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I'm just at the beginning of learning this stuff. I learnt that we can obtain what is called the 'expected mean' of the sample, which is basically the mean that we'd expect to get if we sampled the sample many times.

In the specific case below, we have one sample of an overall population. The book then assumes that this specific case's mean and standard deviation is exactly the same as actual mean of the population. My understanding is that the specific case's mean and standard deviation of a given sample are not necessarily the same as those of the statistic X bar, which is what we usually use as the unbiased estimator of the mean of the population. Even when we look at another unbiased estimator of the mean of the original population, which is E(X), we still end up with the fact that this doesn't necessarily mean that the expected mean of the original population is the same as the mean of the specific case of a given sample

So, WHY is the mean of a specific case sample being used as an unbiased estimator of the mean of the entire population? Someone please help clear my misconceptions :)

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The post is revised with the comments to the original answer taken into account hopefully providing a more clear account of what I was trying to state:

Whether the mean is unbiased is tightly coupled to the sampling design. Provided a simple random sample the sample mean is an unbiased estimator of the population parameter because over many samples the mean does not systematically overestimate or underestimate the true mean of the population.

A biased estimator of a population parameter will on the other hand systematically either over- or underestimate the population parameter.

With probability samples other than a simple random sample y for each element (i) must be weighted against the selection probability in order for the mean to be unbiased (look up for instance the Horvitz-Thompson estimator for a more detailed explanation).

Feel free to correct any incorrect or imprecise statements.

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    $\begingroup$ Unbiasedness is a property of estimators(i.e formulas) not actual estimates/numbers. So don't include confidence intervals in your answer. Also this does not answer the OP's question>> $\endgroup$ – machazthegamer May 5 '17 at 12:12
  • $\begingroup$ Original question has problem. "Calculate unbiased estimates of ..." should be "Use unbiased estimator to estimate ..." $\endgroup$ – user158565 May 5 '17 at 15:12
  • $\begingroup$ It's likely worthwhile to maintain the conventional distinction between a "simple random sample"--in which all elements have equal chances of being chosen and those choices are independent--and a "probability sample," which is a far more general concept in which elements may have difference chances of selection. It would also be more accurate to retain the logical distinction between "only under" (meaning "if and only if") and "under" (meaning "is implied by"), because the sample mean can be unbiased for many other kinds of sampling procedures. $\endgroup$ – whuber May 5 '17 at 15:33
  • $\begingroup$ @machazthegamer indeed. I will remove that section for the purpose of clarity and relevance to the OP's question. $\endgroup$ – MiGH May 6 '17 at 5:42
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    $\begingroup$ @whuber good point. Based on your good input I have revised my answer. $\endgroup$ – MiGH May 6 '17 at 5:46
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The sample mean is an unbiased estimator for the population mean. An estimator is a random variable with a probability distribution of its own. One specific sample with one specific value provides only one possible value of this (estimator) random variable.

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    $\begingroup$ These assertions merely restate what's in the question. What the OP would like is an explanation of why the sample mean is unbiased. $\endgroup$ – whuber May 5 '17 at 15:34
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This is how I think of this: expectation is the value you would expect to get when you sample a 'point' randomly from a population. So, if you approximate the mean using only one sample then, in expectation this should equal the population mean (because, as I said earlier, expectation is the value you expect to get when you sample from a population). Therefore, the mean is an unbiased estimator.

For a mathematical proof see this answer: https://stats.stackexchange.com/a/202669/218083

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