Answer
$617$
Work Step by Step
While the problem could be solved algebraically, the best way to solve this problem is using Microsoft Excel. In order to do this, assign a random number (such as 500) to any cell, and consider this to be cell $n$. As the book recommends, we find the value of $\hat{p}$. To do this, we first must find the standard deviation:
$σ=\sqrt{\frac{(0.5)(0.5)}{n}}$
We also know
$\hat{p}=zσ+μ$
And
$z=\frac{\hat{p}−p}{\sqrt{\frac{-p^2+p}{n}}}$
Doing this, we plug in the known values into the equations, and we plug each equation into excel. Now, we use the Goal Seek command, which can be found by going to Data and then to "What-if-Analysis." Doing this, we find that the value of $n$ so that there is a probability of $80$ percent (found using the necessary value of z from table A-2) is $617$.
