Statistics - Mathematics Vision Project

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 9

Statistics

ALGEBRA II

An Integrated Approach

ALGEBRA II // MODULE 9

STATISTICS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 9 - TABLE OF CONTENTS

STATISTICS

9.1 What Is Normal? – A Develop Understanding Task

Understanding normal distributions and identify their features (S.ID.4)

Ready, Set, Go Homework: Statistics 9.1

9.2 Just ACT Normal – A Solidify Understanding Task

Using the features of a normal distribution to make decisions (S.ID.4)


9.3 Y B Normal? – A Solidify Understanding Task

Introducing z scores to compare normal distributions (S.ID.4)


9.4 Wow, That’s Weird! – A Practice Understanding Task

Comparing normal distributions using z scores and understanding of mean and standard

deviation (S.ID.4)


9.5 Would You Like to Try a Sample – A Develop Understanding Task

Understanding and identifying different methods of sampling (S.IC.1)


9.6 Let’s Investigate – A Solidify Understanding Task

Identifying the difference between survey, observational studies, and experiments (S.IC.1,

S.IC.2)



STATISTICS


9.7 Slacker’s Simulation – A Solidify Understanding Task

Using simulation to estimate the likelihood of an event (S.IC.2, S.IC.3)



STATISTICS – 9.1


9.1 What Is Normal?

A Develop Understanding Task

Oneveryimportanttypeofdatadistributioniscalled

a“normaldistribution.”Inthiscase,theword

“normal”hasaspecialmeaningforstatisticaldistributions.Inthistask,youwillbegivenpairof

datadistributionsrepresentedwithhistogramsanddistributioncurves.Ineachpair,one

distributionisnormalandoneisnot.Yourjobistocompareeachofthedistributionsgivenand

comeupwithalistoffeaturesfornormaldistributions.

1.Thisisapproximatelynormal: Thisisnot:

Whatdifferencesdoyouseebetweenthesedistributions?

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STATISTICS – 9.1


2.Thisisnormal: Thisisnot:




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STATISTICS – 9.1






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STATISTICS – 9.1





Whatdifferencesdoyouseebetweenthese

distributions?

_________________________________________________________________________________________________________________

8.Basedupontheexamplesyouhaveseenin#1-7,whatarethefeaturesofanormaldistribution?

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STATISTICS – 9.1


9. a.Whatdoesthestandarddeviationtellusaboutadistribution?

b.Eachofthedistributionsshownbelowarenormaldistributionswiththesamemeanbut

adifferentstandarddeviation.

Mean=3,StandardDeviation=0.5 Mean=3,StandardDeviation=1

Mean=3,StandardDeviation=0.25

Howdoeschangingthestandarddeviationaffectanormalcurve?Whydoesithavethiseffect?

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STATISTICS – 9.1


10. a.Whatdoesthemeantellusaboutadistribution?

b.Eachofthedistributionsshownbelowarenormaldistributionswiththesamestandard

deviationbutadifferentmean.

Mean=1,StandardDeviation=0.25 Mean=2,StandardDeviation=0.25

Mean=3,StandardDeviation=0.25

Howdoeschangingthemeanaffectanormalcurve?Whydoesithavethiseffect?

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STATISTICS – 9.1


11.Nowthatyouhavefiguredoutsomeofthefeaturesofanormaldistribution,determineifthe

followingstatementsaretrueorfalse.Ineachcase,explainyouranswer.

a.Anormaldistributiondependsonthemeanandthestandarddeviation.

True/FalseWhy?

b.Themean,median,andmodeareequalinanormaldistribution.

True/FalseWhy?

c.Anormaldistributionisbimodal.

True/FalseWhy?

d.Inanormaldistribution,exactly50%ofthepopulationiswithinonestandarddeviationofthe

mean.

True/FalseWhy?

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STATISTICS – 9.1

9.1


Needhelp?Visitwww.rsgsupport.org

READY Topic:Workingwithstandarddeviationandpercentiles

1.Jordanscoresa53onhismathtest.Theclassaverageis57withastandarddeviationof2points.HowmanystandarddeviationsbelowthemeandidJordanscore?

2.InJordan’sscienceclass,hescoreda114.Theclassaveragewasa126withastandarddeviation

of6points.HowmanystandarddeviationsbelowthemeandidJordanscore?Incomparisontohispeers,whichtestdidJordanperformbetteron?

3.Rankthedatasetsbelowinorderofgreateststandarddeviationtosmallest:

! = {1,2,3,4}B = {2,2,2,2, }C = {2,4,6,8}D = {4,5,6,7}E = {1,1.5,2,2.5}

4.Robinmadeittotheswimmingfinalsforherstatechampionshipmeet.Thetimesinthefinals

wereasfollows:

{2: 10.3, 2: 12.5, 2: 12.7, 2: 12.38, 2: 20.45, 2: 21.43}

IfRobin’stimewasa2:12.7,whatpercentofhercompetitorsdidshebeat?

5.Rememberthatinstatistics,MisthesymbolformeanandOisthesymbolforstandarddeviation.

Usingtechnology,identifythemeanandstandarddeviationforthedatasetbelow:

{1.23, 1.3, 1.1, 1.48, 1, 1.14, 5.21, 5.1, 4.63}

M = O =

6.Forthedatainnumber5,whattimewouldfallonestandarddeviationabovethemean?

Threestandarddeviationsbelowthemean?

READY, SET, GO! Name PeriodDate

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STATISTICS – 9.1

9.1



SET Topic:Identifyingpropertiesofthenormalcurve

Foreachdistribution,identifythepropertiesthatmatchwithaNormalDistribution,andthendecideiftheNormalcurvecouldbeusedasamodelforthedistributionandexplainwhy.7.NormalProperties:ModelwithaNormalCurve?YesorNo8.NormalProperties:ModelwithaNormalCurve?YesorNo

9.NormalProperties:ModelwithaNormalCurve?YesorNo10.NormalProperties:ModelwithaNormalCurve?YesorNo

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STATISTICS – 9.1

9.1



11.

