How do I determine the distribution of an additional sample $x_{n+1}$ given sample mean, variance $\bar x, s^2$ of a normally distributed $x$? Suppose I draw $n$ samples $x_1...x_n$ of a random variable $x$, which is normally distributed with an unknown mean $\mu$ and variance $\sigma^2$. From those samples, I compute a sample mean $\bar x$ and a sample variance $s^2$. I wish to compute the distribution of $x_{n+1} \mid \bar x, s^2$, i.e. the distribution of one additional sample $x_{n+1}$, given my measured $\bar x$ and $s^2$.
How should I proceed? This is probably a bad way of proceeding with loads of issues, but here is my thought process so far:

*

*Start with the fact that $\frac{\bar{x} - \mu}{s/\sqrt{n}}$ is $t$-distributed, and then use that to find a likelihood of $\mu$ given $\bar x$.

*Use the fact that $\frac{x_{n+1} - \mu}{\sigma} \sim \mathcal N(0,1)$.

*Use the fact that $\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}$

*Combine these facts to get a distribution of $x_{n+1}$.

An alternative approach I thought of goes as follows:

*

*$x_{n+1}\sim N(\mu, \sigma^2)$

*$\bar x \sim N(\mu, \frac{\sigma^2}{n})$

*$(x_{n+1} - \bar x) \sim N(0, \sigma^2 + \frac{\sigma^2}{n})$

*Find the distribution of $\sigma^2 \mid s^2$ using the fact that $\frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}$

*Integrate $N(0, \sigma^2 + \frac{\sigma^2}{n})$ over all possible $\sigma^2$ weighted by the distribution of $\sigma^2 \mid s^2$

As @whuber points out, this question is much more naturally answered in a Bayesian context. In that case the distribution you are interested in is known as the posterior predictive distribution.
$$
p(x_{n+1}|x_1,\ldots,n) = \int_\theta p(x_{n+1}|\theta,x_1,\ldots,x_n)p(\theta|x_1,\ldots,x_n)d\theta
$$
where $\theta = (\mu,\sigma^2)$. This is about as close as you can get in the general case. If you are willing to accept a particular prior distribution for $\mu$ and $\sigma^2$ (Normal for $\mu$ and Inverse-Gamma for $\sigma^2$) you can get a closed form distribution (it is a $t$ distribution) which you can look up here
