401
9.15Practice3:SingleProportion
15
9.15.1StudentLearningOutcomes
 Thestudentwillconductahypothesistestofasinglepopulationproportion.
9.15.2Given
The National l Institute e of Mental l Health published d an article e stating g that in any y one-year pe-
riod, approximately y 9.5 5 percent of American adults suffer r from m depression or a depressive illness.
(http://www.nimh.nih.gov/publicat/depression.cfm)Supposethatinasurveyof100peopleinacertain
town,sevenofthemsufferedfromdepressionoradepressiveillness. Conductahypothesistesttodeter-
mineifthetrueproportionofpeopleinthattownsufferingfromdepressionoradepressiveillnessislower
thanthepercentinthegeneraladultAmericanpopulation.
9.15.3HypothesisTesting:SingleProportion
Exercise9.15.1
(Solutiononp.424.)
Isthisatestofmeansorproportions?
Exercise9.15.2
(Solutiononp.424.)
Statethenullandalternativehypotheses.
a. H
o
:
b. H
a
:
Exercise9.15.3
(Solutiononp.424.)
Isthisaright-tailed,left-tailed,ortwo-tailedtest?Howdoyouknow?
Exercise9.15.4
(Solutiononp.424.)
WhatsymbolrepresentstheRandomVariableforthistest?
Exercise9.15.5
(Solutiononp.424.)
Inwords,definetheRandomVariableforthistest.
Exercise9.15.6
(Solutiononp.424.)
Calculatethefollowing:
a: x=
b: n=
c: p’=
Exercise9.15.7
(Solutiononp.424.)
Calculates
p
.Makesuretoshowhowyousetuptheformula.
Exercise9.15.8
(Solutiononp.424.)
Statethedistributiontouseforthehypothesistest.
Exercise9.15.9
Sketchagraphofthesituation. Labelthehorizontalaxis. Markthehypothesizedmeanandthe
sampleproportion,p-hat.Shadetheareacorrespondingtothep-value.
15
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402
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
Exercise9.15.10
(Solutiononp.424.)
Findthep-value
Exercise9.15.11
(Solutiononp.424.)
Atapre-conceiveda=0.05,whatisyour:
a. Decision:
b. Reasonforthedecision:
c. Conclusion(writeoutinacompletesentence):
9.15.4DiscusionQuestion
Exercise9.15.12
Doesitappearthattheproportionofpeopleinthattownwithdepressionoradepressiveillness
islowerthangeneraladultAmericanpopulation?Whyorwhynot?
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403
9.16Homework
16
Exercise9.16.1
(Solutiononp.424.)
Someofthestatementsbelowrefertothenullhypothesis,sometothealternatehypothesis.
Statethenullhypothesis,H
o
,andthealternativehypothesis,H
a
,intermsoftheappropriatepa-
rameter(morp).
a. ThemeannumberofyearsAmericansworkbeforeretiringis34.
b. Atmost60%ofAmericansvoteinpresidentialelections.
c. ThemeanstartingsalaryforSanJoseStateUniversitygraduatesisatleast$100,000peryear.
d. 29%ofhighschoolseniorsgetdrunkeachmonth.
e. Fewerthan5%ofadultsridethebustoworkinLosAngeles.
f. Themeannumberofcarsapersonownsinherlifetimeisnotmorethan10.
g. AbouthalfofAmericansprefertoliveawayfromcities,giventhechoice.
h. Europeanshaveameanpaidvacationeachyearofsixweeks.
i. Thechanceofdevelopingbreastcancerisunder11%forwomen.
j. Privateuniversitiesmeantuitioncostismorethan$20,000peryear.
Exercise9.16.2
(Solutiononp.425.)
For(a)-(j)above,statetheTypeIandTypeIIerrorsincompletesentences.
Exercise9.16.3
For(a)-(j)above,incompletesentences:
a. StateaconsequenceofcommittingaTypeIerror.
b. StateaconsequenceofcommittingaTypeIIerror.