Mean=0Median=0.1Mode=0.1

NormalProperties:ModelwithaNormalCurve?YesorNo

12.

Mean:68Median:68Mode:68

NormalProperties:ModelwithaNormalCurve?YesorNo

13.IftwoNormaldistributionshavethesamestandarddeviationof4.9butdifferentmeansof3and6,howwillthetwoNormalcurveslookinrelationtoeachother?DrawasketchofeachNormalcurvebelow. 14.IftwoNormaldistributionshavethesamemeanof3butstandarddeviationsof1and4,howwilltheylookinrelationtoeachother?DrawasketchofeachNormalcurvebelow.

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STATISTICS – 9.1

9.1



15.TheNormalCurvegivenbelowhasbeenlabeledoutthreestandarddeviations.Estimatewhat

onestandarddeviationisforthiscurve.

GO

Topic:Recallinginverses

Writetheinverseofthegivenfunctioninthesameformatasthegivenfunction:

16.T(V) = 3VX + 2 17.Z(V) =X[\]

^ 18.ℎ(V) = 3 + √2V − 1 19.

Determineifthefollowingfunctionsareinversesbyfindingcde(f)ghijedc(f)g.

20.T(V) = 2V + 3andZ(V) = k

XV −

l

X 21.T(V) = 2VX − 3andZ(V) = m

[n

X+ 3

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STATISTICS – 9.2


9.2 Just ACT Normal

A Solidify Understanding Task

Oneofthemostcommonexamplesofanormal

distributionisthedistributionofscoresonstandardized

testsliketheACT.In2010,themeanscorewas21and

thestandarddeviationwas5.2(Source:NationalCenterforEducationStatistics).

1.Usethisinformationtosketchanormaldistributioncurveforthistest.

2.Usetechnologytocheckyourgraph.Didyougetthepointsofinflectionintherightplaces?(Makeadjustments,ifnecessary.)

3.In“WhatIsNormal”,youlearnedthatthe68–95–99.7rule.Usetheruletoanswerthe

followingquestions:

a.Whatpercentageofstudentsscoredbelow21?

b.Aboutwhatpercentageofstudentsscoredabove16?

c.Aboutwhatpercentageofstudentsscoredbetween11and26?

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STATISTICS – 9.2


4.Yourfriend,Calvin,wouldliketogotoaveryselectivecollegethatonlyadmitsthetop1%ofallstudentapplicants.Calvinhasgoodgradesandscored33onthetest.DoyouthinkthatCalvin’sACTscoregiveshimagoodchanceofbeingadmitted?Explainyouranswer.5.ManystudentsliketoeatmicrowavepopcornastheystudyfortheACT.Microwavepopcornproducersassumethatthetimeittakesforakerneltopopisdistributednormallywithameanof120secondsandastandarddeviationof13forastandardmicrowaveoven.Ifyou’readevotedpopcornstudier,youdon’twantalotofun-poppedkernels,butyouknowthatifyouleavethebaginlongenoughtobesurethatallthekernelsarepopped,someofthepopcornwillburn.Howmuchtimewouldyourecommendformicrowavingthepopcorn?Useanormaldistributioncurveandthefeaturesofanormaldistributiontoexplainyouranswer.

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STATISTICS – 9.2

9.2



READY Topic:ThinkingabouttheLawofLargeNumbers

1.Youandyourfriendarerollingonedieoverandoveragain.After6rolls,yourfriendhasrolledfourfives.Areyousurprisedbytheseresults?Explain

2.Afterrollingthedie50times,youknownoticethatyourolledatotalof20fives.Areyousurprisednow?Explain.

3.Yousurvey100peopleinyourschoolandaskthemiftheyfeelyourschoolhasadequateparking.Only30%ofthesamplefeelstheschoolhasenoughparking.Ifyouhave728studentstotalinyourschool,howmanywouldyouexpectoutofallthestudentbodythatfelttherewasenoughparking?

SET Topic:Applyingpropertiesofnormaldistribution

4.ThepopulationofNBAplayersisNormallydistributedwithameanof6’7”andastandarddeviationof3.9inches.(Wikepedia)Gregisconsideredunusuallytallforhishighschoolat6’3”.

a. WhatpercentofNBAplayersaretallerthanGreg?�

b. Whatpercentareshorter?�

c. HowtallwouldGreghavetobeinordertobeinthetop2.5%ofNBAplayerheights? �


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STATISTICS – 9.2

9.2



5.TheaverageheightofboysatGreg’sschoolis5’9”withastandarddeviationof2“.Ifweassumethepopulationisnormallydistributed,

d. WhatpercentofstudentsintheschoolareshorterthanGreg?�

e. Whatpercentofstudentsarebetween5’5”and5’11”?�

6.Jordanisdrinkingacupofhotchocolate.Frompreviousresearch,heknowsthatittakesanaveragetimeof10minutesforthehotchocolatetoreachatemperaturewherehistonguewillnotburn.ThetimeittakesthechocolatetocoolvariesNormallywithastandarddeviationof2minutes.

a. Howlongshouldhewaittodrinkhishotchocolateifhewantstobe84%surethathewon’tburnhimself?�

b. Ifhewaits8minutes,whatpercentofthetimewillheburnhistongue? �

GO Topic:Applyingthepropertiesoflogarithms

Usethepropertiesoflogarithmstoexpandtheexpressionasasumordifferenceand/orconstantmultipleoflogarithms.(Assumeallvariablesarepositive.)

7.log$3' 8.log( )* 9.ln √'- 10.log $./0123/

11.log2 45./625./ 12.log .

