商务统计学（第7版）英文 课件11 Chi-Square Tests、13 Multiple Regression

Chi-SquareTestsChapter11ObjectivesInthischapter,youlearn:Howandwhentousethechi-squaretestforcontingencytablesContingencyTablesContingencyTablesUsefulinsituationscomparingmultiplepopulationproportionsUsedtoclassifysampleobservationsaccordingtotwoormorecharacteristicsAlsocalledacross-classificationtable.DCOVA Left-Handedvs.Gender

DominantHand:Leftvs.Right Gender:Malevs.Female2categoriesforeachvariable,sothisiscalleda2x2tableSupposeweexamineasampleof300childrenContingencyTableExampleDCOVAContingencyTableExampleSampleresultsorganizedinacontingencytable:(continued)GenderHandPreferenceLeftRightFemale12108120Male2415618036264300120Females,12werelefthanded180Males,24werelefthandedsamplesize=n=300:DCOVA

2TestfortheDifferenceBetweenTwoProportionsIfH0istrue,thentheproportionofleft-handedfemalesshouldbethesameastheproportionofleft-handedmalesThetwoproportionsaboveshouldbethesameastheproportionofleft-handedpeopleoverallH0:π1=π2(Proportionoffemaleswhoareleft handedisequaltotheproportionof maleswhoarelefthanded)H1:π1≠π2(Thetwoproportionsarenotthesame)DCOVATheChi-SquareTestStatisticwhere:

fo=observedfrequencyinaparticularcell

fe=expectedfrequencyinaparticularcellifH0istrue

(Assumed:eachcellinthecontingencytablehasexpectedfrequencyofatleast5)TheChi-squareteststatisticis:DCOVADecisionRule

2

2αDecisionRule:If,rejectH0,otherwise,donotrejectH0Theteststatisticapproximatelyfollowsachi-squareddistributionwithonedegreeoffreedom0

RejectH0DonotrejectH0DCOVAComputingthe

OverallProportionHere:

120Females,12werelefthanded180Males,24werelefthandedi.e.,basedonall300childrentheproportionoflefthandersis0.12,thatis,12% Theoverallproportionis:DCOVAFindingExpectedFrequenciesToobtaintheexpectedfrequencyforlefthandedfemales,multiplytheaverageproportionlefthanded(p)bythetotalnumberoffemalesToobtaintheexpectedfrequencyforlefthandedmales,multiplytheaverageproportionlefthanded

(p)bythetotalnumberofmalesIfthetwoproportionsareequal,then

P(LeftHanded|Female)=P(LeftHanded|Male)=.12i.e.,wewouldexpect (.12)(120)=14.4femalestobelefthanded (.12)(180)=21.6malestobelefthandedDCOVAObservedvs.ExpectedFrequenciesGenderHandPreferenceLeftRightFemaleObserved=12Expected=14.4Observed=108Expected=105.6120MaleObserved=24Expected=21.6Observed=156Expected=158.418036264300DCOVAGenderHandPreferenceLeftRightFemaleObserved=12Expected=14.4Observed=108Expected=105.6120MaleObserved=24Expected=21.6Observed=156Expected=158.418036264300TheChi-SquareTestStatisticTheteststatisticis:DCOVADecisionRuleDecisionRule:If>3.841,rejectH0,otherwise,donotrejectH0Here,=0.7576<=3.841,sowedonotrejectH0andconcludethatthereisnotsufficientevidencethatthetwoproportionsaredifferentat=0.05

2

20.05=3.8410

0.05RejectH0DonotrejectH0DCOVAExtendthe

2testtothecasewithmorethantwoindependentpopulations:

2TestforDifferencesAmong

MoreThanTwoProportionsH0:π1=π2=…=πcH1:Notalloftheπjareequal(j=1,2,…,c)DCOVATheChi-SquareTestStatisticWhere:

fo=observedfrequencyinaparticularcellofthe2xctable

fe=expectedfrequencyinaparticularcellifH0istrue

(Assumed:eachcellinthecontingencytablehasexpectedfrequencyofatleast1)TheChi-squareteststatisticis:DCOVAComputingthe

OverallProportionTheoverallproportionis:Expectedcellfrequenciesfortheccategoriesarecalculatedasinthe2x2case,andthedecisionruleisthesame:Whereisfromthechi-squareddistributionwithc–1degreesoffreedomDecisionRule:If,rejectH0,otherwise,donotrejectH0DCOVA

2TestofIndependenceSimilartothe

2testforequalityofmorethantwoproportions,butextendstheconcepttocontingencytableswithrrowsandccolumnsH0:Thetwocategoricalvariablesareindependent (i.e.,thereisnorelationshipbetweenthem)H1:Thetwocategoricalvariablesaredependent (i.e.,thereisarelationshipbetweenthem)DCOVA

