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
Standard Teacher Notes
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)
Ready, Set, Go Homework: Statistics 9.2
9.3 Y B Normal? – A Solidify Understanding Task
Introducing z scores to compare normal distributions (S.ID.4)
Ready, Set, Go Homework: Statistics 9.3
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)
Ready, Set, Go Homework: Statistics 9.4
9.5 Would You Like to Try a Sample – A Develop Understanding Task
Understanding and identifying different methods of sampling (S.IC.1)
Ready, Set, Go Homework: Statistics 9.5
9.6 Let’s Investigate – A Solidify Understanding Task
Identifying the difference between survey, observational studies, and experiments (S.IC.1,
S.IC.2)
Ready, Set, Go Homework: Statistics 9.6
ALGEBRA II // MODULE 9
STATISTICS
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9.7 Slacker’s Simulation – A Solidify Understanding Task
Using simulation to estimate the likelihood of an event (S.IC.2, S.IC.3)
Ready, Set, Go Homework: Statistics 9.7
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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?
CC
BY
Jam
es H
.
http
s://f
lic.k
r/p/
e65L
Ua
1
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.Thisisnormal: Thisisnot:
Whatdifferencesdoyouseebetweenthesedistributions?
3.Thisisapproximatelynormal: Thisisnot:
Whatdifferencesdoyouseebetweenthesedistributions?
2
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4.Thisisnormal: Thisisnot:
Whatdifferencesdoyouseebetweenthesedistributions?
5.Thisisapproximatelynormal: Thisisnot:
Whatdifferencesdoyouseebetweenthesedistributions?
-
3
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6.Thisisapproximatelynormal: Thisisnot:
Whatdifferencesdoyouseebetweenthesedistributions?
7.Thisisnormal: Thisisnot:
Whatdifferencesdoyouseebetweenthese
distributions?
_________________________________________________________________________________________________________________
8.Basedupontheexamplesyouhaveseenin#1-7,whatarethefeaturesofanormaldistribution?
4
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9. a.Whatdoesthestandarddeviationtellusaboutadistribution?
b.Eachofthedistributionsshownbelowarenormaldistributionswiththesamemeanbut
adifferentstandarddeviation.
Mean=3,StandardDeviation=0.5 Mean=3,StandardDeviation=1
Mean=3,StandardDeviation=0.25
Howdoeschangingthestandarddeviationaffectanormalcurve?Whydoesithavethiseffect?
5
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
10. a.Whatdoesthemeantellusaboutadistribution?
b.Eachofthedistributionsshownbelowarenormaldistributionswiththesamestandard
deviationbutadifferentmean.
Mean=1,StandardDeviation=0.25 Mean=2,StandardDeviation=0.25
Mean=3,StandardDeviation=0.25
Howdoeschangingthemeanaffectanormalcurve?Whydoesithavethiseffect?
6
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
isfoundby_____________________________.Themeanis_______________________andisfoundby
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
_______________________.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.
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 9
STATISTICS – 9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Normal
Distribution
Definition Features
Examples Non-Examples
ALGEBRA II // MODULE 9
STATISTICS – 9.1
9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
ALGEBRA II // MODULE 9
STATISTICS – 9.1
9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
SET Topic:Identifyingpropertiesofthenormalcurve
Foreachdistribution,identifythepropertiesthatmatchwithaNormalDistribution,andthendecideiftheNormalcurvecouldbeusedasamodelforthedistributionandexplainwhy.7.NormalProperties:ModelwithaNormalCurve?YesorNo8.NormalProperties:ModelwithaNormalCurve?YesorNo
9.NormalProperties:ModelwithaNormalCurve?YesorNo10.NormalProperties:ModelwithaNormalCurve?YesorNo
9
ALGEBRA II // MODULE 9
STATISTICS – 9.1
9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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.
10
ALGEBRA II // MODULE 9
STATISTICS – 9.1
9.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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
11
ALGEBRA II // MODULE 9
STATISTICS – 9.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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?
