Answer:
a) The probability that a hobbit picked at random is no more than 36.7 in tall is P= 0.56631.
b) The probability that a random sample of 16 hobbits have a mean height of no more than 36.7 in tall is P=0.74857.
c) The probability that a random sample of 100 hobbits have a mean height of no more than 36.7 in tall is P=0.95254.
Step-by-step explanation:
We have this parameters for the population: normally distributed with mean 36 in. and standard deviation 4.2 in.
a) The probability can be computed calculating z and looking up in a table.

The probability that a hobbit picked at random is no more than 36.7 in tall is P= 0.56631.

b) In this case, the sample standard deviation change to
.
We can calculate z with the sample standard deviation

The probability that a random sample of 16 hobbits have a mean height of no more than 36.7 in tall is P=0.74857.

c) We apply the same principle as in pont b.
We can calculate z with the sample standard deviation

The probability that a random sample of 100 hobbits have a mean height of no more than 36.7 in tall is P=0.95254.