/74$.7$8) 13.log227'* 14.log 10)=>

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STATISTICS – 9.3


9.3 Y B Normal?


Asacollegeadmissionsofficer,yougettoevaluatehundredsofapplicationsfromstudentsthatwanttoattendyourschool.Manyofthemhavegoodgrades,haveparticipatedinschoolactivities,havedoneservicewithintheircommunities,andallkindsofotherattributesthatwouldmakethemgreatcandidatesforattendingthecollegeyourepresent.Onepartoftheapplicationthatisconsideredcarefullyistheapplicantsscoreonthecollegeentranceexamination.Atthecollegeyouworkfor,somestudentshavetakentheACTandsomestudentshavetakentheSAT.Youhavetomakeafinaldecisionontwoapplicants.TheyarebothwonderfulstudentswiththeverysameG.P.A.andclassrankings.Itallcomesdowntotheirtestscores.StudentAtooktheACTandreceivedascoreof29inmathematics.StudentBtooktheSATandreceivedascoreof680inmathematics.Sinceyouareanexpertincollegeentranceexams,youknowthatbothtestsaredesignedtobenormallydistributed.AperfectACTscoreis36.TheACTmathematicssectionhasameanof21andstandarddeviationof5.3.(Source:NationalCenterforEducationStatistics2010)AperfectscoreontheSATmathsectionis800.TheSATmathematicssectionhasameanof516andastandarddeviationof116.(Source:www.collegeboard.com2010Profile).1. Basedonlyontheirtestscores,whichstudentwouldyouchooseandwhy?

Thisanalysisisstartingtomakeyouhungry,soyoucallyourfriendintheStatisticsDepartmentattheuniversityandaskhertogotolunchwithyou.Duringlunch,youtellherofyourdilemma.Theconversationgoessomethinglikethis:

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STATISTICS – 9.3


You:I’mnotsurethatI’mmakingtherightdecisionaboutwhichoftwostudentstoadmittotheuniversity.Theirentranceexamscoresseemlikethey’reinaboutthesamepartofthedistribution,butIdon’tknowwhichoneisbetter.It’sliketryingtofigureoutwhichbagoffruitweighsmorewhenoneismeasuredinkilogramsandoneismeasuredinpounds.Theymightlooklikeaboutthesameamount,butyoucan’ttelltheexactdifferenceunlessyouputthemonthesamescaleorconvertthemtothesameunits.Statistician:Actually,thereisawaytomakecomparisonsontwodifferentnormaldistributionsthatislikeconvertingthescorestothesameunit.Thescaleiscalledthe“standardnormaldistribution”.Sinceitwasinventedtomakeiteasytothinkaboutanormaldistribution,theysetitupsothatthemeanis0andthestandarddeviationis1.Here’swhatyourstatisticianfrienddrewonhernapkintoshowyouthestandardnormal

distribution:

You:Well,thatlooksjustlikethewayIalwaysthinkofnormaldistributions.

Statistician:Yes,it’sprettysimple.Whenweusethisscale,wegivethingsaz-score.Az-scoreof1meansthatit’s1standarddeviationabovethemean.Az-scoreof-1.3meansthatitisbetween1and2standarddeviationsbelowthemean.Easy-peasy.What’sevenbetteristhatwhenwehaveaz-scoretherearetablesthatwillshowtheareaunderthecurvetotheleftofthatscore.ForatestscoreliketheACTorSAT,itshowsthepercentageofthepopulation(orsample)thatisbelowthatscore.I’vegotaz-scoretablerighthereinmypurse.See,thez-scoreis-1.3,then9.68%ofthepopulationscoredless.Youcanalsosaythat90.32%ofthepopulationscoredbetter,so-1.3wouldn’tbeaverygoodscoreonatest.

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STATISTICS – 9.3


Tryit:Let’ssayyouhadtwoimaginarytesttakers,JackandJill.Jack’sz-scorewas1.49andJill’sz-scorewas0.89.2. WhatpercentofthetesttakersscoredbelowJack?WhatpercentscoredaboveJack?

3. WhatpercentofthetesttakersscoredbelowJill?WhatpercentscoredaboveJill?

4. WhatpercentofthetesttakersscoredbetweenJackandJill?

5. JackandJill’sfriend,Jason,scored-1.49.Findthenumberoftesttakersthatscoredabovehimwithoutusingatableortechnology.Explainyourstrategy.

You:That’sverycool,butthetwoscoresI’mworkingwitharenotgivenasz-scores.IstheresomewaythatIcantransformvaluesfromsomenormaldistributionlikethescoresontheACTorSATtoz-scores?Statistician:Sure.Thescalewouldn’tbesoamazingifyoucouldn’tuseitforanynormaldistribution.There’salittleformulafortransformingadatapointfromanynormaldistributiontoastandardnormaldistribution:

z-score=!"#"%&'(#)*+"(,#"(!"-!!+.'"#'&(

6. So,ifyouhaveanACTscoreof23.ThemeanscoreontheACTis21andthestandarddeviationis5.2.Whatwouldyouestimatethez-scoretobe?

7. Let’susetheformulatofigureitout:z-score=/0)/12./ .Howwasyourestimate?Explain

whythisvalueisreasonable.

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STATISTICS – 9.3


You:That’sgreat.I’mgoingbacktotheofficetodecidewhichstudentisadmitted.

8. ComparethescoresofStudentAandStudentB.Explainwhichstudenthasthehighestmathematicstestscoreandwhy.

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STATISTICS – 9.3

9.3



READY Topic:Usingprobabilitytopredictfromasample

AtSouthBeachHighSchool,thereare2500studentsattending.Marianasurveys40ofherfriendswheretheyprefertoeatlunch.Shecreatedthefollowingtwo-waytableshowingherresults:

9thGrade 10thGrade 11thGrade 12thGrade TotalsSchoolCafeteria

18 6 2 1 28

OffCampus 2 4 3 4 12Totals 20 10 5 5 40

Marianaplanstouseherdatatoanswerthefollowingquestions:

I. Dostudentsprefertoeatoncampusoroffcampusoverall?II. Isthereadifferencebetweengradelevelsforwherestudentsprefertoeatlunch?

1.InMariana’ssample,whatpercentofstudentspreferschoollunch?

