Stat 401 B – Lecture 321Indicator Variables Response: Highway MPG Explanatory: 2 explanatory variables Indicator variables for types of drive – All Wheel, Rear Wheel, and Front Wheel.2All Wheel Indicator All Wheel = 1 if the vehicle has all wheel drive. All Wheel = 0 if the vehicle does not have all wheel drive.3Rear Wheel Indicator Rear Wheel = 1 if the vehicle has rear wheel drive. Rear Wheel = 0 if the vehicle does not have rear wheel drive.Stat 401 B – Lecture 324Front Wheel Indicator There is no need for a separate indicator variable for front wheel drive. If All Wheel = 0 and Rear Wheel = 0, then the vehicle has front wheel drive.5Multiple Regression Fit a multiple regression model with the two indicator variables as the explanatory variables and Highway MPG as the response.6RSquareRSquare AdjRoot Mean Square ErrorMean of ResponseObservations (or Sum Wgts)0.2541030.2387235.31162627.7100Summary of FitModelErrorC. TotalSource29799DF932.30312736.69693669.0000Sum ofSquares466.15228.213Mean Square16.5224F Ratio<.0001*Prob > FAnalysis of VarianceInterceptAll WheelRear WheelTerm29.983333-7.374638-3.453922Estimate0.6857281.3026481.459395Std Error43.72-5.66-2.37t Ratio<.0001*<.0001*0.0199*Prob>|t|Parameter EstimatesAll WheelRear WheelSource11Nparm11DF904.23720158.02813Sum ofSquares32.05005.6012F Ratio<.0001*0.0199*Prob > FEffect TestsResponse Highway MPGStat 401 B – Lecture 327Prediction EquationPredicted Highway MPG = 29.983 – 7.375*All Wheel – 3.454*Rear Wheel8Summary R2= 0.254 adj R2= 0.239 RMSE = 5.312 Model is useful. F = 16.5224, P-value < 0.00019Interpretation Estimated intercept: When All Wheel = 0 and Rear Wheel = 0 the predicted Highway MPG is 29.983. The average Highway MPG for front wheel drive cars is 29.983.Stat 401 B – Lecture 3210Interpretation Estimated slope for All Wheel: Holding Rear Wheel = 0, if you change All Wheel from 0 to 1 you are comparing the Front Wheel drive to the All Wheel drive. An All Wheel drive vehicle gets, on average, 7.375 mpg less than a Front Wheel drive vehicle.11Interpretation Estimated slope for Rear Wheel: Holding All Wheel = 0, if you change Rear Wheel from 0 to 1 you are comparing the Front Wheel drive to the Rear Wheel drive. An Rear Wheel drive vehicle gets, on average, 3.454 mpg less than a Front Wheel drive vehicle.12Comparisons Are the differences between All Wheel and Front Wheel and Rear Wheel and Front Wheel statistically significant?Stat 401 B – Lecture 3213Comparison All Wheel to Front Wheel. Difference in average Highway mpg is 7.375. t = –5.66, F = 32.05 P-value < 0.0001 The difference is statistically significant because the P-value is so small.14Comparison Rear Wheel to Front Wheel. Difference in average Highway mpg is 3.454. t = –2.37, F = 5.60 P-value < 0.0199 The difference is statistically significant because the P-value is so small.15Comparison All Wheel to Rear Wheel. This comparison is not considered in the JMP output. However, there is enough information to figure it out.Stat 401 B – Lecture 3216Predictions All Wheel: Predicted Highway MPG = 29.983 – 7.375 = 22.608 Rear Wheel: Predicted Highway MPG = 29.983 – 3.454 = 26.529 17Comparison All Wheel to Rear Wheel. Difference in average Highway mpg is 22.608 – 26.529 = –3.921 Is this difference statistically significant?18Test of Significance Difference in means()ErrordfdfnnRMSEyyt=⎟⎟⎠⎞⎜⎜⎝⎛+−=212111Stat 401 B – Lecture 3219Summary6029.983Front Wheel1726.529Rear Wheel2322.608All WheelSample SizeMean20Comparison All Wheel to Rear Wheel() ()97 with 308.26989.1921.31712313116.5529.26608.22112121=−−=⎟⎠⎞⎜⎝⎛+−=⎟⎟⎠⎞⎜⎜⎝⎛+−=dftnnRMSEyyt21Comparison All Wheel to Rear Wheel. Difference in average Highway mpg is 3.921. t = –2.31, F = 5.33 P-value = 0.0231 The difference is statistically significant because the P-value is so small (< 0.05).Stat 401 B – Lecture 3222-25-20-15-10-50510152025Highway MPGResidual15 20 25 30 35 40Highw ay MPG Predicted23Residuals vs. Predicted The predicted values correspond to the three types of drive; All Wheel, Rear Wheel and Front Wheel. Front Wheel drive vehicles show more variation than the other types of drives.24.01.05.10.25.50.75.90.95.99-3-2-10123Normal Quantile Plot10203040Count-15 -10 -5 0 5 10 15 20 25ResidualStat 401 B – Lecture 3225Interpretation Histogram: Mounded to the left of zero and skewed to the right. Box Plot: Skewed right with one potential outlier on the negative side and several potential outliers on the positive side.26Interpretation Normal Quantile Plot: the points start off following the red diagonal (normal model) line but then dip below, curve above and dip below. This indicates a skewed right distribution.27Conditions? Equal standard deviation condition may not be met. Identically distributed condition may not be met. Normally distributed condition may not be
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