Stat 13, Intro. to Statistical Methods for the Life and Health Sciences.

1. Echinacea and rejecting the null example. 2. Sampling, bias, and students example. 3. Estimating the mean, and guessing elapsed time example. 4. Sig. level, Type 1 and type 2 errors.

Read chapter 3. http://www.stat.ucla.edu/~frederic/13/F17 .HW2 is due Thu Oct 19 and is problems 2.3.15, 3.3.18, and 4.1.23.2.3.15 starts "Consider a manufacturing process that is producing hypodermic needles that will be used for blood donations."3.3.18 starts "Reconsider the investigation of the manufacturing process that is producing hypodermic needles. Using the data from the most recent sample of needles, a 90% confidence interval for the average diameter of needles is...."4.1.23 starts "In November 2010, an article titled 'Frequency of Cold Dramatically Cut with Regular Exercise' appeared in Medical News Today."

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1.Failingtorejectthenullvs.acceptingthenull.

– 28inechinacea groupand30inplacebogroup.– "[V]olunteers recruitedfromhospitalpersonnelwererandomlyassignedtoreceive3capsulestwicedailyofeitherplacebo(parsley)orE.purpurea [echinacea]for8weeksduringthewintermonths.Upperrespiratorytractsymptomswerereportedweeklyduringthisperiod.

– "Individualsintheechinacea groupreported9sickdaysperpersonduringthe8-weekperiod,whereastheplacebogroupreported14sickdays(z=-0.42;P=.67)."

1.Failingtorejectvs.accepting– conclusioninOneil etal.(2008),"commerciallyavailableE.purpurea capsulesdidnotsignificantlyalterthefrequencyofupperrespiratorytractsymptomscomparedwithplacebouse."

– Fromsciencebasedmedicine.org,"[Thestudy]addedtotheevidencethat Echinacea isnotusefulforpreventionofcoldsorflus.Theyfoundnodifferenceinincidenceofcoldsymptoms."

– ABCNewsheadline"Study:Echinaceanohelpforcolds".

1.Failingtorejectvs.accepting.

Today, most of the evidence seems to indicate that echinaceadoes boost the immune system a little bit and help to fight colds. From WebMD: "Extracts of echinacea do seem to have an effect on the immune system, your body's defense against germs. Research shows it increases the number of white blood cells, which fight infections. A review of more than a dozen studies, published in 2014, found the herbal remedy had a very slight benefit in preventing colds."

2.Samplingstudents

Example2.1A

• Wewilllookatdatacollectedfromtheregistrar’sofficefromtheCollegeoftheMidwestforALLstudentsforSpring2011bylookingatthetwovariablesinthespreadsheetbelowthatshowsthefirst8students.

SamplingStudents

StudentID CumulativeGPA Oncampus?1 3.92 Yes2 2.80 Yes3 3.08 Yes4 2.71 No5 3.31 Yes6 3.83 Yes7 3.80 No8 3.58 Yes… … …

• Whattypeofvariableis“Oncampus”?• WhattypeisCumulativeGPA?

SamplingStudents

StudentID CumulativeGPA Oncampus?1 3.92 Yes2 2.80 Yes3 3.08 Yes4 2.71 No5 3.31 Yes6 3.83 Yes7 3.80 No8 3.58 Yes… … …

• Herearegraphs(ahistogramandabargraph)representingallofthe2919students attheCollegeoftheMidwestforourtwovariablesofinterest.

SamplingStudents

• Weusuallydon’thaveinformationonanentirepopulationlikewedohere.

• Weusuallyneedtomakeinferencesaboutapopulationbasedonasample.

• Supposearesearcherasksthefirst30studentsshefindsoncampusonemorningwhethertheyliveoncampus.Thiswouldbeaquickandconvenientwaytogetasample.

SamplingStudents

Forthisscenario:• Whatisthepopulation?• Whatisthesample?• Whatistheparameter• Whatisthestatistic?• Doyouthinkthisquickandconvenientsamplingmethodwillresultinasimilarsampleproportiontothepopulationproportion?

SamplingStudents

• Theresearcher’ssamplingmethodmightoverestimatetheproportionofstudentsthatliveoncampusbecauseifitistakenearlyinthemorningmostofthosethatliveoffcampusmightnothavearrivedyet.

• Wecallthissamplingmethodbiased.• Asamplingmethodisbiased ifstatisticsfromsamplesoverorunder-estimatethepopulationparameteronaverage.

SamplingStudents

• Biasisapropertyofasamplingmethod,notthesample– Amethodmustconsistently (onaverage)producenon-representativeresultstobeconsideredbiased

• Samplingbiasalsodependsonwhatismeasured– WouldthemorningsamplingmethodbebiasedinestimatingtheaverageGPAofstudentsatthecollege?

