Combining results from multiple distributions

[mwoldinga]

Suppose I have a result where the outcome is that with 95% confidence interval of the sample is between 27 and 42. With a second method the test result of the same sample gives a 95% confidence interval between 27 and 48. And with a third method 95% confidence interval of the sample is between 35 and 38.5.
The overlap between the three intervals is in this case 35-38.5. How can I find out what the probability is for this interval, using all three test results?
Let's assume each of the three different test methods give an outcome that has a normal distribution, where the mean each time lies in the middle of each interval.
Of course the answer has to lie above 95%, since the probability will increase if you have more results that do not contradict the last one, but how can I calculate this?
Any suggestions?
Monica (The Netherlands)

 

[bpet]

Suppose I have a result where the outcome is that with 95% confidence interval ... The overlap between the three intervals is in this case 35-38.5. How can I find out what the probability is for this interval, using all three test results?...

Interpreting the CI percentages as exact probabilities, we could say for example there's a 0% chance the mean lies between 27 and 35 or 38.5 and 48, and 5% chance that the mean lies outside of 27 to 48 - I hope you agree that's nonsense.

...Let's assume each of the three different test methods give an outcome that has a normal distribution, where the mean each time lies in the middle of each interval.
Of course the answer has to lie above 95%, since the probability will increase if you have more results that do not contradict the last one, but how can I calculate this?
Any suggestions?

With a few additional assumptions we'd get one answer but then slightly different assumptions could give quite a different result.

It may be helpful to write down what facts are known about the 3 methods before translating those into assumptions.

 

[mwoldinga]

I don't think it is very helpful if write down more facts about the methods. The tests are about dating bones. Each of the three test will give an outcome with a different confidence interval, using the same bone each time.
I calculated the probability where the boneresults were outside the overlap in all three cases. 1-this answer is then the probability that the boneresults are not outside the overlap in all three cases. This answer is about 96%. At least this method fits with my intuition, but intuition in statistics is a dangerous thing.