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So far I understand, confidence interval (for mean) is defined by treating the sample mean as a proxy for population mean and then finding a interval where the true mean will be in certain percentage of trials. The smaller the value of C.I., the closer it is to the true mean.

I cannot make sense of the above text when I think about the following situation: imagine that I am collecting data for a random variable, which has a uniform distribution in the range [1,10]. But due to some error in data collection, my collected data ranges from 1 to 5, and I completely ignored data in the upper range. Now the sample mean will be 3, and lets say I have a huge sample size, which yielded very low SE. So I will have really small C.I., but the sample mean will not be close to the true mean, and CI will not contain it even once.

So when we say that smaller CI reflects true parameter, is there any assumption about the soundness of the data collection process? Or I am not understanding CI correctly?

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    $\begingroup$ The calculation of the confidence interval you're talking about is predicated on the data being obtained from a simple random sample from the population of interest. If that's not the case, then the confidence interval will not have the desired coverage properties. Other calculations can be used for other sampling schemes, but biases in sampling (not explicitly designed for in the sampling) will ruin the coverage nonetheless. $\endgroup$ – Glen_b Sep 18 '18 at 6:04
  • $\begingroup$ @Glen_b, so in order for the claims about CI to be valid, the sample must represent the population reasonably well, is that what you are saying? $\endgroup$ – Rakib Sep 18 '18 at 19:12
  • $\begingroup$ No, I didn't say that the sample should be representative -- indeed, making certain of that while still treating it as a random sample for the CI calculation would impact the coverage properties. A single sample obtained from simple random sampling might (by chance, especially if it's small) be a fairly poor representation of the population (at least on one or more criteria of interest): note that CI coverage is not a property of a single sample, but of the process by which a sample and its CI are produced. $\endgroup$ – Glen_b Sep 18 '18 at 22:22
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To compute C.I. you must first assume a distribution for the error term. Here in your example, if your assumption is that your data collection is truncated, your upper bound for CI should not be small however large your sample is.

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