Use the distribution of the sample mean and the central limit theorem to construct confidence intervals.

What this dot point is asking

Statistical inference uses a sample to make justified statements about an entire population. The bridge is the sampling distribution of the sample mean.

So $\bar{X}$ is an unbiased estimator of $\mu$, and it becomes more precise as $n$ grows, since the standard error shrinks like $\tfrac{1}{\sqrt{n}}$.

This is what makes inference possible even when the population itself is not normal. As a rule of thumb $n \geq 30$ is enough for the approximation to be good.

Confidence intervals

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

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 margin of error and choosing a sample size

The term added and subtracted, $E = z\dfrac{\sigma}{\sqrt{n}}$, is the margin of error. It is half the width of the interval. Because it depends on $n$ only through $\dfrac{1}{\sqrt{n}}$, you can solve for the sample size needed to meet a target precision.

Interpreting the interval correctly

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.

To halve the width of an interval you must quadruple the sample size, because the width depends on $\tfrac{1}{\sqrt{n}}$. This square-root law is why doubling precision is expensive: cutting the margin of error to a quarter needs sixteen times the data.

Reading a given interval backwards

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%).