ETHS COMAP HiMCM National Outstanding Paper 11.docx
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ETHS COMAP HiMCM National Outstanding Paper 11.docx
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ETHSCOMAPHiMCMNationalOutstandingPaper11
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2010
13thAnnualHighSchoolMathematicalContestinModeling(HiMCM)SummarySheet
(PleaseattachacopyofthispagetoeachcopyofyourSolutionPaper.)
TeamControlNumber:
2663
ProblemChosen:
B
Pleasetypeasummaryofyourresultsonthispage.Pleaseremembernottoincludethe
nameofyourschool,advisor,orteammembersonthispage.
Whiletheproblemdidnotgiveanyspecificgoalfordecreasingtheincidenceofcrimeinthecity,wedeterminedthataneffectiveandpracticalgoalforthecityistoreduceviolentcrimeby10%bytheyear2020. Inordertoreducetheamountofcrimeinthecity,wefirstcreatedamodelthatcalculatedincidenceofviolenceasafunctionofdifferentdatathatweweregiven. Wedeterminedthatsomeoftheprovideddatahadlittletonoimpactontheamountofviolentcrime. Tryingtoincorporatethisunhelpfulinformationintoourmodelwouldmakeitlessaccurate,sowedevelopedawaytodecideifasetofdatawasworthconsidering. Whenweappliedthismethodtoallofthesetsofdata,wewereleftwithfivethathadanoticeablecorrelationwithviolence. ThesefivevalueswereCityPopulation,HighSchoolGraduationRate,HighSchoolDropoutRate,PrisonPopulation,andPercentofParoleViolations. Withthesefivevariablesdetermined,wecheckedtomakesurethateachrelationshipwithviolencewasroughlylinear,thenranamultivariatelinearregressiontocreateafunctionthatoutputsanapproximateincidenceofviolencegiventheCityPopulation,HighSchoolGraduationRate,HighSchoolDropoutRate,PrisonPopulation,andPercentofParoleViolations. Whenwetestedthismodelagainstdatathatwealreadyhad,therewasrelativelylowerror,withdiscrepanciesinincidenceofviolencerangingfrom15crimestolessthan1crime.
Oncewecouldpredictanapproximateamountofviolentcrimeforanysetofdata,wefoundwhichvariable(s)affectedthefunctionmost. WefoundthatHighSchoolGraduationandDropoutRateshadthebiggesteffect,whichsupportsthecommentthatthecityhadproblemswithgangviolence(weassumedhighschooldropoutsweremuchmorelikelytojoingangs). Becausethesevariablesheldthemostweightinourmodel,wesetouttocreatearelationwhichcouldmodelincreasesingraduationrateanddecreasesindropoutrateasafunctionofthepercentincreaseinschoolfunding. Wefeltthatthismodelwouldbebeneficialinourproposaltothemayor,asitallowsustostatespecificandconcretechangesthatthecitycanmake,insteadoftryingtogenerallyincreasethenumberofgraduatesandstopstudentsfromdroppingout. Therewaserrorinthismodelaswell,whichcompoundedtheerrorofouroriginalmodel. However,wefeltthatitallowedustomakeastrongerandmorerealisticproposaltothemayor.
Aspartofourreport,wecreatedacomputerprogramthatcouldacceptproposedchangesinfunding,convertittoanapproximatechangeingraduationanddropoutrates,producethemultivariablelinearregressionfunction,andinputthemodifieddata(modifiedbythechangeinschoolfunding)tooutputaprojectedincidenceofviolenceinagivenyear,andtheamounteliminated. Byusingthisprogram,wewereabletotestdifferentvaluesfortheproposedchangeinfundingandassesshowwellitreducedcrime. Fromthistesting,wewereconfidentthatbyincreasingschoolfundingby275%bytheyear2020,thecrimewoulddecreaseby10%. However,inthecreationofourmodelsweconsistentlyusedconservativeestimates,sowebelievethatthetruepercentincreaseoffundingnecessarytoreducetheincidenceofviolenceby10%maybelessthanthat. Inaddition,ourproposalincludesqualitativemeasuresforthecitytointroducetocontinuetoreducecrime. Thesemeasurescouldnotbeincludedinourcomputerprogram,sothepercentincreaseinfundingmaybeevenless.
Assumptions:
1.ThiscitycanbeaccuratelymodeledbytheexistingtownofSalinas,California.
ManystatisticsforthetownofSalinas,Californiaanditscounty,MontereyBay,verycloselyreflectthoseofthegivencity.Overthepastnineyears,from2000-2008,thehomicidecountforSalinasisalmostidentical.Salinas’spopulationalsofollowssimilartrendstothatofthegivencityandisidenticalintheyear2000(theyearofthecensus)—thesameistrueofMontereyBayandthegivencountypopulation.BecauseSalinasisnearlyequivalenttoourcityinpopulationandcrimerate,weassumedthatotherstatisticsfromSalinascouldbeusedtomodeldataforourpredictions.
2.Thestatisticsgivenforjuvenileinmatesandprisonpopulation(anddataassociatedwiththese)reflectstatisticsforthestate,notonlyforthecity.
