Improve Your Understanding of College Level Statistics Problems

Accuracy of confidence intervals
Suppose I have iid observations ${\displaystyle X_i}$ (empirical mean ${\displaystyle {\bar{X}}_n}$), drawn from a distribution with unknown mean ${\displaystyle \mu }$ and known variance ${\displaystyle {\sigma }^2}$. To build a confidence interval for ${\displaystyle \mu }$ I can use the central limit theorem that states:

and get the following approximation (if I am not mistaken), with ${\displaystyle \varphi }$ being the quantile function of the standard normal distribution:

I've always been told to just provide this as an answer for an interval with ${\displaystyle 1-\alpha }$ confidence level. But what about the real confidence level? It must be something like ${\displaystyle 1-\alpha -{ϵ}_n}$, right? What about ${\displaystyle {ϵ}_n}$?

A study compared women who viewed high levels of television violence as children with those who did not in order to study the differences with regard to physical abuse of their partners as adults. Use the table shown below to calculate the observed value of the chi-square statistic. The table shows both actual counts and expected counts (in parentheses).
$\begin{array}{|ccc|}\hline & \text{ High TV Violence }& \text{ Low TV Violence }\\ \text{ Yes, Physical Abuse }& 14(8.42)& 26(31.58)\\ \text{ No Physical Abuse }& 22(27.58)& 109(103.42)\\ \hline\end{array}$
The chi-square statistic is ${\chi }^2=\square$