confidence interval when \(\sigma\) is known:

\[\overline{x}-z_\alpha\cdot\left(\frac{\sigma}{\sqrt{n}}\right)<\mu<\overline{x}+z_\alpha\cdot\left(\frac{\sigma}{\sqrt{n}}\right)\]

Degrees of Freedom (for means):

\[n-1\]

confidence interval when the population standard deviation is unknown one uses the formula:

\[\overline{x}-t_\alpha\cdot\left(\frac{s}{\sqrt{n}}\right)<\mu<\overline{x}+t_\alpha\cdot\big(\frac{s}{\sqrt{n}}\big)\]

The formula for the \(z\)-test is:

\[z=\frac{\overline{x}-\mu}{\left(\frac{\sigma}{\sqrt{n}}\right)}\]

The formula for the \(t\)-test is:

\[t=\frac{\overline{x}-\mu}{\left(\frac{s}{\sqrt{n}}\right)}\]