Chapter2_Part3 - Chapter2 Review of Simple Linear...

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Chapter2: Review of Simple Linear Regression Part 3 Yeying Zhu Fall, 2013 STAT331 Applied Linear Regression, Chapter2, Part 3 Chapter2: Review of Simple Linear Regression Part 3
Sampling Distribution of ˆ y 0 After β 0 , β 1 are estimated by ˆ β 0 and ˆ β 1 , the fitted value given an x 0 is: ˆ y 0 = ˆ β 0 + ˆ β 1 x 0 E y 0 ) = E ( y 0 ) Var y 0 ) = σ 2 [ 1 n + ( x 0 - ¯ x ) 2 ( x i - ¯ x ) 2 ] Under normality assumption, ˆ y 0 N ( E ( y 0 ) , σ 2 [ 1 n + ( x 0 - ¯ x ) 2 ( x i - ¯ x ) 2 ]) ˆ y 0 - E ( y 0 ) se y 0 ) t n - 2 where se y 0 ) = q s 2 [ 1 n + ( x 0 - ¯ x ) 2 ( x i - ¯ x ) 2 ]. STAT331 Applied Linear Regression, Chapter2, Part 3 Chapter2: Review of Simple Linear Regression Part 3
Confidence Interval for E ( y 0 ) The confidence interval for E ( y 0 ) is given by (point estimate ± Multiplier t ? × se of the point estimate) ( b y 0 ± t n - 2 ,α/ 2 × se y 0 )) where ˆ y 0 = ˆ β 0 + ˆ β 1 x 0 , and se y 0 ) = q s 2 · ( 1 n + ( x 0 - ¯ x ) 2 ( x i - ¯ x ) 2 ). STAT331 Applied Linear Regression, Chapter2, Part 3 Chapter2: Review of Simple Linear Regression Part 3
Prediction Interval for y p A prediction interval for a new y (PI) is an interval estimate for a new individual observation y p (random variable) given a new x p . The new observation on y p to be predicted is viewed as the result of a new trial, independent of the trials on which the regression analysis is based. The new observation can be written as y p = β 0 + β 1 x p + p . where p is a future unknown random error. After β 0 , β 1 are estimated by ˆ β 0 and ˆ β 1 , the point estimator for new y p given a new x p is: ˆ y p = ˆ β 0 + ˆ β 1 x p . STAT331 Applied Linear Regression, Chapter2, Part 3 Chapter2: Review of Simple Linear Regression Part 3
Properties of y p - ˆ y p E ( y p - ˆ y p ) = 0 Var ( y p - ˆ y p ) = σ 2 · (1 + 1 n + ( x p - ¯ x ) 2 ( x i - ¯ x ) 2 ) Under normality assumption, y p - ˆ y p N (0 , σ 2 · (1 + 1 n + ( x p - ¯ x ) 2 ( x i - ¯ x ) 2 ) y p - ˆ y p se ( y p - ˆ y p ) t n - 2 where se ( y p - ˆ y p ) = q s 2 · (1 + 1 n + ( x p - ¯ x ) 2 ( x i - ¯ x ) 2 ) STAT331 Applied Linear Regression, Chapter2, Part 3 Chapter2: Review of Simple Linear Regression Part 3
CI for E ( Y ) and PI for y : Comparison A prediction interval for y p at x p is calculated as ˆ y p ± t n - 2 ,α/ 2 s s 2 · (1 + 1 n + ( x p - ¯ x ) 2 ( x i - ¯ x ) 2 ) Therefore, P.I. is always wider than the corresponding C.I. STAT331 Applied Linear Regression, Chapter2, Part 3 Chapter2: Review of Simple Linear Regression Part 3
Example Example: Consider a sample of n

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