Lecture22 - lecture 22, Linear Regression III 1 / 26...

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Unformatted text preview: 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 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 inside the experimental region, ˆ y = b + b 1 x , • is the point estimate of the mean value of the response ( μ y | x = β + β 1 x ) when the value of the predictor is x ; • is also the point prediction of an individual value of the response ( y = β + β 1 x + ) when the value of the predictor is x . • 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 of the predictor is Distance value = 1 n + ( x- ¯ x ) 2 SXX . • The distance value is a measure of the distance between the value x and ¯ x : • The further x is from ¯ x , the larger is the distance value. 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 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?...
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This note was uploaded on 12/28/2010 for the course BUSINESS A ISOM 111 taught by Professor Yingyingli during the Fall '10 term at HKUST.

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Lecture22 - lecture 22, Linear Regression III 1 / 26...

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