The following is the percentage share of money income for households before direct taxes for the US$.$

$\backslash$table$[[$Income group,$2015,1975,1960,1947],[$Lowest fifth, $0\mathrm{.}2,0\mathrm{.}032,0\mathrm{.}044,0\mathrm{.}048,0\mathrm{.}051],[$Second fifth, $0\mathrm{.}4,0\mathrm{.}115,0\mathrm{.}149,0\mathrm{.}17,0\mathrm{.}169],[$Third fifth, $0\mathrm{.}6,0\mathrm{.}259,0\mathrm{.}32,0\mathrm{.}348,0\mathrm{.}336],[$Fourth fifth, $0\mathrm{.}8,0\mathrm{.}49,0\mathrm{.}568,0\mathrm{.}588,0\mathrm{.}568],[$Highest fifth, $1,1\mathrm{.}00,1\mathrm{.}00,1\mathrm{.}00,1\mathrm{.}00]]$

a$.)$ Using a quadratic regression equation to find the Lorenz curve for the year $1975.$

b$.)$ Find the Gini index for the Lorenz curve you found in part a$.$