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Week 5 Mon Sept 27

# Week 5 Mon Sept 27 - WEEK 5 Monday Sept 27 Today Cover...

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WEEK 5, Monday, Sept 27 Today: Cover Sections 3-8, 3-10 and 3-11 . Hypergeometric Distributions (Sec 3-8) . The Hypergeometric probability distributions arise when counting Successes when sampling n at random from a dichotomous, finite population. Hypergeometric probability distributions have three parameters: N = # of elements in the population, n = sample size, and S = # of elements in the population of a particular type, which we will call Successes. We will denote the Hypergeometric distribution by HG( n , S , N ). It is sometimes useful to let p denote the ratio S / N , the probability of a Success on a single random selection from the population. The Hypergeometric distribution is the sampling distribution for the number X of S's in a simple random sample of size n selected from the population. Here is the picture of what is going on. S N - S Simple Random Sampling n from N x n - x Successes Others n N Sample Population 1

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Summary on HG( n , S , N ). (p. 122 of textbook) Probability Mass Function - - = n N x n S N x S x p ) ( , x = 0, 1, 2,…, n Expectation (Mean), Variance and Standard Deviation of the HG( n , S , N ) Distribution ) 1 ( 1 ) ( ), 1 ( 1 ) ( , ) ( p np N n N X SD p np N n N X V np X E - - - = - - - = = f inite population correction factor ( fpcf ) where recall here N S p = . When using simple random sampling and a small sample size relative to the population size, i.e., with n / N small, the Hypergeometric distribution is well-approximated by the Binomial with # of Trials equal to n and chance of success on a single trial equal to p = S / N . For this reason, the Binomial distributions arise as sampling distributions in simple random sampling from a dichotomous population. (If the selections were made with replacement, the Binomial is the exact sampling distribution.) The textbook usually ignores the population size N and the fpcf after Chapter 3. We will not. Binomial Hypergeometric_( p = S / N )___________ 2
x n x p p x n x p - - = ) 1 ( ) ( - - = n N x n S N x S x p ) ( x = 0, 1, …, n x = 0, 1, …, n np X E = = ) ( μ np X E = = ) ( μ ) 1 ( ) ( p np X SD - = = σ ) 1 ( 1 ) ( p np N n N X SD - - - = = σ x B(10, .2) HG(10,20,1 00) 0 0.107374 0.095116 1 0.268435 0.267933 2 0.301990 0.318171 3 0.201327 0.209208 4 0.088080 0.084107 5 0.026424 0.021531 6 0.005505 0.003541 7 0.000786 0.000368 8 0.000074 0.000023 9 0.000004 0.000001 10 0.000000 0.000000 E ( X ) = 2.000 2.000 SD ( X ) = 1.265 1.206 fpcf = 0.953463 3

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Example 1. Rotten Eggs .
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• Spring '08
• Anderson
• Probability theory, probability density function, Cumulative distribution function, Probability mass function

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Week 5 Mon Sept 27 - WEEK 5 Monday Sept 27 Today Cover...

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