I’m trying to understand the confidence interval by reference to the diagram below:
• I want to guess a parameter that we know to be 0.88. The vertical line represents the parameter’s value.
• I choose a confidence interval of 95%. The length of each horizontal line is such that 95 out of 100 of them include the population parameter.
• If I choose a 99% confidence interval, then each horizontal line will lengthen until 99 out of 100 of them include the parameter.
• If I choose a 90% confidence interval, then each horizontal line will contract until 90 out of 100 of them include the parameter.
So, what sense is there in saying that we have a 95% confidence interval of, say, 0.86 to 0.91? In my textbook, it says, “We are 95% confident that the actual parameter proportion is between 86% and 91%”.
Shouldn’t we have many confidence intervals, e.g. the 100 confidence intervals represented by each blue line above and in the table below, of which 95 of them will include the actual parameter:
Would it be better to say: “There is a 95% probability that one of the confidence intervals I select using my selection procedure will contain the parameter. An example of one of those confidence intervals that I could have selected is [0.86, 0.91]. (Yet, once I select this confidence interval, it either does (100%) or does not (0%) contain the parameter)". In other words, how do I answer textbook questions without turning it into a philosophical discussion?