CC
BY
Ber
ellia
n
http
s://f
lic.k
r/p/
obde
B6
12
ALGEBRA II // MODULE 9
STATISTICS – 9.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4.Yourfriend,Calvin,wouldliketogotoaveryselectivecollegethatonlyadmitsthetop1%ofallstudentapplicants.Calvinhasgoodgradesandscored33onthetest.DoyouthinkthatCalvin’sACTscoregiveshimagoodchanceofbeingadmitted?Explainyouranswer.5.ManystudentsliketoeatmicrowavepopcornastheystudyfortheACT.Microwavepopcornproducersassumethatthetimeittakesforakerneltopopisdistributednormallywithameanof120secondsandastandarddeviationof13forastandardmicrowaveoven.Ifyou’readevotedpopcornstudier,youdon’twantalotofun-poppedkernels,butyouknowthatifyouleavethebaginlongenoughtobesurethatallthekernelsarepopped,someofthepopcornwillburn.Howmuchtimewouldyourecommendformicrowavingthepopcorn?Useanormaldistributioncurveandthefeaturesofanormaldistributiontoexplainyouranswer.
13
ALGEBRA II // MODULE 9
STATISTICS – 9.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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:
S.ID.4:Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandto
estimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureis
notappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormal
curve.
RelatedStandards:S.ID.1
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP7–Lookforandmakeuseofstructure
TheTeachingCycle:
Launch(WholeClass):
Beginthetaskbytellingstudentsthatscoresonstandardizedtestsareusuallydistributed
normally.Askwhyitmakessensethatstandardizedtestscoresmightbedistributednormally.
Someresponsesmaybe:moststudentswillscorearoundthemean(makingthatthepeakofthe
distribution),aboutthesamenumberofstudentswillscoreaboveandbelowthemean(makingthe
distributionsymmetric),thetestsaredesignedsothatveryfewstudentsgeteverythingrightor
everythingwrong,whichstretchesoutthedistributionandmakesthe68%,95%,99.7%ruleseem
reasonableinthiscontext.Usethisdiscussiontodrawoutsomeofthefeaturesofthenormal
ALGEBRA II // MODULE 9
STATISTICS – 9.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
AlignedReady,Set,Go:Statistics9.2
ALGEBRA II // MODULE 9
STATISTICS – 9.2
9.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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? �
READY, SET, GO! Name PeriodDate
14
ALGEBRA II // MODULE 9
STATISTICS – 9.2
9.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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
ALGEBRA II // MODULE 9
STATISTICS – 9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9.3 Y B Normal?
A Solidify Understanding Task
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:
CC
BY
Tyl
er P
hilli
ps
http
s://f
lic.k
r/p/
rdH
WBj
16
ALGEBRA II // MODULE 9
STATISTICS – 9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
17
ALGEBRA II // MODULE 9
STATISTICS – 9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
18
ALGEBRA II // MODULE 9
STATISTICS – 9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
You:That’sgreat.I’mgoingbacktotheofficetodecidewhichstudentisadmitted.
8. ComparethescoresofStudentAandStudentB.Explainwhichstudenthasthehighestmathematicstestscoreandwhy.
19
ALGEBRA II // MODULE 9
STATISTICS – 9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9.3 Y B Normal? – Teacher Notes A Solidify Understanding Task
Purpose:
Thepurposeofthistaskistobuilduponpreviousworkwithnormaldistributionstounderstand
thestandardnormaldistribution.Inthistask,studentswillbeintroducedtothez-scoreformula
andusingz-scoretablestofindthepercentofthedistributionthatwereaboveorbelowa
particularpointonthedistribution.Studentswillusez-scorestocomparescoresontheACTand
SATtests,whicharebothnormallydistributedwithdifferentmeansandstandarddeviations.
CoreStandardsFocus:
S.ID.4:Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandto
estimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureis
notappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormal
curve.
RelatedStandards:S.ID.1
StandardsforMathematicalPractice:
SMP5–Useappropriatetoolsstrategically
Vocabulary:z-score,standardnormaldistribution
TheTeachingCycle:
Note:Studentswillneedsomeexplanationofhowtouseaz-scoretabletocompletethetask.
Becauseoftheamountoftextandtheproceduresinthetask,thistaskrequiresmoreteacher
guidancethanusual.
Launch(WholeClass):Beginthediscussionbyexplainingthesituationandensuringthatstudents
understandthecontextofACTandSATtestsbeingusedforcollegeentrance.Askstudentstowork
individuallytodecidewhichscorethattheybelieveisbetterandthendiscussargumentsfromthe
ALGEBRA II // MODULE 9
STATISTICS – 9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
AlignedReady,Set,Go:Statistics9.3
ALGEBRA II // MODULE 9
STATISTICS – 9.3
9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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?
