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16 Pages

### 428_3_CLTReview

Course: STAT 428, Summer 2008
School: Ohio State
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Word Count: 524

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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...

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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 ...

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