420Hw08ans - STAT 420(due Friday October 24 by 3:00 p.m...

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STAT 420 Fall 2008 Homework #8 (due Friday, October 24, by 3:00 p.m.) From the textbook: 6.7 (a), (b) Year Y X1 X2 X1 X1 X1 X2 X2 X2 1930 2.31 120 52 14400 6240 2704 1931 2.61 84 23 7056 1932 529 1932 1.94 66 20 4356 1320 400 1933 2.30 52 25 2704 1300 625 1934 2.53 72 17 5184 1224 289 1935 2.35 53 33 2809 1749 1089 1936 2.33 68 40 4624 2720 1600 1937 2.45 80 36 6400 2880 1296 1938 1.93 89 40 7921 3560 1600 1939 2.53 67 36 4489 2412 1296 1940 2.33 68 44 4624 2992 1936 25.61 819 366 64567 28329 13364 Year Y X1 X2 X1 Y X2 Y Y Y 1930 2.31 120 52 277.2 120.12 5.3361 1931 2.61 84 23 219.24 60.03 6.8121 1932 1.94 66 20 128.04 38.8 3.7636 1933 2.30 52 25 119.6 57.5 5.29 1934 2.53 72 17 182.16 43.01 6.4009 1935 2.35 53 33 124.55 77.55 5.5225 1936 2.33 68 40 158.44 93.2 5.4289 1937 2.45 80 36 196 88.2 6.0025 1938 1.93 89 40 171.77 77.2 3.7249 1939 2.53 67 36 169.51 91.08 6.4009 1940 2.33 68 44 158.44 102.52 5.4289 25.61 819 366 1904.95 849.21 60.1113 a) X T X = ° ° ° ± ² ³ ³ ³ ´ µ = ° ° ° ± ² ³ ³ ³ ´ µ 13364 28329 366 28329 64567 819 366 819 11 X X X X X X X X X X 2 2 2 1 2 2 1 2 1 1 2 1 i i i i i i i i i i n .
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det ( X T X ) = 34027693. ( X T X ) 1 = ° ° ° ± ² ³ ³ ³ ´ µ - - - - - - × 39476 11865 430071 11865 13048 576702 430071 576702 60341147 34027693 1 = ° ° ° ± ² ³ ³ ³ ´ µ - - - - - - 00116 . 0 00035 . 0 01264 . 0 00035 . 0 000383 . 0 01695 . 0 01264 . 0 01695 . 0 773295 . 1 . X T Y = ° ° ° ± ² ³ ³ ³ ´ µ = ° ° ° ± ² ³ ³ ³ ´ µ 849.21 1904.95 25.61 Y X Y X Y 2 1 i i i i i . ° ˆ = ( X T X ) 1 X T Y = ° ° ° ± ² ³ ³ ³ ´ µ - 00273 . 0 0.000311 2.395922 . Y ˆ = 2.395922 + 0.000311 X 1 – 0.00273 X 2 . b) Year Y X1 X2 Y ˆ e 1930 2.31 120 52 2.29118548 0.01881452 1931 2.61 84 23 2.35920445 0.25079555 1932 1.94 66 20 2.36180525 -0.42180525 1933 2.30 52 25 2.34379937 -0.04379937 1934 2.53 72 17 2.37186308 0.15813692 1935 2.35 53 33 2.32226056 0.02773944 1936 2.33 68 40 2.30780287 0.02219713 1937 2.45 80 36 2.32245614 0.12754386 1938 1.93 89 40 2.31432777 -0.38432777 1939 2.53 67 36 2.31841692 0.21158308 1940 2.33 68 44 2.29687811 0.03312189 25.61 819 366 25.61 0
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> Pr67.dat <- read.table("http://www.stat.uiuc.edu/~stepanov/Pr67.csv", header=T, sep=",") > Pr67.dat Year Y X1 X2 1 1930 2.31 120 52 2 1931 2.61 84 23 3 1932 1.94 66 20 4 1933 2.30 52 25 5 1934 2.53 72 17 6 1935 2.35 53 33 7 1936 2.33 68 40 8 1937 2.45 80 36 9 1938 1.93 89 40 10 1939 2.53 67 36 11 1940 2.33 68 44 > > Pr67.fit <- lm(Y ~ X1 + X2, data=Pr67.dat) > summary(Pr67.fit) Call: lm(formula = Y ~ X1 + X2, data = Pr67.dat) Residuals: Min 1Q Median 3Q Max -0.42181 -0.01249 0.02774 0.14284 0.25080 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.3959222 0.3259146 7.351 7.98e-05 *** X1 0.0003107 0.0047926 0.065 0.950 X2 -0.0027312 0.0083361 -0.328 0.752 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2447 on 8 degrees of freedom Multiple R-squared: 0.01513, Adjusted R-squared: -0.2311 F-statistic: 0.06147 on 2 and 8 DF, p-value: 0.9408 > Pr67.fit$coefficients (Intercept) X1 X2 2.3959222231 0.0003107096 -0.0027311902 > > Pr67.fit$fitted.values 1 2 3 4 5 6 7 8 2.291185 2.359204 2.361805 2.343799 2.371863 2.322261 2.307803 2.322456 9 10 11 2.314328 2.318417 2.296878 > Pr67.fit$residuals 1 2 3 4 5 6 0.01881452 0.25079555 -0.42180525 -0.04379937 0.15813692 0.02773944 7 8 9 10 11 0.02219713 0.12754386 -0.38432777 0.21158308 0.03312189
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OR > X <- cbind(c(rep(1,11)),Pr67.dat$X1,Pr67.dat$X2) > t(X) %*% X [,1] [,2] [,3] [1,] 11 819 366 [2,] 819 64567 28329 [3,] 366 28329 13364 > C <- solve(t(X) %*% X); C [,1] [,2] [,3] [1,] 1.77329527 -0.0169480194 -0.0126388527 [2,] -0.01694802 0.0003834524 -0.0003486866 [3,] -0.01263885 -0.0003486866 0.0011601139 > t(X) %*% Pr67.dat$Y [,1] [1,] 25.61 [2,] 1904.95 [3,] 849.21 > > C %*% t(X) %*% Pr67.dat$Y [,1] [1,] 2.3959222231 [2,] 0.0003107096 [3,] -0.0027311902 > > Yhat <- X %*% C %*% t(X) %*% Pr67.dat$Y; Yhat [,1] [1,] 2.291185 [2,] 2.359204 [3,] 2.361805 [4,] 2.343799 [5,] 2.371863 [6,] 2.322261 [7,] 2.307803 [8,] 2.322456 [9,] 2.314328 [10,] 2.318417 [11,] 2.296878 > > e <- Pr67.dat$Y – Yhat; e [,1] [1,] 0.01881452 [2,] 0.25079555 [3,] -0.42180525 [4,] -0.04379937 [5,] 0.15813692 [6,] 0.02773944 [7,] 0.02219713 [8,] 0.12754386 [9,] -0.38432777 [10,] 0.21158308 [11,] 0.03312189
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