$($d$)$ In an election, a proportion $p$ will vote for candidate $G$ and a

proportion $1-p$ will vote for candidate B$.$ In an election poll,

a number of voters are asked for whom they will vote. Let $x_i$

be the indicator random variable for the event "the $i^{th}$ person

interviewed will vote for G$."$ A model for the election poll is that

the the people to be interviewed are selected in such a way that

the indicator random variables $x_1,x_2,$dots, are independent and

have a Ber$(p)$ distribution.

i$.$ Suppose ${\stackrel{}{x}}_n$ is used to predict $p$ According to Chebyshev's

inequality, how large should $n$ be such that the probability

that ${\stackrel{}{x}}_n$ is within $0\mathrm{.}2$ of the "true" p ia at least $0\mathrm{.}9?(5mks)$

ii$.$ If $p>\frac{1}{2}$ candidate $G$ wins; if ${\stackrel{}{x}}_n>\frac{1}{2}$ you predict that $G$ will

win. Find an $n($as small as you can$)$ such that the probability

that you predict correctly is at least $0\mathrm{.}9,$ if in fact $p=0\mathrm{.}6(5$

mks$)$