Limit Central Theorem Review
Reading: Section 5.4 Learning Objectives:
Understand the CLT approximation Understand the CLT conditions Be able to apply the CLT appropriately Be able to use the Normal approximation to estimate probabilities/sample sizes
Spring 2009 Statistics 428 1
Properties of the Sample Mean
Suppose X1,...Xn are independent random draws from a population with mean and variance 2. The sample mean is defined: Recall some properties of expected value and variance:
Applying these to the sample mean gives the general result:
Spring 2009
Statistics 428
2
Distribution for Normal Population
Suppose X1,...Xn are independent random draws from a population with mean and variance 2. If the population values have a Normal distribution, then the sample mean also has a Normal distribution:
Spring 2009
Statistics 428
3
Distribution for Non-Normal Population
Suppose X1,...Xn are independent random draws from a population with mean and variance 2. For example, consider the shower flow rates in Perth, Australia. What is the distribution of the sample mean of 129 houses' rates? What does this question mean?
Spring 2009
Statistics 428
4
Continuous Distribution
Spring 2009
Statistics 428
5
The Central Limit Theorem
How large is "sufficiently large"?
Spring 2009
Statistics 428
6
The Perth Flow Rates
Spring 2009
Statistics 428
7
Sample Average # Ones in n Dice Rolls
Suppose we roll a 6-sided die once, and let X be a RV such that:
Notice that the average X over n die rolls:
Spring 2009
Statistics 428
8
Sample Average # Ones in n Dice Rolls
Spring 2009
Statistics 428
9
Sample Average # Ones in n Dice Rolls
The less "normal" the population distribution is, the larger the sample size (n) you need for CLT the approximation to be reasonable.
Spring 2009 Statistics 428 10
Using a Normal Distribution/CLT
Example: Carbon Monoxide (CO) emissions for a certain kind of car vary with a mean 2.9 g/mi and standard deviation 0.4 g/mi. 1. Assume that CO emissions for this kind of car follow a Normal distribution. What is the probability that a randomly chosen car of this type has CO emissions greater than 3.0 g/mi?
Spring 2009
Statistics 428
11
Using a Normal Distribution/CLT
Example: Carbon Monoxide (CO) emissions for a certain kind of car vary with a mean 2.9 g/mi and standard deviation 0.4 g/mi. 2. Do NOT assume that CO emissions for this kind of car follow a Normal distribution. A company has 80 of this kind of car in its fleet, which are approximately a random sample of all of this kind of car. What is the probability that the average CO emissions for these 80 cars is greater than 3.0 g/mi?
Spring 2009
Statistics 428
12
Spring 2009
Statistics 428
13
Using a Normal Distribution/CLT
Example: Carbon Monoxide (CO) emissions for a certain kind of ...