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.

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 $\tfrac{1}{\sqrt{n}}$.