– Whataboutestimatingtheproportionofstudentswearingorangeshirts?

SamplingStudents

• What’sabetterwayofselectingarepresentativesample?

• Usearandommechanismtoselecttheobservationalunits

• Don’trelyonconveniencesamples• A SimpleRandomSample(SRS) iswhereeverycollectionofsizen isequallylikelytobethesampleselectedfromthepopulation.

SamplingStudents

• HowcouldwetakeaSimpleRandomSampleof30studentsfromtheCollegeoftheMidwest?

• RepresenteachstudentbyIDnumbers1to2919• Havethecomputerrandomlyselect30numbersbetween1and2919

SamplingStudents

IDCumGPA

Oncampus? ID

CumGPA

Oncampus? ID

CumGPA

Oncampus?

827 3.44 Y 844 3.59 N 825 3.94 Y1355 2.15 Y 90 3.30 Y 2339 3.07 N1455 3.08 Y 1611 3.08 Y 2064 3.48 Y2391 2.91 Y 2550 3.41 Y 2604 3.10 Y575 3.94 Y 2632 2.61 Y 2147 2.84 Y2049 3.64 N 2325 3.36 Y 2590 3.39 Y895 2.29 N 2563 3.02 Y 1718 3.01 Y1732 3.17 Y 1819 3.55 N 168 3.04 Y2790 2.88 Y 968 3.86 Y 1777 3.83 Y2237 3.25 Y 566 3.60 N 2077 3.46 Y

SamplingStudentsIDsofthe30peopleselected,alongwiththeircumulativeGPAandresidentialstatus

• WhatistheaveragecumGPAforthese30students?– �̅� isthesampleaverage– �̅� =3.24

• Whatproportionliveoncampus?– �̂� isthesampleproportion– �̂� =0.80

• 𝜇isthepopulationmean.• p isthepopulationproportion.

SamplingStudents

• Howdoweknowif�̅� and�̂� areclosetothepopulationvalues,𝜇 andp?

• Adifferentsampleof30studentswouldprobablyhavehaddifferentvalues.

• Howarethesestatisticsusefulinestimatingthepopulationparametervalues?

• Let’stakemoresimplerandomsamplesof30studentstoexaminethenulldistributionofthestatisticsfromothersamples.

SamplingStudents

• Herearegraphs(ahistogramandabargraph)representingallofthe2919students attheCollegeoftheMidwestforourtwovariablesofinterest.

SamplingStudents

IDCumGPA

Oncampus? ID

CumGPA

Oncampus? ID

CumGPA

Oncampus?

827 3.44 Y 844 3.59 N 825 3.94 Y1355 2.15 Y 90 3.30 Y 2339 3.07 N1455 3.08 Y 1611 3.08 Y 2064 3.48 Y2391 2.91 Y 2550 3.41 Y 2604 3.10 Y575 3.94 Y 2632 2.61 Y 2147 2.84 Y2049 3.64 N 2325 3.36 Y 2590 3.39 Y895 2.29 N 2563 3.02 Y 1718 3.01 Y1732 3.17 Y 1819 3.55 N 168 3.04 Y2790 2.88 Y 968 3.86 Y 1777 3.83 Y2237 3.25 Y 566 3.60 N 2077 3.46 Y

SamplingStudentsIDsofthe30peopleselected,alongwiththeircumulativeGPAandresidentialstatus

• WhatistheaveragecumGPAforthese30students?– �̅� isthesampleaverage– �̅� =3.24

• Whatproportionliveoncampus?– �̂� isthesampleproportion– �̂� =0.80

• 𝜇isthepopulationmean.• p isthepopulationproportion.

SamplingStudents

• Howdoweknowif�̅� and�̂� areclosetothepopulationvalues,𝜇 andp?

• Adifferentsampleof30studentswouldprobablyhavehaddifferentvalues.

• Howarethesestatisticsusefulinestimatingthepopulationparametervalues?

• Let’stakemoresimplerandomsamplesof30studentstoexaminethenulldistributionofthestatisticsfromothersamples.

SamplingStudents

• Wetook5differentSRSsof30students• Eachsamplegivesdifferentstatistics• Thisissamplingvariability.• Thevaluesdon’tchangemuch:– AverageGPAsrangefrom3.22to3.40– Sampleproportionsrangefrom0.63to0.83

SamplingStudents

Randomsample 1 2 3 4 5

AverageGPA() 3.22 3.29 3.40 3.26 3.25

proportiononcampus( ) 0.80 0.83 0.77 0.63 0.83

• Populationparameters:– 𝜇 =3.288– 𝜋 ≈0.776(2265/2919).

