What is a confidence interval for a population mean? I have been learning about the central limit theorem and confidence intervals from this video.
The instructor states that when all sample means are compiled it forms a normal distribution (according to the central limit theorem). She also states that the population mean (unknown) lies at the center of this distribution. She states that a confidence interval is constructed by taking a sample mean and adding 2 times the sample error (rule of thumb).
However, near the end of the video, she states that this formula is used for finding the "confidence interval of a population mean". How is this right? Aren't we taking a sample mean from the population? And when we construct a confidence interval from a sample, is it also based on the normal distribution? But how could it be, as a normal distribution is only made when numerous samples are taken?
 A: I understand that you are confused about the term "confidence interval of a population mean" since you only have the sample mean and construct a confidence interval based on the sample.
I would like to raise the question, what would you like to do with a "confidence interval of the sample mean"? 
Once you have the sample, you know the sample mean, i.e. you should be 100% confident that the sample mean is one specific number. There is no need for a confidence interval for your sample mean. 
What you do not know, is the population mean. However, you have at least an idea of it, since you have collected a sample from the population. Your best guess of the population mean is the sample mean, as simple as it is. 
But are we 100% confident that the sample mean is the population mean? Of course, we are not. How confident are we? Well, there are several different approaches to answer that question, but probably the most famous way is to apply a mathematical procedure on your sample mean, which gives you an interval that captures in 95% of the cases the population mean. 
This is called the confidence interval, and the mathematical procedure you have just seen in the video. To make this interval capture the population mean in actually 95% of the cases, you must rely on central limit theorem (or apply other methods).
