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Solution to Tutorial 4 - Solutions to Tutorial 4 1 An...

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Solutions to Tutorial 4 1. An output of a simple linear regression model Y i = β 0 + β 1 X i + ε i , i = 1 , .., 10 is as follows Coefficients: Estimate Std. Error t value P-value (Intercept) -0.07727 0.12005 -0.644 0.537814 x 0.97295 0.14345 6.783 0.000140 Residual standard error: 0.3761 on 8 degrees of freedom Multiple R-squared: 0.8519, Adjusted R-squared: 0.8333 F-statistic: 46 on 1 and 8 DF, p-value: 0.0001403 (a) b 0 = 0 . 07727 , b 1 = 0 . 97295 , s ( b 0 ) = 0 . 12005 , s ( b 1 ) = 0 . 14345 , t ( b 0 ) = 0 . 644 , t ( b 1 ) = 6 . 783 , R 2 = 0 . 8519 , ˆ σ = 0 . 3761 , r XY = + R 2 = + 0 . 8519 , F-statistic=46 (b) SSE = 0 . 3761 2 8 = 1 . 131610; SST = SSE/ (1 R 2 ) = 7 . 640849; SSR = SSR SSE = 6 . 509239 Source SS DF MS F-vale regression 6.509239 1 6.509239 46.01756 error 1.131610 8 0.1414512 Total 7.640849 9 (c) t = 0 . 97295 1 0 . 14345 = 0 . 1885674 < t (1 0 . 05 / 2 , 8) = 2 . 306 we accept H 0 2. Sales Growth (R code) i 1 2 3 4 5 6 7 8 9 10 X i : Year 0 1 2 3 4 5 6 7 8 9 Y i : Sales 98 135 162 178 221 232 283 300 374 395 (a) There is nonlinear pattern; see the first panel in Figure 1 (b) ˆ z = 10.26 + 1.07 x (SE) (0.2129) (0.0399) 1
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(c) Does the regression line appear to be good to the transformed data? Answer: YES (for the transformed data); See the second panel in Figure 1 (d) See the thrid and 4th panels in Figure 1 below. 0 2 4 6 8 100 200 300 400 x y 0 2 4 6 8 10 12 14 16 18 20 x z 0 2 4 6 8 -0.4 0.0 0.2 0.4 x residuals Histogram of residuals
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