statistics...secondary math iii // module 9 statistics mathematics vision project licensed under the...

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

SECONDARY

MATH THREE

An Integrated Approach

Standard Teacher Notes

SECONDARY MATH III // 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?

4


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?

6


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?

7


STATISTICS – 9.1


9.1 What Is Normal? – Teacher Notes A Develop Understanding Task

Purpose:Thepurposeofthistaskistorecognizeanormaldistribution.Inthetask,studentsare

givenmultipleexamplesoffrequencydistributions,somenormalandsomethatarenot.Students

areaskedtocomparetheexamplesandidentifyfeaturesofanormaldistribution.Thepurposeisto

understandthattheshapeofanormaldistributionissymmetric,single-peaked,andbellshaped.

Studentswillrecognizetheeffectonthedistributioncurveofchangingthemeanandthestandard

deviation.Theywillalsolearnthatthemean,median,andmodeareequalinanormaldistribution.

Studentswillalsoseethe68–95–99.7ruleillustrated,andthenwillusetheruleinalatertask.

CoreStandardsFocus:

S.ID.4:Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandto

estimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureis

notappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormal

curve.

RelatedStandards:S.ID.1

StandardsforMathematicalPractice:

SMP6–Attendtoprecision

SMP7–Lookforandmakeuseofstructure

Vocabulary:pointofinflection,distributioncurve,normaldistribution

TheTeachingCycle:

Launch(WholeClass):Beginthelessonbyactivatingstudents’backgroundknowledgebyasking

themtocompletethefollowingsentenceframes:

Threemeasureofthecenterofadistributionaremean,median,andmode.Themedianis

____________________________andisfoundby___________________________.Themodeis__________________and


STATISTICS – 9.1


isfoundby_____________________________.Themeanis_______________________andisfoundby

_______________________.Havestudentswritetheirresponsesandthensharetheirideaswithapartner.

Haveabriefdiscussionoftheseterms.

Displayoneofthedistributionsfromthetaskanduseittodiscussthesefeaturesofadistribution

includingshape,spread,andcenter.Describeforstudentsthedifferencebetweenafrequency

distributionandadistributioncurve.Studentsshouldhaveexperiencewithfrequency

distributionsbutmaynothavediscussedadistributioncurve.Explainthatadistributioncurve

“smoothsoutthebumps”inafrequencydistributionwithatheoreticalcurvethatshowshowoften

anexperimentwillproduceaparticularresult.Manyoftheupcomingexampleshavedistribution

curvesalongwithotherrepresentations.Tellstudentsthattheywillseeseveralpairsofimages

thathavebeengroupedtohighlightparticularfeatures.Allofthedistributionsontheleftsideare

normaldistributions,allofthedistributionsontherightsidearenot.Theirjobistocomparethe

twoimagesinthepairsothattheycanarriveatadescriptionforanormaldistribution.

Explore(SmallGroup):Monitorstudentsastheyareworkingtoseethattheyarecomparing

featuresandidentifyingdifferencesbetweenthetwodistributions.Astheybegintoworkonthe

secondpartofthetask,besuretheyareconnectingtotheirpreviousunderstandingofthemean

andthestandarddeviationtomakesenseoftheeffectofthesefeaturesonthedistribution.

Discuss(WholeClass):Gothrougheachoftheexamplesgivenin1-7,highlightingthefeaturesofa

normaldistributionineachpair.Thefollowinglistsuggestsfeaturesthatcouldbenoticedwith

eachpair.Bytheendofthediscussion,allofthesefeaturesshouldbeidentified.Studentsprobably

won’tusethecorrectvocabularyastheydescribethefeaturesbutshouldbesupportedin

developingandusingthevocabularyasthemoduleproceeds.

1. Anormaldistributionissymmetric.2. Themean,median,andmodeareequalinanormaldistribution.3. Thefrequencycurveofanormaldistributionissymmetric.4. Anormalcurvehaspointsofinflectionat±1standarddeviationfromthemean.5. Anormaldistributionhasasinglemode.


STATISTICS – 9.1


6. 68%ofthedistributionwillbe±1standarddeviationsfromthemean.95%ofthe distributionwillbewithin2standarddeviationsfromthemean.99.7%ofthedistribution willbewithin3standarddeviationsfromthemean.

