So I've come across this question (Exercise $17$) whilst attempting some Probability/ Statistics questions.
Let $X_1, \dots, X_n\sim \mathcal N(\mu, {\sigma}^2)$ be i.i.d. where $\mu$ is known but ${\sigma}^2$ is unknown. Consider the estimate $${\hat\sigma}^2=\frac{1}{n}\sum_{i=1}^n (X_i-\mu)^2$$ for the variance.
a) Show that $$C:=\sum_{i=1}^n\left( \frac{X_i-\mu}{\sigma}\right)^2 \sim {\chi}^2(n).$$
b) For $\alpha \in (0, 1) $, denote the $\alpha$-quantile of ${\chi}^2(n)$ by $q_n(\alpha)$. Using this notation, find the numbers $a_n, b_n \in \mathbb R$ such that $P({\hat\sigma}^2<a_n{\sigma}^2)=2.5\%$ and $P({\hat\sigma}^2>b_n{\sigma}^2)=2.5\%$
c) Using these results, develop a test which can be used to test the hypothesis $H_0:{\sigma}^2={\sigma_0}^2$ against the alternative $H_1:{\sigma}^2\ne {\sigma_0}^2$
d) Assume we have $n=100, \;\mu=0$ and we have observed ${\hat\sigma}=4.81$. Test the hypothesis $H_0:{\sigma}^2=4$ vs. $H_1:{\sigma_0}^2\ne 4$ at significance level $5\%$.
I'm a bit confused about how to approach it. It firstly asks me 'a show that' question, that is related to $C$ being of a $\chi ^2$ distribution. I haven't managed to figure any way of solving it. For the second part of the question (part b), I am essentially stumped! Any help/ pointers/ explanations would be greatly appreciated. Thank you in advance! (By the way help on the second two parts isn't needed!). Here's the image of question.
I will get you started with part (a):
If $X_1, X_2, \dots, X_n$ is a random sample from $\mathsf{Norm}(\mu, \sigma)$ with $\mu$ known and $\sigma$ unknown, then you have $Z_i = \frac{X_i - \mu}{\sigma} \sim \mathsf{Norm}(0,1).$ Hence $Z_i^2 \sim \mathsf{Chisq}(df = 1)$ and $$C = \sum_{i=1}^n Z_i^2 = \sum_{i=1}^n \left(\frac{X_i - \mu}{\sigma}\right)^2 =\frac{1}{\sigma^2}\sum_{i=1}^n(X_i - \mu)^2 = \frac{n\hat\sigma^2}{\sigma^2} \sim \mathsf{Chisq}(n).$$ The sum $C$ of $n$ independent random variables, each distributed $\mathsf{Chisq}(1),$ has $C \sim \mathsf{Chisq}(n).$ One proof uses moment generating functions.
Suppose that $L$ cuts 2.5% of the area from the lower tail of $\mathsf{Chisq}(n)$ and $U$ cuts 2.5% of the area from its upper tail. Then, for example,
$$0.95 = P\left(L < \frac{n\hat\sigma^2}{\sigma^2} < U\right) = P\left(\frac{n\hat\sigma^2}{U} < \sigma^2 < \frac{n\hat\sigma^2}{L}\right),$$ so that a 95% confidence interval for $\sigma^2$ (in the case where $\mu$ is known) is of the form $$\left(\frac{n\hat\sigma^2}{U},\;\frac{n\hat\sigma^2}{L}\right).$$
If a hypothetical value $\sigma_0^2$ of $\sigma^2$ lies outside such a 95% confidence interval, then it can be rejected at the 5% level of significance.
For the other parts, involving tests of hypotheses, manipulations are similar.