The standard for physics reporting on things like this is "five sigma" or that the test statistic in their hypothesis test must be greater than 5. Since this is a one-tailed test, this corresponds to a p-value smaller than 0.00000029, so the chances of being wrong are about 1 in 3.5 million. This isn't a "confidence interval" like you were asking about, but I think that it answers what you meant to ask. Let me know if you wanted something else.

Thanks, that pretty much explained what I was after.