Askstudentsforsomeexamplesofmeasurementdatathatmightbedistributednormally.Some

possibilities:

• Theheightofadultmalesoradultfemales,butnottheheightofalladultsbecausethat distributionwouldprobablyhavetwomodes—oneforwomenandoneformen.

• Theaveragenumberofdaysthatittakesforaparticulartypeofseedtosprout.• Theaverageamountofsleeppernightforadults.

Becarefultodiscusseachexampletoconsiderwhetherornotthedistributionwouldbesymmetric,

single-peaked,andbellshaped.

Movethediscussiontoquestions10and11.Beginbyaskingstudentstoexplainwhatthestandard

deviationdescribes.Thenaskwhateffecttheysawbychangingthestandarddeviationandto

explainwhythatwouldhappen.Repeatwiththeprocesswhendiscussingthemean.

Completethediscussionbyansweringeachofthetrue/falsequestionsandaskingstudentsto

explaintheirreasoning.Usethisasanopportunitytoreinforcevocabulary.Answerswithsample

studentexplanations:

12a.True–Themeantellswherethepeakofthedistributionisandthestandarddeviation

determineshowspreadoutthedistributionis.

12b.True–That’swhynormaldistributionsturnouttobesymmetricwithasinglepeak.

12c.False–Abimodaldistributionwillhavetwopeaks.That’snotnormal.

12d.False–68%ofthedistributioniswithin1standarddeviationofthemeaninanormal

distribution.

AlignedReady,Set,Go:Statistics9.1


STATISTICS – 9.1


Normal

Distribution

Definition Features

Examples Non-Examples


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

8


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.IftwoNormaldistributionshavethesamestandarddeviationof4.9butdifferentmeansof3

and6,howwillthetwoNormalcurveslookinrelationtoeachother?DrawasketchofeachNormalcurvebelow.

14.IftwoNormaldistributionshavethesamemeanof3butstandarddeviationsof1and4,howwilltheylookinrelationtoeachother?DrawasketchofeachNormalcurvebelow.

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STATISTICS – 9.1


9.1


15.TheNormalCurvegivenbelowhasbeenlabeledoutthreestandarddeviations.Estimatewhatonestandarddeviationisforthiscurve.

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) = kX V −

lX 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 Just ACT Normal – Teacher Notes A Solidify Understanding Task

Purpose:

Thepurposeofthistaskisforstudentstousetheirunderstandingofthenormaldistributionto

makeestimatesoffrequenciesusingarea.Inthetask,theywillusetheirknowledgethat1,2,and3

standarddeviationsreferto68%,95%,or99.7%ofthedistribution,respectively,todraw

conclusions.Thistaskdoesnotrequiretheuseoftechnologyortables.

CoreStandardsFocus:




curve.



SMP3–Constructviableargumentsandcritiquethereasoningofothers


TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbytellingstudentsthatscoresonstandardizedtestsareusuallydistributed

normally.Askwhyitmakessensethatstandardizedtestscoresmightbedistributednormally.

Someresponsesmaybe:moststudentswillscorearoundthemean(makingthatthepeakofthe

distribution),aboutthesamenumberofstudentswillscoreaboveandbelowthemean(makingthe

distributionsymmetric),thetestsaredesignedsothatveryfewstudentsgeteverythingrightor

everythingwrong,whichstretchesoutthedistributionandmakesthe68%,95%,99.7%ruleseem

reasonableinthiscontext.Usethisdiscussiontodrawoutsomeofthefeaturesofthenormal


STATISTICS – 9.2


distribution.Havestudentssketchthedistributioncurve(question#1)andthenmodelhowto

checkitusingtechnology(question#2).Besuretodiscusswithstudentsthatthecenterofthe

distributionshouldbeidentifiedonthegraphwiththemeanscore,thatthepointsofinflectionare

±1standarddeviationawayfromthemean,andthat2and3standarddeviationsareappropriately

marked.

Explore(SmallGroup):

Monitorstudentsastheyworkontheremainderofthetask.Noticethatthestandarddeviationis

5.2(notexactly5),sostudentsareestimatingusingthe68%,95%,99.7%rule.Encouragestudents

tolabeltheirdistributioncurvesandusethemtofindthepercentages.Whenstudentsareworking

on#4,listenforstudentsmakinguseoftheruletoestimatethepercentageofkernelsthatwillbe

popped.Don’tletthemgettoocaughtupinhowlongitmighttakeforthepoppedkernelstostart

toburn.Theideaistothinkthatyouwouldwantmostkernelstobepoppedbutcan’twaitforallof

them.

