G different choices of explanatory variable the

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be and is most useful for comparing different models (e.g., different choices of explanatory variable). The coefficient of determination is equal to the square of the correlation coefficient and so is denoted by r 2 or R 2 (and is often pronounced “r - squared.”)
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Chance/Rossman, 2015 ISCAM III Investigation 5.8 373 Another measure of the quality of the fit is s , the standard deviation of the residuals. This is a measure of the unexplained variability about the regression line and gives us an idea of how accurate our predictions should be (the actual response should be within 2 s of the predicted response). If s is much smaller than the variability in the response variable ( s y ) then we have explained a good amount of variability in y . Most statistical packages report s , or it can be found from ) 2 /( ± n SSE . (cc) Determine and interpret the value of s for these data. [What are the units?] Study Conclusions There is a fairly strong positive linear association between the foot length of statistics students and their heights ( r = 0.711). To predict heights from foot lengths, the least-squares regression line is foot eight h 03 . 1 3 . 38 ˆ ² . This indicates that if one person’s foot length measurement is one centimeter longer than another, we will predict that person’s height to be 1.03 inches taller. This regression line has a coefficient of determination of 50.6%, indicating that 50.6% of the variability in heights is explained by this least squares regression line with foot length. The other 49.4% of the variability in heights is explained by other factors (perhaps including gender) and also by natural variation. So although the foot lengths are informative, they will not allow us to perfectly predict the heights of the students in this sample. The value of s is 3.61 inches, meaning we should be able to predict a person’s height within 3.61 inches based only on the size of his or her foot. Technology Detour Determining Least Squares Regression Lines In R To calculate intercept and slope: > lm(response~explanatory) To then superimpose regression line on scatterplot: > abline(lm(response~explanatory)) In Minitab x Choose Stat > Regression > Regression > Fit Regression Model and specify the response variable in the first box and then the explanatory variable in the Predictors box. x To superimpose the regression line on the scatterplot, choose Stat > Regression > Fitted Line Plot . x Minitab reports additional output, but you should be able to find the least-squares regression equation and the value of r 2 . x Click the Storage button and check the Residuals box to store them in their own column.
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Chance/Rossman, 2015 ISCAM III Investigation 5.8 374 Practice Problem 5.8 For the Cat Jumping data set from Investigation 5.6: (a) Calculate and interpret the correlation coefficient between velocity and body mass. (b) Square the correlation coefficient to obtain r 2 . Interpret the coefficient of determination in context.
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