Lecture-9_Simple-Linear-Regression-第九章-简单线性回归分析课件.ppt

Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-1,Chapter 12Simple Linear Regression,Business Statistics:A First CourseFifth Edition,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-2,Learning Objectives,In this chapter,you learn:How to use regression analysis to predict the value of a dependent variable based on an independent variableThe meaning of the regression coefficients b0 and b1How to evaluate the assumptions of regression analysis and know what to do if the assumptions are violatedTo make inferences about the slope and correlation coefficientTo estimate mean values and predict individual values,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-3,Correlation vs.Regression,A scatter plot can be used to show the relationship between two variablesCorrelation analysis is used to measure the strength of the association(linear relationship)between two variablesCorrelation is only concerned with strength of the relationship No causal effect is implied with correlationScatter plots were first presented in Ch.2Correlation was first presented in Ch.3,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-4,Introduction to Regression Analysis,Regression analysis is used to:Predict the value of a dependent variable based on the value of at least one independent variableExplain the impact of changes in an independent variable on the dependent variableDependent variable:the variable we wish to predict or explainIndependent variable:the variable used to predict or explain the dependent variable,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-5,Simple Linear Regression Model,Only one independent variable,XRelationship between X and Y is described by a linear functionChanges in Y are assumed to be related to changes in X,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-6,Types of Relationships,Y,X,Y,X,Y,Y,X,X,Linear relationships,Curvilinear relationships,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-7,Types of Relationships,Y,X,Y,X,Y,Y,X,X,Strong relationships,Weak relationships,(continued),Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-8,Types of Relationships,Y,X,Y,X,No relationship,(continued),Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-9,Linear component,Simple Linear Regression Model,Population Y intercept,Population SlopeCoefficient,Random Error term,Dependent Variable,Independent Variable,Random Error component,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-10,(continued),Random Error for this Xi value,Y,X,Observed Value of Y for Xi,Predicted Value of Y for Xi,Xi,Slope=1,Intercept=0,i,Simple Linear Regression Model,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-11,The simple linear regression equation provides an estimate of the population regression line,Simple Linear Regression Equation(Prediction Line),Estimate of the regression intercept,Estimate of the regression slope,Estimated(or predicted)Y value for observation i,Value of X for observation i,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-12,The Least Squares Method,b0 and b1 are obtained by finding the values of that minimize the sum of the squared differences between Y and:,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-13,Finding the Least Squares Equation,The coefficients b0 and b1,and other regression results in this chapter,will be found using Excel or Minitab,Formulas are shown in the text for those who are interested,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-14,b0 is the estimated mean value of Y when the value of X is zerob1 is the estimated change in the mean value of Y as a result of a one-unit change in X,Interpretation of the Slope and the Intercept,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-15,Simple Linear Regression Example,A real estate agent wishes to examine the relationship between the selling price of a home and its size(measured in square feet)A random sample of 10 houses is selectedDependent variable(Y)=house price in$1000sIndependent variable(X)=square feet,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-16,Simple Linear Regression Example:Data,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-17,Simple Linear Regression Example:Scatter Plot,House price model:Scatter Plot,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-18,Simple Linear Regression Example:Using Excel,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-19,Simple Linear Regression Example:Excel Output,The regression equation is:,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-20,Simple Linear Regression Example:Minitab Output,The regression equation isPrice=98.2+0.110 Square FeetPredictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010S=41.3303 R-Sq=58.1%R-Sq(adj)=52.8%Analysis of VarianceSource DF SS MS F PRegression 1 18935 18935 11.08 0.010Residual Error8 13666 1708Total 9 32600,The regression equation is:,house price=98.24833+0.10977(square feet),Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-21,Simple Linear Regression Example:Graphical Representation,House price model:Scatter Plot and Prediction Line,Slope=0.10977,Intercept=98.248,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-22,Simple Linear Regression Example:Interpretation of bo,b0 is the estimated mean value of Y when the value of X is zero(if X=0 is in the range of observed X values)Because a house cannot have a square footage of 0,b0 has no practical application,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-23,Simple Linear Regression Example:Interpreting b1,b1 estimates the change in the mean value of Y as a result of a one-unit increase in XHere,b1=0.10977 tells us that the mean value of a house increases by 0.10977($1000)=$109.77,on average,for each additional one square foot of size,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-24,Predict the price for a house with 2000 square feet:,The predicted price for a house with 2000 square feet is 317.85($1,000s)=$317,850,Simple Linear Regression Example:Making Predictions,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-25,Simple Linear Regression Example:Making Predictions,When using a regression model for prediction,only predict within the relevant range of data,Relevant range for interpolation,Do not try to extrapolate beyond the range of observed Xs,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-26,Measures of Variation,Total variation is made up of two parts:,Total Sum of Squares,Regression Sum of Squares,Error Sum of Squares,where:=Mean value of the dependent variableYi=Observed value of the dependent variable=Predicted value of Y for the given Xi value,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-27,SST=total sum of squares(Total Variation)Measures the variation of the Yi values around their mean YSSR=regression sum of squares(Explained Variation)Variation attributable to the relationship between X and YSSE=error sum of squares(Unexplained Variation)Variation in Y attributable to factors other than X,(continued),Measures of Variation,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-28,(continued),Xi,Y,X,Yi,SST=(Yi-Y)2,SSE=(Yi-Yi)2,SSR=(Yi-Y)2,_,_,_,Y,Y,Y,_,Y,Measures of Variation,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-29,The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variableThe coefficient of determination is also called r-squared and is denoted as r2,Coefficient of Determination,r2,note:,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-30,r2=1,Examples of r2 Values,Y,X,Y,X,r2=1,r2=1,Perfect linear relationship between X and Y:100%of the variation in Y is explained by variation in X,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-31,Examples of r2 Values,Y,X,Y,X,0 r2 1,Weaker linear relationships between X and Y:Some but not all of the variation in Y is explained by variation in X,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-32,Examples of r2 Values,r2=0,No linear relationship between X and Y:The value of Y does not depend on X.