Discuss(WholeClass):Beginthediscussionwith#3.Askapreviously-selectedstudentto

explaineachoftheitemsinpartsa,b,andc,includinghowhe/sheusedthedistributioncurveand

the68%,95%,99.7%ruletofindtheapproximatevalues.Movethediscussiontoquestion#4.

StudentsshouldbeabletoarguethatCalvin’sscoreisbetween2and3standarddeviationsaway

fromthemean,soitisprobablynotinthetop1%.Askstudentstoestimateaboutwhatpercentof

studentstheythinkscoredbelowCalvin,basedonthedistributioncurve.Discusswhatwouldbea

reasonableanswerandthenshowhowitcanbecalculatedusingtechnology.

Discussquestion#5,beginningwiththedistributioncurve.Expectstudentstohavedifferent

answersdependinguponwhichaspectsofthenormalcurvehasbeentheirfocus.Engagestudents

inadebateabouttheviabilityofvariousanswers,pressingstudentstojustifytheiranswersusing

featuresofnormaldistributions.



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)=>

15


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.

You:That’sgreat.I’mgoingbacktotheofficetodecidewhichstudentisadmitted.

8. ComparethescoresofStudentAandStudentB.Explainwhichstudenthasthehighestmathematicstestscoreandwhy.

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STATISTICS – 9.3


9.2 Y B Normal? – Teacher Notes A Solidify Understanding Task

Purpose:

Thepurposeofthistaskistobuilduponpreviousworkwithnormaldistributionstounderstand

thestandardnormaldistribution.Inthistask,studentswillbeintroducedtothez-scoreformula

andusingz-scoretablestofindthepercentofthedistributionthatwereaboveorbelowa

particularpointonthedistribution.Studentswillusez-scorestocomparescoresontheACTand

SATtests,whicharebothnormallydistributedwithdifferentmeansandstandarddeviations.

CoreStandardsFocus:




curve.



SMP5–Useappropriatetoolsstrategically

Vocabulary:z-score,standardnormaldistribution

TheTeachingCycle:

Note:Studentswillneedsomeexplanationofhowtouseaz-scoretabletocompletethetask.

Becauseoftheamountoftextandtheproceduresinthetask,thistaskrequiresmoreteacher

guidancethanusual.

Launch(WholeClass):Beginthediscussionbyexplainingthesituationandensuringthatstudents

understandthecontextofACTandSATtestsbeingusedforcollegeentrance.Askstudentstowork


STATISTICS – 9.3


individuallytodecidewhichscorethattheybelieveisbetterandthendiscussargumentsfromthe

class.Besuretobringouttheideathatbothofthescoresareverygood(morethan2standard

deviationsabovethemean),buttheyareontotallydifferentscales.Afterthisbriefdiscussion,read

throughthescenariowiththeclass(uptoproblem#2)andspendsometimebeingsurethey

understandwhataz-scoreisandhowtouseatabletolookupareas(thepercentofthedistribution

belowthatpoint.Introducetheterm“standardnormaldistribution”forthedistributioncurve.

Modeldrawingaverticallineonthedistributioncurvethroughthepointunderconsiderationand

thenlookingupthenumberinthetable.Besurethatstudentsseetherelationshipbetweenthe

areaunderthecurveandthenumberinthetable.

Explore(SmallGroup):Letstudentsworkontherestoftheproblems.Astheyareworking,be

surethattheyareusingthetablecorrectlyandinterpretingtheareasonthedistributioncurve.As

theymoveon,checkonhowtheyareusingtheformulatoconverttoz-scores.Identifystudentsfor

thediscussionthatarereadytousethez-scorestomakeanargumentaboutwhichstudentshould

beadmitted.

Discuss(WholeClass):Beginthediscussionwithquestions3,4,and5,whicharedirect

applicationsofusingthez-scoretable.Foreachquestion,havethestudentsharehowtousethe

distributioncurvetovisualizeandinterprettheareasthattheyarelookingupinthetable.Move

thediscussionto#8.Haveastudentshowhowtoconverteachofthestudents’scorestoz-scores.

