Finding the line thats closest to the data is the

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Unformatted text preview: (m = 1), regression is essentially the same as correlation, and the regression equation is the same as the regression line we would draw on a ˆ scatterplot. The equation for a line always has the form Y = b0 + b1⋅X, where b0 and b1 are numbers representing the intercept and the slope. Finding the line that’s closest to the ˆ data is the same as choosing b0 and b1 so that the predicted Y values are as close as possible to the true Y values. When there are more than one variable, we take the basic equation for a line and extend it in a natural way. ˆ Y = b0 + b1 X1 + b2 X 2 + ... + bm X m = b0 + " bi X i (1) i =1 to m The two lines of Equation 1 mean the same thing. The first line writes everything out, and the second line combines things into a sum (you can choose to remember either one). In the s! the index i takes on all values from 1 to m. When i = 1, the summand is b1X1; when um, i is 2, the summand is b2X2; and so on until bmXm. Notice that the contribution of each predictor (biXi) is the same as in the equation for a simple line with one predictor. This is why we say that each predictor has a linear effect on the predicted outcome. If we held all of the predictors except one fixed, then the relationship between the remaining predictor and the prediction would be a line. The goal of regression is to find the values of b0 through bm that lead to the best predictions. Therefore, the bs are a main focus of regression. Each bi is called the regression coefficient for its corresponding predictor (e.g., b1 is the regression coefficient for X1). The regression coefficient tells what kind of influence each predictor has on the predicted outcome. When a regression coefficient is positive, that means the predictor has a positive effect, just like with a positive...
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This document was uploaded on 02/25/2014 for the course PSYC 3101 at Colorado.

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