Section 11.1 Notes - Complete

Observed

Counts

Take your bag of M&Ms and the complete the table below showing the frequency of the colors and how they were distributed .

The M&M website advertises that plain M&M colors SHOULD be distributed by the following percentages.

Expected

Rel.Freq.

Expected

Counts

IfyouassumethatthenumberofM&Msinyourbagdoesnotchange,howmanyofeachcolorwouldyouhaveEXPECTEDtoGind,basedonM&Msclaim?Pleasedonotroundtoomuch...

Section 11.1 Notes - Complete

Ultimatelywewouldliketodetermineifthedifferencesweseebetweentheobservedcountsofcolorsandtheexpectedcountsofcolors(assumingM&M'sclaimistrue)aresigniGicant.

Wecoulddothisbyrunning6differentone-proportionz-testsbut...

Performingone-proportionztestsforeachcolorwouldn’ttellushowlikelyitistogetarandomsampleofthesamenumbercandieswithacolordistributionthatdiffersasmuch(ormore)fromtheoneclaimedbythecompanyasthisbagdoes(takingallthecolorsintoconsiderationatonetime).

Forthat,weneedanewkindofsigniGicancetest,calledachi-squaregoodness-of-1ittest.

Theideaofthechi-squaregoodness-of-Gittestisthis:wecomparetheobservedcountsfromoursamplewiththecountsthatwouldbeexpectedifH0istrue.Themoretheobservedcountsdifferfromtheexpectedcounts,themoreevidencewehaveagainstthenullhypothesis.

Section 11.1 Notes - Complete

Hypotheses:

Thenullhypothesisinachi-squaregoodness-of-Gittestshouldstateaclaimaboutthedistributionofasinglecategoricalvariableinthepopulationofinterest.Inourexample,theappropriatenullhypothesisis:

Thealternativehypothesisinachi-squaregoodness-of-GittestisthatthecategoricalvariabledoesnothavethespeciGieddistribution.Inourexample,thealternativehypothesisis

TestStatisticDe1inition:

Thechi-squarestatisticisameasureofhowfartheobservedcountsarefromtheexpectedcounts.Theformulaforthestatisticis

wherethesumisoverallpossiblevaluesofthecategoricalvariable.

Section 11.1 Notes - Complete

Observed

Counts

Expected

Counts

Chi-Square

Contribution

CalculatetheX2statisticbycopyingyourobservedandexpectedcountshereanddeterminingtheX2contributionforeachvalueofthevariablecolor.

ThinkofX2asameasureofthedistanceoftheobservedcountsfromtheexpectedcounts.

LargevaluesofX2arestrongerevidenceagainstH0becausetheysaythattheobservedcountsarefarfromwhatwewouldexpectifH0weretrue.

SmallvaluesofX2suggestthatthedataareconsistentwiththenullhypothesis.

Section 11.1 Notes - Complete

P-ValueInordertocalculatethep-valueweneedtounderstandtheX2Distribution.

Thesamplingdistributionofthechi-squarestatisticisNOTaNormaldistribution.

Thechi-squaredistributionsareafamilyofdistributionsthattakeonlypositivevaluesandareskewedtotheright.Aparticularchi-squaredistributionisspeciGiedbygivingitsdegreesoffreedom.Thechi-squaregoodness-of-Gittestusesthechi-squaredistributionwithdegreesoffreedom=thenumberofcategories-1.

Findingthep-valueusingTableC:• Locatethecorrectrowusingthedegreesoffreedom

• ReadacrosstherowtoGindapairofX2valuesthatcreateanintervalthatcontainsYOURX2teststatistic

• Looktothetopoftheserowstoreadoftwotailprobabilities

• Yourp-valueliesbetweenthesetwoprobabilities.

Findyourp-valuefromtheM&MexampleusingTableC.

Section 11.1 Notes - Complete

Findingthep-valueusingyourcalculator:• Gotothedistributionmenu• SelectX2cdf(thisshouldseemfamiliar...)• Lower:yourX2teststatistic• Upper:10^99• df:numberofcategories-1Findyourp-valuefromtheM&Mexampleusingyourcalculator.

Conditions

Random:datacamefromarandomsample,randomizedexperiment,orrandomphenomenon

LargeSampleSize:ThesamplesizemustbelargeenoughsothatALLEXPECTEDcountsaregreaterthanorequalto5.

Independent:Individualsshouldbeindependent.Ifsamplingwithoutreplacement,checkthe10%condition.

Section 11.1 Notes - Complete

Thingstokeepinmind...1.Thechi-squareteststatisticcomparesobservedandexpectedcounts.Don’ttrytoperformcalculationswiththeobservedandexpectedproportionsineachcategory.

2.WhencheckingtheLargeSampleSizecondition,besuretoexaminetheexpectedcounts,nottheobservedcounts.

Arebirthsevenlydistributedacrossthedaysoftheweek?Theone-waytablebelowshowsthedistributionofbirthsacrossthedaysoftheweekinarandomsampleof140birthsfromlocalrecordsinalargecity.DothesedatagivesigniGicantevidencethatlocalbirthsarenotequallylikelyonalldaysoftheweek?

Section 11.1 Notes - Complete

Section11.1Homework:p.692#s1,3,5,7,9,11,17