• Whatdotheparametersdescribe?– ThetrueaveragecumulativeGPAandthetrueproportiononcampusofthe2919students

• Thestatisticstendtobeclosetotheparameters.

SamplingStudents

Randomsample 1 2 3 4 5

AverageGPA() 3.22 3.29 3.40 3.26 3.25

proportiononcampus( ) 0.80 0.83 0.77 0.63 0.83

• Wetook1000SRSsandhavegraphsofthe1000samplemeans(fortheGPAs)and1000sampleproportions(forlivingoncampus).

• Themeanofeachdistributionfallsnearthepopulationparameter.

SamplingStudents

• Whatwouldhappenifwetookallpossiblerandomsamplesof30studentsfromthispopulation?– Theaveragesofthestatisticswouldmatchtheparameters

exactly• StatisticscomputedfromSRSsclusteraroundtheparameter.• Whyisthisanunbiasedsamplingmethod?

– Thereisnotendencytooverorunderestimatetheparameter.• Thesamplingmethodandstatisticyouchoosedetermineifa

samplingmethodisbiased.• Asamplemeanofasimplerandomsampleisanunbiasedestimate

ofthepopulationmean.Sameforproportionsinsteadofmeans.

SamplingStudents

• Wecangeneralizewhenweusesimplerandomsamplingbecauseitcreates:– Asamplethatisrepresentativeofthepopulation.– Asamplestatisticthatisunbiasedandthusclosetotheparameterforlargen.

SamplingStudents

• IftheresearcherattheCollegeoftheMidwestuses75studentsinsteadof30withthesameearlymorningsamplingmethodwillitbelessbiased?

• Selectingmorestudentsinthesamemanner doesn’tfixthetendencytooversamplestudentswholiveoncampus.

• Asmallersamplethatisrandomisactuallymoreaccurate.

SamplingStudents

• Whatisanadvantageofalargersamplesize?– Lesssampletosamplevariability.– Statisticsfromdifferentsamplesclustermorecloselyaroundthecenterofthedistribution.

SamplingStudents

3.InferenceforaSingleQuantitativeVariable

Section2.2

Example2.2:EstimatingElapsedTime

• Studentsinastatsclass(fortheirfinalproject)collecteddataonstudents’perceptionoftime

• Subjectsweretoldthatthey’dlistentomusicandbeaskedquestionswhenitwasover.

• 10secondsoftheJackson5’s“ABC”andsubjectswereaskedhowlongtheythoughtitlasted

• Canstudentsaccuratelyestimatethelength?

Hypotheses

NullHypothesis:Peoplewillaccuratelyestimatethelengthofa10second-songsnippet,onaverage.(μ=10seconds)AlternativeHypothesis:Peoplewillnotaccuratelyestimatethelengthofa10second-songsnippet,onaverage.(μ≠10seconds)

EstimatingTime

• Asampleof48studentsoncampusweresubjectsandsonglengthestimateswererecorded.

• Whatdoesasingledotrepresent?• Whataretheobservationalunits?Variable?

Skewed,mean,median

• Thedistributionobtainedisnotsymmetric,butisrightskewed.

• Whendataareskewedright,themean getspulledouttotherightwhilethemedian ismoreresistanttothis.

MeanvsMedian• Themeanis13.71andthemedianis12.• Howwouldthesenumberschangeifoneofthe

peoplethatgaveananswerof30secondsactuallysaid300seconds?

• Thestandarddeviationis6.5sec.Alsonotresistanttooutliers.

Inference• H0:μ=10seconds• Ha:μ≠10seconds• Ourproblemnowis,howdowedevelopanull

distribution?• Flippingcoinswillnotworktomodelwhatwould

happenunderatruenullhypothesis.– Herewedon’thavepopulationdatathatreflectsournullhypothesiswhereμ =10seconds.

– Allwehaveisoursampleof48.

Population?

• Weneedtocomeupwithalargedatasetthatwethinkourpopulationoftimeestimatesmightlooklikeunderatruenull.

• Wemightassumethepopulationisskewed(likeoursample)andhasastandarddeviationsimilartowhatwefoundinoursample,buthasameanof10seconds.

• Thebookrecommendsusinganappletforthis.WecoulduseR,ordoa(theory-based)t-test.

Theory-BasedTest

• Usingsimulationstocreateapopulationeachtimewewanttorunatestofsignificanceisextremelytimeconsumingandcumbersome.(Sothiswillbetheonlytimewewilldoit.)

