Probability

Here is a not so important point about confidence intervals but at the same time perhaps a bit interesting.

It is tempting to calculate a 95% confidence interval for \(\mu\), say (23,45) and interpret it as there is a probability of .95 that \(\mu\) is between 23 and 45. That is not a correct interpretation. The probability is either 0 or 1.

Here is the idea. Probability is branch of mathematics that allows us to assign numbers (i.e. probabilities) to events that have yet to occur! Meaning it makes sense to assign probabilities to events BEFORE you do the random experiment.

Imagine your random experiment is the proverbial coin toss. BEFORE you toss the coin it makes perfect sense to assign a probability of .5 to the event the coin lands on Tails.

After the coin has been tossed, it really does not make sense to assign probabilities to the outcome, though you can. If the coin landed Tails the Pr(Tails) = 1. If the coin landed on Heads, the Pr(Tails) = 0. So by necessity after the experiment is over your probality is 0 or 1.

If you look at our confidence interval formulas, \(\hat{P}\) is akin to our coin toss. After we get a \(\hat{p}\) the random experiment is over, and our interval either contains the parameter or it does not. It is doubly tricky because in this random experiment even though it is over, we really don’t see the outcome, unlike the coin toss where we can clearly see the outcome.

In summary 99.9% of people interpret confidence intervals as a range of likely values. Just understand that they are not the probability a parameter is contained in the interval, and hopefully above is an explanation of that.