Editing Prediction interval (section)

==== Known mean, unknown variance ====
Conversely, given a normal distribution with known mean ''μ'' but unknown variance <math>\sigma^2</math>, 
the sample variance <math>s^2</math> of the observations <math>X_1,\dots,X_n</math> has, up to scale, a [[chi-squared distribution|<math>\chi_{n-1}^2</math> distribution]]; more precisely:
:<math>\frac{(n-1)s^2}{\sigma^2} \sim \chi_{n-1}^2.</math>
On the other hand, the future observation <math>X_{n+1}</math> has distribution <math>N(\mu,\sigma^2).</math>
Taking the ratio of the future observation residual <math>X_{n+1}-\mu</math> and the sample standard deviation ''s'' cancels the ''σ,'' yielding a [[Student's t-distribution]] with ''n''&nbsp;–&nbsp;1 [[degrees of freedom (statistics)|degrees of freedom]] (see its [[Student%27s_t-distribution#Derivation|derivation]]):

: <math>\frac{X_{n+1}-\mu} s \sim T_{n-1}.</math>

Solving for <math>X_{n+1}</math> gives the prediction distribution <math>\mu \pm sT_{n-1},</math> from which one can compute intervals as before.

Notice that this prediction distribution is more conservative than using a normal distribution with the estimated standard deviation <math>s</math> and known mean ''μ'', as it uses the t-distribution instead of the normal distribution, hence yields wider intervals. This is necessary for the desired confidence interval property to hold.