05 Confidence Intervals

Learning Objectives:

1. Use z-tables tp compute a confidence interval on the mean, given the population standard deviation

Central Limit Theorem

In short, CLT says:

1. If a sample size is large enough, it is normally distributed
2. Criteria to know if a data set is large enough:
   1. If pop normal ⇒ sample is normal
   2. If pop not normal but close ⇒ $N>30$
   3. If pop not normal but not even close ⇒ $N\gg 500$ Lots of samples can make prediction on the population.

When you have a large enough sample:

1. Sampling mean ${\mu }_{\bar{x}}=\mu$
2. Sampling Standard Deviation ${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{N}}$ You can then find the mean of the population with a confidence interval

Confidence Interval

Important formula:

1. $z=\frac{\bar{x}-{\mu }_{\bar{x}}}{{\sigma }_{\bar{x}}}=\frac{\bar{x}-\mu }{\frac{\sigma }{\sqrt{N}}}$
2. $\bar{x}-z_{\alpha /2}\frac{\sigma }{\sqrt{N}}<\mu <\bar{x}+z_{\alpha /2}\frac{\sigma }{\sqrt{N}}$