551
12.11Summary
11
BivariateData:Eachdatapointhastwovalues.Theformis(x,y).
LineofBestFitorLeastSquaresLine(LSL):
^
y
=a+bx
x=independentvariable;y=dependentvariable
Residual:Actualyvalue predictedyvalue=y
^
y
CorrelationCoefficientr:
1. Usedtodeterminewhetheralineofbestfitisgoodforprediction.
2. Between-1and1inclusive.Thecloserristo1or-1,theclosertheoriginalpointsaretoastraightline.
3. Ifrisnegative,theslopeisnegative.Ifrispositive,theslopeispositive.
4. Ifr=0,thenthelineishorizontal.
SumofSquaredErrors(SSE):ThesmallertheSSE,thebettertheoriginalsetofpointsfitsthelineofbest
fit.
Outlier:Apointthatdoesnotseemtofittherestofthedata.
11
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552
CHAPTER12. LINEARREGRESSIONANDCORRELATION
12.12Practice:LinearRegression
12
12.12.1StudentLearningOutcomes
 Thestudentwillevaluatebivariatedataanddetermineifalineisanappropriatefittothedata.
12.12.2Given
BelowarerealdataforthefirsttwodecadesofAIDSreporting. (Source:CentersforDiseaseControland
Prevention,NationalCenterforHIV,STD,andTBPrevention)
AdultsandAdolescentsonly,UnitedStates
Year
#AIDScasesdiagnosed
#AIDSdeaths
Pre-1981
91
29
1981
319
121
1982
1,170
453
1983
3,076
1,482
1984
6,240
3,466
1985
11,776
6,878
1986
19,032
11,987
1987
28,564
16,162
1988
35,447
20,868
1989
42,674
27,591
1990
48,634
31,335
1991
59,660
36,560
1992
78,530
41,055
1993
78,834
44,730
1994
71,874
49,095
1995
68,505
49,456
1996
59,347
38,510
1997
47,149
20,736
1998
38,393
19,005
1999
25,174
18,454
2000
25,522
17,347
2001
25,643
17,402
2002
26,464
16,371
Total
802,118
489,093
Table12.4
12
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553
NOTE
: We e willusethecolumns“year”and“#AIDScasesdiagnosed”forallquestionsunless
otherwisestated.
12.12.3Graphing
Graph“year”vs. “#AIDScasesdiagnosed.” ” Plotthepointsonthegraphlocatedbelowinthesection
titled"Plot".Donotincludepre-1981.Labelbothaxeswithwords.Scalebothaxes.
12.12.4Data
Exercise12.12.1
Enteryourdataintoyourcalculatororcomputer.Thepre-1981datashouldnotbeincluded.Why
isthatso?
12.12.5LinearEquation
Writethelinearequationbelow,roundingto4decimalplaces:
NOTE
: Foranypredictionquestions,theanswersarecalculatedusingtheleastsquares(bestfit)
lineequationcitedinthesolution.
Exercise12.12.2
(Solutiononp.579.)
Calculatethefollowing:
a. a=
b. b=
c. corr.=
d. n=(#ofpairs)
Exercise12.12.3
(Solutiononp.579.)
equation:
^
y=
12.12.6Solve
Exercise12.12.4
(Solutiononp.579.)
Solve.
a. Whenx=1985,
^
y
=
b. Whenx=1990,
^
y
=
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554
CHAPTER12. LINEARREGRESSIONANDCORRELATION
12.12.7Plot
Plotthe2abovepointsonthegraphbelow.Then,connectthe2pointstoformtheregressionline.
Obtainthegraphonyourcalculatororcomputer.
12.12.8DiscussionQuestions
Lookatthegraphabove.
Exercise12.12.5
Doesthelineseemtofitthedata?Whyorwhynot?
Exercise12.12.6
Doyouthinkalinearfitisbest?Whyorwhynot?
Exercise12.12.7
Handdrawasmoothcurveonthegraphabovethatshowstheflowofthedata.
Exercise12.12.8
Whatdoesthecorrelationimplyabouttherelationshipbetweentime(years)andthenumberof
diagnosedAIDScasesreportedintheU.S.?
Exercise12.12.9
Whyis“year”theindependentvariableand“#AIDScasesdiagnosed.”thedependentvariable
(insteadofthereverse)?
Exercise12.12.10
(Solutiononp.579.)
Solve.
a. Whenx=1970,
^
y
=:
b. Whydoesn’tthisanswermakesense?
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555
12.13Homework
13
Exercise12.13.1
(Solutiononp.579.)
