Math Help - T distributions

1. 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!!

2. 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

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