** 1. ** Show that

has the

distribution with

degrees of freedom. The population of interest in normal, so that

constitutes a random sample from a normal distribution with both

and

unknown.

Now a

distribution

with n - 1 degrees of freedom (Mr F adds) is defined as

where

is a standard normal variable.

Mr F says: No. has a t-distribution with n - 1 degrees of freedom.
So

is a standard normal variable,

Mr F says: No. is normally distributed with mean and variance so is a standard normal variable.
and

is the square root of a chi-squared

Mr F says: No. How did you get this from the (correct) theorem quoted below?
(we used the fact that

). Is this correct?