READY, SET, GO! Name PeriodDate
20
ALGEBRA II // MODULE 9
STATISTICS – 9.3
9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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$16,400inordertobreakeven,howlikelyinagivenmonthisthecompanytomakeaprofit?
OntheWechslerAdultIntelligenceScale,anaverageIQis100withastandarddeviationof15units.(Source:http://en.wikipedia.org/wiki/Intelligence_quotient)
8.IQscoresbetween90and109areconsideredaverage.AssumingIQscoresfollowaNormaldistribution,whatpercentofpeopleareconsideredaverage?
9.OnemeasureofGeniusisanIQscoreofabove135.Whatpercentofpeopleareconsideredgenius?
10.EinsteinhadanIQscoreof160.Whatishisz-score?
11.WhatistheprobabilityofanindividualhavingahigherIQthanEinstein?
21
ALGEBRA II // MODULE 9
STATISTICS – 9.3
9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
GO Topic:SketchingPolynomials
Withoutusingtechnology,sketchthegraphofthepolynomialfunctionwiththegivencharacteristics.Iftheequationisnotgiven,writeitinstandardform.
12.Degree4
Roots:-1multiplicity2,5,-2
!–intercept:20
Equation:
13.$(&) = (& + 2)(& − 3)-
22
ALGEBRA II // MODULE 9
STATISTICS – 9.3
9.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
14.1(&) = −&(& − 3)2(& + 5)(& − 5)
15.Degree3$(−1) = 10$(25) = 0$(3) = 0Equation:
23
ALGEBRA II // MODULE 9
STATISTICS – 9.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
Rank_________Explanation:_________________________________________________________________________________
C. Thelifetimeofabatteryisnormallydistributedwithameanlifeof40hoursandastandarddeviationof1.2hours.Ijustboughtabatteryanditdiedafterjust20hours
Rank_________Explanation:_________________________________________________________________________________
D. Theamountthatahumanfingernailgrowsinayearisnormallydistributedwithameanlengthof3.5cmandastandarddeviationof0.63cm.Myneighbor’sthumbnailgrewallyearwithoutbreakinganditis4.6cmlongwithstarsandstripespaintedonit.
Rank_________Explanation:_________________________________________________________________________________
CC
BY
Sha
wn
Car
pent
er
http
s://f
lic.k
r/p/
x9zW
w
24
ALGEBRA II // MODULE 9
STATISTICS – 9.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
E. Mylittlebrotherwasdigginginthegardenandfoundagiantearthwormthatwas35cmlong.Thelengthofearthwormsisnormallydistributedwithameanlengthof14cmandastandarddeviationof5.3cm.
Rank_________Explanation:_________________________________________________________________________________
F. Themeanlengthofahumanpregnancyis268dayswithastandarddeviationof16days.Myauntjusthadaprematurebabydeliveredafteronly245days.
Rank_________Explanation:_________________________________________________________________________________
G. IQscoresforyoungadultsonafamousIQtestaredistributednormallywithameanof110andastandarddeviationof25.I’mprettysmartandmyIQis135.
Rank_________Explanation:_________________________________________________________________________________
H. Thearmymeasuredheadsizesamongmalesoldiersandfoundthatthedistributionisprettyclosetonormalwithameanof22.8inchesandstandarddeviationof1.1inches.LittleJoewasalmosttoosmalltogetintothearmybecausehisheadsizewasonly20.6inches.
Rank_________Explanation:_________________________________________________________________________________
25
ALGEBRA II // MODULE 9
STATISTICS – 9.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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:
S.ID.4:Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandto
estimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureis
notappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormal
curve.
RelatedStandards:S.ID.1
StandardsforMathematicalPractice:
SMP5–Useappropriatetoolsstrategically
SMP6–Attendtoprecision
TheTeachingCycle:
Launch(WholeClass):Introducethetaskbytellingstudentsthatwhensomethinghappensthatis
outoftheordinary,weoftentendtothinkthatitisstrangerthanitactuallyis.Tellthemthatthis
taskisaboutofanumberof“strange”occurrences.Theneedtousetheirknowledgeaboutnormal
distributionstorankeachoftheseoccurrencesfrommostunusualtoleastunusual.Tellthemthat
theymaywanttousetechnologyforsomeoftheproblems,butitisn’tnecessaryforallofthem.
