Central Limit Theorem

Dr. V.R. Bencivenga and Dr. Stephen Donald Economics 329

SAMPLING DISTRIBUTIONS – ICQ’S IQ scores

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)?