A random sample of $80$ observations produced a mean of $\overline {x} = 31.5$ from a population with a normal distribution and a standard deviation $\sigma = 2.55$.

(a) Find a $99$% confidence interval for $\mu$
$\leq \mu \leq$

(b) Find a $90$% confidence interval for $\mu$
$\leq \mu \leq$

(c) Find a $95$% confidence interval for $\mu$
$\leq \mu \leq$