Lecture 5_RegressionInference-2012

Lecture 5_RegressionInference-2012 - Lecture 5 Stat 102...

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1 Lecture 5 Stat 102 Spring, 2012 Inference in Simple Regressions Read Chapter 3.3 Tests and CI s for ! 1 , 0 . Example: 2003 house prices in ZIP 30062
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2 Regression Model (review) Y = ! 0 + 1 x + e μ Y x = 0 + 1 x , a straight line the model for the mean of Y at given values of x Y = dependent variable x = independent variable 0 = y-intercept 1 = slope of line e = error normally distributed with mean = 0 and Var e ( ) = e 2 The errors are assumed to be independent of each other.
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3 Sampling Distribution of b 0 , b 1 b 0 , b 1 are the least squares estimates of ! 0, 1 The “sampling distribution” of b 0 , b 1 is the probability distribution of these estimated values over repeated samples y 1 ,.., y n from the ideal linear regression model with fixed values of 0, 1 and e 2 and x 1 ,.., x n . b 0 and b 1 have normal distributions, with nice formulas for their expectation and variance. (See later pages.) These important facts can be mathematically proved. This is done in more advanced courses. The appendix of these notes describes a Monte-Carlo demonstration to illustrate these facts.
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4 Sampling Distribution (Details) Sampling distribution of b 0 is normal with E b 0 ( ) = ! 0 Var b 0 ( ) = e 2 1 n + x 2 n " 1 ( ) s x 2 # $ % ' ( where s x 2 ! 1 n ! 1 x i ! x ( ) 2 i " Sampling distribution of b 1 is normal with E b 1 ( ) = 1 Var b 1 ( ) = e 2 n " 1 ( ) s x 2
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Hence: If we knew ! e 2 we could produce a confidence interval for 1 as b 1 ± z 2 " e 2 n # 1 ( ) s x 2 , where z 2 comes from the normal tables. Here, the term e 2 n " 1 ( ) s x 2 is the standard deviation of b . So we must first estimate e 2 .
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This note was uploaded on 04/04/2012 for the course STAT 102 taught by Professor Shaman during the Spring '08 term at UPenn.

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Lecture 5_RegressionInference-2012 - Lecture 5 Stat 102...

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