Student's t-distribution - Confidence Intervals

Notes:

In the previous section we discovered how to make a confidence interval for estimating population mean. However we knew what the population standard deviation ($\sigma$) was. However it is not always the case that $\sigma$ is known.

If population standard deviation ($\sigma$) is unknown then to make a confidence interval to estimate population mean we cannot our old formula for error: $E=Z_\frac{\sigma}{2}*\frac{\sigma}{\sqrt{n}}$ as it requires a knowledge of $\sigma$. So instead we are required to use a thing called t-scores ($t_{\frac{\alpha}{2}})$.

Once we find the t-scores for particular values (this is done in a similar way to finding z-scores) we have a new formula for the Margin of Error:

$E=Z_\frac{\sigma}{2}*\frac{S}{\sqrt{n}}$