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Lecture 2, assignment 2

# Lecture 2, assignment 2 - Lecture 2 Notes Central Limit...

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Lecture 2 Notes , Central Limit Theorem – Certain statistics (sample means, for example) are normally distributed, If the distribution is symmetrical, a small sample the larger the sample size the more normal the distribution will become. An Exercise to demonstrate the Central Limit Theorem : X P(X) (X - µ x ) 2 •P(Xi) P() 1 1/3 4/3 3 1/3 0 3/9 5 1/3 4/3 2/9 µ= Σ (Xi•P(Xi)) = 3 σ 2 = E(X-µ x ) 2 = Σ (X - µ x ) 2 •P(Xi) = 8/3 1/9 Sample Size 2 1 2 3 4 5 Notice that with a sample size of only two, the distribution of sample means can be approximated by the normal distribution. That is to say, sample means quickly take on the shape of the normal even when the sample size is very small. Notice, however, that the original distribution of the variable was symmetrical. If the underlying distribution is highly skewed, then it would take a much larger sample size for the distribution of sample means to become normal. P() (P) ( - µ) 2 ( - µ) 2 •P() 1, 1 1 1 1/9 1/9 4 4/9 1, 3 2 2 2/9 4/9 1 2/9 1, 5 3 3 3/9 9/9 0 0 3, 1 2 4 2/9 8/9 1 2/9 3, 3 3 5 1/9 5/9 4 4/9 3, 5 4 µ = 3 σ 2 = 4/3 5, 1 3 5, 3 4 5, 5 5 σ 2 = Σ (- µ ) 2 P() σ 2 = (8/3)/2 = 4/3 1

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E() = µ x (the population mean, regardless of population size) E() = Σ (Xi•P(Xi)) and Σ •P() = µ
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Lecture 2, assignment 2 - Lecture 2 Notes Central Limit...

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