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# Regression Analysis STA 108

UCD

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This 3 page Class Notes was uploaded by Carmen Mayer on Tuesday September 8, 2015. The Class Notes belongs to STA 108 at University of California - Davis taught by Staff in Fall. Since its upload, it has received 65 views. For similar materials see /class/191918/sta-108-university-of-california-davis in Statistics at University of California - Davis.

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Date Created: 09/08/15

Information Chapter 1 Simple linear regression model The data consist of 71 pairs of observations X1Y1 Xn Y The model is Yi 051Xi ii ln where 81En are independent 0 and Var i 0392 From this it follows that g lXi Va7 Yl 0392 and that 31Yn are independent Estimation of 5 0 and 5 1 least squares method For this method we nd the straight line equivalent to nding its intercept b0 and its slope 31 which ts the data the closest Let Q 7 2 K 7 be 7 bps2 199 The least squares method minimizes Q with respect to be and b1 Calculus tells us this is achieved by setting the derivatives of Q with respect to be and b1 to zero and then solving the simultaneous equations ie solve 8Q 7 8Q 7 8120 70781 70 lt can be shown these lead to the equations nbo XXIb1 7 Zn ZXm 229 2X32 These equations are called the normal equations and the solutions are given by 51 ZXi XXE 3be 7 7 517 XXXz39 XV Note that be estimates g and b1 estimates 31 So the tted regression line is l7 be le For the Housing data we have n 19 7 15719 7 75211 X 772 40805 20 772 556078 X 7 7Y 7 Y 120001 80 51 14209800051 2941 and 51 75211 7 294115719 28981 So the tted line is 17 28981 2941X So when X 183 estimated selling price is 17 28981 2941183 8339 Fitted response and residuals Before we can estimate the variance 0392 Y values are bg lei i 1 n The residuals are de ned by 61 Y 7 Y 7 bg lei Note that Ei Y 7 g lXi Now Ei is unknown since we do not know g and 1 However if we substitute g and 1 by its estimates bg and b1 we get the residual el Y 7 bg lei So 61 is we need concepts of tted Yevalues and residuals The tted an estimate of 81 The fact that 61 estimate is a very useful fact and we will see this again in chapter 3 Here are some properties of the estimated regression line and the residuals i Z 5 0 ii 6 S 7 U0 7 ulXi2 for any real numbers 11g and 111 iii 2 Y 7 z is iv 2 Xiel 0 v 2 e 7 0 vi the regression line passes thorough the point of averages Residual sum of squares The residual sum of squares is de ned to be SSE 2612 Here are two important computing formulas SSE 20 7W 7 b1209 77 SSE 13 7 ngY 7 b1 ZXui Here is an important fact ESSE n 7 2a392 De ne MSE Then EMSE 0 2 80 MSE is an estimate of 0 2 For the Housing data using the computing formula we have SSE 2312 772 7 bf X 772 20317 and MSE 21093127 1195 So an estimate of 0392 here is 1195 Normal regression model This normal regression model is Y g le 81 i 1 n where 81En are assumed to be independent N0 0392 This means Y1 Yn are independent and for each i Y is normally distributed with mean g le and variance 0392 HMM 4 2 l W WWW 1 l nzjj a m 2quot 4 x 1 wSJIi Yribj ll 21 y4033 Emmi 554472 2 x nym 1241 00 i i P1732 Wm M Y 2239 283732 2393 Hgtlt 555 zc3 7 H31 9512 1 1 1 3972 s L X i i u Alba 2 zfg gar50 H5 2 7 23372 Ma 39593 7quot w mi 2 23233 5 1 1 307 Wu mi Eh g I bgitl ai nay5150 My i 312r 740 84344 m 23 146 9 9243 393 4i 9574 Wu ah A Liitf 3 5f37 25CEJ114 z iraif wloM lzz 563 Qg ifi ldfzx 53967 5908 43953 1 1 5 Xk ST vsJ g 19 Unix 2333 234jl8 583a330 I I i 39 Aquot V A L 4 KS 2 9 a T t 1 r g a 1M ZMKHA sh 3 9 gas 3332 AHA 70 9 39W thu n39XQAZZ I 16 2 IOoCgt H May Won of m Mm M mac1mm wt was 7 A i U I quot WWquot U YR 739 LariBra 1quot 7 ALHMQ HSE 1quot 4512 3 1351392 paw 148450 A W3e8523 3 57 1am 15L AMI P u a 1 A l 7k 2 H375 nszsama vQg83331L2110N38573 219593 7318w3 f m 5225 ai52 3 4 9f y 7quot gr 1 0 quot 5 1quot H55 1 i 9 1

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