Central Limit Theorem

In the population, “intelligence quotient” scores (IQ’s) are approximately normally distributed 
with a mean of 100 and a standard deviation of 15. Suppose we plan to obtain IQ scores of a 
random sample of n = 30 individuals, and then we’ll compute the sample mean and sample 
standard deviation. Find each of the following. 
1) The probability that the sample mean will be larger than 105. 
2) Values a and b (where a < b) such that the sample standard deviation is between a and b with 
probability 0.8, and the probability that the sample standard deviation is larger than b is 0.1. 
3) Suppose you were unsure of the value of the population standard deviation, and so you plan 
to use only the sample standard deviation. Find the exact probability that the sample mean 
will be within one-third of one sample standard deviation of the population mean. 
4) What would your answer to the question in (3) be if instead you use an approximation based 
on large sample theory (i.e., based on the Central Limit Theorem)?