Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf ·...
Transcript of Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf ·...
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Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.
0.Handinhw4.1.Calculatingcorrelation.2.Slopeoftheregressionline.3.Goodnessoffit.4.Commonproblemswithregression.
a.Inferringcausation.b.Extrapolation.c.Curvature.
5.Testingsignificanceofcorrelationorslope.6.ANOVAandF-test,ch9.7.Reviewandexamples.
ThefinalisThuinclassandwillbeonch 1-7,10,andatmost1questiononch9.BringaPENCILandCALCULATORandanybooksornotesyouwant.Nocomputers.http://www.stat.ucla.edu/~frederic/13/sum17.
1
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1.Calculatingcorrelation,r.
ρ =rho=correlationofthepopulation.SupposethereareNpeopleinthepopulation,X=temperature,Y=heartrate,themeanandsd oftempinthepop.areµ" and𝜎" ,andthepop.meanandsd ofheartrateareµ$and𝜎$.
ρ =*+∑ "-./0
10+23*
$-./414
.
Givenasampleofsizen,weestimateρ using
r= *5.*
∑"-.060
523*
$-.464
.
ThisisinAppendixA.
2
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2.Slopeofregressionline.• Suppose𝑦8 =a+bx istheregressionline.
• Theslopeboftheregressionlineisb=r6460.
Thisisusuallythethingofprimaryinteresttointerpret,asthepredictedincreaseinyforeveryunitincreaseinx.• Bewareofassumingcausationthough,esp.withobservationalstudies.Bewaryofextrapolationtoo.
• Theintercepta=$ - b " .
• TheSDoftheresidualsis 1 − 𝑟<� 𝑠$.Thisisagoodestimateofhowmuchtheregression
predictionswilltypicallybeoffby.
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3.Howwelldoesthelinefit?• 𝑟< isameasureoffit.Itindicatestheamountofscatteraroundthebestfittingline.
• 1 − 𝑟<� 𝑠$isusefulasameasureofhowfaroffpredictionswouldhavebeenonaverage.• Residualplotscanindicatecurvature,outliers,orheteroskedasticity.
• Notethatregressionresidualshavemeanzero,whetherthelinefitswellorpoorly.
-2 -1 0 1 2 3 4 5
-4-2
02
46
8
x
y
-2 -1 0 1 2 3 4 5-10
-8-6
-4-2
02
x
residual
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• Heteroskedasticity:whenthevariabilityinyisnotconstantasxvaries.
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4.Commonproblemswithregression.• a.Correlationisnotcausation.Especiallywithobservationaldata.
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Commonproblemswithregression.
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Commonproblemswithregression.Holmes and Willett (2004) reviewed all prospective studies on fat consumption and breast cancer with at least 200 cases of breast cancer. "Not one study reported a significant positive association with total fat intake.... Overall, no association was observed between intake of total, saturated, monounsaturated, or polyunsaturated fat and risk for breast cancer."
They also state "The dietary fat hypothesis is largely based on the observation that national per capita fat consumption is highly correlated with breast cancer mortality rates. However, per capita fat consumption is highly correlated with economic development. Also, low parity and late age at first birth, greater body fat, and lower levels of physical activity are more prevalent in Western countries, and would be expected to confound the association with dietary fat."
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Commonproblemswithregression.• b.Extrapolation.
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Commonproblemswithregression.• b.Extrapolation.• Oftenresearchersextrapolatefromhighdosestolow.
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Commonproblemswithregression.• b.Extrapolation.Therelationshipcanbenonlinearthough.Researchersalsooftenextrapolatefromanimalstohumans.
Zaichkina etal.(2004)onhamsters
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Commonproblemswithregression.• c.Curvature.Thebestfittinglinemightfitpoorly.Portetal.(2005).
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Commonproblemswithregression.• c.Curvature.Thebestfittinglinemightfitpoorly.Wongetal.(2011).
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Commonproblemswithregression.• d.Statisticalsignificance.Couldtheobservedcorrelationjustbeduetochancealone?
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5.InferencefortheRegressionSlope:Theory-BasedApproachSection10.5
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Dostudentswhospendmoretimeinnon-academicactivitiestendtohavelowerGPAs?Example10.4
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Dostudentswhospendmoretimeinnon-academicactivitiestendtohavelowerGPAs?
