# 511_Chap7 - 1 Simple Linear Regression A Model for the Mean...

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Unformatted text preview: 1 Simple Linear Regression: A Model for the Mean Chapter 7 2 An Intermediate Model (if the groups are defined by values of a numeric variable) Separate Means Model Equal Means Model Means fall on a straight line function of the group values 3 Meaning of the Word Regression • statistical meaning not related to the usual English definition of regression • misnomer, but there ’ s a historical reason • refers to finding the best fitting (straight line) relationship between the mean of Y and values of the variable defining the groups (X) 4 Galton ’ s data 75 70 65 60 75 70 65 60 height of father height of son 5 Case Study 1 • Hubble data (from 1920 ’ s) • observational data • each point represents a nebula • Y – distance of the nebula • X – recession velocity of the nebula 6 Case Study 1 • don ’ t worry about the geometry, but Big Bang theory implies that Y = const. * X (i.e. a straight line relationship between Y and X, with intercept 0) • furthermore, const. should be an estimate of the age of the universe 7 Case Study 1 0.5 1 1.5 2 DISTANCE-250 250 500 750 1000 1250 VELOCITY 8 Case Study 1 Regression methods can help answer big cosmological questions: • Is the Big Bang theory correct? – is the intercept of the straight-line relationship 0? • How old is the universe? – what is the slope of the straight-line relationship? 9 Case Study 2 • designed experiment • response variable (Y) – pH of steer carcass • explanatory variable (X) – log time (hours) after slaughter • what is the approximate pH of a particular steer carcass 3 hours after slaughter? • about how long do you have to wait after slaughter for the pH to reach 6.0? 10 Case Study 2-0.20-0.10 0.00 0.10 0.20 Residual 1 2 3 4 5 6 7 8 9 TIME 5.5 6 6.5 7 PH 1 2 3 4 5 6 7 8 9 TIME 11 Case Study 2 5.5 6 6.5 7 PH 0.5 1 1.5 2 logtime-0.10-0.05 0.00 0.05 0.10 Residual 0.5 1 1.5 2 logtime 12 Regression Analysis • statistical methods based on describing the distribution of values of one variable (Y – the response variable ) as a function of the other variable (X – the explanatory variable ) – simple linear regression: the function is a straight line function 13 Regression Analysis • specifically, – the mean of Y is a straight line function of X μ {Y|X} = β + β 1 X where β is the intercept and β 1 is the slope...
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## This note was uploaded on 11/30/2011 for the course STAT 380 taught by Professor Stevens during the Spring '11 term at Brigham Young University, Hawaii.

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511_Chap7 - 1 Simple Linear Regression A Model for the Mean...

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