D
IRECTIONS
: Foreachofthewordproblems,useasolutionsheettodothehypothesistest. The
solutionsheetisfoundin14. Appendix(onlinebookversion: : thelinkis"SolutionSheets";PDF
bookversion: lookunder14.5SolutionSheets). . Pleasefeelfreetomakecopiesofthesolution
sheets.Fortheonlineversionofthebook,itissuggestedthatyoucopythe.docorthe.pdffiles.
NOTE
:Ifyouareusingastudent’s-tdistributionforahomeworkproblembelow,youmayassume
that theunderlyingpopulationisnormallydistributed. . (Ingeneral,youmust t firstprovethat
assumption,though.)
Exercise9.16.4
Aparticularbrandoftiresclaimsthatitsdeluxetireaveragesatleast50,000milesbeforeitneeds
tobereplaced.Frompaststudiesofthistire,thestandarddeviationisknowntobe8000.Asurvey
ofownersofthattiredesignisconducted. Fromthe28tiressurveyed,themeanlifespanwas
46,500mileswithastandarddeviationof9800miles. Dothedatasupporttheclaimatthe5%
level?
Exercise9.16.5
(Solutiononp.425.)
Fromgenerationtogeneration,themeanagewhensmokersfirststarttosmokevaries.However,
thestandarddeviationofthatageremainsconstantofaround2.1years. Asurveyof40smokers
ofthisgenerationwasdonetoseeifthemeanstartingageisatleast19. Thesamplemeanwas
18.1withasamplestandarddeviationof1.3.Dothedatasupporttheclaimatthe5%level?
16
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404
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
Exercise9.16.6
Thecostofadailynewspapervariesfromcityto city. . However, , thevariationamongprices
remainssteadywithastandarddeviationof20¢.Astudywasdonetotesttheclaimthatthemean
costofadailynewspaperis$1.00.Twelvecostsyieldameancostof95¢withastandarddeviation
of18¢.Dothedatasupporttheclaimatthe1%level?
Exercise9.16.7
(Solutiononp.425.)
AnarticleintheSanJoseMercuryNewsstatedthatstudentsintheCaliforniastateuniversity
systemtake4.5years,onaverage,tofinishtheirundergraduatedegrees.Supposeyoubelievethat
themeantimeislonger.Youconductasurveyof49studentsandobtainasamplemeanof5.1with
asamplestandarddeviationof1.2.Dothedatasupportyourclaimatthe1%level?
Exercise9.16.8
Themeannumberofsickdaysanemployeetakesperyearisbelievedtobeabout10. Members
ofapersonneldepartmentdonotbelievethisfigure. Theyrandomlysurvey8employees. . The
numberofsickdaystheytookforthepastyearareasfollows:12;4;15;3;11;8;6;8.Letx=the
numberofsickdaystheytookforthepastyear.Shouldthepersonnelteambelievethatthemean
numberisabout10?
Exercise9.16.9
(Solutiononp.425.)
In1955,LifeMagazinereportedthatthe25year-oldmotherofthreeworked,onaverage,an80
hourweek. Recently,manygroupshavebeenstudyingwhetherornotthewomen’smovement
has,infact,resultedinanincreaseintheaverageworkweekforwomen(combiningemployment
andat-homework).Supposeastudywasdonetodetermineifthemeanworkweekhasincreased.
81womenweresurveyedwiththefollowingresults. Thesamplemeanwas83;thesamplestan-
darddeviationwas10. Doesitappearthatthemeanworkweekhasincreasedforwomenatthe
5%level?
Exercise9.16.10
Yourstatisticsinstructorclaimsthat60percentofthestudentswhotakeherElementaryStatistics
classgothroughlifefeelingmoreenriched.Forsomereasonthatshecan’tquitefigureout,most
peopledon’tbelieveher. Youdecidetocheckthisoutonyourown. . Yourandomlysurvey64of
herpastElementaryStatisticsstudentsandfindthat34feelmoreenrichedasaresultofherclass.
Now,whatdoyouthink?
Exercise9.16.11
(Solutiononp.425.)
