stat400lec22 - Lecture 22 Fall 2005- 2 -Let X be the sample...

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Statistics 400 Section 6.4 and 6.5 Central limit theorem Central Limit Theorem (CLT) If X1, X2, …, Xn are observations of a random sample of size n from a distribution with mean μ and variance σ 2 , Then we have W= n X / σ μ - = n n X i - N(0,1) as n →∞ In another word, W be can approximated by normal distribution when sample size n is sufficiently large. “sufficiently large”: n is greater than 20 Illustration: Continuous distribution http://www.gams.com/~erwin/cenlim/cenlim.html Discrete Distribution http://www.stat.sc.edu/~west/javahtml/CLT.html A notation Φ (z)=P(Z z) Ping Ma Lecture 22 Fall 2005 - 1 -
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Example: If X1, X2, …, X20 are observations of a random sample of size 20 from the uniform distribution U(0,1) , Find (1) P( = 20 1 i i X <9.1) (2) P(8.5 < = 20 1 i i X <11.7) Ping Ma
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Unformatted text preview: Lecture 22 Fall 2005- 2 -Let X be the sample mean of a random sample of size 32 from a distribution with mean 40 and variance 8. Approximate P(39.75&lt; X &lt;41.25) Approximation of Binomial distribution If X1, X2, , Xn are observations of a random sample of size n from the Bernoulli distribution with success probability p , Y= = n i i X 1 is b(n,p) W= n n X i - = ) 1 ( p np np Y--= n p p p X / ) 1 (-- N(0,1) sufficiently large: np 5 and n(1-p) 5 Very tricky. P(Y k) ( npq np k-+ 5 . ) Ping Ma Lecture 22 Fall 2005- 3 -Example: Let Y have a binomial distribution b(10,0.5). Calculate P(3 Y&lt;6) and approximate it using CLT Ping Ma Lecture 22 Fall 2005- 4 -...
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stat400lec22 - Lecture 22 Fall 2005- 2 -Let X be the sample...

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