Two-SampleTestsofHypothesisand

AnalysisofVarianceChapters11&12

Dr.RichardJerz

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GOALS

• Conductatestofahypothesisaboutthedifferencebetweentwoindependentpopulationmeans

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Comparingtwopopulations–SomeExamples

• IsthereadifferenceintheaveragevalueofresidentialrealestatesoldbymaleagentsandfemaleagentsinsouthFlorida?

• IsthereadifferenceinthemeannumberofdefectsproducedbyfirstshiftandsecondshiftsatKimbleProducts?

• Isthereadifferenceinthemeannumberofdaysabsentbetweenworkersunder21yearsoldandworkersolderthan60yearsoldinthefast-foodindustry?

• Isthereanincreaseintheproductionrateifmusicispipedintotheproductionarea?

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FormatofHypotheses

• Thehypothesesforcomparingtwoindependentpopulationmeansμ1 andμ2 are:

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PossibleScenarios

1. Populationσ2’sareknown2. Populationσ2’sareunknown

1. Areσ2’sequal?2. Areσ2’sunequal?

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StatisticalMethods

• Historicalmethod(usingtables)• Computea“special”teststatistic,eitherzort

• Compareittotheteststatisticofthedesiredconfidencelevel(oralpha).

• Modernmethod• Computethep-value(probability)andcomparethistotheminimumacceptableerror(α).Ifthep-valueissmallerthanα,thereisadifference(rejectnullthattheyarethesame.)

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PossibleScenarios

• Populationσ2’sareknown:Zstatistic• Populationσ2’sareunknown:Tstatistic

1. Ares2’sequal?2. Ares2’sunequal?

Saidalittledifferently• Ifthepopulationvariancesσ12 andσ22areknown,thenusethenormaldistribution.

• Ifpopulationvariancesareunknownandestimatedusings12 ands22,thenusetheStudent-tdistribution.

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“Special”TestStatistics10-8

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Case3:UnknownVariances,AssumedUnequal

• Welch-Satterthwaite test

• with degrees of freedom

• A Quick Rule for degrees of freedom is to use min(n1 – 1, n2 – 1).

10-9

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AnalysisMethodology

• MSExcel• Testforequalmeans,knownpopulationvariances

• Testforequalvariances• Testforequalmeans,unknownpopulationvariances

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AlternativeExcelMethod–DataAnalysisAdd-in

• Excel:Data|Data Analysis

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ComparingTwoVariances– FStatistic

• Formatofhypotheses• Totestwhethertwopopulationmeansareequal,wemayalsoneedtotestwhethertwopopulationvariancesareequal.

• Thehypothesesmaybestatedas

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CharacteristicsofF-Distribution

• Thereisa“family”ofFDistributions.

• Eachmemberofthefamilyisdeterminedbytwoparameters:thenumeratordegreesoffreedomandthedenominatordegreesoffreedom.

• Fcannotbenegative,anditisacontinuousdistribution.

• TheFdistributionispositivelyskewed.

• Itsvaluesrangefrom0to¥• AsF®¥ thecurveapproaches

theX-axis.© 2017 rjerz.com13

TheFTest

• Theteststatisticistheratioofthesamplevariances:

• If the variances are equal, this ratio should be near unity: F » 1

10-14

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TheFTest

• Iftheteststatisticisfarbelow1orabove1,wewouldrejectthehypothesisofequalpopulationvariances.

10-15

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AssumptionsoftheFTest

• Itissensitivetonon-normalityofthesampledpopulations.

• TheFtestassumesthatthepopulationsbeingsampledarenormal.

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Examples

• P368• P377• P380,problem17• P381• P384,problem22• P386• P405

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