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Let's pretend I'm conducting a poll/survey. It's a simple yes/no poll (i.e. everyone only gives 1 of 2 answers). I have asked N people so far, and X of them have said "yes".

I would like to stop asking people (i.e. stop the poll) when I can be sure to some high level of statistical confidence (e.g. 95% sure) that I have something that's accurate.

Does this question even make sense?

This is not a proper stats problem, i.e. "good enough" answers are alright. I don't need a high level of mathematical rigour. Right now I have nothing and I would like something that would at least point me down the path of knowledge. What would I need to do/know/compute/provide to you to figure out what's going on?

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  • $\begingroup$ The question makes sense, but in general it depends on your sampling procedure. Polling firms have to develop very robust sampling procedures precisely to ensure the confidence and margin of error of their results. In the absence of them, you need to make some possibly unwarranted assumptions, e.g. that your sampling is perfectly uniform (the probability of getting a "yes" is the same as the proportion of "yes"es in the general population). $\endgroup$ – Arturo Magidin Nov 22 '11 at 20:55
  • $\begingroup$ Well let's assuming my samplying is perfectly uniform then. What would be the formulas then? $\endgroup$ – Rory Nov 22 '11 at 20:58
  • $\begingroup$ en.wikipedia.org/wiki/Sequential_Probability_Ratio_Test $\endgroup$ – whuber Nov 22 '11 at 21:18
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That is a very basic (and essential !) question in statistics. The maths behind the answer to this is the central limit theorem. It tells you that no matter what the law of probability is, the averages of N samples behave like a gaussian (the bound is not explicit unless you know the variance of your law).

In the problem you are mentionning you can do something more explicit since the law is rather simple (the law of one answer is called a Bernoulli, and the law of the sum is called a binomial. If p is the probability of "yes", then the variance for a N-sample is N p (1-p), and you can compute explicitly the N you need in order to make a mistake of less than say 5% with a 95% probability (you need both a margin of error and a trust interval for the questino to be well-posed).

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    $\begingroup$ Wow! (I have learnt so much on this site). Thanks @Glougloubarbaki. An example will be great - a simple one at that! $\endgroup$ – Adhesh Josh Nov 22 '11 at 22:08
  • $\begingroup$ @Adhesh Notice that this answer is appropriate in a subtly different setting where no people have yet been surveyed. It will give a sample size in the current setting where some results are available, but by ignoring the existing results it may overestimate how many more are needed. Note, too, an apparent logical problem: the purpose of the survey is to estimate $p$, so how exactly does one use $p$ to design the survey? :-) (One answer is, you guess its value, figure out the survey size, and increase that a little bit to allow for uncertainty in the guess.) $\endgroup$ – whuber Nov 22 '11 at 23:00

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