09_13ans - STAT 420 Examples for Fall 2007 x 1 x 2 y 0 1 11...

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Unformatted text preview: STAT 420 Examples for 09/13/2007 Fall 2007 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 + 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 = & & & ¡ ¢ £ £ £ ¤ ¥ 88 200 24 200 496 56 24 56 8 , X T Y = & & & ¡ ¢ £ £ £ ¤ ¥ 336 740 96 , C = ( X T X ) – 1 = & & & ¡ ¢ £ £ £ ¤ ¥------ 1625 . 05 . 1375 . 05 . 025 . 025 . 1375 . 025 . 7125 . , a) Obtain the least-squares estimates & ˆ , 1 & ˆ , and 2 & ˆ . & ˆ = ( X T X ) – 1 X T Y = & & & ¡ ¢ £ £ £ ¤ ¥------ 1625 . 05 . 1375 . 05 . 025 . 025 . 1375 . 025 . 7125 . & & & ¡ ¢ £ £ £ ¤ ¥ 336 740 96 = & & & ¡ ¢ £ £ £ ¤ ¥- 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 : β 1 = β 2 = . H a : at least one of β 1 and β 2 is significantly different from 0. 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 if F > 5.79. Decision: Do NOT Reject H ....
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09_13ans - STAT 420 Examples for Fall 2007 x 1 x 2 y 0 1 11...

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