2TestofIndependencewhere:

fo=observedfrequencyinaparticularcelloftherxctable

fe=expectedfrequencyinaparticularcellifH0istrue

(Assumed:eachcellinthecontingencytablehasexpectedfrequencyofatleast1)TheChi-squareteststatisticis:(continued)DCOVAExpectedCellFrequenciesExpectedcellfrequencies:Where: rowtotal=sumofallfrequenciesintherow columntotal=sumofallfrequenciesinthecolumn n=overallsamplesizeDCOVADecisionRuleThedecisionruleisWhereisfromthechi-squaredistributionwith(r–1)(c–1)degreesoffreedomIf,rejectH0,otherwise,donotrejectH0DCOVAExampleThemealplanselectedby200studentsisshownbelow:ClassStandingNumberofmealsperweekTotal20/week10/weeknoneFresh.24321470Soph.22261260Junior1014630Senior14161040Total708842200DCOVAExampleThehypothesistobetestedis:(continued)H0:Mealplanandclassstandingareindependent (i.e.,thereisnorelationshipbetweenthem)H1:Mealplanandclassstandingaredependent (i.e.,thereisarelationshipbetweenthem)DCOVAClassStandingNumberofmealsperweekTotal20/wk10/wknoneFresh.24321470Soph.22261260Junior1014630Senior14161040Total708842200ClassStandingNumberofmealsperweekTotal20/wk10/wknoneFresh.24.530.814.770Soph.21.026.412.660Junior10.513.26.330Senior14.017.68.440Total708842200Observed:ExpectedcellfrequenciesifH0istrue:Exampleforonecell:Example:

ExpectedCellFrequencies(continued)DCOVAExample:TheTestStatisticTheteststatisticvalueis:(continued)=12.592fromthechi-squaredistributionwith(4–1)(3–1)=6degreesoffreedomDCOVAExample:

DecisionandInterpretation(continued)DecisionRule:If>12.592,rejectH0,otherwise,donotrejectH0Here,=0.709<=12.592,sodonotrejectH0

Conclusion:thereisnotsufficientevidencethatmealplanandclassstandingarerelatedat=0.05

2

20.05=12.5920

0.05RejectH0DonotrejectH0DCOVAChapterSummaryInthischapterwediscussed:Howandwhentousethechi-squaretestforcontingencytablesIntroductiontoMultipleRegressionChapter13ObjectivesInthischapter,youlearn:

HowtodevelopamultipleregressionmodelHowtointerprettheregressioncoefficientsHowtodeterminewhichindependentvariablestoincludeintheregressionmodelHowtousecategoricalindependentvariablesinaregressionmodelTheMultipleRegressionModelIdea:Examinethelinearrelationshipbetween1dependent(Y)&2ormoreindependentvariables(Xi)MultipleRegressionModelwithkIndependentVariables:Y-interceptPopulationslopesRandomErrorDCOVAMultipleRegressionEquationThecoefficientsofthemultipleregressionmodelareestimatedusingsampledataEstimated(orpredicted)valueofYEstimatedslopecoefficientsMultipleregressionequationwithkindependentvariables:EstimatedinterceptInthischapterwewilluseExcelandMinitabtoobtaintheregressionslopecoefficientsandotherregressionsummarymeasures.DCOVATwovariablemodelYX1X2SlopeforvariableX1SlopeforvariableX2MultipleRegressionEquation(continued)DCOVAAdistributoroffrozendessertpieswantstoevaluatefactorsthoughttoinfluencedemandDependentvariable:Piesales(unitsperweek)Independentvariables:Price(in$)

Advertising($100’s)Dataarecollectedfor15weeksExample:

2IndependentVariablesDCOVAPieSalesExampleSales=b0+b1(Price) +b2(Advertising)WeekPieSalesPrice($)Advertising($100s)13505.503.324607.503.333508.003.044308.004.553506.803.063807.504.074304.503.084706.403.794507.003.5104905.004.0113407.203.5123007.903.2134405.904.0144505.003.5153007.002.7Multipleregressionequation:DCOVAExcelMultipleRegressionOutputRegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888DCOVAMinitabMultipleRegressionOutputTheregressionequationisSales=307-25.0Price+74.1Advertising

