X1, . . . ,X9 are independent and identically distributed random variables?

Question

X1, . . . ,X9 are independent and identically distributed random variables
each with a N(4, 25) distribution. Write down the distribution of
the sample mean ¯X and hence find the probability it is
(a) less than 5,
(b) greater than 5.8,
(c) between 3.5 and 4.4.

Answer 1

since the underlying distribution is normal, you can use a normal distribution:

E(X) is estimated by the mean of the underlying distribution
V(X) is estimated by Var(X)/n = 25/9

Then you run it like standard normal...