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Math Help - T distributions

  1. #1
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    Exclamation T distributions

    Suppose X1, X2,...Xn come from a random sample of a normal distribution with mean and standard deviation unknown. let mu hat and sigma hat denote maximum likelihood estimators of the mean and standard deviation. for a sample size of n=17, find the value of k such that:
    P(Mu hat> Mu + k*sigmahat)= .95

    I am just wondering if anyone could kind of walk me through how to do a problem like this. Thanks so much!!
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  2. #2
    MHF Contributor matheagle's Avatar
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    well, with out calculating them, I believe that

    \hat \mu=\bar X and \hat \sigma^2={\sum (X_i-\bar X)^2\over n}

    You want k such that .95=P\left({\bar X-\mu\over\hat\sigma}>k\right)

    you need to make that term on the left look like a t_{16}

    by using the fact that {\bar X-\mu\over S/\sqrt{n}}\sim t_{16}

    Just multiply and divide, using the fact that

    \hat \sigma^2={\sum (X_i-\bar X)^2\over n}={n-1\over n}S^2
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