ANissanMotorCorporationadvertisementread,“Theaverageman’sI.Q.is107. Theaverage
browntrout’sI.Q.is4.Sowhycan’tmancatchbrowntrout?”Supposeyoubelievethatthebrown
trout’smeanI.Q.isgreaterthan4. Youcatch12browntrout. Afishpsychologistdeterminesthe
I.Q.sasfollows:5;4;7;3;6;4;5;3;6;3;8;5.Conductahypothesistestofyourbelief.
Exercise9.16.12
Refertothepreviousproblem. Conductahypothesistesttoseeifyourdecisionandconclusion
wouldchangeifyourbeliefwerethatthebrowntrout’smeanI.Q.isnot4.
Exercise9.16.13
(Solutiononp.425.)
AccordingtoanarticleinNewsweek,thenaturalratioofgirlstoboysis100:105. InChina,the
birthratiois100:114(46.7%girls). Supposeyoudon’tbelievethereportedfiguresofthepercent
ofgirlsborninChina.Youconductastudy.Inthisstudy,youcountthenumberofgirlsandboys
bornin150randomlychosenrecentbirths. Thereare60girlsand90boysbornofthe150. Based
onyourstudy,doyoubelievethatthepercentofgirlsborninChinais46.7?
Exercise9.16.14
ApolldoneforNewsweekfoundthat13%ofAmericanshaveseenorsensedthepresenceofan
angel. Acontingentdoubtsthatthepercentisreallythathigh. Itconductsitsownsurvey. Out
of76Americanssurveyed,only2hadseenorsensedthepresenceofanangel.Asaresultofthe
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405
contingent’ssurvey,wouldyouagreewiththeNewsweekpoll?Incompletesentences,alsogive
threereasonswhythetwopollsmightgivedifferentresults.
Exercise9.16.15
(Solutiononp.425.)
Themeanworkweekforengineersinastart-upcompanyisbelievedtobeabout60hours. A
newlyhiredengineerhopesthatit’sshorter. Sheasks10engineeringfriendsinstart-upsforthe
lengthsoftheirmeanworkweeks.Basedontheresultsthatfollow,shouldshecountonthemean
workweektobeshorterthan60hours?
Data(lengthofmeanworkweek):70;45;55;60;65;55;55;60;50;55.
Exercise9.16.16
Usethe“Laptime”dataforLap4(seeTableofContents)totesttheclaimthatTerrifinishesLap
4,onaverage,inlessthan129seconds.Usealltwentyracesgiven.
Exercise9.16.17
Usethe“InitialPublicOffering”data(seeTableofContents)totesttheclaimthatthemeanoffer
pricewas$18pershare.Donotuseallthedata.Useyourrandomnumbergeneratortorandomly
survey15prices.
NOTE
:Thefollowingquestionswerewrittenbypaststudents.Theyareexcellentproblems!
Exercise9.16.18
18."AsianFamilyReunion"byChauNguyen
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406
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
Exercise9.16.19
(Solutiononp.425.)
"TheProblemwithAngels"byCyndyDowling
Exercise9.16.20
"BlowingBubbles"bySondraPrull
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407
Exercise9.16.21
(Solutiononp.426.)
21."DalmatianDarnation"byKathySparling
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408
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
Exercise9.16.22
"MacaroniandCheese,please!!"byNeddaMisherghiandRachelleHall
AsapoorstarvingstudentIdon’thavemuchmoneytospendforeventhebarenecessities. So
myfavoriteandmainstaplefoodismacaroniandcheese. It’shighintasteandlowincostand
nutritionalvalue.
Oneday,asIsatdowntodeterminethemeaningoflife,Igotaseriouscravingforthis,oh, so
important,foodofmylife. SoIwentdownthestreettoGreatwaytogetaboxofmacaroniand
cheese,butitwasSOexpensive! $2.02!!! Canyoubelieveit? Itmademestopandthink. . The
worldischangingfast.Ihadthoughtthatthemeancostofabox(thenormalsize,notsomesuper-
gigantic-family-value-pack)wasatmost$1,butnowIwasn’tsosure.However,Iwasdetermined
tofindout. Iwentto53oftheclosestgrocerystoresandsurveyedthepricesofmacaroniand
cheese.HerearethedataIwroteinmynotebook:
PriceperboxofMacandCheese:
 5stores@$2.02
 15stores@$0.25
 3stores@$1.29
 6stores@$0.35
 4stores@$2.27
 7stores@$1.50
 5stores@$1.89
 8stores@0.75.
