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420Hw09ans - STAT 420 Fall 2008 Homework#9(due Friday...

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Unformatted text preview: STAT 420 Fall 2008 Homework #9 (due Friday, October 31, by 3:00 p.m.) From the textbook: 1. 5.12 (d) x y y ˆ = 5 x e = y – y ˆ 1.0 3.4 5 – 1.6 1.2 7.0 6 1.0 1.4 6.0 7 – 1.0 1.6 9.0 8 1.0 1.8 8.0 9 – 1.0 Recall Homework 4: & ˆ = & & = = n i i n i i i x y x 1 2 1 = 5. 2.0 11.0 10 1.0 Without β in the model, SSTotal = Y T Y = Σ y 2 , dfTotal = n . SSE = Σ e 2 . Without β in the model, RegrSS = Y Y ˆ ˆ T = & 2 y ˆ = Σ 2 & ˆ x 2 . > x <- c(1.0, 1.2, 1.4, 1.6, 1.8, 2.0) > y <- c(3.4, 7.0, 6.0, 9.0, 8.0, 11.0) > fit <- lm(y ~ x + 0) or > fit <- lm(y ~ x - 1) > summary(fit) Call: lm(formula = y ~ x - 1) Residuals: 1 2 3 4 5 6 -1.6 1.0 -1.0 1.0 -1.0 1.0 Coefficients: Estimate Std. Error t value Pr(>|t|) x 5.0000 0.3263 15.32 2.15e-05 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 1.23 on 5 degrees of freedom Multiple R-Squared: 0.9791, Adjusted R-squared: 0.975 F-statistic: 234.8 on 1 and 5 DF, p-value: 2.148e-05 OR Without β in the model, SSTotal = Y T Y = Σ y 2 = 362.56. Without β in the model, dfTotal = n = 6. SSError = Σ e 2 = 7.56. Without β in the model, SSRegr = Y Y ˆ ˆ T = & 2 ˆ y = 355. Source SS df MS F Regression 355 1 355 234.7884 Error (Residual) 7.56 5 1.512 Total 362.56 6 F 0.01 ( 1 , 5 ) = 16.26 . Reject H : β = 0 at α = 0.01. OR > sum(fit$residuals^2) [1] 7.56 > anova(fit) Analysis of Variance Table Response: y Df Sum Sq Mean Sq F value Pr(>F) x 1 355.00 355.00 234.79 2.148e-05 *** Residuals 5 7.56 1.51 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 F 0.01 ( 1 , 5 ) = 16.26 . Reject H : β = 0 at α = 0.01. 2. 6.12 E ( Y | X ) = β 1 X + β 2 X 2 V ( Y | X ) = I σ 2 , σ 2 > 0 a) Y = & & & & & & & & ¡ ¢ £ £ £ £ £...
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420Hw09ans - STAT 420 Fall 2008 Homework#9(due Friday...

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