# Lecture 5 - TA Office Hours Vadiraj Hombal Thursdays...

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TA Office Hours: Vadiraj Hombal: Thursdays 4:30-6:30 PM JEC 6027 Max Henderson: Sundays 4:00-7:00 PM McNeil Room RU

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The Hypergeometric Distribution computes the number of successes (x) in a completely random sample of size n drawn from a population consisting of M “successes” and (N-M) “non-successes”. The random variable can take on values from max(0, n-N+M) and min(n, M), i.e., max(0, n-N+M) x min(n, M) (see the probability mass function, parameter definitions on pp. 117 of the text).
Hypergeometric N – lot size ( population size ) M – number of “successes” in N n – sample size ) , min( ) , 0 max( M n X M N n +

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N M n x E = ) ( = N M N M n N n N x V 1 1 ) ( Sampling without replacement ( ) , defined for, max(0, ) min( , ) NM M nx x p xn N M x n M N n ⎛⎞ ⎜⎟ ⎝⎠ =− +
A three parameter distribution used in many QC applications: N – population or lot size M – number of “successes” in N n – sample size Sampling without “replacement”, i.e., a finite population version of the binomial. Problem 3-69 (page 120) Illustration of hypergeometric calculations N=12, M=7, n=6 compute p(x=5), p(x 4), p(x ≥μ + σ )

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3-69 (page 120) N = 12, M = 7, n = 6 114 . 0 6 12 1 5 5 7 ) 5 ( = = = x p ) 4 ( ) 3 ( ) 2 ( ) 1 ( ) 0 ( 1 ) 4 ( = = = = = x p x p x p x p x p x p 879 . 0 = ( ) ( 4.392) ( 5) (5 ) . 0 0 7 px μ σ + =≥ ≥=
(5 ) ( 1) x nx n px p p x ⎛⎞ == ⎜⎟ ⎝⎠ 58 . 0 12 7 = = p 51 6 0.58 (0.42) 5 = 0.114 vs. 165 . 0 = Hence, binomial is a poor approximation due to small population size If we were to try this approximation for part a.) of the problem.

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Lecture 5 - TA Office Hours Vadiraj Hombal Thursdays...

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