IcouldseethatthecostsvariedbutIhadtositdowntofigureoutwhetherornotIwasright. If
itdoesturnoutthatthismouth-wateringdishisatmost$1,thenI’llthrowabigcheesypartyin
ournextstatisticslab,withenoughmacaroniandcheeseforjustme.(Afterall,asapoorstarving
studentIcan’tbeexpectedtofeedourclassofanimals!)
Exercise9.16.23
(Solutiononp.426.)
"WilliamShakespeare:TheTragedyofHamlet,PrinceofDenmark"byJacquelineGhodsi
THECHARACTERS(inorderofappearance):
 HAMLET,PrinceofDenmarkandstudentofStatistics
 POLONIUS,Hamlet’stutor
 HOROTIO,friendtoHamletandfellowstudent
Scene:Thegreatlibraryofthecastle,inwhichHamletdoeshislessons
ActI
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409
(Thedayisfair,butthefaceofHamletisclouded.Hepacesthelargeroom.Histutor,Polonius,is
reprimandingHamletregardingthelatter’srecentexperience.Horatioisseatedatthelargetable
atrightstage.)
POLONIUS:MyLord,howcans’tthouadmitthatthouhastseenaghost! Itisbutafigmentof
yourimagination!
HAMLET:Ibegtodiffer;Iknowofacertaintythatfive-and-seventyinonehundredofus,con-
demnedtothewhipsandscornsoftimeasweare,havegazeduponaspiritofhealth,orgoblin
damn’d,betheirintentswickedorcharitable.
POLONIUSIfthoudoestinsistuponthywretchedvisionthenletmeinvestyourtime;betrue
tothyworkandspeaktomethroughthereasonofthenullandalternatehypotheses. (Heturns
toHoratio.)DidnotHamlethimselfsay,“Whatpieceofworkisman,hownobleinreason,how
infiniteinfaculties?Thenletnotthisfoolishnesspersist.Go,Horatio,makeasurveyofthree-and-
sixtyanddiscoverwhatthetrueproportionbe.Formypart,Iwillneversuccumbtothisfantasy,
butdeemmantobedevoidofallreasonshouldthyproposalofatleastfive-and-seventyinone
hundredholdtrue.
HORATIO(toHamlet):Whatshouldwedo,myLord?
HAMLET:Gotothypurpose,Horatio.
HORATIO:Towhatend,myLord?
HAMLET:Thatyoumustteachme.Butletmeconjureyoubytherightsofourfellowship,bythe
consonanceofouryouth,buttheobligationofourever-preservedlove,beevenanddirectwith
me,whetherIamrightorno.
(Horatioexits,followedbyPolonius,leavingHamlettoponderalone.)
ActII
(Thenextday,Hamletawaitsanxiouslythepresenceofhisfriend,Horatio. Poloniusentersand
placessomebooksuponthetablejustamomentbeforeHoratioenters.)
POLONIUS:So,Horatio,whatisitthoudidstrevealthroughthydeliberations?
HORATIO:Inarandomsurvey,forwhichpurposethouthyselfsentmeforth,Ididdiscoverthat
one-and-fortybelieveferventlythatthespiritsofthedeadwalkwithus.BeforemyGod,Imight
notthisbelieve,withoutthesensibleandtrueavouchofmineowneyes.
POLONIUS:Givethineownthoughtsnotongue,Horatio. (PoloniusturnstoHamlet.)Butlook
to’tIchargeyou,myLord.ComeHoratio,letusgotogether,forthisisnotourtest.(Horatioand
Poloniusleavetogether.)