Whatpercentprefertoeatoffcampus?

2.Foreachgradelevelinhersample,determinethepercentofstudentsthatpreferschoollunchandthepercentthatpreferoffcampuslunch.Doyounoticeanythingunusual?

3.Basedonhersample,MarianaconcludesthatstudentsatSouthBeachHighschooloveralllikeschoollunch.Doyouagreeordisagree? Why?


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STATISTICS – 9.3

9.3



SET Topic:Exploringz-scores

Acompanymakesameanmonthlyincomeof$20,300withastandarddeviationof$3,200.Inonegivenmonththecompanymakes$29,500.

4.Findthez-score.

5.AssumingthecompaniesmonthlyincomeisNormallydistributed,whatpercentofthetimedoesthecompanymakemorethanthisamount?Lessthan?

6.Whatpercentofthetimedoesthecompanymakebetween$15,000and$25,000?

7.Ifthecompanyneedstomake$16,400inordertobreakeven,howlikelyinagivenmonthisthecompanytomakeaprofit?

OntheWechslerAdultIntelligenceScale,anaverageIQis100withastandarddeviationof15units.(Source:http://en.wikipedia.org/wiki/Intelligence_quotient)

8.IQscoresbetween90and109areconsideredaverage.AssumingIQscoresfollowaNormaldistribution,whatpercentofpeopleareconsideredaverage?

9.OnemeasureofGeniusisanIQscoreofabove135.Whatpercentofpeopleareconsideredgenius?

10.EinsteinhadanIQscoreof160.Whatishisz-score?

11.WhatistheprobabilityofanindividualhavingahigherIQthanEinstein?

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STATISTICS – 9.3

9.3



GO Topic:SketchingPolynomials

Withoutusingtechnology,sketchthegraphofthepolynomialfunctionwiththegivencharacteristics.Iftheequationisnotgiven,writeitinstandardform.

12.Degree4

Roots:-1multiplicity2,5,-2

!–intercept:20

Equation:

13.$(&) = (& + 2)(& − 3)-

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STATISTICS – 9.3

9.3



14.1(&) = −&(& − 3)2(& + 5)(& − 5)

15.Degree3$(−1) = 10$(25) = 0$(3) = 0Equation:

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STATISTICS – 9.4


9.4 Wow, That's Weird!

A Practice Understanding Task

Eachofthestoriesbelowarebaseduponnormaldistributions.Rankorderthesestoriesfrommostunusualtomostaverage.(1isthemostunusual,8isthemostaverage.)Ineachcase,explainyourranking.A. ThenumberofredloopsinaboxofTutti-Frutti-O’sisnormallydistributedwithmeanof800loopsandstandarddeviation120.Tonyboughtanewbox,openedit,andcounted1243redloops.(Itdidn’treallymatterbecauseallthecolorsarethesameflavoranyway.)

Rank_________Explanation:_________________________________________________________________________________

B. Theweightofhousecatsisnormallydistributedwithameanof10poundsandstandarddeviation2.1pounds.Mycat,BigBoy,weighs6pounds.


C. Thelifetimeofabatteryisnormallydistributedwithameanlifeof40hoursandastandarddeviationof1.2hours.Ijustboughtabatteryanditdiedafterjust20hours


D. Theamountthatahumanfingernailgrowsinayearisnormallydistributedwithameanlengthof3.5cmandastandarddeviationof0.63cm.Myneighbor’sthumbnailgrewallyearwithoutbreakinganditis4.6cmlongwithstarsandstripespaintedonit.


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STATISTICS – 9.4


E. Mylittlebrotherwasdigginginthegardenandfoundagiantearthwormthatwas35cmlong.Thelengthofearthwormsisnormallydistributedwithameanlengthof14cmandastandarddeviationof5.3cm.


F. Themeanlengthofahumanpregnancyis268dayswithastandarddeviationof16days.Myauntjusthadaprematurebabydeliveredafteronly245days.


G. IQscoresforyoungadultsonafamousIQtestaredistributednormallywithameanof110andastandarddeviationof25.I’mprettysmartandmyIQis135.


H. Thearmymeasuredheadsizesamongmalesoldiersandfoundthatthedistributionisprettyclosetonormalwithameanof22.8inchesandstandarddeviationof1.1inches.LittleJoewasalmosttoosmalltogetintothearmybecausehisheadsizewasonly20.6inches.


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STATISTICS – 9.4

9.4



READY Topic:Fillingintwo-waytables

ThedatabelowisthedatafromMrs.Hender’sclass.Studentsneededtoscore60%orbettertopassthetest.

1sthour: 2ndhour: 3rdhour:72,83,56,63,89,92,92,67,88,84,67,97,96,100,84,82

80,83,81,81,67,90,70,71,72,77,81,85,86,77,74,51

51,45,67,83,99,100,94,52,48,46,100,59,65,56,72,63

1.Makeatwo-wayfrequencytableshowinghowmanystudentspassedthetestandhowmanyfailedeachclass.

1st 2nd 3rd TotalPassed Failed 2.WhatpercentofstudentspassedMrs.Hender’stestineachclass?Whatisthetotalpercentofallclassesthatpassed?

3.Combinethedatafromallthreeclassestocreateahistogramusingtechnology.Sketchyourhistogrambelow.WhatfeaturesoftheNormalcurvedoesyourhistogramhave?

4.IfMrs.Hender’sweregoingtopredicthertotalpassrateusingonly2ndhour,wouldshemakeanaccurateprediction?Explainwhyorwhynot.


