Question

What is the mathematical relationship between α and the confidence level? For example, if the α level = 05, what then is the confidence level?

Answer 1

Let's suppose we have some normally distributed random variable (just to take advantage of drawing a simple bell curve). The probability that the random variable, call it \(X\), is less than some value \(k\) is given by the area under the curve \(\textbf{to the left}\) of \(X=k\), as you can see in the sketch. (For continuous random variables, \(P(X<k)=P(X\le k)\).) Similarly, \(P(X>k)\) is the area to the right.


The confidence interval gives you the endpoints \(a\) and \(b\) such that \(P(a<X<b)\) is (confidence level)% of the sample/population. The significance level \(\alpha\) represents the proportion of the sample/population that doesn't lie in the confidence interval.

For any probability distribution, the total area under the curve should add up to \(1\).