# Where two confidence intervals meet?

I have two means that refers to two different samples of different populations. Let's say, for example, that the two means are 1.0 and 3.15. I can compute confidence intervals about these two cases and, logically, higher is the confidence interval, larger are the two intervals (they tend to infinite width). So, the question is: is there a way to find out the point in which these two confidence intervals meet?

• Depending on how the confidence intervals are computed, the answer may be easy indeed. But I am curious about the application: what do you suppose this point would tell you? (You might be interested in our thread on comparing confidence intervals.) – whuber Jan 27 '16 at 16:19
• I would use it nextly. I will receive some points and I will see if they belongs to the first or the second confidence interval. In this way I'll be able to say, for each of these point, if it has the character of the first sample otherwise follows the second one character. – foolcool Jan 27 '16 at 16:31
• Unfortunately I'm not so good with statistics thoery. So, I don't see formulas or approaches that could help me to find the point in which two CI meet. Isn't it? – foolcool Jan 28 '16 at 9:28

The exact answer depends on what are the intervals about and other circumstances but I'm going to made a couple of assumptions in the hope they fit your problem:

• You are computing confidence intervals on the mean.
• Your samples are large (let's say, at least over 100). If that assumption is false, then the answer here is just an approximation.

(For statisticians reading this, the second assumption means that for sake of simplicity I'm going to use normal distribution instead of t-Students and using standard deviation of sample).

You can compute means and standard deviation of both samples, and let's denote them mean1, mean2, sd1 and sd2.

Now you can use the following formula:

$$z=\frac{(mean2-mean1)}{sd1+sd2}$$

That $z$, in a normal standard distribution, is the one that contains between -z and +z the same probability of the level of confidence of your intervals when they start overlapping.

If you are interested in the point of overlapping, you just need to do:

$$point=mean1+z*sd1$$

or

$$point=mean2-z*sd2$$

and both should yield the same point.

If you need to know the confidence level of your intervals, you just need to check $z$ in a normal standard distribution table (like this one). If $p$ is the probability you get in the table you can find the confidence level by:

$$CL=1-2*(1-p)$$

If your samples are not large, and especially if they are small and of different sizes, the problem is a little different and it needs to be solved numerically - although an Excel spreadsheet can be enough to handle it.