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# 09_13 - STAT 420 Examples for x1 0 11 11 Fall 2007 1...

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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 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 , ( 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 . 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. c) Test H 0 : β 1 = 0 vs. H a : β 1 0 at α = 0.10. Find the p-value. d) Test H 0 : β 2 = 0 vs. H a : β 2 0 at α = 0.05. Find the p-value. e) Test H 0 : β 0 = β 2 vs. H a : β 0 β 2 at α = 0.10. Find the p-value. f) Test H 0 : 2 β 1 + β 2 = 0 vs. H a : 2 β 1 + β 2 0 at α = 0.05. Find the p-value. g) Construct a 90% prediction interval for the value of Y at x 0 1 = 2 and x 0 2 = 3. h) Construct a 90% prediction interval for the value of Y at x 0 1 = 8 and x 0 2 = 5.

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Y i = β 0 + β 1 x i 1 + … + β p x i p
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09_13 - STAT 420 Examples for x1 0 11 11 Fall 2007 1...

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