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Unformatted text preview: it? We will fit a line that goes through the data in the best way possible, based on the least squares criterion. Definition: The residual sum of squares (a.k.a. SS(resid) or SSE) is n SS resid SSE yi ‐yi 2 i1 The least squares criterion states that the optimal fit of a model to data occurs when the SS(resid) is as small as possible. Note that under our model n n yi ‐yi SS resid SSE i1 2 yi ‐ bO b1 xi 2 i1 Refer to the applet at http://standards.nctm.org/document/eexamples/chap7/7.4/ Regression and Correlation Page 2 Using calculus to minimize the SSE, we find the coefficients for the regression equation. ∑n 1 xi ‐x yi ‐y
b1 ∑n 1 xi ‐x 2
i b0 y‐ b1 x TI‐83/84 Enter the data into two lists. STAT ‐> TESTS ‐> LinRegTTest We’ll go over the options in class. Example 12.3 Find the linear regression of weight (Y) on Length (X). Scatterplot of Weight vs Length with Fitted Regression Line 200
180 Weight (g) 160
100 55.0 57.5 60.0
Length (cm) 65.0 67.5 70.0 Interpret the slope (b1) in the context of the setting. Can we interpret the meaning of the Y intercept (bO) in this setting? Definition: An extrapolation occurs when one uses the model to predict a y value corresponding to an x value which is not within the range of the observed x’s. Regression and Correlation Page 3 A Measure of Variability – sY|X Once we fit a line to our data and use it to make predictions, it is natural to ask the question of how far off our predictions are in general. Definition: The residual standard deviation is sY|X SS resid n‐2 ∑n
i yi ‐yi 2 n‐2 1 Caution: This is not to be confused with sY! Recall, sY ∑n
i yi ‐y 2 n‐1 1 Scatterplot of Weight vs Length with Fitted Regression Line
200 180 Weight (g) 160
100 55.0 57.5 60.0
Length (cm) 65.0 67.5 70.0 Determine and interpret sY|X for the regression of female Vipera bertis weight on length. Regression and Correlation Page 4 The Linear Statistical Model Definition: A conditional mean is the expected value of a variable conditional on another variable....
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- Fall '09