\problemdata                                                             
        {Polling} % latex-friendly label for the prob.
        {Binomial Estimation} % The topic(s) of the problem content.  
        {?} % Source (if known)                                          
        {S01 Tutorial 12, 1} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Suppose you are heading a scientific polling organization. Candidate
Al Gore wants you poll a group of voters in Florida to determine his
popularity.  Furthermore, he would like a margin of error of 1.5\%.  How
large a sample size do you need to be able to tell him with 95 percent
confidence that the error attributable to sampling and other random
effects is within plus or minus 1.5 percentage points?

\solution{ 
Let $p$ be the probability that a person will vote for Al
Gore, $\epsilon$ be the margin of error and let $\delta$ be the
probability that our result lies outside this margin.  Therefore we
have $\epsilon = 0.015$, and $\delta = 0.05$.  Since we get the
maximum value of $\delta$ when $p=0.5$ 

\begin{eqnarray*}
\delta & \leq & F_{n,\frac{1}{2}}((\frac{1}{2} - \epsilon) n) +
                F_{n,1-\frac{1}{2}}((1-\frac{1}{2}-\epsilon) n) \\
        & = &   2 F_{n\frac{1}{2}}((\frac{1}{2} - \epsilon) n) \\
	& = &   2 F_{n\frac{1}{2}}((\frac{1}{2} - 0.015) n)\\
	& = &   2 F_{n\frac{1}{2}}(0.485n)\\
	& \leq & 2 \frac{1 - 0.485}{1 - \frac{0.485}{0.5}} f_{n,0.5}(0.485n)\\
        & = &   2 \cdot 17.167 f_{n,0.5}(0.485n)
\end{eqnarray*}

Solving for $n$, we get $n=300,455$. Therefore, if the sample size is $n\geq 300,455$ we get the desired property.
}

\end{problem}
                                 