IPS6eCh02_3bb - Looking at Data Relationships Least-Squares...

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    Looking at Data - Relationships   Least-Squares Regression IPS Chapter 2.3
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Objectives (IPS Chapter 2.3) Least-squares regression Regression lines Prediction and Extrapolation Correlation and r 2 Transforming relationships
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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?
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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.
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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 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 as small as possible.
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Properties is the predicted y value (y hat) b 1 is the slope b 0 is the y -intercept ˆ y The least-squares regression line can be shown to have this equation: x b b y 1 0 ˆ + =
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x y s s r b = 1 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.
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