Lecture-9_Simple-Linear-Regression-第九章-简单线性回归分析课件.ppt
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1、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 analy
2、sis 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 coefficien
3、tTo 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(li
4、near 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
5、,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
6、 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
7、 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,
8、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 Re
9、gression 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,Predic
10、ted 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),Estim
11、ate 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 minimi
12、ze 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
13、 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,
14、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)=hous
15、e 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:
16、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
17、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 Varia
18、nceSource 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,H
19、ouse 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 ob
20、served 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 increas
21、e 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 f
22、or 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 with
23、in 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 Squa
24、res,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
25、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 Co
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