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# stat400lec21 - Lecture 21 Fall 2005 2-Example X1~N(693.2...

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Statistics 400 Section 6.3 and 6.4 Central limit theorem If X1, X2, …, Xn are observations of a random sample of size n from a normal distribution N( μ , σ 2 ), then distribution of the sample mean X is N( μ , σ 2 /n). If X1, X2, …, Xn are observations of a random sample of size n from a normal distribution N( μ , σ 2 ), then we have (a) X and S 2 are independent (b) 2 2 ) 1 ( σ S n - is χ 2 (n-1) Note that we have U= = - n i i X 1 2 2 ) ( μ ~ χ 2 (n) W = 2 2 ) 1 ( S n - = = - n i i X X 1 2 2 ) ( ~ χ 2 (n-1) Example: If X1, X2, …, X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find (1) P(796.2 < = - 16 1 2 ) 50 ( i i X <2630) Ping Ma Lecture 21 Fall 2005 - 1 -

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(2) P(726.1 < = - 16 1 2 ) ( i i X X <2500) If X1, X2, …, Xn are n independent normal random variables with respective means μ 1, μ 2, …, μ n and variances 2 1 σ , 2 2 , …, 2 n , then Y= n i Xi a 1 ~N( i n i a μ 1 , n i i a 1 2 2 ) Ping Ma

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Unformatted text preview: Lecture 21 Fall 2005- 2 -Example: X1~N(693.2, 22820) and X2~N(631.7,19205), and X1 and X2 are independent. Find P(X1>X2). Central Limit Theorem 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 Ping Ma Lecture 21 Fall 2005- 3 -Example : Let X be the mean of a random sample size 36 from an exponential distribution with mean 3. Approximate P(2.5< X <4) Ping Ma Lecture 21 Fall 2005- 4 -...
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stat400lec21 - Lecture 21 Fall 2005 2-Example X1~N(693.2...

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