Predictor

Coef

SECoef

T

PConstant 306.50

114.302.68

0.020Price -24.98

10.83

-2.31

0.040Advertising 74.13

25.97

2.85

0.014

S=47.4634

R-Sq=52.1%

R-Sq(adj)=44.2%

AnalysisofVariance

Source

DF

SS

MS

F

PRegression

2

29460

14730

6.54

0.012ResidualError

12

27033

2253Total

14

56493DCOVATheMultipleRegressionEquationb1=-24.975:saleswilldecrease,onaverage,by24.975piesperweekforeach$1increaseinsellingprice,netoftheeffectsofchangesduetoadvertisingb2=74.131:saleswillincrease,onaverage,by74.131piesperweekforeach$100increaseinadvertising,netoftheeffectsofchangesduetopricewhere SalesisinnumberofpiesperweekPriceisin$Advertisingisin$100’s.DCOVAUsingTheEquationtoMakePredictionsPredictsalesforaweekinwhichthesellingpriceis$5.50andadvertisingis$350:Predictedsalesis428.62piesNotethatAdvertisingisin$100s,so$350meansthatX2=3.5DCOVAPredictionsinExcelusingPHStatPHStat|regression|multipleregression…Checkthe“confidenceandpredictionintervalestimates”boxDCOVAInputvaluesPredictionsinPHStat(continued)

PredictedYvalue<ConfidenceintervalforthemeanvalueofY,giventheseXvaluesPredictionintervalforanindividualYvalue,giventheseXvaluesDCOVAPredictionsinMinitabInputvaluesPredictedValuesforNewObservations

NewObs

Fit

SEFit

95%CI

95%PI

1

428.6

17.2

(391.1,466.1)

(318.6,538.6)

ValuesofPredictorsforNewObservations

NewObs

Price

Advertising

1

5.50

3.50ConfidenceintervalforthemeanvalueofY,giventheseXvaluesPredictionintervalforanindividualYvalue,giventheseXvaluesDCOVATheCoefficientofMultipleDetermination,r2ReportstheproportionoftotalvariationinYexplainedbyallXvariablestakentogetherDCOVARegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.7088852.1%ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertisingMultipleCoefficientof

DeterminationInExcelDCOVAMultipleCoefficientof

DeterminationInMinitabTheregressionequationisSales=307-25.0Price+74.1Advertising

Predictor

Coef

SECoef

T

PConstant 306.50

114.302.68

0.020Price -24.98

10.83

-2.31

0.040Advertising 74.13

25.97

2.85

0.014

S=47.4634

R-Sq=52.1%

R-Sq(adj)=44.2%

AnalysisofVariance

Source

DF

SS

MS

F

PRegression

2

29460

14730

6.54

0.012ResidualError

12

27033

2253Total

14

5649352.1%ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertisingDCOVAAdjustedr2r2neverdecreaseswhenanewXvariableisaddedtothemodelThiscanbeadisadvantagewhencomparingmodelsWhatistheneteffectofaddinganewvariable?WeloseadegreeoffreedomwhenanewXvariableisaddedDidthenewXvariableaddenoughexplanatorypowertooffsetthelossofonedegreeoffreedom?DCOVAShowstheproportionofvariationinYexplainedbyallXvariablesadjustedforthenumberofX

variablesused

(wheren=samplesize,k=numberofindependentvariables)PenalizesexcessiveuseofunimportantindependentvariablesSmallerthanr2UsefulincomparingamongmodelsAdjustedr2(continued)DCOVARegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.7088844.2%ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertising,takingintoaccountthesamplesizeandnumberofindependentvariablesAdjustedr2inExcelDCOVAAdjustedr2inMinitabTheregressionequationisSales=307-25.0Price+74.1Advertising

Predictor

Coef

SECoef

T

PConstant 306.50

114.302.68

0.020Price -24.98

10.83

-2.31

0.040Advertising 74.13

25.97

2.85

0.014

S=47.4634

R-Sq=52.1%

R-Sq(adj)=44.2%

AnalysisofVariance

Source

DF

SS

MS

F

PRegression

2

29460

14730

6.54

0.012ResidualError

12

27033

2253Total

14

5649344.2%ofthevariationinpiesalesisexplainedbythevariationinpriceandadvertising,takingintoaccountthesamplesizeandnumberofindependentvariablesDCOVAFTestforOverallSignificanceoftheModelShowsifthereisalinearrelationshipbetweenalloftheXvariablesconsideredtogetherandYUseF-teststatisticHypotheses:H0:β1=β2=…=βk=0(nolinearrelationship)H1:atleastoneβi≠0(atleastoneindependent variableaffectsY)

IstheModelSignificant?DCOVAFTestforOverallSignificanceTeststatistic:

whereFSTAThasnumeratord.f.=kand denominatord.f.=(n–k-1)

DCOVARegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888(continued)FTestforOverallSignificanceInExcelWith2and12degreesoffreedomP-valuefortheFTestDCOVAFTestforOverallSignificanceInMinitabTheregressionequationisSales=307-25.0Price+74.1Advertising

Predictor

Coef

SECoef

T

PConstant 306.50

114.302.68

0.020Price -24.98

10.83

-2.31

0.040Advertising 74.13

25.97

2.85

0.014

S=47.4634

R-Sq=52.1%

R-Sq(adj)=44.2%

AnalysisofVariance

Source

DF

SS

MS

F

PRegression

2

29460

14730

6.54

0.012ResidualError

12

27033

2253Total

14

56493With2and12degreesoffreedomP-valuefortheFTestDCOVAH0:β1=β2=0H1:β1andβ2notbothzero

