The central limit theorem is important in statistics because it allows us to use the Normal distribution to find probabilities involving the sample mean

(a) if the sample size is reasonably large (for any population).

(b) if the population is Normally distributed and the sample size is reasonably large.

(c) if the population is Normally distributed (for any sample size).

(d) if the population is Normally distributed and the population standard deviation is known (for any sample size).

(e) if the population size is reasonably large (whether the population distribution is known or not).