chapter4

# chapter4 - Relationships Regression PSLS chapter 4 2009 W.H...

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Relationships Regression PSLS chapter 4 © 2009 W.H. Freeman and Company

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Objectives (PSLS chapter 4) Regression Regression lines The least-squares regression line Using technology Facts about least-squares regression Residuals Influential observations Cautions about correlation and regression Association does not imply causation
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. And we would like to make predictions based on the observed association. But which line best describes our data?

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Distances between the points and line are squared so all are positive values. This is done so that distances can be properly added (Pythagoras). The least-squares regression line The least-squares regression line is the unique line such that the sum of the total vertical (y) distances is zero and sum of the squared vertical (y) distances between the data points and the line is the smallest possible.
Facts about least-squares regression 1. The distinction between explanatory and response variables is essential in regression. 2. There is a close connection between correlation and the slope of the least-squares line. 3. The least-squares regression line always passes through the point 4. The correlation r describes the strength of a linear relationship. The square of the correlation, r 2 , is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x . ( 29 , x y

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is the predicted y value ( y hat) b is the slope a is the y -in tercept ˆ y = ( y - r x s y s x ) + r s y s x x , or ˆ y = a + bx Properties ˆ y "a" is in units of y "b" is in units of y / units of x The least-squares regression line can be shown to have this equation:
b = r s y s x First we calculate the slope of the line, b , 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 , the slope, we can calculate a, the y -intercept : a = y - b x where x and y are the sample means of the x and y variables How to: This means that we don’t have to calculate a lot of squared distances to find the least- squares regression line for a data set. We can instead rely on the equation. But typically, we use a 2-var stats calculator or a stats software.

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BEWARE !!! Not all calculators and software use the same convention: Some use instead: ˆ y = a + bx ˆ y = ax + b Make sure you know what YOUR calculator gives you for a and b before you answer homework or exam questions.
Software output Intercept Slope R 2 r R 2 Intercept Slope

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The equation completely describes the regression line.
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