Table $1$ below shows a set of patterns P$1-$ P$20,$ drawn from two classes $C1$ and $C2,$ and of known class membership. Each pattern is defined in terms of two pattern features $x_1$ and $x_2,$ whose measured values are shown. The linear discriminant decision rule below has been suggested for the classification task.

$d(x)=8x_1-7x_2-10\mathrm{.}8$

if $d(x)0,$ assign $x$ to $C1$;

otherwise, assign $x$ to $C2$

$($a$)$ Using the samples $($P$1-$ P$20)$ from the Table as your test data, graphically or otherwise, determine the classification error rate of this classifier.

$($b$)$ Suggest a different linear discriminant decision rule that will reduce the error rate.

Table $1$

$\backslash$table$[[\backslash$table$[[$Pattern$],[$number$]],$Class,x$1,$x$2,\backslash$table$[[$Pattern$],[$number$]],$Class,x$1,$x$2],[$P$1,$C$1,3\mathrm{.}2,1\mathrm{.}4,$P$11,$C$2,3\mathrm{.}3,2\mathrm{.}1],[$P$2,$C$1,3\mathrm{.}2,1\mathrm{.}7,$P$12,$C$2,2\mathrm{.}7,1\mathrm{.}9],[$P$3,$C$1,3\mathrm{.}1,1\mathrm{.}5,$P$13,$C$2,3\mathrm{.}0,2\mathrm{.}1],[$P$4,$C$1,2\mathrm{.}3,1\mathrm{.}3,$P$14,$C$2,2\mathrm{.}9,1\mathrm{.}8],[$P$5,$C$1,2\mathrm{.}9,1\mathrm{.}5,$P$15,$C$2,3\mathrm{.}0,2\mathrm{.}2],[$P$6,$C$1,2\mathrm{.}8,1\mathrm{.}3,$P$16,$C$2,3\mathrm{.}0,2\mathrm{.}1],[$P$7,$C$1,3\mathrm{.}3,1\mathrm{.}8,$P$17,$C$2,2\mathrm{.}5,1\mathrm{.}6],[$P$8,$C$1,2\mathrm{.}4,1,$P$18,$C$2,2\mathrm{.}9,1\mathrm{.}8],[$P$9,$C$1,2\mathrm{.}9,1\mathrm{.}3,$P$19,$C$2,2\mathrm{.}5,1\mathrm{.}8],[$P$10,$C$1,2\mathrm{.}7,1\mathrm{.}4,$P$20,$C$2,3\mathrm{.}6,2\mathrm{.}5]]$