Iam trying to calculate upper and lower confidence levels for a parameter, but i can't get it straight (in this case $\sigma^2$):
the reference variable: $R_{\sigma^2} := \frac{n-1s^2}{\sigma^2} \sim \chi^2(n-1)$
where $s^2 = \frac{1}{n-1}\sum\limits_{i=1}^{n} (x_i-\bar{x})^2 $
now for the confidence interval we get something like this
$1-\alpha = P\big(\chi^2_{1-\alpha/2}(n-1) < R_{\sigma^2} < \chi^2_{\alpha/2}(n-1) \big)= P\big( \frac{(n-1)s^2}{\chi^2_{\alpha/2}(n-1)} \big) < \sigma^2 < \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}(n-1)}\big)$
ok that was the context. now i want to find the one-sided upper and lower confidence level and Iam thinking something like this: (for the one sided upper confidence level)
$ \alpha= P\big(R_{\sigma^2} > \chi^2_{\alpha}(n-1)\big) = P\big(\frac{(n-1)s^2}{ \chi^2_{\alpha}(n-1)}\ > \sigma^2 \big) $ so the upper confidence level is given by $\frac{(n-1)s^2}{ \chi^2_{\alpha}(n-1)} = \bar{\sigma^2}$ but in my book it says $\frac{(n-1)s^2}{ \chi^2_{\textbf{1-$\alpha$} }(n-1)} = \bar{\sigma^2}$ Iam a big confused about this can someone help me to understand (observe $\chi^2_{1-\alpha}(n-1)$)
$\alpha=P(R_{\sigma ^2}>\chi_{\alpha,n-1}^2)=P(\frac{(n-1)s^2}{\sigma ^2}>\chi_{\alpha,n-1}^2)=P(\frac{\sigma ^2}{(n-1)s^2}>\frac{1}{\chi_{1-\alpha,n-1}^2})$
$=P(\sigma ^2 > \frac{(n-1)s^2}{\chi_{1-\alpha,n-1}^2})\implies\bar{\sigma ^2} = \frac{(n-1)s^2}{\chi_{1-\alpha,n-1}^2}.$ A good way to check your answer is that you know when $\alpha$ becomes small, $\chi_{\alpha,n-1}^2$ becomes large and $\chi_{1-\alpha,n-1}^2$ becomes small for any value of $n>1$. This means that $\dfrac{1}{\chi_{1-\alpha,n-1}^2}$ becomes increasingly large as $\alpha\to 0,$ so you know that $\dfrac{1}{\chi_{1-\alpha,n-1}^2}$ is the upper confidence level.