This week we will be creating confidence intervals for the mean. 

First we will do this when \(\sigma\) is known, as well as the following two assumptions are met:

When these assumptions are met we know from the Central Limit Theorem that we are "safe" to assume that the sample means are normally distributed with mean \(\mu\) (the population mean) and standard deviation \(\frac{\sigma}{\sqrt{n}}\) (the standard error). 

This theorem then lets us know the area underneath a normal curve will tell us the probability of choosing a random sample with an sample mean in the corresponding interval.

So if we want to know (for example) where we would have 90% probability of choosing

Recall that the \(z\)-score tells us the number of standard deviations from the mean, and hence we can look at our table to find \(-z\) value corresponding to 5% i.e. -1.65.

Finally the formula for a confidence interval:

\[\overline{x}-z_{\frac{\alpha}{2}}\cdot\left(\frac{\sigma}{\sqrt{n}}\right)<\mu<\overline{x}+z_{\frac{\alpha}{2}}\cdot\left(\frac{\sigma}{\sqrt{n}}\right)\]