chapter06_mine - Chapter 6 Simple Linear Regression...

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Chapter 6 Simple Linear Regression Remember that most of this course can be boiled down to examining the as- sociation 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 we’ll talk about regression with a single explanatory variable, which we’ll call simple linear regression . (The textbook sometimes calls this bivariate regression, since there are two variables—one explanatory and one response.) Later we’ll extend the idea to situations with multiple explanatory variables, using what we’ll 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. Let’s begin by considering a quantitative response variable Y and quantitative explanatory variable X .
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6.1 Concepts and Setup 90 Z Population Each individual in the population has a value of X and a value of Y . It’s easiest to think about the relationship between X and Y with an example. Let’s say our population is American adults, and that X is height, in inches, and Y is weight, in pounds. Let’s consider only those people who are 65 inches tall ( X   65 ). They won’t all weigh the same amount, but they’ll vary around some mean, which we’ll call μ Y ˆ 65  . (We use μ to indicate it’s a mean, Y to indicate what it’s 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 μ Y ˆ 70  , which we expect to be greater than μ Y ˆ 65  . Linear Relationship What we’re going to assume about the population is that X and μ Y ˆ 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 μ Y ˆ X  . The dotted line represents μ Y ˆ X  . We write this relationship as μ Y ˆ X    α βX, where β is the slope and α is the y -intercept (for the population—we’ll
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6.1 Concepts and Setup 91 also see slopes and y -intercepts that come from samples). Since α and β are population parameters, we seldom know what their values actually are. Notice that this formula can represent any type of linear association between X and Y : ˆ If Y tends to be larger when X is larger, then β is positive ( β A 0 ). ˆ If
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This note was uploaded on 01/08/2012 for the course STA 3024 taught by Professor Ta during the Summer '08 term at University of Florida.

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chapter06_mine - Chapter 6 Simple Linear Regression...

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