t-value for Price is t = -2.306, with p-value .0398t-value for Advertising is t = 2.855, with p-value .0145(continued)Are Individual Variables Significant?19d.f. = 15-2-1 = 12= .05t/2 = 2.1788Inferences about the Slope: t Test ExampleH0: βi= 0HA: βi0The test statistic for each variable falls in the rejection region (p-values < .05)There is evidence that both Price and Advertising affect pie sales at = .05From Excel output: Reject H0 for each variableCoefficientsStandard Errort StatP-valuePrice-24.9750910.83213-2.305650.03979Advertising74.1309625.967322.854780.01449Decision:Conclusion:Reject H0Reject H0/2=.025-tα/2Do not reject H00tα/2/2=.025-2.17882.178820Confidence Interval Estimate for the SlopeConfidence interval for the population slope β1 (the effect of changes in price on pie sales):Example:Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling priceib2/istbCoefficientsStandard Error…Lower 95%Upper 95%Intercept306.52619114.25389…57.58835555.46404Price-24.9750910.83213…-48.57626-1.37392Advertising74.1309625.96732…17.55303130.70888where t has (n – k – 1) d.f.21Standard Deviation of the Regression ModelThe estimate of the standard deviation of the regression model is:MSEknSSEs1Is this value large or small? Must compare to the mean size of y for comparison22Regression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15ANOVAdfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888The standard deviation of the regression model is 47.46 (continued)Standard Deviation of the Regression Model23The standard deviation of the regression model is 47.46A rough prediction range for pie sales in a given week isPie sales in the sample were in the 300 to 500 per week range, so this range is probably too large to be acceptable. The analyst may want to look for additional variables that can explain more of the variation in weekly sales(continued)Standard Deviation of the Regression Model94.22(47.46)24
5MulticollinearityMulticollinearity: High correlation exists between two independent variablesThis means the two variables contribute redundant information to the multiple regression model 25Detect Collinearity (Variance Inflationary Factor)VIFjis used to measure collinearity:If VIFj> 5, xjis highly correlated with the other explanatory variablesR2jis the coefficient of determination when the jthindependent variable is regressed against the remaining k – 1 independent variables211jjRVIF26Detect Collinearity in PHStatOutput for the pie sales example:Since there are only two explanatory variables, only one VIF is reported VIF is < 5 There is no evidence of collinearity between Price and AdvertisingRegression AnalysisPrice and all other XRegression StatisticsMultiple R0.030437581R Square0.000926446Adjusted R Square-0.075925366Standard Error1.21527235Observations15VIF1.000927305
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- Fall '09