Multiple hypothesis testing

Question

Let's suppose I have 10 independent measurements with results close to zero. How can I claim that they are in agreement with the theory, them being zero?

The errors of these 10 results are not equal, but almost (could be approximated by a single number). I now like to say that all measurement results are in good agreement with the theory, i.e. with the claim that they all vanish.
Do I have to give 10 different hypotheses or just a single one that combines all results? Can test all values together in a single test of normal distribution? Up to now, I just said that all results are zero within two standard deviations. But is this already sufficient?

So let's say my data are [-0.0445, -0.0237, 0.0047, -0.0002, 0.0020, 0.0059, -0.0017, -0.0239, 0.0093, 0.0083], each with an error between 0.0235 and 0.0238, let's say, all have an error of 0.0236. How can I claim that this outcome is in agreement with the theory of all being zero?

Thanks!

Answer 1

Revised per OP Comment

If you only have 10 measurements and an assumed standard error, then you can use a Bonferroni Correction to develop simultaneous confidence intervals with a guaranteed maximum family-wise error rate (assuming that your errors are well approximated by a normal distribution):

1. Select your desired overall Type I error rate $\alpha$
2. We have 10 intervals, so you want to calculate the $\left(1-\frac{\alpha}{10}\right)\times 100\%$ CI around each measurement using the normal distribution whose standard deviation is equal to your assumed errors for each measurement.
3. If all of these intervals contain 0, then you have reason to believe that your data are consistent with your hypothesis.

Note: Failure to reject $\nRightarrow$ Accept the null (or that the null is correct). The Bonferroni correction is fairly conservative, and since "agreement with theory" is implied by the null hypothesis, it will take rather drastic deviations from the theoretical predictions to refute the theory.