Say I picked 10 random samples from a normal distribution with unknown parameters: 1.7, 2.6, 3.0, 4.4, 1.6, 2.1, 2.4, 2.7, 5.2, 3.3
What is the probability that I will pick a value larger than 100.0 from this distribution?
Here is the process that I have in mind, but I don’t know how to do the second step:
Step 1. Find out the unbiased estimators of the population mean and the standard deviation by calculating the mean and the appropriate standard deviation of the sample.
Step 2. Evaluate the following integral: $$ \int_{\sigma=0}^{+\infty} \int_{\mu=-\infty}^{+\infty} P(X > 100 | \mu, \sigma, X\tilde~ N(\mu, \sigma) )f(\mu)g(\sigma) \,d\mu d\sigma $$
where $f$ and $g$ are the probability distribution functions of $\mu$ and $\sigma$ respectively.
$\frac{\mu - \bar X}{\sqrt{s^2/n}}$ follows Student's t-distribution with n-1 degrees of freedom: $t_{n-1}$ (In my example n=10, $\bar X$ is the sample mean, and $s$ is the unbiased standard deviation of the sample.)
So my questions are:
1. What distribution does $\sigma$ follow?
2. How to evaluate the integral shown above? Is there a closed form solution?
Thanks!