Markthescoresonthestandardnormaldistribution.Askastudenttoshowhowtheyusedthe

scorestofindthepercentageoftest-takersthatscoredaboveandbelowStudentAandStudentB.



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?


19


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?

20


STATISTICS – 9.3


9.3


GO Topic:SketchingPolynomials

Withoutusingtechnology,sketchthegraphofthepolynomialfunctionwiththegivencharacteristics.Iftheequationisnotgiven,writeitinstandardform.

12.Degree4

Roots:-1multiplicity2,5,-2

"–intercept:20

Equation:

13.$(&) = (& + 2)(& − 3)-

21


STATISTICS – 9.3


9.3


14.1(&) = −&(& − 3)2(& + 5)(& − 5)

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

22


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.


24


STATISTICS – 9.4


9.4 Wow, That's Weird! – Teacher Notes A Practice Understanding Task

Purpose:

Thepurposeofthistaskistopracticeusingthefeaturesofanormaldistributiontodecidehow

unusualaparticulareventis.Studentswillrankordertheseeventsfrommostunusualtoleast

unusualusingthe68-95-99.7rule,z-scores,andlogic.Technologyand/orz-scoretableswillbe

necessarytodistinguishbetweensomeoftheevents.

CoreStandardsFocus:




curve.



SMP5–Useappropriatetoolsstrategically

SMP6–Attendtoprecision

TheTeachingCycle:

Launch(WholeClass):Introducethetaskbytellingstudentsthatwhensomethinghappensthatis

outoftheordinary,weoftentendtothinkthatitisstrangerthanitactuallyis.Tellthemthatthis

taskisaboutofanumberof“strange”occurrences.Theneedtousetheirknowledgeaboutnormal

distributionstorankeachoftheseoccurrencesfrommostunusualtoleastunusual.Tellthemthat

theymaywanttousetechnologyforsomeoftheproblems,butitisn’tnecessaryforallofthem.

Explore(SmallGroup):Monitorstudentsastheyareworkingtohearhowtheyareusingthe

informationgiventoranktheevents.Itispossible,butnotnecessary,tocomeupwiththeranking


STATISTICS – 9.4


byconvertingtoz-scoresandusingatableorcalculatortofindthepercentofthedistributiontothe

leftofz.Encouragestudentstodoasmanyastheycanwithoutresortingtoaprocedure.Listenfor

strategiessuchascategorizingbasedonthenumberofstandarddeviationsawayfromthemean,so

thatthiscanbeusedinthediscussion.Theremaybesomediscussionabouttheideathatsomeof

the“weird”thingsareonthefarrightofthedistributionandothersareonthefarleft.Thismaybe

agreatwaytostartthediscussion.

Discuss(WholeClass):Startthediscussionbyaskingstudentstosharetheirinitialstrategiesfor

ranking.Sortthroughtheideathataneventthatisthreestandarddeviationstotheleftofthemean

isjustasunusualasaneventthatisthreestandarddeviationstotheright.Askastudenttoshare

thatusedthestrategyofgroupingtheeventsthatarewithin1standarddeviation,between1and2

standarddeviations,between2and3,andgreaterthan3.Then,askstudentshowtheycanrank

withinthosecategories.Finally,havestudentssharehowtheyusedz-scorestodeterminetheones

thatweretooclosetotellotherwise.



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.


25


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

26


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

28


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.

29


STATISTICS – 9.5


N. O.

4. Whatmightbesomeoftheadvantagesanddisadvantagesofeachtype?

5. Apersonyouknowownsasmalltheaterthatshowslocaldramaticproductions.Shewantstoknowtheaverageageofthepeoplethatbuyticketstotheseetheshowssothatshecanbetterselectwhichplaystostage.Explaintotheownerwhyselectingthefirst20peoplethatarrivefortheshowmaynotbearepresentativesample.

6.Describeaprocessforselectingarepresentativesampleofthetheaterpatrons.

30


STATISTICS – 9.5


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

Purpose:

Thepurposeofthistaskistointroducestudentstotheideaofusingarandomsamplefroma

populationtoinvestigateandmakeaninferenceaboutaparameterofinterest.Inthetask,students

sortvarioussamplingmethodsandconsiderstrengthsandweaknessesofeachtype.Theideathat

asamplemustbetrulyrandomtoberepresentativeofthepopulationisdeveloped.

CoreStandardsFocus:

Understandandevaluaterandomprocessesunderlyingstatisticalexperiments.

