a. Explanation Calculation: Linear regression model: A linear regression model is given as y = b 0 + b 1 x where y ^ be the predicted values of response variable and x be the predictor variable. b 1 be the slope and b 0 be the intercept of the line. The estimated least square line is y ^ = β ^ 0 + β ^ 1 x where y ^ be the predicted values of response variable and x be the predictor variable. The β ^ 1 be the estimate of slope and β ^ 0 be the estimate of intercept of the line. Expectation of y i : The expectation of least square line y i = β 0 + β 1 x i is, E ( y i ) = E ( β 0 + β 1 x i ) = β 0 + β 1 x i Thus, the expectation of y i is β 0 + β 1 x i . Expectation of y ¯ : The expectation of averaged values of least square line y ¯ = β 0 + β 1 x ¯ is, E ( y ¯ ) = E ( β 0 + β 1 x ¯ ) = β 0 + β 1 x ¯ Thus, the expectation of y ¯ is β 0 + β 1 x ¯ . Expectation of ( y i − y ¯ ) : The expectation of the term E ( y i − y ¯ ) is, E ( y i − y ¯ ) = E ( y i ) − E ( y ¯ ) = ( β 0 + β 1 x i ) − ( β 0 + β 1 x ¯ ) = ( β 1 x i − β 1 x ¯ ) = β 1 ( x i − x ¯ ) Thus, the expectation of ( y i − y ¯ ) is β 1 ( x i − x ¯ ) . Least squares estimate of slope β ^ 1 : By the principle of estimation of least squares, the value of b 1 which minimizes the error function f ( b 0 , b 1 ) = ∑ i = 1 n ( y i − ( b 0 + b 1 x i ) ) 2 is, β ^ 1 = ∑ i ( x i − x ¯ ) ( y i − y ¯ ) ∑ i ( x i − x ¯ ) 2 b. To determine Prove that V ( β ^ 1 ) = σ 2 ∑ i ( x i − x ¯ ) 2 .

Probability and Statistics for Engineering and the Sciences

9th Edition

ISBN: 9781305251809

Author: Jay L. Devore

Publisher: Cengage Learning

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Chapter 12.3, Problem 41E

To determine

Prove that β^1 is an unbiased estimator of β1.

To determine

Prove that V(β^1)=σ2∑i(xi−x¯)2.

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Chapter 12.3, Problem 40E

Chapter 12.3, Problem 42E

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Probability and Statistics for Engineering and the Sciences

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