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sol4 - PSTAT 120C Solutions for Assignment#4 1 A plot of...

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PSTAT 120C: Solutions for Assignment #4 May 15, 2006 1. A plot of the data is in Figure 1. Figure 1: Plot of the original data. (a) The results of the linear regression for the original data is Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -21.858339 5.743089 -3.806 0.0126 * Distance 0.164069 0.001343 122.210 6.96e-10 *** --- Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 12.14 on 5 degrees of freedom Multiple R-Squared: 0.9997, Adjusted R-squared: 0.9996 F-statistic: 1.494e+04 on 1 and 5 DF, p-value: 6.958e-10 The equation is Olympic Record = - 21 . 86 + 0 . 164 Distance 1
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(b) The standard error for ˆ β 0 , s . d . ( ˆ β 0 ) = s e n x 2 i x 2 i - n ¯ x 2 = 12 . 14324 7 128100000 81814286 = 5 . 74309 Therefore, the t statistic is t = - 21 . 86 5 . 74309 = - 3 . 806 which is larger than the α = 0 . 025 critical value with 5 degrees of freedom t 5 ,. 025 = 2 . 57, and is therefore significantly less than 0. This is a result of the curvature in the model. The 0 point is too far away from the center of the data and the straight line estimate does not fit well there. (c) The computer output is Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.93671 0.09101 -32.27 5.37e-07 *** Log Distance 1.12611 0.01305 86.32 3.95e-09 *** --- Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 0.05341 on 5 degrees of freedom Multiple R-Squared: 0.9993, Adjusted R-squared: 0.9992 F-statistic: 7451 on 1 and 5 DF, p-value: 3.954e-09 log Time = - 2 . 93671 + 1 . 12611 log Distance (d) The standard error for the ˆ β 1 is 0.01305 so that the t statistic is t = ˆ β 1 - 1 s e / ( x i - ¯ x ) 2 = 1 . 12611 - 1 0 . 01305 = 9 . 663602 Therefore, this is a significant statistic (versus, again, t 5 ,. 025 = 2 . 57). (e) The best model is one using the log data, and the time goes up as a function Time = 0 . 053(Distance) 1 . 12 . The linear model does not fit the data as well especially around a distance of 0. 2. (a) The fitted model is Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 22.16269 7.08948 3.126 0.00556 ** Money Spent 0.36317 0.09712 3.739 0.00139 ** --- Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 23.5 on 19 degrees of freedom Multiple R-Squared: 0.424, Adjusted R-squared: 0.3936 F-statistic: 13.98 on 1 and 19 DF, p-value: 0.001389 2
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Figure 2: A plot of the residuals versus the independent variable shows some curvature and the two largest residuals.
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