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STATISTICS – 9.4

9.4



SET Topic:Usingfeaturesofthenormalcurvetorankdata

5.FivetrackathletesareintherunningfortheAthleticPerformanceoftheYearaward.Apanelofcoachesistryingtodecidewhichathleteisthemostdeservingtowintheaward.Rankeachathletebelowbythegiveninformation.AssumealldistributionsfollowaNormalCurve.

a. JavierthrewtheJavelin215ft.TheaverageJavelinthrowis152.08ft.withastandarddeviationof15.85ft.

b. Chancerana400mtimeof46.99seconds.Theaverage400mtimewas52.6,withastandarddeviationof1.01seconds.

c. Derickrana36.26inthe300mHurdles.Theaveragetimewas41.77withastandarddeviationof1.49seconds.

d. Chadrana100mtimeof10.59seconds.Theaveragetimewas11.603secondswithastandarddeviationof.29seconds.

e. Kaydenthrewthediscus180ft.Theaveragethrowwas122.4ft.withastandarddeviationof14.38ft.

GO Topic:Solvinglogarithmicequations

Solveeachequationbelowforxbyapplyingpropertiesforexponentsandlogarithms.

6.2%&' = 128 7., -./01

%= 27

8.3%3. = 27%&0 9.log(29 + 4) − log(39) = 0

10.log.(29. + 49 − 2) − log. 10 = 0 11. ?@(%3A)?@(.%&0) = 1

12.?BC(/%3.)?BC -' = 1 13.?BCD(0%3E)?BCD F-= 1

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STATISTICS – 9.5


9.5 Would You Like to Try a Sample?

A Develop Understanding Task

InthetaskWow!That’sWeird!,yousawanumberof

statisticsforthingsliketheaverageweightofahouse

cat.Youknowitwouldbeimpossibletomeasureallthehousecatstofindtheiraverageweights,

butscientistsstillclaimtoknowit.

You’veprobablyhearditmanytimesbefore:“Surveyresultsshowthat54%ofAmericansbelieve

that...”You’resurethatyoudidn’tparticipateinthesurveyandneitherdidanyoneyouknow,and

yet,theresearchersclaimthatthesurveyrepresentsthebeliefsofallAmericans.

Howcanthisbepossible?Inthenextfewtasks,we’llexplorehowstatisticsallowustodraw

conclusionsaboutanentiregroupwithoutactuallyworkingwiththeentiregroup.Sometimesthe

resultsmakesenseandothertimesyoumightthinkthattheyjustcan’tberight.Wewilllearnhow

tomakejudgmentsaboutstatisticalstudies,basedonthemethodsthathavebeenused.

First,weneedtogetourtermsstraight.Whenwetalkabouttheentiregroupthatweare

interestedin,thatiscalledthepopulation.Whensomemembersofthegroupareselectedto

representtheentiregroup,thatiscalledasample.Thethingweareinterestedinknowingabout

thepopulationistheparameterofinterest.

Foreachofthescenariosbelow,identifythepopulation,thesampleandthepopulationparameter

ofinterest.

1. Agrocerystorewantstoknowtheaveragenumberofitemsthatshopperspurchaseineachvisittothestore.Theydecidetocounttheitemsinthecartofeverytwentiethpersonthroughthecheckstand.Population_______________________________________________________________________________________Sample___________________________________________________________________________________________Parameterofinterest___________________________________________________________________________

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STATISTICS – 9.5


2. Ateamofbiologistswantstoknowtheaverageweightoffishinalake.Theydecidetodropanet

andmeasureallthefishcaughtinthreedifferentlocationsinthelake.Population_______________________________________________________________________________________Sample___________________________________________________________________________________________Parameterofinterest___________________________________________________________________________

3.Therearelotsofdifferentwaysthatasamplecanbechosenfromapopulation.Groupthe

followingexamplesofwaystoselectasampleintosixcategories.

A. Youareinchargeofschoolactivities.Youwanttoknowwhatactivitiesstudentswouldprefertoparticipateinduringtheschoolyear.Youdecidetoputthenameofeachstudentintheschoolintoabigbowl.Youdraw100namesandaskthosestudentstorespondtoasurveyabouttheactivitiestheyprefer.B. Youareinchargeofschoolactivities.Youwanttoknowwhatactivitiesstudentswouldprefertoparticipateinduringtheschoolyear.Youassigneachstudentintheschoolanumber.Yourandomlyselectastartingnumberamongthefirst10numbersandthenselecteverytenthstudentinthelistfromthatpointforward.C. Youareinchargeofschoolactivities.Youwanttoknowwhatactivitiesstudentswouldprefertoparticipateinduringtheschoolyear.Youusetherollsfromeachhomeroomclass.Yougothrougheachhomeroomclass,drawing2namesfromeachclass.Youaskthosestudentstorespondtoasurveyabouttheactivitiestheyprefer.

D. Youareinchargeofschoolactivities.Youwanttoknowwhatactivitiesstudentswouldprefertoparticipateinduringtheschoolyear.Yougetthelistofallthehomeroomclassesandrandomlyselect5classes.Yougotoeachoftheclassesselectedandsurveyallthestudentsinthatclass.E. Youareinchargeofschoolactivities.Youwanttoknowwhatactivitiesstudentswouldprefertoparticipateinduringtheschoolyear.Youstandinthecafeteriaduringyourlunchbreakandaskstudentsintheywouldbewillingtoparticipateinyoursurveyastheywalkby.

F. Youareinchargeofschoolactivities.Youwanttoknowwhatactivitiesstudentswouldprefertoparticipateinduringtheschoolyear.Youmakealotofcopiesofthesurveyaboutthe

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STATISTICS – 9.5


activitiesthatstudentspreferandyouputthemonatableoutsidethecafeteria.Studentscanchoosetotakethesurveyanddroptheirresponsesintoabigboxonthetable.G. Youareinterestedinfindingoutthepercentofresidentsinthecitythathaveexperiencedarobberyinthepastyear.Usingthecitypropertyrecords,youassigneachresidenceanumber.Youusearandomnumbergeneratortogiveyoualistofnumbers.Youlookupthepolicereportsforeachresidenceselected.H. Youwanttoknowtheaveragenumberofhoursthathighschoolseniorsspendplayingvideogamesinyourstate.Yourandomlyselect20highschoolsinthestateandthenaskalltheseniorsateachofthe20highschoolsabouttheirvideogamehabits.

