelv isv142(0.282)v=v11.1328.
c. SSErrorvwouldvprobablyvbevlargervbecausevtherevwouldvprobablyv
bevmorevvariabilityvinvthevtestvscoresvwithinveachvgroupvduevtovthevv
ariabilityvinvages.
1.2.14 Thevvariancevofvthevscoresvinvthevdistractingvenvironmentvisv2.
5vandvthevvariancevofvthevscoresvinvthevdistractingve n v i r o_
n m e n t visv6.v
Thevsquarevrootvofvthevaveragevofvthesevtwovvariancesvisv√ 4.25_ v =v2.0
6.vThevSSErrorvisv34,vsovthevstandardverrorvofvthevresidualsvisv√34/8v
=v2.06.
1.2.15
a. Thevexplanatoryvvariablyvisvwhethervthevnamevofvthevhurricanevis
, i=1 i=1
Takingvthevsquarevroot,vwevget v
.
v
Sectionv1.3
1.3.1 D.
1.3.2 A.
1.3.3 D.
1.3.4 A.
1.3.5 A.
1.3.6 Thev validityv conditionsv arev notv metv becausev the
v malev sample vsizevis vsmallvand vthevdistributionvofvthe vnu
mbervofvflip-
flopsvownedvbyvthevmalesvisvquitevskewedvtovthevright.
1.3.7
a. √(24.v382v +v 36.v992)/2v =v 31.33.
b. tv=vv v 92.16v−v60.34v v v=v4.06.
31.33v√ 1/32v +v 1/32
Sectionv1.1 1.1.10v Colorvofvavsignvisvthevexplanatoryvvariablevwithvwhite,vyellow
,vandvredvbeingvthevlevels.
1.1.1 B.
1.1.11
1.1.2 Bv&vC.
1.1.3 A. ObservedvVariatio Sourcesvof Sourcesvofvu
1.1.4 C. nvin: vexplaine nexplainedvv
1.1.5 E. f.vwhethervthevstudent dvvariatio ariation
vobeyedvthevsign n
1.1.6 B.
60.34vifvrigidvlibrarian Inclusionv criteria a.vcolorvofvthevsi b.vwhethervthevsubject
1.1.7 predictedvnumbervofv usesvforvitemsv=
{92.19vifv eccentricv poet gn vwasvleft-
• c.vtimevofvday
1.1.8 handedvorvright-
• e.vagevofv subject handed
a. Thevinclusionvcriteriavarevhavingvavclinicalvdiagnosisvofvmildvtov
d.vattitudevofvstudent
moderatevdepressionvwithoutvanyvtreatmentvfourvweeksvpriorvandvdu
ringvthevstudy. e.vagevofv subject
1.1.12
b. Thevpurposevofvrandomlyvassigningvsubjectsvtovthevgroupsvisvtov
makevgroupsvveryvsimilarvexceptvforvthevonevvariablev(swimmingvwit a. Thevvaluev6.21vrepresentsvthevoverallvmeanvquizvscore,v5.50vrepresent
hvdolphinsvorvnot)vthatvthevresearchersvimpose.vVolunteeringvforvavgro svthevgroupvmeanvquizvscorevforvpeoplevwhovusedvcomputervnotes,va
upvcouldvintroducevavconfoundingvvariable. nd
6.92vrepresentsvthevgroupvmeanvscorevforvpeoplevwhovusedvpapervnotes.
c. Itvwasvimportantvthatvthevsubjectsvinvthevcontrolvgroupvswimvever
yvdayvwithoutvdolphinsvsovthatvthisvcontrolvgroupvdoesveverythingv(i b. Wevlookvtovseevhowvfarv6.92vandv5.50varevfromvonevanothervorvfro
n- mvthevoverallvmeanvofv6.21vtovdeterminevwhethervthevnote-
vcluding vswimming)vthat vthe vexperimental vgroup vdoes vexceptvthatvw
takingvmethodvmightvaffectvthevscore.
henvtheyvswimvtheyvdon’tvdovitvinvthevpresencevofvdolphins.vWithoutv c. Thevnumberv1.76vrepresentsvthevtypicalvdeviationvofvanvobserva-
thisvwevwouldn’tvknowvwhethervjustvswimmingvcausesvthevdifferenc vtion vfromvthe vexpected vvalue, vinvthisvcase, vfromvthe voverallvmean. vT
evinvthevreductionvofvdepressionvsymptoms. hevnumberv1.61vrepresentsvthevtypicalvdeviationvofvanvobservationvaf
d. Yes,vthisvisvanvexperimentvbecausevthevsubjectsvwerevrandomlyvas tervcreatingvavmodelvthatvtakesvintovaccountvwhethervthevpersonvisvus
-vsignedvtovthevtwovgroups. ingvcomputervorvpapervnotes.
d. Becausevthevstandardvdeviationvofvthevresidualsvrepresentsvthevleft
1.1.9.