=.05df1=2df2=12TestStatistic:Decision:Conclusion:SinceFSTATteststatisticisintherejectionregion(p-value<.05),rejectH0ThereisevidencethatatleastoneindependentvariableaffectsY0

=.05F0.05=3.885RejectH0DonotrejectH0CriticalValue:F0.05=3.885FTestforOverallSignificance(continued)FDCOVATwovariablemodelYX1X2YiYi<x2ix1iThebestfitequationisfoundbyminimizingthesumofsquarederrors,e2SampleobservationResidualsinMultipleRegressionResidual=ei=(Yi–Yi)<DCOVAMultipleRegressionAssumptionsAssumptions:TheerrorsarenormallydistributedErrorshaveaconstantvarianceThemodelerrorsareindependentei=(Yi–Yi)<Errors(residuals)fromtheregressionmodel:DCOVAResidualPlotsUsed

inMultipleRegressionTheseresidualplotsareusedinmultipleregression:Residualsvs.YiResidualsvs.X1iResidualsvs.X2iResidualsvs.time(iftimeseriesdata)<UsetheresidualplotstocheckforviolationsofregressionassumptionsDCOVAUsettestsofindividualvariableslopesShowsifthereisalinearrelationshipbetweenthevariableXjandYholdingconstanttheeffectsofotherXvariablesHypotheses:H0:βj=0(nolinearrelationship)H1:βj≠0(linearrelationshipdoesexist betweenXjandY)AreIndividualVariablesSignificant?DCOVAH0:βj=0(nolinearrelationshipbetweenXjandY)H1:βj≠0(linearrelationshipdoesexist betweenXjandY)TestStatistic: (df=n–k–1)AreIndividualVariablesSignificant?(continued)DCOVARegressionStatisticsMultipleR0.72213RSquare0.52148AdjustedRSquare0.44172StandardError47.46341Observations15ANOVA

dfSSMSFSignificanceFRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandardErrortStatP-valueLower95%Upper95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888tStatforPriceistSTAT=-2.306,withp-value.0398tStatforAdvertisingistSTAT=2.855,withp-value.0145(continued)AreIndividualVariablesSignificant?ExcelOutputDCOVAAreIndividualVariablesSignificant?MinitabOutputTheregressionequationisSales=307-25.0Price+74.1Advertising

Predictor

Coef

SECoef

T

PConstant 306.50

114.302.68

0.020Price -24.98

10.83

-2.31

0.040Advertising 74.13

25.97

2.85

0.014

S=47.4634

R-Sq=52.1%

R-Sq(adj)=44.2%

AnalysisofVariance

Source

DF

SS

MS

F

PRegression

2

29460

14730

6.54

0.012ResidualError

12

27033

2253Total

14

56493tStatforPriceistSTAT=-2.31,withp-value.040tStatforAdvertisingistSTAT

=2.85,withp-value.014DCOVAd.f.=15-2-1=12=.05t/2=2.1788InferencesabouttheSlope:

t

TestExampleH0:βj=0H1:βj

0Theteststatisticforeachvariablefallsintherejectionregion(p-values<.05)ThereisevidencethatbothPriceandAdvertisingaffectpiesalesat

=.05FromtheExceloutput:RejectH0foreachvariableDecision:Conclusion:RejectH0RejectH0a/2=.025-tα/2DonotrejectH00tα/2a/2=.025-2.17882.1788ForPricetSTAT=-2.306,withp-value.0398ForAdvertisingtSTAT

=2.855,withp-value.0145DCOVAConfidenceIntervalEstimate

fortheSlopeConfidenceintervalforthepopulationslopeβjExample:Forma95%confidenceintervalfortheeffectofchangesinprice(X1)onpiesales:-24.975±(2.1788)(10.832)Sotheintervalis(-48.576,-1.374)(Thisintervaldoesnotcontainzero,sopricehasasignificanteffectonsales)

CoefficientsStandardErrorIntercept306.52619114.25389Price-24.9750910.83213Advertising74.1309625.96732wherethas(n–k–1)d.f.Here,thas(15–2–1)=12d.f.DCOVAConfidenceIntervalEstimate

fortheSlopeConfidenceintervalforthepopulationslopeβjExample:Exceloutputalsoreportstheseintervalendpoints:Weeklysalesareestimatedtobereducedbybetween1.37to48.58piesforeachincreaseof$1inthesellingprice,holdingtheeffectofadvertisingconstant

CoefficientsStandardError…Lower95%Upper95%Intercept306.52619114.25389…57.58835555.46404Pric