Lecture02

Lecture02 - Correlation tells us about strength (scatter)...

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Examining Relationships Least-Squares Regression & Cautions about Correlation and Regression Sections 2.3 and 2.4 © 2009 W. H. Freeman and Company Correlation tells us about strength (scatter) and direction of the linear relationship between two quantitative variables. In addition, we would like to have a numerical description of how both variables vary together. For instance, is one variable increasing faster than the other one? And we would like to make predictions based on that numerical description. But which line best describes our data? The regression line ! A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. " We often use a regression line to predict the value of y for a given value of x . " In regression, the distinction between explanatory and response variables is important. Distances between the points and line are squared so all are positive values. The regression line The least-squares regression line is the unique line such that the sum of the squared vertical ( y ) distances between the data points and the line is the smallest possible.
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Equation of the Regression Line ! “y -hat” is the predicted response for any x ! b 1 is the slope ! b 0 is the intercept The least-squares regression line is the line: First we calculate the slope of the line, b 1 ; from statistics we already know: r is the correlation. s y is the standard deviation of the response variable y. s x is the the standard deviation of the explanatory variable x. Once we know b 1 , the slope, we can calculate b 0 , the y -intercept: where x and y are the sample means of the x and y variables How to: The computation of these coefficients can be done in Excel. Software output Example Regression Analysis: Selling Price ($) vs. Square Footage of houses The regression equation is Selling Price = 4795 + 92.8 Square Footage ($) Predictor Coef SE Coef T P Constant 4795 13452 0.36 0.723 Square F 92.802 8.844 10.49 0.000 S = 30344 R-Sq = 69.6% R-Sq(adj) = 69.0% " Slope: What is the change in selling price for a unit increase in square footage? " Intercept: Is the intercept meaningful? " Prediction: If the square footage of a house is 2500, what do we predict as the selling price?
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Facts about least-squares regression " If we reverse the roles of the explanatory and response variables, we will get a different regression line " The slope, b 1 is related to the correlation coefficient, r " The least-squares line passes through the means of the x and y variables. " The fraction of the variation in the values of y that is explained by the regression of y on x is r 2 Coefficient of determination, r 2 r 2 represents the percentage of the variance in y (vertical scatter from the regression line) that can be explained by changes in x . r
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Lecture02 - Correlation tells us about strength (scatter)...

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