Statistics is challenging enough before getting into confidence intervals. I use an analogy to help make the concept of confidence intervals accessible. Here’s the analogy. You are looking for a female student who is in a classroom in a hallway. You don’t know which room but you do know that her boyfriend is well trained and is waiting close by. You look in the hallway for him and this gives you a good estimate for the room she is in. The girl is the population average, which you are estimating. The boyfriend is the sample average, which you compute from the sample data. Your estimate is not a single room but an intervals of rooms, e.g. 104-106.
The confidence interval works on the same premise as the Empirical Rule. In the Empirical Rule if you go up and down 2 σ you will have 95% of x-bars in that interval under the bell curve. Using the analogy, if you go up and down a couple rooms from the girl friend there’s a high probability probability the boyfriend is in that interval.
The formula for the confidence interval mirrors the Empirical Rule.