I'm reading about Linear Regression in Introduction to Statistical Learning (Chapter 3) I see the confidence interval defined as
A 95% confidence interval is defined as range of values such that with 95% probability, the range will contain the true unknown value of the parameter
First off, this seems incorrect by various definitions I have seen, specifically wikipedia says
A 95% confidence level does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).
Unless authors of ISLR meant
... ranges of values ... instead of
... range of values ..., although that would be relying on reader noticing the plural word and being mindful of interpreting it that way.
It further goes on to interpret the computed confidence interval for the example advertising-sales data as (paraphrasing)
the 95% confidence interval for intercept ($\beta_0$) is [6.130, 7.935] . Therefore in the absence of any advertising, sales will on average fall somewhere between 6,130 and 7,940 units.
This again seems to be the interpretation that is incorrect.
More generally, I would also like to understand the utility of calculating a confidence interval (both for a general statistic and well as in the context of regression co-efficient)
The general interpretation seems to be that if we had 100 samples from population, and we calculated confidence intervals for the statistic using all 100 samples (independently), 95% of those intervals would have true the population statistic. i.e 95% denotes to the strength of the estimation method rather than that of the data.
Given that interpretation and the example interval for the sales-advertising example ([6.130, 7.935] ), what conclusions can I make about the interval?, I certainly couldn't say the true value of intercept is in that specific range 95% of the time (its either in that range or its not)? What insight does that specific range offer about the modeling procedure?