I'm struggling with understanding confidence intervals and the common fallacies that I'm obviously not alone in doing. For example, why can't I say that I'm 95% confidence that the true value lies within my 95% confidence interval if I've made a measurement with a device that has a normal distributed error with known standard deviation?
In an attempt to understand this better I read the submarine example in the article The Fallacy of Placing Confidence in Confidence Intervals by Morey, R.D., Hoekstra, R., Lee, M.D., Rouder, J.N., Wagenmakers, E-J, the example goes like this:
"A 10-meter-long research submersible with several people on board has lost contact with its surface support vessel. The submersible has a rescue hatch exactly halfway along its length, to which the support vessel will drop a rescue line. Because the rescuers only get one rescue attempt, it is crucial that when the line is dropped to the craft in the deep water that the line be as close as possible to this hatch. The researchers on the support vessel do not know where the submersible is, but they do know that it forms two dis- tinctive bubbles. These bubbles could form anywhere along the craft’s length, independently, with equal probability, and float to the surface where they can be seen by the support vessel."
They go on by showing several different ways of calculating a 50% confidence interval which highlights the fallacies of confidence and precision very well. However I can't really translate the example into a real-world scenario. Say that I'm performing a measurement with a device that has a normal distributed error with known standard deviation. Are there other ways to calculate the 95% confidence interval besides $[x-1.96\sigma, x+1.96\sigma]$ in that scenario as well?