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

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.

r 2 is the coefficient of determination r 2 tells us the percentage of the variation in y that is explained by our regression equation. If r 2 =.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 r 2 is to 1, the better the fit of our equation.

Calculating r 2

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

r 2 is algebraically equivalent to (r) 2

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

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 r 2 ) We can create a residual plot (** Very important for AP Exam **) Plot the x-values vs residuals

The manatee data should still be in L 1 and L 2 Go to the heading of L 3 Press 2 ND ; 2; -; VARS; Y-VARS; 1: FUNCTION; 1: Y 1; (L 1 ) Press ENTER Now L 3 is the residuals OR, after you run the regression, press 2 nd STAT; and RESID will be the last choice. You could put this into the heading of L 3.

Plot 1: x-list should be L 1 y-list should be L 3 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

Data may not be linear:

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

Line looks like a good fit:

Do p.180 #61