I do not know what to do with the following exercise:
A polling company asks $n$ random people which party they will vote for in the next election. How big does $n$ need to be so that the spread for each party in a $95\% -$confidence interval is $\pm 1\%$?
Remark: We assume that there are $> 2$ parties
My problems with this exercise start with that I do not understand what we are estimating here. (I mean I get that we try to predict the outcome of an election, but I can not relate this to a concept from stochastics like say expected value or something.)
The only thing somehow related to this from my lecture ist that if $X \sim N(\mu,\sigma^2)$ and $\sigma^2$ is known we can compute a lower bound for the sample size by the equation
$$n_{min} = \frac{u_{1-\frac{\alpha}{2}}^2 \sigma^2}{b^2},$$ where $b$ is the width of the confidence interval of level $1-\alpha$.
However, I do not see why we can assume that the population is normally distributed here or why we should know $\sigma^2$, so this probably does not work here.
Could you please give me a hint?