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?