Can someone please help me with the attached problem, I really don't know where to start with it. I'd really appreciate some pointers in the right direction.
Thanks.
note that $\displaystyle Y = \frac{1}{\sigma^2} (X_i-\mu)^2 $ has a chi-squared distribution of $\displaystyle n $ degrees of freedom since each term in the sum is a squared normal random variable and is independent of other random variables in the sum. Then show that $\displaystyle Y - \frac{(n-1)S^2}{\sigma^2} = \left(\frac{\bar{X} - \mu}{\sigma/ \sqrt{n}} \right)^2 $. The RHS is a chi-squared random variable with one degree of freedom. Since df's add, this implies that $\displaystyle \frac{(n-1)S^2}{\sigma^2} $ is a chi-squared distribution with $\displaystyle n-1 $ degrees of freedom provided that $\displaystyle \frac{(n-1)S^2}{\sigma^2} $ and $\displaystyle \left(\frac{\bar{X} - \mu}{\sigma/ \sqrt{n}} \right)^2 $ are independent.