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Let \(X_1, X_2, \dots\) be independent random variables with mean \(\mu=50\) and variance \(\sigma^2=100\).Let \(\overline{X}_n\) be the mean of the first \(n\) variables.
Calculate the standard deviation of the sample mean, \(\sigma(\overline{X}_n)\), for \(n=1\), \(n=100\), and \(n=10,000\).
What happens to \(\sigma(\overline{X}_n)\) as \(n \to \infty\)?
How does this result support the Law of Large Numbers?
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