How to estimate parameters of a normal distribution?

Question

Suppose one knew that 105 workers were evaluated by their boss. Such evaluation is distributed according to a normal distribution with mean $\mu$ and std. deviation $\sigma$. We also know that 20 workers received score less than 2, 25 were evaluated between 2 and 5, and 60 with evaluations bigger than 5. How can I form reasonable estimates for $\mu$ and $\sigma$?

Answer 1

You know two proportions, i.e. two probabilities. So you know two areas under the normal curve for $N(\mu,\sigma)$.

There is a correspondence between $x$-values on the curve for $N(\mu,\sigma)$ and $z$-values on the curve for the standardized $N(0,1)$, for the same areas under the curves.

Hint: two equations and two unknowns.