Multiply the numerator and the denominator with a factor that gives us:
It follows from the central limit theorem now that has a standard normal distribution.
2. has a chi square distribution with degrees of freedom.
3. If follows from that
If we substitute this in the epression for we obtain:
, with the information that and if follows that has a student distribution with degrees of freedom.
4. and thus by definition
5. , we know that and thus where (for each ) therefore