-
Observedvvariationvi Sourcesvof Sourcesvofvu vover vvariation,vwe vcanvsee vthatvaftervincluding vthe vtype vofvnotesvas v
n: vexplaine nexplainedv anvexplanatoryvvariablevinvourvmodelvthevunexplainedvvariationvhasvb
d.vsubstantialvreductionv dvvariatio variation eenvreducedv(downvtov1.61vfromv1.76).vThisvtellsvusvthatvknowingvthe
vtype vof vnote-takingvmethod venablesvus vtovbettervpredictvscores.
invdepressionvsymptoms n
1.1.13vRandomv assignmentv shouldv makev thev twov groupsv veryvsi
Inclusionv criteria a.vswimmingvwithvd • g.vproblemsvinvthe milarvwithvregardvtovvariablesvlikevintelligence,vpreviousvknowl-
• b.vmildvtovmoderate olphinsvorvnot vpersonalvlivesvofvt
vedge, vor vany vothervvariablevand vthusvlikelyveliminatevpossible vconfo
depression hevsubjectsvduringv undingvvariables.
• c.vnovusevofvantidepre thevstudy
1.1.14
ssantvdrugsvorvpsycho • h.villnessvofvsu
therapyvfourvweeksvpr bjects vduringvt a. Thisvtablevshowsvusvpossiblevconfounding vvariablesvbutvthenvsho
iorvtovthevstudy hevstudy wsv thatv subjectsv inv thev twov groupsv arev quitev similarv withvregar
Design dvtovthesevcharacteristics,vthusvrulingvoutvthesevpossiblevconfounding
• e.vswimming vvariables.
• f.vstayingvonvanvislan b. Wev wouldv wantv thev p-
dvforvtwovweeksvduri valuesv tov bev large,v sov wev couldv sayv thatvwevhavevlittlevtovnovevide
ngvthevstudy ncevthatvtherevisvavdifferencevinvmeanvage,vproportionvofvmales,vetc.vb
etweenvthevtwovgroups. vWevwantvourvgroupsvtovbevveryvsimilarvgoingv
intovthevstudy,vsovavcausalvconclusionvisvpossi-
vble vifvwevfind va vsmallvp-value vaftervapplying vthevtreatment(s).
, 3
v 1
,4v CHAPTER v 1v v SourcesvofvVariation
1.1.15vItvisvlikelyvthatv3-vtov5-year- c. R2v=v11.1328/199.62 v=v0.0558. vWevcanvinterpretvthisvbyvsayingvth
oldsvmightvhavevdifferentvpreferenc- atv5.58%vofvthevvariationvinvthevperceivedvlevelvofvriskvisvexplainedv
ves vwhen vitvcomes vtovtoy vor vcandy vthan v12- vtov14-year- byvwhethervthevnamevofvthevhurricanevisvmalevorvfemale.
olds.vThevoldervgroupvisvprobablyvmuchvmorevlikelyvtovprefervthevca d. SSErrorv =v199.62v−v11.13v=v188.49.
ndyvovervthevtoyvandvthevoppositevcouldvbevtruevwithvthevyoungervgr
oup.vWevwouldvnot
seevthisvdifferencevifvthevresultsvofvallvthevagesvarevcombinedvtogether.
v
e.v √ 188.4872/140v=v1.16 0.28v ifvmalevname
.
Sectionv1.2 f.v v predictedv hurricanev riskv ratingv =v 5.29v +v
{−0.28v ifv femalev namev
,
1.2.1 B. SEvofvresidualsv=v1.16.
1.2.2 A,vD. 1.2.16
1.2.3 C. a. Thevexplanatoryvvariablevisvthevnote-takingvmethodvandvthevre-
v sponse vvariablevis vthevquizvscore.
1.2.4 A.
b. Theveffectvofvtakingvnotesvonvpapervisv0.71vandvtheveffectvofvtaki
1.2.5 C.
ngvnotesvonvthevcomputervisv−0.71.
1.2.6 D.
c. SSModelv =v 40v ×v (0.712)v =v 20.164.
1.2.7 B.
d. R2v =v20.164/120.92v=v0.16675.vWevcanvinterpretvitvbyvsayingvtha
1.2.8 Usingvtheveffectsvmodel,vbecausev4.48v+v0.65v=v5.13v(thevmean tv16.675%vofvthevvariationvofvquizvscorevisvexplainedvbyvthevnote-
vof vthevscent vgroup) vand v4.48 v− v0.65v= v3.83 v(thevmean vof vthe vnon-
takingvmethod.
scentvgroup),vthevmodelsvarevequivalent.
e. 120.92v–v20.164v=v100.756.