• Thesubjectswere34undergraduatestudentsfromtheUniversityofMinnesota.
• Theywereaskedquestionsabouthowmuchtimetheyspentinactivitieslikework,watchingTV,exercising,non-academiccomputeruse,etc.aswellaswhattheircurrentGPAwas.
• WearegoingtotesttoseeifthereisanegativeassociationbetweenthenumberofhoursperweekspentonnonacademicactivitiesandGPA.
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Hypotheses• NullHypothesis:ThereisnoassociationbetweenthenumberofhoursstudentsspendonnonacademicactivitiesandstudentGPAinthepopulation.
• AlternativeHypothesis:ThereisanegativeassociationbetweenthenumberofhoursstudentsspendonnonacademicactivitiesandstudentGPAinthepopulation.
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DescriptiveStatistics• GPAU = 3.60 − 0.0059(nonacademichours).• Istheslopsignificantlydifferentfrom0?
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ShuffletoDevelopNullDistribution
• Wearegoingtoshufflejustaswedidwithcorrelationtodevelopanulldistribution.
• Theonlydifferenceisthatwewillbecalculatingtheslopeeachtimeandusingthatasourstatistic.
• atestofassociationbasedonslopeisequivalenttoatestofassociationbasedonacorrelationcoefficient.
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Betavs Rho• Testingtheslopeoftheregressionlineisequivalenttotestingthecorrelation (samep-value,butobviouslydifferentconfidenceintervalssincethestatisticsaredifferent)
• Hencethesehypothesesareequivalent.• Ho:β =0Ha:β <0(Slope)• Ho:ρ =0Ha:ρ < 0(Correlation)
• Sampleslope(b)Population(β:beta)• Samplecorrelation(r)Population(ρ:rho)
• Whenwedothetheorybasedtest,wewillbeusingthet-statisticwhichcanbecalculatedfromeithertheslopeorcorrelation.
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Introduction• Ournulldistributionsareagainbell-shapedandcenteredat0(foreithercorrelationorslopeasourstatistic).
The book on p549 finds a p value of 3.3% by simulation.
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ValidityConditions• Undercertainconditions,theory-basedinferenceforcorrelationorslopeoftheregressionlineuset-distributions.
• Wecouldusesimulationsorthetheory-basedmethodsfortheslopeoftheregressionline.
• Wewouldgetthesamep-valueifweusedcorrelationasourstatistic.
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PredictingHeartRatefromBodyTemperatureExample10.5A
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HeartRateandBodyTemp• Earlierwelookedattherelationshipbetweenheartrateandbodytemperaturewith130healthyadults
• PredictedHeartRate = −166.3 + 2.44 Temp• r=0.257
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HeartRateandBodyTemp
• Wetestedtoseeifwehadconvincingevidencethatthereisapositiveassociationbetweenheartrateandbodytemperatureinthepopulationusingasimulation-basedapproach.(Wewillmakeit2-sidedthistime.)
• NullHypothesis:Thereisnoassociationbetweenheartrateandbodytemperatureinthepopulation. β=0
• AlternativeHypothesis:Thereisanassociationbetweenheartrateandbodytemperatureinthepopulation. β≠0
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HeartRateandBodyTemp
Wegetaverysmallp-value(0.0036).Anythingasextremeasourobservedslopeof2.44happeningbychanceisveryrare
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HeartRateandBodyTemp
• Wecanalsoapproximatea95%confidenceintervalobservedstatistic+ multiplierx SE2.44± 1.96x 0.842=0.790to4.09
• Whatdoesthismean?We’re95%confidentthat,inthepopulationofhealthyadults,each1° increaseinbodytempisassociatedwithanincreaseinheartrateofbetween0.790to4.09beatsperminute.
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HeartRateandBodyTemp
• Thetheory-basedapproachshouldworkwellsincethedistributionoftheslopeshasanicebellshape
• Alsocheckthescatterplot
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HeartRateandBodyTemp
• Wewillusethet-statistictogetourtheory-basedp-value.
• Wewillfindatheory-basedconfidenceintervalfortheslope.
