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ACTM–StatisticsRegionalExamMarch5,2016 ForQuestions1through25recordyouranswerontheanswersheetprovided.Besuretouseapencil tomarkyouranswers.Aftercompletingthemultiplechoicesection,gototheconstructedresponse items.Theseareusedastiebreakeritemsonlyanddonotcountintheoverallstudentscore.Students shouldanswertheconstructedresponseitemsinnumericalorder.Besurethatyournameisoneach sheet. 1.Themeansalaryforcollegestudentsis$45,000.Isthisvalueastatisticoraparameter? a. Thevalueisastatisticbecauseitisanumericalmeasurementdescribingsomecharacteristicofa population. b. Thevalueisaparameterbecauseitisanumericalmeasurementdescribingsomecharacteristic ofasample. c. Thevalueisaparameterbecauseitisanumericalmeasurementdescribingsomecharacteristic ofapopulation. d. Thevalueisastatisticbecauseitisanumericalmeasurementdescribingsomecharacteristicofa sample. 2.Atownobtainscurrentemploymentdatabypolling10,000ofitscitizensthismonth.Identifythe typeofobservationalstudy. a. Cross-sectional b. Prospective c. Retrospective d. Noneofthese 3.ThefrequencydistributionbelowsummarizesthehomesalepricesinthecityofSummerhillforthe monthofJune.Determinethewidthofeachclass. Salespriceinthousand$ Frequency 80.0-110.9 2 111.0-141.9 5 142.0-172.9 7 173.0-203.9 10 204.0-234.9 3 235.0-265.9 1 a. 30 b. 28 c. 61 d. 31 ACTM–StatisticsRegionalExamMarch5,2016 4.Anursemeasuredthebloodpressureofeachpersonwhovisitedherclinic.Belowisarelativefrequencybargraphforthesystolicbloodpressurereadingsforthosepeopleagedbetween25and40. Thebloodpressurereadingsweregiventothenearestwholenumber.Identifythecenterofthethird class. Rela]veFrequency a. 130 b. 124 0.4 c. 120 0.3 d. 125 0.2 0.1 0 100-110 110-120 120-130 130-140 140-150 150-160 SystolicBloodPressure(mmHg) 5.Themanagerofabankrecordedtheamountoftimeeachcustomerspentwaitinginlineduringpeak businesshoursoneMonday.Thefrequencydistributionbelowsummarizestheresults.Estimatethe meanwaitingtime.Roundtoonedecimalplace. WaitingTime(inminutes) Numberofcustomers 0-3 9 4-7 9 8-11 13 12-15 6 16-19 6 20-23 1 24-27 2 a. 6.6minutes b. 9.9minutes c. 9.7minutes d. 45minutes 6.Michaelgetstestgradesof71,76,80,and86.Hegetsa90onhisfinalexam.Findtheweighted meanifthetestseachcountfor10%andthefinalexamcountsfor60%ofthefinalgrade.Roundtoone decimalplace. a. 85.3 b. -71.2 c. 241.8 d. 80.6 ACTM–StatisticsRegionalExamMarch5,2016 7.Theagesofthemembersofagymhaveameanof46yearsandastandarddeviationof12years. WhatcanyouconcludefromChebyshev’stheoremaboutthepercentageofgymmembersaged between14.8and77.2? a. Thepercentageisatmost85.2% b. Thepercentageisatleast85.2% c. Thepercentageisatleast61.5% d. Thepercentageisapproximately61.5% 8.Mario’sweeklypokerwinningshaveameanof$301andastandarddeviationof$60.Lastweekhe won$182.Howmanystandarddeviationsfromthemeanisthat? a. 0.99standarddeviationsbelowthemean b. 1.98standarddeviationsbelowthemean c. 1.98standarddeviationsabovethemean d. 0.99standarddeviationsabovethemean 9.Thetablesummarizestheresultsoftestingforacertaindisease.Ifoneoftheresultsisrandomly selected,whatistheprobabilitythatitisafalsenegative(testindicatesthepersondoesnothavethe diseasewheninfacttheydo)?Whatdoestheprobabilitysuggestaboutthetest? PositiveTest NegativeTest Result Result Subjecthasthedisease 115 5 Subjectdoesnothavethedisease 11 180 a. 0.595;Theprobabilityofthiserrorishighsothetestisnotveryaccurate. b. 0.035;Theprobabilityofthiserrorislowsothetestisfairlyaccurate. c. 0.016;Theprobabilityofthiserrorislowsothetestisfairlyaccurate. d. 0.042;Theprobabilityofthiserrorislowsothetestisfairlyaccurate. 