Foreachsituationbelow,statetheindependentvariableandthedependentvariable.
a. Astudyisdonetodetermineifelderlydriversareinvolvedinmoremotorvehiclefatalities
thanallotherdrivers.Thenumberoffatalitiesper100,000driversiscomparedtotheageof
drivers.
b. Astudyisdonetodetermineiftheweeklygrocerybillchangesbasedonthenumberoffamily
members.
c. Insurancecompaniesbaselifeinsurancepremiumspartiallyontheageoftheapplicant.
d. Utilitybillsvaryaccordingtopowerconsumption.
e. Astudyisdonetodetermineifahighereducationreducesthecrimerateinapopulation.
NOTE
: Foranypredictionquestions,theanswersarecalculatedusingtheleastsquares(bestfit)
lineequationcitedinthesolution.
Exercise12.13.2
Recently,
the
annual
number
of
driver
deaths
per
100,000
for
the
selected
age
groups
was
as
follows
(Source:
http://
http://www.census.gov/compendia/statab/cats/transportation/motor_vehicle_accidents_and_fatalities.html
14
):
Age
NumberofDriverDeathsper100,000
16-19
38
20-24
36
25-34
24
35-54
20
55-74
18
75+
28
Table12.5
a. Foreachagegroup,pickthemidpointoftheintervalforthexvalue. (Forthe75+group,use
80.)
b. Using“ages”astheindependentvariableand“Numberofdriverdeathsper100,000”asthe
dependentvariable,makeascatterplotofthedata.
c. Calculatetheleastsquares(best–fit)line.Puttheequationintheformof:
^
y
=a+bx
d. Findthecorrelationcoefficient.Isitsignificant?
e. Picktwoagesandfindtheestimatedfatalityrates.
f. Usethetwopointsin(e)toplottheleastsquareslineonyourgraphfrom(b).
g. Basedontheabovedata,istherealinearrelationshipbetweenageofadriveranddriverfatality
rate?
h. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
13
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14
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556
CHAPTER12. LINEARREGRESSIONANDCORRELATION
Exercise12.13.3
(Solutiononp.579.)
Theaveragenumberofpeopleinafamilythatreceivedwelfareforvariousyearsisgivenbelow.
(Source:HouseWaysandMeansCommittee,HealthandHumanServicesDepartment)
Year
Welfarefamilysize
1969
4.0
1973
3.6
1975
3.2
1979
3.0
1983
3.0
1988
3.0
1991
2.9
Table12.6
a. Using“year”astheindependentvariableand“welfarefamilysize”asthedependentvariable,
makeascatterplotofthedata.
b. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
c. Findthecorrelationcoefficient.Isitsignificant?
d. Picktwoyearsbetween1969and1991andfindtheestimatedwelfarefamilysizes.
e. Usethetwopointsin(d)toplottheleastsquareslineonyourgraphfrom(b).
f. Basedontheabovedata,istherealinearrelationshipbetweentheyearandtheaveragenumber
ofpeopleinawelfarefamily?
g. Usingtheleastsquaresline,estimatethewelfarefamilysizesfor1960and1995.Doestheleast
squareslinegiveanaccurateestimateforthoseyears?Explainwhyorwhynot.
h. Arethereanyoutliersintheabovedata?
i. Whatistheestimatedaveragewelfarefamilysizefor1986?Doestheleastsquareslinegivean
accurateestimateforthatyear?Explainwhyorwhynot.
j. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.4
UsetheAIDSdatafromthepracticeforthissection(Section12.12.2:Given),butthistimeusethe
columns“year#”and“#newAIDSdeathsinU.S.”Answerallofthequestionsfromthepractice
again,usingthenewcolumns.
Exercise12.13.5
(Solutiononp.580.)
Theheight(sidewalktoroof)ofnotabletallbuildingsinAmericaiscomparedtothenumberof
storiesofthebuilding(beginningatstreetlevel).(Source:MicrosoftBookshelf)
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557
Height(infeet)
Stories
1050
57
428
28
362
26
529
40
790
60
401
22
380
38
1454
110
1127
100
700
46
Table12.7
a. Using“stories”astheindependentvariableand“height”asthedependentvariable,makea
scatterplotofthedata.
b. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?
c. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
d. Findthecorrelationcoefficient.Isitsignificant?
e. Findtheestimatedheightsfor32storiesandfor94stories.
f. Usethetwopointsin(e)toplottheleastsquareslineonyourgraphfrom(b).
g. Basedontheabovedata,istherealinearrelationshipbetweenthenumberofstoriesintall
buildingsandtheheightofthebuildings?