• Onp554,thebooknotestheformulat= cdefgheg
�.
• Herethetstatisticis2.97.• Thep-valueis0.36%.Sothecorrelationisstatisticallysignificantlygreaterthanzero.
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SmokingandDrinkingExample10.5B
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ValidityConditionsRememberourvalidityconditionsfortheory-basedinferenceforslopeoftheregressionequation.
1. Thescatterplotshouldfollowalineartrend.2. Thereshouldbeapproximatelythesamenumber
ofpointsaboveandbelowtheregressionline(symmetry).
3. Thevariabilityofverticalslicesofthepointsshouldbesimilar.Thisiscalledhomoskedasticity.
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ValidityConditions• Let’slookatsomescatterplotsthatdonotmeettherequirements.
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SmokingandDrinkingTherelationshipbetweennumberofdrinksandcigarettesperweekforarandomsampleofstudentsatHopeCollege.
Thedotat(0,0)represents524students
Aretheconditionsmet?Hardtosay.Thebooksaysno.
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SmokingandDrinking• Whentheconditionsarenotmet,applyingsimulation-basedinferenceispreferabletotheory-basedt-testsandCIs.
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ValidityConditions
• Whatdoyoudowhenvalidityconditionsaren’tmetfortheory-basedinference?• Usethesimulated-basedapproach.
• Anotherstrategyisto“transform”thedataonadifferentscalesoconditionsaremet.• Thelogarithmicscaleiscommon.
• Onecanalsofitadifferentcurve,notnecessarilyaline.
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6.ANOVAandF-test.
Section9.2
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ANOVA• ANOVAstandsforANalysis OfVAriance.• Usefulwhencomparingmorethan2means.• IfIhave2 meanstocompare,I justlookattheirdifferencetomeasurehowfaraparttheyare.
• SupposeI wantedtocomparethreemeans.IhavethemeanforgroupA,themeanforgroupB,andthemeanforgroupC.
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Fteststatistic
• TheanalysisofvarianceF teststatisticis:
groupsy within variabilitgroupsbetween y variabilit
=F
• Thisissimilartothet-statisticwhenwewerecomparingjusttwomeans. 𝑡 = "̅d."̅g
kdg
hdlkd
g
hg
�
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RecallingAmbiguousProseExample9.2
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ComprehensionExample
(Don’t follow along in your book or look ahead on the PowerPoint until after I read you the passage.) • Studentswerereadanambiguousprosepassageunderoneofthefollowingconditions:• Studentsweregivenapicturethatcouldhelptheminterpretthepassagebefore theyheardit.
• Studentsweregiventhepictureafter theyheardthepassage.
• Studentswerenot shownanypicturebeforeorafterhearingthepassage.
• Theywerethenaskedtoevaluatetheircomprehensionofthepassageona1to7scale.
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ComprehensionExample• ThisexperimentisapartialreplicationdoneatHopeCollegeofastudy
donebyBransfordandJohnson(1972).• Studentswererandomlyassignedtooneofthe3groups.• Listentothepassageandseeifitmakessense.Wouldapicturehelp?Iftheballoonspopped,thesoundwouldn’tbeabletocarrysinceeverythingwouldbetoofarawayfromthecorrectfloor.Aclosedwindowwouldalsopreventthesoundfromcarrying,sincemostbuildingstendtobewellinsulated.Sincethewholeoperationdependsonasteadyflowofelectricity,abreakinthemiddleofthewirewouldalsocauseproblems.Ofcourse,thefellowcouldshout,butthehumanvoiceisnotloudenoughtocarrythatfar.Anadditionalproblemisthatastringcouldbreakontheinstrument.Thentherecouldbenoaccompanimenttothemessage.Itisclearthatthebestsituationwouldinvolvelessdistance.Thentherewouldbefewerpotentialproblems.Withfacetofacecontact,theleastnumberofthingscouldgowrong.
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Hypotheses
• Null:Inthepopulationthereisnoassociationbetweenwhetherorwhenapicturewasshownandcomprehensionofthepassage
• Alternative:Inthepopulationthereisanassociationbetweenwhetherandwhenapicturewasshownandcomprehensionofthepassage
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Hypotheses
• Null: All three of the long term mean comprehension scores are the same.
