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# 420Hw02ans - STAT 420 Homework#2 Fall 2007 1 Sometimes it...

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STAT 420 Fall 2007 Homework #2 1. Sometimes it is known in advance that the least-squares regression line must go through the origin, i.e., the regression model is of the form Y i = β x i + ε i , i = 1, 2, … , n , where ε i ’s are i.i.d. N ( 0, σ 2 ) , and the equation of the regression line is y ˆ = ° ˆ x . In this case, finding the least-squares line reduces to finding the value ° ˆ that minimizes the expression ( ) [ ] ° = - = n i i i x y f 1 2 ° ° . Use the derivative of f with respect to β to derive the formula for the slope of the least-squares regression line in this case. We want to minimize ( ) [ ] ° = - = n i i i x y b f 1 2 ° . ( ) [ ] ( ) ° ° ° = = = - = - - = n i i i n i i n i i i i y x x x x y b f 1 1 2 1 2 2 2 ° ° ' . To find the extremum points: ( ) 0 ° ' = f . Therefore, ° ° = = = n i i n i i i x y x 1 2 1 ° ˆ . ( ) ° = = n i i x f 1 2 2 ° " > 0. ± ( ) ° f has minimum at ° ˆ . OR ( ) [ ] [ ] ° ° = = + - = - = n i i i i i n i i i x y x y x y b f 1 2 2 2 1 2 ° ° ° 2 = 2 2 1 2 ° ° 1 1 2 ° = ° = ² ² ³ ´ µ µ · + ² ² ³ ´ µ µ · - ² ² ³ ´ µ µ · ° = n i i x n i i y i x y n i i – parabola. The vertex (minimum) is at ° ° = = = = - n i i n i i i x y x a b 1 2 1 2 ° ˆ .

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2. It has been proposed that the brightness measured in some unit of color or a commercial product is proportional to the time it is in a certain chemical reaction during the production process, or Y i = β x i + ε i , i = 1, 2, … , n , where ε i ’s are i.i.d.
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420Hw02ans - STAT 420 Homework#2 Fall 2007 1 Sometimes it...

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