An unbiased estimator of the sampling variance of x

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An unbiased estimator of the sampling variance of X ̄ , Var X ̄ 2 / n , is S 2 / n . 73
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A consistent estimator of is S , the sample standard deviation. When we replace with S in sd X ̄ / n we get the standard error of X ̄ : se X ̄ S n Notice that what we call the sample standard deviation, S , is an estimator of the population standard deviation, . The standard error of X ̄ is an estimator of sd X ̄ / n . 74
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Now again consider testing H 0 : 0 against any of the three alternatives. (Remember, if H 1 : 0 then the null is effectively H 0 : 0; if H 1 : 0 the null is effectively H 0 : 0.) When we replace with S in the test statistic we get T X ̄ se X ̄ n X ̄ S , which is easily computed given a random sample X i : i 1,..., n . 75
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