Microsoft PowerPoint - IntroductionTstatistic

TRANSCRIPT:

ThisisDr.Chumneywithan introductiontothet statistic.

TRANSCRIPT:

Before wetalkaboutthetstatistic,itishelpfultohaveaquickreminderaboutzscores,

andtheirassumptionsinparticular.Firstofall,thesamplemeanisassumedto

approximatethepopulationmean.

Second,thestandarderrorofthemeanestimateshowwellthesamplemeanapproximates

thepopulationmean,andtellsushowmuchdifferenceisreasonabletoexpectbetween

thesamplemeanandthepopulationmean.

Thethirdassumptionisthatthez‐statisticquantifiesinferencesaboutthepopulationthat

wemakefromthesample.

Theprimaryproblemwithz‐scoresisthatthecomputationofz‐scoresrequiresknowledge

ofthepopulationstandarddeviation.Thisisaproblembecausemostofthetimewedonot

knowenoughaboutthepopulationwearesamplingfromtoknowwhatitsstandard

deviationis.So,thisisacircularproblem:weneedthepopulationvaluestocomputez‐

statistics,

butthepurposeofaz‐statisticistomakeinferencesaboutapopulationfroma

sampleofit.

TRANSCRIPT:

Becausewedonotknowthepopulationstandarddeviation,wecannotcalculatethe

standarderrorofthemean.Instead,weestimatethestandarderrorofthemean.The

estimateofthestandarderrorisanestimateofwhattheerroriswhenthepopulation

standarddeviationisnotknown.Itisanestimate ofthestandarddifferencebetweenthe

samplemeanandthepopulationmean.

Theformula,shownhere,isthesquarerootofthesamplevariancedividedbythesample

size.

TRANSCRIPT:

Themoredegreesoffreedomwehavetoworkwith,thebetterjobthesamplevariance

doesofrepresentingthepopulationvariancebecausebiggersamplesarebetter

representativesofthepopulationingeneral.

TRANSCRIPT:

Thetstatisticisastatisticweusetotesthypothesesaboutunknownpopulationmeans

whenthevalueofthepopulationstandarddeviationisalsounknown.

Theformulaforthetstatistichasthesamestructureastheformulaforthez‐statistic,

exceptthatitusestheestimatedstandarderrorofthemeaninthedenominator.

TRANSCRIPT:

Thet‐distributionapproximatesanormalcurveinthesamewaythatthet‐statistic

approximatesaz‐statistic.

Howwellthet‐distributionapproximatesanormaldistributionisdeterminedbydf.The

largerthesamplesize,thelargerthedf,andthelargerthedf,thebetterjobthet‐

distributiondoesofapproximatinganormaldistribution.

So,asdf increases,thetdistributionlooksmoreandmore“normal”inshape.

TRANSCRIPT:

Withthez‐statistic,weusedtheUnitNormalTable.Withthet‐statistic,wewillusethet‐

distributiontable.

Inthet‐distributiontable,weidentifythepvalueinthetop2rows,lookforthecorrect

numberofdf inthefirstcolumn,and thenlookforthecriticalvalueoft–thevalueoft

thatservesastheboundaryforthecriticalregion.

Forexample,ifdf =3,5%ofthet‐distributionislocatedinthetailbeyondavalueoft =

2.353.

TRANSCRIPT:

There areafewassumptionsassociatedwiththet‐statistic.First,itisassumedthatallobservationsareindependent.

Second,thet‐statisticassumesthatthedatacomefromapopulationforwhichthedatawouldbenormallydistributedif

datawereobtainedfromallmembersofthepopulation.

Wedohypothesistestswiththet‐statisticthesameaswiththez‐statistic.Totesthypotheseswiththet‐statistic,westart

withthepopulationforwhichwedonotknowthemeanandvariance.Usually,thispopulationisatreatmentgroupof

somesort,fromwhichwehavedataonasample.

Thegoalistouseasampleofthetreatedpopulationtodetermineifthetreatmenthadanyeffectonthepopulation.