• Thenulldistributionthatwedevelopedcanbepredictedwiththeory-basedmethods.

• Weknowitwillbecenteredonthemeangiveninthenullhypothesis.

• Wecanalsopredictitsshapeanditsstandarddeviation.

t-distribution• Theshapeisverymuchlikeanormaldistribution,butslightlywider

inthetailsandiscalledat-distribution.• Thet-statisticisthestandardizedstatisticweusewithasingle

quantitativevariableandcanbefoundusingtheformula:

𝑡 =�̅� − 𝜇𝑠

𝑛�<The= >�< (standarddeviationofoursampledividedbythesquarerootofthesamplesize)iscalledthestandarderrorandisanestimateforthestandarddeviationofthenulldistribution.

Here𝑡 = ?@.A?B?C.CD.E

FG�<= 3.95.

p-value=2*(1-pt(3.95,df=47))=0.000261.

ValidityConditions

• Theobservationsmustbeindependent.• Thepopulationmustbenormallydistributed.• Thebooksaysyouneedthesamplesizetobeatleast20forthet-test,butthisisnotright.However,itisoftenhardtohaveanyideaifthepopulationisnormalwithouthavingatleast20observations.

EstimatingTime

FormulateConclusions.• Basedonoursmallp-value,wecanconcludethatpeopledon’taccuratelyestimatethelengthofa10-secondsongsnippetandinfacttheysignificantlyoverestimateit.

• Towhatlargerpopulationcanwemakeourinference?

Summary

• Whenwetestasinglequantitativevariable,ourhypothesishasthefollowingform:– H0:μ=somenumber– Ha:μ≠somenumber,µ<somethingorµ>something.

• Wewillgetourdata(ormean,samplesize,andSDforourdata)andusetheTheory-BasedInferencetodeterminethep-value.

• Thep-valuewegetwiththistesthasthesamegeneralmeaningasthosefromatestforasingleproportion.

4.Significancelevel,Type1andType2errors

Section2.3

SignificanceLevel

• Wethinkofap-valueastellingussomethingaboutthestrengthofevidencefromatestofsignificance.

• Thelowerthep-valuethestrongertheevidence.• Somepeoplethinkofthisinmoreblackandwhiteterms.Eitherwerejectthenullornot.

SignificanceLevel

• Thevaluethatweusetodeterminehowsmallap-valueneedstobetoprovideconvincingevidencewhetherornottorejectthenullhypothesisiscalledthesignificancelevel.

• Werejectthenullwhenthep-valueislessthanorequalto(≤)thesignificancelevel.

• ThesignificancelevelisoftenrepresentedbytheGreekletteralpha,α.

SignificanceLevel

• Typicallyweuse0.05foroursignificancelevel.Thereisnothingmagicalabout0.05.Wecouldsetupourtesttomakeit– hardertorejectthenull(smallersignificancelevelsay0.01)or

– easier(largersignificancelevelsay0.10).

TypeIandTypeIIerrors

• Inmedicaltests:– AtypeIerrorisafalsepositive.(Theyconcludesomeonehasadiseasewhentheydon’t.)

– AtypeIIerrorisafalsenegative.(Theyconcludesomeonedoesnothaveadiseasethentheyactuallydo.)

• Thesetypesoferrorscanhaveverydifferentconsequences.

TypeIandTypeIIErrors

•

TypeIandTypeIIerrors

TheprobabilityofaTypeIerror

• TheprobabilityofatypeIerroristhesignificancelevel.

• Supposethesignificancelevelis0.05.Ifthenullistruewewouldrejectit5%ofthetimeandthusmakeatypeIerror5%ofthetime.

• Ifyoumakethesignificancelevellower,youhavereducedtheprobabilityofmakingatypeIerror,buthaveincreasedtheprobabilityofmakingatypeIIerror.

TheprobabilityofaTypeIIerror• TheprobabilityofatypeIIerrorismoredifficulttocalculate.

• Infact,theprobabilityofatypeIIerrorisnotevenafixednumber.Itdependsonthevalueofthetrueparameter.

• TheprobabilityofatypeIIerrorcanbeveryhighif:– Thetruevalueoftheparameterandthevalueyouaretestingareclose.

– Thesamplesizeissmall.

Power• Theprobabilityofrejectingthenullhypothesiswhenitisfalseiscalledthepower ofatest.

• Powercanalsobethoughtofas1minustheprobabilityofatypeIIerror.

• Wewantatestwithhighpowerandthisisaidedby– Alargeeffectsize,i.e.asamplemeanfarfromtheparameterinthenullhypothesis.

– Alargesamplesize.– Asmallstandarddeviation.