%BEGIN Ivan's problems (spring '98)

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%pset10 problem9 spring98 (not used)
% Fall 01, problem set 12, problem 3
\begin{problem}{\bf [5 points]}

We want to determine the percentage $r$ of people who intend to watch
the final episode of {\em Seinfeld.} Our polling method is the same as
that discussed in lecture: we sample the opinion of $n$ people chosen
uniformly and at random, with replacement.

How many people should be polled so that $r$ is within 3 points of the
true percentage, with probability at least .96?

\solution{
{\bf Solution}

$\epsilon = .03, \delta = .04$

\begin{eqnarray*}
\delta \geq 2 F_{1/2}((1/2- \epsilon)n) & \Rightarrow &
.05 \geq 2  F_{1/2}(0.49n)\\
& \Rightarrow & 0.025 \geq {0.51 \over 0.02}f_{1/2}(0.49n)\\
& \Rightarrow & 0.025 \geq {25.5 \over\sqrt{2n\pi (.49)(.51)}}2^{(H(.49)-1)n}\\
& \Rightarrow & 0.0012285 \geq {2^{(H(.49)-1)n} \over \sqrt{n}}\\
& \Rightarrow & 0.0012285 \geq {2^{-0.000288574n} \over \sqrt{n}}\\
& \Rightarrow & 0.0012285 \geq {(.9998)^n \over \sqrt{n}}\\
\end{eqnarray*}

Choosing $n=10388$ will do it.
}

\end{problem}