S.IC.1Understandthatstatisticsallowsinferencestobemadeaboutpopulationparametersbased

onarandomsamplefromthatpopulation.


SMP3–Constructviableargumentsandcritiquethereasoningofothers


Vocabulary:population,sample,parameter,inference,simplerandomsample,systematic

sample,clustersample,stratifiedrandomsample,conveniencesample,volunteersample

TheTeachingCycle:

Launch(WholeClass):

Beginthelessonbyexplainingtostudentsthatalotofstatisticsareabouttryingtodrawvalid

conclusionsaboutsometopicofinterest.Usually,thepopulationistoobigtoincludeevery

memberofthepopulationintheinvestigation,whateverthedesign.So,investigatorsusuallytakea

samplethattheyhoperepresentstheentirepopulation.Workthroughquestions1-3withtheclass


STATISTICS – 9.5


tobesurethattheyunderstandthattherecanbemanywaystoselectasample,withprosandcons

toeachmethod.Tellstudentsthatforthenextpartofthetask,theywillbegivenexamplesof

severaldifferentsamplemethods.Theywillneedtopayattentiontohowtheparticipantsinthe

samplearechosenandtosortthesemethodsintocategories.Itmaybeusefultocutupthe

examplessothattheycaneasilybemovedaroundasstudentssortandre-sort.


Monitorstudentsastheyareworking,makingsurethattheyareattendingtosomeofthe

distinguishingdetailsintheexamples.Thereareactuallysixcategories,althoughstudentsmay

justifysortinginvariousways.Watchforstudentsthathaveidentifiedtwobasiccategories–

randomandnotrandom,becausethisisafundamentalidea.Therearefourrandomtypeshere–

simple,systematic,cluster,andstratified.Therearetwonon-randomtypes–convenienceand

volunteer.Encouragestudentstousethediagramstohelpsortthetwomajorgroupsintothe

smallercategories.

Discuss(WholeClass):

Beginthediscussionwithastudentthathasthetwobasiccategories,evenifalltheirsortingisnot

correct.Asktheclasshowtheymightnamethetwocategoriesbasedonthefeaturesofeach.

Recordthefeaturesofthetwocategories.Thenaskstudentstopresentthatcansuccessivelybreak

downthecategories.Continuetonotethefeaturesofthecategoriesuntilallsixhavebeen

identified.Then,givestudentsthenamesforthecategoriesanddiscusssomeofthemeritsofeach.

Askwhatmightbetheproblemsassociatedwithconvenienceandvolunteersamples,emphasizing

thatneitherofthesetwotypesareactuallystrictlyrandom.

Thesamplingmethodsandtheirexamplesare:

Simplerandomsample(A,G,L)

Systematicrandomsample(B,K,M)

Stratifiedrandomsample(C,I,O)

Clusterrandomsample(D,H,N)

Conveniencesample(E,J)

Volunteersample(F)


STATISTICS – 9.5


Turnthediscussiontoquestion#5.Discusswhythisisnotarandomsampleandhowusingthis

methodmayskewtheresults.Askstudentswhichoftherandomsamplingmethodsmightbeused

inthiscaseandhowparticipantscouldbeselected.



STATISTICS – 9.5


SamplingMethods

SimpleRandomSample

Description:Examples:

SystematicRandomSample



STATISTICS – 9.5


StratifiedRandomSample


ClusterRandomSample



STATISTICS – 9.5


ConvenienceSample


VolunteerSample



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.



31


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.


32


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.

33


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.

35


STATISTICS – 9.6


3.Describehowyoumightselectasampleanduseasurveytoinvestigatewhichsoftdrinkpeople

prefer:FizzyPoporKookyKola.

4.Describehowyoumightselectasampleanduseanobservationalstudytoinvestigatewhichsoft

drinkpeopleprefer:FizzyPoporKookyKola.

5.Describehowyoumightselectasampleanduseanexperimenttoinvestigateifconsuminglarge

quantitiesofKookyKolaisassociatedwithhavingheadaches.

6.Describethemethodyouwouldusetodetermineifexcessivetextingisassociatedwithbad

grades.Explainwhyyouchosethatmethodandwhatconclusionscouldbedrawnfromthestudy.