HAMLET:Toreject,ornotreject,thatisthequestion:whether‘tisnoblerinthemindtosufferthe
slingsandarrowsofoutrageousstatistics,ortotakearmsagainstaseaofdata,and,byopposing,
endthem.(Hamletresignedlyattendstohistask.)
(Curtainfalls)
Exercise9.16.24
"Untitled"byStephenChen
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410
CHAPTER9. HYPOTHESISTESTING:SINGLEMEANANDSINGLE
PROPORTION
I’veoftenwonderedhowsoftwareisreleasedandsoldtothepublic.Ironically,Iworkforacom-
panythatsellsproductswithknownproblems.Unfortunately,mostoftheproblemsaredifficult
tocreate,whichmakesthemdifficulttofix.IusuallyusethetestprogramX,whichteststheprod-
uct,totrytocreateaspecificproblem. Whenthetestprogramisruntomakeanerroroccur,the
likelihoodofgeneratinganerroris1%.
So,armedwiththisknowledge,IwroteanewtestprogramYthatwillgeneratethesameerrorthat
testprogramXcreates,butmoreoften. Tofindoutifmytestprogramisbetterthantheoriginal,
sothatIcanconvincethemanagementthatI’mright,Iranmytestprogramtofindouthowoften
Icangeneratethesameerror. WhenIranmytestprogram50times,Igeneratedtheerrortwice.
Whilethismaynotseemmuchbetter,IthinkthatIcanconvincethemanagementtousemytest
programinsteadoftheoriginaltestprogram.AmIright?
Exercise9.16.25
(Solutiononp.426.)
JapaneseGirls’Names
byKumiFuruichi
It usedto beverytypicalforJapanesegirls’namestoendwith“ko.” ” (Thetrendmighthave
startedaroundmygrandmothers’generationanditspeakmighthavebeenaroundmymother’s
generation.)“Ko”means“child”inChinesecharacter.Parentswouldnametheirdaughterswith
“ko”attachingtootherChinesecharacterswhichhavemeaningsthattheywanttheirdaughters
tobecome,suchasSachiko–ahappychild,Yoshiko–agoodchild,Yasuko–ahealthychild,and
soon.
However,InoticedrecentlythatonlytwooutofnineofmyJapanesegirlfriendsatthisschoolhave
nameswhichendwith“ko.”Moreandmore,parentsseemtohavebecomecreative,modernized,
and,sometimes,westernizedinnamingtheirchildren.
Ihaveafeelingthat,while70percentormoreofmymother’sgenerationwouldhavenameswith
“ko”attheend,the proportionhasdroppedamongmypeers. . IwrotedownallmyJapanese
friends’,ex-classmates’,co-workers,andacquaintances’namesthatIcouldremember.Beloware
thenames.(Somearerepeats.)Testtoseeiftheproportionhasdroppedforthisgeneration.
Ai,Akemi,Akiko,Ayumi,Chiaki,Chie,Eiko,Eri,Eriko,Fumiko,Harumi,Hitomi,Hiroko,Hi-
roko,Hidemi,Hisako,Hinako,Izumi,Izumi,Junko,Junko,Kana,Kanako,Kanayo,Kayo,Kayoko,
Kazumi,Keiko,Keiko,Kei,Kumi, Kumiko, Kyoko,Kyoko,Madoka,Maho,Mai, Maiko, Maki,
Miki, Miki, Mikiko, , Mina, Minako, Miyako, Momoko, Nana, Naoko, Naoko, Naoko, Noriko,
Rieko, Rika, Rika, Rumiko,Rei, Reiko, Reiko, Sachiko,Sachiko,Sachiyo,Saki,Sayaka,Sayoko,
Sayuri,Seiko,Shiho,Shizuka,Sumiko,Takako,Takako,Tomoe,Tomoe,Tomoko,Touko,Yasuko,
Yasuko,Yasuyo,Yoko,Yoko,Yoko,Yoshiko,Yoshiko,Yoshiko,Yuka,Yuki,Yuki,Yukiko,Yuko,
Yuko.
Exercise9.16.26
Phillip’sWishbySuzanneOsorio
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