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# Lecture14 - lecture 14 Sampling Distributions II lecture 14...

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lecture 14, Sampling Distributions II 1 / 21 Review Mean and Variance of Sample Mean Sample with Replacement Sample without Replacement Unbiased Estimates Sampling Distribution of the Sample Proportion lecture 14, Sampling Distributions II Outline 1 Review 2 Mean and Variance of Sample Mean Sample with Replacement Sample without Replacement 3 Unbiased Estimates 4 Sampling Distribution of the Sample Proportion

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lecture 14, Sampling Distributions II 2 / 21 Review Mean and Variance of Sample Mean Sample with Replacement Sample without Replacement Unbiased Estimates Sampling Distribution of the Sample Proportion lecture 14, Sampling Distributions II Review CLT (review) 1 If the population of individual items is normal, then the population of all sample means is also normal 2 Even if the population of individual items is not normal, as the sample size gets large, the population of all sample means is approximately normal 3 The mean of the sample mean equals the population mean, i.e., μ ¯ X = μ X 4 The standard deviation σ ¯ X of the sample mean is less than the standard deviation of the population, i.e., σ ¯ X < σ X If the population size N is infinity or is much larger than the sample size n (say, N / n 20), then σ ¯ X σ n .
lecture 14, Sampling Distributions II 3 / 21 Review Mean and Variance of Sample Mean Sample with Replacement Sample without Replacement Unbiased Estimates Sampling Distribution of the Sample Proportion lecture 14, Sampling Distributions II Review CLT (review), ctd

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lecture 14, Sampling Distributions II 4 / 21 Review Mean and Variance of Sample Mean Sample with Replacement Sample without Replacement Unbiased Estimates Sampling Distribution of the Sample Proportion lecture 14, Sampling Distributions II Mean and Variance of Sample Mean Sample with Replacement Sample with Replacement Suppose in a population the possible values of measurements are a 1 , a 2 , . . . , a , with frequencies k 1 , k 2 , . . . , k Population size N := k 1 + k 2 + . . . + k Relative frequencies = k 1 / N , k 2 / N , . . . , k / N Mean μ = ( k 1 · a 1 + . . . + k · a ) / N = a 1 · k 1 / N + . . . + a · k / N Variance σ 2 = ( a 2 1 · k 1 / N + . . . + a 2 · k / N ) - μ 2 Now suppose we draw a random sample of size n with replacement . Let X 1 , X 2 , . . . , X n be the outcomes of the n draws in order. What’s the distribution of X 1 ? X 2 ?
lecture 14, Sampling Distributions II 5 / 21 Review Mean and Variance of Sample Mean Sample with Replacement Sample without Replacement Unbiased Estimates Sampling Distribution of the Sample Proportion lecture 14, Sampling Distributions II Mean and Variance of Sample Mean Sample with Replacement Sample with Replacement, ctd X 1 Random sampling P ( X 1 = a 1 ) = k 1 / N = relative frequency of a 1 Similarly, P ( X 1 = a i ) = k i / N = relative frequency of a i The distribution of X 1 is the same as the distribution of the population X 2 Sample with replacement the distribution of X 2 is also the same as the distribution of the population Are X 1 and X 2 independent?

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Lecture14 - lecture 14 Sampling Distributions II lecture 14...

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