36


STATISTICS – 9.6


9.6 Let's Investigate – Teacher Notes A SolidifyUnderstanding Task

Purpose:

Thepurposeofthetaskisforstudentstolearntoidentifythreemethodsofinvestigatinga

parameterofinterest:survey,observationalstudy,andexperiment.Thetaskbeginswithan

explanationofthethreemethodsincludinganexampleofeach.Studentsarethenaskedtoidentify

whichmethodisillustratedbysomeotherexamplesandtoconsidertheadvantagesand

disadvantagesofeachmethod.Finally,studentsaregivenasituationandaskedtodesignan

investigationincludingselectingasamplingmethodandamethodofinvestigatingtheparameterof

interest.

CoreStandardsFocus:

S.IC.3:Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,and

observationalstudies;explainhowrandomizationrelatestoeach.

RelatedStandards:S.IC.1



Vocabulary:survey,observationalstudy,experiment,controlgroup,experimentalgroup

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbyreadingthroughthedescriptionofthethreeinvestigationmethodswiththeclass

anddiscussingtheexamples.Givesstudentsachancetoaskquestionsandclarifythedifferences

betweeneachofthethreetypes.


STATISTICS – 9.6


Givestudentsafewminutestothinkabouttheadvantagesanddisadvantagesofeachtypeandthen

discusstheiranswers.Duringthisdiscussion,includetheideathatthevalidityoftheresultswill

partiallydependonhowthesamplewasselectedaswellasthemethodofthestudy.Forinstance,

surveyshavetheadvantageofbeingeasytogaugeopinions.Adisadvantagemaybethedifficultyin

writingquestionsthataccuratelymeasuretheparameterofinterest.Nomatterhowgoodthe

surveyquestionsare,thesurveymaynotyieldusefulresultsifitisnotbasedonarepresentative

sampleofthepopulation.Afterdiscussingtheideasofeachmethod,thenaskstudentstocomplete

thetask.


Listentostudentsastheyareworkingtoseethattheyarenoticingthedistinguishing

characteristicsofeachmethod.Studentsoftenhavedifficultywiththedifferencebetweenan

observationalstudyandanexperiment.Bepreparedtohelpthemunderstandthatitisnotan

experimentaldesignifthereisnotacontrolgroupandanexperimental(treatment)group.

Trytoidentifystudentstopresenteachofthethreemethodsasastudydesignfor#6.


Beginthediscussionbyquicklyidentifyingthemethodforeachoftheexamplesin#2.Foreach

example,clarifythedistinguishingfeaturesofthatmethod.

Turnthediscussiontoquestion#3.Askastudenttopresenthowtheymightselectasampleand

designasurvey.Asktheclasswhatquestionstheymightasktoobtainanunbiasedresult.

Haveastudentpresenttheirdesignforquestion#4.Askstudentshowtheycanensurethatthe

studyparticipants’preferencesarenotinfluencedbytheobserverorotherfactors.

Askstudentstopresenttheirexperimentaldesignforquestion#5.Howwillstudentsselectthe

twogroupssotheyarecomparable?Whatwillbethetreatment?Howwilltheymaintainthe

controlgroup?


STATISTICS – 9.6


Finally,discussquestion#6.Ifyouhaveidentifiedstudentstopresenteachofthethreemethods,

thenaskthemtosharetheirthinkingandhavetheclasscritiquetheirdesignincludingthe

samplingmethod.Ifthereisamethodthatwasnotusedbyanystudentintheclass,facilitatea

discussionwiththewholeclasstodesignastudytogetherusingthatmethod.



STATISTICS – 9.6


StudyDesignsSurvey

Description:

Advantages:

Disadvantages:

Observation

Description:

Advantages:

Disadvantages:

Experiment

Description:

Advantages:

Disadvantages:


STATISTICS – 9.6


9.6


READY Topic:Findingprobabilitiesfromatwo-waytable

Thefollowingdatarepresentsarandomsampleofboysandgirlsandhowmanyprefercatsordogs.Usetheinformationtoanswerthequestionsbelow. Cats Dogs TotalBoys 32 68 100Girls 41 11 52Total 73 79 152

1.!(#) = 2.!(&) = 3.!(') = 4.!(() =5.!('|&) = 6.!('+,#) = 7.!((|#) = 8.!(# ∩ () =9.Ifthisisarandomsamplefromaschool,whattotalpercentofboysinthisschooldoyouthinkwouldpreferdogs?

