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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)

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    $\begingroup$ It takes effort to look at every single account to get the exact answer with certainty, so taking a sample saves effort at the cost of some possible sampling error. The bigger the sample is, the smaller the sampling error is likely to be, but requires greater effort. $\endgroup$ – Henry Mar 28 '15 at 0:21
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    $\begingroup$ The expected value of the sample mean may be the population mean, but more often than not the sample mean will not be exactly equal to the population mean, and different samples are likely to have different sample means. $\endgroup$ – Henry Mar 28 '15 at 0:23
  • $\begingroup$ @Henry Thanks. So, when people say that the mean of the sampling distribution of the sample mean is exactly equal to the population mean, are they assuming that there are infinite samples taken? I.e. if you took, say, 100 samples (or any number of samples less than infinity), then the mean of the sampling distribution of the sample mean would not equal the population mean? $\endgroup$ – randomUser47534 Mar 28 '15 at 0:26
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    $\begingroup$ @amoeba In some corners (mainly medical ones) where the word "sample" is used also outside of a statistical context (say, "take a sample of blood"), the world "sample" means one item (one observation/measurement) (hence, "take 3 samples of blood" means simply "take blood three times"), not as in "sample = set of many observations/measurements". This difference in the meaning assigned to the word "sample" is a recurring source of confusion in communication. $\endgroup$ – Alecos Papadopoulos Mar 28 '15 at 2:13
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    $\begingroup$ I wonder if the OP thinks that taking the mean of multiple sample means is what is meant by "mean of the sampling distribution of sample means". It isn't -- that's just the mean of a bigger sample! $\endgroup$ – Glen_b Mar 28 '15 at 2:20
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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.

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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.

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