14 - Least-squares regression line: The line that makes the...

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Least-squares regression line: The line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. Fact 1. The distinction between explanatory and response variables is essential in regression. Least-squares regression makes the distances of the data points from the line small only in the y direction. If we reverse the roles of the two variables, we get a different least-squares regression line. Fact 2. There is a close connection between correlation and the slope of the least-squares line. The slope is As the correlation grows less strong, the prediction ŷ moves less in response to changes in x. Fact 3. The least-squares regression line always passes through the point (x-bar, y-bar) on the graph of y against x. Fact 4. The correlation r describes the strength of a straight-line relationship. In the regression setting, this description takes a specific form: the square of the correlation, r2, is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x. One of the first principles of data analysis is to look for an overall pattern and also for
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