Why do we need confidence intervals? I am following a video lecture on Statistics, which introduces the concept of confidence intervals in the following way:
"A bank vice president is interested in the average checking account balance for all personal accounts. A random sample of 500 accounts is selected, and the average is calculated. What level of "confidence" for the mean will the VP be satisfied with?"
What I don't understand is, why are confidence intervals even necessary? I learned previously that the mean of the sampling distribution of the sample mean is exactly equal to the population mean. So, wouldn't it be better for the VP to do that i.e. take many samples and calculate the mean of the sampling distribution of the sample mean (and thus get the exact population mean), instead of just using one sample to calculate confidence intervals (which will not give you the exact population mean)?
Thanks.
(Question source: https://www.youtube.com/watch?v=9GtaIHFuEZU)
 A: You forget that accuracy comes at the cost of effort. He'd need to gather all the data from thousands of accounts. And what if five accounts have billions while the rest are in the hundreds? The confidence interval is a faster way to give you a reasonable answer in a reasonable amount of time.
Confidence interval is even more apparent with surveys of a city or country. You can't put a gun to everyone's head and yell at them to complete the survey.
A: One answer is that the exact population may not be so easily reachable. The example is kind of weird in that the population of bank accounts is easily knowable to the banker and thus he should/would indeed just use that information. A better question would be what is the average sum a person is expected to deposit to the bank, although a random sampling would not be available.
Even more classic example would be the banker surveying the population what they think about his bank. In this case it would be very difficult to reach everyone and thus the concept of a confidence interval makes more sense.
