Quick questions on Statistical inference - TCE Mathematics Specialised (Tasmania)

A confidence interval gives a range of plausible values for $\mu$. Using the normal model for $\bar{X}$, a confidence interval for the population mean is $$\bar{x} \pm z \,\frac{\sigma}{\sqrt{n}},$$where$z$is the critical value for the chosen confidence level. The common values are$z = 1.96$for 95 percent and$z = 2.576$ for 99 percent confidence.

The confidence level describes the long-run reliability of the method: if we repeated the sampling many times and built an interval each time, about 95 percent of those intervals would contain the true $\mu$. It is not a probability statement about $\mu$ for one fixed interval.

Exam questions often hand you a completed interval and ask you to recover the sample mean, the standard error, or the confidence level. Two facts make this routine. The sample mean is the midpoint of the interval, $\bar{x} = \dfrac{L + U}{2}$, and the margin of error is the half-width, $E = \dfrac{U - L}{2}$. Once you have $E$ and $\dfrac{\sigma}{\sqrt{n}}$, the implied critical value is $z = \dfrac{E}{\sigma/\sqrt{n}}$, which tells you the confidence level ($z = 1.96$ for 95%, $z = 2.576$ for 99%).