Lesson 14_002

# Lesson 14_002 - Lesson 14 Distribution of the Sample Means...

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Lesson 14: Distribution of the Sample Means and the Central Limit Theorem

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Goals of this lesson: 1) Understand why we need to use a distribution of sample means 2) Understand what the distribution of sample means is 3) Understand the properties of the distribution of sample means and the Central Limit Theorem 4) Application of the distribution of sample means and the Central Limit Theorem: how to use the distribution of sample means in practice.
A. Notation! Notation! Notation! Why we need a distribution of sample means Gas prices continue to fluctuate. Prices soared after Hurricane Katrina, topping \$3 per gallon in many parts of the country. Then they came down for awhile but skyrocketed above \$3 per gallon again in the spring. As of this writing (end of June), they’ve finally fallen below \$3 for a gallon of regular unleaded gasoline. Suppose you are researching gas prices and want to find out how likely it is that gas prices are above \$3 per gallon (regular unleaded gasoline). There are two approaches that you could take: 1. You’re driving down the freeway and notice that your gas tank is almost empty. The next exit off the freeway is 10 miles up the road and you know there is one gas station there. What is the probability the price of a gallon of regular unleaded gasoline at that station is more than \$3? a. Notice that the question is about the price of gas at one gas station. Write a probability statement for this question.

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a. Notice that the above probability statement is in terms of X. What does X represent? c. There are two key points to this example: i] We are trying to find the probability of one (and only one) observation. When doing so, the probability statement is in the form of P(X). (In this case, it’s P(X>\$3).) ii] In order to find this probability, we need to know how X is distributed . That is, X represents the gas price at this one gas station, but it is one of thousands of gas stations in the population (population: gas stations across the country). Each gas station has a price for a gallon of regular unleaded gasoline. We need to know the distribution of these prices. That is, we need to know how the data (gas prices) in the population (all gas stations) are distributed!
a. Let’s suppose that gas prices are normally distributed (which seems reasonable). What two parameters are associated with data that are normally distributed? e. Do we know these two values? How much effort would it take to find these two parameters?

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1. You are planning on taking a road trip across the country. You are going to buy gas at many gas stations across the country. Think of the gas stations you stop at as a sample of all gas stations around the country. What is the probability that the average price (from your sample) of a gallon of regular unleaded gasoline that you’ll pay is at least \$3?
a. The important idea: It’s too costly and time-consuming to collect data on every single observation in the population

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Lesson 14_002 - Lesson 14 Distribution of the Sample Means...

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