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Lecture3 - Feb 1st lecture Sampling from the Normal...

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Feb 1 st lecture Sampling from the Normal Population * Example : We wish to estimate the distribution of heights of adult US male. It is believed that the height of adult US male follows a normal distribution 2 ( , ) N μ σ Def. Simple random sample: A sample in which every subject in the population has the same chance to be selected. X : The Random Variable denote the height of a adult male we will choose randomly from the population So 2 ~ ( , ) X N μ σ : the distribution of a randomly selected subject is the population distribution. Theorem 1 Sampling from the normal population Let i.i.d. 2 1 2 , ,..., ~ ( , ) n X X X N μ σ , where i.i.d stands for independent and identically distributed 1. 2 ~ ( , ) X N n σ μ 2. 2 2 2 1 1 2 2 (X ) ( 1) ~ (Chi Square distribution with ( 1) degrees of freedom), n i i n X n S n χ σ σ = - - - = - *Reminder: The Sample variance 2 S is defined as: 2 2 1 (X ) 1 n i i X S n = - = - *Def 1: The Chi-square distribution is a special gamma distribution (*** Please find out which one it is.) *Def 2: Let i.i.d.
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