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Lecture22 - lecture 22 Linear Regression III lecture 22...

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lecture 22, Linear Regression III 1 / 26 Confidence and Prediction Intervals The Simple Coefficient of Determina- tion lecture 22, Linear Regression III Outline 1 Confidence and Prediction Intervals 2 The Simple Coefficient of Determination

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lecture 22, Linear Regression III 2 / 26 Confidence and Prediction Intervals The Simple Coefficient of Determina- tion lecture 22, Linear Regression III Confidence and Prediction Intervals Confidence and Prediction Intervals Recall that for an x 0 inside the experimental region, ˆ y = b 0 + b 1 x 0 , is the point estimate of the mean value of the response ( μ y | x 0 = β 0 + β 1 x 0 ) when the value of the predictor is x 0 ; is also the point prediction of an individual value of the response ( y = β 0 + β 1 x 0 + ) when the value of the predictor is x 0 . There will always be estimation/prediction errors! And to provide information about the likely size of the errors, just as before, we will give interval estimates/predictions : We will compute confidence interval (CI) for the mean value of the response when the predictor is equal to a given value; and prediction intervals (PI) for an individual value of the response when the predictor is equal to a given value.
lecture 22, Linear Regression III 3 / 26 Confidence and Prediction Intervals The Simple Coefficient of Determina- tion lecture 22, Linear Regression III Confidence and Prediction Intervals Distance Value Both the CI for the mean value of the response and the PI for an individual value of the response employ a quantity called the distance value The distance value for a particular value x 0 of the predictor is Distance value = 1 n + ( x 0 - ¯ x ) 2 SXX . The distance value is a measure of the distance between the value x 0 and ¯ x : The further x 0 is from ¯ x , the larger is the distance value.

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lecture 22, Linear Regression III 4 / 26 Confidence and Prediction Intervals The Simple Coefficient of Determina- tion lecture 22, Linear Regression III Confidence and Prediction Intervals Formulas for CI and PI Assume that the model assumptions hold The formula for a 100 ( 1 - α )% CI for the mean value of y is as follows: ˆ y ± t α/ 2 s distance value , (1) The formula for a 100 ( 1 - α )% PI for an individual value of y is as follows: ˆ y ± t α/ 2 s 1 + distance value , (2) where t α/ 2 is based on n - 2 degrees of freedom. The further x 0 is from ¯ x , the wider are the CI and PI! The PI is always wider than the CI!
lecture 22, Linear Regression III 5 / 26 Confidence and Prediction Intervals The Simple Coefficient of Determina- tion lecture 22, Linear Regression III Confidence and Prediction Intervals Why is PI Wider than CI? The mean value of y given x = x 0 is μ y | x 0 = β 0 + β 1 x 0 The population of all possible values of ˆ y is normally distributed with mean μ y | x 0 and variance σ 2 · distance value.

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Lecture22 - lecture 22 Linear Regression III lecture 22...

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