1.

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