ThegivenstatisticsforjuvenileinmatesexactlyreflectthoseforthestateofCalifornia.Inaddition,thedataforprisonpopulationintheentirestateofCaliforniaareanalogoustothosegiven.ConsideringthatthecitygivencanbeaccuratelymodeledbythecityofSalinasinCalifornia,itseemslogicalthatthestatisticsgivenforjuvenileinmatesandprisonpopulationarereflectiveofthestateandnotthetown(therearealsomoreprisoninmatesthanresidentsofthecity,soitwouldnotmakesensefortheprisontoonlyholdinmatesfromtheimmediatecity).However,weassumethatthetrendinthesenumbersreflectthecrimerateinourtown.
3.Highschooldropoutsaremorelikelytobecomeinvolvedingangactivity.
Weassumethathighschoolstudentswhodropoutofschoolareatahigherriskforjoininggangs,whichisreflectedinthedata.
4.Factorsforwhichwearegivenstatisticsintheproblemwillcontinuetofollowcurrenttrends.
Inourmodel,weuseCityPopulation,HighSchoolGraduationRate,HighSchoolDropoutRate,PercentParoleViolations,andPrisonPopulation.Ourmodelisusedtoprojectthesefactors,alongwiththeeffectofincreasedschoolfunding,inthefuturetopredictincidenceofviolenceinagivenyear.Forsomeofthesedata,therewasnoapparenttrendovertheeightyearsgiven(theyessentiallyremainconstant)andweassumethatthiswillcontinuetobetrueinthefuture.Forpopulation,whichhasfollowedatrendinrecentyears,weusedafunctionfittingthedatatoprojectthepopulationintothefutureandfactoranestimatedpopulationintothepredictedincidenceofviolence(SeeAppendixB).
RestatementofProblem/GoalStatement:
Acityhasalargeamountofgangviolenceandcrime.Weneedtoidentifyvaluesthatarepredictorsofviolenceandminimizeormaximizetheminordertoreduceviolence.Wewillcreateaproposalthatdescribeswaystochangethesepredictorsoverthenexttenyearssothatviolentcrimewillbereducedby10%bytheyear2020.Inordertoensurethatourmethodsareeffective,wewillmodelviolenceasafunctionofothergivendataaswellasthechangesoutlinedintheproposalandusethattocomputeanestimateamountofviolentcrimeintheyear2020.
Proposal:
Tothemayor,chiefofpolice,andcitycouncil:
Withtheprovidedstatisticsfromtheyears2000-2008,wehavecompletedareportthatdetailsouranalysisofthedataandrecommendationfordecreasingviolentcrimeinyourcity.Wehavedeterminedthatincidenceofcrimecanbeeffectivelypredictedbyanumberofcontrollablefactorsinyourcommunity.Thepopulationofthecityandhighschoolgraduationrateareinverselyrelatedtotheincidenceofviolentcrime.Thehighschooldropoutrate,prisonpopulation,andpercentofparoleviolationsincreaseareproportionaltothecrime—astheyincrease,sodoesthecrimerate.Itwouldthereforebebeneficialtoreducethedropoutrate,prisonpopulation,andpercentofparoleviolations,andtoraisethehighschoolgraduationrateandcitypopulation.However,tobebothrealisticandeffective,wehavecarefullyexaminedwhichfactorsaremoststronglycorrelatedtotheincidenceofviolenceandwehaveproposedchangesthatwillmaximizeoureffectivenessbyaddressingthesefactors.Ourconclusionisthatthroughcarefulbudgetingandimplementationofourproposal,thecitycancutviolentcrimeby10%bytheyear2020.
Ofthesefivefactors,thetwothataremoststronglycorrelatedtoviolentcrime,andthereforethetwothatreducecrimemosteffectively,arethehighschoolgraduationanddropoutrates.Therefore,ourproposalisfocusedlargelyonincreasingthegraduationrateandloweringthedropoutrate.Thesetwoquantitiesareofgreatconcernnationwide,sothereisalargeamountofinformationpublishedonthem.
Inadditiontochangesinthehighschoolgraduationanddropoutrates,thereshouldbenoticeablechangesinthepopulationofthecitybytheyear2020.Whilethepopulationofthecitydoesnotfollowawelldefinedfunctionwithrespecttotime,thepopulationofthecountydoes.Moreover,thepopulationofthecityisproportionaltothepopulationofthecounty.Wewerethereforeabletoexpressthecity’spopulationasafunctionoftimebyfirstusingafunctionthatrelatesthecounty’spopulation.Fromthisprojection,wefoundthatthepopulationofthecitywillcontinuetoincreasethrough2020andareabletomodelthisgrowthandestimatethecity’spopulationforspecificyears.Becauseviolentcrimeisinverselyrelatedwithpopulation,wecanexpecttoseedropsinviolenceaslongasthecity’strendofgrowthcontinues.
Inordertotakeadvantageofthisgrowth,werecommendthatyoucreateacampaigntoattractcitizens
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