Slide 1Bivariate DataCoefficient of CorrelationCoefficient of Correlation PropertiesSum of SquaresSlide 6Scatter Diagram and Correlation CoefficientVertical DistancesLeast Squares LineSlide 10Sum of Squares of ErrorLeast Squares Line for Real Estate DataAssumptions for the Simple Regression ModelAssumption 1 for the Simple Regression ModelViolation of Assumption 3Assumptions 1, 2, 3 for the Simple Regression ModelEstimating the Error Variance, e2Three Possible PopulationsHypothesis Test on the Slope of the Regression LineSlide 20t Curve with 8 dfReal Estate ExampleSlide 23Slide 24Slide 25Scatter DiagramSlide 27Confidence Interval for 1Curvilinear RelationshipMeasuring the Strength of the ModelDanger of Assuming CausalityCoefficient of DeterminationTotal Variation, SSYSlide 34Estimation and Prediction Using the Simple Linear ModelConfidence Interval for µY|xConfidence and Prediction IntervalsSlide 38Confidence and Prediction Intervals95% Confidence IntervalsPrediction Interval for YXSlide 42Checking Model AssumptionsExamination of ResidualsSlide 45Autocorrelation and the Durbin-Watson StatisticSlide 47Checking for OutliersIdentifying Outlying ValuesSlide 50Real Estate ExampleSlide 52Slide 53Identifying Influential ObservationsLeverages, Standardized Residuals, and Cook’s Distance MeasuresSummary of Figures 13.26 and 13.28Engine Capacity and MPGSlide 58Slide 59Slide 60Slide 61©2006 Thomson/South-Western 1Chapter 13 –Chapter 13 –Correlation andCorrelation andSimple Simple RegressionRegressionSlides prepared by Jeff HeylLincoln University©2006 Thomson/South-WesternConcise Managerial StatisticsConcise Managerial StatisticsKVANLIPAVURKEELINGKVANLIPAVURKEELING©2006 Thomson/South-Western 2Bivariate DataBivariate DataFigure 13.1Figure 13.135 –35 –30 –30 –25 –25 –20 –20 –15 –15 –10 –10 –5 –5 –Square footage (hundreds)Square footage (hundreds)||2020||3030||4040||5050||6060||7070||8080YYXXIncome (thousands)Income (thousands)(a)(a)35 –35 –30 –30 –25 –25 –20 –20 –15 –15 –10 –10 –5 –5 –Square footage (hundreds)Square footage (hundreds)||2020||3030||4040||5050||6060||7070||8080YYXXIncome (thousands)Income (thousands)(b)(b)©2006 Thomson/South-Western 3Coefficient of CorrelationCoefficient of CorrelationThe strength of the linear relationship The strength of the linear relationship between two variables is called the between two variables is called the coefficient of correlation, r.coefficient of correlation, r.rr = =∑∑((xx - - xx)()(yy - - yy)) ∑ ∑((xx - - xx))22 ∑( ∑(yy - - yy))22==∑∑xyxy - (∑ - (∑xx)(∑)(∑yy) / ) / nn ∑ ∑xx22 - (∑ - (∑xx))22 / / nn ∑ ∑yy22 - (∑ - (∑yy))22 / / nn©2006 Thomson/South-Western 4Coefficient of Correlation Coefficient of Correlation PropertiesProperties1.1.r ranges from r ranges from -1.0-1.0 to to 1.01.02.2.The larger |r | is, the stronger the linear The larger |r | is, the stronger the linear relationshiprelationship3.3.The sign of r tells you whether the The sign of r tells you whether the relationship between X and Y is a positive relationship between X and Y is a positive (direct) or a negative (inverse) relationship(direct) or a negative (inverse) relationship4.4.r r = 1= 1 or or -1-1 implies that a perfect linear implies that a perfect linear pattern exists between the two variables, pattern exists between the two variables, that they are perfectly correlatedthat they are perfectly correlated©2006 Thomson/South-Western 5Sum of SquaresSum of SquaresSSSSXX= sum of squares for = sum of squares for XX= ∑(= ∑(xx - - xx))22= ∑= ∑xx22 - - (∑(∑xx))22nnSSSSYY= sum of squares for = sum of squares for YY= ∑(= ∑(yy - - yy))22= ∑= ∑yy22 - - (∑(∑yy))22nnSCPSCPXYXY= sum of cross products for = sum of cross products for XYXY= ∑(= ∑(xx - - xx)()(yy - - yy))= ∑= ∑xyxy - - (∑(∑xx) (∑) (∑yy))nn©2006 Thomson/South-Western 6Sum of SquaresSum of SquaresSSSSXX= sum of squares for = sum of squares for XX= ∑(= ∑(xx - - xx))22= ∑= ∑xx22 - - (∑(∑xx))22nnSSSSYY= sum of squares for = sum of squares for YY= ∑(= ∑(yy - - yy))22= ∑= ∑yy22 - - (∑(∑yy))22nnSCPSCPXYXY= sum of cross products for = sum of cross products for XYXY= ∑(= ∑(xx - - xx)()(yy - - yy))= ∑= ∑xyxy - - (∑(∑xx) (∑) (∑yy))nnrr = =SCPSCPXYXYSSSSXX SS SSYY©2006 Thomson/South-Western 7Scatter Diagram and Scatter Diagram and Correlation CoefficientCorrelation CoefficientFigure 13.2Figure 13.2©2006 Thomson/South-Western 8Vertical DistancesVertical Distancesdd11dd22dd33dd44dd55dd66dd77dd88dd99dd1010Line Line LLFigure 13.3Figure 13.3||2020||3030||4040||5050||6060||7070||8080XXYYSquare footageSquare footageIncomeIncome©2006 Thomson/South-Western 9Least Squares LineLeast Squares LineThe least squares line is the line The least squares line is the line through the data that minimizes the through the data that minimizes the sum of the differences between the sum of the differences between the observations and the lineobservations and the line∑∑dd22 = = dd1122 + + dd2222 + + dd3322 + … + + … + d dnn22bb11 = = bb00 = = yy - - bb11xxSCPSCPXYXYSSSSXX©2006 Thomson/South-Western 10Least Squares LineLeast Squares LineFigure 13.6Figure 13.6dd11dd22YY = = bb00 + + bb11XX^^YY for for XX = 50 = 50YY for for XX = 50 = 50^^YYXX5050IncomeIncomeSquare footageSquare footageDistance is Distance is YY −− YY^^©2006 Thomson/South-Western 11Sum of Squares of ErrorSum of Squares of ErrorSSE = SSSSE = SSYY - -(SCP(SCPXYXY))22SSSSXXSSE = ∑SSE = ∑dd22 = ∑( = ∑(yy - - yy))22^^©2006 Thomson/South-Western 12Least Squares Line Least Squares Line for Real Estate Datafor Real Estate DataFigure 13.5Figure 13.5YYXX5050IncomeIncomeSquareSquarefootagefootageYY = 4.915 + .3539 = 4.915 + .3539XX^^YY = 20 = 20YY = 22.67 = 22.67^^©2006 Thomson/South-Western 13Assumptions for theAssumptions for the Simple Regression Model Simple Regression Model1.1.The mean of each error component is zeroThe mean of each error component is zeroYY = = 00 + + 11XX + + ee2.2.Each error component (random variable) Each error component (random variable) follows an approximate normal distributionfollows an approximate normal distribution3.3.The variance of the error component is the The variance of the error component is the same for each value of Xsame for each
View Full Document
Unlocking...