10.Whatpercentofstudentsattheschoolwouldprefercats?11.Ifyousampledadifferent152students,wouldyougetthesamepercentages?Explain.12.Whatwouldhappentoyourpercentagesifyouusedalargersamplesize?

SET Topic:Distinguishingbetweensurveys,observationalstudies,andexperiments

Forthefollowingscenarios,identifyeachsituationasasurvey,observationalstudy,oranexperiment.

13.Todetermineifanewpainmedicationiseffective,researchersrandomlyassigntwogroupsof

peopletousethepainmedicationingroup1andaplaceboingroup2.Bothgroupsareaskedtoratetheirpainandtheresultsarecompared.


37


STATISTICS – 9.6


9.6


14.Officialswanttodetermineifraisingthespeedlimitfrom75mphto80mphwillhaveanimpactonsafety.Todeterminethis,theywatchastretchofthehighwaywhenthespeedlimitis75andseehowmanyaccidentsthereare.Thentheyobservethenumberofaccidentsoveraperiodoftimeonthesamestretchofhighwayforaspeedlimitof80mph.Theythencomparethedifference.

15.Todetermineifanewsandwichonthemenuispreferredmorethantheoriginal,themanagerof

therestauranttakesarandomsampleofcustomersthathavetriedbothsandwichesandasksthemwhichsandwichtheylikebest.

16.Anewspaperwantstoknowwhatitscustomersatisfactionis.Itrandomlyselects500

customersandasksthem.Mrs.Goodmorewantstoknowifdoinghomeworkactuallyhelpsstudentsdobetterontheirunitexams.17.DescribehowMrs.Goodmorecouldcarryoutasurveytodetermineifhomeworkactuallyhelps.

Explaintheroleofrandomizationinyourdesign.18.DescribehowMrs.Goodmorecouldcarryoutanobservationalstudytodetermineifhomework

helpstestscores.19.DescribehowMrs.Goodmorecouldcarryoutanexperimenttodetermineifhomeworkhelps

testscores.Explainhowyouwilluserandomizationinyourdesignandhowyouwilluseacontrol.

20.IfMrs.Goodmorewantstodetermineifhomeworkcausestestscorestorise,whichmethod

wouldbebest?Why?

38


STATISTICS – 9.6


9.6


GO Topic:Recallingnormalcurves

Theaveragerestingheartrateofayoungadultisapproximately70beatsperminutewithastandarddeviationof10beatsperminute.AssumingrestingheartratefollowsaNormalDistribution,answerthefollowingquestions.21.DrawandlabeltheNormalcurvethatdescribesthisdistribution.Besuretolabelthemean,and

themeasurements1,2,and3standarddeviationsoutfromthemean.22.Whatpercentofpeoplehaveaheartratebetween55and80beatsperminute?Labelthese

pointsonyourNormalcurveaboveandshadeintheareathatrepresentsthepercentofpeoplewithheartbeatsbetween55and80beatsperminute.

23.Ifarestingheartrateabove80beatsperminuteisconsideredunhealthy,whatpercentof

peoplehaveanunhealthyheartrate?

39


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

Test1

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40


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?

41


STATISTICS – 9.7


9.7 Slacker's Simulation – Teacher Notes A SolidifyUnderstanding Task

Purpose:Thepurposeofthistaskisforstudentstodecideifaparticularresultfromamodelis

likelyusingasimulation.Inthistask,studentssimulatetheresultsofatrue/falsetestusingacoin

flip.Theresultsarecollectedfortheentireclassanddiscussedtodeveloptheideathatthe

particularresultispossible,butunlikely.

CoreStandardsFocus:

Understandandevaluaterandomprocessesunderlyingstatisticalexperiments.

S.IC.2Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,

e.g.,usingsimulation.Forexample,amodelsaysaspinningcoinfallsheadsupwithprobability0.5.

Wouldaresultoffivetailsinarowcauseyoutoquestionthemodel?



Vocabulary:simulation

TheTeachingCycle:

Launch(WholeClass):

BeginthetaskbyreadingthescenarioandaskingstudentsiftheythinkitispossibleforSlackerto

get80%.Somestudentswillarguethathewouldjustget5/10right,sincetheoddsofgettinga

questionrightis50%.Askthem,iftheoddsare50%,andSlackergets5questionsinarowcorrect,

doesthismakeitlesslikelythathewillgetthesixthquestioncorrect?Reinforcetheideathateach

questionisanindependentevent.


