How to test $\mu_1 = \mu_2 = \mu_3$ when the covariance matrix is known/unknown? I need to verify the hypothesis H0: $\mu_1 = \mu_2 = \mu_3$ vs the alternative(at least one mean doesn't equal to the others). Where $N=20$ and the estimated mean $\overline{x} = (1, 0, -1)^T$. Popluation comes from normal distribution $N_3(μ,Σ)$.
a) with known covariance matrix
$ \begin{bmatrix}
3 & 2 & 1 \\
2 & 3 & 1 \\
1 & 1 & 4 
\end{bmatrix}  $. I wrote the hypothesis in the form of $C\mu$ where $ C = \begin{bmatrix}
1 & -1 & 0 \\
1 & 0 & -1 \\
\end{bmatrix}  $. Then I calculated the statistics (I found the formula in the book, is it ok?) in such a way
$$ T^2 = N(C\overline{x})^T(CSC^T)^{-1}(C\overline{x})$$ but then I cannot find one good critical value to compare it to (I found this formula, but again could you please confirm/deny is it correct):
$$ T^2_{n-1,N-1,\alpha} = \frac{(N-1)(n-1)}{N-n+1}, \quad \text{where $n$ is the dimension of the mean distribution, here it's 3.}$$
Is this formula correct? (I saw other formulas as well and I am confused.)
B) What will be the difference in the calculations if we assume that the covariance matrix is unknown?
Note: I don't need any coding solutions (R, python, etc.); I want to understand it on the theoretical level.
 A: You want to test a null hypothesis of the form
$$H_0:C\begin{pmatrix}
\mu_1\\
\mu_2\\
\vdots\\
\mu_q
\end{pmatrix}=c\in\mathbb{R}^p, C\in\mathbb{R}^{p\times q},\mathrm{rank}(C)=p.
$$
Under $H_0$, and assuming that the sample mean $\bar{x}$ is calculated from an i.i.d. random sample from a $\mathcal{N}_q(\mu,\Sigma)$ population, we have $$N(C\bar{x}-c)^\mathsf{T}(C\Sigma C^\mathsf{T})^{-1}(C\bar{x}-c)\mathrel{=:}T_a\sim\chi^2(p)$$ if $\Sigma$ is known, as well as
$$
(N-1)(C\bar{x}-c)^\mathsf{T}(CSC^\mathsf{T})^{-1}(C\bar{x}-c)\mathrel{=:}T_b\sim T^2(p,N-1)
$$ if $\Sigma$ is unknown (and estimated based on $\frac{N}{N-1}S$, the unbiased version of the sample covariance matrix $S$) with
$$S=\frac{1}{N}\sum_{i=1}^{N}(x_i-\bar{x})(x_i-\bar{x})^\mathsf{T},
$$
where $x_i$ denotes the $i$-th observation of the $q$-variate normal random variable.
In your question we have $p=2$, $c=(0,0)^\mathsf{T}$, and $N=20$.
Therefore, at a $(\alpha\cdot100)\%$ significance level, your critical value in a) is the $(1-\alpha)$-quantile of the chi-squared distribution with $2$ degrees of freedom and you want to compare the realization of $T_a$ to this critical value.
Your critical value in b) is the $(1-\alpha)$-quantile of the Hotelling's T-squared distribution with degrees of freedom $2$ and $19$.
Since
$$\frac{N-p}{(N-1)p}T_b\mathrel{=:}T_b^*\sim F(p,N-p) \iff T_b\sim T^2(p,N-1),$$
you can equivalently compare the realization of $T_b^*$ to the $(1-\alpha)$-quantile of the $F$-distribution with degrees of freedom $2$ and $18$, which should be easy to get in a statistical software of your choice.