10.Findtheprobabilitythatfourrandomlyselectedpeopleallhavedifferentbirthdays.Assumeonly 365possiblebirthdays. a. 0.9891 b. 0.9918 c. 0.9729 d. 0.9836 11.Inabatchof8,000clockradios,5%aredefective.Asampleof15clockradiosisrandomlyselected withoutreplacementfromthe8,000andtested.Theentirebatchwillberejectedifatleastoneof thosetestedisdefective.Findtheprobabilitythattheentirebatchwillberejected. a.0.537b.0.050 c.0.067 d.0.463 ACTM–StatisticsRegionalExamMarch5,2016 12.Eightbasketballplayersaretobeselectedtoplayinaspecialgame.Theplayerswillbeselected fromalistof27players.Iftheplayersareselectedrandomly,whatistheprobabilitythattheeight tallestplayerswillbeselected? a. 1/2,220,075 b. 1/40,320 c. 1/213,127,200 d. 8/27 13.Refertothetable,whichsummarizestheresultsofasample.Asubjectisrandomlyselected,what istheprobabilitythesubjecthasbrownhairgiventhattheyhaveblueeyes? GreenEyes BlueEyes BrownHair 82 7 BlondHair 26 308 a. 0.079 b. 0.978 c. 0.022 d. 0.210 14.Anarcherisabletohitthebull’seye57%ofthetime.Ifsheshootseightarrows,whatisthe probabilitythatshegetsexactlyfourbull’s-eyes?Assumeeachshotisindependentoftheothers. a. 0.152 b. 0.106 c. 0.253 d. 0.004 15.Theincomesoftraineesatalocalmillarenormallydistributedwithameanof$1,100andastandard deviationof$150.Whatpercentageoftraineesearnlessthan$900amonth? a. 40.82% b. 90.82% c. 35.31% d. 9.12% 16.SupposethattheIQscoresofadultsarenormallydistributedwithameanof100andastandard deviationof15.FindtheIQscorethatwouldbethecutoffscoreforhavinganIQinthetop10%ofthe population. a. 100.5 b.119.2 c.108.1 d.80.8 ACTM–StatisticsRegionalExamMarch5,2016 17.Inapopulationof150women,theheightsofthewomenarenormallydistributedwithameanof 64.1inchesandastandarddeviationof3.10inches.If49womenareselectedatrandom,findthemean andstandarddeviationofthepopulationofsamplemeans.Assumethatthesamplingisdonewithout replacementanduseafinitepopulationcorrectionfactor. a. 64.1inches,3.10inches b. 64.1inches,2.26inches c. 64.1inches,0.36inches d. 52.8inches,2.55inches 18.Anewspaperarticleabouttheresultsofapollstates:“Intheory,theresultsofsuchapoll,in99 casesoutof100shoulddifferbynomorethan3percentagepointsineachdirectionfromwhatwould havebeenobtainedbyinterviewingallvotersintheUnitedStates.”Findthesamplesizesuggestedby thisstatement. a. 1,068 b. 1843 c. 1509 d. 74 19.Asurveyof300unionmembersinNewYorkStaterevealsthat112favortheRepublicancandidate forgovernor.Constructa98%confidenceintervalforthetruepopulationproportionpofallNewYork StateunionmemberswhofavortheRepublicancandidate.Usethenormalapproximationmethod. a. 0.304<p<0.442 b. 0.316<p<0.430 c. 0.308<p<0.438 d. Noneofthese 20.Agroupof67randomlyselectedstudentshaveameanscoreof21.9onaplacementtest.The populationstandarddeviationis 𝜎 = 5.Whatisthe90%confidenceintervalforthemeanscoreµofall studentstakingthetest? a. 20.5<µ<23.3 b. 20.3<µ<23.5 c. 20.7<µ<23.1 d. 20.9<µ<22.9 21.Findthep-valueforatwo-tailedtestofmeanswithn=29andateststatistict=2.743. a. 0.0105 b. 0.0210 c. 0.0053 d. Noneofthese ACTM–StatisticsRegionalExamMarch5,2016 22.Theaccuracyofverbalresponsesistestedinanexperimentinwhichindividualsreporttheirheights andthenaremeasured.Thedataconsistofthereportedheightandmeasuredheightforeach individual.Whattypeofsampleisthis? a. Independentsample b. Dependentsample c. Codependentsample d. Clustersample 23.Thepaireddatabelowconsistofthetemperaturesonrandomlychosendaysandtheamounta