Thenullhypothesisfortestingwiththet‐statisticisthesameaswiththez‐statistic.Thenullhypothesisisalwaysthatthe

treatmenthadnoeffect;

or,thatthepopulationmeanisunchanged.Justlikeinunit2,thenullhypothesisprovides

specificvaluesforthepopulationmean.

Differentfromz‐statistichypothesistests,isthatt‐statistictestsusethesampledatatoprovideavalueforthesample

mean,andthevarianceandestimatedstandarderrorarecomputedfromthesampledatainsteadoffromthepopulation

parameters.

So,fromjustlookingattheformulaforthet‐statistic,wecanmakeafewinferencesabouttherelationshipbetweenthe

differenceinthemeans(thenumerator)andtheestimatedstandarderrorofthemean.Forinstance,

whenthe

differenceinmeansismuchlargerthantheestimatedstandarderror,wegetalargertvalue(positiveornegative).

Ifthedifferencebetweenthesampleandpopulationmeansislargerelativetotheestimatedstandarderror,weare

morelikelytorejectthenullhypothesisbecausethedifferencesindicatesthesamplemeanisverydifferentfromthe

populationmean.

TRANSCRIPT:

Wedohypothesistestswiththet‐statisticthesameaswedidinchapter8withthez‐statistic.Totest

hypotheseswiththet‐statistic,westartwiththepopulationforwhichwedonotknowthemeanand

variance.Usually,thispopulationisatreatmentgroupof

somesort,fromwhichwehavedataonasample.

Thegoalistouseasampleofthetreatedpopulationtodetermineifthetreatmenthadanyeffectonthe

population.

Thenullhypothesisfortestingwiththet‐statisticisthesameaswiththez‐statistic.Thenullhypothesis

is

alwaysthatthetreatmenthadnoeffect;or,thatthepopulationmeanisunchanged.Justlikeinunit2,the

nullhypothesisprovidesspecificvaluesforthepopulationmean.

Differentfromz‐statistichypothesistests,isthatt‐statistictestsusethesampledatatoprovideavaluefor

the

samplemean,andthevarianceandestimatedstandarderrorarecomputedfromthesampledatainstead

offromthepopulationparameters.

So,fromjustlookingattheformulaforthet‐statistic,wecanmakeafewinferencesabouttherelationship

betweenthedifferenceinthemeans(thenumerator)andthe

estimatedstandarderrorofthemean.For

instance,whenthedifferenceinmeansismuchlargerthantheestimatedstandarderror,wegetalargert

value(positiveornegative).

Ifthedifferencebetweenthesampleandpopulationmeansislargerelativetotheestimatedstandarderror,

wearemorelikely

torejectthenullhypothesisbecausethedifferencesindicatesthesamplemeanisvery

differentfromthepopulationmean.

TRANSCRIPT:

InadditiontoanestimationofCohen’sd,r‐squaredcanalsobeusedasaneff ectsizefort‐

tests.R2determineshowmuchofthevariabilityinscoresisexplainedbythetreatment

effect.Becausethetreatmentcausesscorestoincrease/decrease,thetreatmentiscausing

thevaryingscores.

Ifweidentifyhowmuchofthatvariationisexplainedbythetreatment,wethenhavea

measureofthesizeofthetreatmenteff ect.

Theamountofvariabilityaccountedforisusuallyreportedasaproportionorpercentageof

thetotalvariability.

TRANSCRIPT:

Theestimateofthestandarderrorhasalottodowiththemagnitude ofat statistic,and

whetherornotwedecidetorejectanullhypothesis.Understandinghowthesamplesize

andstandarddeviationcanimpactthet statisticisimportantbecauseithelpstoidentify

potentialexplanationsforsurprisingresults.

Largervaluesofthestandarderrortypicallyresultint statisticsclosertozero.Thelarger

thevarianceofasample,thelesslikelythetstatisticwillbesignificant,andthesmallerthe

effectsizewillbe.

Finally,withalargesample,thestandarderroristypicallygoingtobesmaller,whichmeans

thestatisticismorelikelytobesignificant.