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cltans - Outline Sampling Distributions and the Central...

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Unformatted text preview: Outline Sampling Distributions and the Central Limit Theorem October 25,2011 Sampling and CLT Outline Outline 1 Introduction 2 Sampling Distributions related to the Normal 3 Law of Large Numbers 4 Central Limit Theorem Sampling and CLT Introduction Sampling Distributions related to the Normal Law of Large Numbers Central Limit Theorem Outline 1 Introduction 2 Sampling Distributions related to the Normal 3 Law of Large Numbers 4 Central Limit Theorem Sampling and CLT Introduction Sampling Distributions related to the Normal Law of Large Numbers Central Limit Theorem A change of direction Up to now we’ve focused on probability distributions What will data look like if we have a given probability distribution (with some parameters) This is important, and has some immediate applications Decision analysis, life tables, Las Vegas We haven’t touched statistics yet! Sampling and CLT Introduction Sampling Distributions related to the Normal Law of Large Numbers Central Limit Theorem We want to estimate the mean SBP of female graduate students at UMN Enroll a (hopefully) random sample of n women and measure their SBPs X i What do we mean by a random sample? Lets call X 1 ... X n a random sample of size n from a distribution f if x i ∼ f and the samples are all independent When all the samples come from the same distribution we can write X 1 ... X n iid ∼ f How do we know that X 1 . . . X n iid ∼ f ? What should f be? N ( μ, σ 2 ) ? What do our data tell us about μ and σ 2 ? Sampling and CLT Introduction Sampling Distributions related to the Normal Law of Large Numbers Central Limit Theorem A statistic: a function of the observed random variables in a sample Mean: ¯ X = ∑ X i n Variance: S 2 = ∑ ( X i- ¯ X ) 2 n- 1 Since a statistic is simply a function of R.V.’s, they too are R.V.’s They also have distributions: sampling distributions Its natural to consider the properties of these statistics Is ¯ X a good estimate of μ ? What do we mean by “good”? How large should n be to make it “good”? Sampling and CLT Introduction Sampling Distributions related to the Normal Law of Large Numbers Central Limit Theorem Where we were, are and are going Were: Probability Specify parameters and a distribution to generate data Are: Sampling Distributions, CLT Given data, what can we say about the mean, variance Going: General methods of Inference Frequentist, Bayesian inference - figure out why we care about ¯ X , S 2 Sampling and CLT Introduction Sampling Distributions related to the Normal Law of Large Numbers Central Limit Theorem Rest of this Lecture 1 Distribution of the sample mean of draws from a normal dist’n with known variance (Normal dist’n) 2 Distribution of the sample variance of draws from a normal dist’n ( χ 2 dist) 3 Distribution of the sample mean of draws from a normal dist’n with unknown variance (t-dist) 4 Distribution of the sample mean of draws from a non-normal dist’n Sampling and CLT Introduction Sampling Distributions related to the Normal...
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