certainkindofplantgrew(inmillimeters).Findthevalueofthelinearcorrelationcoefficientr. Temp. 62 76 50 51 71 46 51 44 79 Growth 36 39 50 13 33 33 17 6 16 a. -0.210 b. 0 c. 0.196 d. 0.256 24.Sixpairsofdatayieldr=0.444andtheregressionequation𝑦 = 5𝑥 + 2.Also,𝑦 = 18.3.Whatisthe bestpredictedvalueofyforx=5?Useasignificancelevelof0.05andreferencethetableofcriticalr valuesfortestingHo:𝜌 = 0. Samplesizen Criticalrvaluesforα=0.05 4 0.950 5 0.878 6 0.811 7 0.754 a. 93.5 b. 4.22 c. 18.3 d. 27 25.Completethesentence.Thetimeittakesforthebustoarriveatthebusstopis________________. a.adiscreterandomvariable. b.acontinuousrandomvariable. c.notarandomvariable. d.arandomvariable,butitisnotdiscreteorcontinuous. ACTM–StatisticsRegionalExamMarch5,2016 Name:_________________________________________ TIEBREAKERS 1. Fourteenpeoplewhoseteethwerethoroughlycleanedandpolishedwererandomlyassigned totwogroupsofsevensubjectseach.Bothgroupswereassignedtouseoralrinses(no brushing)fora2-weekperiod.Group1usedarinsethatcontainedanantiplaqueagent.Group 2,thecontrolgroup,receivedasimilarrinseexceptthat,unknowntothesubjects,therinse containednoantiplaqueagent.Aplaqueindexx,ameasureofplaquebuildup,wasrecordedat 4,7and14days.Themeanandstandarddeviationforthe14dayplaquemeasurementsare showninthetableforthetwogroups. Control Antiplaque Group SampleSize 7 7 Mean 1.26 0.78 Standard 0.32 0.32 Deviation a. Statethenullandalternativehypothesesthatshouldbeusedtotesttheeffectivenessof theantiplaqueoralrinse. b. Dothedataprovideevidencetoindicatethattheoralantiplaquerinseiseffective?Test using𝛼 = 0.05. ACTM–StatisticsRegionalExamMarch5,2016 Name:_____________________________________________ 2. Supposeyouwishtocomparethemeansoffourpopulationsbasedonindependentrandom samples,eachofwhichcontainssixobservations. A. CompletetheANOVAtablebelow. Source df SS MS F Treatments 339.8 Error - Total 473.2 - - B. HowmanydegreesoffreedomareassociatedwiththeFstatisticfortesting𝐻! : 𝜇! = 𝜇! = 𝜇! = 𝜇! ? C. Dothedataprovidesufficientevidencetoindicatedifferencesamongthepopulation means?(Use𝛼 = 0.05.) 3. PreviousgeologicalstudiesindicatethatexploratoryoilwellsdrilledintheFayettevilleshale regionhavea40%chanceofstrikingoil.Assumethewellsbeingdrilledareindependentofone another.LetXbethenumberofwellsdrilleduptoandincludingthewellwherestrikingoilfirst occurs.Usingthisinformation,answerthequestionsbelow. A. WhatistheprobabilityfunctionforX?(Specifythedistributionandtherangeofpossiblevalues ofX.) B. Whatistheprobabilityofstrikingoilforthefirsttimeafterthesecondwellisdrilled? ACTM–StatisticsRegionalExamMarch5,2016 ACTMStatisticsExamSolutions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. C A D D C A B B C D A A C C D B C B C D A B C C B TieBreakers 1. a. Iftheantiplaquerinseiseffective,theplaquebuildupshouldbelessforthegroupusing therinse.Thus𝐻! : 𝜇! − 𝜇! = 0, 𝐻! : 𝜇! − 𝜇! > 0Where1isfromthecontrolgroupand2is fromtheantiplaquegroup. b. TestStatistic:𝑡 = 2.806 CriticalValue:𝑡! = 1.782(Reject𝐻! ) p-value=0.0079 Reject𝐻! .Thereisevidencetoindicatetherinseiseffective. 2. a. Source Treatments Error Total df 3 20 23 SS 339.8 133.4 473.2 MS 113.2667 6.67 F 16.981 ACTM–StatisticsRegionalExamMarch5,2016 b.Thedegreesoffreedomforthistestare𝑑𝑓! = 3and𝑑𝑓! = 20. c.Therejectionregionforthistestis𝐹!" > 3.10Yes,Theteststatistic,16.981fallsintherejection regionsoreject𝐻! .Thereisevidencetoconcludetherearedifferencesinthepopulationmeans. 3. a.P(X=x)=f(x)=0.4(0.6)x-1forx≥1. B.P(X>2)=1-P(X≤2)=1-[f(1)+f(2)]=1-[0.4+0.4*0.6]=0.36or36%chance