Let $X$ be a random variable with $E(X)=\mu$ and $Var(X)=\sigma^2$ and let $\overline{X}=\frac{X_1+X_2+\dots+X_8}{8}$. We have the following hypotheses. $$H_0:\mu=\mu_0$$ $$H_1:\mu<\mu_0$$ My question is, which of the following assumptions is correct in order to conduct a $Z$-test?
Assumption 1: In order to perform a $Z$-test, we assume that $X$ is normal and the distribution of the test statistic $$Z=\frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{8}}}\sim N(0,1)$$ is known, and we can proceed with conducting the $Z$-test.
Assumption 2: In order to perform a $Z$-test, we assume that $\overline{X}$ is normal and the distribution of the test statistic $$Z=\frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{8}}}\sim N(0,1)$$ is known, and we can proceed with conducting the $Z$-test.