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Exploring r2 and Residual Plots in AP Statistics HW

Learn about the meaning of r2 and how to use residual plots in regression analysis. Calculate r and r2 and interpret their values. Understand the significance of residuals in assessing the fit of a regression equation. Analyze residual plots to evaluate linearity and accuracy of predictions.

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Exploring r2 and Residual Plots in AP Statistics HW

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  1. AP StatisticsHW: p. 165 #42, 44, 45Obj: to understand the meaning of r2 and to use residual plots Do Now: On your calculator select: 2ND; 0; DIAGNOSTIC ON; ENTER: ENTER Now re-run the regression on the Manatee data from yesterday C3 D5

  2. Now r is given. • This is the correlation coefficient. • It gives the strength and direction of the relationship between x and y. • If |r| is close to 1  strong linear relationship. • If |r| is close to 0  weak or no linear relationship. • The numerical value of r basically tells us how close the point are to a line. • If r is positive  positive relationship (increase in x means an increase in y). • If r is negative  negative relationship (increase in x mean a decrease in y). • The sign of r tells us the sign of the slope of the line that the points are close to.

  3. r2 is the coefficient of determination • r2 tells us the percentage of the variation in y that is explained by our regression equation. • If r2 = .8, then we have come up with an equation that accounts for 80% of the variation in y. The remaining variation would be due to random chance or possible another variable or variables that we have not included in our equaion. • The closer r2 is to 1, the better the fit of our equation.

  4. Calculating r2

  5. SST = “Total Sum of Squares about the Mean” - gives the sum of the squares of the differences from the mean • SSE = “Sum of Squares for Error” - gives the sum of the squares of the residuals

  6. r2 is algebraically equivalent to (r)2

  7. Residuals Residual = observed y – predicted y (error) (data) (value from eqn) = y - yhat • The sum of the residuals for a least-squares regression line will always be 0.

  8. Residual Plot • We can use residuals as another way to check the fit of a regression equation (in addition to looking at a scatterplot, r, and r2) • We can create a residual plot (** Very important for AP Exam **) • Plot the x-values vs residuals

  9. The manatee data should still be in L1 and L2 • Go to the heading of L3 • Press 2ND; 2; -; VARS; Y-VARS; 1: FUNCTION; 1: Y1; (L1) • Press ENTER • Now L3 is the residuals • OR, after you run the regression, press 2nd STAT; and RESID will be the last choice. You could put this into the heading of L3.

  10. Plot 1: x-list should be L1 y-list should be L3 • You are looking for a uniform scattering of points – no pattern. • Any sort of patter would indicate the equation may not be a good fit

  11. Data may not be linear:

  12. The predicted y is less accurate for large values of x:

  13. Line looks like a good fit:

  14. Do p.180 #61

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