ALGEBRA II // MODULE 9
STATISTICS – 9.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Explore(SmallGroup):Monitorstudentsastheyareworkingtohearhowtheyareusingthe
informationgiventoranktheevents.Itispossible,butnotnecessary,tocomeupwiththeranking
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.
AlignedReady,Set,Go:Statistics9.4
ALGEBRA II // MODULE 9
STATISTICS – 9.4
9.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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.
READY, SET, GO! Name PeriodDate
26
ALGEBRA II // MODULE 9
STATISTICS – 9.4
9.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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
27
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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___________________________________________________________________________
CC
BY
Mr.
Tin
DC
http
s://f
lic.k
r/p/
4NW
QA
2
28
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
29
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
30
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
N. O.
4. Whatmightbesomeoftheadvantagesanddisadvantagesofeachtype?
5. Apersonyouknowownsasmalltheaterthatshowslocaldramaticproductions.Shewantstoknowtheaverageageofthepeoplethatbuyticketstotheseetheshowssothatshecanbetterselectwhichplaystostage.Explaintotheownerwhyselectingthefirst20peoplethatarrivefortheshowmaynotbearepresentativesample.
6.Describeaprocessforselectingarepresentativesampleofthetheaterpatrons.
31
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP7–Lookforandmakeuseofstructure
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
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
tobesurethattheyunderstandthattherecanbemanywaystoselectasample,withprosandcons
toeachmethod.Tellstudentsthatforthenextpartofthetask,theywillbegivenexamplesof
severaldifferentsamplemethods.Theywillneedtopayattentiontohowtheparticipantsinthe
samplearechosenandtosortthesemethodsintocategories.Itmaybeusefultocutupthe
examplessothattheycaneasilybemovedaroundasstudentssortandre-sort.
Explore(SmallGroup):
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)
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Turnthediscussiontoquestion#5.Discusswhythisisnotarandomsampleandhowusingthis
methodmayskewtheresults.Askstudentswhichoftherandomsamplingmethodsmightbeused
inthiscaseandhowparticipantscouldbeselected.
AlignedReady,Set,Go:Statistics9.5
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
SamplingMethods
SimpleRandomSample
Description:Examples:
SystematicRandomSample
Description:Examples:
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
StratifiedRandomSample
Description:Examples:
ClusterRandomSample
Description:Examples:
ALGEBRA II // MODULE 9
STATISTICS – 9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
ConvenienceSample
Description:Examples:
VolunteerSample
Description:Examples:
ALGEBRA II // MODULE 9
STATISTICS – 9.5
9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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.
Population: Sample: Parameter:
7.AgovernmentofficialisinterestedinthepercentofpeopleatJFKairportthataresearchedbysecurity.Hewatches300peoplegothroughsecurityandobserves42thataresearched.
Population: Sample: Parameter:
READY, SET, GO! Name PeriodDate
32
ALGEBRA II // MODULE 9
STATISTICS – 9.5
9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
Foreachscenario,identifywhattypeofsamplingwasusedtoobtainthesample.Explainwhetherornotyouthinkthesamplewillberepresentativeofthepopulationitwassampledfrom:
8.Elvirasurveysthefirst60studentsinthelunchlinetodetermineifstudentsattheschoolaresatisfiedwithschoollunch.
Typeofsample:Representative?Explain.
9.Elviraselectsevery5thstudentinthelunchlinetodetermineifstudentsattheschoolaresatisfiedwithschoollunch.
Typeofsample:Representative?Explain.
10.Elvirarandomlyselects7differenttablesinthelunchroomandsurveyseverystudentonthetabletodetermineifstudentsattheschoolaresatisfiedwithschoollunch.
Typeofsample:Representative?Explain.
11.Elviraassignseverystudentintheschoolanumberandrandomlyselects60studentstosurveytodetermineifstudentattheschoolaresatisfiedwithschoollunch.
Typeofsample:Representative?Explain.
12.Elvirawantstodetermineifstudentsaresatisfiedwithschoollunch.Sheleavessurveysonatableforstudentstoanswerasthewalkby.
Typeofsample:Representative?Explain.
13.Elvirawantstodetermineifstudentsaresatisfiedwithschoollunch.Shewantstoincludeinputfromeachgradelevelatthehighschool.Sherandomlysurveys25freshman,25sophomores,25juniors,and25seniors.