µnopicture =µpicturebefore =µpictureafter
• Alternative: At least one of the mean comprehension scores is different.
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RecallScore• Studentsratedtheircomprehension,andtheresearchersalsohadthestudentsrecallasmanyideasfromthepassageastheycould.Theywerethengradedonwhattheycouldrecallandtheresultsareshown.
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ValidityConditions• Justaswiththesimulation-basedmethod,weareassumingwehaveindependentgroups.
• TwoextraconditionsmustbemettousetraditionalANOVA:• Normality:Ifsamplesizesaresmallwithineachgroup,datashouldn’tbeveryskewed.Ifitis,usesimulationapproach.(Samplesizesofatleast20isagoodguidelineformeans.)
• Equalvariation:Largeststandarddeviationshouldbenomorethantwicethevalueofthesmallest.
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ANOVAOutput• ThisisthekindofoutputyouwouldseeinmoststatisticspackageswhendoingANOVA.
• Thevariabilitybetweenthegroupsismeasuredbythemeansquaretreatment(40.02).
• Thevariabilitywithinthegroupsismeasuredbythemeansquareerror(3.16).
• TheFstatisticis40.02/3.16=12.67.
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Conclusion• Sincewehaveasmallp-valuewehavestrongevidenceagainstthenullandcanconcludeatleastoneofthelong-runmeanrecallscoresisdifferent.
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Reviewlist.1.MeaningofSD. 19.Randomsamplingandrandom2.Parametersandstatistics. assignment.3.Zstatisticforproportions. 20.TwoproportionCIsandtesting.4.Simulationandmeaningofpvalues. 21.IQRand5number summaries.5.SEforproportions. 22.CIs for 2means andtesting.6.Whatinfluencespvalues. 23.Paired data.7.CLTandvalidityconditionsfortests. 24.Placeboeffect,adhererbias,8.1-sidedand2-sidedtests. andnonresponsebias.9.Rejectthenullvs.acceptthealternative. 25.Predictionandcausation.10.Samplingandbias. 26.Multipletestingandpublicationbias11.Significancelevel. 27.Regression.12.TypeI,typeIIerrors,andpower. 28.Correlation.13.CIsforaproportion. 29.Calculate&interpreta&b.14.CIsforamean. 30.Goodnessoffitforregression.15.Marginoferror. 31.Commonregressionproblems16.Practicalsignificance. (causation,extrapolation,curvature,heterskedasticity).17.Confounding. 32.ANOVAandF-test.18.Observationalstudiesandexperiments.
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exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
1.Whatdoesthecorrelationof0.82imply?a.82%ofthevariationinweightisexplainedbyheight.b.Thetypicalvariationinpeople'sheightsis82%aslargeasthetypicalvariationintheirweights.c.Thereisstrongassociationbetweenheightandweightinthissample.d.Foreveryinchofincreaseinone'sheight,wewouldpredicta0.82lb.increaseinweight.e.Ifapersonweighs100pounds,thenwetypicallywouldexpectthepersontobeabout82inchestall.
![Page 52: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/52.jpg)
exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
1.Whatdoesthecorrelationof0.82imply?a.82%ofthevariationinweightisexplainedbyheight.b.Thetypicalvariationinpeople'sheightsis82%aslargeasthetypicalvariationintheirweights.c.Thereisstrongassociationbetweenheightandweightinthissample.d.Foreveryinchofincreaseinone'sheight,wewouldpredicta0.82lb.increaseinweight.e.Ifapersonweighs100pounds,thenwetypicallywouldexpectthepersontobeabout82inchestall.
![Page 53: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/53.jpg)
exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
2.Whatistheestimatedslope,inlbs/inch,oftheregressionlineforpredictingweightfromheight?a.6.56.b.7.12.c.8.04.d.9.92.e.10.2.f.11.4.
![Page 54: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/54.jpg)
exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
2.Whatistheestimatedslope,inlbs/inch,oftheregressionlineforpredictingweightfromheight?a.6.56.b.7.12.c.8.04.d.9.92.e.10.2.f.11.4.
rsy/sx =.82x 40/5=6.56.