(None of the variation in Y is explained by variation in X),Y,X,r2=0,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-33,Simple Linear Regression Example:Coefficient of Determination,r2 in Excel,58.08%of the variation in house prices is explained by variation in square feet,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-34,Simple Linear Regression Example:Coefficient of Determination,r2 in Minitab,The regression equation isPrice=98.2+0.110 Square FeetPredictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010S=41.3303 R-Sq=58.1%R-Sq(adj)=52.8%Analysis of VarianceSource DF SS MS F PRegression 1 18935 18935 11.08 0.010Residual Error8 13666 1708Total 9 32600,58.08%of the variation in house prices is explained by variation in square feet,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-35,Standard Error of Estimate,The standard deviation of the variation of observations around the regression line is estimated by,WhereSSE=error sum of squares n=sample size,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-36,Simple Linear Regression Example:Standard Error of Estimate in Excel,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-37,Simple Linear Regression Example:Standard Error of Estimate in Minitab,The regression equation isPrice=98.2+0.110 Square FeetPredictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010S=41.3303 R-Sq=58.1%R-Sq(adj)=52.8%Analysis of VarianceSource DF SS MS F PRegression 1 18935 18935 11.08 0.010Residual Error8 13666 1708Total 9 32600,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-38,Comparing Standard Errors,Y,Y,X,X,SYX is a measure of the variation of observed Y values from the regression line,The magnitude of SYX should always be judged relative to the size of the Y values in the sample data,i.e.,SYX=$41.33K is moderately small relative to house prices in the$200K-$400K range,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-39,Assumptions of RegressionL.I.N.E,LinearityThe relationship between X and Y is linearIndependence of ErrorsError values are statistically independentNormality of ErrorError values are normally distributed for any given value of XEqual Variance(also called homoscedasticity)The probability distribution of the errors has constant variance,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-40,Residual Analysis,The residual for observation i,ei,is the difference between its observed and predicted valueCheck the assumptions of regression by examining the residualsExamine for linearity assumptionEvaluate independence assumption Evaluate normal distribution assumption Examine for constant variance for all levels of X(homoscedasticity)Graphical Analysis of ResidualsCan plot residuals vs.X,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-41,Residual Analysis for Linearity,Not Linear,Linear,x,residuals,x,Y,x,Y,x,residuals,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-42,Residual Analysis for Independence,Not Independent,Independent,X,X,residuals,residuals,X,residuals,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-43,Checking for Normality,Examine the Stem-and-Leaf Display of the ResidualsExamine the Boxplot of the ResidualsExamine the Histogram of the ResidualsConstruct a Normal Probability Plot of the Residuals,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-44,Residual Analysis for Normality,Percent,Residual,When using a normal probability plot,normal errors will approximately display in a straight line,-3-2-1 0 1 2 3,0,100,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-45,Residual Analysis for Equal Variance,Non-constant variance,Constant variance,x,x,Y,x,x,Y,residuals,residuals,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-46,Simple Linear Regression Example:Excel Residual Output,Does not appear to violate any regression assumptions,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-47,Inferences About the Slope,The standard error of the regression slope coefficient(b1)is estimated by,where:=Estimate of the standard error of the slope=Standard error of the estimate,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-48,Inferences About the Slope:t Test,t test for a population slopeIs there a linear relationship between X and Y?Null and alternative hypotheses H0:1=0(no linear relationship)H1:1 0(linear relationship does exist)Test statistic,where:b1=regression slope coefficient 1=hypothesized slope Sb1=standard error of the slope,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-49,Inferences About the Slope:t Test Example,Estimated Regression Equation:,The slope of this model is 0.1098 Is there a relationship between the square footage of the house and its sales price?,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-50,Inferences About the Slope:t Test Example,H0:1=0H1:1 0,From Excel output:,b1,Predictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010,From Minitab output:,b1,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-51,Inferences About the Slope:t Test Example,Test Statistic:tSTAT=3.329,There is sufficient evidence that square footage affects house price,Decision:Reject H0,Reject H0,Reject H0,a/2=.025,-t/2,Do not reject H0,0,t/2,a/2=.025,-2.3060,2.3060,3.329,d.f.=10-2=8,H0:1=0H1:1 0,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-52,Inferences About the Slope:t Test Example,H0:1=0H1:1 0,From Excel output:,p-value,There is sufficient evidence that square footage affects house price.,Decision:Reject H0,since p-value,Predictor Coef SE Coef T PConstant 98.25 58.03 1.69 0.129Square Feet 0.10977 0.03297 3.33 0.010,From Minitab output:,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-53,F Test for Significance,F Test statistic:where,where FSTAT follows an F distribution with 1 numerator and(n 2)denominator degrees of freedom,Business Statistics:A First Course,5e 2009 Prentice-Hall,Inc.,Chap 12-54,F-Test for SignificanceExcel Output,With 1 and 8 degrees of freedom,p-value for the F-Test,Business S