Statistical test for vector of expected values Two characteristics are tested on population elements $X_1$ and $X_2$.
We assume that $X = (X_1, X_2)\sim N_2 (\mu, \Sigma)$ where $\mu$ and $\Sigma$ are parameters. The sample size is equal to $20$. It was calculated that (for the sample)
$$
\bar x = \begin{bmatrix}0 \newline 1
         \end{bmatrix}
$$
$$
s = \begin{bmatrix}
5&-3 \newline
-3&9
         \end{bmatrix}
$$
The above values are the mean and standard deviation calculated from the sample.
At the $\alpha = 0.05$ level, verify the hypothesis system:
$H_0: $ $\mu_1 = 2\mu_2$
$H_1: $ $\mu_1 \neq 2\mu_2$.
I don't know how to get down to this task. I have never encountered such an arrangement of hypotheses.

The problem can be solved using a slightly modified paired $t$-test in which, the null is $H_0:\mu_1 -2\mu_2 = 0$ and the alternative is $H_1:\mu_1 -2\mu_2 \neq 0$. The procedure is thus to apply a $t$ to the difference data $Y_1,\ldots,Y_n$, where $Y_i = X_{1i}-2X_{2i}$, as noted by @whuber.
The test statistic is
$$\frac{\sqrt{n}\bar Y}{\sqrt{s_Y^2}}\sim t_{n-1}.$$
For this test thus you only require $s_Y^2$, the sample variance of $Y_1,\ldots,Y_n$, and of course, its sample average $\bar Y = \bar{X_1}-2\bar{X_2}$.