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exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
3.Howmuchwouldapredictionusingthisregressionlinetypicallybeoffby?a.12.7lbs. b.13.5lbs. c.14.4lbs. d.20.2lbs.e.22.9lbs.
![Page 56: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/56.jpg)
exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
3.Howmuchwouldapredictionusingthisregressionlinetypicallybeoffby?a.12.7lbs. b.13.5lbs. c.14.4lbs. d.20.2lbs.e.22.9lbs.
√(1-r2)sy =√(1-.822)x40=22.9.
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exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
4.Ifweweretorandomlytakeoneadultfromthissample,howmuchwouldhis/herheighttypicallydifferfrom65by?a.0.05in.b.0.1in.c.0.5in.d.1.0in.e.2.5in.f.5.0in.
![Page 58: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/58.jpg)
exampleproblems.
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
4.Ifweweretorandomlytakeoneadultfromthissample,howmuchwouldhis/herheighttypicallydifferfrom65by?a.0.05in.b.0.1in.c.0.5in.d.1.0in.e.2.5in. f.5.0in.
![Page 59: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/59.jpg)
Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,themedianheightis64.5inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
5.Whyshouldn'tonetrustthisregressionlinetopredicttheweightofsomeonewhois25inchestall?a.Thesamplesizeisinsufficientlylarge.b.ThesampleSDofweightistoosmall.c.Thevalueof25inchesistoofaroutsidetherangeofmostobservations.d.ThecorrelationoftheANOVAisat-testconfidenceintervalwithstatisticalsignificance.e.Thedatacomefromanobservationalstudy,sotheremaybeconfoundingfactors.f.Theheightvaluesareheavilyrightskewed,sothepredictionerrorsarelarge.
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Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,themedianheightis64.5inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.
5.Whyshouldn'tonetrustthisregressionlinetopredicttheweightofsomeonewhois25inchestall?a.Thesamplesizeisinsufficientlylarge.b.ThesampleSDofweightistoosmall.c.Thevalueof25inchesistoofaroutsidetherangeofmostobservations.d.ThecorrelationoftheANOVAisat-testconfidenceintervalwithstatisticalsignificance.e.Thedatacomefromanobservationalstudy,sotheremaybeconfoundingfactors.f.Theheightvaluesareheavilyrightskewed,sothepredictionerrorsarelarge.
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Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,themedianheightis64.5inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.Theestimatedslopeinpredictingweightfromheightis6.56lbs/in.
6.Howshouldoneinterprettheestimatedslopeof6.56?a.Eachextrainchyougrowcausesyoutoincreaseyourweightby6.56lbs onaverage.b.Eachextralb.youweighcausesyoutogrow6.56inches.c.TheamountofweightAmericansaverageis6.56standarderrorsabovethemean.d.TheZ-scorecorrespondingtothecorrelationbetweenheightandweightis6.56.e.Foreachextrainchtalleryouare,yourpredictedweightincreasesby6.56lbs.f.Theproportionofvarianceinweightexplainedbytheregressionequationis6.56%.
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Supposethatamongasampleof100adultsinagiventown,thecorrelationbetweenheight(inches)andweight(lbs.)is0.82,andthemeanheightis65inches,themedianheightis64.5inches,thesd ofheightis5inches,themeanweightis160lbs.,andthesd ofweightis40lbs.Theestimatedslopeinpredictingweightfromheightis6.56lbs/in.
6.Howshouldoneinterprettheestimatedslopeof6.56?a.Eachextrainchyougrowcausesyoutoincreaseyourweightby6.56lbs onaverage.b.Eachextralb.youweighcausesyoutogrow6.56inches.c.TheamountofweightAmericansaverageis6.56standarderrorsabovethemean.d.TheZ-scorecorrespondingtothecorrelationbetweenheightandweightis6.56.e.Foreachextrainchtalleryouare,yourpredictedweightincreasesby6.56lbs.f.Theproportionofvarianceinweightexplainedbytheregressionequationis6.56%.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
7.Whatisthepooledsamplepercentagewithacne,inbothgroupscombined?a.16.1%.b.17.2%.c.18.4%.d.19.7%.e.21.2%.f.23.0%.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
7.Whatisthepooledsamplepercentagewithacne,inbothgroupscombined?a.16.1%.b.17.2%.c.18.4%.d.19.7%. e.21.2%.f.23.0%.
