Correct hypotesis test for showing multiple temperature profiles are the same I have data from an experiment where I heated 3 different objects. During the experiment I measured the temperature inside all 3 objects at 3 different depths from the heat source (but the same 3 depths for all the objects). I plotted the temperature profiles for each object at a relevant time and I want to see (and prove), with an hypotesis test, that the profiles are equal for the three objects. My question is if this is possible and significant with this amount of data, and if so, which is the best test for this. Thanks.

You cannot prove that means are exactly equal. You can, however, establish with specified confidence that two means do not differ by more than a specified threshold $\tau$, using an equivalence test. A common method is the "two one-sided test" (TOST) procedure of Schuirmann. One way to describe that procedure is to construct a $1 - 2\alpha$ confidence interval for the difference of means. If that interval lies entirely within the interval $(-\tau,+\tau)$, then you reject the null hypothesis of non-equivalence with significance level $\alpha$.
