2018fall-exam1-solutions.pdf - ISyE4031-B Regression and...

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1 ISyE4031 - B Regression and Forecasting Exam 1 Solutions Fall 2018 1. a. Since 0 ˆ = y - 1 ˆ x => => y = 8.25 + (2.923)(4.6) = 21.696. b. 1 ˆ = SS xy / SS xx => SS xy = (2.923)(44.4) = 129.7812 7495 . 52 ) 129.7812 )( 923 . 2 ( 1 . 432 ˆ 1 xy yy S SS SSE 2 s = MSE = SSE /( n - 2) = 52.7495/8 = 6.594. c. 8779 . 0 ) 1 . 432 )( 4 . 44 ( 7812 . 129 2 2 2 yy xx xy SS SS SS R (87.79%) or 8779 . 0 1 . 432 7812 . 129 923 . 2 ˆ 1 2 yy xy SS SS R or find R and square. 2. a. 1 ˆ 1 / ˆ s T => T s / ˆ 1 ˆ 1 = 2.34/20.387 = 0.1148. b. 981 . 0 2 yy SS SSR R => 981 . 0 43 . 912 2 R SSR SS yy = 930.102 = Total variation. c. SSE = 930.102 912.43 = 17.67 => MSE = SSE /8 = 17.67/8 = 2.208. 3. a. Least squares estimator: Upon setting the derivative to zero, we obtain ) ( ) ˆ ( 2 ) ˆ ( ˆ 1 1 1 2 1 1 i n i i i n i i i x x y y x y y = 0 => 0 ˆ 2 1 i i i i x x y x y => x y n x y x i i i 2 1 ˆ => n i i n i i i x x y n x y 1 2 1 1 ˆ (Also, 2 1 ) ( ˆ i i i x y y x and 2 1 ˆ i xy x SS ).
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