When does a confidence interval lead to a probabilistic interpretation?

Question

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?

Answer 1

You had me at the first paragraph of your reconciliation and lost me in the second. Let me try to be clear.

When we say we are 95% confident that the parameter is in the interval we are (as you said) not saying that the probability that the parameter is in the interval is 95%, but that the probability that this procedure yields an interval containing the parameter is 95%.

The distribution that this probability is coming from is the sampling distribution. 95% of samples will lead to an interval containing the (completely fixed) parameter.

edit

Everything about the parameter is deterministic. However, when you relate it to a random variable you can make probabilistic statements that are based on the distribution of that random variable.

The sample you draw is a random variable. Therefore the confidence interval you get is a random variable. So the answer to the question "Will the parameter be in the interval?" is a random variable. But the answer to "is the parameter in this interval?" is not.

To make an analogy. If I produce a coin that is heads 95% and tails 5%. And tell you it is 95-5 but I don't tell you which side is more likely. The following statement is true "If I flip the coin the probability it will land on the face it is biased toward is 95%." Once you flip the coin, the following is false "There is a 95% chance it is biased toward this face." You could declare "I say, with 95% confidence that the coin is biased to this face" because the word confidence in this field means precisely that!