Given $\bar x = 150$, $X \sim (\mu , 20^2)$ distributed normally with unknown mean and $\mu \sim (180, 40^2) $ distributed normally, find the posterior distribution for $\mu$ in terms of $n$.
I already have $f(\mu) = \frac{1}{40\sqrt{2\pi}} e^{\frac{-1}{2} (\frac{\mu - 80}{40})^2}$ for the prior and have calculated $f(x \vert \mu) = (\frac{1}{20 \sqrt{2\pi}})^n \exp{\frac{-1}{2*20^2} \sum_{i=1}^{n}(x_i-\mu )^2}$ for the likelihood
Then I understand that I have to use Bayes' Theorem, but I do not know where to go with the likelihood function to get to this.
Any help would be appreciated