1234436925[1] - Topic One Sampling Concepts and...

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MSOR221 1 / 4 Topic One Sampling Concepts and Distributions Sampling Concepts Sample statistics are used to make inferences (estimates or decisions) about unknown population parameters θ . Because a statistic T is a function of the random variables, X 1 , X 2 , …, X n , observed in a sample ( T = f ( X 1 , X 2 , …, X n )), the statistic itself is a random variable. Typical of what we mean by “statistic” are the sample mean, the sample variance, and the sample proportion. Since statistics are random variables, their values will vary from sample to sample, and it is customary to refer to their distribution as sampling distribution . Consider all possible samples of size n that can be drawn from a given population. For each sample, we can compute a statistic T (such as the mean, the variance, and the proportion) that will vary from sample to sample. In this manner we obtain a distribution of the possible values of the statistic that is called its sampling distribution. From a practical point of view, the sampling distribution for a statistic provides a theoretical probability distribution of the possible values of the statistic. Population Parameter Sample Statistic Mean N X = μ n x x = Variance N N X N X 2 2 2 2 ) ( μ μ σ - = - = 1 1 ) ( 2 2 2 2 - - = - - = n x n x n x x s Proportion N X p = n x p = ˆ Sampling Distribution of the Mean The sampling distribution of the mean is the distribution of all possible sample means if all possible samples of a certain size are selected from the population.
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