Suppose we want to know if there is a significant difference in the mean weight change for different types of treatments. We have three treatments $A$, $B$, $C$ and denote the mean weight change by $\overline{A}$, $\overline{B}$ and $\overline{C}$ respectively. Define $M=\frac{1}{3}(\overline{A}+\overline{B}+\overline{C})$. We have 21 patients that we divide in 3 groups of 7. Now I have to show that the two test statistics $T=(\overline{A}-M)^2+(\overline{B}-M)^2+(\overline{C}-M)^2$ and $S=\overline{A}^2+\overline{B}^2+\overline{C}^2$ are equivalent in the sense that they result in the same testing procedure.
I don't know what is meant by this question. I can't find a definition of equivalency between two test statistics. I thought maybe that I should show that the p-value of both tests is the same, but there are $\frac{21!}{7!7!7!}$ possibilities. So how should I interpret this question?
Hints:
The expansion of $T$ will involve $\overline{A}^2$, $\overline{B}^2$ and $\overline{C}^2$ terms so try to express $T$ as $S+f(M)$ for some function of $M$
Consider what might be counted as extreme values of $T$. Very high ones? Very low ones? Similarly for $S$
Given $M$, is there a direct correspondence between extreme values of $T$ and extreme values of $S$? What does that suggest about testing the two statistics?