Validate a hypotesis using a test and real data I don't know how to start with that, but I wanted, to given a set of Gaussian data $\{x_i\}$, and a given a test function,
$$Z^2=\frac{n(\bar{x}-\mu)^2}{\sigma^2}$$
And i want to test with a certain $CL\%$ of confidence level, the hypothesis $\mu=\mu_0$, where you know exactly $\sigma$.
I started computing $Z^2(X)$, but I don't see how to continue... If someone can give me some reference or some tip on how to start I will be thankful.

Assuming your data is not only Gaussian but also independent, you can use a two-tailed test. More formally, let
$$ Z = \frac{\sqrt{n}(\bar{x}-\mu_0)}{\sigma}$$
where $\bar{x}$ is the arithmetic mean of your data. Under this setting $Z \sim \mathcal{N}(0,1)$ and therefore you reject the null hypothesis
$$H_0: \mu = \mu_0$$
under a significance of $\alpha $ (for a $100(1-\alpha)$% confidence interval) if
$$ |Z| > \mathcal{z}_{\alpha/2}  .$$
For a concrete example, if $\alpha = 0.05$ then $\mathcal{z}_{\alpha/2} = 1.96$.