h. Arethereanyoutliersintheabovedata?Ifso,whichpoint(s)?
i. Whatistheestimatedheightofabuildingwith6stories? Doestheleastsquareslinegivean
accurateestimateofheight?Explainwhyorwhynot.
j. Basedontheleastsquaresline,addinganextrastoryispredictedtoaddabouthowmanyfeet
toabuilding?
k. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.6
BelowisthelifeexpectancyforanindividualbornintheUnitedStatesincertainyears.(Source:
NationalCenterforHealthStatistics)
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558
CHAPTER12. LINEARREGRESSIONANDCORRELATION
YearofBirth
LifeExpectancy
1930
59.7
1940
62.9
1950
70.2
1965
69.7
1973
71.4
1982
74.5
1987
75
1992
75.7
2010
78.7
Table12.8
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Drawascatterplotoftheorderedpairs.
c. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
d. Findthecorrelationcoefficient.Isitsignificant?
e. Findtheestimatedlifeexpectancyforanindividualbornin1950andforonebornin1982.
f. Whyaren’ttheanswerstopart(e)thevaluesontheabovechartthatcorrespondtothoseyears?
g. Usethetwopointsin(e)toplottheleastsquareslineonyourgraphfrom(b).
h. Basedontheabovedata,istherealinearrelationshipbetweentheyearofbirthandlifeex-
pectancy?
i. Arethereanyoutliersintheabovedata?
j. Usingtheleastsquaresline,findtheestimatedlifeexpectancyforanindividualbornin1850.
Doestheleastsquareslinegiveanaccurateestimateforthatyear?Explainwhyorwhynot.
k. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.7
(Solutiononp.580.)
Thepercentoffemalewageandsalaryworkerswhoarepaidhourlyratesisgivenbelowforthe
years1979-1992.(Source:BureauofLaborStatistics,U.S.Dept.ofLabor)
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559
Year
Percentofworkerspaidhourlyrates
1979
61.2
1980
60.7
1981
61.3
1982
61.3
1983
61.8
1984
61.7
1985
61.8
1986
62.0
1987
62.7
1990
62.8
1992
62.9
Table12.9
a. Using“year”astheindependentvariableand“percent”asthedependentvariable, makea
scatterplotofthedata.
b. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
c. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
d. Findthecorrelationcoefficient.Isitsignificant?
e. Findtheestimatedpercentsfor1991and1988.
f. Usethetwopointsin(e)toplottheleastsquareslineonyourgraphfrom(b).
g. Basedontheabove data, istherealinearrelationshipbetweentheyearand thepercent of
femalewageandsalaryearnerswhoarepaidhourlyrates?
h. Arethereanyoutliersintheabovedata?
i. Whatistheestimatedpercentfortheyear2050? Doestheleastsquareslinegiveanaccurate
estimateforthatyear?Explainwhyorwhynot?
j. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Exercise12.13.8
ThemaximumdiscountvalueoftheEntertainment®cardforthe“FineDining”section,Edition
10,forvariouspagesisgivenbelow.
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560
CHAPTER12. LINEARREGRESSIONANDCORRELATION
Pagenumber
Maximumvalue($)
4
16
14
19
25
15
32
17
43
19
57
15
72
16
85
15
90
17
Table12.10
a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependent
variable.
b. Drawascatterplotoftheorderedpairs.
c. Calculatetheleastsquaresline.Puttheequationintheformof:
^
y
=a+bx
d. Findthecorrelationcoefficient.Isitsignificant?
e. Findtheestimatedmaximumvaluesfortherestaurantsonpage10andonpage70.
f. Usethetwopointsin(e)toplottheleastsquareslineonyourgraphfrom(b).
g. Doesitappearthattherestaurantsgivingthemaximumvalueareplacedinthebeginningof
the“FineDining”section?Howdidyouarriveatyouranswer?
h. Supposethattherewere200pagesofrestaurants. Whatdoyouestimatetobethemaximum
valueforarestaurantlistedonpage200?
i. Istheleastsquareslinevalidforpage200?Whyorwhynot?
j. Whatistheslopeoftheleastsquares(best-fit)line?Interprettheslope.
Thenexttwoquestionsrefertothefollowingdata: Thecostofaleadingliquid laundrydetergentin
differentsizesisgivenbelow.
Size(ounces)
Cost($)
Costperounce
16
3.99
32
4.99
64
5.99
200
10.99
Table12.11
Exercise12.13.9
(Solutiononp.580.)
a. Using“size”astheindependentvariableand“cost”asthedependentvariable,makeascatter
plot.
b. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyorwhy
not?
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