Relevance of opinion poll (500 people from 10.5 millions)

Question

In local news [1] comes with information that 58% of people is for something. This information is based on option pool that ask 500 people from 10.5 million about that question. What is confidence interval for people that are for that something?

To solve this I try following. The $n$ people from $10^7$ is for something. I randomly select 500 of them and I will try to find $n_1,n_2$ for which:

$$ n_1: \frac{\binom{n_1}{290}\binom{10^7-n_1}{210}}{\binom{10^7}{500}}<0.05 $$

$$ n_2: \frac{\binom{n_2}{290}\binom{10^7-n_2}{210}}{\binom{10^7}{500}}>0.95 $$

But this is not easy to solve. So I am looking for some solvable algorithm.

[1] link

Physicist's back-of-the-envelope calculation: $500\ll10.5M\approx\infty$, so it is a Poisson distribution, then count error is like $\sqrt{500}\approx22$ people, so 4.4 in per cent points. Thus CI is like 54%-62%. — , Jul 12 '13 at 11:31
@mbq I urge you to re-think that comment, because it appears to address a different problem. The value of 500 is not random but is selected by the researcher. It is not the realization of a Poisson variate and your answer has no meaning in this context. — whuber, Jul 12 '13 at 14:27
This question is extensively discussed and answered here, in many guises. This [search](http://stats.stackexchange.com/search?tab=votes&q=binomial%20confidence%20interval) will reveal a few hundred relevant posts. Enjoy! — whuber, Jul 12 '13 at 14:31
@whuber I agree, I posted that as a math trick how to use binomial->Poisson asymptotic to quickly approximate solution to such a problem. That's why its a comment and starts with a disclaimer. — , Jul 12 '13 at 17:50
@mbq I appreciate that: but your approach, as you have outlined it, is conceptually wrong. The question asks for a confidence limit for $p$ in a sample of $n=500$ where $\hat{p}=0.58$. It is an accident that $\sqrt{n}$ is even the right order of magnitude in this case. There is no Poisson distribution in sight, either: it's a Binomial distribution problem. For $\hat{p} \approx 1/2$ and a $95$% CI it *is* the case that two SEs is approximately equal to $\sqrt{n}$, but that is *not* because $n$ (a *constant*) has a Poisson distribution! — whuber, Jul 12 '13 at 18:18

score 3 · Accepted Answer · answered Jul 12 '13 at 08:19

Computing confidence intervals for binomial proportions is an interesting problem that has attracted quite a bit of research. None of the proposed confidence intervals take the population size into account. You can start here: http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval

Some of the proposed confidence intervals for your problem are (computed using Stata):

. cii 500 290, wald

                                                         -- Binomial Wald ---
    Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |        500         .58    .0220726        .5367385    .6232615

. cii 500 290, exact

                                                         -- Binomial Exact --
    Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |        500         .58    .0220726        .5353716    .6236769

. cii 500 290, wilson

                                                         ------ Wilson ------
    Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |        500         .58    .0220726        .5362895    .6224906

. cii 500 290, agresti

                                                         -- Agresti-Coull ---
    Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |        500         .58    .0220726        .5362852    .6224949

. cii 500 290, jeffreys

                                                         ----- Jeffreys -----
    Variable |        Obs        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |        500         .58    .0220726        .5363799    .6226969

It would be nice to see all prerequisites that must be fulfilled to make this approach to be correct for actual opinion poll analysis. — boucekv, Jul 15 '13 at 09:59
@boucekv : a discussion of **all** prerequisites would require a book. The short answer is: it depends on what you exactly want to do, as that defines what _correct_ means. — Maarten Buis, Aug 05 '13 at 07:06

Relevance of opinion poll (500 people from 10.5 millions)

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