Question

The Correlation and Regression Applet allows you to animate the given figure. Click to create a group of 10 points in the lower-left corner of the scatterplot with a strong straight-line pattern (correlation approximately 0.9). Click the "Show least-squares line" box to display the regression line. Move the outlier down Subject 16 Removing Subject 16 moves the regression line only a little. and the least-squares line chases it down 20 40 60 80 100 20 40 Empathy score 60 80 100 Empathy score (a) Add one point at the upper right that is far from the other 10 points but exactly on the regression line. Why does this outlier have no effect on the line even though it changes the correlation? O The new point on the existing line exactly matches the prediction; thus, it has no residual. Since the regression line results in the least sum of residual squares, adding a point with zero residual does not change the line. O The new point on the existing line does not change the standard deviations of x and y, meaning that the slope of the regression line cannot change.

Answer 1

$(a)~For~adding~one~point,~\bar{x},~\bar{y}~are~changed,~standard~deviations~of~x,~y\\ are~changed~and~correlation~coefficient~is~also~changed.~However~we~get~intercept~and~slope\\ ~by~minimizing~residual~sum~of~square~and~since~this~point~falls~on~regression~line~so~it~has~no\\ ~effect~on~residual~sum~of~square~i.e.~the~regression~line~is~unchanged.~Hence\\ the~new~point~on~the~existing~line~exactly~matches~the~prediction;~thus,~it~has~no~residual.\\ Since~the~regression~lines~in~the~least~sum~of~residual~squares,~adding~point~with~zero~residual\\ does~not~change~the~line.\\ (b)~If~\bar{x}~and~\bar{y}~are~changed~then~intercept~is~changed~but~slope~is~unchanged.\\ ~So~the~first~and~2nd~options~are~not~correct.\\ The~standard~deviations~of~x~and~y~and~these~affect~the~value~of~intercept~and~slope\\ hence~Option~3~i.e.\\ "The~outlier~is~far~from~the~ group~in~both~the~x~and~y~directions,~and,~therefore,~has~a~strong~\\$$influence~on~the~standard~deviations~of~both~variables~x~and~y.~The~change~in~the~standard~deviations\\ standard~deviations~strongly~influences~the~parameters~of~the~regression~line" is~correct.$

$Option~4~i.e.~"the~line~always~passes~through~(\bar{x},\bar{y}),~and~the~group~of~10~points\\ dominates~the~average.~Thus~the~line~has~to~pass~through~the~group~in~the~lower~left~\\ corner,~i.e.~the~group~acts~as~a~pivot~point~for~the~line"`~is~correct.\\ Option~5~i.e.~"The~outlier~is~far~from~the~group~in~both~the~x~and~y~directions,~and,~\\ therefore,~has~a~strong~influence~on~the~correlation,~which,~in~turn,~affects~the~slope\\ ~of~the~line"~is~correct.$