Explanation Given info: The appropriate model for the regression line of the variables shelf space ( x ) and sales revenue ( y ) to pass through the point ( 0 , 0 ) is Y = β 1 x + ε . Calculation: Least squares estimate: In a linear equation y = b 0 + b 1 x i the constant b 1 be the slope and b 0 be the y -intercept form and x is the independent variable and y is the independent variable. The error function for the equation is, f ( b 0 , b 1 ) = ∑ i = 1 n ( y i − ( b 0 + b 1 x i ) ) 2 In the linear equation the point estimates of the b 0 and b 1 are β ^ 0 and β ^ 1 . These point estimates are the values that minimize the equation f ( b 0 , b 1 ) . That is, f ( β ^ 0 , β ^ 1 ) ≤ f ( b 0 , b 1 ) . Then the values of β ^ 0 and β ^ 1 are the least squares estimates and the equation y ^ = β ^ 0 + β ^ 1 x i Here, in the equation Y = β 1 x + ε the least squares estimate of β 1 is β ^ 1 . From the concept of least squares, the value of β ^ 1 is the value of β 1 which minimizes the equation ∑ i = 1 n ( Y i − ( β 1 x i ) ) 2 Minima: Let y be an objective function in terms of x. To obtain the value of x that minimizes the objective function, the first derivative of y with respect to x must be equalized to 0. Then the obtained value of x is considered as the value which minimizes the objective function y

WEBASSIGN ACCESS FOR PROBABILITY & STATS

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ISBN: 9780357893111

Author: DEVORE

Publisher: CENGAGE L

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Chapter 12.2, Problem 27E

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Derive the least squares estimate of β1.

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Chapter 12 Solutions

WEBASSIGN ACCESS FOR PROBABILITY & STATS

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Derive the least squares estimate of β 1 .

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Chapter 12 Solutions