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# 19 - When we choose many SRSs from a population the...

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When we choose many SRSs from a population, the sampling distribution of the sample means is centered at the mean of the original population and is less spread out than the distribution of individual observations. Here are the facts. Suppose that is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ. Then the sampling distribution of has mean μ and standard deviation σ/√ n . # The mean of the statistic is always equal to the mean μ of the population. That is, the sampling distribution of is centered at μ. In repeated sampling, will sometimes fall above the true value of the parameter μ and sometimes below, but there is no systematic tendency to overestimate or underestimate the parameter. This makes the idea of lack of bias in the sense of “no favoritism” more precise. Because the mean of is equal to μ, we say that the statistic is an unbiased estimator of the parameter μ. # An unbiased estimator is “correct on the average” in many samples. How close the estimator falls to the parameter in most samples is determined by the spread of the sampling distribution. If

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