The earnings per share $($in dollars$)$ and the dividends per share $($in dollars$)$ for $6$ medical supply companies in a recent year are shown in the data set below.

$\backslash$table$[[$Earnings per share, $x,2\mathrm{.}78,5\mathrm{.}08,4\mathrm{.}51,3\mathrm{.}05,3\mathrm{.}78,2\mathrm{.}25],[$Dividends per share, $y,0\mathrm{.}57,2\mathrm{.}42,1\mathrm{.}46,0\mathrm{.}82,1\mathrm{.}07,0\mathrm{.}28]]$

a$)$ Determine the correlation coefficient $r=$

b$)$ Does this represent a positive correlation or a negative correlation?

c$)$ State a model linear equation, hat$(y)=mx+b,$ for the regression line.

d$)$ Use the model to predict the dividends per share for an earning share of $4.$