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**Unformatted text preview: **STA 3024 Introduction to Statistics 2 Chapter 5: Simple Linear Regression Analysis As stated in chapter 3 and chapter 4, the table below summarizes the major materials that we need to cover Table 1: Methods to Investigate the Association between Variables Explanatory Variable(s) Response Variable Method Chapter 3 Categorical Categorical Contingency Tables Chapter 4 Categorical Quantitative Analysis of Variance (ANOVA) Chapter 5 and 6 Quantitative Quantitative Regression Analysis Quantitative Categorical (not discussed) This chapter deals with cases where both explanatory and response variables are quanti- tative where we’ll use regression analysis to study the association between the two variables. The regression methods that we’re studying in this chapter restricted to the linear regression family (as opposed to nonlinear regression analysis). If there’s only one quantitative explanatory variable, then we’ll study simple linear re- gression . If there are more than one explanatory variables, then we’ll introduce multiple linear regression . This chapter corresponds to chapter 12 in our textbook. 1 PART I - BACKGROUND 1.1 Background and Remarks Given two quantitative variables, we’d like to find the association between them. For instance, we might wonder if height and weight have any kind of association that knowing one might predict the other. (Certainly, both height and weight are quantitative variables) Now, we go out and collect data. Each person/subject in our experiment should be identified by the information that we are interested in; that is, each subject is described by a pair of data (weight, height). In fact, all data in simple linear regression are decribed by a pair of number ( X,Y ) which we can think of them as points on a 2D plane. A usual data set for two quantitative variables looks like Table 2: Data for Weights and Heights X (= Weight) Y (= Height) x 1 = 130 y 1 = 63 x 2 = 133 y 2 = 65 x 3 = 157 y 3 = 70 x 4 = 180 y 4 = 72 x 5 = 160 y 5 = 74 x 6 = 177 y 6 = 73 x 7 = 193 y 7 = 73 x 8 = 126 y 8 = 61 x 9 = 179 y 9 = 63 x 10 = 143 y 10 = 77 x 11 = 157 y 11 = 68 x 12 = 122 y 12 = 65 x 13 = 145 y 13 = 68 x 14 = 135 y 14 = 67 x 15 = 117 y 15 = 61 x 16 = 186 y 16 = 73 x 17 = 170 y 17 = 71 x 18 = 173 y 18 = 71 x 19 = 192 y 19 = 74 x 20 = 155 y 20 = 64 x 21 = 177 y 21 = 77 x 22 = 107 y 22 = 59 x 23 = 138 y 23 = 69 x 24 = 155 y 24 = 70 x 25 = 201 y 25 = 73 2 Figure 1: A Scatterplot for a Sample of People where Each Individual is Identified by a Pair of Number ( X = Weight, Y = Height). Recall: a 2D line can be mathematically described by the equation y = m + n * x Figure 2: A Straight Line y = m + n * x where m Indicates the y-Interception and n Indicates the Slope Let X be the quantitative explanatory variable and let Y be the quantitative response variable. If the relationship between X and Y can be described by a straight line μ Y ( X ) = α + βX (where μ Y ( X ) is the mean for Y at X ) then simple linear regression is the statistical inference method that analyses the associa-...

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