I. Anautoanalystisconductingasatisfactionsurvey,samplingfromalistof10,000newcarbuyers.Thelistincludes2,500Fordbuyers,2,500GMbuyers,2,500Hondabuyers,and2,500Toyotabuyers.Theanalystselectsasampleof400carbuyers,byrandomlysampling100buyersofeachbrand.J. Ashoppingmallmanagementcompanywouldliketoknowtheaverageamountthatshoppersinthemallspendduringtheirvisit.Theyposttwosurveytakersnearoneoftheexitswhoaskshopperstotellthemwhattheyspentastheyleavethemall.

K. ArestaurantownerwantstofindouttheaveragenumberofdishesorderedateachtableservedonFridayevenings,theirbusiesttime.Shedecidestocollectandanalyzeeveryfifthreceiptofthenight,startingat6:00p.m.

L. M.

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STATISTICS – 9.5


N. O.

4. Whatmightbesomeoftheadvantagesanddisadvantagesofeachtype?

5. Apersonyouknowownsasmalltheaterthatshowslocaldramaticproductions.Shewantstoknowtheaverageageofthepeoplethatbuyticketstotheseetheshowssothatshecanbetterselectwhichplaystostage.Explaintotheownerwhyselectingthefirst20peoplethatarrivefortheshowmaynotbearepresentativesample.

6.Describeaprocessforselectingarepresentativesampleofthetheaterpatrons.

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STATISTICS – 9.5

9.5



READY Topic:Contrastingassociationandcausation

Whencollectingdata,statisticiansareofteninterestedinmakingpredictions.Sometimestheysimplywanttoknowifonevariableisrelatedorisassociatedwithanothervariable.(Canyoupredictonevariablegiveninformationontheotherone).Othertimes,theywanttodetermineifonevariableactuallycausesachangeinanothervariable.Foreachexamplebelow,decidewhetherthevariablessimplyexplaineachother,orifyouthinkonevariablewouldcausetheothertochange.

1.AstheamountoffoodOllietheelephanteatsincreasesherweightalsoincreases.(Associated/Causes)

2.AsPopsiclesalesgoupinthesummer,thenumberofpeopledrowningalsoincreases.(Associated/Causes)

3.AsErika’sfeetgrowlonger,shegrowstaller.(Associated/Causes)

4.AsTabathagetsolder,herreadingscoreimprovesinschool.(Associated/Causes)

SET Topic:Identifyingpopulation,sample,andparameter

Foreachscenariobelow,identifythepopulation,sampleandparameterofinterest.

5.Thelocalschoolboardwantstogetparentstoevaluateteachers.Theyselect100parentsandfindthat89%approveoftheirchild’steacher.

Population: Sample: Parameter:

6.Jarretwantstoknowtheaverageheightofthestudentsinhisschool.Thereare753studentsinhishighschool;hefindstheheightsof52ofthem.


7.AgovernmentofficialisinterestedinthepercentofpeopleatJFKairportthataresearchedbysecurity.Hewatches300peoplegothroughsecurityandobserves42thataresearched.



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STATISTICS – 9.5

9.5



Foreachscenario,identifywhattypeofsamplingwasusedtoobtainthesample.Explainwhetherornotyouthinkthesamplewillberepresentativeofthepopulationitwassampledfrom:

8.Elvirasurveysthefirst60studentsinthelunchlinetodetermineifstudentsattheschoolaresatisfiedwithschoollunch.

Typeofsample:Representative?Explain.

9.Elviraselectsevery5thstudentinthelunchlinetodetermineifstudentsattheschoolaresatisfiedwithschoollunch.


10.Elvirarandomlyselects7differenttablesinthelunchroomandsurveyseverystudentonthetabletodetermineifstudentsattheschoolaresatisfiedwithschoollunch.


11.Elviraassignseverystudentintheschoolanumberandrandomlyselects60studentstosurveytodetermineifstudentattheschoolaresatisfiedwithschoollunch.


12.Elvirawantstodetermineifstudentsaresatisfiedwithschoollunch.Sheleavessurveysonatableforstudentstoanswerasthewalkby.


13.Elvirawantstodetermineifstudentsaresatisfiedwithschoollunch.Shewantstoincludeinputfromeachgradelevelatthehighschool.Sherandomlysurveys25freshman,25sophomores,25juniors,and25seniors.


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STATISTICS – 9.5

9.5



GO Topic:Graphingtrigfunctions

Foreachfunctionidentifytheamplitude,period,horizontalshift,andtheverticalshift.

14.!(#) = 120 cos ,-. (# − 3)1 + 30 15.!(#) = 3.5 sin7-8 (# +9:); + 7

Amplitude: Amplitude:

Period: Period:

HorizontalShift: HorizontalShift:

VerticalShift: VerticalShift:

16.Graphonefullperiodof!(=) = 8 sin(= − ?) − 2.

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STATISTICS – 9.6


9.6 Let's Investigate


Whenwewanttodrawconclusionsaboutsomepopulation,thereareatleasttwodifferentstatisticalideastoconsider.WelearnedaboutsamplinginWouldYouLiketoTryaSample,sinceitisusuallymorepracticaltosamplethepopulationratherthansomehowmeasureeveryoneoreverythinginthepopulation.Thesecondthingtoconsiderishowtomeasuretheparameterofinterest,thethingwewanttoknowaboutthepopulation.Sometimesit’sobvious,likeifyouwanttoknowtheaverageweightofapopulation,youdetermineasampleandthenputeachofthesubjectsonascale.Threeothertechniquesarethefollowing:

• Surveys:Whentheywanttoknowhowpeoplefeel,whattheirpreferencesare,whattheyown,howmuchtheymake,etc.,researchersoftenconstructasurveytoaskthepeopleinthesampleabouttheparameterofinterest.

