Confidence Intervals

Population Statistic	Population Distribution	Conditions	Confidence Interval
$μ$	Normal	You know what $σ^{2}$ is $n$ is large or small $\bar{x}$ is the sample mean	$$(\bar{x}-c\frac{\sigma }{\sqrt{n}},\bar{x}+c\frac{\sigma }{\sqrt{n}})$$
$μ$	Non-normal	You know what $σ^{2}$ is $n$ is large (at least 30) $\bar{x}$ is the sample mean	$$(\bar{x}-c\frac{\sigma }{\sqrt{n}},\bar{x}+c\frac{\sigma }{\sqrt{n}})$$
$μ$	Normal or Non-normal	You don’t know what $σ^{2}$ is $n$ is large (at least 30) $\bar{x}$ is the sample mean $s^{2}$ is the sample variance	$$(\bar{x}-c\frac{s}{\sqrt{n}},\bar{x}+c\frac{s}{\sqrt{n}})$$
$p$	Binomial	$n$ is large $p_{s}$ is the sample proportion $q_{s}$ is $1 - p_{s}$	$$(p_{s}-c\sqrt{\frac{p_{s}q_{s}}{n}},p_{s}+c\sqrt{\frac{p_{s}q_{s}}{n}})$$
$μ$	Normal or Non-normal	You don’t know what $σ^{2}$ is $n$ is small (less than 30) $\bar{x}$ is the sample mean $s^{2}$ is the sample variance	$$(\bar{x}-t(v)\frac{s}{\sqrt{n}},\bar{x}+t(v)\frac{s}{\sqrt{n}})$$
where $c$ is:

statistic ± (margin of error)
margin of error = c * (standard deviation of statistic)

Sources: 1