In an econometrics text book, the author writes that an incorrect interpretation of a confidence interval (CI) would be to say that the population parameter is within an estimated CI with 95% probability. Once the interval is estimated, the parameter is either in the CI or not in the CI. Probability plays no role.
This is a standard definition of a CI and one that I am comfortable with.
On the next page, referring to a different example, he writes "The value zero is excluded from this (confidence) interval, so we can conclude that, with 95% confidence, the average change in scrap rates in the population is not zero."
I am trying to reconcile these two ideas in my head because at first glance I thought they were contradictory. My interpretation of the distinction is below.
Whether a parameter is in an estimated CI is determined by that specific sample. It may be that the parameter is in the estimated CI for the first sample from the population, in the estimated CI for the second sample from the population, but NOT in the estimated CI for the third sample from the population. Therefore we say that the population parameter is either in the CI or not in the CI and have no need for a probabilistic interpretation.
However, whether a population parameter is equal to zero has nothing to do with the sample that the CI is estimated from. For example, if it is greater than zero, it will be greater than zero in the first sample, second sample, third sample, etc. So, because whether the parameter is equal to zero is not determined by the sample, we can still talk in probabilistic terms regarding whether it is above/below certain values?
Does this interpretation make sense?