財(cái)經(jīng).管理mba統(tǒng)計(jì)學(xué)第10章

1、1Relationship among variables Functional relationship Statistical relationship(correlation) Y depends on X, but isnt merely determined by X. Example: price and sales daily high temperaturethe demand for air-conditioning RegressionAccording to observed data, establish regression equation and make sta

2、tistical reference (predict) .Chapter 10 (P 227) Correlation and Regression Analysis2What does regression do?Solve the following problems:Determine whether there is statistical relationship among variables, if does, give the regression equation. Forecast the value of another variable (dependent) acc

3、ording to one variable or a group of variables (independent).3Example:X-price，Y-sales for a kind of productWe collect data:1. Scatter plot2. Regression equation(the Least Square Estimation)3. Correlation coefficient (Testing the regression model)4.Forecasting (point and interval forecasting )Simple

4、Linear RegressionX(Yuan)708090100110Y(thousand)11.2511.2811.6511.7012.144Linear Regression ModelVariables consist of a linear function. YXiii01SlopeY-InterceptIndependent (Explanatory) VariableDependent (Response) Variable Random Error5Sample Linear Regression Modelei = random errorYXYbbXeiii01YbbXi

5、i01Sampled Observed Value6Sample Linear Regression ModelThe least squares method provides an estimated regression equation that minimizes the sum of squared deviations between the observed values of the dependent variable yi and the estimated values of the dependent variable . 7Least Squares estimat

6、ione2YXe1e3e4YbbXeiii01YbbXii01OLS Min eeeeeii2112223242Predicted Value8Coefficient & EquationYbXbXYnXYXnXbYbXiiiiiniin011122101Sample regression equationSlope for the estimated regression equationP 238 (10.17)Intercept for the estimated regression equationb9Evaluating the ModelSignificance TestTest

7、 Coefficient of Determination and Standard Deviation of EstimationResidual AnalysisYbbXii0110Measures of Variation in Regression SST = SSR + SSE 1. Total Sum of Squares (SST) P 239 (10.20)Measure the variation between the observed value Yi and the mean Y. 2. Sum of Squares due to Regression (SSR) Va

8、riation caused by the relationship between X and Y. 3. Sum of Squares due to Error (SSE) Variation caused by other factors.11Variation MeasuresYXYXiSST (Yi - Y)2 SSE (Yi -Yi)2 SSR (Yi - Y)2 Yi YbbXii0112Coefficient of Determination 0 r2 1rbYbXYnYYnYiiiininiin201211212Explained variation Total variat

9、ionSSRSSTA measure of the goodness of fit of the estimated regression equation. It can be interpreted as the proportion of the variation in the dependent variable y that is explained by the estimated regression equation.13Correlation CoefficientA numerical measure of linear association between two v

10、ariables that takes values between 1 and +1. Values near +1 indicate a strong positivelinear relationship, values near 1 indicate a strong negative linear relationship, and values near zero indicate lack of a linear relationship.14Coefficients of Determination (r2) and Correlation (r)r2 = 1,r2 = 0,Y

11、Yi = b0 + b1XiXYYi = b0 + b1XiXYYi = b0 + b1XiXYYi = b0 + b1XiXr = +1r = -1r = +0.9r = 015Test of Slope Coefficient for Significance1. Tests a Linear Relationship Between X & Y 2.Hypotheses H0: 1 = 0 (No Linear Relationship) H1: 1 0 (Linear Relationship) 3.Test Statistic16Example Test of Slope Coeff

12、icientH0: 1 = 0H1: 1 0 .05df 5 - 2 = 3Critical value:Statistic: Determine:Conclusion:tbSb1110700019153655.Reject at = 0.05There is evidence of a relationship.t03.1824-3.1824.025RejectReject.02517Multiple Regression ModelThere exists linear relationship among an dependent variable and two or more tha

13、n two independent variables.YXXXiiiPPii01122slope of populationintercept of population Yrandom errorDependent VariableIndependent Variables18Example: New York Times You work in the advertisement department of New York Times(NYT). You will find to what extent do ads size(square inch ) and publishing

14、volume (thousand) influence the response to ads(hundred). You have collected the following data: response size volume112488131357264410619Example (NYT) Computer Output Parameter Estimates Parameter Standard T for H0:Variable DF Estimate Error Param=0 Prob|T|INTERCEP 1 0.0640 0.2599 0.246 0.8214ADSIZ

15、E 1 0.2049 0.0588 3.656 0.0399CIRC 1 0.2805 0.0686 4.089 0.0264 b2b0bPb120Interpretation of Coefficients 1.Slope (b1)If the publishing volume remains unchanged,when ads sizeincreases one square inch, the response is expected to increase 0.2049 hundred times. 2.Slope (b2)If ads size remains unchanged

16、, when publishing volume increases one thousand, the response is expected to in- crease 0.2805 hundred times. 21Evaluating the Model1.How does the model describe the relationship among variables? 2.Closeness of Best Fit3.Assumptions met4.Significance of estimates5.Correlation among variables6.Outlie

17、rs (unusual observations)22Testing Overall SignificanceTest whether there is linear relationship between Y and all the independent variables. 2.Use F statistic.Hypothesis H0: 1 = 2 = . = P = 0 There is no linear relationship between Y and independent variables. H1: At least there is a coefficient isnt equal to 0. At least there is an independent variable influences Y23Testing Overall Significance Computer OutputAnalysis of Variance Sum of Mean Source DF Squares Square F Value ProbFModel 2 9.2497 4.6249 55.440 0.0043Error 3 0.2503 0.0834C Total 5 9.5000Pn - P -1n -