ex04ans - STAT 420 Examples#4 x1 0 11 11 Spring 2008 1...

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STAT 420 Examples #4 Spring 2008 x 1 x 2 y 0 1 11 11 5 15 11 4 13 7 3 14 4 1 0 10 4 19 5 4 16 1. Consider the following data set: Consider the model Y i = β 0 + β 1 x i 1 + β 2 x i 2 + e i ., i = 1, … , 8. where e i ’s are i.i.d. N ( 0, 2 e σ ) . 8 2 8 Then X T X = ° ° ° ± ² ³ ³ ³ ´ µ 88 200 24 200 496 56 24 56 8 , X T Y = ° ° ° ± ² ³ ³ ³ ´ µ 336 740 96 , C = ( X T X ) 1 = ° ° ° ± ² ³ ³ ³ ´ µ - - - - - - 1625 . 0 05 . 0 1375 . 0 05 . 0 025 . 0 025 . 0 1375 . 0 025 . 0 7125 . 0 , a) Obtain the least-squares estimates 0 ° ˆ , 1 ° ˆ , and 2 ° ˆ . ° ˆ = ( X T X ) 1 X T Y = ° ° ° ± ² ³ ³ ³ ´ µ - - - - - - 1625 . 0 05 . 0 1375 . 0 05 . 0 025 . 0 025 . 0 1375 . 0 025 . 0 7125 . 0 ° ° ° ± ² ³ ³ ³ ´ µ 336 740 96 = ° ° ° ± ² ³ ³ ³ ´ µ - 4.4 0.7 3.7 . SYY = Σ ( y y ) 2 = 240, RSS = Σ ( y y ˆ ) 2 = 76.4, b) Perform the significance of the regression test at a 5% level of significance. H 0 : β 1 = β 2 = 0 . H a : at least one of β 1 and β 2 is significantly different from 0.
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Source SS DF MS F Regression 163.6 p – 1 = 2 81.8 5.3534 Error (Residual) 76.4 n p = 5 15.28 Total 240 n – 1 = 7 Critical Value: F 0.05 ( 2 , 5 ) = 5.79 . Reject H 0 if F > 5.79.
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