• ObservationalStudies:Inthistypeofstudy,researchersobservethebehaviorofthe

participants/subjectswithouttryingtoinfluenceitinanywaysotheycanlearnabouttheparameterofinterest.

• Experiments:Inanexperiment,researchersmanipulatethevariablestotrytodetermine

causeandeffect.

1.Imaginethatyouwanttoknowwhetheranewdietplaniseffectiveinhelpingpeoplelose

weight.Youmightchooseanyofthethreemethodstodeterminethis.

• Ifyouusedasurvey,youcouldsimplyaskpeoplethathadtriedthedietplaniftheylostweight.

• Ifyouusedanobservationalstudy,youmightmonitorvolunteersthattrythedietplanandmeasurehowmuchweighttheylost(orgained).

• Ifyouusedanexperiment,youmightrandomlyassignparticipantstotwogroups.Onegroup(thecontrolgroup)eatsastheynormallywouldandtheothergroup(theexperimentalgroup)eatsaccordingtothedietplan.Attheendoftwomonths,thetwogroupsarecomparedtoseetheaverageweightgainorlossineachgroup.

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STATISTICS – 9.6


Basedonthesethreeexamples,

a. Whataresomepossibleadvantagesanddisadvantagesofsurveys?

b. Whataresomepossibleadvantagesanddisadvantagesofobservationalstudies?c. Whataresomepossibleadvantagesanddisadvantageofexperiments?

2.Identifywhichmethodisillustratedbyeachexample:a. Todeterminewhetherdrinkingorangejuicepreventscolds,researchersrandomlyassigned

participantstoagroupthatdranknoorangejuiceoragroupthatdranktwoglassesof orangejuiceaday.Theymeasuredthenumberofcoldsthateachgrouphadoverthecourse oftheyearandcomparedtheresultsofthetwogroups.b. Todeterminewhetherexercisereducesthenumberofheadaches,researchersrandomly

selectedagroupofparticipantsandrecordedthenumberofhourseachparticipant exercisedandthenumberofheadacheseachparticipantexperienced.c. Todeterminetheeffectivenessofanewadvertisingcampaign,arestaurantaskedevery

tenthcustomeriftheyhadseentheadvertisement,andifithadinfluencedtheirdecisionto visittherestaurant.d. Todetermineifanewdrugisaneffectivetreatmentfortheflu,researchersrandomly

selectedtwogroupsofpeoplethathadtheflu.Onegroupwasgivenaplacebo(asugarpill thathasnophysicaleffect)andonegroupwasgiventhenewdrug.Researchersmeasured thenumberofdaysthatparticipantsexperiencedflusymptomsandcomparedthetwo groupstoseeiftheyweredifferent.e. Todetermineifhigherspeedlimitscausemoretrafficfatalities,researcherscomparedthe

numberoftrafficdeathsonrandomlyselectedstretchesofhighwaywith65mphspeed limitstothenumberoftrafficdeathsonanequalnumberofrandomlyselectedstretchesof highwaywith75mphspeedlimits.

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STATISTICS – 9.6


3.Describehowyoumightselectasampleanduseasurveytoinvestigatewhichsoftdrinkpeople

prefer:FizzyPoporKookyKola.

4.Describehowyoumightselectasampleanduseanobservationalstudytoinvestigatewhichsoft

drinkpeopleprefer:FizzyPoporKookyKola.

5.Describehowyoumightselectasampleanduseanexperimenttoinvestigateifconsuminglarge

quantitiesofKookyKolaisassociatedwithhavingheadaches.

6.Describethemethodyouwouldusetodetermineifexcessivetextingisassociatedwithbad

grades.Explainwhyyouchosethatmethodandwhatconclusionscouldbedrawnfromthestudy.

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STATISTICS – 9.6

9.6



READY Topic:Solvingsystemsoflinearequations

1.Solvethesystemofequations

a.Bygraphing b.Bysubstitution

c.Byelimination

SET Topic:Distinguishingbetweensurveys,observationalstudies,andexperiments

Forthefollowingscenarios,identifyeachsituationasasurvey,observationalstudy,oranexperiment.

2.Todetermineifanewpainmedicationiseffective,researchersrandomlyassigntwogroupsof

peopletousethepainmedicationingroup1andaplaceboingroup2.Bothgroupsareaskedtoratetheirpainandtheresultsarecompared.

€

5x − 3y = 32x + y = 10# $ %


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STATISTICS – 9.6

9.6



3.Officialswanttodetermineifraisingthespeedlimitfrom75mphto80mphwillhaveanimpactonsafety.Todeterminethis,theywatchastretchofthehighwaywhenthespeedlimitis75andseehowmanyaccidentsthereare.Thentheyobservethenumberofaccidentsoveraperiodoftimeonthesamestretchofhighwayforaspeedlimitof80mph.Theythencomparethedifference.

4.Todetermineifanewsandwichonthemenuispreferredmorethantheoriginal,themanagerof

therestauranttakesarandomsampleofcustomersthathavetriedbothsandwichesandasksthemwhichsandwichtheylikebest.

5.Anewspaperwantstoknowwhatitscustomersatisfactionis.Itrandomlyselects500customers

andasksthem.Mrs.Goodmorewantstoknowifdoinghomeworkactuallyhelpsstudentsdobetterontheirunitexams.6.DescribehowMrs.Goodmorecouldcarryoutasurveytodetermineifhomeworkactuallyhelps.

Explaintheroleofrandomizationinyourdesign.7.DescribehowMrs.Goodmorecouldcarryoutanobservationalstudytodetermineifhomework

helpstestscores.8.DescribehowMrs.Goodmorecouldcarryoutanexperimenttodetermineifhomeworkhelpstest

scores.Explainhowyouwilluserandomizationinyourdesignandhowyouwilluseacontrol.9.IfMrs.Goodmorewantstodetermineifhomeworkcausestestscorestorise,whichmethod

wouldbebest?Why?

