# Confidence Intervals for not integers numbers ratio

Tags:
1. Jan 18, 2016

### fatgianlu

Hi, I’m having a problem with a particular case of binomial proportion.
I want calculate a confidence Intervals for a binomial proportion for an efficiency. This kind of intervals are usually defined for ratios between integers numbers but in my case I had to subtract from both numerators and denominators some decimals numbers. I’d like to use Clopper Pearson method and I’m also able to extract the limits for these decimals numbers but I don’t know how to legitimise this calculation theoretically and if I can. Do you know if there is some way to threat confidence intervals properly using decimals numbers? Or some paper that talks about this issue?

2. Jan 19, 2016

### Staff: Mentor

It's not clear to me what you're asking. It might be helpful if you told us the problem you're trying to solve.

3. Jan 20, 2016

### fatgianlu

Ok,
I'm calculating an efficiency to pass a selection of some data. In this data there is some background that I want to subtract. In order to do so I have a simulated sample of the background. This simualted sample is scaled with several weights. (normalization, efficiency correction, etc..) that leads to have not integers numbers. When i want to calculate the efficency, i.e. ratio between the successes in the trials, I have to cope with not integers numbers. When I want to calculate an uncertainty of a proportion I usually use the Cloipper-Pearson method that is derived form a binomial distribution. The binomial distribution is a discrete probability distribtion and so is not correct in my case where i have not integers but continuous numbers. What I'd like to have is a method to calculate a confidence-level in case of a proportion between continuous numbers. Or much better, a way to take into account a subtraction of weighted event. I read something about using the beta distribution but probably I should improve my stats knowledge... I don't know if the problem is clear.

4. Jan 20, 2016

### Staff: Mentor

That will be necessary, so I don't think the non-integer values are the main issue.
You can still use the basic approach: calculate efficiency values where the probability to observe more/fewer events than observed is below some threshold (2.5% or 5% or whatever). The probability will also have to take the background subtraction and its uncertainty into account.

How many events do you have? If you have enough, you can probably ignore the discrete nature of the observed events, and use conventional continuous methods.

Do you have proper uncertainties for the MC background sample?