118/600=19.7%.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
8.Usingthispooledsamplepercentage,underthenullhypothesisthatthetwogroupshavethesameacnerate,whatisthestandarderrorforthedifferencebetweenthetwopercentages?a.1.42%. b.1.88%. c.2.02%. d.2.99%.e.3.08%. f.3.44%.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
8.Usingthispooledsamplepercentage,underthenullhypothesisthatthetwogroupshavethesameacnerate,whatisthestandarderrorforthedifferencebetweenthetwopercentages?a.1.42%. b.1.88%. c.2.02%. d.2.99%.e.3.08%. f.3.44%.
√(.197*(1-.197)/400+.197*(1-.197)/200)=.0344.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
9.WhatistheZstatisticforthedifferencebetweenthetwogrouppercentages?a.1.02. b.1.23. c.1.55. d.1.88. e.2.03. f.2.43.
![Page 68: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/68.jpg)
Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
9.WhatistheZstatisticforthedifferencebetweenthetwogrouppercentages?a.1.02. b.1.23. c.1.55. d.1.88. e.2.03. f.2.43.
(88/400– 30/200)÷ 0.0344=2.03.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
10.Usingtheunpooled standarderrorforthedifferencebetweenthetwopercentages,finda95%confidenceintervalforthepercentagewithacneamongthoseage21-30minusthepercentageofacneamongthoseage31-40.a.7%+/- 5.02%.b.7%+/- 5.54%.c.7%+/- 5.92%.d.7%+/- 6.03%.e.7%+/- 6.41%.
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Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
10.Usingtheunpooled standarderrorforthedifferencebetweenthetwopercentages,finda95%confidenceintervalforthepercentagewithacneamongthoseage21-30minusthepercentageofacneamongthoseage31-40.a.7%+/- 5.02%.b.7%+/- 5.54%.c.7%+/- 5.92%.d.7%+/- 6.03%.e.7%+/- 6.41%.
88/400– 30/200+/- 1.96*√(88/400*(1-88/400)/400+30/200*(1-30/200)/200)=7%+/- 6.41%.
![Page 71: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/71.jpg)
Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
11.Whatcanweconcludefromthe95%CIintheproblemabove?a.Thereisnostatisticallysignificantdifferenceinthepercentagewithacneamongthoseage21-30andthoseage31-40.b.Thereisstatisticallysignificantcorrelationbetweenthesamplesizeandtheeffectsizefortheconfoundingfactorsofarandomizedcontrolledexperiment.c.Ofthosewithacne,thereisnostatisticallysignificantdifferencebetweenthepercentagewhoareage21-30andthepercentagewhoare31-40.d.Astatisticallysignificantlyhigherpercentageofpeopleage21-30haveacnethanthoseage31-40.e.Thesamplesizesaretoosmallforaconclusiontobevalidhere.
![Page 72: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/72.jpg)
Supposearesearcherstudyingacne,oroilyskin,takesasimplerandomsampleof400Americansage21-30andcallsthemgroupA,andasimplerandomsampleof200Americansage31-40andcallsthemgroupB.IngroupA,88peoplehaveacne,andingroupB,30peoplehaveacne.
11.Whatcanweconcludefromthe95%CIintheproblemabove?a.Thereisnostatisticallysignificantdifferenceinthepercentagewithacneamongthoseage21-30andthoseage31-40.b.Thereisstatisticallysignificantcorrelationbetweenthesamplesizeandtheeffectsizefortheconfoundingfactorsofarandomizedcontrolledexperiment.c.Ofthosewithacne,thereisnostatisticallysignificantdifferencebetweenthepercentagewhoareage21-30andthepercentagewhoare31-40.d.Astatisticallysignificantlyhigherpercentageofpeopleage21-30haveacnethanthoseage31-40.e.Thesamplesizesaretoosmallforaconclusiontobevalidhere.
