Confidence interval for the conversion on site I am the developer of web service and I'm trying to to build some plots for the inner dashboard. I raised two questions that I can not solve on their own. 
Suppose n visitors went to the site for some period of time. During this time, m visitors were successfully registered and became users. Conversion to registration is the ratio of registered users to all visitors or the probability that a visitor registers on the site - m / n. 
The questions are:


*

*What is the (minimum) range of the true value of the conversion to registration with defined probability p? 

*What is the probability that the true value of the conversion are greater than a predefined value c? 


If I understand correctly, it is Bernoulli process, but I do not understand what to do with this knowledge. Just I have gaps in terminology, so I will be grateful to you if you tell me how to mathematically call those things I'm looking for.
 A: The first question refers to the fact that the proportion $\frac{m}{n}$ of registered visitors observed in a finite sample is not necessarily the "true" rate that we could observe, in theory, considering an infinite sample. If my interpretation is right, the answer might be given by the $95\%$ confidence interval for the observed proportion. This interval can be interpreted as a a range of values around the observed proportion, where the "true" proportion falls with $95\%$ probability. 
As correctly stated in the OP, the  context of this problem is that of a Bernoulli trial and the observed proportion follows a binomial distribution. There are a variety of methods to calculate the $95\%$ confidence interval for a proportion within a binomial distribution. One of the  most used is the normal approximation. The formula for the $95\%$ confidence interval with this method is
$$\hat p \pm 1.96 \sqrt{\frac{\hat p (1-\hat p)}{n}}$$
where $\hat p$ is the observed proportion in a given sample of observations, $\pm 1.96$ is the $z$-value corresponding to the $2.5\%$ and $97.5\%$ percentile of a standard normal distribution, and $n$ is the sample size. So, the lowest value of this confidence interval can be used as the "minimal" value of the "true" proportion with $95\%$ probability. Note that the normal approximation, although widely used, may be not accurate in cases with  small sample sizes or proportions that are very close to $0$ or $1$. In these cases, more robust methods to calculate the confidence interval can be used, such as Wilson score, Jeffreys,
Clopper-Pearson, Agresti-Coull, arcsin transformation, and so on. 
As regards the second question, the probability $P(p > c)$ that the "true" proportion $p$ is higher than a predefined value $c$ can be computed again using the normal approximation. For instance, if $c$ is lower than the inferior bound of the $95\%$ interval of our observed proportion, we can conclude that the true proportion is higher than $c$ with $95\%$ probability. If $c$ is included in the interval, we can write
$$\hat p + z \sqrt{\frac{\hat p (1-\hat p)}{n}}=c$$
From this equation, we can calculate the value of $z$ (positive or negative), which directly leads to the corresponding probability $P(p > c)$ using standard conversion tables or online z-to-p conversion tools.
