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LCT22 - Introduction to Business Statistics Lecture 22...

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1 Introduction to Business Statistics L e c t u r e 2 2 Regression Analysis II A ( α 1 ) confidence interval for ) | ( X Y E when p X X = : Y S t Y ˆ 2 / ˆ ± , where p X b b Y 1 0 ˆ + = and SSX X X n s S p Y 2 ˆ ) ( 1 + = The test statistics for 0 0 ) | ( : E X Y E H p = is Y S E Y t ˆ 0 ˆ = Explanation : Since SSX X X n p Y 2 2 2 ˆ ) ( 1 [ + = σ σ ], we use SSX X X n s S p Y 2 ˆ ) ( 1 + = for the standard error of Y ˆ .

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2 Note that 1 1 1 1 0 ) ( ˆ b X X Y X b X b Y X b b Y p p p + = + = + = . Because ) var( ) ( ] ) ( , cov[ 2 ) var( ] ) ( var[ ) ˆ var( 1 2 1 1 b X X b X X Y Y b X X Y Y p p p + + = + = , and SSX b b Y n Y 2 1 1 2 ) var( , 0 ) , cov( , ) var( σ = = = , we found that the variance of Y ˆ equals ] ) ( 1 [ 2 2 2 ˆ SSX X X n p Y + = σ . Do the same for all possible values of p X within the range of X ’s, we then obtain a band that encloses the true population regression line with high confidence. Next we will construct a band (wider than the previous one), which will enclose a future observation of Y with high probability.
3 P rediction interval for a future value of Y when p X X = : ) ˆ ( 2 / ˆ Y Y S t Y ± α Explanation: Using the estimated conditional mean Y ˆ to predict a future value of Y , two errors will result. In more details, 4 43 4 42 1 43 42 1 Error Estimation Error n Observatio ˆ ) | ( ˆ

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LCT22 - Introduction to Business Statistics Lecture 22...

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