Typeofsample:Representative?Explain.
33
ALGEBRA II // MODULE 9
STATISTICS – 9.5
9.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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.
34
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9.6 Let's Investigate
A Solidify Understanding Task
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.
CC
BY
Sim
on B
lack
ley
http
s://f
lic.k
r/p/
aj4H
Wg
35
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
36
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3.Describehowyoumightselectasampleanduseasurveytoinvestigatewhichsoftdrinkpeople
prefer:FizzyPoporKookyKola.
4.Describehowyoumightselectasampleanduseanobservationalstudytoinvestigatewhichsoft
drinkpeopleprefer:FizzyPoporKookyKola.
5.Describehowyoumightselectasampleanduseanexperimenttoinvestigateifconsuminglarge
quantitiesofKookyKolaisassociatedwithhavingheadaches.
6.Describethemethodyouwouldusetodetermineifexcessivetextingisassociatedwithbad
grades.Explainwhyyouchosethatmethodandwhatconclusionscouldbedrawnfromthestudy.
37
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
StandardsforMathematicalPractice:
SMP7–Lookforandmakeuseofstructure
Vocabulary:survey,observationalstudy,experiment,controlgroup,experimentalgroup
TheTeachingCycle:
Launch(WholeClass):
Beginthetaskbyreadingthroughthedescriptionofthethreeinvestigationmethodswiththeclass
anddiscussingtheexamples.Givesstudentsachancetoaskquestionsandclarifythedifferences
betweeneachofthethreetypes.
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Givestudentsafewminutestothinkabouttheadvantagesanddisadvantagesofeachtypeandthen
discusstheiranswers.Duringthisdiscussion,includetheideathatthevalidityoftheresultswill
partiallydependonhowthesamplewasselectedaswellasthemethodofthestudy.Forinstance,
surveyshavetheadvantageofbeingeasytogaugeopinions.Adisadvantagemaybethedifficultyin
writingquestionsthataccuratelymeasuretheparameterofinterest.Nomatterhowgoodthe
surveyquestionsare,thesurveymaynotyieldusefulresultsifitisnotbasedonarepresentative
sampleofthepopulation.Afterdiscussingtheideasofeachmethod,thenaskstudentstocomplete
thetask.
Explore(SmallGroup):
Listentostudentsastheyareworkingtoseethattheyarenoticingthedistinguishing
characteristicsofeachmethod.Studentsoftenhavedifficultywiththedifferencebetweenan
observationalstudyandanexperiment.Bepreparedtohelpthemunderstandthatitisnotan
experimentaldesignifthereisnotacontrolgroupandanexperimental(treatment)group.
Trytoidentifystudentstopresenteachofthethreemethodsasastudydesignfor#6.
Discuss(WholeClass):
Beginthediscussionbyquicklyidentifyingthemethodforeachoftheexamplesin#2.Foreach
example,clarifythedistinguishingfeaturesofthatmethod.
Turnthediscussiontoquestion#3.Askastudenttopresenthowtheymightselectasampleand
designasurvey.Asktheclasswhatquestionstheymightasktoobtainanunbiasedresult.
Haveastudentpresenttheirdesignforquestion#4.Askstudentshowtheycanensurethatthe
studyparticipants’preferencesarenotinfluencedbytheobserverorotherfactors.
Askstudentstopresenttheirexperimentaldesignforquestion#5.Howwillstudentsselectthe
twogroupssotheyarecomparable?Whatwillbethetreatment?Howwilltheymaintainthe
controlgroup?
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Finally,discussquestion#6.Ifyouhaveidentifiedstudentstopresenteachofthethreemethods,
thenaskthemtosharetheirthinkingandhavetheclasscritiquetheirdesignincludingthe
samplingmethod.Ifthereisamethodthatwasnotusedbyanystudentintheclass,facilitatea
discussionwiththewholeclasstodesignastudytogetherusingthatmethod.
AlignedReady,Set,Go:Statistics9.6
ALGEBRA II // MODULE 9
STATISTICS – 9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
StudyDesignsSurvey
Description:
Advantages:
Disadvantages:
Observation
Description:
Advantages:
Disadvantages:
Experiment
Description:
Advantages:
Disadvantages:
ALGEBRA II // MODULE 9
STATISTICS – 9.6
9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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# $ %
READY, SET, GO! Name PeriodDate
38
ALGEBRA II // MODULE 9
STATISTICS – 9.6
9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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?