Givestudentstencoinsorchipstotossforthesimulation(oruseatechnologytosimulatetossing

tencoins)andaskthemtocompletequestions2and3.Bepreparedwithawaytocollectand

displaythedatafromtheentireclass.


STATISTICS – 9.7



Basedonthedatacollectedfromtheclass,askstudentsiftheybelievethatSlackerhasagood

chanceofgetting8/10correct.Thedistributionmaynotlookparticularlynormal,butconsider

howtousethedistributionfromthesimulationtothinkaboutSlacker’sidea.Thendiscuss

question5.Whymightweexpectthatthedistributionwouldbesymmetrical?Whymightwe

expectthedistributiontocenterat5/10?Whatisthestandarddeviationofthedistribution?

Whereis8/10onthedistribution?Aboutwhatpercentofthedistributionisbelow8/10?Make

thepointthatthenormaldistributionthatoccursforalargenumberofsamplesinthiscase,helps

ustounderstandwhetherornotaparticularsamplemightrepresenttheentirepopulation.



STATISTICS – 9.7


9.7


READY Topic:Reviewinghistograms

1.Takeacoinandflipit5times.Recordthenumberoftimesthecoinlandswithheadsup.Repeatthisprocess20times,eitherbyhandorbysimulationusingtechnology,eachtimerecordingyourresultsinthetablebelow.(Everytimeyouperformthesimulation,countthenumberofheadsyouhaveandrecordtheresultintheTallycolumnbelow.Forexample,ifyouflipthecoin5timesandget3heads,putatallymarkbythe3headsor60%row.)http://www.rossmanchance.com/applets/CoinTossing/CoinToss.html

2.Createahistogramofyourresults.Describetheshapeofthehistogram(Shape,Center,Spread)

#Heads %Heads Tally0 0% 1 20% 2 40% 3 60% 4 80% 5 100%


42


STATISTICS – 9.7


9.7


3.Flipacoin20times.Recordthenumberoftimesheadslandssideup.Repeatthisprocess20timeseitherbyhandorbysimulationusingtechnology.http://www.rossmanchance.com/applets/CoinTossing/CoinToss.html

Recordyourresultsinthetablebelow.

4.Createahistogramofyourresultsbelow.Describetheshapeofthehistogram(Shape,Center,Spread)

#Heads

%Heads

Frequency

#Heads

%Heads

Frequency

0 0% 11 55% 1 5% 12 60% 2 10% 13 65% 3 15% 14 70% 4 20% 15 75% 5 25% 16 80% 6 30% 17 85% 7 35% 18 90% 8 40% 19 95% 9 45% 20 100% 10 50%

43


STATISTICS – 9.7


9.7


5.Comparetheshapecenterandspreadofeachdistribution.Whatdoyounotice?

6.Ifyourepeatedthisprocesswith500flipsinsteadof5or20,predictwhatwouldhappentotheshape,spread,andcenterofthenewhistogram.

SET Topic:Usingasimulation

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

7.Whichstrategydoyouthinkwouldresultinthebestchanceofselectingthewinningdoor?Shouldthecontestantswitchdoors,orstickwiththefirstonetheychose?

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

Selecttheappletstickorswitch.

8.Playthegame20timesusingthestickmethodand20timesusingtheswitchmethod.Recordyourwinsandlossesinthetablebelow:

Stick Switch Total

Win

Lose

Total

9.Basedonthesimulation,whatis!(#$%%$%&|()$*+) =

10.Basedonthesimulation,whatis!(#$%%$%&|(#$)*ℎ) =

44


STATISTICS – 9.7


9.7


11.Clickonthemultiplegamestab.Simulate100gamesforeachstrategy.Whatistheprobabilityofwinningusingeachmethod?

GO Topic:Reviewingprobability

12.Foryourtwo-waytableinproblem8,createaVenndiagramandatreediagrambelow.

13.!(#$%%$%&) = 14.!(#$%%$%& ∩ ()$*+$%&) =

15.!(#$%%$%& ∪ ()$*+$%&) = 16.!(122($%&|()$*+$%&) =

17.!(#$%%$%&24122($%&) =

18.Aretheeventswinningandstickingindependentofeachother?Justifyyouranswerusingprobabilities.

45

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