1.2.9 f. √100.756/38 v =v 1.628.
a. SSModel. 0.71vifvusingvpapervnotes
g. predictedvquizvscorev=v6.21v+v .
b. SSError. {−0.71vifvusingvcomputervnotesv
1.2.17
1.2.10
a. Becausevthevsamplevsizesvofveachvgroupvarevthevsame,vthevsample
a. R2v =v SSModel/SSTotalv =v 0.4651. vsizevofveachvgroup visvjustvhalfvof vthevtotalvsamplevsize.
b. R2v =v 1v −v SSError/SSTotalv =v 0.7111. ∑ (xv −vx)2 ∑ (yv −vy)2
allvobsv v i ̅ v v ̅ _1
b. + allvobsv
vi
v
1.2.11 nv−v1 nv−v1
( _
2 _
2 )2
a. 8. ∑allvobs(xiv −vx̅) +v∑allvobs(yiv −v y̅)2v v _1
2v
b. v68v–
v=v8 v= v–2, v10v–
v2. n
=
( _v−v1
)2
2
c. 74.
∑allvobs(xiv −vx)̅ +v∑allvobs(yiv −v y̅)
2v 2
d. 40. =v( )
nv−v2
e. 34.
Takingv thev squarev rootv wev getv ∑allvobs(xiv −vx̅) +v∑allvobs(yiv −v y̅
2v
f. 0.5405.
√ )2
nv−v2
⎛v n n
2⎞
⎜ ⎟
1.2.12 2
a. Thev explanatoryv variablev isv thev typev ofv testingv environment;v it Usevsumvfromv1vtovn:v v∑v(vxivv −vx)̅ v + v∑v(yv viv−v y
1
_v i=1 i=1
vn vn
⎝ ̅)2v v −v1
2
2v −v1v v ⎠
isvcategorical.
b. Thevresponsevvariablevisvthevtestvscore;vitvisvquantitative. ⎛v n n 2⎞ n
2
n
2
2
c. Thevtwovlevelsvarevquietvenvironmentvandvdistractingvenvironment. v∑(x iv−v x̅)v +v ∑(y iv−v y )̅v ∑(xiv −v x)̅ +v ∑(yiv −v y̅)
1.2.13
1v
⎝
⎜
=v_2v i=1 vnv
i=1
−v1
⎟⎠
v=
vi=1 i=1
nv−v2
2
√
n n
a. SSTotalv wouldv probablyv bev largerv withv thesev 10v subjectsv because ∑(xiv −v x)̅ 2 + ∑(yiv−v y)̅ 2
withvthevwidevvarietyvofvagesvtherevwouldvprobablyvbevmorevvariabilit vmalevor vfemalevandvthe vresponse visvthevperceived vriskvlevel.
nv−v
yvinvthevtestvscores. b. Thev effectv ofv namingv thev hurricanev Christina v isv 5.01v −v 5.29v =
2
b. SSModelvwouldvprobablyvbevthevsamevbecausevitvwouldvstillvrepre −0.28v andv thev effectv ofv namingv thev hurricanev Christopherv isv 5.57v −
-vsentvthevdifferencevbetweenvtestingvenvironments. 5.29v=v0.28.vThevSSModelv isv142(0.282)v=v11.1328.
c. SSErrorvwouldvprobablyvbevlargervbecausevtherevwouldvprobablyv
bevmorevvariabilityvinvthevtestvscoresvwithinveachvgroupvduevtovthevv
ariabilityvinvages.
1.2.14 Thevvariancevofvthevscoresvinvthevdistractingvenvironmentvisv2.
5vandvthevvariancevofvthevscoresvinvthevdistractingve n v i r o_
n m e n t visv6.v
Thevsquarevrootvofvthevaveragevofvthesevtwovvariancesvisv√ 4.25_ v =v2.0
6.vThevSSErrorvisv34,vsovthevstandardverrorvofvthevresidualsvisv√34/8v
=v2.06.
1.2.15
a. Thevexplanatoryvvariablyvisvwhethervthevnamevofvthevhurricanevis
, i=1 i=1
Takingvthevsquarevroot,vwevget v
.
v
Sectionv1.3
1.3.1 D.
1.3.2 A.
1.3.3 D.
1.3.4 A.
1.3.5 A.
1.3.6 Thev validityv conditionsv arev notv metv becausev the
v malev sample vsizevis vsmallvand vthevdistributionvofvthe vnu
mbervofvflip-
flopsvownedvbyvthevmalesvisvquitevskewedvtovthevright.
1.3.7
a. √(24.v382v +v 36.v992)/2v =v 31.33.
b. tv=vv v 92.16v−v60.34v v v=v4.06.
31.33v√ 1/32v +v 1/32
