# 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.

• 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. Mar 28 '15 at 0:21
• 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. Mar 28 '15 at 0:23
• @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? Mar 28 '15 at 0:26
• @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. Mar 28 '15 at 2:13
• 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! Mar 28 '15 at 2:20