39
ALGEBRA II // MODULE 9
STATISTICS – 9.6
9.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
GO Topic:Recallingnormalcurves Theaveragerestingheartrateofayoungadultisapproximately70beatsperminutewithastandarddeviationof10beatsperminute.AssumingrestingheartratefollowsaNormalDistribution,answerthefollowingquestions.10.DrawandlabeltheNormalcurvethatdescribesthisdistribution.Besuretolabelthemean,and
themeasurements1,2,and3standarddeviationsoutfromthemean.11.Whatpercentofpeoplehaveaheartratebetween55and80beatsperminute?Labelthese
pointsonyourNormalcurveaboveandshadeintheareathatrepresentsthepercentofpeoplewithheartbeatsbetween55and80beatsperminute.
12.Ifarestingheartrateabove80beatsperminuteisconsideredunhealthy,whatpercentof
peoplehaveanunhealthyheartrate?
40
ALGEBRA II // MODULE 9
STATISTICS – 9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9.7 Slacker's Simulation
A Solidify Understanding Task
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
CC
BY
Mar
co A
rmen
t
http
s://f
lic.k
r/p/
5MY
VT
6
41
ALGEBRA II // MODULE 9
STATISTICS – 9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Test2
Test3
Test4
Test5
Didyouget8outof10correctinanyofyourtrials?
3. Basedonyourtrials,doyouthinkSlackerhasagoodchanceofgetting80%correct?
4. Collectthedatafromtheentireclassanddisplayitusingtechnology.NowwhatdoyouthinkofSlacker’schancesofgetting80%correct?Explainwhy.
5. Whatwouldyouexpectthegraphtolooklikeifyoucontinuedtocollectsamples?Why?
6. Baseduponyourunderstandingofthisdistribution,whatwouldyouestimatethelikelihoodofSlackergetting80%onthetestwithoutstudying?
42
ALGEBRA II // MODULE 9
STATISTICS – 9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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?
StandardsforMathematicalPractice:
SMP7–Lookforandmakeuseofstructure
Vocabulary:simulation
TheTeachingCycle:
Launch(WholeClass):
BeginthetaskbyreadingthescenarioandaskingstudentsiftheythinkitispossibleforSlackerto
get80%.Somestudentswillarguethathewouldjustget5/10right,sincetheoddsofgettinga
questionrightis50%.Askthem,iftheoddsare50%,andSlackergets5questionsinarowcorrect,
doesthismakeitlesslikelythathewillgetthesixthquestioncorrect?Reinforcetheideathateach
questionisanindependentevent.
Explore(SmallGroup):
Givestudentstencoinsorchipstotossforthesimulation(oruseatechnologytosimulatetossing
tencoins)andaskthemtocompletequestions2and3.Bepreparedwithawaytocollectand
displaythedatafromtheentireclass.
ALGEBRA II // MODULE 9
STATISTICS – 9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Discuss(WholeClass):
Basedonthedatacollectedfromtheclass,askstudentsiftheybelievethatSlackerhasagood
chanceofgetting8/10correct.Thedistributionmaynotlookparticularlynormal,butconsider
howtousethedistributionfromthesimulationtothinkaboutSlacker’sidea.Thendiscuss
question5.Whymightweexpectthatthedistributionwouldbesymmetrical?Whymightwe
expectthedistributiontocenterat5/10?Whatisthestandarddeviationofthedistribution?
Whereis8/10onthedistribution?Aboutwhatpercentofthedistributionisbelow8/10?Make
thepointthatthenormaldistributionthatoccursforalargenumberofsamplesinthiscase,helps
ustounderstandwhetherornotaparticularsamplemightrepresenttheentirepopulation.
AlignedReady,Set,Go:Statistics9.7
ALGEBRA II // MODULE 9
STATISTICS – 9.7
9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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
�
16
�
6 6 45.003 2 19.00⎡
⎣ ⎢
⎤
⎦ ⎥ ⇒
1 1 7.503 2 19.00⎡
⎣ ⎢
⎤
⎦ ⎥
READY, SET, GO! Name PeriodDate
43
ALGEBRA II // MODULE 9
STATISTICS – 9.7
9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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?
44
ALGEBRA II // MODULE 9
STATISTICS – 9.7
9.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
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.
45