From statements in J. R. R. Tolkien's "The Fellowship of the Ring" (Prologue I, paragraph 4), I construe that the heights of hobbits are approximately normally distributed with mean 36 inches and standard deviation 4.2 inches. Find the following probabilities (be as accurate as our process allows).
Find the probability that one hobbit picked at random is no more than 36.7 inches tall.
Find the probability that a simple random sample of 16 hobbits have a mean height of no more than 36.7 inches tall.
Find the probability that a simple random sample of 100 hobbits have a mean height of no more than 36.7 inches tall.
Find the probability that a simple random sample of 400 hobbits have a mean height of no more than 36.7 inches tall.