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notes509fall11sec36

# notes509fall11sec36 - STAT 509 Section 3.6 Sampling...

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STAT 509 Section 3.6: Sampling Distributions Definition: Parameter = a number that characterizes a population (example: population mean ) it’s typically unknown . Statistic = a number that characterizes a sample (example: sample mean Y ) we can calculate it from our sample data. Y = We use the sample mean Y to estimate the population mean . Suppose we take a sample and calculate Y . Will Y equal ? Will Y be close to ? Suppose we take another sample and get another Y . Will it be same as first Y ? Will it be close to first Y ? • What if we took many repeated samples (of the same size) from the same population, and each time, calculated the sample mean? What would that set of Y values look like? The sampling distribution of a statistic is the distribution of values of the statistic in all possible samples (of the same size) from the same population.

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Consider the sampling distribution of the sample mean Y when we take samples of size n from a population with mean and variance 2 . Picture: The sampling distribution of Y has mean and standard deviation n / . Notation: Central Limit Theorem We have determined the center and the spread of the sampling distribution of Y . What is the shape of its sampling distribution? Case I: If the distribution of the original data is normal , the sampling distribution of Y is normal. (This is true no matter what the sample size is.)
Case II: Central Limit Theorem : If we take a random sample (of size n ) from any population with mean and standard deviation , the sampling distribution of Y is approximately normal, if the sample size is large .

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