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Unformatted text preview: Chapter 6 Simple Linear Regression Remember that most of this course can be boiled down to examining the association between different kinds of variables. We classified the possibilities like this: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical Quantitative ANOVA Quantitative Quantitative Regression Quantitative Categorical (not discussed) This part of the course focuses on regression. First well talk about re gression with a single explanatory variable, which well call simple linear regression . (The textbook sometimes calls this bivariate regression, since there are two variablesone explanatory and one response.) Later well ex tend the idea to situations with multiple explanatory variables, using what well call multiple linear regression . This first chapter on simple linear regression corresponds to Chapters 3 and 12 of the textbook. 6.1 Concepts and Setup To properly understand concepts about regression, we first need to under stand how populations and samples relate to each other in the context of regression. Lets begin by considering a quantitative response variable Y and quantitative explanatory variable X . 6.1 Concepts and Setup 89 Population Each individual in the population has a value of X and a value of Y . Its easiest to think about the relationship between X and Y with an example. Lets say our population is American adults, and that X is height, in inches, and Y is weight, in pounds. Lets consider only those people who are 65 inches tall ( X = 65). They wont all weigh the same amount, but theyll vary around some mean, which well call Y ( 65 ) . (We use to indicate its a mean, Y to indicate what its the mean of, and 65 to indicate that it only refers to individuals with X = 65.) We could instead consider only those people who are 70 inches tall ( X = 70). Their weights will also vary around some mean Y ( 70 ) , which we expect to be greater than Y ( 65 ) . Linear Relationship What were going to assume about the population is that X and Y ( X ) are related by a straight line, as shown in Figure 6.1. This is why the technique is called linear regression. X Y Figure 6.1: Linear relationship between X and Y ( X ) . The dotted line represents Y ( X ) . We write this relationship as Y ( X ) = + X, where is the slope and is the yintercept (for the populationwell also see slopes and yintercepts that come from samples). Since and are population parameters, we seldom know what their values actually are. 6.1 Concepts and Setup 90 Notice that this formula can represent any type of linear association be tween X and Y : If Y tends to be larger when X is larger, then is positive ( > 0)....
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This note was uploaded on 02/08/2011 for the course STA 3024 taught by Professor Ta during the Spring '08 term at University of Florida.
 Spring '08
 TA
 Linear Regression

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