Chapter 11

# Chapter 11 - Chapter 11 Simple Linear Regression 1...

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1 Chapter 11 Simple Linear Regression

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2 Multivariate Data In engineering studies involving multivariate data, often the objective is to determine the relationships among the variables or to build an empirical model. Methods to determine if the relationship is linear: 1) Scatter diagram or scatterplot 2) Correlation analysis
3 In many situations, we have two or more variables of interest that are related, but the model relating these variables is unknown. In these cases, it is necessary to build a model relating the variables based on observed data. This is known as an empirical model. Simple linear regression is used to model one response (or dependent) variable, Y, using one predictor (or independent or regressor) variable, X. Simple Linear Regression

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4 In looking for a simple linear regression model to explain the relationship between X and Y, we need to make sure the data has a linear relationship. We can do this using a scatterplot. 0.0475 0.05 0.0525 0.055 0.0575 0.06 0.0625 Thermal Conductivity 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Product Density Linear Fit Bivariate Fit of Thermal Conductivity By Product Density
5 Based on the scatter diagram, it is probably reasonable to assume that the expected value of the random variable Y is related to X by the following straight-line relationship where the slope and the y- intercept are the unknown parameters:

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6 (Theoretical Model)
7 The points on the regression line are the predicted values we obtain from the fitted regression model and are referred to as y-hat ( ). The distance between each of the observed values (y) and the predicted values ( ) is known as the prediction errors or residuals. y y y

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8 Obtaining Estimates for Slope and Intercept Suppose that we have n pairs of observations ( x 1 , y 1 ), ( x 2 , y 2 ), …, ( x n , y n ). The method of least squares is used to estimate the parameters, β 0 and β 1 , by minimizing the sum of the squares of the vertical deviations in Figure 6-6 (next slide).
9 In least squares estimation, we wish to minimize the sum of the squared residuals. Finding formulas for estimates using least

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Chapter 11 - Chapter 11 Simple Linear Regression 1...

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