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STATISTICS – 9.6

9.6



GO Topic:Recallingnormalcurves Theaveragerestingheartrateofayoungadultisapproximately70beatsperminutewithastandarddeviationof10beatsperminute.AssumingrestingheartratefollowsaNormalDistribution,answerthefollowingquestions.10.DrawandlabeltheNormalcurvethatdescribesthisdistribution.Besuretolabelthemean,and

themeasurements1,2,and3standarddeviationsoutfromthemean.11.Whatpercentofpeoplehaveaheartratebetween55and80beatsperminute?Labelthese

pointsonyourNormalcurveaboveandshadeintheareathatrepresentsthepercentofpeoplewithheartbeatsbetween55and80beatsperminute.

12.Ifarestingheartrateabove80beatsperminuteisconsideredunhealthy,whatpercentof

peoplehaveanunhealthyheartrate?

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STATISTICS – 9.7


9.7 Slacker's Simulation


Iknowastudentwhoforgotabouttheupcominghistorytestanddidnotstudyatall.Toprotecthisidentity,I’lljustcallhimSlacker.WhenIremindedSlackerthatwehadatestinthenextclass,hesaidthathewasn’tworriedbecausethetesthas10true/falsequestions.Slackersaidthathewouldtotallyguessoneveryquestion,andsincehe’salwayslucky,hethinkshewillgetatleast8outof10.That’swhathedidonthelastquizanditworkedgreat.I’mskeptical,butSlackersaid,“Hey,sometimesyouflipacoinanditseemslikeyoujustkeepgettingheads.Youmayonlyhavea50/50chanceofgettingheads,butyoustillmightgetheadsseveraltimesinarow.Ithinkthisisjustaboutthesamething.Icouldgetlucky.”

1. WhatdoyouthinkofSlacker’sclaim?Isitpossibleforhimtoget8outof10questionsright?Explain.

Ithoughtaboutitforaminuteandsaid,“Slacker,Ithinkyou’reontosomething.I’mnotsurethatyouwillget80%onthetest,butIagreethatthesituationisjustlikeacoinflip.It’seitheronewayortheotherandtheyarebothequallylikelyifyou’rejustguessing.”MyideaistouseacoinfliptosimulatetheT/Ftestsituation.Wecantryitmanytimesandseehowoftenweget8outof10questionsright.I’mgoingtosaythatifthecoinlandsonheads,thenyouguessedtheproblemcorrectly.Ifitlandsontails,thenyougotitwrong.

2. Tryitafewtimesyourself.Tosavealittletime,justflip10coinsatonceandcountupthenumberofheadsforeachtest.

#Correct(Heads) #Incorrect(Tails) %Correct

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STATISTICS – 9.7


Test2

Test3

Test4

Test5

Didyouget8outof10correctinanyofyourtrials?

3. Basedonyourtrials,doyouthinkSlackerhasagoodchanceofgetting80%correct?

4. Collectthedatafromtheentireclassanddisplayitusingtechnology.NowwhatdoyouthinkofSlacker’schancesofgetting80%correct?Explainwhy.

5. Whatwouldyouexpectthegraphtolooklikeifyoucontinuedtocollectsamples?Why?

6. Baseduponyourunderstandingofthisdistribution,whatwouldyouestimatethelikelihoodofSlackergetting80%onthetestwithoutstudying?

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STATISTICS – 9.7

9.7



READY Topic:RecallingrowoperationsinsystemsoflinearequationsForeachcombinationofmatrices,writeoutthejustificationthatallowsyoutowritethenewmatrixbasedonthethreemanipulationswecanperformontheequationsinasystem.

• Replaceanequationinthesystemwithaconstantmultipleofthatequation

• Replaceanequationinthesystemwiththesumordifferenceofthetwoequations

• Replaceanequationwiththesumofthatequationandamultipleoftheother

Forexample,thefollowingmatrixtransformationcanbejustifiedbywriting“Ireplacedthefirstrowofthematrixbymultiplyingthefirstrowby .”

1.!−5 2 9

10 4 8* ⟹ !−15 6 2730 12 24*

2.!8 12 51 −6 2* ⟹ !10 0 9

1 −6 2*

4.!1 2 51 4 10* ⟹ !1 2 5

0 2 5*

5.!16 22 1910 44 38* ⟹ !16 22 19

11 0 0 *

SET Topic:Usingasimulation

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STATISTICS – 9.7

9.7



In1963,NBCstartedtohostagamecalledLet’sMakeaDeal!Contestantsweregiventhreedoorstochoosefrom.Behindonedoorwasaprize.Afterselectingonedoor,thecontestantwasshownwhatwasbehindoneofthedoorstheydidnotselect.Thecontestantisthenaskediftheywouldliketostickwiththedoortheyfirstselected,orswitchtotheremainingone.

6.Whichstrategydoyouthinkwouldresultinthebestchanceofselectingthewinningdoor?Shouldthecontestantswitchdoors,orstickwiththefirstonetheychose?

Gotothefollowingwebsite:http://nlvm.usu.edu/en/nav/category_g_3_t_2.html

Selecttheappletstickorswitch.

7.Playthegame20timesusingthestickmethodand20timesusingtheswitchmethod.Recordyourwinsandlossesinthetablebelow:

Stick Switch Total

Win

Lose

Total

8.Basedonthesimulation,whatis/(1233234|67289) =

9.Basedonthesimulation,whatis/(1233234|61278ℎ) =

10.Clickonthemultiplegamestab.Simulate100gamesforeachstrategy.Whatistheprobabilityofwinningusingeachmethod?

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STATISTICS – 9.7

9.7



GO Topic:Reviewingprobability 7.Foryourtwo-waytableinproblem3,createaVenndiagramandatreediagrambelow.

8./(1233234) = 14./(1233234 ∩ 67289234) =

9./(1233234 ∪ 67289234) = 16./(?@@6234|67289234) =

10./(1233234@B?@@6234) =

11.Aretheeventswinningandstickingindependentofeachother?Justifyyouranswerusingprobabilities.

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