# 420Pr3 - 1 Consider the following data set x y a 0 110 2...

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1. Consider the following data set: x 0 2 4 6 8 y 110 123 119 86 62 a) Construct a scatter plot. Does the plot suggest that a linear relationship is appropriate? Consider the model Y i = β 0 + β 1 x i + e i , i = 1, 2, 3, 4, 5, where e i ’s are i.i.d. N ( 0 , σ 2 ). Σ x = 20, Σ y = 500, Σ x 2 = 120, Σ y 2 = 52,630, Σ x y = 1,734, Σ ( x x ) 2 = 40, Σ ( y y ) 2 = 2,630, Σ ( x x ) ( y y ) = Σ ( x x ) y = – 266. OR X T X = ± ² ³ ´ µ 120 20 20 5 ( X T X ) 1 = ± ² ³ ´ µ - - 025 . 0 1 . 0 1 . 0 6 . 0 X T Y = ± ² ³ ´ µ 1,734 500 b) Find the equation of the least-squares regression line. Add the regression line to the scatter plot. Σ ( y y ˆ ) 2 = 861.1. c) What proportion of the observed variation in y values is explained by a straight-line relationship with x ? d) Is regression significant at a 5% level of significance? e) Test H 0 : β 1 = 0 vs. H 1 : β 1 < 0 at a 5% level of significance. f) Test H 0 : β 0 = 100 vs. H 1 : β 0 > 100 at a 5% and at a 10% level of significance. g) Construct a 90% prediction interval for the future value of y corresponding to x = 10.

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Consider the model Y i = β 0 + β 1 x i + β 2 x i 2 + e i , i = 1, 2, 3, 4, 5, where e i ’s are i.i.d. N ( 0 , σ 2 ). X T X = ± ² ³ ³ ³ ´ µ 664 , 5 800 120 800 120 20 120 20 5 ( X T X ) 1 = ± ² ³ ³ ³ ´ µ - - - - 0045 . 0 0357 . 0 0357 . 0 0357 . 0 3107 . 0 3857 . 0 0357 . 0 3857 . 0 8857 . 0 X T Y = ± ² ³ ³ ³ ´ µ 9,460 1,734 500 , ˆ = ± ² ³ ³ ³ ³ ´ µ 2 1 0 ˆ ˆ ˆ = ± ² ³ ³ ³ ´ µ - 8393 . 1 0643 8 8857 111 . . . h) Add the quadratic regression curve to the scatter plot. Does a quadratic regression function provide a better fit to the data than a straight line does? Σ
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## This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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420Pr3 - 1 Consider the following data set x y a 0 110 2...

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