# hw4_soln4 - Homeworks 10-12 Solutions 3.18(a To obtain the...

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Homeworks 10-12 - Solutions 3.18 (a) To obtain the least squares regression equation, first we will compute the slope and the vertical intercept using the equations provided in Section 3.3. ( 29 ( 29 ( 29 ( 29 62615 . 14 517 652289939 . 14 346 : Also 652289939 . 93 . 002 , 20 71 . 047 , 13 , 93 . 002 , 20 14 517 095 , 39 71 . 047 , 13 14 346 517 825 , 25 2 = - = - = = = = = - = = - = x b y a S S b So S S xx xy xx xy Thus, the equation for the least squares line is: x y 6523 . 0 626 . 0 ˆ + = (b) ( 29 46 . 23 35 65228939 . 62615 . ˆ , 35 When = + = = y x The corresponding residual is: ( 29 ( 29 46 . 2 46 . 23 21 ˆ - = - = - = y y residual 3.21 (c) The least squares line is y ˆ = 3.2925 + .10748x. For a beam whose modulus of elasticity is x = 40, the predicted strength would be y ˆ = 3.2925 + .10748(40) = 7.59. The value x = 100 is far beyond the range of the x values in the data, so it would be dangerous (i.e., potentially misleading) to extrapolate the linear relationship that far. (d) From the printout, SSResid = 18.736, SSTo = 71.605, and the coefficient of determination is r 2 = .738 (or, 73.8%) . The r 2 value is large, which suggests that the linear relationship is a useful approximation to the true relationship between these two variables.

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3.22 Using Minitab to run this regression, we obtain the following least squares regression line: x y 00882 . 0 86 . 2 ˆ + = The anova function in R gives: SS Exp = 0.2273, SS Resid = 0.00911. 3.23 (a) From the 'Parameter Estimate' column of the printout, the least squares line is y ˆ = 3.620906 - 0.014711x, where x = fracture strength. Substituting the value x = 50 into the equation gives a predicted attenuation of y ˆ = 3.620906 - 0.014711(50) = 2.8854, or, about 2.89. (b) From the 'Sum of Squares' column of the printout, SSResid = .26246 and SSTo = 2.55714. The r 2 value is .8974, or 89.74%. s ε is called the 'Root MSE' (where MSE stands for 'mean square error') in the printout, so s ε = .14789. The high vale of r 2 and the small value of s ε (compared to the typical size of the y data values) indicate
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hw4_soln4 - Homeworks 10-12 Solutions 3.18(a To obtain the...

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