Help with visualising the confidence interval.

Question

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?

Answer 1

The text book seems to give a wrong definition of confidence interval. Like you remark correctly, the confidence interval is a interval such that when an experiment is repeated a sufficiently large number of times(*), x% of these confidence intervals contain the population parameter.

The problem with the definition as given in the book is that in frequentist statistics you do not assign probabilities to population parameters, so it does not make sense to talk about the probability (and even less to talk about "confidence", because what does that mean if it does not mean "probability"?).