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Suppose there is a population, with goods and bads. The bad rate of the population(=bads/(bads+goods)) is of course unknown.

Now, I have a sample of $N$ from the population and I know the bad rate of this sample as $b$. The question is can I calculate the confidence interval based on $N$ and $b$ ONLY? In other words, can I calculate the confidence interval $x$ such that with, say, 95% confidence the population bad rate falls in $[\text{range}_1,\text{range}_2]$, where $\text{range}_1$ will be $b-x$ and $\text{range}_2$ will be $b+x$.

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    $\begingroup$ Under random sampling, yes; this is simply a confidence interval for a binomial proportion, with the usual caveats about the assumptions of and interpretation of confidence intervals. $\endgroup$ – Glen_b Oct 13 '13 at 6:24
  • $\begingroup$ As long as $np(1-p)$ is not small (bigger than 10 is usually plenty), you can use the normal interval described here. If the $n$ is small and $p$ is very near 0 or 1, you may need to consider one of the other binomial approximate confidence intervals. Many other posts here cover aspects of CIs for binomial proportions; the search bar turns many up. $\endgroup$ – Glen_b Oct 13 '13 at 6:36
  • $\begingroup$ Further examples of previous posts on this topic: e.g. 1, e.g. 2 $\endgroup$ – Glen_b Oct 13 '13 at 6:39
  • $\begingroup$ Just as a side note - if you want to control your confidence level, then the size of the interval also depends on that. $\endgroup$ – Glen_b Oct 13 '13 at 6:45
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Exactly as Glen_b said. Under random sampling the confidence interval for a binomial proportion can be easily calculated with, e.g., using the normal approximation. The formula can be found from Wikipedia, among other sources (http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval).

As an example, the 95% confidence interval for a sample of 1000 with the proportion of bad of 0.5, the confidence interval would be from 0.5-sqrt((1/1000)*0.5*0.5)*1.96 to 0.5+sqrt((1/1000)*0.5*0.5)*1.96. In other words, in this case the 95% confidence interval would be 0.469-0.530.

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