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Forthefollowingtwoproblems,supposearesearcherstudiesatreatmentforacertaincondition.Theresearcherdivides230subjectsrandomlyintothreegroups.GroupAreceivesaplacebo.GroupBreceivesthetreatmentatalowdose,andgroupCreceivesthetreatmentatahighdose.ThetablebelowshowstheoutputfromanANOVAF-testonthemeanlevelsoftheconditioninthe3groups.
AnalysisofVarianceTable
Response:outcomeDf SumSq MeanSq Fvalue Pr(>F)
group 20.467 0.233 0.221 0.803Residuals 22728.5 1.06
12.Inthetable,whatnumberisameasureofthevariabilitybetweengroups?a.2.b.0.233c.0.221d.0.803e.28.5 f.1.06.
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Forthefollowingtwoproblems,supposearesearcherstudiesatreatmentforacertaincondition.Theresearcherdivides230subjectsrandomlyintothreegroups.GroupAreceivesaplacebo.GroupBreceivesthetreatmentatalowdose,andgroupCreceivesthetreatmentatahighdose.ThetablebelowshowstheoutputfromanANOVAF-testonthemeanlevelsoftheconditioninthe3groups.
AnalysisofVarianceTable
Response:outcomeDf SumSq MeanSq Fvalue Pr(>F)
group 20.467 0.233 0.221 0.803Residuals 22728.5 1.06
12.Inthetable,whatnumberisameasureofthevariabilitybetweengroups?a.2.b.0.233 c.0.221d.0.803e.28.5 f.1.06.
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Forthefollowingtwoproblems,supposearesearcherstudiesatreatmentforacertaincondition.Theresearcherdivides230subjectsrandomlyintothreegroups.GroupAreceivesaplacebo.GroupBreceivesthetreatmentatalowdose,andgroupCreceivesthetreatmentatahighdose.ThetablebelowshowstheoutputfromanANOVAF-testonthemeanlevelsoftheconditioninthe3groups.
AnalysisofVarianceTableResponse:outcome
Df SumSq MeanSq Fvalue Pr(>F)group 20.467 0.233 0.221 0.803Residuals 22728.5 1.06
13.WhatcanweconcludefromtheresultoftheFtest?a.Thetreatmenthasastatisticallysignificanteffect,thoughwecannotbesurewhichdoseisattributabletothissignificanteffectorinwhichdirectiontheeffectgoes.b.Thetreatmentseemstohavesignificantlygreatereffectatlargedosesthanatsmalldoses.c.Thetreatmentdoesnotseemtohaveastatisticallysignificanteffect.d.Wefailtorejectthenullhypothesisthatthetreatmenthasasignificanteffectonthecondition.e.Thecorrelationbetweendoseandoutcomeappearstobestatisticallysignificant.
![Page 76: Stat 13, Intro. to Statistical Methods for the Life and Health …frederic/13/sum17/day09.pdf · 2017-07-31 · Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.](https://reader034.fdocuments.us/reader034/viewer/2022042022/5e790d1c1b016958c36db153/html5/thumbnails/76.jpg)
Forthefollowingtwoproblems,supposearesearcherstudiesatreatmentforacertaincondition.Theresearcherdivides230subjectsrandomlyintothreegroups.GroupAreceivesaplacebo.GroupBreceivesthetreatmentatalowdose,andgroupCreceivesthetreatmentatahighdose.ThetablebelowshowstheoutputfromanANOVAF-testonthemeanlevelsoftheconditioninthe3groups.
AnalysisofVarianceTableResponse:outcome
Df SumSq MeanSq Fvalue Pr(>F)group 20.467 0.233 0.221 0.803Residuals 22728.5 1.06
13.WhatcanweconcludefromtheresultoftheFtest?a.Thetreatmenthasastatisticallysignificanteffect,thoughwecannotbesurewhichdoseisattributabletothissignificanteffectorinwhichdirectiontheeffectgoes.b.Thetreatmentseemstohavesignificantlygreatereffectatlargedosesthanatsmalldoses.c.Thetreatmentdoesnotseemtohaveastatisticallysignificanteffect.d.Wefailtorejectthenullhypothesisthatthetreatmenthasasignificanteffectonthecondition.e.Thecorrelationbetweendoseandoutcomeappearstobestatisticallysignificant.