231hw10solutionsF11

# 231hw10solutionsF11 - Will Landau STAT 231 Homework 10...

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Will Landau November 14, 2011 STAT 231 Homework 10 Solutions Exercise 1.1 (Prof. Vardeman’s Ex 1) . a. I use the least squares line, b y i = β 0 + β 1 x i . I estimate β 0 and β 1 using formulas 12.2 and 12.3 on page 479. To use the formulas, I ﬁrst calculate the following: n X i =1 x i = - 1 + 0 + 1 + 2 + 3 = 5 n X i =1 y i = 4 + 4 + 3 + 3 + 2 = 16 n X i =1 x 2 i = ( - 1) 2 + 0 2 + 1 2 + 2 2 + 3 2 = 15 n X i =1 y 2 i = 4 2 + 4 2 + 3 2 + 3 2 + 2 2 = 54 n X i =1 x i y i = ( - 1)(4) + (0)(4) + (1)(3) + (2)(3) + (3)(2) = 11 x = n i =1 x i n = 5 5 = 1 y = n i =1 y i n = 16 5 = 3 . 2 S xy = n X i =1 x i y i - ( n X i =1 x i )( n X i =1 y i ) /n = 11 - (5)(16) / 5 = - 5 S xx = n X i =1 x 2 i - ( n X i =1 x i ) 2 /n = 15 - 5 2 / 5 = 10 S yy = n X i =1 y 2 i - ( n X i =1 y i ) 2 /n = 54 - 16 2 / 5 = 2 . 8 1

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Now, I can estimate the regression line parameters: b β 1 = S xx S yy = - 5 10 = -0.5 b β 0 = y - b β 1 x = 3 . 2 - ( - 0 . 5)(1) = 3.7 And I get the sample correlation using formula 12.8 on page 509: r xy = S xy S xx p S yy = - 5 10 2 . 8 = -0.9449 For the correlation between b y and y , I need ﬁnd b y i , S b yy , and S b y b y : b y 1 =3 . 7 + ( - 0 . 5)( - 1) = 4 . 2 b y 2 =3 . 7 + ( - 0 . 5)(0) = 3 . 7 b y 3 =3 . 7 + ( - 0 . 5)(1) = 3 . 2 b y 4 =3 . 7 + ( - 0 . 5)(2) = 2 . 7 b y 5 =3 . 7 + ( - 0 . 5)(3) = 2 . 2 n X i =1 b y i = 4 . 2 + 3 . 7 + 3 . 2 + 2 . 7 + 2 . 2 = 16 n X i =1 b y 2 i = 4 . 2 2 + 3 . 7 2 + 3 . 2 2 + 2 . 7 2 + 2 . 2 2 = 53 . 7 n X i =1 b y i y i = (4 . 2)(4) + (3 . 7)(4) + (3 . 2)(3) + (2 . 7)(3) + (2 . 2)(2) = 53 . 7 S b yy = n X i =1 b y i y i - ( n X i =1 b y i )( n X i =1 y ) /n = 53 . 7 - (16)(16) / 5 = 2 . 5 S b y b y = n X i =1 b y 2 i - ( n X i =1 b y i ) 2 /n = 53 . 7 - 16 2 / 5 = 2 . 5 r b yy = S b yy p S b y b y p S yy = 2 . 5 2 . 5 2 . 8 Now, the coeﬃcient of determination r 2 is just r 2 b yy : r 2 = r 2 b yy = ( 2 . 5 2 . 5 · 2 . 8 ) 2 = 0.8928571
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231hw10solutionsF11 - Will Landau STAT 231 Homework 10...

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