420Hw06ans - STAT 420 (due Friday, October 10, by 3:00...

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STAT 420 Fall 2008 Homework #6 (due Friday, October 10, by 3:00 p.m.) From the textbook: 5.6 a) x y x x - y y - ( ) 2 x x - ( ) ( ) y y x x - - ( ) 2 y y - 2 20 –2 –12.2 4 24.4 148.84 3 27 –1 –5.2 1 5.2 27.04 4 33 0 0.8 0 0 0.64 5 38 1 5.8 1 5.8 33.64 6 43 2 10.8 4 21.6 116.64 20 161 0 0 10 57 326.8 SXX SXY SYY 5 20 = x = 4, 5 161 = y = 32.2. 10 57 SXX SXY ˆ = = β = 5.7 . x y ˆ ˆ α - = = 32.2 – 5.7 4 = 9.4 . b) In class we showed that since ( ) ( ) x x y x x y x y i i i i - + = + - = + = ˆ ˆ ˆ ˆ ˆ ˆ , RSS = ( ) ( ) ( ) 2 2 2 2 2 2 ˆ ˆ ˆ ˆ = - = - = - x x x x y y i i i SXX = 324.9. OR x y y ˆ e e 2 2 20 20.8 -0.8 0.64 3 27 26.5 0.5 0.25 4 33 32.2 0.8 0.64 5 38 37.9 0.1 0.01 6 43 43.6 -0.6 0.36 20 161 161 0 1.9 SSE
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Source SS df MS F Regression 324.9 1 324.9 513 Residual (Error) 1.9 3 0.63333 Total 326.8 4 F 0.05 ( 1, 3 ) = 10.13. Reject H 0 : β = 0. ( p-value 0.000188 ) c) X 0 = 7. 0 ˆ ˆ y ˆ x α + = = 9.4 + 5.7 7 = 49.3. 3 9 . 1 2 SSE 2 = - = n s e = 0.63333. s e = 0.7958. ( ) ( ) ( ) & - - + + - ± 2 2 0 2 1 1 2 t y ˆ x x x x n s n i e t 0.025 ( 3 ) = 3.182. ( ) 10 4 7 5 1 1 7958 . 0 182 . 3 3 . 49 2 - + + ± 49.3 ± 3.67 . X 0 = 7 is outside of the range of the available data. OR > x = c(2,3,4,5,6) > y = c(20,27,33,38,43) > fit = lm(y ~ x) > summary(fit) Call: lm(formula = y ~ x) Residuals: 1 2 3 4 5 -0.8 0.5 0.8 0.1 -0.6 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 9.4000 1.0677 8.804 0.003088 ** x 5.7000 0.2517 22.650 0.000188 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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Residual standard error: 0.7958 on 3 degrees of freedom Multiple R-squared: 0.9942, Adjusted R-squared: 0.9922 F-statistic: 513 on 1 and 3 DF, p-value: 0.0001885 > summary(aov(fit)) Df Sum Sq Mean Sq F value Pr(>F) x 1 324.90 324.90 513 0.0001885 *** Residuals 3 1.90 0.63
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This note was uploaded on 05/09/2010 for the course MATH 420 taught by Professor Stepanov during the Spring '10 term at University of Illinois, Urbana Champaign.

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420Hw06ans - STAT 420 (due Friday, October 10, by 3:00...

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