Generalization of one-sample t-test for multiple groups

If I have some data*, and want to test the hypothesis that it has a given mean $$\mu_0$$, I know of the possibility of using a one-sample t-test. Here, $$\mu_0=0$$ could be the null-hypothesis of a treatment having no effect, for example. (*let’s assume the data is normally distributed around its mean)

Now, suppose I now have several such datasets, and am interested in the question “Which of these groups have a mean different to $$\mu_0$$”. I could run several of these t-tests against $$\mu_0$$, but then I increase Type-I errors due to the multiple comparisons. A Bonferroni correction seems like a valid but very inelegant and too conservative solution.

I learned of the ANOVA as a generalization of the two-sample t-tests to compare their means, but here the question is different: “Do any two of these groups have different means?”

My question is: are there any generalizations of one-sample t-tests that control for multiple comparisons, other than a Bonferroni correction?

• Thanks! I had not considered the possibility of multivariate testing. This now begs the next question: if there are differences, how do I find out which groups have different means from $\mu_0$ without increasing my error rate? – Purple Rover Dec 10 '18 at 15:45